Obtaining a Converged Solution with Abaqus

Transcription

1 Obtaining a Converged Solution with Abaqus Day 1 Lecture 1 Workshop 1 Lecture 2 Lecture 3 Workshop 2 Lecture 4 Workshop 3 Lecture 5 Workshop 4 Workshop 5 Introduction to Nonlinear FEA Nonlinear Spring Nonlinear FEA with Abaqus/Standard Solution of Unstable Problems Reinforced Plate Under Compressive Loads Why Abaqus Fails to Find a Converged Solution Crimp Forming Analysis Convergence Problems: Contact Simulations Contact: Beam Lift-Off Contact: Stabilization 1

2 Day 2 Lecture 6 Workshop 6 Lecture 7 Lecture 8 Workshop 7 Workshop 8 Convergence Problems: Element Behavior Element Selection Convergence Problems: Constraints and Loading Convergence Problems: Materials Limit Load Analysis Ball Impact Legal Notices The Abaqus Software described in this documentation is available only under license from Dassault Systèmes and its subsidiary and may be used or reproduced only in accordance with the terms of such license. This documentation and the software described in this documentation are subject to change without prior notice. Dassault Systèmes and its subsidiaries shall not be responsible for the consequences of any errors or omissions that may appear in this documentation. No part of this documentation may be reproduced or distributed in any form without prior written permission of Dassault Systèmes or its subsidiary. Dassault Systèmes, Printed in the United States of America Abaqus, the 3DS logo, SIMULIA and CATIA are trademarks or registered trademarks of Dassault Systèmes or its subsidiaries in the US and/or other countries. Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerning trademarks, copyrights, and licenses, see the Legal Notices in the Abaqus 6.10 Release Notes and the notices at: 2

3 Revision Status Lecture 1 5/10 Updated for 6.10 Lecture 2 5/10 Updated for 6.10 Lecture 3 5/10 Updated for 6.10 Lecture 4 5/10 Updated for 6.10 Lecture 5 5/10 Updated for 6.10 Lecture 6 5/10 Updated for 6.10 Lecture 7 5/10 Updated for 6.10 Lecture 8 5/10 Updated for 6.10 Workshop Answers 1 5/10 Updated for 6.10 Workshop Answers 4 5/10 Updated for 6.10 Workshop Answers 5 5/10 New for 6.10 Workshop Answers 6 5/10 Updated for 6.10 Workshop Answers 7 5/10 Updated for 6.10 Workshop Answers 8 5/10 Updated for 6.10 Workshop 1 5/10 Updated for 6.10 Workshop 2 5/10 Updated for 6.10 Workshop 3 5/10 Updated for 6.10 Workshop 4 5/10 Updated for 6.10 Workshop 5 5/10 New for 6.10 Workshop 6 5/10 Updated for 6.10 Workshop 7 5/10 Updated for 6.10 Workshop 8 5/10 Updated for

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7 Introduction to Nonlinear FEA Lecture 1 L1.2 Overview Why Use FEA to Solve Mechanics Problems? What is Convergence? When is a Problem Nonlinear? Properties of Linear Problems in Mechanics Properties of Nonlinear Problems in Mechanics Numerical Techniques for Solving Nonlinear Problems 7

8 Why Use FEA to Solve Mechanics Problems? L1.4 Why Use FEA to Solve Mechanics Problems? Understand the behavior of a design Finite element analysis (FEA) is a useful tool for studying the behavior of various mechanical designs. Example: Burst load analysis of a micro-channel tube. Tube must withstand three times the maximum working pressure. 8

9 Internal Pressure (psi) L1.5 Why Use FEA to Solve Mechanics Problems? 2.0 [x10 3 } 1.5 Operating Pressure Overload Pressure XMIN 1.463E-05 XMAX 1.354E-02 YMIN 1.562E+02 YMAX 1.725E Bulge Displacement (in) [x10-3 } A B A) Comparison of Mises stress in tube B) Predicted burst pressure in tube L1.6 Why Use FEA to Solve Mechanics Problems? Reduce product costs and development time FEA can reduce product costs and development time by: Identifying forming problems prior to tooling fabrication. Minimizing tooling rework (see the following figure). Reducing overall prototyping effort while identifying shortcomings in the design. Minimizing the amount of material used during fabrication. 9

10 L1.7 Why Use FEA to Solve Mechanics Problems? initial design numerical simulation fabricate soft tooling stamp pre-prototype parts rework soft tooling design modifications numerical simulation stamp production parts stamp prototype parts rework hard tooling fabricate hard tooling Incorporating FEA in development cycle L1.8 Why Use FEA to Solve Mechanics Problems? Only way to get an answer FEA can be used to predict the ability of a design to withstand extreme loading conditions that cannot be duplicated in an experiment. Hopefully these extreme loading conditions will be considered early in the design process. An example of such a finite element analysis is the simulation of the ability of an offshore platform to withstand the forces produced by a hundred-year storm. 10

11 L1.9 Why Use FEA to Solve Mechanics Problems? Unfortunately some extreme loading conditions are never imagined during the design process. Consider the case of the buckling of a solar panel under a severe thermal transient. The panel never returned to its original configuration when the normal operating temperature was restored, rendering it useless. It would have been extremely difficult to simulate this loading on the structure, which is over 120-inches long, in a laboratory; FEA was the only tool available to investigate the proposed design modifications for subsequent panels that were produced. What is Convergence? 11

12 L1.11 What is Convergence? In FEA convergence can imply multiple meanings Mesh convergence Time integration accuracy Convergence of nonlinear solution procedure Solution accuracy L1.12 What is Convergence? Mesh convergence Increasing the number of elements in the model will cause the solution to approach the analytical solution of the equations that govern the response. Applies both to linear and nonlinear analysis. Applies for h-based element technology used in Abaqus. At some point further mesh refinement yields little or no change in solution, and the mesh is assumed to have converged. 12

13 L1.13 What is Convergence? A few exceptions to the mesh convergence rule: Singular solutions (e.g., fracture mechanics). Localization problems where material damage can accumulate in particular regions of the model. Lodygowski, T., Shear Bands and Failure in Adiabatic and Fracture Tests, ABAQUS Users Conference, 1996, pp Abaqus provides special techniques to reduce mesh dependence of localization effects for softening materials, such as concrete. What is Convergence? Abaqus provides tools to evaluate mesh convergence Strain jumps at nodes (SJP) written to printed output (.dat) and results (.fil) files Contouring options in Abaqus/Viewer: Quilt plots Discontinuity plots Error estimates and adaptive remeshing Not discussed here; see Adaptive Remeshing with Abaqus/Standard lecture notes. L

14 L1.15 What is Convergence? Mesh convergence example: load bearing bracket pulled Twisted about the 1-direction fixed L1.16 What is Convergence? Discontinuity comparison between coarse and refined meshes Discontinuity ~85% of averaged stress Discontinuity < 30% of averaged stress Discontinuities in maximum principal stress 14

15 L1.17 What is Convergence? Time integration accuracy for transient problems For transient problems with a physical time scale, Abaqus/Standard provides user-specified parameters to control the accuracy of integrating the relevant equations forward in time. DYNAMIC, HALFINC SCALE FACTOR = factor, HAFTOL = value Half-increment residual tolerance Estimated maximum out-ofbalance force at the half-way point of the current increment What is Convergence? L1.18 HEAT TRANSFER, DELTMX = value Maximum temperature change allowed in an increment VISCO, CETOL = value Maximum difference in creep strain increment calculated from rates based on conditions at beginning and end of increment 15

16 L1.19 What is Convergence? Convergence of nonlinear solution procedure The bulk of this seminar will focus on this topic. Solution accuracy An accurate solution requires all the above: Converged mesh Accurate time integration for transient problems Proper convergence for nonlinear solution procedure In addition, an accurate solution requires good engineering judgment to create a proper finite element model, including materials, loads, boundary conditions, and solution procedure(s). When is a Problem Nonlinear? 16

17 L1.21 When is a Problem Nonlinear? Always! The behavior of all physical structures is nonlinear. However, the response of a structure can often be approximated as linear if its deformations/motions are small. Thus, when you start a simulation and you are unsure whether it should be a linear or nonlinear analysis, do not think, Do I need to make this nonlinear? Instead, ask yourself, Can I approximate the response of the structure in this analysis as linear? For example, a geometrically nonlinear structural analysis is one in which the structure s stiffness changes as it deforms. Ultimately, this implies that equilibrium is calculated in the current configuration. L1.22 When is a Problem Nonlinear? What is equilibrium? The concept of mechanical equilibrium is: Real P I 0, Nonlinear Equilibrium PDEs where P represents the externally applied loads and I the internal forces due to stresses in the structure: T I β σ dv. V is the relationship between displacement and strain increments. 17

18 L1.23 When is a Problem Nonlinear? General sources of nonlinearity Geometry Large deflections/rotations, large deformations Material Nonlinear elasticity, plasticity, damage, failure,... Boundary Contact, friction L1.24 When is a Problem Nonlinear? V T β σ dv P Geometric nonlinearity is characterized by: Nonlinear relationship between displacement increments and strain increments Integration over current (unknown) volume Effects: Stress-stiffening Bifurcation, buckling, and collapse Snap-through 18

19 L1.25 When is a Problem Nonlinear? V T β σ dv P Material nonlinearity is characterized by: Dependence of stress on current strain Effects: Plasticity Plastic hinge formation and plastic collapse Necking Rubber nonlinear elasticity Cracking, crushing L1.26 When is a Problem Nonlinear? V T β σ dv P Boundary nonlinearity is characterized by: Dependence of P on current displacements. Effects: Contact across body surfaces (by means of load transfer) Nonlinear external loads Pressure load nonlinearities 19

20 L1.27 When is a Problem Nonlinear? True mechanical equilibrium is: Real V T β σ dv P Real Linearized equilibrium is: Nonlinear Equilibrium PDEs K0u P Nonlinear Equilibrium PDEs How do you linearize the equilibrium equation? Expand in a Taylor series and throw away the higher order terms. Linearization is about the undeformed position. K 0 V 0 T β Dβ σ Dε dv Linearized Equilibrium PDEs L1.28 When is a Problem Nonlinear? Linear approximations of a problem The behavior of a structural problem can be approximated adequately as linear if all three sources of nonlinearity can be safely ignored: The material properties must not change during the simulation. The contact conditions must not change during a linear analysis. The strains and displacements/rotations must be small. In Abaqus/Standard a linear analysis can be performed about the current configuration of the structure (obtained at the end of a previous nonlinear analysis). The current configuration is known as the base state. Linear simulations about such a base state are called linear perturbation analyses. 20

21 Properties of Linear Problems in Mechanics L1.30 Properties of Linear Problems in Mechanics EXISTENCE For each load F there will always be at least one solution u. UNIQUENESS For each F there will always be only one solution u! SCALING If F causes a displacement u, then af causes a displacement au. SUPERPOSITION If F causes a displacement u and G causes a displacement v, then F + G causes a displacement u + v. Real Nonlinear Equilibrium PDEs Linearized Equilibrium PDEs 21

22 Properties of Nonlinear Problems in Mechanics L1.32 Features of Nonlinear Problems in Mechanics Nonexistence and Nonuniqueness For some load F there may be none, one, many, or an infinite number of solutions u! No Scaling If F causes a displacement u, then af might not cause a displacement au. No Superposition If F causes a displacement u and G causes a displacement v, then F G might not cause a displacement u v. Real Nonlinear Equilibrium PDEs 22

23 L1.33 Features of Nonlinear Problems F 0 in Mechanics Example of nonunique solutions v How many displacement solutions are there to bending a cocktail stirrer (the elastic-plastic cantilever beam)? F = 0 Real v In elastic regime, only one solution. In plastic regime, an infinite number of solutions exist depending on previous load history. F v Nonlinear Equilibrium PDEs L1.34 Features of Nonlinear Problems in Mechanics History dependence Real In a linear problem, the solution u is determined by the current value of the external load P. Real K0u P Nonlinear Equilibrium PDEs In a nonlinear problem, the unique solution u at time t is determined by the entire load history of P(t). Nonlinear Equilibrium PDEs V T β σ dv P Linearized Equilibrium PDEs 23

24 L1.35 Features of Nonlinear Problems in Mechanics Example of history dependence How many displaced shapes are there for a pop top lid (the shallow arch problem)? F >0 v Real F Nonlinear Equilibrium PDEs v Numerical Techniques for Solving Nonlinear Problems 24

25 L1.37 Numerical Techniques for Solving Nonlinear Problems For nonlinear, static, structural mechanics problems, the system of equations is the statement of static equilibrium: where V T I β σ dv. P I = 0, Many different techniques have been proposed for solving such nonlinear systems of equations. In all cases the total applied load is broken down into small increments. An approximate solution is obtained for each load increment. It is likely that several iterations will be needed to obtain an approximate solution that is sufficiently accurate. L1.38 Numerical Techniques for Solving Nonlinear Problems Two of the more robust iterative methods are the Newton-Raphson technique and the quasi-newton technique. Both are incremental/iterative methods. Both are available in Abaqus/Standard. 25

26 L1.39 Numerical Techniques for Solving Nonlinear Problems Newton-Raphson technique With this technique, which is described in detail in Lecture 2, each iteration involves the formulation and solution of linearized equilibrium equations. Each iteration consists of defining the terms in the equilibrium equations (forming the stiffness matrix) and solving the resulting system. The solution from an iteration is deemed sufficiently accurate if the error in the equilibrium equation is smaller than certain tolerances: P I R tolerances. Numerical Techniques for Solving Nonlinear Problems Quasi-Newton method The quasi-newton method differs from the full Newton-Raphson method in how frequently the stiffness matrix is recalculated. In the full Newton-Raphson method the stiffness is recalculated in every iteration. In the quasi-newton method it is not recalculated in every iteration. Thus, the quasi-newton method can provide substantial savings of computational effort if the number of iterations does not increase. In Abaqus the quasi-newton method reforms the stiffness matrix every eight iterations. This default value can be modified by the user. L

27 L1.41 Numerical Techniques for Solving Nonlinear Problems The quasi-newton method is most successful when the system of equations is large and the stiffness matrix is not changing much from iteration to iteration. This can be the case in a dynamic analysis using implicit time integration or in a small-displacement analysis with local plasticity. The quasi-newton method in Abaqus/Standard has the following limitations: It cannot be used with unsymmetric problems, which includes fully coupled temperature-displacement analyses and problems with high friction coefficients. L1.42 Numerical Techniques for Solving Nonlinear Problems Usage: *SOLUTION TECHNIQUE, TYPE=QUASI-NEWTON 27

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31 Nonlinear FEA with Abaqus/Standard Lecture 2 L2.2 Overview Equilibrium Revisited Nonlinear Solution Methods Abaqus/Standard Convergence Criteria: An Overview Automatic Time Incrementation Contact Convergence 31

32 Equilibrium Revisited L2.4 Equilibrium Revisited The basic statement of static equilibrium is that the internal (I) and external (P) forces should balance: P I 0. 32

33 L2.5 Equilibrium Revisited In a nonlinear problem the internal forces in the model, I(u,, t,, f i, ), may be a nonlinear function of: u (displacement) (strain) t (time) (temperature) f i (user-defined field variables) Other variables The history of any of the above variables In a nonlinear problem the external forces in the model, P, may also be a nonlinear function of such independent variables as u and t. Nonlinear Solution Methods 33

34 L2.7 Nonlinear Solution Methods Consider an analysis in which you know the total load applied and the (initial) stiffness of the structure. The goal is to find the final displacement. In a linear analysis the final displacement could be found with one calculation. Load In a nonlinear problem this is not possible because the structure s stiffness changes as it deforms. P P 0 u 0 u? Displacement Nonlinear force vs. displacement curve L2.8 Nonlinear Solution Methods The solution of such a nonlinear problem requires an incremental/iterative technique. The solution provided with such a technique is an approximation of the actual solution to the nonlinear problem: P I 0. There is generally no exact solution to these equations so Abaqus solves the equation iteratively by using the Newton-Raphson method to find an approximate solution that minimizes the residuals! 34

35 L2.9 Nonlinear Solution Methods Using Abaqus The load history versus time is defined as a sequence of steps. Each step is broken up into a sequence of time increments. You specify a guess for the initial time increment *static Δt-initial, t-step Abaqus determines all other time increment sizes using its automatic time incrementation algorithm. At the end of each increment the current load magnitude is calculated by Abaqus as a function of time using the concept of load amplitude. Nonlinear Solution Methods The total solution at the end of each increment is solved iteratively using the Newton-Raphson procedure. Convergence of the Newton-Raphson procedure is judged using convergence tolerances. If iteration is not successful, the increment size is reduced. A new attempt using a smaller increment size is begun. L

36 BEGIN Solution procedure without contact L2.11 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Compute residuals Yes Compute new t Reduce t No Convergence likely? No Converged? No No Yes Analysis finished? Yes Step finished? Output results Yes DONE! BEGIN Solution procedure without contact L2.12 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Change contact constraints if necessary Contact changes? Compute residuals Yes Compute new t Reduce t No Convergence likely? No Converged? No No Yes Analysis finished? Yes Step finished? Output results 36 Yes DONE!

37 BEGIN Nonconvergence L2.13 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Change contact constraints if necessary Contact changes? Compute residuals Yes Compute new t Reduce t Yes Continue? No Convergence likely? No Converged? No No No Yes Analysis finished? Yes Step finished? Output results Yes DONE! NONCONVERGENCE! L2.14 Nonlinear Solution Methods Steps, increments, and iterations Analysis steps The load history for a simulation consists of one or more steps. Increments An increment is part of a step. Iterations In static problems the total load applied in a step is broken into smaller increments so that the nonlinear solution path may be followed. An iteration is an attempt at finding the equilibrium solution in an increment. 37

38 L2.15 Nonlinear Solution Methods Newton-Raphson method Abaqus/Standard uses an incremental-iterative solution technique based on the Newton-Raphson method. The method is unconditionally stable (any size increment can be used). Accuracy in dynamic analysis is affected by the increment size. Each increment usually requires several iterations to achieve convergence, and each step is usually made up of several increments. Newton-Raphson locates a displaced shape in which the residuals are zero. Nonlinear Solution Methods The basics of the Newton-Raphson method Equilibrium is a nonlinear equation in u: P I 0. Newton-Raphson iteration solves a linear equation in c u : where c u is the correction to u. Ktangent cu P I, L

39 L2.17 Nonlinear Solution Methods What are residuals? To obtain the linear system of equations, re-write the equilibrium equations as follows: R u P I, where R(u) are the residuals at u. The residuals represent the out-of-balance force at u. In general, R(u) 0 but, in equilibrium, R(u) = 0. Remember: R(u) is nonlinear! L2.18 Nonlinear Solution Methods Physically, residuals represent the magnitude and distribution of extra external force at each degree of freedom needed to bring the structure into equilibrium at u: R( u) P I. u IS NOT an equilibrium position for P : 0 ( P R( u)) I. u IS an equilibrium position for P R(u). 39

40 L2.19 Nonlinear Solution Methods Assume you are NOT in equilibrium at displaced position u so that R(u) 0: Find c u so that u + c u IS in equilibrium R(u+ c u ) = 0. Expand R(u) in a Taylor series about current displacement u: R R( u cu) R( u) cu 0. u u Throw away higher-order terms and solve the resulting equations for c u. R c u R ( u ) u u c u is known as the displacement correction. Nonlinear Solution Methods Substitute the equilibrium equation into the Newton-Raphson scheme: L2.20 R c u R ( u ) u u Ktangent cu P I This equation is LINEAR in c u! Known at u! 40

41 L2.21 Nonlinear Solution Methods Properties of the Newton-Raphson method Highly convergent if: System starts in equilibrium System is stable Ktangent cu P I Tangent stiffness matrix is consistent L2.22 Nonlinear Solution Methods Once c u is calculated using the Newton-Raphson formula, update u using ui 1 ui c u. Are you done? Equilibrium check: Test to see if u i+1 is an equilibrium position (i.e., check whether a converged solution has been found). How? 41

42 L2.23 Nonlinear Solution Methods Testing for convergence Two criteria are used to test for convergence: the sum of all forces acting on each node the displacement correction Both should be small. If not, iterate again (i.e., calculate another solution for the incremental load). Iterations are continued until the solution converges. Often several iterations are needed to find a converged solution. Nonlinear Solution Methods If Abaqus cannot find a solution for a given increment of applied load, another attempt will be made to obtain a converged solution. When a new attempt is made, Abaqus reduces (cuts back) the magnitude of the load increment. Several attempts may be used in any increment of the analysis. If too many attempts are made in a single increment, Abaqus terminates the analysis the model has failed to converge. L

43 L2.25 Nonlinear Solution Methods The goal of Abaqus is to use approximately 4 6 equilibrium iterations to find a converged solution to the load increment, although occasionally the number of iterations may be as high as 10. Although it may be possible to obtain a solution for a larger load increment if more iterations are used, the extra cost of the additional iterations will usually outweigh the cost savings due to the reduced number of increments in the analysis. The solution at the end of an increment is, by definition, converged. Nonlinear Solution Methods Simple single degree-of-freedom example L2.26 P (external load) K K u (displacement) 43

44 L2.27 Nonlinear Solution Methods Apply load increment, P, and solve for displacement correction, c 1, based on tangent stiffness, K 0, and internal forces, I 0 : K 0 c 1 = P TOTAL I 0. Update current state of model: u 1 = u 0 + c 1 ; I 1 = I(u 1 ). Calculate force residual, R 1, and test to see if it is small : R 1 = P TOTAL I 1. Nonlinear Solution Methods L2.28 Form new tangent stiffness, K 1, based on current configuration (u 1 ). Solve for new displacement correction, c 2 : K 1 c 2 = P TOTAL I 1. Update current state of model: u 2 = u 1 + c 2 ; I 2 =I(u 2 ). Calculate force residual, R 2, and test to see if it is small : R 2 = P TOTAL I 2. 44

45 L2.29 Nonlinear Solution Methods Newton-Raphson graph P 1 P R(u 0 ) K(u 0 ) R(u 1 ) K(u 1 ) R(u 2 ) K(u 2 ) R(u 3 ) V β T σ dv P 0 u 0 u 1 u 2 u 3 u 1 u 3 CONVERGED! but u 3 u 1! u u 1 u 2 u 3 L2.30 Nonlinear Solution Methods Once an iteration produces a converged solution, Abaqus applies a new increment of load and the process is repeated. A flowchart summarizing the Newton- Raphson method as implemented in Abaqus is shown at right. The basic steps include: Form LHS (stiffness) and RHS (residual) Solve Update Check Iterate (if needed) Begin increment Iteration loop No Form K tangent, R Solve for c u Update u Form K tangent, R Check residuals Converged? Yes Output results Next Increment (or Done) 45

46 L2.31 Nonlinear Solution Methods Quadratic convergence For relatively smooth nonlinear response Newton-Raphson exhibits quadratic convergence: Relative error between successive iterations decreases by relative error itself: 2 e new e old e old e new Quadratic convergence is achieved only once the solution estimate is within the N-R method s radius of convergence. If the solution to the initial iterations is not within this radius of convergence, Abaqus may not converge. Discontinuous nonlinearities, such as contact or failure, typically have an adverse impact on convergence behavior. L2.32 Nonlinear Solution Methods Extrapolating the previous solution By default, Abaqus/Standard will extrapolate the u calculated in the previous increment and use it as the estimate for u in the current increment. This method usually helps improve the rate of convergence in the analysis and often speeds up the simulation. The extrapolation of u can be controlled by the user by using the EXTRAPOLATION parameter on the STEP option. 46

47 Abaqus/Standard Convergence Criteria: An Overview L2.34 Abaqus/Standard Convergence Criteria: An Overview Local versus Global convergence criteria Local convergence Requires that one or more convergence criteria must be met at every node in the model to accept an iteration as converged Global convergence Examines prescribed quantities (such as energy balance) summed over the entire model to ascertain convergence Abaqus uses local convergence criteria More conservative Ensures that correct solution will be attained 47

48 L2.35 Abaqus/Standard Convergence Criteria: An Overview When is the solution accepted? Determining when the force residual, R i, is sufficiently small is key to accepting the solution for the current load increment, P, and proceeding to the next increment. The magnitude of the tolerances used to determine if an iteration has produced a converged solution is very important. The tolerances must be small enough to ensure that the approximate solution is close to the exact mathematical solution of the problem. However, they must be large enough that only a reasonable number of iterations are performed. Abaqus/Standard Convergence Criteria: An Overview Average force Abaqus determines an average nodal force, q, for the model in each iteration. Internal forces are summed over all active degrees of freedom in the model. Active external forces are also summed. L2.36 q (Sum of internal forces) (Sum of external forces) (No. of active dofs in model) (No. of active external forces) Abaqus automatically determines the "active" regions in the model. 48

49 L2.37 Abaqus/Standard Convergence Criteria: An Overview How is the average force computed? The average force (in increment i ) is the average of all of the nodal force component magnitudes in all of the elements in the structure: N Nnodes elem N elems dofs node Nloads I n external favg fi fi Nsums e 1 n 1 i 1 i 1 f y 2 + ij f x L2.38 Abaqus/Standard Convergence Criteria: An Overview Time average force Abaqus also determines a time average force, q ~, for the model throughout each step: ~ (Sum of prior q for converged iterations in this step) q (No. of increments so far inthis step) q ~ is recalculated at each iteration of the current increment. For multi-step analyses q ~ is passed to the next step as an initial value for that step. 49

50 L2.39 Abaqus/Standard Convergence Criteria: An Overview Residual tolerance By default, Abaqus/Standard requires that the maximum residual (out-ofbalance force) is less than or equal to 0.5 of the time average force for every node in the model to accept an iteration as converged: Selection of 0.5 R q. max Provides sufficient accuracy for most nonlinear problems Represents a reasonable trade-off between accuracy and efficiency for quadratic convergence Abaqus/Standard Convergence Criteria: An Overview Concepts of average force and time average force adapt automatically for all possible modeling scenarios: L2.40 Cardiac stent under deployment loads Concrete arch dam subjected to selfweight and hydrostatic pressure ~q ~q Cardiac stent model courtesy Nitinol Devices & Components of the Cordis Corporation and SIMULIA Western Region 50

51 L2.41 Abaqus/Standard Convergence Criteria: An Overview Additional convergence criteria Maximum displacement correction check By default, this is limited to less than or equal to 1 maximum displacement increment: of the c 0. 01Δu max max L2.42 Abaqus/Standard Convergence Criteria: An Overview Constraint equation checks Hybrid elements The volumetric deformation of the element is treated as an active degree of freedom in the analysis. The volumetric degree of freedom must be compatible with the actual element volume. By default, the compatibility error must be less than This error is a residual-type quantity and not an analysis runtime ERROR. Other convergence checks for distributed coupling, contact, etc. 51

52 L2.43 Abaqus/Standard Convergence Criteria: An Overview Visual diagnostics in Abaqus/Viewer Force equilibrium checks L2.44 Abaqus/Standard Convergence Criteria: An Overview The message (.msg) file also provides detailed convergence information. INCREMENT umax 6 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT 8.750E-02 EQUILIBRIUM ITERATION 1 AVERAGE FORCE 632. TIME AVG. FORCE 329. LARGEST RESIDUAL FORCE 18.1 AT NODE 11 DOF 1 LARGEST INCREMENT OF DISP AT NODE 11 DOF 2 LARGEST CORRECTION TO DISP E-02 AT NODE 11 DOF 1 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE. c max EQUILIBRIUM ITERATION 2 AVERAGE FORCE 629. TIME AVG. FORCE 329. LARGEST RESIDUAL FORCE AT NODE 11 DOF 1 LARGEST INCREMENT OF DISP AT NODE 11 DOF 2 LARGEST CORRECTION TO DISP E-03 AT NODE 11 DOF 2 THE FORCE EQUILIBRIUM EQUATIONS HAVE CONVERGED q R max R q~ max q ~

53 L2.45 Abaqus/Standard Convergence Criteria: An Overview Modifying Abaqus/Standard convergence criteria CONTROLS This option provides access to all solution controls used in Abaqus/Standard. Default values are usually appropriate. Have been determined based upon many years of accumulated experience. L2.46 Abaqus/Standard Convergence Criteria: An Overview Relaxing default convergence criteria may lead to: Acceptance of solutions that are not sufficiently accurate Nonconvergence Generally recommend examination of modeling issues before altering convergence criteria. 53

54 L2.47 Abaqus/Standard Convergence Criteria: An Overview Though default convergence values are usually appropriate, certain cases may require user intervention. Problems with strongly localized stresses (large unstressed regions of model) Some fracture mechanics applications Problems that do not exhibit quadratic convergence due to approximate stiffness Problems with nonsymmetric tangent stiffness for which symmetric solver is used (friction, pressure-dependent plasticity) User elements (UEL) or user materials (UMAT) that use an approximate tangent stiffness For problems without history dependence in which only the final solution is important, solution cost can sometimes be reduced by relaxing convergence controls during initial analysis phases. L2.48 Abaqus/Standard Convergence Criteria: An Overview Example: Thermally coupled disc brake (Example Problem 5.1.1) Brake pad pressed against rotor, which is then rotated 45º Coupled temperature-displacement procedure brake pad Active field Displacement, u Rotation, Temperature, Conjugate flux Force, F Moment, M Heat flux, q rotor Disc brake assembly 54

55 L2.49 Abaqus/Standard Convergence Criteria: An Overview Localized stresses and temperature gradients Default flux convergence criteria are too strict, resulting in additional unnecessary iterations. path Mises stress along circumferential path L2.50 Abaqus/Standard Convergence Criteria: An Overview Addressed by specifying nondefault solution controls Estimate time average force and heat flux in regions of interest Specify flux convergence criteria *CONTROLS,PARAMETERS=FIELD, FIELD=DISPLACEMENT 0.01,,,300. ( Out-of-balance force) max *CONTROLS,PARAMETERS=FIELD, FIELD=TEMPERATURE 0.01,,,900. ( Out-of-balance heat flux) max

56 Total number of iterations L2.51 Abaqus/Standard Convergence Criteria: An Overview Savings in solution cost Modified Controls Default Controls Increments L2.52 Abaqus/Standard Convergence Criteria: An Overview Example: Jounce bumper (Example Problem ) Axisymmetric, hyperfoam energy-absorbing structure used for shock isolation Subjected to significant axial compression Jounce bumper 56

57 L2.53 Abaqus/Standard Convergence Criteria: An Overview Attempt to relax check on displacement corrections *CONTROLS,PARAMETERS=FIELD, FIELD=DISPLACEMENT,0.1 cmax 0. 1 umax Analysis terminates prematurely Modified convergence controls usually do not work well for all problems. Recommend investigating modeling issues before modifying convergence controls Always recommend starting new problem type with default convergence controls Automatic Time Incrementation 57

58 BEGIN Solution procedure without contact L2.55 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Compute residuals Yes Compute new t Reduce t No Convergence likely? No Converged? No No Yes Analysis finished? Yes Step finished? Output results Yes DONE! L2.56 Automatic Time Incrementation The efficient solution of nonlinear problems relies on using an appropriate increment size. When a solution is obtained easily, the increment size should be increased. When it is difficult or impossible to obtain a solution, the increment size should be decreased. The optimal increment size is one in which, on average, 4 6 equilibrium iterations are needed to obtain a converged solution. While it might be possible to take a larger increment size and use many more iterations to find a solution, the solution with more increments will usually be cheaper and also provide more useful information (because the model is in equilibrium) than that available in intermediate iterations. 58

59 L2.57 Automatic Time Incrementation The basics of the algorithm Abaqus uses empirical algorithms to control the size of increments during an analysis. These algorithms are based on the problem-solving experience gained over the past 30+ years. The basic algorithm is based on the number of iterations required to find a converged solution. When a converged solution is obtained in four iterations or less in two consecutive increments, the increment size is increased by a factor of 1.5. When a converged solution is obtained in more than 10 iterations, the next increment size is reduced by 25. Automatic Time Incrementation When convergence is not obtained, Abaqus reduces the increment size (by a factor of 0.25) and tries to find a solution again. In other cases the increment size remains unchanged. There are many other reasons why Abaqus may cut back the increment size (see Lecture 4 through Lecture 8). L

60 L2.59 Automatic Time Incrementation The initial increment size The user should provide an initial increment size for every step in a nonlinear analysis. By default, Abaqus will try to apply the entire load in a single increment, which may result in one or more cutbacks this approach is inefficient and should be avoided. Try to provide a reasonable initial increment size. For most nonlinear static problems an initial increment size that is a small fraction e.g., between 0.01 and 0.1 of the total step size is appropriate. Automatic Time Incrementation Maximum and minimum increment size Abaqus will calculate a minimum increment size based on the total time specified for the step. If the increment size is reduced because of cutbacks to a value below this minimum value, the analysis will terminate. Occasionally the minimum allowable time increment is defined directly by users. There is no default maximum allowable increment size; the entire load can be applied in a single increment. However, by specifying a maximum allowable increment size, a more efficient solution can sometimes be obtained because cutbacks can be avoided. L

61 L2.61 Automatic Time Incrementation Time incrementation in transient problems In transient simulations such as dynamic, creep, or heat transfer problems, the accuracy of the integration of the transient equations must be considered in addition to the equilibrium accuracy. Thus, there are additional time incrementation control parameters for these types of problems. Abaqus does not calculate default values for the time integration accuracy tolerances for transient problems; the user must specify them. In transient problems a cutback will occur if these time integration accuracy tolerances are not satisfied even if the equilibrium criteria are satisfied. The cutback factor will depend on the degree to which the integration accuracy tolerance has been violated. Contact Convergence 61

62 L2.63 Contact Convergence Many Abaqus analyses involve contact interactions. The automatic time-incrementation scheme used in Abaqus is based on tracking and predicting the equilibrium iteration convergence rate. Contact, however, causes kinks in the load vs. displacement curve. There is a slope discontinuity upon change in contact status. As a result, contact changes interrupt overall convergence rate tracking. Deformed shape (Mises stress contours) P 3. Compress tip 2. Contact rigid surface 1. Bend beam Challenging for Newton method! L2.64 Contact Convergence Severe discontinuity iterations (SDIs) are used to filter out contact effects from convergence rate tracking. An SDI is an iteration during which contact constraints change state (open/closed, stick/slip) SDIs are clearly distinguished from equilibrium iterations. The logic to adjust the increment size treats SDIs separately. 62

63 L2.65 Contact Convergence Recall hard contact: Contact pressure Hard contact Compliant Non-compliant Gap distance Penetration distance Penetration for open contacts Tensile stress for closed contacts Default behavior: SDIs do not block convergence Convert SDI : small penetrations/tensile stresses trigger contact status changes (& SDI s) but do not necessarily block convergence Without Convert SDI Contact status changes (SDI s) block convergence L2.66 Contact Convergence Schematic of behavior within an increment (default behavior) Begin increment 1 Identify initially active contact constraints 2 Form and solve system of equations Identify changes 3 in contact constraint status Yes Newton iterations No 5 Determine if tending toward convergence No 4 Check if solution has converged Yes End increment (Reduce increment size and try again) (At least one convergence criterion is not satisfied) (Within convergence tolerances) 63

64 L2.67 Contact Convergence Contact diagnostics in ABAQUS/Viewer Constrained nodes want to open: incompatible contact state Toggle on to see the locations in the model where the contact state is changing. L2.68 Contact Convergence The message (.msg) file also provides detailed contact convergence information: INCREMENT 6 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT 1.266E-02 CONTACT PAIR (ASURF,BSURF) NODE 167 IS NOW SLIPPING. CONTACT PAIR (ASURF,BSURF) NODE 171 IS NOW SLIPPING. : : : : CONTACT PAIR (ASURF,BSURF) NODE 153 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (ASURF,BSURF) NODE 161 OPENS. CONTACT PRESSURE/FORCE IS E+006. CONTACT PAIR (ASURF,BSURF) NODE 165 OPENS. CONTACT PRESSURE/FORCE IS E+006. CONTACT PAIR (CSURF,DSURF) NODE 363 OPENS. CONTACT PRESSURE/FORCE IS E+006. CONTACT PAIR (ESURF,FSURF) NODE 309 IS NOW SLIPPING. Due to slip reversal *PRINT, CONTACT=YES causes this detailed printout. (Useful for troubleshooting) 5 SEVERE DISCONTINUITIES OCCURRED DURING THIS ITERATION. 4 POINTS CHANGED FROM CLOSED TO OPEN 1 POINTS CHANGED FROM STICKING TO SLIPPING Slave nodes that slip; stick/slip messages cause SDIs only if Lagrange friction is used or if slip reversal occurs. Incompatibilities detected in the assumed contact state SDI 64

65 L2.69 Contact Convergence CONVERGENCE CHECKS FOR SEVERE DISCONTINUITY ITERATION 1 MAX. PENETRATION ERROR E-009 AT NODE 331 OF CONTACT PAIR (ESURF,FSURF) MAX. CONTACT FORCE ERROR AT NODE 363 OF CONTACT PAIR (CSURF,DSURF) THE ESTIMATED CONTACT FORCE ERROR IS LARGER THAN THE TIME-AVERAGED FORCE. Convergence checks for contact state AVERAGE FORCE 5.350E+03 TIME AVG. FORCE 3.137E+03 LARGEST RESIDUAL FORCE E+04 AT NODE 333 DOF 2 LARGEST INCREMENT OF DISP E-04 AT NODE 329 DOF 2 LARGEST CORRECTION TO DISP E-05 AT NODE 337 DOF 2 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE. AVERAGE MOMENT 110. TIME AVG. MOMENT 89.0 ALL MOMENT RESIDUALS ARE ZERO LARGEST INCREMENT OF ROTATION 1.847E-33 AT NODE 100 DOF 6 LARGEST CORRECTION TO ROTATION 6.454E-34 AT NODE 300 DOF 6 THE MOMENT EQUILIBRIUM EQUATIONS HAVE CONVERGED Convergence checks for equilibrium Not only is the contact incompatibility too large, but force equilibrium has not been achieved either Contact Convergence Contact controls: Automatic tolerances Abaqus can calculate allowable tolerances for overclosure and separation pressure Automatically adapts tolerances to current modeling situation Derived from force and displacement correction convergence criteria (R max, c max ) Can alleviate contact chattering Chattering occurs when contact points switch between open and overclosed states in successive iterations Insignificant overclosures and/or negative contact pressures can preclude convergence Discussed further in Lecture 5. L

66 66

67 Notes 67

68 68 Notes

69 Solution of Unstable Problems Lecture 3 L3.2 Overview Unstable Quasi-Static Problems Globally Unstable Problems Stabilization of Local Instabilities Symptoms of Local Instability Automated Viscous Damping Implicit Dynamics Examples Stabilization of Initial Rigid Body Motion 69

70 Unstable Quasi-Static Problems L3.4 Unstable Quasi-Static Problems When a quasi-static problem goes unstable, part or all of the model accelerates from one stable configuration to the next. In reality, the problem becomes dynamic in nature: local or global strain energy is transformed into kinetic energy. Usually the unstable transient dynamic response in the problem is not of interest. Instead, the goal is to analyze the problem as a quasi-static process. 70

71 Globally Unstable Problems L3.6 Globally Unstable Problems Globally unstable problems, for which sequences of equilibrium configurations can be associated with varying levels of a global loading, are usually addressed well with arc length methods such as the Riks capability in Abaqus/Standard. Local instabilities, however, may not influence the global load significantly. Contact changes may also prevent the use of global load control. 71

72 L3.7 Globally Unstable Problems Riks method for globally unstable problems Consider this loaddisplacement curve: Load P 2 P 1 Converged solution for increment 1 Displacement Unstable load-displacement curve Globally Unstable Problems At the top of the curve the slope of the load-displacement curve equals zero and the stiffness is singular. This can occur when structures snapthrough, buckle, or collapse. The structure s instability may be the result of geometric or material effects. In an unstable problem the structure must release energy to remain in equilibrium. In reality, this energy is converted to kinetic energy. A way of studying a buckling problem is to use displacement control rather than force control; i.e., you prescribe the motion of a particular part of the model and look at the reaction forces to understand the loaddisplacement behavior. Even with displacement control the structure may buckle dynamically. L3.8 72

73 L3.9 Globally Unstable Problems An alternative is to use the modified Riks method. The basic solution method is still the Newton-Raphson method, so the usual convergence rules apply. It is the method by which the analysis progresses along the solution path that is changed. The Riks method solves for both the displacements and the applied loads to find the equilibrium path. The method can calculate solutions even when the slope of the load-displacement curve is negative. The magnitude of the load must be expressed in terms of a load proportionality factor (LPF), l. The method uses the concept of arc length (l) to track the size of the increment and how far the analysis has progressed. Globally Unstable Problems L3.10 RIKS is an arc-length control procedure The solution is advanced along the load-displacement curve by solving for the equilibrium position a particular arc-length away from the last position. Load l l Displacement 73

74 L3.11 Globally Unstable Problems In a Riks analysis the loading forms part of the solution, and the analysis will not simply finish when the load specified in the model is reached. An alternative method of halting the analysis must be used. Specify either: The maximum value of the LPF. A maximum displacement at a given node. If neither is specified, Abaqus will continue until the maximum number of increments has been reached. The Riks method will usually work well only for globally unstable problems. The Riks method is described in detail in the Abaqus Analysis User's Manual as well as the Buckling, Postbuckling and Collapse Analysis with Abaqus seminar. Globally Unstable Problems L3.12 Example: Global collapse analysis of frame structure The stability of structures is a problem that analysts face frequently. In this example we consider the postbuckling behavior of the rectangular frame shown at right under the application of a point load at the corner. Rectangular cross-section Linear elastic material Pinned at ends Follower force 74

75 L3.13 Globally Unstable Problems Static analysis fails because of snap-through behavior. Primary symptom is divergence: ***WARNING: THE SOLUTION APPEARS TO BE DIVERGING. Static analysis fails at ~30% of applied load L3.14 Globally Unstable Problems Solution is obtained using the Riks method. *STEP, NLGEOM, INC=100, UNSYMM=YES *STATIC, RIKS 0.02, 1.0,, 0.02,

76 L3.15 Globally Unstable Problems A follower force is applied to the frame. *CLOAD, FOLLOWER 13, 2, Postbuckling shapes of frame L3.16 Globally Unstable Problems Static analysis fails here Global postbuckling: Trajectory of corner of frame Global postbuckling: Load versus displacement 76

77 L3.17 Globally Unstable Problems Example: Buckling of a rubber keyboard dome Rubber dome is compressed using a pressure load. Outer boundary is fixed. Contact interaction defined between the inside of the dome and the floor. The unstable postbuckling response is investigated using the Riks method. Load may increase or decrease as the solution progresses. floor Cutaway view of the model pressure load fixed BCs L3.18 Globally Unstable Problems The picture at the right shows a series of deformed configurations. The dome collapses unstably at the limit load. After contact is established with the floor, the dome stiffens and the response is stable (see load-displacement curve). The final deformed state is shown below. Limit load Contact established 77

78 Stabilization of Local Instabilities L3.20 Stabilization of Local Instabilities Abaqus tries to address two other classes of unstable problems: Problems that develop local instabilities in the course of deformation. Contact problems that have initial rigid body motions. The origin of local instabilities can be threefold: 1. Geometric, such as local buckling. 2. Contact, such as separation of a body that applies loads to other portions of the model. 3. Frictional stick-slip behavior. Localization due to material softening is also a form of local instability. This kind of problem leads to very large local strains and requires special modeling techniques not discussed in this course. 78

79 L3.21 Stabilization of Local Instabilities One approach to solving quasi-static problems with local instabilities is to use damping. Local strain energy is transformed into viscous dissipated energy until a new stable configuration is found. Two approaches for transforming local strain energy in viscous dissipated energy are found in the literature: A dynamic relaxation -like approach. Direct application of damping. Abaqus follows the second approach. It offers the advantage that unstable shapes are obtained, which provide insight to the user. The problems addressed are quasi-static; therefore, the lower structural modes need to be damped, and volume proportional damping is applied. Another approach is to use a dynamic analysis procedure. Abaqus offers a specialized solution technique for quasi-static problems based on the implicit dynamics procedure. Symptoms of Local Instability 79

80 L3.23 Symptoms of Local Instability Sudden convergence difficulty with very small time increments Status (.sta) file excerpt: e e e e e : : e e e e e Sudden convergence difficulty Continued convergence difficulty No. of cutbacks Time increment L3.24 Symptoms of Local Instability Message (.msg) file excerpt: EQUILIBRIUM ITERATION 4 AVERAGE FORCE 147. TIME AVG. FORCE 95.8 LARGEST RESIDUAL FORCE 2.315E-06 AT NODE 1487 DOF 2 LARGEST INCREMENT OF DISP E-09 AT NODE 4131 DOF 1 LARGEST CORRECTION TO DISP E-10 AT NODE 1263 DOF 1 DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREMENT AVERAGE MOMENT 72.9 TIME AVG. MOMENT 42.2 LARGEST RESIDUAL MOMENT 8.555E-05 AT NODE 4131 DOF 5 LARGEST INCREMENT OF ROTATION 6.594E-11 AT NODE 1344 DOF 5 LARGEST CORRECTION TO ROTATION 1.455E-11 AT NODE 1344 DOF 5 THE MOMENT EQUILIBRIUM EQUATIONS HAVE CONVERGED ***WARNING: THE SOLUTION APPEARS TO BE DIVERGING. ***NOTE: THE INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-11 ***ERROR: TOO MANY ATTEMPTS MADE FOR THIS INCREMENT: ANALYSIS TERMINATED 80

81 L3.25 Symptoms of Local Instability Example: Safety loads on a luggage retention system Model represents the bottom of a car trunk. Holes are luggage strap attachment points. Experimental data: Inertia transferred via attachment points when go over a bump or in an accident. Localized buckling in vicinity of holes; shape of buckling changes during loading. attachment points Courtesy of Ford Motor Company, Germany L3.26 Symptoms of Local Instability Large contact overclosures with numerical singularity warnings INCREMENT 1 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT ***WARNING: SOLVER PROBLEM. NUMERICAL SINGULARITY WHEN PROCESSING NODE 111 D.O.F. 2 RATIO = E+13 ***WARNING: SOLVER PROBLEM. NUMERICAL SINGULARITY WHEN PROCESSING NODE 111 D.O.F. 6 RATIO = E+09 ***WARNING: Solver problem. Numerical singularity when processing D.O.F. 6 of 1 nodes. The nodes have been identified in node set WarnNodeSolvProbNumSing_6_1_1_1_1. ***WARNING: Solver problem. Numerical singularity when processing D.O.F. 2 of 1 nodes. The nodes have been identified in node set WarnNodeSolvProbNumSing_2_1_1_1_1. ***WARNING: OVERCLOSURE OF CONTACT SURFACES _G13 and _G12 IS TOO SEVERE -- CUTBACK WILL RESULT. YOU MAY WANT TO CHANGE THE VALUE OF HCRIT ON THE *CONTACT PAIR OPTION. ***WARNING: OVERCLOSURE OF CONTACT SURFACES _G13 and _G12 IS TOO SEVERE -- CUTBACK WILL RESULT. YOU MAY WANT TO CHANGE THE VALUE OF HCRIT ON THE *CONTACT PAIR OPTION. 81

82 L3.27 Symptoms of Local Instability This indicates unconstrained rigid body motion Example: Sheet forming operation Pressure applied to blank to deform it into shape of the die. Contact between the blank and die prevents rigid body motion only after contact has been established. Need to control initial rigid body motion. Automated Viscous Damping 82

83 L3.29 Automated Viscous Damping Designed to help obtain solutions for unstable, quasi-static problems Automatically applies damping forces to local regions that develop sudden instabilities Smoothes the severe discontinuity in response Automated Viscous Damping L3.30 A mass matrix is constructed with unit density. Damping is then applied to the equilibrium equations, where u = nodal displacement vector, M * = mass matrix with unit density, I = internal force vector, P = external load vector, c = damping factor, and u * cm u I u P, = nodal velocity vector. 83

84 L3.31 Automated Viscous Damping The equations solved in each Newton-Raphson iteration become where K t R du c t M * u c * * u Kt M du R - cm, t t = static tangent stiffness matrix = force residual vector (P-I) = nodal displacement correction = damping factor = time increment = mass matrix with unit density = nodal displacement increment. If the structure is stable, K t will be much larger than c t M. L3.32 Automated Viscous Damping As a problem turns unstable, large increments (in terms of displacements and strains) will be obtained, usually leading to divergence. Abaqus responds by using smaller and smaller time increments. The mass component of the operator will increase in size and eventually dominate the stiffness component, thus stabilizing the structure. The static stabilization procedure always uses a unit density, regardless of whether a real density is provided. No damping is associated with point masses or rotary inertias; discrete dashpots must be used if such damping contributions should be added. 84

85 L3.33 Automated Viscous Damping Automatic selection of the damping factor Abaqus automatically calculates the damping factor c. By default, it varies in space and with time. Abaqus automatically adapts the damping factor based on the convergence history and the ratio of energy dissipated by viscous damping to the total energy. Alternatively, the damping factor can be held constant over the duration of a step. The initial damping factor is based on the following premises: The model s response in the first increment of a step to which damping is applied is stable. Under stable circumstances the amount of dissipated energy should be very small. Automated Viscous Damping The initial damping factor is calculated as follows: During the first increment of the step, calculations are made of strain energy and dissipation energy. These energies are extrapolated to the time length of the step, as if the solution were to be scaled linearly in time. The damping factor c is determined in such a way that the viscous dissipation energy is a small fraction of the model s strain energy. This small fraction, called the dissipation intensity, is controlled by the user. It has a default value of L

86 L3.35 Automated Viscous Damping Direct specification of the damping factor Alternatively, the user may specify the damping factor directly. For adaptive automatic stabilization, the user-specified damping factor serves as an initial value. Since c is related to the model size and material stiffness in nonobvious ways, it may be difficult to choose a proper value. Initiate a run without explicit declaration of a damping factor. Abaqus prints out the value of the damping factor, which can then be used as a guideline for selecting appropriate values. Automated Viscous Damping Unstable or singular first increment There are cases where the first increment is either unstable or singular (due to a rigid body mode). It is not possible to obtain a solution in the first increment without damping. Abaqus precomputes the damping factor based on a sampling of the average element stiffnesses. L3.36 If the calculated strain energy change in the first increment indicates that a solution without damping is stable, the damping factor is recalculated as described earlier; otherwise, the initially calculated factor is maintained. Warning: The damping factor may be stronger than desired; critically review the solution. If necessary, follow the stabilized step with another step in which stabilization is not used or with a step in which a much smaller damping factor is used. 86

87 L3.37 Automated Viscous Damping Abaqus usage Automatic stabilization can be added in the following quasi-static procedures in Abaqus: STATIC VISCO COUPLED TEMPERATURE-DISPLACEMENT SOILS, CONSOLIDATION L3.38 Automated Viscous Damping For example, or or *STATIC, STABILIZE *STATIC, STABILIZE=..., ALLSDTOL=... *STATIC, STABILIZE, FACTOR=..., ALLSDTOL=... To suppress adaptive automatic stabilization, set ALLSDTOL to zero. This parameter defines the maximum allowable ratio of the stabilization energy to the total strain energy. 87

88 L3.39 Automated Viscous Damping Damping factors are not carried over from one step to another. Exception: adaptively computed damping factors can be propagated to subsequent steps. *STATIC, STABILIZE=..., ALLSDTOL=..., CONTINUE=YES Automated Viscous Damping Output variables The total amount of viscous energy dissipated by this process is reported separately from other viscous dissipation energies by means of the element output variables ELSD and ESDDEN and the global energy variable ALLSD. Use the ENERGY OUTPUT, ENERGY PRINT, or ENERGY FILE options to request this output. The reported energy can be limited to a group of elements. The nodal viscous forces and moments are available as nodal output variable VF (VFn and VMn). The damping factor calculated by Abaqus is reported in the message (.msg) file. L

89 L3.41 Automated Viscous Damping By default, not written to.odb file. ELSD ELSDDN ALLSD VF c Element stabilization dissipation energy Element stabilization dissipation energy density Element set or model stabilization dissipation energy Nodal viscous forces Damping factor (message file) L3.42 Automated Viscous Damping If you want to know how much you altered a problem by adding stabilization, look at: 1 Energy dissipation due to stabilization Look at whole model energies. Here, the total energy dissipated due to stabilization is very small compared to the total energies involved in deformation. 89

90 L3.43 Automated Viscous Damping 2 Viscous forces during deformation The figures at right show the applied and viscous forces at the load application point as functions of displacement VF varies significantly in time; its order-of-magnitude is very small compared to the global load, however. 1.5 Can conclude the presence of stabilization has not changed the problem significantly. L3.44 Automated Viscous Damping Usage hints The automatic calculation of the dissipation factor, c, is based on information obtained during the first increment of a step. The first increment should be representative of the deformation pattern that follows. If the character of the deformation changes during the step, split the step to force a re-evaluation of damping. If the first part of the step can be completed without stabilization, it is better to split the step and introduce stabilization in the latter steps. Ensures stable response is the basis for computing the dissipation factor. 90

91 Implicit Dynamics L3.46 Implicit Dynamics Abaqus/Standard uses implicit time integration to calculate the transient dynamic or quasi-static response of a system. Three application types: dynamic responses requiring transient fidelity and involving minimal energy dissipation; dynamic responses involving nonlinearity, contact, and moderate energy dissipation; and quasi-static responses in which considerable energy dissipation provides stability and improved convergence behavior for determining an essentially static solution. This offers a simple alternative approach to stabilizing unstable quasi-static problems. 91

92 L3.47 Implicit Dynamics *Dynamic, Application = Bouncing disc example: Moderate Dissipation Transient Fidelity Quasi-static 1 st setting (default) 2 nd setting 3 rd setting Default for contact models Default for noncontact models Intended for quasi-static modeling 234 solver passes 1277 solver passes 168 solver passes Kinetic Energy Comparison L3.48 Implicit Dynamics Quasi-static applications Intended for cases in which a static solution is desired but stabilizing effects of inertia are beneficial Upon convergence difficulty with static procedure Default amplitude type is ramp instead of step Like the general static procedure High numerical dissipation Backward Euler time integrator u u v v t t t t t t v a t t t t t t t t Ma t t - I t t - R P Automatic time incrementation scheme identical to that used for static procedures Adjust the density (or loading rate) to ensure the ratio of kinetic to internal energy is small Crimping prediction Wire crimping example Original configuration 92

93 Examples L3.50 Examples Bending of a thin tube The material model is elasticplastic. Rigid end caps are attached to the nodes at each end of the tube. Apply rotation to one end of the tube; pin the other end. Local buckling of tube walls occurs when bending is severe. 93

94 L3.51 Examples Approach 1: Automatic static stabilization *Step, nlgeom=yes, inc=1000 *Static, stabilize 0.1, 1., 1e-08, 1. ** *Controls, parameters=time incrementation,,,,,,, 10 Minimum allowable time increment reduced to allow the damping to take effect; not uncommon for problems requiring stabilization Allow 10 attempts per increment; not uncommon for this class of problem Approach 2: Implicit dynamics (quasi-static application type) *Step, nlgeom=yes, inc=1000 *Dynamic, application=quasi-static 0.1, 1.0 L3.52 Examples Incrementation details Static stabilization Implicit dynamics (quasi-static app) Number of increments Number of iterations Smallest time increment required Max number of attempts required in any given increment 1.2e-8 1.0e

95 L3.53 Examples Results When local buckling occurs, a large fraction of the strain energy in the tube is released. The kinetic and stabilization energies remain a small fraction of the internal energy. Examples Mises stress distribution Reaction moments, deformed shapes, and stress contours show excellent agreement In the dynamic solution, a brief period of oscillation occurs at the onset of instability static stabilization L3.54 dynamic 95

96 L3.55 Examples Local buckling of reinforced plate Plate with small and large reinforcements Linear elastic material The plate represents part of a larger structure: the two longitudinal sides have symmetry boundary conditions, and the two transverse sides have pinned boundary conditions Spring connections to the rest of the structure Compressive axial (i.e., longitudinal) loading symmetry pinned springs pinned symmetry Courtesy of IRCN-France L3.56 Examples First localized buckling occurs in the plate sections between the small reinforcements. Then, buckling of a line of sections and small reinforcements occurs corresponding to the maximum load carrying capacity. Total collapse of the plate follows. global buckling Contours of localized plate section buckling displacements normal to structure Buckling of a line of sections in the structure 96

97 L3.57 Examples Approach 1: Automatic static stabilization *Step, nlgeom=yes, inc=1000 *Static, stabilize 0.1, 1.0 Approach 2: Implicit dynamics (quasi-static application type) *Step, nlgeom=yes, inc=1000 *Dynamic, application=quasi-static 0.01, 1.0 L3.58 Examples Results Buckling of line of sections (loss of load-carrying capacity) Local plate buckling Axial force vs. axial displacement Energy history plots 97

98 L3.59 Examples Weatherseal model Plane-strain, half-symmetric model CPE4RH elements Rubber material Frictional contact Two-part analysis: 1. Seal to surrounding sheet metal assembly 2. Window closing effort glass insertion We will focus on the first part only Exhibits (strong) energy release during assembly glass surface rubber weatherseal vehicle window frame Model courtesy of Advanced Elastomer Systems and Manta Corporation; example courtesy of SIMULIA Great Lakes Region L3.60 Examples Static analysis dies at this point. Why? All Force-Deflection response plots show RF2 vs. U2 for the window frame rigid surface 98

99 L3.61 Examples Static stabilization This analysis can be run successfully using automatic stabilization. When stabilization is used, the minimum allowable increment size may need to be reduced to allow the damping to take effect. The default minimum is 1.E-05 times the step time. Minimum allowable increment size reduced ** Step with stabilization *STEP,INC=100,NLGEOM *STATIC, STABILIZE 0.02,1.0,1.E-10,,1.0 *BOUNDARY 9999,2,2,12.5 *END STEP Excerpt from status file U e U e e e-06 L3.62 Examples Static stabilization (cont d) The analysis with stabilization runs successfully. The solution path prior to the onset of instability is nearly identical to that obtained with a regular static analysis. 99

100 L3.63 Examples Implicit dynamics (quasi-static application type) The static instability disappears in a dynamic analysis. A brief period of oscillation occurs at the point of instability but is much less noisy than would occur in a true dynamic analysis Stabilization of Initial Rigid Body Motion 100

101 L3.65 Stabilization of Initial Rigid Body Motion Unconstrained rigid body motion (statics only) Many mechanical assemblies rely on contact between bodies to prevent unconstrained rigid body motion. Often it is impractical or impossible to model such systems with contact initially established. Example with initial play between pin and other components Without user intervention, Abaqus may report solver singularities in the message (.msg) file : ***WARNING: SOLVER PROBLEM. NUMERICAL SINGULARITY WHEN PROCESSING NODE 17 D.O.F. 2 RATIO = E+16 Typically leads to convergence failure Displacements may be extremely large if a solution is obtained L3.66 Stabilization of Initial Rigid Body Motion Simple example with a singular system of equations prior to establishing contact F 1 k 2 F 1-D representation k -k -k k u 1 F = u 2 0 Determinant is 0 (singular) Workaround #1: Displacement-controlled loading prior to establishing contact Will discuss more generally-applicable strategies Observation: Once contact is established, the system of equations is also stable for force-controlled loading k u f k -k 0 k 1 2 u 2 = ku 1 1 k 2 -k k u 1 Nonsingular, sol n: u2=u1=u F u 2 = 0 l 0 Nonsingular Sol n: u 1 =F/k, u 2 =0, l=f 101

102 L3.67 Stabilization of Initial Rigid Body Motion Automated stabilization of contact Abaqus/Standard offers two capabilities that automatically control rigid body motions in static problems before contact closure and friction restrain such motions: Stabilization based on the stiffness of the underlying elements (recommended method) Stabilization based on the initial opening distance The automated stabilization of contact may be activated during any step of an analysis as an aid to modeling Additional forces are introduced, which will influence the solution during the step. These forces are ramped to zero at the end of the step. The automated stabilization of contact is available for both contact pairs (global or pair-wise basis) and general contact (global assignment only) A different technique for local assignment with general contact is available; it is discussed later. Stabilization of Initial Rigid Body Motion Stabilization based on the stiffness of the underlying elements This method addresses more general situations. By default, the damping coefficient is: L3.68 based on the stiffness of the underlying elements and the step time, applied to all contact pairs equally in the normal and tangential directions, ramped down linearly over the step, and active only when the distance between the contact surfaces is smaller than a characteristic surface dimension. This is the preferred stabilization method, and has a beneficial effect in many cases with loose contact, even if no true rigid body modes are present. For example, in some situations, it may help with chattering (even though stabilization is not designed to address chattering). 102

103 L3.69 Stabilization of Initial Rigid Body Motion Abaqus usage: Use the default damping coefficient: *CONTACT CONTROLS, STABILIZE Scale the default damping coefficient: *CONTACT CONTROLS, STABILIZE=<factor> Specify the damping coefficient directly: *CONTACT CONTROLS, STABILIZE <damping factor> Specify a ramp-down factor at the end of the step: *CONTACT CONTROLS, STABILIZE, <ramp-down factor> Decrease or increase the tangential damping or set it to zero: *CONTACT CONTROLS, STABILIZE, TANGENT FRACTION=<value> L3.70 Stabilization of Initial Rigid Body Motion Example: Joint with pin and spacer Four bodies, connected by contact pairs Without stabilization, this analysis requires a special analysis sequence, many SDIs, and nondefault contact controls With the stabilization procedure, only a small increase in maximum number of SDIs (to 16) is needed in the first increment. Stabilization No Yes Wallclock time (min) # Increments # Iterations Mises stress in pin 103

104 L3.71 Stabilization of Initial Rigid Body Motion Example: Front loader bucket 28 rigid body modes Including 4 free rings Only option added: CONTACT CONTROLS, STABILIZE=1 Model courtesy of Caterpillar Inc. SUMMARY OF JOB INFORMATION: STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF ITERS FREQ Stabilization of Initial Rigid Body Motion Local stabilization for general contact As noted earlier, the *CONTACT CONTROLS option applies stabilization globally when used with general contact To apply local stabilization with general contact, use the *CONTACT STABILIZATION option Can be used to specify either local or global contact stabilization controls for general contact Some differences with *CONTACT CONTROLS No tangential stabilization by default Stabilization is more aggressively ramped down over increments L3.72 This option is not currently supported by Abaqus/CAE Use the Keywords Editor to include it in your model 104

105 L3.73 Stabilization of Initial Rigid Body Motion Stabilization based on the initial opening distance Addresses situations where a single rigid body mode exists normal to the contact surface. Damping is applied only in the contact direction to a specific contact pair. Primarily used for problems where the initial separation is large. Abaqus usage: *CONTACT CONTROLS, APPROACH, MASTER=master-name, SLAVE=slave-name Stage 2 Stage 1 The positioning of a sheet in a 2-stage stamping operation is representative of the type of problem suited for stabilization based on the initial opening distance. L3.74 Stabilization of Initial Rigid Body Motion Output variables The total amount of viscous energy dissipated by this process is included in the global energy variable ALLSD. The contact damping stresses CDSTRESS can be compared to the true contact stresses CSTRESS. 105

106 L3.75 Stabilization of Initial Rigid Body Motion Usage hints This capability is not meant to stabilize unstable contact problems. The automatic stabilization option is more adequate for such problems. It is not intended for making small adjustments in the initial positioning of bodies. In such cases, adjust the surfaces in the contact pair definition. 106

107 Notes 107

108 108 Notes

109 Why Abaqus Fails to Find a Converged Solution Lecture 4 L4.2 Overview The Basic Problems Understanding the Warning Messages Helping Abaqus Find a Converged Solution 109

110 The Basic Problems L4.4 The Basic Problems Nonconvergence happens! There are NO good, general methods for solving systems of more than one nonlinear equation. Furthermore, there NEVER WILL BE any good, general methods. Numerical Recipes: The Art of Scientific Computing (FORTRAN version), by Press, Flannery, Teukolsky, and Vetterling, Cambridge University Press,

111 L4.5 The Basic Problems Levels of approximation (linear) Real 1 = Idealization Nonlinear equilibrium PDEs 2 = Linearization Linearized equilibrium PDEs 3 = Discretization FEA (mesh) Exact Algebraic solution of Ku = F L4.6 The Basic Problems Levels of approximation (nonlinear) Real 1 = Idealization Nonlinear equilibrium PDEs 2 = Discretization FEA (mesh) 3 = Iteration Newton Raphson 111

112 L4.7 The Basic Problems Nonlinear Levels of approximation: Idealization Discretization Solution procedure is approximate. Many linear solutions are needed to minimize the residual. We can t predict beforehand how many linear solutions will be required. Nonconvergence is possible. Linear Levels of approximation: Idealization Linearization Discretization One linear solution is needed. The cost can be predicted in advance. Nonconvergence never occurs. L4.8 The Basic Problems There are almost as many reasons why an Abaqus simulation may fail to find a converged solution as there different types of problems being studied with Abaqus. The majority of convergence problems can be explained and easily corrected. 112

113 L4.9 The Basic Problems Symptoms of convergence problems The symptoms of almost all convergence problems can be found in the message file. The printed output (.dat) and status (.sta) files also may contain the symptoms of a problem. The following message in the message file, ***WARNING: THE SOLUTION APPEARS TO BE DIVERGING is not the cause of a convergence problem it is a reason for Abaqus to cut back the increment size. However, such a warning might be the symptom of a convergence problem in the model. It also might be because too large an increment was used. The Basic Problems The following is a list of warning messages that appear in the message file and may be symptoms of a convergence problem: L4.10 ***WARNING: THE STRAIN INCREMENT HAS EXCEEDED FIFTY TIMES THE STRAIN TO CAUSE FIRST YIELD AT 7 POINTS ***WARNING: THE SYSTEM MATRIX HAS 3 NEGATIVE EIGENVALUES ***WARNING: ELEMENT 441 IS DISTORTING SO MUCH THAT IT TURNS INSIDE OUT ***WARNING: OVERCLOSURE OF CONTACT SURFACES ARM1-A and PIPE IS TOO SEVERE -- CUTBACK WILL RESULT ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING NODE 1 D.O.F

114 L4.11 The Basic Problems Causes of convergence problems The most common cause of convergence problems in nonlinear simulations is inadequate FE modeling. Examples of inadequate FE modeling include: Defining conflicting constraints between boundary conditions, contact conditions, and/or multi-point constraints Not adequately constraining the model allowing rigid body motions Having incomplete (or inadequate) material data Using an inappropriate element type for the problem The Basic Problems Another common cause of convergence problems is that the physical system is very unstable, making it very difficult to find an equilibrium solution. Examples include: Buckling of thin-walled, cylindrical shell structures Snap-through with contact changes Compression of highly confined, incompressible materials (rubber) Development of local instabilities occur In these cases it is important that the correct element type and analysis techniques be used; otherwise, it can be very difficult to obtain a solution. L

115 Understanding the Warning Messages L4.14 Understanding the Warning Messages There are many types of messages in the message (.msg) file that tell you something about nonconvergence of the N-R and the load incrementation algorithm: Algorithmic messages FEA messages Numerical messages No physically-based messages! 115

116 L4.15 Understanding the Warning Messages Reasons for cutback Algorithmic cutbacks Divergence Too slow convergence Too many iterations Stiffness/residual calculation Element distortion Material model calculation problems Contact Too many Severe Discontinuity Iterations Too severe penetration L4.16 Understanding the Warning Messages Algorithm messages These messages refer to the behavior or response of the Abaqus load incrementation algorithm. Examples include: ***NOTE: THE RATE OF CONVERGENCE IS VERY SLOW. ***NOTE: SOLUTION FAILS TO CONVERGE IN MAXIMUM EQUILIBRIUM ITERATIONS ALLOWED. ***WARNING: THE SOLUTION APPEARS TO BE DIVERGING. ***WARNING: CONVERGENCE JUDGED UNLIKELY. INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-02 ***WARNING: FORCE EQUILIBRIUM ACCEPTED USING ALTERNATE TOLERANCE ***ERROR: TOO MANY INCREMENTS NEEDED TO COMPLETE THE STEP ***ERROR: TOO MANY ATTEMPTS MADE FOR THIS INCREMENT: ANALYSIS TERMINATED ***ERROR: TIME INCREMENT REQUIRED IS LESS THAN MINIMUM SPECIFIED. ANALYSIS ENDS 116

117 L4.17 Understanding the Warning Messages More on convergence check During iteration, Abaqus monitors the rate of convergence and checks for Divergence Log rate of convergence Too many iterations If the rate of convergence is too slow, Abaqus cuts back and starts the increment over again. Begin new iteration Iteration loop Form K tangent Solve for Du Update u Compute residuals Yes No Convergence likely? No Converged? Understanding the Warning Messages L4.18 Divergence check Starting after iteration 4, the smallest max residual in the current or previous iteration must be less than the max residual before the previous iteration. That is, if both the current and previous max residual are larger than the 2nd-previous residual, divergence is occurring and the increment is cut back. If divergence occurs, the increment size is reduced ***WARNING: THE SOLUTION APPEARS TO BE DIVERGING. ***NOTE: THE INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-04 VERY COMMON! 117

118 L4.19 Understanding the Warning Messages Divergence check in iteration i max Ru ( ) all dofs Assume the max residual in iteration i-1 is less than the max residual in i-2. Therefore, the max residual in iteration i may be greater or less than the max residual in iteration i-2 to pass. Iteration # 4 i-2 i-1 i L4.20 Understanding the Warning Messages Divergence check in iteration i max Ru ( ) all dofs However, if the max residual in iteration i-1 is greater than the max residual in i-2 The max residual in iteration i must be less than the max residual in i-2 to pass. Iteration # 4 i-2 i-1 i 118

119 L4.21 Understanding the Warning Messages Divergence check in i+1 max Ru ( ) all dofs Assume the max residual in iteration i is less than the max residual in i-2 (and therefore i-1). Thus, the max residual in iteration i+1 may be greater or less than the max residual in iteration i-1. Iteration # 4 i-2 i-1 i i+1 L4.22 Understanding the Warning Messages Divergence check in i+2 max Ru ( ) all dofs Assume the max residual in iteration i+1 is less than the max residual in i-1 but greater than the max residual in i. The max residual in iteration i+2 must be less than max residual in i to pass. 4 i-2 i-1 i i+1 i+2 Iteration # 119

120 L4.23 Understanding the Warning Messages So, divergence is... max Ru ( ) all dofs The max residuals in both iteration i and i-1 are greater than the max residual in i-2. Iteration # 4 i-2 i-1 i L4.24 Understanding the Warning Messages Log rate of convergence check After iteration 8 Abaqus estimates how many iterations will be required for eventual convergence using the actual logarithmic rate of convergence in the increment If more than 16 iterations are estimated to be required, the increment size is reduced 0.5. ***NOTE: THE RATE OF CONVERGENCE IS VERY SLOW. THE INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-05 ***NOTE: THE RATE OF CONVERGENCE SUGGESTS 18 ITERATIONS ARE NEEDED. ***NOTE: THE INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-04 UNCOMMON IN STRESS ANALYSIS! 120

121 L4.25 Understanding the Warning Messages Log rate of convergence check max Ru ( ) all dofs If N > 16, test fails! 0.02q Iteration # 8 i N L4.26 Understanding the Warning Messages Alternate residual In some convergent problems force convergence to 0.5% is not possible. After 9 iterations, Abaqus widens the force convergence tolerance and accepts solutions where max Ru ( ) 0.02q. all dofs EQUILIBRIUM ITERATION 10 AVERAGE FORCE 1.366E+03 TIME AVG. FORCE 506. LARGEST RESIDUAL FORCE 6.68 AT NODE 685 DOF 1 LARGEST INCREMENT OF DISP..812 AT NODE 408 DOF 2 LARGEST CORRECTION TO DISP E-03 AT NODE 284 DOF 2 ***WARNING: FORCE EQUILIBRIUM ACCEPTED USING ALTERNATE TOLERANCE 121

122 L4.27 Understanding the Warning Messages Alternate tolerance max Ru ( ) all dofs Alternate tolerance!!!! Displacement convergence tolerance remains unchanged!! 0.02q 8 10 i N Iteration # L4.28 Understanding the Warning Messages Too many iterations No more than 16 iterations are allowed per increment. If convergence is not attained in 16 iterations, the current increment size is decreased ***NOTE: SOLUTION FAILS TO CONVERGE IN MAXIMUM EQUILIBRIUM ITERATIONS ALLOWED. VERY RARE! Because analysis usually fails due to divergence or log rate check first! 122

123 Helping Abaqus Find a Converged Solution L4.30 Helping Abaqus Find a Converged Solution Build up the model slowly The most important way to help Abaqus find a converged solution is to build up your model piece by piece. Do not put every complexity and detail of your problem into your first model it probably will not work. If you begin with the simplest model possible perhaps one with contact but no plasticity, friction, or nonlinear geometry you will gain valuable insight into how the model behaves. When you then add more complexities (friction or plasticity), do so one at a time. Doing this will limit the number of questions you have to consider if a convergence problem arises. Although it might seem as if this process will increase the time needed to perform the analysis, in fact it often reduces the time because debugging a large model with convergence problems can take days or weeks. 123

124 L4.31 Helping Abaqus Find a Converged Solution Provide reasonable values Give reasonable values for the minimum increment size and the maximum increment size (or use the defaults). Make sure that the units of the material properties (especially density) are consistent with the geometry and loads in the model. Make sure that the material properties provide sufficient stiffness to resist the applied loads, or plan accordingly and use the appropriate analysis techniques. Helping Abaqus Find a Converged Solution How to begin solving a convergence problem If an analysis fails to complete successfully, you will need the information in the message file, the output database (.odb) file, the printed output (.dat) file, the restart (.res) file, and possibly the status (.sta) file to try to get an understanding of why the problem has occurred. Do not delete these files. Do not limit the data written to the message file. L4.32 Use the Job Diagnostics tools available in Abaqus/Viewer to identify the regions in the model with the largest residuals, solution corrections, contact changes, etc. In a contact analysis, you can also use *PRINT, CONTACT=YES to get detailed contact information in the message (.msg) file. 124

125 L4.33 Helping Abaqus Find a Converged Solution Where to look in the Job Diagnostics dialog box or the message file Nonconvergence often occurs suddenly but is carried out over many increments as Abaqus cuts back on the increment size. Distinguish the two types of nonconvergence: Nonconvergence of the Newton-Raphson loop Nonconvergence of the load incrementation algorithm To resolve nonconvergence, always look at the first attempt in the first increment of the cutback sequence! To diagnose nonconvergence, you need to figure out why Newton- Raphson is not converging. Helping Abaqus Find a Converged Solution Status file (edited) L4.34 SUMMARY OF JOB INFORMATION: STEP INC ATT SDI EQUIL TOTAL TOTAL STEP INC OF ITERS ITERS TIME TIME TIME THE ANALYSIS HAS NOT BEEN COMPLETED 125

126 L4.35 Helping Abaqus Find a Converged Solution Job Diagnostics / Message file (edited) L4.36 Helping Abaqus Find a Converged Solution Use this information. For example, the node with the maximum residual in the increment/iteration where convergence problems start to occur is often a good place to start looking when the symptom is a diverging solution. Consider the following: On a contact surface: might be unstable separation or stick-slip behavior. Attached to a reduced-integration element: might be hourglassing instability. Attached to a long slender beam structure: might need to use hybrid beam elements. Repeated warning messages provide an obvious place to start looking: ***WARNING: THE STRAIN INCREMENT HAS EXCEEDED FIFTY TIMES THE STRAIN TO CAUSE FIRST YIELD AT 192 POINTS If this message is the cause of every cutback in a model, most likely either the load is too great or the ultimate yield stress is too low. 126

127 Notes 127

128 128 Notes

129 Convergence Problems: Contact Simulations Lecture 5 L5.2 Overview Unstable Separation of Contacting Surfaces Chattering Between Contact Surfaces Contact with Quadratic Elements Poorly Defined Master Surfaces Friction 129

130 BEGIN Solution procedure without contact L5.3 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Compute residuals Yes Compute new t Reduce t No Convergence likely? No Converged? No No Yes Analysis finished? Yes Step finished? Output results Yes DONE! BEGIN Solution procedure without contact L5.4 Begin new step Begin new increment Begin new attempt Begin new iteration Form K tangent Step loop Increment loop Attempt loop Iteration loop Solve for u Update u Change contact constraints if necessary Contact changes? Compute residuals Yes Compute new t Reduce t No Convergence likely? No Converged? No No Yes Analysis finished? Yes Step finished? Output results 130 Yes DONE!

131 Unstable Separation of Contacting Surfaces L5.6 Unstable Separation of Contacting Surfaces A large quantity of strain energy can be generated in two contacting bodies. Should the two bodies separate suddenly, the release of the elastic strain energy causes unstable response. During this unstable phase it is difficult for Abaqus to find a converged solution. Usually Abaqus will cut back the increment size because the solution diverges. Problems in which contact between two bodies occurs at a single node are especially vulnerable to unstable separation. Snap-through or snap-in problems are the most common type of simulations where unstable separation occurs. 131

132 L5.7 Unstable Separation of Contacting Surfaces Unstable separation of contact surfaces is one of many possible causes of diverging solution warnings in Abaqus. The presence of severe discontinuity iterations (SDIs, where a slave node opens) before the diverging solution warnings is one symptom that contact separation is the cause of the problem. Another symptom of unstable contact is that the node with the largest displacement correction and the magnitude of the correction does not change as Abaqus cuts back the increment size this is a typical symptom of unstable behavior regardless of whether contact is involved or not. There are two techniques for overcoming unstable contact conditions: Adding inertia to the problem (making the problem dynamic). Adding viscous damping to the (static) problem. Unstable Separation of Contacting Surfaces Adding inertia When inertia is added to an unstable contact problem, the inertial forces counterbalance the forces created by the unstable contact conditions. However, the addition of inertia to a contact problem in Abaqus/Standard makes the analysis more complex and more expensive to perform. If implicit dynamics is required, use the quasi-static application type (see Lecture 3). Alternatively, consider using Abaqus/Explicit. This technique, however, is not an ideal option for many practical simulations. L

133 L5.9 Unstable Separation of Contacting Surfaces Adding viscous damping Viscous damping can be used to control unstable problems, even in a static simulation. Abaqus/Standard calculates the nodal velocities as the increment of displacement, u, divided by the increment of time, t. The easiest way to include viscous damping in a model is to use the automated stabilization capability available in Abaqus/Standard (see Lecture 3): *STATIC, STABILIZE Local viscous damping can be added to a model by defining DASHPOT1 elements at selected nodes in the model. Unstable Separation of Contacting Surfaces For unstable contact conditions, viscous damping can be added to the behavior of a contact interaction rather than to all the nodes. The viscous forces will be applied normal to the master surface and will be proportional to the relative approach velocity of the surfaces. The viscous damping coefficient, 0, is defined as a function of the clearance, c, between the surfaces. u slave u master n t L5.10 slave master Vrel n ( u u ) Relative approach velocity of two surfaces 133

134 L5.11 Unstable Separation of Contacting Surfaces Viscous surface damping L5.12 Unstable Separation of Contacting Surfaces Add damping to the behavior of a surface interaction model. *surface interaction, name=intprop-1 *contact damping, definition=damping coefficient 0.01, 0.12, c0 The following example illustrates a technique to determine appropriate values for the contact damping parameters. c 0 c 0 134

135 L5.13 Unstable Separation of Contacting Surfaces Example: Reinforced medical tubing Polymer tube reinforced internally with a series of metallic coils, or filars Beam elements Tube outer radius = 1.0 Tube inner radius = 0.76 Spring radius = 0.03 Linear elastic material ITT elements used to model: Filar-to-filar contact Filar-to-tube contact Lateral load applied to wrap tube around cylinder tube filars F Rigid cylinder Detail of finite element model L5.14 Unstable Separation of Contacting Surfaces Abaqus begins having trouble finding a converged solution at about 3.5 of the applied load. Difficulty determining the contact state Filars are long, flexible, curved wires Behavior is highly unstable as the tube bends around the cylinder. SUMMARY OF JOB INFORMATION: STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF ITERS FREQ U U : Deformed state in early stages of analysis Difficulties begin in this increment and continue throughout analysis 135

136 L5.15 Unstable Separation of Contacting Surfaces The analysis ultimately completes successfully with default controls 144 increments, 604 iterations SUMMARY OF JOB INFORMATION: STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF ITERS FREQ : U THE ANALYSIS HAS COMPLETED SUCCESSFULLY ANALYSIS SUMMARY: TOTAL OF 144 INCREMENTS 39 CUTBACKS IN AUTOMATIC INCREMENTATION 604 ITERATIONS INCLUDING CONTACT ITERATIONS IF PRESENT 604 PASSES THROUGH THE EQUATION SOLVER OF WHICH 604 INVOLVE MATRIX DECOMPOSITION Use contact damping to stabilize the contact and improve convergence. L5.16 Unstable Separation of Contacting Surfaces Estimating the damping factor 1 In the converged increment immediately prior to the onset of cutbacks it is observed in the Job Diagnostics dialog box (or the message file) that u max 7. 5e 3 for t = 1. 5e 3. Thus, estimate u max Look at the Job Diagnostics dialog box (or the message file) to estimate the average force in the model. For the converged increment immediately preceding the onset of cutbacks : CONVERGENCE CHECKS FOR EQUILIBRIUM ITERATION 1 q AVERAGE FORCE 4.765E-03 TIME AVG. FORCE 2.498E-03 LARGEST RESIDUAL FORCE 1.263E-10 AT NODE 7036 DOF 3 LARGEST INCREMENT OF DISP E-03 AT NODE 7036 DOF 1 LARGEST CORRECTION TO DISP E-06 AT NODE 7036 DOF 3 THE FORCE EQUILIBRIUM EQUATIONS HAVE CONVERGED Thus, estimate q

137 L5.17 Unstable Separation of Contacting Surfaces 3 The typical contact area (A c ) for this problem is The damping forces should reach the same order of magnitude as the average force, but at a much lower velocity (e.g., 100 slower). Therefore, calculate based on q 0.005, A c = 1.0, and u max q A c u max 0.1. L5.18 Unstable Separation of Contacting Surfaces 5 Estimate the clearance at which damping drops off to zero. For filar-to-filar and filar-to-tube contact, set equal to a small fraction (e.g., 5 ) of the minimum clearance between the filars: c For tube-to-cylinder contact, set equal to a small fraction (e.g., 1 the tube outer radius: ) of c Assume the damping coefficient drops off linearly:

138 L5.19 Unstable Separation of Contacting Surfaces With contact damping convergence is much easier 37 increments, 173 iterations SUMMARY OF JOB INFORMATION: STEP INC ATT SEVERE EQUIL TOTAL TOTAL STEP INC OF DISCON ITERS ITERS TIME/ TIME/LPF TIME/LPF ITERS FREQ U : : THE ANALYSIS HAS COMPLETED SUCCESSFULLY L5.20 Unstable Separation of Contacting Surfaces Viscous dissipation energy is small relative to the internal energy. Adequate viscous forces were provided during unstable behavior, which had a minimal influence on the model during more stable response. Final deformed shape 138

139 L5.21 Unstable Separation of Contacting Surfaces Additional comments on using viscous damping When viscous damping is added to a model to control unstable behavior, it is best if damping parameters are specified so that Abaqus immediately obtains a converged solution. If the parameters are not sufficient to control the unstable behavior, much time will be spent running simulations that fail to converge. Unstable Separation of Contacting Surfaces Ideally you will perform a simulation that converges with the damping parameters. Then you must ask, Is the damping too large? Is it influencing the model s behavior in the stable regime? To answer these questions, you should reduce the damping parameters (for example, by a factor of 10) and run the simulation again. If this analysis converges and there is no appreciable difference in the solution obtained by Abaqus (determined by comparing force versus deflection curves), you can have some confidence that the damping is not influencing the model in the stable regime. L5.22 If this second analysis fails to converge, you will know that your first set of damping parameters was close to the minimum values needed in the model. 139

140 Chattering Between Contact Surfaces L5.24 Chattering Between Contact Surfaces Chattering is a phenomenon in which Abaqus has difficulty determining which nodes on a slave surface are supposed to be in contact. The symptoms of chattering are repeated SDIs that involve the same nodes. Without detailed contact diagnostics (Job Diagnostics dialog box or PRINT, CONTACT=YES), it is impossible to detect when and where chattering occurs. An example of the output seen in the message (.msg) file when chattering occurs is shown on the next slide. 140

141 L5.25 Chattering Between Contact Surfaces CONTACT PAIR (SLAVE, RIGID1) NODE 2 IS OVERCLOSED BY E SEVERE DISCONTINUITY OCCURRED DURING THIS ITERATION. 1 POINT CHANGED FROM OPEN TO CLOSED same node in most SDIs CONTACT PAIR (SLAVE, RIGID1) NODE 2 OPENS. CONTACT PRESSURE/FORCE IS E SEVERE DISCONTINUITY OCCURRED DURING THIS ITERATION. 1 POINT CHANGED FROM CLOSED TO OPEN CONTACT PAIR (SLAVE, RIGID1) NODE 2 IS OVERCLOSED BY E SEVERE DISCONTINUITY OCCURRED DURING THIS ITERATION. 1 POINT CHANGED FROM OPEN TO CLOSED CONTACT PAIR (SLAVE, RIGID1) NODE 2 OPENS. CONTACT PRESSURE/FORCE IS E SEVERE DISCONTINUITY OCCURRED DURING THIS ITERATION. 1 POINT CHANGED FROM CLOSED TO OPEN L5.26 Chattering Between Contact Surfaces When the (default) node-to-surface contact discretization is used, chattering often occurs at the edge of two bodies that are in contact. As the slave node makes contact with the master surface, it slides off the edge of the master surface (1), but it regains contact one or a few iterations later (2). trimmed master surface slave node Automatically extended master surface For the finite-sliding, node-to-surface contact discretization, Abaqus/Standard automatically extends deformable master surfaces to try to minimize this problem. This problem is less likely with the surface-to-surface contact discretization, because each contact constraint is based on a region of the slave surface rather than individual slave nodes. 141

142 L5.27 Chattering Between Contact Surfaces Chattering can also occur when the normal force between the two contacting bodies is very small. It can be very hard for Abaqus to find an equilibrium configuration for the model in this situation. When chattering is caused for this reason, use the automatic contact tolerances feature to help achieve convergence. Chattering Between Contact Surfaces Automatic contact tolerances The automated contact tolerances allow some slight penetration at a slave node that was not previously in contact and some slight tensile force at a slave node that is predicted to be in contact without causing a contact iteration. The penetration and tensile force tolerances are based on the magnitude of the displacement solution correction and the time-average force at a node. These automatic tolerances are generally recommended, especially in cases where chattering is observed. L

143 L5.29 Chattering Between Contact Surfaces These contact tolerances are computed as follows: The allowable penetration is set to twice the maximum displacement correction. During the first two iterations the allowable tensile contact pressure is equal to 10 times the maximum allowable force residual divided by the contact area of a node. After the second iteration the allowable tensile contact pressure is equal to the maximum allowable force residual divided by the contact area of a node. If convergence occurs during the first two iterations, at least one more iteration is performed with the tighter tolerance. Chattering Between Contact Surfaces The syntax for using these automatic tolerances is: *CONTACT CONTROLS, AUTOMATIC TOLERANCES The SLAVE and MASTER parameters can be used to limit the controls to a specific contact pair. The RESET parameter is used to remove the automatic tolerances. L

144 L5.31 Chattering Between Contact Surfaces Example: Clip insertion problem Pipe ARM2 ARM1 Mesh for clip insertion simulation L5.32 Chattering Between Contact Surfaces The clip is made out of a pliable polymer material. For the purposes of this example, it is modeled as linear elastic. The goals of the simulation are to find the rotation of the pipe as it is pushed into the clip (with an applied displacement in the 2-direction) and to find the force-deflection curve for the pipe. Fourteen analysis steps in the simulation. The pipe is modeled as a rigid surface. The rigid body motions of the pipe, in the 1- and 6-directions, are initially controlled by soft springs attached to the reference node. They are removed early in the simulation. Dashpot elements in the 1- and 6-directions are used to control the unstable motion of the pipe as it snaps into the clip. 144

145 L5.33 Chattering Between Contact Surfaces Abaqus has problems finding a converged solution in the later stages of the simulation. Abaqus/Standard terminates the analysis prematurely in Step 12 because of chattering. The node highlighted in the figure has the most difficulty establishing a stable contact state in the final (failed) increment (# 111) of Step 12. Node 607 L5.34 Chattering Between Contact Surfaces Focus on SDI history in Step 12 of node

146 L5.35 Chattering Between Contact Surfaces An edited summary from the.msg file is shown below: INCREMENT 111 STARTS. ATTEMPT NUMBER 2, TIME INCREMENT 1.000E-08 SDI #2: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 IS OVERCLOSED BY E-008. SDI #3: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 OPENS BY E-009 WITH A CONTACT PRESSURE/FORCE OF E-005. SDI #5: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 IS OVERCLOSED BY E-008. SDI #6: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 OPENS BY E-009 WITH A CONTACT PRESSURE/FORCE OF E-005. SDI #8: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 IS OVERCLOSED BY E-008. SDI #9: CONTACT PAIR (ARM1-B,ARM2-B) NODE 607 OPENS BY E-009 WITH A CONTACT PRESSURE/FORCE OF E-005. : : ***NOTE: A REPETITIVE SDI PATTERN OCCURS. CONVERGENCE IS JUDGED UNLIKELY. Chattering Between Contact Surfaces Using automatic contact tolerances, convergence difficulties are overcome. L5.36 Fully inserted pipe Force vs. deflection curve for the fully inserted clip simulation 146

147 L5.37 Chattering Between Contact Surfaces Alternatives to automatic contact tolerances The following may also be used to control chattering: Penalty enforcement of hard contact Contact stabilization The softened contact model *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=[EXPONENTIAL LINEAR TABULAR] Contact with Quadratic Elements 147

148 L5.39 Contact with Quadratic Elements Transmission of pressure across element faces is basic to contact problems Pressure is applied to element faces by using element shape functions to calculate the equivalent consistent nodal loads Constant pressure on an element face generates consistent nodal loads which are: equal for linear elements 2D example shown q ½ pa p q q vary across the element face for quadratic elements p r q r q 2/3 pa 2D example shown r 1/6 pa L5.40 Contact with Quadratic Elements For some element types, consistent nodal loads for a uniform pressure act in opposite directions or are equal to zero at certain nodes p q r q 1 3 pa 3-D, quadratic, serendipity element No midface node C3D20 r 1 12 pa Forces act in opposite direction at corner nodes p q q q q 1 3 pa 3-D, quadratic, tetrahedral element C3D10(I) Zero force at corner nodes 148

149 L5.41 Contact with Quadratic Elements Zero or negative consistent nodal forces are problematic for traditional contact formulations Not likely to converge, due to difficulty determining contact status (active or inactive) for slave nodes at corners of C3D20 elements q r q r pa pa Forces act in opposite direction at corner nodes In the case of slave nodes at corners of C3D10(I) elements, the surface area associated with the constraint is 0 Often results in convergence problems or very noisy contact pressures q q q q 1 3 pa Zero force at corner nodes Uniaxial pressure load of 5.0 (large contact pressure noise!) Contact with Quadratic Elements These problems are avoided with the surface-to-surface formulation L5.42 Uniaxial pressure load of 5.0 Slave: C3D10 Contact pressure on the slave surface Node-to-surface formulation: From previous page Master: C3D8 Surface-to-surface formulation: Desired solution! 149

150 Poorly Defined Master Surfaces L5.44 Poorly Defined Master Surfaces There are many different ways that a master surface can be poorly defined; some of them include: It can be incorrectly oriented. It can have a seam or crack. It can be poorly discretized (e.g., it has kinks). 150

151 L5.45 Poorly Defined Master Surfaces Surface orientations Master surfaces must have consistent orientations and must point toward the slave surface. For the surface-to-surface contact discretization, slave surfaces must also have consistent orientations and must point toward the master surface. Poorly Defined Master Surfaces L5.46 Definition of normals for various surfaces: Analytical rigid surfaces: defined by the order in which the surface is defined. Rigid elements: defined by element connectivity. Structural elements: defined by element connectivity. Continuum elements: always point out of the element. Unique normals cannot be defined for three-dimensional beams. Therefore, they cannot be used as master surfaces. In Abaqus/CAE, these surfaces are chosen interactively; the element connectivity that is generated will be consistent with your surface selection. 151

152 L5.47 Poorly Defined Master Surfaces Symptoms of incorrect normals: Inconsistent normals from one element to the next: Invalid Valid In the invalid case this error message is printed in the printed output (.dat) file: ***ERROR: SURFACE TEST HAS FACETS THAT ARE NOT ORIENTED PROPERLY WITH RESPECT TO EACH OTHER. CHECK THE ELEMENT CONNECTIVITIES FOR UNDERLYING ELEMENT 3 AND 6 (SHARING THE COMMON NODE 4), AS WELL AS THE *SURFACE Poorly Defined Master Surfaces Normals pointing in the wrong direction for the entire master surface: L5.48 Data in the printed output file due to *PREPRINT, CONTACT=YES reflects severe initial overclosure: SLAVE SURFACE ASURF MASTER SURFACE BSURF NODE NUMBER 11 INITIALLY OVERCLOSED BY Convergence difficulties will usually follow. 152

153 L5.49 Poorly Defined Master Surfaces To check normals before doing the analysis, use the following procedure: 1. Run a datacheck analysis: abaqus job=contact datacheck 2. Start an Abaqus/Viewer session. 3. Open the contact.odb output database file. 4. Open the Common Plot Options dialog box. 5. Choose the Normals folder. Toggle on Show normals and select On surfaces. 6. Click Apply. L5.50 Poorly Defined Master Surfaces Seams in a master surface Avoid defining a threedimensional master surface using coincident surface nodes; it results in a crack or seam in the surface. The effect of the crack in a finitesliding analysis is that slave nodes can fall through and become stuck under the surface, particularly if the surface is concave. Another effect is that the surface is not smoothed at the crack, which may also cause convergence problems. Both vertices have the same coordinates. They are separated to show the crack in the surface. 153

154 L5.51 Poorly Defined Master Surfaces Perimeter plots in Abaqus/Viewer can help detect such cracks: Perimeter plots are wire frame plots in which only element edges belonging to just one element are shown. Cracks in the surface will be plotted as extra perimeter lines. crack A perimeter plot can help identify seams in master surfaces L5.52 Poorly Defined Master Surfaces Snagging Corners or small protrusions of a jagged master surfaces can penetrate the spaces between slave nodes causing them to snag. Simplistic representation of customer model Node-to-surface Surface-to-surface slave master In general, slave nodes get snagged easily as they transverse a corner. The averaged penetration alleviates the tendency of slave surface to snag. 154

155 L5.53 Poorly Defined Master Surfaces Abaqus/Standard automatically smoothes the master surface for contact calculations utilizing the node-to-surface discretization to minimize snagging. Master surface smoothing ensures that master surfaces have continuous surface normals at all points. This minimizes the tendency of slave nodes to snag. Snagging is not a problem for the surface-to-surface contact discretization. Abaqus accounts for the spaces between nodes on both the master and slave surfaces. Thus, no smoothing of the master surface occurs when using surface-to-surface contact discretization. Poorly Defined Master Surfaces Master surface smoothing Abaqus/Standard automatically smoothes the following types of master surfaces for node-to-surface finite sliding: Two-dimensional deformable Three-dimensional deformable Surfaces defined on rigid elements Abaqus/Standard does not automatically smooth analytical rigid surfaces. Smoothing has no effect on slave surfaces. L5.54 Smoothing is done only when two adjoining surface facets have different normals. 155

156 Friction L5.56 Friction Adding friction to a model generally makes convergence more difficult. Penalty friction is used by default. Approximates ideal stick-slip behavior by allowing a small amount of elastic slip. Provides a balance between accuracy and efficiency for most problems (e.g., metal forming). Lagrange friction enforces exact stick-slip behavior. Much more difficult to obtain convergence. Sometimes it is the only way to obtain convergence. YOU MUST FOLLOW RIGID BODY CONSTRAINT RULES! 156

157 L5.57 Friction Example: Insertion/removal of a metallic press-on clip Explore different attachment strategies by: Simulating installation of the clip Moving the pin sideways and back to center Removing the clip Study how the removal force decreases with the amount of sideways motion. In this example focus on clip insertion and removal. L5.58 Friction Pin and clip modeled with shell elements For efficiency, the pin is assumed rigid. Linear elastic material assumed Proof-of-concept analysis. Expect small strains, large displacements/rotations. Rigid-deformable contact High friction ( = 0.4) between the clip and the pin Three-step analysis Insertion Reversal Snap Away Fix pin reference node Step 1: Insert clip Move edges of clip Steps 2+3: Remove clip 157

158 L5.59 Friction Step definitions *boundary pinref, 1, 6 handle, 1, 6 *amplitude, name=disp, time=total 0., 0., 1., 9., 2., -5. ** *step, name=insert, nlgeom *static 0.1, 1.0 *boundary, op=mod, amp=disp handle, 3, 3, -1.0 *end step ** *step, name=reverse, nlgeom *static , 0.75 *end step ** * step, name=snap, nlgeom *static, stabilize 0.05, 0.25 *end step Break removal into two steps to use appropriate static stabilization in each. Reversal step: no stabilization necessary Snap-away step: use stabilization absorb energy release associated with loss of contact L5.60 Friction Boundary conditions *boundary pinref, 1, 6 handle, 1, 6 *amplitude, name=disp, time=total 0., 0., 1., 9., 2., -5. ** *step, name=insert, nlgeom *static 0.1, 1.0 *boundary, op=mod, amp=disp handle, 3, 3, -1.0 *end step ** *step, name=reverse, nlgeom *static , 0.75 *end step ** * step, name=snap, nlgeom *static, stabilize 0.05, 0.25 *end step BCs need only be edited in the first step because total-time amplitude curve is used. Splitting removal phase into two steps may require trial-and-error; total-time amplitude curve facilitates this without having to redefine BCs 158

159 L5.61 Friction Insertion Relatively straightforward No convergence difficulties L5.62 Friction Reversal High coefficient of friction causes clip to stick upon load reversal. Friction-driven snap-through behavior is induced. Problem statically stable. Global bending due to contact forces. Penalty friction: convergence difficulties upon load reversal. Lagrange friction: easy resolution of load reversal. Friction-driven snap-through End step just prior to snap-away 159

160 L5.63 Friction Snap away Eventually the clip snaps away from the pin Dynamic event: large energy release occurs Implicit static analysis encounters convergence difficulties No surprise! Can either simulate as a dynamic step (costly, can be difficult to set parameters) or use automatic stabilization (easy, inexpensive solution) L5.64 Friction Load displacement curve Friction-driven snapthrough: contact points stick, force decreases! Snapaway End insertion; begin removal Begin insertion Sliding begins Clip sticks to pin 160

161 Notes 161

162 162 Notes

163 Convergence Problems: Element Behavior Lecture 6 L6.2 Overview Hourglassing in Reduced-Integration Elements Checkerboarding Ill-Conditioning 163

164 Hourglassing in Reduced-Integration Elements L6.4 Hourglassing in Reduced-Integration Elements Full integration vs. reduced-integration elements In elastic finite element analysis the strain energy density must be integrated over the element volume to obtain the element stiffness matrix. Full integration refers to the minimum Gauss integration order required for exact integration of the strain energy (if the element is not distorted). Reduced integration refers to a Gauss integration rule of one order less than full integration. The reduced integration method can be used only in quadrilateral and hexahedral elements. 164

165 L6.5 Hourglassing in Reduced-Integration Elements What is hourglassing? The use of the reduced-integration scheme has a drawback: it can result in mesh instability, commonly referred to as hourglassing. Consider a rectangular plate simply supported along two edges. The hourglass mode does not cause any strain and, hence, does not contribute to the energy integral. It behaves in a manner that is similar to that of a rigid body mode. L6.6 Hourglassing in Reduced-Integration Elements The hourglass mode in second-order serendipity (8-node) elements (CPS8R, CPE8R) is nonpropagating. Neighboring elements cannot share the mode, so the mode cannot occur in a mesh with more than two elements. There is no real danger of hourglassing in these elements. 165

166 L6.7 Hourglassing in Reduced-Integration Elements Hourglass modes of first-order reduced-integration quadrilateral and hexahedral elements can propagate; therefore, hourglassing can be a serious problem in those elements. To suppress hourglassing, an artificial hourglass control stiffness must be added. Abaqus uses hourglass control in all first-order reduced-integration elements. L6.8 Hourglassing in Reduced-Integration Elements General comments Hourglassing is mainly an issue with first-order reduced-integration quad/hex elements. Regular triangular and tetrahedral elements in Abaqus always use full integration and, hence, are not susceptible to hourglassing. Hourglassing can occur in geometrically linear and geometrically nonlinear problems, including finite-strain problems. In geometrically linear problems hourglassing usually does not affect the quality of the calculated stresses. 166

167 load disk roll drum load disk roll drum L6.9 Hourglassing in Reduced-Integration Elements In geometrically nonlinear analysis the hourglass modes tend to interact with the strains at the integration points, leading to inaccuracy and/or instability. Hourglassing is particularly troublesome for problems involving finite-strain elasticity (hyperelasticity) or very large (incompressible) plastic deformations. Fully integrated elements are strongly recommended, whenever feasible, for finite-strain elasticity analysis. When hourglassing is creating convergence problems, the simulation will often have many diverging solution cutbacks. Hourglassing in Reduced-Integration Elements L6.10 When is hourglassing a problem? Hourglassing is almost never a problem with the enhanced hourglass control available in Abaqus. More robust than other schemes No user-set parameters Based on enhanced strain methods Rubber disk rolling against rigid drum Combined hourglass control scheme ALLIE Enhanced hourglass control scheme ALLIE ALLAE ALLAE Comparison of energy histories 167

168 L6.11 Hourglassing in Reduced-Integration Elements Currently, enhanced hourglass control is not the default scheme for most elements. The following table summarizes the hourglass control methods currently available in Abaqus, including the default schemes for most elements: Abaqus/Standard Stiffness (default) Enhanced strain Abaqus/Explicit Relax stiffness (default) Enhanced strain Stiffness Viscous Combined (stiffness+viscous) But enhanced strain hourglass control is the default for: All modified tri and tet elements All elements modeled with finite-strain elastic materials (hyperelastic, hyperfoam, and hysteresis) L6.12 Hourglassing in Reduced-Integration Elements To activate enhanced hourglass control, use the option *SOLID SECTION, CONTROLS=name, ELSET=elset *SECTION CONTROLS, NAME=name, HOURGLASS=ENHANCED No user parameters Abaqus/CAE usage: Mesh module: Mesh Element Type 168

169 L6.13 Hourglassing in Reduced-Integration Elements Detecting and controlling hourglassing Hourglassing can usually be seen in deformed shape plots. Example: Coarse and medium meshes of a simply supported beam with a center point load. Excessive use of hourglass control energy can be verified by looking at the energy histories. Verify that the artificial energy used to control hourglassing is small ( 1 ) relative to the internal energy. Same load and displacement magnification (1000 ) L6.14 Hourglassing in Reduced-Integration Elements Use the X Y plotting capability in Abaqus/Viewer to compare the energies graphically. Internal energy Internal energy Artificial energy Artificial energy Two elements through the thickness: Ratio of artificial to internal energy is 2. Four elements through the thickness: Ratio of artificial to internal energy is

170 L6.15 Hourglassing in Reduced-Integration Elements Example: Engine mount Consider two forms of hourglass control: Stiffness-based Enhanced strain steel rubber Outer rim moves up under load control DISPLACEMENT MAGNIFICATION FACTOR = 1.00 RESTART FILE = a STEP 1 INCREMENT 20 TIME COMPLETED IN THIS STEP.267 TOTAL ABAQUS VERSION: DATE: 15-MAY-97 TIME L6.16 Hourglassing in Reduced-Integration Elements Results with stiffness hourglass control Nonconvergence at 27 of load DISPLACEMENT MAGNIFICATION FACTOR = 1.00 RESTART FILE = a STEP 1 INCREMENT 20 TIME COMPLETED IN THIS STEP.267 TOTAL ACCUMULATED TIME.267 Severe hourglassing occurs ABAQUS VERSION: DATE: 15-MAY-97 TIME: 13:16:10 NIFICATION FACTOR = 1.00 ORIGINAL MESH Obtaining a DISPLACED Converged MESH Solution with Abaqus STEP 1 INCREMENT N THIS STEP.267 TOTAL ACCUMULATED TIME.267

171 L6.17 Hourglassing in Reduced-Integration Elements Results with enhanced hourglass control Deformation at 100 of load; rubber uses default enhanced hourglass control DISPLACEMENT MAGNIFICATION FACTOR = 1.00 No hourglassing RESTART FILE = a2 STEP 1 INCREMENT 13 TIME COMPLETED IN THIS STEP 1.00 TOTAL ACCUMULATED TIME 1.00 ABAQUS VERSION: DATE: 15-MAY-97 TIME: 13:26:19 TOR = 1.00 ORIGINAL MESH Obtaining a DISPLACED Converged MESH Solution with Abaqus CREMENT 13 L6.18 Hourglassing in Reduced-Integration Elements Elastic bending problems and coarse mesh accuracy For elastic bending problems, improved coarse mesh accuracy may be obtained using the enhanced hourglass control method. The enhanced hourglass control formulation is tuned to give accurate results for regularly shaped elements undergoing elastic bending. Where these conditions apply, a coarse mesh may give acceptable results despite the artificial energy being greater than a few percent of the internal energy. An independent check of the results should be made to determine if they are acceptable. 171

172 L6.19 Hourglassing in Reduced-Integration Elements Plastic bending problems When plasticity is present, the stiffness-based hourglass control causes elements to be less stiff than in the enhanced control case. This may give better results with plastic bending; enhanced hourglass control may cause delayed yielding or excessive springback. In using enhanced hourglass control in this case, the usual rule-ofthumb regarding the acceptable level of artificial energy should be followed. Recall that C3D10M elements use enhanced hourglass control by default. Use alternative hourglass control for problems involving yielding. Checkerboarding 172

173 L6.21 Checkerboarding Whereas hourglassing is a behavior where large, oscillating displacements occur without significant stresses, checkerboarding is a behavior where large, oscillating stresses occur without significant displacements. Checkerboarding typically occurs for hydrostatic stresses in (almost) incompressible materials that are highly confined. It can occur in first- and second-order elements but is most notable in first-order elements. It is more likely to occur in regular meshes than in irregular meshes. Checkerboarding Checkerboarding is related to but is not the same as volumetric locking. Volumetric locking occurs when incompressible material behavior puts more constraints on the deformation field then there are displacement degrees of freedom. For example, in a refined, three-dimensional mesh of 8-node hexahedra, there is on average 1 node with 3 degrees of freedom per element. L6.22 The volume at each integration point must remain fixed. Since full integration uses 8 points per element, we have as many as 8 constraints per element but only 3 degrees of freedom. Consequently, the mesh is overconstrained it locks. Volumetric locking can be avoided by using the proper element type; for a more detailed discussion of this topic see the Element Selection in Abaqus lecture notes. 173

174 L6.23 Checkerboarding Checkerboarding does not always manifest itself clearly. The displacement field may initially be unaffected, and stress contour plots may not show the checkerboarding because of smoothing of element stresses during postprocessing. Discontinuous or quilt plots will show the checkerboard pattern, however. In linear analyses checkerboarding rarely causes convergence difficulties. However, in nonlinear analyses the high hydrostatic stress oscillations can eventually interact with the displacements and cause sudden, usually catastrophic, convergence problems. Checkerboarding can be eliminated by introducing some local mesh irregularities. L6.24 Checkerboarding Example: Rubber bushing Consider a cylindrical rubber bushing made of an (almost) incompressible rubber. The bushing is modeled with first-order, generalized plane strain elements. Both the inner and outer radius of the bushing are fully constrained. This constraint severely limits the deformations that can occur in the model. A compressive axial load is applied to the bushing through the element reference node. 174

175 L6.25 Checkerboarding In this model the element indicated in the figure is given a bulk modulus that is one order-of-magnitude smaller than that assigned to the rest of the elements in the mesh. This should lead to a smaller hydrostatic pressure in this element. smaller K L6.26 Checkerboarding A quilt contour plot (without averaging between elements) clearly shows a checkerboard pattern with a significant pressure variation. Quilt (nonaveraged) contour plot of hydrostatic pressure 175

176 Ill-Conditioning L6.28 Ill-Conditioning When elements or materials show large stiffness differences, conditioning problems may occur. In linear analyses problems typically occur only when the stiffness differences are extreme (factors of 10 6 or more). In such cases the solution of the linear equation system becomes inaccurate. In nonlinear analyses problems occur at a much earlier stage. The stiffness differences may cause poor convergence or even divergence if the increment size is not very small. 176

177 L6.29 Ill-Conditioning Long, slender or rigid structures Large differences in stiffness occur in long, slender structures (such as very long pipes or cables) or very stiff structures (such as a link in a vehicle s suspension system). If such structures undergo large motions in geometrically nonlinear analyses, convergence can be very difficult to obtain. Slight changes in nodal positions can cause very large (axial) forces that, in turn, cause incorrect stiffness contributions. This makes it very difficult or impossible for the usual finite element displacement method to converge. Convergence problems in these simulations usually manifest themselves in very slow or irregular convergence rates or in diverging solutions. Ill-Conditioning L6.30 Use hybrid beam elements (types B21H, B31H, B31OSH) or hybrid truss elements to model such problems. In these hybrid elements the axial and, in the case of hybrid beams, the transverse shear forces in the elements are included as primary variables in the element formulation Because the forces are primary variables, they remain reasonably accurate during iteration, and the elements usually converge faster. Even though the additional primary variables make these elements more expensive per iteration, they are usually much more efficient because the improved convergence rate reduces the number of iterations. 177

178 L6.31 Ill-Conditioning Example: Near bottom pipeline pull-in and tow Simulating a seabed pipeline installation. Model: Drag chains used to offset buoyancy effects. Pipeline modeled using beam elements. Pipeline is very slender. One end of the pipeline is winched into an anchor point. The other end is built in for the pull-in and free for the tow. Pipeline dimensions: Length = 1000 ft Outer diameter = 0.75 ft Wall thickness = ft L6.32 Ill-Conditioning Pull-in analysis The job with hybrid elements converges significantly faster than the job without hybrid elements. Element type Number of increments Number of cutbacks Number of iterations B33H B

179 L6.33 Ill-Conditioning Tow analysis The pipeline has no restraint (and is, therefore, singular) until the drag chain extends sufficiently to stabilize the pipeline. INCREMENT 1 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT 1.000E-02 ***WARNING: SOLVER PROBLEM. NUMERICAL SINGULARITY WHEN PROCESSING NODE 3 D.O.F. 6 RATIO = E+15 ***WARNING: THE SYSTEM MATRIX HAS 1 NEGATIVE EIGENVALUES. To overcome numerical difficulties in the early stages of the analysis, a small initial stress is applied to the pipeline: *INITIAL CONDITIONS,TYPE=STRESS BEAMS,1.E-8 Element type Number of increments Number of cutbacks Number of iterations B33H B Ill-Conditioning Approximately incompressible material behavior If the bulk modulus, K, is much larger than the shear modulus, G, large stiffnesses occur inside an element. Slight changes in nodal positions can cause very large volumetric strains and, as a result, large hydrostatic stresses. The large hydrostatic stresses cause incorrect stiffness contributions, which seriously hamper convergence. This effect is particularly seen with hyperelastic materials. L

180 L6.35 Ill-Conditioning Use hybrid solid elements (types CPE4H, C3D20H, CAX4H, etc.) in such cases. In these elements the hydrostatic pressure (or in some cases the volume change) is included as a primary variable in the element formulation. Consequently the hydrostatic stresses (and, thus, the effective stiffness) remain reasonably accurate during the iteration process. Although the cost per iteration increases due to the additional degrees of freedom, the overall analysis cost typically is reduced because a smaller number of iterations will be needed. An exception is the modified 10-node tetrahedral elements (C3D10MH). For these elements the cost per iteration increases significantly. 180

181 Notes 181

182 182 Notes

183 Convergence Problems: Constraints and Loading Lecture 7 L7.2 Overview General Remarks Overconstraints Detected in the Analysis Preprocessor Overconstraints Detected/Resolved During the Analysis Controlling the Overconstraint Checks Example: Four-bar Linkage Nonconservative Loads 183

184 General Remarks L7.4 General Remarks An overconstraint means applying multiple consistent or conflicting kinematic constraints. TIE and symmetry boundary conditions (node 141 or 151) FRICTION, ROUGH and symmetry boundary conditions (node 101) Intersecting TIE definitions TIE and contact nodes (node 801 or 901) symmetry boundary conditions rigid punch TIE fixed rigid surface contact with FRICTION, ROUGH 184

185 L7.5 General Remarks Consistent overconstraints Two or more compatible constraints are applied at the same node. Also referred to as redundant constraints. All overconstraints in the previous slide are consistent. Conflicting overconstraints Two or more incompatible (inconsistent) constraints applied at the same node. Example: the boundary conditions and the contact constraint at the marked slave node are conflicting. BOUNDARY rigid indenter L7.6 General Remarks The Abaqus/Standard equation solver If the model has consistent overconstraints: The solver cannot guarantee the correct solution of such a system. In some (lucky) cases the expected solution is computed. Results can be computer platform-dependent. If the model has conflicting overconstraints: A correct solution does not exist. The model is ill-posed. If convergence is achieved, the solution has no meaning. Symptoms ( solution or convergence) can be computer platformdependent. 185

186 L7.7 General Remarks For an Abaqus/Explicit analysis If the model has consistent overconstraints not removed in the analysis preprocessor: A solution will be found, and the displacement solution is correct. Forces through the constraints are not unique distribution based approximately on stiffnesses and masses of connected components. Results are not computer platform-dependent due to overconstraint. If the model has conflicting overconstraints not detected in the analysis preprocessor: A correct solution does not exist. The model is ill-posed. Results will not satisfy conflicting constraints but are not computer platform-dependent. General Remarks Abaqus automatically resolves a limited (but frequently encountered) set of consistent overconstraints: Check for overconstraints caused by combinations of: Single-point constraints: BASE MOTION BOUNDARY Multi-point constraints: EQUATION MPC KINEMATIC COUPLING RIGID BODY TIE Contact: CONTACT PAIR Connector elements L

187 L7.9 General Remarks Two categories of detected overconstraints Overconstraints detected in the analysis preprocessing: If consistent, the unnecessary constraints are eliminated (warning message). If conflicting, an error message is issued. Overconstraints detected and resolved during the analysis: Involve contact and either the TIE option or the BOUNDARY option. The analysis may become overconstrained because of contact status changes. General Remarks Zero pivot warning An overconstraint that Abaqus has not been able to resolve will usually cause a zero pivot message to be issued in the message file: L7.10 Constraint chain; discussed in detail later ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING ELEMENT INTERNAL NODE 1 D.O.F. 4 OVERCONSTRAINT CHECKS: An overconstraint was detected at one of the Lagrange multipliers associated with element There are multiple constraints applied directly or chained constraints that are applied indirectly to this element. The following is a list of nodes and chained constraints between these nodes that most likely lead to the detected overconstraint. LAGRANGE MULTIPLIER: 2321 <-> 863: connector element type SLOT ALIGN constraining 2 translations and 3 rotations > 10007: *RIGID BODY (or *COUPLING - KINEMATIC) > 3159: *RIGID BODY (or *COUPLING - KINEMATIC) 187

188 L7.11 General Remarks The effect of a zero pivot is hard to predict. If the overconstraint involves a contact interaction: Abaqus may not be able to establish the proper contact conditions, which will lead to repeated cutbacks as a result of an excessive number of SDIs. If MPCs are involved: The solution may diverge. Occasionally the zero pivot will cause the time average force in the model to increase by orders of magnitude without otherwise causing convergence problems. In this case the convergence tests are effectively disabled, which is a problem in itself because incorrect solutions will likely be created. For example, this situation can occur if a boundary condition conflicts with a TIE constraint. Overconstraints Detected in the Analysis Preprocessor 188

189 Overconstraints Detected in the Analysis Preprocessor Intersecting tie constraints (*TIE) L7.13 TIE At the marked location: There are three constraints tying together the three nodes. Only two constraints are needed. One constraint will be eliminated automatically. three nodes in the same location Overconstraints Detected in the Analysis Preprocessor Boundary conditions applied to tied nodes ( TIE + BOUNDARY) L7.14 TIE between gasket (red) and cover (blue) symmetry boundary conditions In general, BOUNDARY at TIE dependent nodes will be ignored. If different boundary conditions are specified at the paired nodes, inconsistent overconstraint: error message is issued. 189

190 Overconstraints Detected in the Analysis Preprocessor Overconstraints related to rigid bodies In many cases regions of the model that were originally deformable are declared rigid in subsequent runs. This can lead to overconstrained models if: Surfaces tied together (*TIE) belong to rigid bodies. L7.15 Connector elements included in rigid bodies or used to connect rigid bodies. Element sets that are used to define the rigid bodies overlap. Boundary conditions were specified at nodes of the rigid bodies. Overconstraints Detected in the Analysis Preprocessor L7.16 Tie constraints and rigid bodies TIE Original model Both left and right regions are deformable. TIE used between regions. Modified model 1 User makes everything rigid using RIGID BODY. TIE will be ignored. Modified model 2 User creates two adjacent rigid bodies. BEAM connector placed between reference nodes. Modified model 3 User keeps brick mesh region deformable and makes tet mesh region rigid. Surfaces on the TIE data line are switched if necessary, so that the surface associated with the rigid region is the master. 190

191 Overconstraints Detected in the Analysis Preprocessor Connector elements and rigid bodies (engine cradle example) L7.17 WELD connectors Original model (courtesy of GM) All deformable components. Joined with WELD/BEAM connectors. BEAM connectors Modified model 1: BEAM connectors One rigid body definition for the entire cradle. Overconstrained because WELDs/BEAMs are inside the RIGID BODY. Abaqus deactivates the WELDs/BEAMs. Modified model 2: One rigid body definition for each colored part. Overconstrained because too many WELDs/BEAMs between RIGID BODYs. Abaqus deactivates some WELDs/BEAMs. Overconstraints Detected in the Analysis Preprocessor Intersecting rigid bodies L7.18 Original model All elements are deformable. The defined element sets overlap. Modified model 1 ELSET 1 Two rigid bodies created from the two element sets. The two element sets behave as one rigid body. BEAM connector is placed between the two reference nodes. ELSET 2 Modified model 2 User creates a rigid cradle using an additional *RIGID BODY definition including all elements. One additional BEAM connector is placed between two reference nodes. 191

192 Overconstraints Detected in the Analysis Preprocessor Boundary conditions and rigid bodies Original model Deformable turbine. L7.19 Boundary conditions in vertical direction allow only for rotation about the vertical axis. Modified model Turbine shaft (blue) made rigid with RIGID BODY. Boundary conditions transferred to the reference node. Two additional rotational boundary conditions are generated to allow only for rotations about the vertical axis. Reaction forces appear only at the reference node. Overconstraints Detected/Resolved During the Analysis 192

193 Overconstraints Detected/Resolved During the Analysis Overconstraints due to contact interactions L7.21 Y-symmetry conditions transmission pan Example: Transmission pan gasket analysis The pan is bolted to a flat rigid surface. X-symmetry conditions compression stopper elastomer steel backing Overconstraints Detected/Resolved During the Analysis Contact interactions and tie constraints L7.22 gasket TIE Mismatched meshes are tied together (using *TIE) near the compression stopper. Contact slave nodes are constrained both by the contact and the tie constraints. The contact constraint will not be enforced for the slave nodes of the tie constraint. pan 193

194 Overconstraints Detected/Resolved During the Analysis Contact interactions and boundary conditions L7.23 The friction formulation is switched from ROUGH to PENALTY only at overconstrained nodes. FRICTION, ROUGH X-symmetry boundary conditions Overconstraints Detected/Resolved During the Analysis Contact interactions and boundary conditions L7.24 punch The marked contact slave nodes have boundary conditions in the same direction as the contact constraint. blank contact slave nodes with fixed boundary conditions The contact constraint will not be applied at the overconstrained slave node. die 194

195 Controlling the Overconstraint Checks Controlling the Overconstraint Checks Default overconstraint checks: Abaqus will attempt to remove as many overconstraints as possible. If a conflicting overconstraint is detected, an error message is issued and the analysis is stopped. If a consistent overconstraint is identified but cannot be removed, a detailed message is issued. L

196 L7.27 Controlling the Overconstraint Checks CONSTRAINT CONTROLS CONSTRAINT CONTROLS, PRINT=YES Prints the constraint chains to the message file (illustrated shortly). CONSTRAINT CONTROLS, NO CHANGES No automatic elimination of consistent overconstraints is performed. Error messages are issued if overconstraints are encountered. CONSTRAINT CONTROLS, NO CHECKS No overconstraints checks are performed. Not recommended! Controlling the Overconstraint Checks Best practice: L7.28 The model must be changed to remove all identified overconstraints that could not be removed automatically. Scan the message and data files (or use Abaqus/Viewer) for messages starting with the string OVERCONSTRAINT CHECKS. These messages contain useful information about the overconstraints in the model. Search the message file for zero-pivot warnings. Do not ignore them. They are almost always an indication of overconstraint. 196

197 Example: Four-bar Linkage L7.30 Example: Four-bar Linkage A four-bar linkage is modeled using connector elements. The mechanism is actuated by prescribing a connector motion to one of the hinges HINGE = JOIN + REVOLUTE RIGID BODY Number of constraints: JOIN: 4x3= 12 constraints REVOLUTE: 4x2= 8 constraints Boundary: 6 constraints Connector motion: 1 constraint Total constraints: 27 constraints Total number of DOFs: 4x6 = 24 DOFs We have: = 3 constraints too many Connector motion Reference node completely fixed 197

198 L7.31 Example: Four-bar Linkage Model details 3 2 top 2 1 left Connector element number Rigid body reference node 3 bottom Part instance name HINGE = JOIN + REVOLUTE RIGID BODY 2 3 right 2 Element node number 1 1 *Part, name=bar *Node 1, 0.0, 0.0 2, 1.0, 0.0 *Element, type=b31, elset=bar 1, 1, 2 *Node, nset=ref 3, 0.5, 0.0 *Rigid Body, ref node=ref, elset=bar *End Part *Assembly *Instance, name=bottom, part=bar *End Instance : : *End Assembly *Element, type=conn3d2, elset=hinges 1, bottom.2, right.1 2, right.2, top.1 3, top.2, left.1 4, left.2, bottom.1 *Connector Section, elset=hinges Hinge, hingeori, *Orientation, name=hingeori 0., 0., 1., 0., 1., 0. : : L7.32 Example: Four-bar Linkage When Abaqus/Standard attempts to find a solution for this model, three zero pivots are identified in the first increment of the analysis suggesting that there are 3 too many constraints in the model (as expected). ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING ELEMENT 2 (ASSEMBLY) INTERNAL NODE 20 D.O.F. 1 ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING ELEMENT 2 (ASSEMBLY) INTERNAL NODE 20 D.O.F. 4 ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING ELEMENT 2 (ASSEMBLY) INTERNAL NODE 20 D.O.F

199 L7.33 Example: Four-bar Linkage The constraint chains top 1 2 Requires *CONSTRAINT CONTROLS, PRINT=YES 2 Consider the first zero pivot warning: ***WARNING: SOLVER PROBLEM. ZERO PIVOT WHEN PROCESSING ELEMENT 2 (ASSEMBLY) INTERNAL NODE 20 D.O.F. 1 right Constraint Chains LAGRANGE MULTIPLIER: 2 INSTANCE RIGHT <-> 1 INSTANCE TOP: connector element 2 type HINGE..2 INSTANCE RIGHT -> 3 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE RIGHT -> 1 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE RIGHT -> 2 INSTANCE BOTTOM: connector element 1 type HINGE...2 INSTANCE BOTTOM -> 3 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE BOTTOM -> pre-defined type *BOUNDARY...3 INSTANCE BOTTOM -> 1 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE BOTTOM -> 2 INSTANCE LEFT: connector element 4 type HINGE...2 INSTANCE LEFT -> 3 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE LEFT -> 1 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE LEFT -> 2 INSTANCE TOP: connector element 3 type HINGE...2 INSTANCE TOP -> 3 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE TOP -> 1 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...2 INSTANCE LEFT -> 1 INSTANCE BOTTOM: connector element 4 with *CONNECTOR MOTION in components 4 Example: Four-bar Linkage L7.34 LAGRANGE MULTIPLIER: 2 INSTANCE RIGHT <-> 1 INSTANCE TOP: connector element 2 ASSEMBLY_NAME ASSEMBLY type HINGE constraining 3 translations and 2 rotations top Indicates the zero pivot is associated with the connection between the RIGHT and TOP bars. right Subsequent lines of the message trace how the nodes identified in the zero pivot warning are connected to the rest of the model. Use this as a starting point when trying to identify the overconstraint. 199

200 L7.35 Example: Four-bar Linkage Detailed analysis of constraint chain for node right.2 First chain of constraints terminates in a boundary condition: right.2 right.3 right.1 bottom.2 bottom.3 *BOUNDARY..2 INSTANCE RIGHT -> 3 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE RIGHT -> 1 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE RIGHT -> 2 INSTANCE BOTTOM: connector element 1 type HINGE...2 INSTANCE BOTTOM -> 3 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE BOTTOM -> pre-defined type *BOUNDARY...3 INSTANCE BOTTOM -> 1 INSTANCE BOTTOM: *RIGID BODY (or First *COUPLING chain of constraints - KINEMATIC)...1 INSTANCE BOTTOM -> 2 INSTANCE LEFT: connector element terminates 4 type in a HINGE boundary condition...2 INSTANCE LEFT -> 3 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC) Since indentation level is the same, this...3 INSTANCE LEFT -> 1 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC) indicates the end of the first chain...1 INSTANCE LEFT -> 2 INSTANCE TOP: connector element 3 type HINGE...2 INSTANCE TOP -> 3 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE TOP -> 1 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...2 INSTANCE LEFT -> 1 INSTANCE BOTTOM: connector element 4 with *CONNECTOR MOTION Example: Four-bar Linkage The second chain forms a closed loop: right.2 right.3 right.1 bottom.2 bottom.3 bottom.1 left.2 left.3 left.1 top.2 top.3 top.1 <-> right.2 L INSTANCE RIGHT -> 3 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE RIGHT -> 1 INSTANCE The links RIGHT: that form *RIGID the first BODY chain (or until *COUPLING the - KINEMATIC) indentation level is repeated also form...1 INSTANCE RIGHT -> 2 part INSTANCE of the second BOTTOM: chain connector element 1 type HINGE...2 INSTANCE BOTTOM -> 3 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE BOTTOM -> pre-defined type *BOUNDARY...3 INSTANCE BOTTOM -> 1 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE BOTTOM -> 2 INSTANCE LEFT: connector element 4 type HINGE...2 INSTANCE LEFT -> 3 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE LEFT -> 1 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE LEFT -> 2 INSTANCE TOP: connector element 3 type HINGE...2 INSTANCE TOP -> 3 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE TOP -> 1 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...2 INSTANCE LEFT -> 1 INSTANCE BOTTOM: connector element 4 with *CONNECTOR MOTION Repeated indentation: end of the second chain 200

201 L7.37 Example: Four-bar Linkage The third chain ends in a prescribed connector motion: right.2 right.3 right.1 bottom.2 bottom.3 bottom.1 left.2 bottom.1 *Connector Motion..2 INSTANCE RIGHT -> 3 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE RIGHT -> 1 INSTANCE RIGHT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE RIGHT -> 2 INSTANCE BOTTOM: connector element 1 type HINGE...2 INSTANCE BOTTOM -> 3 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE BOTTOM -> pre-defined type *BOUNDARY...3 INSTANCE BOTTOM -> 1 INSTANCE BOTTOM: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE BOTTOM -> 2 INSTANCE LEFT: connector element 4 type HINGE...2 INSTANCE LEFT -> 3 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE LEFT -> 1 INSTANCE LEFT: *RIGID BODY (or *COUPLING - KINEMATIC)...1 INSTANCE LEFT -> 2 INSTANCE TOP: connector element 3 type HINGE...2 INSTANCE TOP -> 3 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...3 INSTANCE TOP -> 1 INSTANCE TOP: *RIGID BODY (or *COUPLING - KINEMATIC)...2 INSTANCE LEFT -> 1 INSTANCE BOTTOM: connector element 4 with *CONNECTOR MOTION L7.38 Example: Four-bar Linkage Removing the overconstraint No unique way to remove the overconstraint in this model. left 4 bottom The constraint chains indicate there are 3 overconstraints (1 translational and 2 rotational). Analyze each constraint in isolation and track the overall impact on the model. Starting point is arbitrary; here we start with the hinge connection between the bottom and left bars: Recall that a hinge connection defines a join and a revolute constraint Join: endpoints of the bars remain coincident Revolute about the Z-axis: the bars cannot rotate about their own axes or out of the X-Y plane (this forces the bars to remain co-planar) 201

202 L7.39 Example: Four-bar Linkage Removing the overconstraint (cont d) 3 left top Adding a hinge connection between the left and top bars forces these two bars to remain joined at the ends and co-planar. Note that now all three bars in the assembly are co-planar. 4 bottom L7.40 Example: Four-bar Linkage Removing the overconstraint (cont d) 3 left top 2 right Adding a hinge connection between the top and right bars now forces these two bars to remain joined at the ends and co-planar. All four bars in the assembly are co-planar. All four bars are prevented from rotating about their respective axes 4 bottom 202

203 L7.41 Example: Four-bar Linkage Removing the overconstraint (cont d) 3 left 4 top bottom 2 right 1 So what to do about the last connection? If add a hinge it introduces 3 redundant constraints on the right bar : Out-of-plane translation and rotation are already prevented because the bars are coplanar from previous constraints. Rotation about its axis is already prevented by the previous revolute constraint. These are the three overconstraints detected by the solver (1 translation and 2 rotations) Only the in-plane translations between the bottom and right bars need to be constrained at this point Use a SLOT connection instead of a hinge to enforce only the two translation constraints in the plane of the mechanism L7.42 Example: Four-bar Linkage Removing the overconstraint (cont d) 3 top 2 Of course this resolution is not unique Recall that a hinge is made up of 2 types of connectors: join and a revolute Can move the revolute constraint from the top-right to the bottom-right left right Option 1 slot hinge hinge hinge Option 2 slot + revolute join hinge hinge 4 bottom 1 Which constraints should be changed is also not unique (can modify 2 and 3 instead of 1 and 4, for example) but it is not arbitrary (cannot remove any 3 constraints; can only remove the constraints that do not affect the kinematics of the model) An alternative solution for overconstraints involving connector elements is to convert the rigid bodies into elastic bodies or to use flexible connections. 203

204 Nonconservative Loads L7.44 Nonconservative Loads Most applied mechanical loads are conservative. The work done by the applied load is determined by the initial and final position only it is independent of the path followed. Such loads cause zero or symmetric load-stiffness contributions. Examples of conservative mechanical loads include: Point loads in a fixed direction, Gravity loads, centrifugal loads, and Pressure loads (internal and external). 204

205 L7.45 Nonconservative Loads Certain loads are nonconservative. The work done by the load is a function of the path followed. Such loads cause unsymmetric load stiffness terms. Examples of nonconservative mechanical loads include: follower forces, Coriolis loads, and pressure loads that do not correspond to internal or external pressure. L7.46 Nonconservative Loads Although nonconservative loads cause nonsymmetric stiffness terms, they need to be taken into account only in certain situations: With follower forces, the force should be of the same order of magnitude as the force-rotation coupling stiffness of the point to which the load is applied. With pressure loads, the pressure should be of the same order of magnitude as the modulus of the material. In these cases using the unsymmetric solver should prevent any convergence problems related to such loads. *STEP, UNSYMM=YES : 205

206 206

207 Notes 207

208 208 Notes

209 Convergence Problems: Materials Lecture 8 L8.2 Overview Large Strains and Linear Elasticity Unstable Material Behavior Example: Plate with a Hole Unsymmetric Material Stiffness Example: Concrete Slump Test 209

210 Large Strains and Linear Elasticity L8.4 Large Strains and Linear Elasticity Linear elasticity should never be used for elastic large strains in Abaqus. The linear elastic material model is formulated for small (< 5 ) elastic strains. The program will provide poor convergence and will fail when the elastic strains become large say 30 or more because the change in configuration as a result of elastic straining is not taken into account. Even if your model converges, your results will be wrong! When modeling structures that undergo large strains, do not approximate the material response as linear elastic to simplify the model. If the material is expected to yield, use an elastic-plastic material. The large strains will be inelastic. If the response is elastic, use hyperelasticity. 210

211 L8.5 Large Strains and Linear Elasticity Example: Compression of a rubber ball Solid rubber ball compressed between two rigid plates. Symmetry is used. Both linear elasticity and hyperelasticity are considered. The hyperelastic coefficients are chosen such that E 0 and 0 correspond to the linear elastic material properties. L8.6 Large Strains and Linear Elasticity 72 strain >> 5 Linear elasticity: Fails to converge at 76 applied displacement of Hyperelasticity 211

212 Unstable Material Behavior L8.8 Unstable Material Behavior Unstable material behavior occurs when the stiffness of the material becomes zero or negative (strain softening). Many materials can exhibit unstable behavior: Metals (when perfectly plastic or when the hardening modulus becomes less than the stress) Rubber Foams Concrete (when it is completely cracked) Soils (when they become perfectly plastic) 212

213 L8.9 Unstable Material Behavior When an Abaqus simulation encounters unstable material behavior, it will often fail to converge. The symptoms in these nonconvergence problems are most often divergence warnings. For some material models, such as hyperelasticity and hyperfoam, Abaqus can test for material stability before the analysis begins and issue warnings about what magnitude of deformation is needed to cause a material instability: *HYPERELASTIC, POLYNOMIAL,N=2,TEST DATA INPUT ***WARNING: UNSTABLE HYPERELASTIC MATERIAL FOR UNIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN FOR UNIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN FOR BIAXIAL TENSION WITH A NOMINAL STRAIN LARGER THAN FOR BIAXIAL COMPRESSION WITH A NOMINAL STRAIN LESS THAN FOR PLANE TENSION WITH A NOMINAL STRAIN LARGER THAN FOR PLANE COMPRESSION WITH A NOMINAL STRAIN LESS THAN Unstable Material Behavior Having perfectly plastic material behavior is generally not a problem in an Abaqus simulation. However, if an entire load bearing portion of the model becomes perfectly plastic under an applied force or pressure load, convergence problems may occur. L8.10 If the model is loaded with applied displacement or velocity boundary conditions, convergence problems are less likely to occur. Perfect plasticity is more likely to cause convergence problems in geometrically nonlinear analyses, in particular if the material is loaded in tension. SAFE RULE: Always extrapolate your plasticity data so that the slope is positive over the entire range of strain. Never allow Abaqus to fall off the curve! 213

214 Stress (ksi) Example: Plate with a Hole L8.12 Example: Plate with a Hole Thin square plate Plane stress elements Symmetry conditions Tensile load length=5 40 ksi Material Modeled as elastic-perfectly plastic. Young's modulus = 1.e4 ksi Poisson's ratio = 0.3 Initial yield stress = 35 ksi r= Strain (%) 214

215 L8.13 Example: Plate with a Hole The analysis terminates prematurely! Mises stress ~ 35 ksi Only 70% of load applied at termination Monitor displacement of this node L8.14 Example: Plate with a Hole Status file (edited to remove failed attempts) Total Time Step Time Time Inc. DOF Monitor e e e e e e e THE ANALYSIS HAS NOT BEEN COMPLETED 215

216 L8.15 Example: Plate with a Hole Message file: INCREMENT 5 STARTS. ATTEMPT NUMBER 1, TIME INCREMENT ***WARNING: THE SYSTEM MATRIX HAS 7 NEGATIVE EIGENVALUES. Onset of perfect plasticity. ***WARNING: ELEMENT 298 INSTANCE PLATE-1 IS SUFFERING EXTREME DEFORMATION. INCREMENT WILL BE SUBDIVIDED. ***WARNING: THE STRAIN INCREMENT HAS EXCEEDED FIFTY TIMES THE STRAIN TO CAUSE FIRST YIELD AT 189 POINTS ***WARNING: THE STRAIN INCREMENT IS SO LARGE THAT THE PROGRAM WILL NOT ATTEMPT THE PLASTICITY CALCULATION AT 5 POINTS ***WARNING: CONVERGENCE JUDGED UNLIKELY. INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-02 These warnings indicate that Abaqus is having trouble in forming the stiffness matrix or calculating the internal forces. L8.16 Example: Plate with a Hole The messages indicate extreme element deformation. Mesh refinement? The refined mesh model at right fails at almost exactly the same load level as the coarse mesh model. Mises stress ~ 35 ksi 216

217 L8.17 Example: Plate with a Hole The warning messages also indicate difficulties with the plasticity calculations! The plate has reached its limit load. The plate has no stiffness to resist further deformation because of the perfectly plastic post-yield behavior of the material. Lack of stiffness indicated by the presence of NEGATIVE EIGENVALUE warnings. Stiffness can be estimated by looking at status file data. Estimate material stiffness: Time Inc. DOF Monitor Stiffness t/ u e e e e e e e L8.18 Example: Plate with a Hole In this model it is quite clear that the convergence problems are closely linked to the plasticity calculations. The question then becomes Are the model data correct? In particular, are the material data correct? Referring to the stress-strain curve, the material exhibits very little (if any) hardening after yield. Thus, the material model approximation is reasonable. The loading then comes into question. In this model the load level is too high given the load carrying capacity of the plate. The plate either needs to be redesigned to sustain the applied load or the load level needs to be reduced. 217

218 Unsymmetric Material Stiffness L8.20 Unsymmetric Material Stiffness Certain plasticity models generate unsymmetric stiffness matrices. The Drucker-Prager material model will generate unsymmetric stiffness matrices if the dilation angle,, is not equal to the friction angle,. The gray cast iron, Mohr-Coulomb, cap plasticity, and crushable foam models always (or almost always) produce unsymmetric stiffness matrices. If the unsymmetric equation solver is not used with these material models, it is possible that Abaqus will be unable to find a converged solution. The symptoms in these cases are diverging solution warnings and warnings about very slow convergence rates (and a subsequent cutback in the increment size). 218

219 Example: Concrete Slump Test L8.22 Example: Concrete Slump Test The cone of wet concrete slumps under its own weight. The material is modeled with a Drucker-Prager model, which assumes incompressible plastic flow ( ). The simulation terminates after only 19% of the load is applied because too many attempts are made in an increment. 219

220 L8.23 Example: Concrete Slump Test An excerpt of the message file follows: EQUILIBRIUM ITERATION 8 AVERAGE FORCE 8.791E-07 TIME AVG. FORCE 7.315E-07 LARGEST RESIDUAL FORCE 1.860E-09 AT NODE 25 DOF 2 LARGEST INCREMENT OF DISP E-07 AT NODE 31 DOF 2 LARGEST CORRECTION TO DISP E-08 AT NODE 6 DOF 1 DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREMENT EQUILIBRIUM ITERATION 9 AVERAGE FORCE 8.791E-07 TIME AVG. FORCE 7.315E-07 LARGEST RESIDUAL FORCE 6.242E-09 AT NODE 16 DOF 1 LARGEST INCREMENT OF DISP E-07 AT NODE 101 DOF 2 LARGEST CORRECTION TO DISP E-07 AT NODE 21 DOF 2 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE. EQUILIBRIUM ITERATION 10 AVERAGE FORCE 8.790E-07 TIME AVG. FORCE 7.315E-07 LARGEST RESIDUAL FORCE E-09 AT NODE 25 DOF 2 LARGEST INCREMENT OF DISP E-07 AT NODE 101 DOF 2 LARGEST CORRECTION TO DISP E-08 AT NODE 6 DOF 1 DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREMENT L8.24 Example: Concrete Slump Test EQUILIBRIUM ITERATION 11 AVERAGE FORCE 8.790E-07 TIME AVG. FORCE 7.315E-07 LARGEST RESIDUAL FORCE E-09 AT NODE 16 DOF 1 LARGEST INCREMENT OF DISP E-07 AT NODE 31 DOF 2 LARGEST CORRECTION TO DISP E-07 AT NODE 21 DOF 2 DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREMENT EQUILIBRIUM ITERATION 12 AVERAGE FORCE 8.791E-07 TIME AVG. FORCE 7.315E-07 LARGEST RESIDUAL FORCE 1.576E-09 AT NODE 25 DOF 2 LARGEST INCREMENT OF DISP E-07 AT NODE 101 DOF 2 LARGEST CORRECTION TO DISP E-08 AT NODE 6 DOF 1 DISP. CORRECTION TOO LARGE COMPARED TO DISP. INCREMENT ***NOTE: THE RATE OF CONVERGENCE IS VERY SLOW. THE INCREMENT WILL BE ATTEMPTED AGAIN WITH A TIME INCREMENT OF E-05 INCREMENT 24 STARTS. ATTEMPT NUMBER 5, TIME INCREMENT 3.129E

221 L8.25 Example: Concrete Slump Test The cause is that the material model is producing unsymmetric stiffness matrices but the symmetric solver is being used. Using the unsymmetric solver, Abaqus completes the simulation in 67 increments. Models with high friction coefficients ( behavior in Abaqus. > 0.2) can exhibit similar Abaqus will use the unsymmetric solver in all models with > 0.2. Example: Concrete Slump Test L8.26 A B Concrete slump simulation (A) Model with the symmetric solver (B) Model with the unsymmetric solver 221

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223 Notes 223

224 224 Notes

225 Workshop Preliminaries Setting up the workshop directories and files If you are taking a public seminar, the steps in the following section have already been done for you: skip to Basic Operating System Commands, (p. WP.2). If everyone in your group is familiar with the operating system, skip directly to the workshops. The workshop files are included on the Abaqus release CD. If you have problems finding the files or setting up the directories, ask your systems manager for help. Note for systems managers: If you are setting up these directories and files for someone else, please make sure that there are appropriate privileges on the directories and files so that the user can write to the files and create new files in the directories. Workshop file setup (Note: UNIX is case-sensitive. Therefore, lowercase and uppercase letters must be typed as they are shown or listed.) 1. Find out where the Abaqus release is installed by typing UNIX and Windows NT: abqxxx whereami where abqxxx is the name of the Abaqus execution procedure on your system. It can be defined to have a different name. For example, the command for the release might be aliased to abq6101. This command will give the full path to the directory where Abaqus is installed, referred to here as abaqus_dir. 2. Extract all the workshop files from the course tar file by typing UNIX: abqxxx perl abaqus_dir/samples/course_setup.pl Windows NT: abqxxx perl abaqus_dir\samples\course_setup.pl Note that if you have Perl and the compilers already installed on your machine, you may simply type: UNIX: abaqus_dir/samples/course_setup.pl Windows NT: abaqus_dir\samples\course_setup.pl 3. The script will install the files into the current working directory. You will be asked to verify this and to choose which files you wish to install. Choose y for the appropriate lecture series when prompted. Once you have selected the lecture series, type q to skip the remaining lectures and to proceed with the installation of the chosen workshops. Dassault Systèmes, 2010 Preliminaries for Abaqus Workshops 225

226 Basic operating system commands (You can skip this section and go directly to the workshops if everyone in your group is familiar with the operating system.) Note: The following commands are limited to those necessary for doing the workshop exercises. Working with directories 1. Start in the current working directory. List the directory contents by typing UNIX: Windows NT: ls dir Both subdirectories and files will be listed. On some systems the file type (directory, executable, etc.) will be indicated by a symbol. 2. Change directories to a workshop subdirectory by typing Both UNIX and Windows NT: cd dir_name 3. To list with a long format showing sizes, dates, and file, type UNIX: ls -l Windows NT: dir 4. Return to your home directory: UNIX: cd Windows NT: cd home-dir List the directory contents to verify that you are back in your home directory. 5. Change to the workshop subdirectory again. 6. The * is a wildcard character and can be used to do a partial listing. For example, list only Abaqus input files by typing UNIX: ls *.inp Windows NT: dir *.inp Working with files Use one of these files, filename.inp, to perform the following tasks: 1. Copy filename.inp to a file with the name newcopy.inp by typing UNIX: cp filename.inp newcopy.inp Windows NT: copy filename.inp newcopy.inp 2. Rename (or move) this new file to newname.inp by typing UNIX: mv newcopy.inp newname.inp Windows NT: rename newcopy.inp newname.inp (Be careful when using cp and mv since UNIX will overwrite existing files without warning.) WP.2 Dassault Systèmes, 2010 Preliminaries for Abaqus Workshops 226

227 WP.3 3. Delete this file by typing UNIX: rm newname.inp Windows NT: erase newname.inp 4. View the contents of the files filename.inp by typing UNIX: more filename.inp Windows NT: type filename.inp more This step will scroll through the file one page at a time. Now you are ready to start the workshops. Dassault Systèmes, 2010 Preliminaries for Abaqus Workshops 227

228 228

229 Notes 229

230 230 Notes

231 Workshop 1 Nonlinear Spring Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Introduction In this workshop you will study different aspects of Abaqus convergence behavior using two different models. The first is a simple model of a single nonlinear spring. You will define the nonlinear spring properties, run an analysis to stretch the spring, and consider the influence of increment size on the results. The second model contains a uniaxial tension test. You will manipulate this model to observe the effects of the tangent stiffness matrix on convergence. Nonlinear spring problem description For this analysis you will define a spring with nonlinear stiffness. The spring is modeled using an axial connector element. The spring stiffness has the force versus deflection data shown in Table W1 1. The goal of the analysis is to stretch the nonlinear spring with 190 units of force. Table W1 1 Force Deflection Question W1 1: What is the initial spring stiffness (the stiffness at zero deflection)? Dassault Systèmes,

232 W1.2 Question W1 2: Question W1 3: What is the spring stiffness between 5.0 and 6.0 units of deflection? Can you estimate what the final deflection will be when the full load of 190 units of force is applied to the spring? Desired results The desired results are a force-deflection curve like the one shown in Figure W1 1. You do not have to use as many increments as shown in the figure. Abaqus can find the solution to this problem in just one or two increments. Figure W1 1 Force-displacement curve for the nonlinear spring. Nonlinear spring model In this section you will run a script to setup the nonlinear spring model. The script creates the spring connector and the load history for the analysis. You must define the spring force versus deflection data. 1. Enter the working directory for this workshop:../nonconvergence/interactive/spring 2. Run the script ws_nonconv_nlspring.py using the following command: abaqus cae startup=ws_nonconv_nlspring.py The above command creates the Abaqus/CAE database spring.cae in the current directory. 3. The axial type connector section named ConnProp-1 is associated with the connector modeled in spring.cae. Edit the connector property (in the Model tree, expand the Connector Sections container and double-click ConnProp-1) Dassault Systèmes,

233 W1.3 to add nonlinear elasticity (Add Elasticity then choose Nonlinear in the elasticity data form). Enter the axial force (F1) versus deflection data provided in Table W Create a set named right containing the reference point at the right end of the connector element. 5. Request history output for CF1 and U1 for this set. 6. Submit the job nlspring. 7. Once the analysis completes, open nlspring.odb in the Visualization module. In order to view the connector you must activate the display of connectors using the ODB Display Options (View ODB Display Options). 8. Compare the analysis results to Figure W1 1. Tip: You can create the force-deflection plot in Abaqus/Viewer as follows. a. In the Model Tree, double-click XYData. b. In the dialog box that appears, select ODB field output as the source and click Continue. c. In the Variables tabbed page of the dialog box that appears, select Unique Nodal as the position and CF1 and U1 and the variables. d. In the Elements/Nodes tabbed page, choose Pick from viewport as the method and click Edit Selection. e. In the viewport, select the node at the right end of the spring and click Done in the prompt area. f. Click Save. g. In the Model Tree, double-click XYData. h. In the dialog box that appears, select Operate on XY data and click Continue. i. From the list of operators, select combine(x,x). j. Double-click the data object for the displacement then do the same for the force. k. At the bottom of the dialog box, click Plot Expression. Question W1 4: Do you think the initial increment size will influence whether or not Abaqus finds a converged solution in this simulation? Dassault Systèmes,

234 W1.4 Effect of the tangent stiffness matrix on convergence Next, you will investigate the effect of the tangent stiffness matrix accuracy on convergence. Recall that the tangent stiffness matrix plays a critical role in the Newton-Raphson algorithm. Errors in the formulation of the tangent stiffness matrix will result in analyses that require more iterations or, in some cases, that diverge. We illustrate the effect of the tangent stiffness accuracy by analyzing a small problem under uniaxial tension. Both displacement and load control are considered. The material response is elastic-plastic; Mises plasticity is assumed. The analysis can be carried out in one of two ways: by modifying the material stiffness calculated with user subroutine UMAT or by using the quasi-newton solver method. Both approaches will effectively introduce an error into the tangent stiffness matrix. To build confidence in user subroutine UMAT, we first solve the problem using the Mises plasticity algorithm that is built into Abaqus. Desired results You will compare the convergence history and results of several analysis jobs. Some of the jobs will include an error in tangent stiffness matrix. The loading method will alternate between a prescribed displacement and a pressure. Uniaxial tension test In this section you will run the provided script to create two uniaxial tension test models. Both models contain an axisymmetric cylindrical specimen pulled in tension. 1. Create a new model database (File New). 2. Make sure the PLATFORM variable is set on your machine. 3. Run the script ws_nonconv_usermat.py (File Run Script). The above script creates an Abaqus/CAE database named umat.cae. This database contains the models usermat and builtin. In the model usermat the cylinder's material is defined by the user subroutine UMAT. In the model builtin the cylinder's material is defined using the Mises plasticity model that is built into Abaqus. 4. Submit the job builtinmaterial to run the model builtin, which is set up as a displacement-controlled analysis and uses the Mises plasticity routine that is built into Abaqus. Monitor the job progress (Job Monitor builtin). Dassault Systèmes,

235 W In the Displacement Control portion of Table W1 2, enter the number of increments and the total number of iterations required to complete the analysis. You can find this information at the end of the analysis message file (in the Job Monitor click the Message File tab to view the contents of the message file and scroll to the bottom of the file). Under the table heading Flag Setting write built in. Table W1 2 Displacement Control Flag Setting Number of Increments Number of Iterations Load Control Flag Setting Number of Increments Number of Iterations 6. Edit the model builtin so that a pressure load is applied: suppress the displacement boundary condition named pull, and apply a pressure load with a magnitude of to the surface named pull. 7. Submit the job for analysis. Add the incrementation information from this analysis to the Load Control portion of Table W1 2. Dassault Systèmes,

236 W1.6 User subroutine approach Note: if you do not have the necessary compilers installed on your system, skip to the Quasi-Newton approach. Next, you will solve the same problem using the user subroutine UMAT included in the file iso_mises_umat.f (or iso_mises_umat.for if you are working on a Windows system). The mechanical constants used in the user subroutine are specified in the material definition included in the model usermat. Seven mechanical constants are specified in this problem. The seventh mechanical constant is used as a flag in user subroutine UMAT: when it is set to 1, UMAT returns the actual material stiffness; when it is set to 0, UMAT returns only the elastic stiffness. 1. Submit the job usermaterial to run the model usermat. This model is set up for a displacement-control analysis; the seventh mechanical constant is currently set to Compare the results of this analysis with those obtained for the displacementcontrol analysis using the built-in Mises plasticity routine. 3. Enter the flag setting, the number of increments, and the total number of iterations for this analysis in the Displacement Control portion of Table W Edit the material usermat (in the Model tree, double-click usermat underneath the Materials container of the model usermat). Set the seventh constant for the user material to 0. Note that you must select the User Material option in the Edit Material dialog box to modify the mechanical constants. 5. Submit the job for analysis, compare the results with the previous analysis jobs, and add the incrementation information from this analysis to the Displacement Control portion of Table W1 2. Question W1 5: Are the results obtained with the modified stiffness matrix correct? 6. Now repeat the analysis under a state of load control. Modify the model as follows: a. Suppress the displacement boundary condition named pull. b. Apply a pressure load with a magnitude of to the surface named pull. c. Set the user material's seventh mechanical constant to Submit the job. Enter the flag setting, the number of increments, and the total number of iterations to the Load Control portion of Table W Edit the material definition in model usermat so that the seventh constant for user material is 0. Submit the job and use the results to complete Load Control portion of Table W1 2. Dassault Systèmes,

237 W1.7 Question W1 6: What can you say about the difference in the convergence behavior of this problem when a pressure loading is applied instead of a boundary condition? Quasi-Newton approach The effect on convergence can also be seen by adopting a quasi-newton method to solve the problem. In this case, the tangent stiffness matrix is recalculated after a specified number of iterations within a given increment. The method reduces to the full Newton method when the tangent stiffness matrix is recalculated after every iteration. The quasi-newton approach will yield different convergence behavior than the usersubroutine approach. With the quasi-newton method, the tangent stiffness is recalculated at least once per increment: at the beginning of the increment. With the user subroutine approach described earlier, the tangent stiffness is effectively never updated if the flag is set to 0 (in this case, the linear elastic properties are always used). To see the effect on convergence using the quasi-newton method, edit the model named builtin. In the Model Tree, expand the Steps container and double-click Step-1. In the Other tabbed page of the step editor, select Quasi-Newton as the solution technique, and set the number of iterations allowed before reforming the kernel matrix to 999. Setting this parameter to 999 means the tangent stiffness matrix will be recalculated only after 999 iterations have been performed in a given increment; choosing such a large number effectively suppresses the operation. Solve the problem under both displacement and load control (by suppressing/resuming the boundary condition/load) and enter the corresponding data in Table W1 2. Under Flag Setting, enter Quasi-Newton. Also, answer questions W1 5 and W1 6 posed earlier. Note: Scripts that create the models described in these instructions are available for your convenience. Run these scripts if you encounter difficulties following the instructions outlined here or if you wish to check your work. The scripts are named ws_nonconv_nlspring_answer.py ws_nonconv_usermat_answer.py and are available using the Abaqus fetch utility. Dassault Systèmes,

238 238

239 Notes 239

240 240 Notes

241 Answers 1 Nonlinear Spring Interactive Version Question W1 1: What is the initial spring stiffness (the stiffness at zero deflection)? Answer: The initial spring stiffness is 100 K f u Question W1 2: What is the spring stiffness between 5.0 and 6.0 units of deflection? Answer: The spring stiffness is Question W1 3: Answer: Can you estimate what the final deflection will be when the full load of 190 units of force is applied to the spring? A rough estimate is that the spring will deflect 4.4 units when the full 190 units of force are applied. Question W1 4: Answer: Do you think the initial increment size will influence whether or not Abaqus finds a converged solution in this simulation? The nonlinearity in this problem is relatively mild and is smooth; therefore, you would expect that Abaqus could find a converged solution easily and that the initial increment size is not important. Question W1 5: Answer: Are the results obtained with the modified stiffness matrix correct? Yes. Errors in the tangent stiffness will only affect the convergence behavior, not the final result. When the elastic stiffness is used instead of the true stiffness, the number of iterations is increased. The results (when obtained) are still correct. Dassault Systèmes,

242 WA1.2 Question W1 6: Answer: What can you say about the difference in the convergence behavior of this problem when a pressure loading is applied instead of a boundary condition? Displacement-control problems are more stable than loadcontrol problems. When the elastic stiffness is used instead of the true stiffness, the displacement-control analysis ran successfully but the load-control analysis failed. Table WA1 1 Displacement Control Flag Setting Number of Increments Number of Iterations Built-In Quasi-Newton 6 8 Load Control Flag Setting Number of Increments Number of Iterations Built-In * 59* Quasi-Newton 6 17 *Did not run to completion. Dassault Systèmes,

243 Notes 243

244 244 Notes

245 Workshop 2 Reinforced Plate Under Compressive Loads Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description This problem models a reinforced plate structure subjected to in-plane compressive loading that produces localized buckling. It is a rectangular plate reinforced with beams in its two principal directions (see Figure W2 1). The plate represents part of a larger structure: the two longitudinal sides have symmetry boundary conditions, and the two transverse sides have pinned boundary conditions. In addition, springs at two major reinforcement intersections represent flexible connections to the rest of the structure. The mesh consists of S4 shell elements for both the plate and larger reinforcements and additional S3 shell and B31 beam elements for the remaining reinforcements. The entire structure is made of the same construction steel, with an initial flow stress of MPa. Gravity loads are applied followed by an in-plane load to one of the pinned sides, which compresses the plate. The plate buckles under the load. The buckling is initially localized within each of the sections bounded by the reinforcements. At higher load levels the plate experiences global buckling in a row of sections closest to the applied load. To provide stability to the numerical solution upon the anticipated buckling, the problem is solved in two different ways: once using automatic static stabilization and once using implicit dynamics (with the quasi-static application type). This workshop is based on Abaqus Example Problem As noted there, SIMULIA thanks IRCN (France) for providing this example. Dassault Systèmes,

246 W2.2 Figure W2 1 Reinforced plate model. Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/interactive/unstableplate 2. Run the script ws_nonconv_unstable.py using the following command: abaqus cae startup=ws_nonconv_unstable.py The above command creates an Abaqus/CAE database named unstableplate.cae in the current directory. The model static contains the model geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the loading, boundary conditions, and analysis procedure. Static analysis (without stabilization) In this section you will complete the model and run a static analysis without stabilization. Step definition You will begin by defining two general static analysis steps. In the first step gravity loading will be applied; in the second a compressive force will be applied to the plate. No stabilization will be activated for the time being. 1. In the Model Tree, double-click Steps. 2. In the Create Step dialog box, accept Static, General as the procedure type and click Continue. 3. In the Edit Step dialog box that appears, toggle on Nlgeom to consider the effects of geometric nonlinearity in the analysis. Accept all other default settings, and click OK. 4. Create a second general static step. In the Incrementation tabbed page of the step editor, set the maximum number of increments to 1000 and the initial time increment size to 0.1. Dassault Systèmes,

247 W2.3 Constraints A multi-point constraint will be defined at the end of the plate to which the load will be applied. Sets containing the control point as well as the region to be constrained have been predefined (load and front, respectively, as shown in Figure W2 2). front load Figure W2 2 Sets used in MPC constraint. 1. In the Model Tree, double-click Constraints. 2. In the Create Constraint dialog box, select MPC Constraint as the type and click Continue. 3. In the prompt area, click Sets to open the Region Selection dialog box. 4. From the list of available sets, select load (toggle on Highlight selections in viewport to identify this point). 5. Click Continue. 6. Select the set named front as the region to be constrained. Note that this set contains all regions on the front of the plate with the exception of the point corresponding to the control point. 7. Click Continue. 8. In the constraint editor, accept Beam as the MPC type and click OK. Dassault Systèmes,

248 W2.4 Loads You will now apply the gravity load to the entire structure and a compressive force to the MPC control point. 1. In the Model Tree, double-click Loads. 2. In the Create Load dialog box that appears: a. Name the load gravity. b. Select Step-1 as the step in which it will be applied. c. Accept Mechanical as the category, and select Gravity as the type. d. Click Continue. 3. By default, a gravity load is applied to the entire model. In the load editor, specify 9.81 for Component 2, and click OK to apply the load. 4. In the Model Tree, double-click Loads. 5. In the Create Load dialog box that appears: a. Name the load axial. b. Select Step-2 as the step in which it will be applied. c. Accept Mechanical as the category, and select Concentrated force as the type. d. Click Continue. 6. Select the set load as the region where the force will be applied. 7. Set CF3 to -646.e4, and click OK to apply the load. Boundary conditions Here you will assign the symmetry conditions described earlier. 1. In the Model Tree, double-click BCs. 2. In the Create Boundary Condition dialog box that appears: a. Name the boundary condition front. b. Select Initial as the step in which it will be applied. c. Accept Mechanical as the category, and select Displacement/Rotation as the type. d. Click Continue. 3. Select the set load as the region where the boundary condition will be applied and constrain U2, UR2, and UR3. 4. Repeat the above steps to create three additional displacement/rotation boundary conditions in the initial step as indicated in Table W2 1. The sets listed in Table W2 1 have been predefined and are shown in Figure W2 2 and Figure W2 3. Dassault Systèmes,

249 W2.5 Table W2 1 BC name Set containing region DOFs to be constrained back back U2, U3, UR2, UR3 back-ctr back-ctr U1 sides sides U1, UR2, UR3 sides back-ctr back Figure W2 3 Sets used for boundary conditions. Job Create and submit the analysis job. 1. Create a job named static and submit it for analysis (click mouse-button 3 on the job name and select Submit from the menu that appears). 2. Monitor the progress of the job (click mouse-button 3 on the job name and select Monitor from the menu that appears). You will find that the analysis has convergence difficulties in the second step. These coincide with the onset of local instability (localized buckling). The analysis terminates prematurely at approximately 38% of the compressive loading. Dassault Systèmes,

250 Postprocessing Open the output database file created by this job in the Visualization module. 1. Click mouse-button 3 on the job name and select Results from the menu that appears. 2. Contour the U2 component of displacement, as shown in Figure W2 4. W2.6 Figure W2 4 Transverse displacement contours(without stabilization). The transverse displacements alternate between positive and negative near the loaded region, indicating the onset of local buckling. 3. Create a force-displacement plot for the loaded node (using field output, extract CF3 and U3 data and combine them using the techniques described earlier). The plot appears as shown in Figure W2 5. The response is linear. Some form of stabilization (viscous damping or inertia) is required to proceed past the linear regime in this problem. Figure W2 5 Force-displacement at loaded node (without stabilization). Dassault Systèmes,

251 W2.7 Static analysis (with automatic stabilization) Now, you will add automatic stabilization to the second step. Modifications to the model Edit the step definition to add automatic stabilization. 1. In the Model Tree, expand the Steps container and double-click Step In the step editor: a. Select Specify dissipated energy fraction as the form of automatic stabilization. Accept the default value of dissipation intensity. b. Toggle off adaptive stabilization. Job and postpresssing Resubmit the job for analysis and monitor its progress. In this case the job completes successfully. As before, postprocess the results in the Visualization module. Initially local out-of-plane buckling develops throughout the plate in an almost checkerboard pattern inside each one of the sections delimited by the reinforcements, as shown in Figure W2 6. Figure W2 6 Transverse displacement contours (96% of load). Dassault Systèmes,

252 Later, global buckling develops along a front of sections closer to the applied load, as shown in Figure W2 7. W2.8 Figure W2 7 Global buckling pattern. The evolution of the displacements produced by the applied load is very smooth, as shown in Figure W2 8, and does not reflect the early local instabilities in the structure. However, when the global instability develops, the curve becomes almost flat, indicating the complete loss of load carrying capacity. Figure W2 8 Force-displacement at loaded node (with stabilization). Dassault Systèmes,

253 W2.9 Implicit dynamics (for quasi-static applications) A quasi-static solution to this problem can be obtained using the implicit dynamics procedure. The quasi-static application type for implicit dynamics provides an alternative approach to solving unstable quasi-static problems. 1. Copy the model named static to one named dynamic. 2. Replace the second step (in the Model Tree, click mouse button 3 on Step-2 and select Replace from the menu that appears). 3. In the Replace Step dialog box, select Dynamic, Implicit as the procedure and click Continue. 4. In the Basic tabbed page of the step editor, select Quasi-static as the application type. 5. In the Incrementation tabbed page of the step editor, set the maximum number of increments to 1000, the initial time increment size to Create a job named dynamic and submit it for analysis. 7. Monitor the job s progress. Once the analysis completes, click mouse button 3 on the job dynamic and select Results from the menu that appears to open the file dynamic.odb in the Visualization module. Postprocess the results as before. Figure W2 9 shows a comparison of the force-displacement curve obtained with the (stabilized) static and implicit dynamics procedures. The agreement between the results is excellent. Figure W2 9 Comparison of force-displacement at loaded node. Dassault Systèmes,

254 W2.10 An inspection of the model's energy content (Figure W2 10) reveals that while the load is increasing, the amount of dissipated/kinetic energy is negligible. As soon as the load flattens out, the dissipated/kinetic energy increases dramatically to absorb the work done by the applied loads. Figure W2 10 Comparison of model energies. Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_nonconv_unstable_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

255 Notes 255

256 256 Notes

257 Workshop 3 Crimp Forming Analysis Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description This model simulates crimp forming. Modern automobiles contain several thousand crimp joints. In a crimp joint a multi-strand wire bundle is mechanically joined to an end terminal to provide electrical continuity across the joint. The portion of the terminal that is folded over and into the wire bundle during crimping is called the grip. Figure W3 1 shows the model geometry. The grip is 0.36 mm thick and has a 60% coin at the tips. Coining is done to help the grip arms curl over the wire bundle as they are pushed against the roof of the punch during crimping. A seven-strand wire bundle is used, with each strand having a 0.31 mm diameter. A plane stress representation of the cross section is used. In reality, out-of-plane extrusion of the wire bundle and grip is a significant factor in crimp formation. To properly represent these effects, three-dimensional models are required (see Abaqus Example Problem for an example). In this workshop, this effect is not considered. The grip is formed from a half hard copper alloy; the wires are made from copper. The grip is meshed with CPS4I elements while the wires are meshed with CPS4 elements. The punch and the anvil are modeled as discrete rigid surfaces. A static analysis is difficult to achieve (even with automatic stabilization) because the model has no static stability (due to the free rigid body motion of the grip and wires) and during crimping the grip arms buckle as they are turned by the punch downward into the bundle. Thus, an implicit dynamic simulation (using the quasi-static application type) is performed. The static analysis is left as an optional exercise. Dassault Systèmes,

258 W3.2 punch wires grip anvil Figure W3 1 Crimp forming model. Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/interactive/wirecrimp 2. Run the script ws_nonconv_crimp.py using the following command: abaqus cae startup=ws_nonconv_crimp.py The above command creates an Abaqus/CAE database named crimp.cae in the current directory. The model dynamic contains the model geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the contact interactions, boundary conditions, and analysis procedure. Dassault Systèmes,

259 W3.3 Implicit dynamic analysis (quasi-static application) In this section you will complete the model definition and run an implicit dynamic analysis with a quasi-static application type. Note that a small amount of beta damping has been predefined to aid convergence. Step definition You will begin by defining two implicit dynamic steps. In the first step the crimp operation will be performed; in the second step a springback analysis will be performed (the punch will be moved back to its original position). 1. In the Model Tree, double-click Steps. 2. In the Create Step dialog box, select Dynamic, Implicit as the procedure type and click Continue. 3. In the Basic tabbed page of the step editor, toggle on Nlgeom to consider the effects of geometric nonlinearity in the analysis and set the time period to 0.2. Select Quasi-static as the application type. 4. In the Incrementation tabbed page of the step editor, set the maximum number of increments to 1000 and the initial time increment size to Create a second implicit dynamic step using the quasi-static application type with a time period of 0.01 and an initial time increment size of Set the frequency for the field and history output to 1. Contact properties Coulomb friction is assumed between the individual wires ( = 0.15), between the grip and wires ( = 0.15), between the grip and anvil ( = 0.3), and between the punch and the grip ( = 0.3); frictionless contact is assumed between the two grip arms. Thus, multiple contact properties are required. 1. In the Model Tree, double-click Interaction Properties. 2. In the Create Interaction Property dialog box, name the property wire, select Contact as the type and click Continue. 3. In the contact property editor, select Mechanical Tangential Behavior. 4. Choose Penalty as the friction formulation and enter a friction coefficient of Click OK. 5. Create two additional contact properties named grip-int and global with friction coefficients of 0.3 and 0.15, respectively, each with a shear stress limit of 300 MPa (specified in the Shear Stress tabbed page of the contact property editor). 6. Create a frictionless contact property named gripself. Dassault Systèmes,

260 W3.4 Contact interactions General contact will be used to define the contact interactions. 1. In the Model Tree, double-click Interactions. 2. In the Create Interaction dialog box, select Initial as the step, General contact (Standard) as the type, and click Continue. 3. In the interaction editor, select global as the global property assignment. 4. Next to Individual property assignments, click Edit to specify different contact properties for different regions of contact. The surfaces representing the different regions of the model have been predefined: grip-1.grip is the entire exterior surface of the grip; wires is the union of all exterior surfaces on the wires; punch-1.punch is the surface of the punch; and anvil-1.anvil is the surface of the anvil. The property assignments are summarized in Figure W3 2. Figure W3 2 Contact property assignments. 5. Once the contact property assignments have been made, click OK in the contact property assignment editor. 6. Click OK in the interaction editor. Dassault Systèmes,

261 Boundary conditions The punch is displaced downward a distance of mm and then returned to its original position. The anvil is held motionless during the entire analysis. 1. In the Model Tree, double-click BCs. 2. In the Create Boundary Condition dialog box that appears: a. Name the boundary condition punch. b. Select Step-1 as the step in which it will be applied. W3.5 c. Accept Mechanical as the category, and select Displacement/Rotation as the type. d. Click Continue. 3. A set containing the punch reference point has been predefined. In the prompt area, click Sets to open the Region Selection dialog box. 4. From the list of available sets, select punch-1.refpt as the region where the boundary condition will be applied (toggle on Highlight selections in viewport to identify this point). 5. Set U1 and UR3 to 0 and U2 to Repeat the above steps to create a displacement/rotation boundary condition for the anvil. Name the boundary condition anvil and select anvil-1.refpt as the region where the boundary condition will be applied. Constrain all degrees of freedom (i.e., set all to 0). 7. Modify the value of the applied punch displacement in the second step and set it to zero. This returns the punch to its original position. Tip: In the Model Tree, expand the States container underneath the punch boundary condition and double-click Step-2, as shown in Figure W3 3. Figure W3 3 Boundary condition modification. Job Create and submit the analysis job. 1. Create a job named crimp-2d and submit it for analysis. 2. Monitor the progress of the job. Dassault Systèmes,

262 W3.6 Postprocessing When the analysis completes, open the output database file created by this job in the Visualization module. 1. Click mouse-button 3 on the job name and select Results from the menu that appears. 2. Plot the Mises stress distribution on the deformed model shape at different stages of the analysis. Figure W3 4 shows the assembly after the grip arms have nearly reached the roof of the punch (126 milliseconds). The wire bundle has already been disturbed by this point. Figure W3 4 Deformed model shape at sec. Figure W3 5 shows the deformed shape of the crimp assembly after the grip arms have begun to curve around the roof of the punch and have partially folded over (142 milliseconds). Dassault Systèmes,

263 W3.7 Figure W3 5 Deformed model shape at sec. The grip arms buckle between the states indicated in Figure W3 4 and Figure W3 5. Figure W3 6 shows the final deformed shape after the forming process. The grip arms have fully folded over into the wire bundle, and the punch has made its complete downward stroke. Figure W3 6 Final deformed shape after forming process. Figure W3 7 shows the final shape of the wire bundle after springback. This figure shows that the originally round wires have been distorted during crimp formation. This distortion is essential for the correct formation of the crimp joint. The bare copper wires are actually covered by a thin layer of brittle copper oxide that forms on exposure of the copper to air. The goal of crimp forming is to break this oxide layer and expose the copper to the surface of the grip by inducing significant surface strains in each wire. Dassault Systèmes,

264 W3.8 Figure W3 7 Final deformed shape of the wire bundle after springback 3. Plot the internal and kinetic energy histories in the first analysis step. Figure W3 8 shows that the kinetic energy remains a small fraction of the internal energy confirming the quasi-static nature of the simulation. Figure W3 8 Model energies Dassault Systèmes,

265 W Plot the force-displacement curve for the punch (extract the data for RF2 and U2 from field output and combine the curves). Figure W3 9 shows the result. Figure W3 9 Force-displacement curve Optional: Static analysis (with stabilization) In this section you will perform the analysis using the general static analysis procedure with automatic stabilization. Step definition You will replace the implicit dynamic steps with general static ones. 1. Copy the model named dynamic to one named static. 2. Replace the first step (in the Model Tree, click mouse button 3 on Step-1 and select Replace from the menu that appears). 3. In the Replace Step dialog box, select Static, General as the procedure and click Continue. 4. In the Basic tabbed page of the step editor: a. Toggle on Nlgeom to consider the effects of geometric nonlinearity in the analysis. b. Set the time period to 0.2. c. Select Specify dissipated energy fraction as the form of automatic stabilization. d. Accept the default value of dissipation intensity and the default setting for automatic stabilization. Dassault Systèmes,

266 W In the Incrementation tabbed page of the step editor, set the maximum number of increments to 1000 and the initial time increment size to Replace the second step with a general static step. Specify a time period of 0.01 and an initial time increment size of Create a job named crimp-2d-static and submit it for analysis. Monitor its progress. You will find that the analysis does not complete successfully. It terminates at approximately 40% of the forming step, corresponding to the point where the wires are about to be disturbed. This is illustrated in Figure W3 10. Recall that the wires are not restrained; thus, their free rigid body motion causes convergence difficulties for the static analysis. Figure W3 10 Configuration at termination of static analysis In order to circumvent the problem caused by the unrestrained wires you can try adding soft springs to hold them in place. Alternatively (or in addition to) you can try increasing the stabilization damping factor. 8. Figure W3 11 shows the result of a static analysis with the static stabilization factor equal to 5 (the default value is determined by stabilization intensity factor; in this case the default initial value for the damping factor was approximately 3). The analysis completed nearly 75% of the forming step. Dassault Systèmes,

267 W3.11 Figure W3 11 Configuration at termination (damping factor = 5) 9. Figure W3 12 shows the result of a static analysis with the following non-default settings and controls: a. Stabilization damping factor = 100 b. Adaptive stabilization tolerance = 0.2 (default is 0.05) c. Minimum allowable time increment = 1.e 12 (default is 1.e 5 times the total time period, or 2.e 6 in this case) d. Number of attempts allowed in an increment = 10 (default is 5) e. Force residual tolerance = 0.25% (default is.5%) f. Displacement correction tolerance = 10% (default is 1%) Dassault Systèmes,

268 W3.12 The analysis completed nearly 90% of the forming step. Figure W3 12 Configuration at termination (damping factor = 100) Clearly it is very difficult to obtain a converged solution to this problem with a static analysis. As this problem demonstrates, the quasi-static application type for implicit dynamics represents a powerful complement to automatic static stabilization techniques and in some cases represents the only way to obtain a quasi-static solution using an implicit solution method. Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_nonconv_crimp_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

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270 270 Notes

271 Workshop 4 Contact: Beam Lift-Off Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to predict the final configuration of a beam as it is bent over a much stiffer body. Other results of interest are the reaction forces at the constrained nodes and the contact pressure between the beam and the body. The beam, at this stage, is modeled as a simple linear elastic material (E and 0.3). The beam is modeled in two dimensions using CPE4I elements (4-node, bilinear, plane strain quadrilateral with incompatible modes). Node-to-surface contact discretization with direct enforcement is considered (this is the default contact formulation). Question W4 1: What are the advantages of this element type over the other available plane strain solid elements? Forty elements are used along the length of the beam. Figure W4 1 Illustration of the beam. Dassault Systèmes,

272 W4.2 Desired results The desired results are shown in the following figures: Figure W4 2: the deformed shape of the model Figure W4 4: the reaction forces at the fixed end of the beam Figure W4 5: the contact pressure between the beam and the body Figure W4 2 Deformed shape of model with boundary conditions shown. Node B Node A Node C Figure W4 3 Deformed shape of model showing nodes A, B and C. Dassault Systèmes,

273 W4.3 Figure W4 4 History of reaction forces at constrained nodes A and B. Peak contact pressure at node C Figure W4 5 Contour of contact pressure on the beam. Note: Contours of CPRESS may be difficult to visualize on two-dimensional models. In Figure W4 5 the plane strain elements have been extruded for display purposes using the ODB Display Options in the Abaqus/CAE Visualization module (View ODB Display Options). Dassault Systèmes,

274 W4.4 The model In this section you will set up the beam model using the provided script, request contact diagnostic output, and run the analysis. 1. Enter the working directory for this workshop:../nonconvergence/interactive/contact 2. Run the script ws_nonconv_beamcontact.py using the following command: abaqus cae startup=ws_nonconv_beamcontact.py The above command creates the Abaqus/CAE database contact.cae in the current directory. 3. Activate contact diagnostics for both analysis steps (Output Diagnostic Print in the Step module). 4. Submit the job beamcontact. Question W4 2: Why does job beamcontact fail to converge? What changes must be made to make the model converge and give the desired results? Dassault Systèmes,

275 Notes 275

276 276 Notes

277 Answers 4 Contact: Beam Lift-Off Interactive Version Question W4 1: What are the advantages of this element type over the other available plane strain solid elements? Answer: The principal advantages of the CPE4I element are that it can model bending effectively (because it uses incompatible modes an enhanced deformation gradient formulation), it is a first-order element (and, thus, works well in contact problems), and it is fully integrated (no hourglassing problems). Question W4 2: Why does job beamcontact fail to converge? What changes must be made to make the model converge and give the desired results? Answer: The model beamcontact does not converge because Abaqus cannot establish the correct contact state at the start of the analysis. Five attempts are made in the first increment of Step 2, but each fails because ultimately the solution diverges. A portion of the message file follows: ***WARNING: OVERCONSTRAINT CHECKS: NODE 1 INSTANCE BEAM-1 ON THE SLAVE SURFACE AND CORRESPONDING NODE 1 INSTANCE RIGID-1 ON THE MASTER SURFACE HAVE EQUAL PRESCRIBED DISPLACEMENTS NORMAL TO THE CONTACT SURFACE. SINCE THIS MAKES THE CONTACT CONSTRAINT REDUNDANT, THE CONTACT STATUS AT THE SLAVE NODE IS CHANGED FROM CLOSE TO OPEN. CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.2 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.4 IS OVERCLOSED BY E-006. CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.5 IS OVERCLOSED BY E-006. CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.6 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.9 OPENS. CONTACT PRESSURE/FORCE IS Dassault Systèmes,

278 WA4.2 CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.11 IS OVERCLOSED BY CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) NODE BEAM-1.13 OPENS. CONTACT PRESSURE/FORCE IS SEVERE DISCONTINUITIES OCCURRED DURING THIS ITERATION. 3 POINTS CHANGED FROM OPEN TO CLOSED 4 POINTS CHANGED FROM CLOSED TO OPEN CONVERGENCE CHECKS FOR SEVERE DISCONTINUITY ITERATION 2 MAX. PENETRATION ERROR AT NODE BEAM-1.11 OF CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) MAX. CONTACT FORCE ERROR AT NODE BEAM-1.13 OF CONTACT PAIR (ASSEMBLY_BOTBEAM,ASSEMBLY_RIGID-1_RIGIDSURF) THE ESTIMATED CONTACT FORCE ERROR IS LARGER THAN THE TIME-AVERAGED FORCE. AVERAGE FORCE 1.954E+03 TIME AVG. FORCE 1.954E+03 LARGEST RESIDUAL FORCE 1.513E+04 AT NODE 13 DOF 2 INSTANCE: BEAM-1 LARGEST INCREMENT OF DISP AT NODE 82 DOF 2 INSTANCE: BEAM-1 LARGEST CORRECTION TO DISP E-02 AT NODE 68 DOF 2 INSTANCE: BEAM-1 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE. AVERAGE MOMENT 1.509E+04 TIME AVG. MOMENT 1.509E+04 ALL MOMENT RESIDUALS ARE ZERO LARGEST INCREMENT OF ROTATION E-31 AT NODE 1 DOF 6 INSTANCE: RIGID-1 LARGEST CORRECTION TO ROTATION E-31 AT NODE 1 DOF 6 INSTANCE: RIGID-1 THE MOMENT EQUILIBRIUM EQUATIONS HAVE CONVERGED : : CONVERGENCE CHECKS FOR SEVERE DISCONTINUITY ITERATION 13 : : ***NOTE: THE SOLUTION APPEARS TO BE DIVERGING. CONVERGENCE IS JUDGED UNLIKELY. Dassault Systèmes,

279 WA4.3 It can be difficult sometimes to sort out all the information in the message file. The job diagnostics tool available in the Visualization module (Tools Job Diagnostics) may help you interpret some of this information. Use the job diagnostics tool to highlight the nodes that open or overclose in each SDI, as shown in Figure WA4 1. Try to identify those nodes whose contact state changes most frequently. 2. Select Contact. 1. Select an SDI. 3. Select nodes. 4. Highlight nodes in the viewport. Figure WA4 1 Slave node openings in the second iteration of the second step. In this case there is no single node that is involved in every SDI. However, looking at all SDIs in a given attempt shows that at least one of five nodes is involved in every SDI. These nodes are marked in Figure WA4 2. At least one of these nodes is involved in every SDI. Figure WA4 2 Nodes that are chattering in the model. Dassault Systèmes,

280 There are many methods that can be used to allow the model to overcome its initial convergence difficulties. Any one of the following methods may be used: WA Use automatic contact tolerances (in the Model Tree, double-click Contact Controls to define the contact control; then expand the Interactions container and double-click Int-1 to assign the newly created contact control to Int-1). 2. Use penalty enforcement of the hard contact constraints. 3. Use the surface-to-surface contact discretization. 4. Round the corner of the rigid body (in reality some amount of rounding is present; here a corner radius of 0.05 is sufficient to permit convergence). 5. Perturb the beam mesh near the corner of the rigid body so that the beam node does not lie exactly at the corner. The results using the automatic contact tolerances are provided below. The other methods produce results that are nearly identical to the results with automatic contact tolerances. Another method that will permit convergence is to use softened contact. However, choosing the values of the softened contact parameters can be difficult. Even if Abaqus obtains a converged solution, it will be very, very difficult to produce the desired results with softened contact alone because it fundamentally changes the contact pressures in the model. Automatic contact tolerances The automatic contact tolerances are designed for exactly this type of model: one in which establishing the initial contact state is very difficult. These automatic tolerances do not affect the final results that you are interested in with this model the only results they should affect in most simulations are those in the first increment. Automatic contact tolerances are discussed in detail in Lecture 5. The reaction forces and the contact pressures for a model using the automatic tolerances are shown in Figure WA4 3 and Figure WA4 4. Note: A script that adds automatic contact tolerances to the model beamcontact and runs the analysis job is available for your convenience. The script is named ws_nonconv_beamcontact_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

281 WA4.5 Figure WA4 3 Reaction forces when the automatic contact tolerances are used. (Refer to Figure W4 3 for the locations of nodes A and B.) Peak contact pressure Figure WA4 4 Contact pressure contours when the automatic contact tolerances are used. Dassault Systèmes,

282 282

283 Notes 283

284 284 Notes

285 Workshop 5 Contact: Stabilization Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Introduction In many structural problems, contact is a critical factor in transferring load from one region to another. In FEA often the most challenging aspect of resolving contact-related issues is establishing stable contact conditions and hence the load path required for static equilibrium. This can be particularly difficult if the exact contact conditions are not known in advance. A loading condition is required to establish contact while a stable contact condition is required for the structure to carry load. Such a finely balanced system can be difficult to solve; however, there are a number of techniques that can be used to assist with this class of problem. The purpose of this workshop is to: 1. Introduce a particular (common) form of contact convergence problem loose/rattling contact 2. Use the job diagnostics tools to identify the cause of convergence difficulties 3. Apply modeling techniques to avoid such problems Model Description The model to be analyzed in this workshop is a bolt-up of wheel rim/hub assembly (see Figure W5 1). Symmetry has been used to reduce the model size. The wheel nuts are to be tightened to a tension of 10kN (5kN for the half nut on the symmetry plane). Dassault Systèmes,

286 W5.2 Figure W5 1 Wheel assembly Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/interactive/contact_stab 2. Run the script ws_nonconv_wheel_disk.py using the following command: abaqus cae startup= ws_nonconv_wheel_disk.py The above command creates the Abaqus/CAE database wheel.cae in the current directory. The model contains the geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the loading, boundary conditions, interactions, and analysis procedure. Dassault Systèmes,

287 W5.3 Basic Model Setup At this stage you will complete the basic model definition. 1. Define a general static step considering geometrically nonlinear effects. Set the initial time increment size to 0.05 s. 2. Coulomb friction is assumed between all parts ( = 0.1). Define a contact property named fric. Specify Penalty enforcement of the normal contact constraints. Also define a friction coefficient of 0.1 using the Penalty friction formulation. 3. Define the contact pairs indicated in Table W5 1. For each contact pair select the Surface to surface discretization method and fric as the interaction property. Interaction name Master surface Slave surface hub-rim rim-hub hub-rim nut1-rim rim-nuts nut1-rim nut2-rim rim-nuts nut2-rim nut3-rim rim-nuts nut3-rim Table W5 1 Contact pairs 4. A coupling constraint will be used to constrain the motion of the hub. Define a Distributing coupling constraint between the surface couplingsurf and the set refpoint. Constrain all available degrees of freedom. 5. The thread engagement between the nuts and the hub will be modeled with tie constraints. Define the tie constraints indicated in Table W5 2. Constraint name Master surface Slave surface nut1tie hubtie1 nut1tie nut2tie hubtie2 nut2tie nut3tie hubtie3 nut3tie Table W5 2 Tie constraints 6. The hub reference point is completely constrained. Apply an ENCASTRE boundary condition to the set refpoint. 7. As indicated earlier only half the structure is modeled due to symmetry. Apply a ZSYMM boundary condition to the set zsymm. Dassault Systèmes,

288 W5.4 First attempt Load Control At this stage you will apply the bolt loads and run the analysis. 1. For each bolt define a Bolt load using the Apply force method (see Figure W5 2). Specify a magnitude of for the full pin regions and 5000 for the pin region on the symmetry plane. Note that surfaces with the prefix pre_tension and datum axes along each bolt axis have been predefined to facilitate the load definition. Figure W5 2 Bolt load definition 2. Edit the default field output requests to include the contact state CSTATUS. 3. Create additional history output requests for the contact normal force CFN for each of the nut-rim contact interactions. 4. Create and submit a job called load_control. 5. Monitor the progress of this analysis using the job monitor tools (see Figure W5 3). NOTE: This analysis is expected to fail do not panic!!! Dassault Systèmes,

289 W5.5 Figure W5 3 Job monitor Results for first attempt The analysis should fail to complete. You will now review the information in the Abaqus output database file to see if there are any clues as to why the analysis fails. 1. Open the output database file in the Visualization module (click Results in the Job Manager). It is good practice to review the contact conditions as defined in the Abaqus results file to check for potential modeling errors. 2. Use the Display Group tools to review the contact surfaces defined in the model (see Figure W5 4). Dassault Systèmes,

290 W5.6 Figure W5 4 Using display groups to review contact surfaces Question W5 1: Do the surfaces created seem reasonable and cover the potential/expected contact conditions? 3. Review the initial contact state (are contacting surfaces open or closed?). If contact pairs are initially overclosed, Abaqus/Standard will by default attempt to resolve the overclosure in the first increment. This often results in large unbalanced contact forces and can cause convergence problems. Question W5 2: Are any regions of the model initially overclosed? Tip: Create a contour plot of COPEN. A negative value of COPEN implies overclosure. Conversely, some surfaces will require contact to be established in order to transfer load from one region of the model to another. Question W5 3: Are there any regions in the model where you might expect contact but that are not initially in contact? Tip: Check CSTATUS=OPEN/COPEN=0 for the different components in the model (use the Display Group tools to help). It should be clear from this initial review of the contact state that the contact load path is not initially established. Contact has not been initiated between the nuts and Dassault Systèmes,

291 W5.7 the rim (see Figure W5 5). Therefore, initially there is no path defined to transfer the bolt tension loads from the nuts and hub to the wheel rim. Minimum gap is 0.005mm. No initial contact between the nuts and the rim. Figure W5 5 Contour plot of COPEN (Nuts - Rim) You will now use the Job Diagnostics tools to determine the reasons for cutbacks/convergence failure. 4. Open the Job Diagnostics dialog box (Tools Job Diagnostics). 5. Check for warnings during the analysis phase using the Warnings and Errors tabbed pages (see Figure W5 6). Dassault Systèmes,

292 W5.8 Figure W5 6 Job Diagnostics dialog box Question W5 4: Are there any analysis warnings? Typical warnings for these problems refer to numerical singularities. Numerical singularities can often indicate that there are unconstrained degrees of freedom in the model resulting in rigid body modes. In many contact problems, the only constraint preventing rigid body modes is contact between the separate regions. If contact is not established when a load is applied then there is the potential that loading is applied to unconstrained degrees of freedom, resulting in severe instability (no structural stiffness to carry the load). This often results in numerical problems and convergence failure. Question W5 5: Are there any Numerical Singularity warnings in the model? 6. Toggle on Highlight selections in viewport to show the regions of the model associated with any numerical singularity warnings. Question W5 6: Does the position of the numerical singularity warnings tie in with the initial contact state? Dassault Systèmes,

293 W Review the information on the Residuals, Contact, and Element tabbed pages for a selection of the attempted iterations. Question W5 7: Were any increments completed? The job diagnostics provide valuable information as to exactly why an attempted increment has failed and required a cutback. Figure W5 7 shows the job diagnostics for the first attempted increment for the current model. Cutback reason given on Summary tab Residuals, Contact, and Element tabs allow identification of the offending mesh regions Figure W5 7 Job Diagnostics: First increment attempt Dassault Systèmes,

294 W For each attempted increment identify the reason given by the solver for the failure/cutback and enter the information into Table W5 3 below. Increment Attempt No. Iterations Reason for failure Severe Overclosures Table W5 3 Iteration history Question W5 8: On the evidence of the analysis history and the discussion so far do you think that resubmitting the load at a reduced load level would help the convergence? Potential Solutions It is clear that for this model, applying the pre-tension load before contact between the nut faces and the wheel rim is established will not permit a solution to be obtained without some assistance. There are a number of techniques that could be employed to avoid these convergence issues: 1) Ensure that contact is initially closed: Reposition the components in the assembly Adjust the slave nodes to lie exactly on the master surface Define contact clearance/interference This method is often effective but is not appropriate for every case; for example, it may not be possible to predict the precise contact condition of the model in advance. 2) Replace loads with prescribed displacements. In some cases it may be possible to replace the loads on the structure with conjugate displacements (for example, this would be straightforward for concentrated loads). Using displacement control instead of load control is often more stable as the application of the fixed displacement boundary conditions remove the rigid body modes and numerical singularities, making the solution inherently more stable. Dassault Systèmes,

295 W5.11 This method is often used in two stages: Step 1: Prescribe sufficiently large dummy displacement boundary conditions to bring the components in the model into contact. Step 2: Deactivate the dummy boundary conditions and replace them with the required load. This method is applicable only in certain situations. In particular, for complex loading patterns (pressures, body loads, multiple loads, etc.) it may not be possible to simply replace loads with displacement boundary conditions. 3) Apply contact stabilization to resist the rigid body modes until contact is established. This method applies viscous pressures to the contact surfaces to react the applied loads until contact is established. Contact stabilization does not suffer from the limitations of the other methods and is very easy to activate; however, it should be used with care as there is the potential to over-damp the solution. For more information refer to section Adjusting contact controls in Abaqus/Standard" in the Abaqus Analysis User's Manual. Attempt 2 Displacement Control Either of the methods discussed above could be used to attempt to resolve the convergence issues of this model. In this workshop you will try methods 2 and 3 from above (feel free to try adjusting the components in the assembly if time permits). 1. Copy the model load_control to a new model named displacement_control. 2. Rename the step to disp. 3. Edit the step disp and specify an initial time increment size of 0.2 s. 4. Create a second general static step named load. Set the initial time increment size to 0.2 s. 5. Edit the bolt load definitions in the step disp to use the Adjust Length method. Change the magnitude to (just large enough to close the gap). 6. In the load step, edit the bolt load definitions. Use the Apply load method and set the magnitude for the full bolts to (for the bolt on the symmetry plane set the magnitude to 5000), as shown in Figure W5 8. Dassault Systèmes,

296 W5.12 Figure W5 8 Modified bolt load definitions 7. Create a new job named disp_control and submit it for analysis. 8. Monitor the progress of this job using the job monitor. Question W5 9: Has loading using fixed displacements improved the convergence? Does this analysis run to completion? 9. Open the Abaqus output database and use the Job Diagnostics to examine the convergence behavior for this latest run. Question W5 10: Are there any analysis warnings produced during this attempt? Question W5 11: Has applying the displacement control method avoided any of the warnings from the previous attempt? Dassault Systèmes,

297 W5.13 Attempt 3 Contact stabilization 1. Copy the model load_control to a new model named load_control_stab. 2. Enter the Interaction module and define a non-default contact controls property (Interaction Contact Controls). 3. In the Edit Contact Controls dialog box, switch to the Stabilization tabbed page and toggle on Automatic Stabilization. Accept the default settings, as shown in see Figure W5 9. Figure W5 9 Contact controls editor 4. Edit each of the nut-rim contact interactions to reference the contact controls property created above, as shown in Figure W5 10. Dassault Systèmes,

298 W5.14 Figure W5 10 Interaction editor 5. Edit the field output requests to include CDPRESS (contact damping pressure). 6. Create a new job named load_control_stab and submit it for analysis. 7. Monitor the progress of this job using the job monitor. Question W5 12: Has the application of contact stabilization allowed the analysis to complete? 8. Open the Abaqus output database and use the Job Diagnostics to examine the convergence behavior for this latest run. Question W5 13: Are there any analysis warnings produced during this attempt? Question W5 14: Has applying the stabilization method avoided any of the warnings from the first attempt? As mentioned earlier, applying viscous damping/stabilization forces should be used with care as over-damping the solution may lead to inaccuracies in the results. Therefore, you will check to see if applying viscous damping forces in this model have adversely affected the results. 9. Create a contour plot of the contact damping pressure CDPRESS. Dassault Systèmes,

299 W5.15 Question W5 15: What is the peak value of CDPRESS? TIP: You will need to review the complete time history to determine the peak value. The Contact Stabilization Factor is ramped down to zero by the end of the step so the peak CDPRESS value will not be at the end of the step. Question W5 16: How does the peak value of CDPRESS compare to the peak contact pressure (CPRESS)? The output variable ALLSD represents the energy dissipated by viscous forces such as contact stabilization. This can be used to evaluate the level of impact of the artificial stabilization forces on the solution. As a general rule of thumb, the stabilization energy should be a small percentage (say < 5%) of the model s internal energy (ALLSE if purely elastic; otherwise, ALLIE). 10. Using history output, plot the time histories of the static dissipation energy ALLSD and the total elastic strain energy, ALLSE. The plot appears as shown in Figure W5 11. Figure W5 11 Energy histories Question W5 17: Is ALLSD small compared to ALLSD? 11. Edit the contact controls to apply a scale factor of 0.1 to the default value (see Figure W5 12) and resubmit the analysis. Dassault Systèmes,

300 W Figure W5 12 Automatic stabilization option 13. Finally, review the results (stresses, strains, contact forces, etc.) from the disp_control and load_control_stab jobs. Question W5 18: How do the results from the two different approaches compare? Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_nonconv_wheel_disk_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

301 Notes 301

302 302 Notes

303 Answers 5 Contact: Stabilization Interactive Version Question W5 1: Do the surfaces created seem reasonable and cover the potential/expected contact conditions? Answer: The main contact areas are all covered. Although in the absence of friction, an extra set of contact pairs between the hub pins and the wheel rims may be required (see Figure WA5 1) Figure WA5 1 Additional contacts Dassault Systèmes,

304 WA5.2 Question W5 2: Are any regions of the model initially overclosed? Answer: No. Question W5 3: Are there any regions in the model where you might expect contact that are not initially in contact? Answer: Yes. You would expect that the underside of the nuts will eventually contact the wheel rim. Question W5 4: Are there any analysis warnings? Answer: Yes. There are three analysis warnings during each increment attempt. Question W5 5: Are there any Numerical Singularity warnings in the model? Answer: Yes. All of the warnings relate to numerical singularities. Question W5 6: Does the position of the numerical singularity warnings tie in with the initial contact state? Answer: Yes. The highlighted nodes refer to regions of the hub where the bolt pre-tension is applied. In order to carry the bolt preload, contact must be established between the underside of the nuts and the wheel rim. Recall that this contact interface is initially open; hence, there is no way to react the pre-tension force, resulting in the numerical singularity warnings. Question W5 7: Were any increments completed? Answer: No. Each attempt fails after a single iteration. Question W5 8: On the evidence of the analysis history and the discussion so far do you think that resubmitting the load at a reduced load level would help the convergence? Dassault Systèmes,

305 WA5.3 Answer: There is no evidence to suggest that reducing the load level will help with convergence. Note that automatic time incrementation has already reduced the load level five times, with the same convergence behavior for each attempt, before completely giving up. Question W5 9: Has loading using fixed displacements improved the convergence? Does this analysis run to completion? Answer: Yes. The analysis completes with no cutbacks. Question W5 10: Are there any analysis warnings produced during this attempt? Answer: No. Question W5 11: Has applying the displacement control method avoided any of the warnings from the previous attempt? Answer: Yes. Establishing contact via displacement control has avoided the numerical singularity warnings. Question W5 12: Has the application of contact stabilization allowed the analysis to complete? Answer: Yes. The analysis completes with no cutbacks. Question W5 13: Are there any analysis warnings produced during this attempt? Answer: No. Question W5 14: Has applying the stabilization method avoided any of the warnings from the first attempt? Answer: Yes. Establishing contact using contact stabilization has avoided the numerical singularity warnings. Question W5 15: What is the peak value of CDPRESS? Answer: MPa Dassault Systèmes,

306 WA5.4 Tip: Create a maximum envelope plot of CDPRESS over all frames (Tools Create Field Output From Frames). Question W5 16: How does the peak value of CDPRESS compare to the peak contact pressure (CPRESS)? Answer: Even with the default contact stabilization the peak CDPRESS is small compared to the CPRESS for this analysis (peak CDPRESS=2.616MPa vs. peak CPRESS=125.4MPa). Question W5 17: Is ALLSD small compared to ALLSD? Answer: ALLSD is a little high (~6% of ALLSE); thus, reduction of the stabilization damping factor is recommended. Question W5 18: How do the results from the two different approaches compare? Answer: The results of both techniques match very well, with very little differences in the stress, strain, CPRESS, etc. Dassault Systèmes,

307 Notes 307

308 308 Notes

309 Workshop 6 Element Selection Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to study the performance of different element types. We consider two problems: the bending of a long, slender cantilever beam and the uniaxial tension of a nearly incompressible elastic solid. Each problem is solved with displacement formulation elements as well as hybrid elements. The difference in the performance of each element formulation is assessed. Desired results You will compare the convergence history of each problem when solved with the displacement and hybrid elements. Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/interactive/element 2. Run the script ws_nonconv_elemselection.py using the following command: abaqus cae startup=ws_nonconv_elemselection.py The above command creates an Abaqus/CAE database named elemselection.cae in the current directory. This database contains two complete models, one for the cantilever beam analysis (slender_beam) and one for the uniaxial tension analysis (elasticsolid). Dassault Systèmes,

310 W6.2 Cantilever beam In this section you will model the bending of a long, slender cantilever beam. The beam is initially 1000 units long and meshed with linear beam elements (B21). First, study the performance of displacement formulation beam elements for beams of different length. 1. Submit the job slender_beam1a to analyze the model slender_beam. Monitor the job progress (Job Monitor slender_beam1a). 2. Enter the number of increments and the total number of iterations required to complete the analysis in Table W6 1. You can find this information at the end of the analysis message file (in the Job Monitor click the Message File tab to view the contents of the message file and scroll to the bottom of the file). 3. Edit the sketch of the beam so that the beam length is 5000 units: In the Model Tree, expand the model named slender_beam. Underneath the Parts container, expand the part named beam. In the list that appears, expand the Features container and double-click Wire-1. In the Edit Feature dialog box, click Edit Section Sketch. A convenient way to modify the beam length is to dimension the beam between its endpoints (Add Dimension) and set the distance appropriately. 4. Mesh the part instance: In the Model Tree, expand the Assembly. Underneath the Instances container, expand the part instance named beam-1. In the list that appears, double-click Mesh. In the Mesh module, select Mesh Instance. 5. Copy the job slender_beam1a to slender_beam1b and submit the new job. 6. Repeat Step Edit the sketch of the beam so that the beam length is units. Mesh the part instance. 8. Copy the job slender_beam1b to slender_beam1c and submit the new job. 9. Repeat Step 2. Dassault Systèmes,

311 W6.3 Next, you will study the performance of hybrid formulation beam elements. 1. Change the beam element type (Mesh Element Type) so that the hybrid formulation is used (B21H). 2. Copy the job slender_beam1c to slender_beam2c and submit the new job. 3. Enter the number of increments and the total number of iterations required to complete the analysis in Table W Change the beam length to 5000 units and mesh the part instance. 5. Copy the job slender_beam2c to slender_beam2b and submit the new job. 6. Repeat Step Change the beam length to 1000 units and mesh the part instance. 8. Copy the job slender_beam2b to slender_beam2a and submit the new job. 9. Repeat Step 3. Question W6 1: Do the results (displacements, stresses, etc.) differ between the displacement element model and the hybrid element model? Question W6 2: Why does the hybrid element model converge more easily than the displacement element model as the beam length increases? Table W6 1 B21 Elements Length Number of Increments Number of Iterations B21H Elements Length Number of Increments Number of Iterations Dassault Systèmes,

312 W6.4 Now, you will study the iteration history for the long beam models (i.e., length =12000). 1. Open the message file slender_beam1c.msg in a text editor or open slender_beam1c.odb in the Visualization module and access the job diagnostic tool (Tools Job Diagnostics). Locate the iteration information for increment Enter the largest residual force for each iteration of increment 4 in Table W Repeat Step 2 for increment 5. Question W6 3: What do you notice about the iteration history for the model with displacement elements? Table W6 2 B21 Elements (Length = 12000) Increment Iteration Largest Residual Force Dassault Systèmes,

313 W Open the message file slender_beam2c.msg in a text editor or open slender_beam2c.odb in the Visualization module and access the job diagnostic tool. Locate the iteration information printed for increment Enter the largest residual force for each iteration of increment 4 in Table W Repeat Step 5 for increment 5. Question W6 4: How does the iteration history for the hybrid element model compare with that for the displacement element model? Table W6 3 B21H Elements (Length = 12000) Increment Iteration Largest Residual Force Dassault Systèmes,

314 W6.6 Incompressibility in an elastic solid This task investigates the convergence behavior of an analysis of a nearly incompressible elastic solid under uniaxial tension. You will analyze one-element models consisting of reduced-integration plane strain elements. A parametric study will show the change in convergence behavior as Poisson s ratio approaches a value of Analyze the elasticsolid model by submitting the job solid. The elastic material used for this job has a Poisson s ratio of The element type is CPE4R (4-node, bilinear, reduced integration, plane strain quadrilateral with hourglass control). 2. Enter the number of increments and the total number of iterations required to complete the analysis in Table W Modify the material elastic so that Poisson s ratio is set closer to 0.5. Create and submit new jobs for the following Poisson s ratio values: 0.499, , and Enter the number of increments and the total number of iterations required to complete each analysis in Table W Modify the mesh element type in the model so that the hybrid formulation is used (CPE4RH). Set Poisson s ratio to 0.49 for the material elastic. Analyze the model. 6. Repeat Steps 2 4 to examine the effects of incompressibility on convergence when using hybrid elements. Question W6 5: How does the iteration history for the hybrid element model compare with that for the displacement element model? Table W6 4 CPE4R Elements ν Number of Increments Number of Iterations CPE4RH Elements ν Number of Increments Number of Iterations Dassault Systèmes,

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317 Answers 6 Element Selection Interactive Version Question W6 1: Do the results (displacements, stresses, etc.) differ between the displacement element model and the hybrid element model? Answer: The displacements and stresses are essentially same for the two element formulations (differences, if any, are less than 0.1%). Question W6 2: Why does the hybrid element model converge more easily than the displacement element model as the beam length increases? Answer: As the beam length increases, the ratio between its axial stiffness and bending stiffness becomes greater and greater. Hybrid elements are specially designed for use in such cases. As a result, the models with hybrid elements converge more rapidly than the models using displacement-based elements. Table WA6 1 B21 Elements Length Number of Increments Number of Iterations B21H Elements Length Number of Increments Number of Iterations Dassault Systèmes,

318 WA6.2 Question W6 3: What do you notice about the iteration history for the model with displacement elements? Answer: The force residual does not decrease monotonically. In fact, the largest force residual fluctuates between very large and very small numbers. The large axial forces produce large displacement corrections. Table WA6 2 B21 Elements (Length = 12000) Increment Iteration Largest Residual Force E E E E E E 03 Dassault Systèmes,

319 WA6.3 Question W6 4: How does the iteration history for the hybrid element model compare with that for the displacement element model? Answer: The model with the hybrid elements requires fewer iterations to complete the analysis than the model with the displacementbased elements. This is particularly noticeable as the beam gets longer and more inextensible. In addition, the peak residuals are more stable from iteration to iteration, which usually results in fewer cutbacks. Table WA6 3 B21H Elements (Length = 12000) Increment Iteration Largest Residual Force E E E E 03 Dassault Systèmes,

320 WA6.4 Question W6 5: How does the iteration history for the hybrid element model compare with that for the displacement element model? Answer: As the material becomes more incompressible (i.e., Poisson s ratio approaches 0.5), the models with the displacement-based elements require more iterations to complete the analysis than the models with hybrid elements. The models with the hybrid elements require the same number of iterations regardless of the value chosen for ν. Table WA6 4 CPE4R Elements ν Number of Increments Number of Iterations CPE4RH Elements ν Number of Increments Number of Iterations Dassault Systèmes,

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323 Workshop 7 Limit Load Analysis Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to study the response of a pressurized fluid channel when different forms of nonlinearity are present, including geometric and material nonlinearity. The loading consists of an internally applied pressure of 2000 psi. monitor Desired results Figure W7 1 Fluid channel model. The desired results include load-deflection curves that compare the internal pressure against the vertical displacement (bulging) of the structure for each of the runs described below. Dassault Systèmes,

324 W7.2 Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/interactive/riks 2. Run the script ws_nonconv_limitload.py using the following command: abaqus cae startup=ws_nonconv_limitload.py The above command creates an Abaqus/CAE database named limit.cae in the current directory. The model linear within this database is complete. It contains the model geometry, material properties, and loading history for the linear analysis of the pressurized structure. Linear analysis In this section you will analyze the linear elastic response of the structure. The results serve as a baseline for the subsequent nonlinear analyses. 1. From the Model Tree, submit the job linear. 2. When the job completes, open linear.odb in the Visualization module (click mouse button 3 on the job name and select Results from the menu that appears). Create the load-deflection curve as follows: a. In the Results Tree, expand the History Output container for the output database file named linear.odb. b. Click mouse button 3 on the data named Spatial Displacement: U2 at Node x in NSET MONITOR. From the menu that appears, select Save As. Name the X Y data linear-dsp and click OK. c. In the Results Tree, double-click the XYData container. d. In the Create XY Data dialog box, choose Operate on XY data and click Continue. e. Exchange the X- and Y-values of linear-dsp using the swap operator and multiply the swapped curve with the magnitude of the pressure load (i.e., 2000 in this case). f. Click Save As. Name the X Y data linear-loaddsp and click OK. g. Click Cancel to close the Operate on XY Data dialog box. h. Expand the XYData container, and double-click the curve named linear- LoadDsp. i. Double-click the Y-axis to open the Axis Options dialog box; change the Y-axis title to Load. Dassault Systèmes,

325 W7.3 Nonlinear analysis I (Nlgeom) Now, you will include the effects of geometric nonlinearity in this problem. 1. Copy the model linear to a model named nlgeom. 2. Activate nonlinear geometric effects in Step-1 (In the Model Tree, expand the Steps container and click mouse button 3 on Step-1; in the menu that appears, select Nlgeom; in the Edit Nlgeom dialog box that appears, toggle on Nlgeom and click OK). 3. Create a job named nlgeom for the model nlgeom. Submit the job for analysis. 4. When the analysis completes, open nlgeom.odb in the Visualization module and create the load-deflection curve using the method described for the linear analysis. Name the curve nlgeom-loaddsp. Compare the load-deflection curves for each analysis run thus far. Tip: Use [Ctrl]+Click in the XYData container to select more than one data object for plotting. Click mouse button 3 and select Plot from the menu that appears. Question W7 1: Do nonlinear geometric effects play an important role in this analysis when the material response is linear? Nonlinear analysis II (Plasticity) Next, you will include the effects of combined geometric and material nonlinearity. 1. Copy the model nlgeom to a model named plastic Modify the material AL3102 to include the effects of classical Mises plasticity. (in the Edit Material dialog box select Mechanical Plasticity Plastic.) The true stress plastic strain data pairs for the isotropic hardening curve are given below: Yield Stress Plastic Strain Create a job named plastic-1 for the model plastic-1 and submit the job for analysis. Create the load-deflection curve plastic1-loaddsp. Compare the load-deflection curves for each analysis run thus far. Question W7 2: Why does this analysis terminate before the total load is applied? Dassault Systèmes,

326 W7.4 Nonlinear analysis III (Plasticity) In this analysis we artificially extend the stress plastic strain curve so that the slope of the curve at psi is maintained until the plastic strain is 1.0. Question W7 3: What is the effect of the additional hardening? 1. Copy the model plastic-1 to a model named plastic Modify the material AL3102 so that the slope of the true stress plastic strain curve at psi is maintained until the plastic strain is 1.0. Tip: You can use the command line interface (CLI) in Abaqus/CAE as a calculator. For example, to determine slope of the curve at psi and then the stress when the plastic strain is equal to 1.0 enter the commands below: 3. Create a job for the model plastic-2, run the job, and postprocess as before. 4. Compare the load-deflection curve with those obtained previously. Question W7 4: Why was this analysis able to run to completion when the previous one didn t? Nonlinear analysis IV (Riks) Usually the response of the structure in the vicinity of its limit load is of interest. In such instances we use an alternative solution scheme known as the Riks method. In this section you will consider the problem with the original plastic hardening curve (plastic-1). 1. Copy the model plastic-1 to a model named riks. 2. Replace the general static step in the model riks with a Riks analysis step: In the Model Tree, expand the Steps container and click mouse button 3 on Step-1; in the menu that appears, select Replace; choose Static, Riks as the new analysis procedure. Include nonlinear geometric effects in the Riks analysis step. Specify stopping criteria so that the analysis will terminate when the vertical displacement (DOF 2) of the node region monitor is in. 3. Activate the degree of freedom monitor for degree of freedom 2 of the set monitor (Output DOF Monitor). Dassault Systèmes,

327 4. Create and submit a new job for the riks model. W Create the load-deflection curve named riks-loaddsp using the Riks analysis results. In the previous analysis jobs the load magnitude was ramped linearly during the analysis. In a Riks analysis the variation of the load magnitude is found as part of the solution. The load proportionality factor (LPF), which multiplies the load defined in the step, is provided as history data in the output database file. To create the load-deflection curve for this analysis, combine the displacement history data with the load proportionality factor data multiplied by the applied pressure magnitude. 6. Compare the load-deflection curve with those obtained previously. Tip: Edit the curve riks-loaddsp (click mouse button 3 on it in the XYData container and select Edit from the menu that appears) and set the Y-axis type to Time (to be consistent with the previously created curves). Otherwise, multiple Y- axes will be displayed. The load-deflection curves for all the workshop analyses are shown in Figure W7 2. Saved X Y data objects are available only for the duration of your Abaqus/CAE session. However, you can copy X Y data to output database files (Tools XY Data Copy to ODB) as long as you opened the output database files with write privileges. Alternatively, you can report X Y data to a text file (Report XY). Figure W7 2 Load-deflection curves for fluid channel analyses. Dassault Systèmes,

328 W7.6 Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_nonconv_limitload_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

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331 Answers 7 Limit Load Analysis Interactive Version Question W7 1: Do nonlinear geometric effects play an important role in this analysis when the material response is linear? Answer: No. The load-deflection curve is linear even when nonlinear geometric effects are included in the analysis. Question W7 2: Why does this analysis terminate before the total load is applied? Answer: The analysis terminates when the limit load of the structure is reached. Special solution techniques are required to advance the solution past this point. Question W7 3: What is the effect of the additional hardening? Answer: Abaqus assumes perfect plasticity beyond the last data pair specified for the plastic material option. Extending the yield curve until a plastic strain of 1.0 delays the onset of perfect plasticity (zero stiffness). Question W7 4: Why was this analysis able to run to completion when the previous one didn t? Answer: The additional (artificial) hardening gives the material some stiffness beyond the previous point of perfect plasticity, thus allowing a greater pressure load to be applied. Dassault Systèmes,

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335 Workshop 8 Ball Impact Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the Keywords version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description A rigid sphere representing the surface of a sports ball impacts a plate as shown in Figure W8 1. The diameter of the rigid sphere is 28 mm. The ball consists of three components: a solid core, a shell cover, and an outer surface. In an effort to simplify the modeling process, the properties of the shell cover and the solid core are reduced to a lumped mass, rotary inertia, stiffness, and damping. Rigid bodies with mass and rotary inertia properties are used to model the solid core and shell cover. The rigid body reference points for these parts are linked with a Cartesian-Cardan connector. The connector represents the translational and torsional stiffness and damping between the core and the shell cover. The outer surface of the ball (also modeled as a rigid body) will come into contact with the plate. A Cartesian-Cardan connector links the shell cover reference point to the ball (outer surface) reference point. It represents the translational and torsional stiffness and damping between the shell cover and the outer surface of the ball. This discrete modeling technique is adopted because meshing the shell cover and core makes the simulation too expensive. The deformable plate is modeled as a linear elastic material. Incompatible mode elements (C3D8I) are used to model the plate because they perform well in bending. The dimensions of the plate are 120 mm by 50 mm. The material properties are: Elastic modulus: MPa Poisson s ratio: 0.29 Density: tonne/mm 3 The consistent set of units used in the model is mm, tonne, N, MPa. The edges of the plate are pinned. The initial velocity of the ball is V z = 30 V y = mm/s mm/s; Dassault Systèmes,

336 W8.2 Figure W8 1 Ball impact model. Desired results The goal of this analysis is to develop and test this discrete technique for modeling the core and the shell cover of the ball. If a valid and accurate model can be developed, it will simplify the design of future ball models. The data that will be needed in the future are the velocity histories of the core, shell cover, and ball reference nodes, such as those shown in Figure W8 2. Figure W8 2 Velocity history of core and shell cover of the ball when = Dassault Systèmes,

337 W8.3 Model The ball strikes the plate at an oblique angle; thus, it is suspected that the presence of friction will impart significant rotation to the ball. The script ws_nonconv_ballimpact.py produces an impact analysis model where the friction between the ball and the plate during impact is assumed to be Enter the working directory for this workshop:../nonconvergence/interactive/impact 2. Run the script ws_nonconv_ballimpact.py using the following command: abaqus cae startup=ws_nonconv_ballimpact.py The above command creates an Abaqus/CAE database named impact.cae in the current directory. 3. Examine the model; note that displacement (U), velocity (V), and acceleration (A) history output has been requested at every increment for the sets named coreref, shellref, and ballref. 4. Submit the job ballimpact to analyze the ball impact model. When you submit this job, Abaqus/CAE will issue a warning because the surfaces associated with the core and shell cover parts are not used. As a result your Abaqus job will not contain the surfaces associated with these parts; it will, however, contain the point mass and rotary inertia properties that have been assigned to reference points of these parts. To see the ball in the Visualization module, you will need to sweep the analytical rigid surfaces (View ODB Display Options). The job fails in the first increment. Question W8 1: Why did the analysis fail? Look in the message file to understand why. Try reducing the size of the fixed time increment by one order of magnitude (from 2.5e-4 s to 2.5e-5 s) and rerun the analysis. In this case the job makes slight progress before failing again. Clearly fixed time incrementation is inadequate for high-speed dynamic impact problems. Very small time increments will be required during the impact phase. Specifying a small fixed time increment appropriate for the impact phase may resolve the convergence difficulties but will result in an inefficient analysis since the post-impact portion of the analysis will be required to continue using the small increment size. Automatic time incrementation is preferred as it will adapt the time increment size as necessary, thus allowing for an accurate and efficient analysis. Invoke automatic time incrementation and specify 1.e8 as the value for the half-step residual tolerance. Rerun the analysis and evaluate the results. Specifically, plot the velocity histories for the shell and core reference nodes to compare with Figure W8 2. Dassault Systèmes,

338 W8.4 Question W8 2: What can you tell about the results obtained with such a high value for HAFTOL? Question W8 3: How can you improve the accuracy of your results? Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_nonconv_ballimpact_answer.py and is available using the Abaqus fetch utility. Dassault Systèmes,

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341 Answers 8 Ball Impact Interactive Version Question W8 1: Why did the analysis fail? Look in the message file to understand why. Answer: The analysis failed because the fixed time increment size was too large to permit convergence. Question W8 2: What can you tell about the results obtained with such a high value for HAFTOL? Answer: While automatic time incrementation may have resolved the convergence difficulties associated with the nonlinear system of equations, the value of the half step residual tolerance was too large to permit an accurate integration (in time) of the equations of motion. This is indicated by the oscillations in the velocity histories of the core and cover compared to the expected results shown in Figure W8 2. Thus, it is fair to say that the time integration has not converged. Dassault Systèmes,

342 WA8.2 Question W8 3: How can you improve the accuracy of your results? Answer: There are a number of ways to increase the integration accuracy. For example, you can continue to reduce the value of the half step residual tolerance (e.g., set to 1.e4): Alternatively, you can use the default automatic time incrementation invoked by the moderate dissipation or transient fidelity application types for implicit dynamics. With the former, the default size of the maximum time increment in this problem is equal to 1.e-4 (10% of the step time). You will need to limit the size of the maximum time increment further in order to obtain sufficient accuracy (e.g., set equal to the initial time increment 2.5e-5): Dassault Systèmes,

343 WA8.3 With the latter the default settings are sufficient to obtain results that are in agreement with those presented in Figure W8 2: Dassault Systèmes,

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347 Workshop 1 Nonlinear Spring Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Introduction In this workshop you will study different aspects of Abaqus convergence behavior using two different models. The first is a simple model of a single nonlinear spring. You will define the nonlinear spring properties, run an analysis to stretch the spring, and consider the influence of increment size on the results. The second model contains a uniaxial tension test. You will manipulate this model to observe the effects of the tangent stiffness matrix on convergence. Problem description The goal of this analysis is to define a nonlinear spring stiffness for a SPRINGA element and to apply 190 units of force to stretch the element. The spring stiffness has the following force versus deflection data: Table W1 1 Force Deflection Dassault Systèmes,

348 W1.2 Question W1 1: Question W1 2: Question W1 3: What is the initial spring stiffness (the stiffness at zero deflection)? What is the spring stiffness between 5.0 and 6.0 units of deflection? Can you estimate what the final deflection will be when the full load of 190 units of force is applied to the spring? Desired results The desired results are a force-deflection curve like the one shown in Figure W1 1. You do not have to use as many increments as shown in the figure. Abaqus can find the solution to this problem in just one or two increments. Nonlinear spring model In this section you will complete the nonlinear spring definition and analyze the spring model. 1. Enter the working directory for this workshop:../nonconvergence/keywords/spring/ and open the input file spring_test.inp in a text editor. 2. Add the spring force versus deflection data to the input file. Use the online manuals if you need help with input syntax. Figure W1 1 Force-displacement curve for the nonlinear spring. Dassault Systèmes,

349 Question W1 4: W1.3 Do you think the initial increment size will influence whether or not Abaqus finds a converged solution in this simulation? 3. Save the input file and run the analysis. 4. Once the analysis completes, compare the analysis results to Figure W1 1. Tip: You can create the force-deflection plot in Abaqus/Viewer as follows. a. In the Model Tree, double-click XYData. b. In the dialog box that appears, select ODB field output as the source and click Continue. c. In the Variables tabbed page of the dialog box that appears, select Unique Nodal as the position and CF1 and U1 and the variables. d. In the Elements/Nodes tabbed page, choose Pick from viewport as the method and click Edit Selection. e. In the viewport, select the node at the right end of the spring and click Done in the prompt area. f. Click Save. g. In the Model Tree, double-click XYData. h. In the dialog box that appears, select Operate on XY data and click Continue. i. From the list of operators, select combine(x,x). j. Double-click the data object for the displacement then do the same for the force. k. At the bottom of the dialog box, click Plot Expression. Note: A complete input file is available for your convenience. You may consult the file if you encounter difficulties completing the spring model or if you wish to check your work. The input file is named spring_test_complete.inp and it is available using the Abaqus fetch utility. Dassault Systèmes,

350 W1.4 Effect of the tangent stiffness matrix on convergence Next, you will investigate the effect of the tangent stiffness matrix accuracy on convergence. Recall that the tangent stiffness matrix plays a critical role in the Newton-Raphson algorithm. Errors in the formulation of the tangent stiffness matrix will result in analyses that require more iterations or, in some cases, that diverge. We illustrate the effect of the tangent stiffness accuracy by analyzing a small problem under uniaxial tension. Both displacement and load control are considered. The material response is elastic-plastic; Mises plasticity is assumed. The analysis can be carried out in one of two ways: by modifying the material stiffness calculated with user subroutine UMAT or by using the quasi-newton solver method. Both approaches will effectively introduce an error into the tangent stiffness matrix. To build confidence in user subroutine UMAT, we first solve the problem using the Mises plasticity algorithm that is built into Abaqus. Desired results You will compare the convergence history and results of several analysis jobs. Some of the jobs will include an error in tangent stiffness matrix. The loading method will alternate between a prescribed displacement and a pressure. Uniaxial tension test In this section you will run uniaxial tension test analyses of an axisymmetric cylindrical specimen pulled in tension. 1. Open axi1.inp in a text editor and review its contents. This input file is set up for a displacement-control analysis and uses the Mises plasticity routine that is built into Abaqus. 2. Submit the input file for analysis. 3. In the Displacement Control portion of Table W1 2, enter the number of increments and the total number of iterations required to complete the analysis. Under the heading Flag Setting enter built in. Dassault Systèmes,

351 W1.5 Table W1 2 Displacement Control Flag Setting Number of Increments Number of Iterations Load Control Flag Setting Number of Increments Number of Iterations 4. Copy axi1.inp into another file name, and modify it as follows: remove the nonzero displacement boundary conditions, and add a negative pressure loading of on the element in the axial direction. 5. Submit the input file for analysis. Add the incrementation information from this analysis to the Load Control portion of Table W1 2. Dassault Systèmes,

352 W1.6 User subroutine approach Note: if you do not have the necessary compilers installed on your system, skip to the Quasi-Newton approach. Next, you will solve the same problem using the user subroutine UMAT included in the file iso_mises_umat.f (or iso_mises_umat.for if you are working on a Windows system). The material constants used in the user subroutine are specified with the USER MATERIAL option. Seven material constants are specified in this problem. The seventh material constant is used as a flag in user subroutine UMAT: when it is set to 1, UMAT returns the actual material stiffness; when it is set to 0, UMAT returns only the elastic stiffness. 1. Open the input file axi2.inp in a text editor and review its contents. This input file is set up for a displacement-control analysis; the seventh material constant is currently set to Submit the input file with the command abaqus job=axi2 user=iso_mises_umat 3. Compare the results of this analysis with those obtained for the displacementcontrol analysis using the built-in Mises plasticity routine. 4. Enter the flag setting, the number of increments, and the total number of iterations for this analysis in the Displacement Control portion of Table W Modify the material constants so that the seventh constant under the USER MATERIAL option is set to Submit the job for analysis, compare the results with the previous analysis jobs, and add the incrementation information from this analysis to the Displacement Control portion of Table W1 2. Question W1 5: Are the results obtained with the modified stiffness matrix correct? Now repeat the analysis under a state of load control. 1. Copy axi2.inp into another file name, and modify it as follows: remove the nonzero displacement boundary conditions, and add a negative pressure loading of on the element in the axial direction. 2. Set the seventh material constant to Submit the job. Enter the flag setting, the number of increments, and the total number of iterations to the Load Control portion of Table W Repeat Step 3 when the seventh material constant is set to 0. Question W1 6: What can you say about the difference in the convergence behavior of this problem when a pressure loading is applied instead of a boundary condition? Dassault Systèmes,

353 W1.7 Quasi-Newton approach The effect on convergence can also be seen by adopting a quasi-newton method to solve the problem. In this case, the tangent stiffness matrix is recalculated after a specified number of iterations within a given increment. The method reduces to the full Newton method when the tangent stiffness matrix is recalculated after every iteration. The quasi-newton approach will yield different convergence behavior than the usersubroutine approach. With the quasi-newton method, the tangent stiffness is recalculated at least once per increment: at the beginning of the increment. With the user subroutine approach described earlier, the tangent stiffness is effectively never updated if the flag is set to 0 (in this case, the linear elastic properties are always used). To see the effect on convergence using the quasi-newton method, copy axi1.inp to a new file named axi3.inp and add the following option to the step definition in axi3.inp: *SOLUTION TECHNIQUE, TYPE=QUASI-NEWTON, REFORM KERNEL=999 The REFORM KERNEL parameter in this case is set to 999, which means the tangent stiffness matrix will be recalculated only after 999 iterations have been performed; choosing such a large number effectively suppresses the operation. Solve the problem under both displacement and load control and enter the corresponding data in Table W1 2. Under Flag Setting, enter Quasi-Newton. Also, answer questions W1 5 and W1 6 posed earlier. Dassault Systèmes,

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357 Answers 1 Nonlinear Spring Keywords Version Question W1 1: What is the initial spring stiffness (the stiffness at zero deflection)? Answer: The initial spring stiffness is 100 K f u Question W1 2: What is the spring stiffness between 5.0 and 6.0 units of deflection? Answer: The spring stiffness is Question W1 3: Answer: Can you estimate what the final deflection will be when the full load of 190 units of force is applied to the spring? A rough estimate is that the spring will deflect 4.4 units when the full 190 units of force are applied. Question W1 4: Answer: Do you think the initial increment size will influence whether or not Abaqus finds a converged solution in this simulation? The nonlinearity in this problem is relatively mild and is smooth; therefore, you would expect that Abaqus could find a converged solution easily and that the initial increment size is not important. Question W1 5: Answer: Are the results obtained with the modified stiffness matrix correct? Yes. Errors in the tangent stiffness will only affect the convergence behavior, not the final result. When the elastic stiffness is used instead of the true stiffness, the number of iterations is increased. The results (when obtained) are still correct. Dassault Systèmes,

358 WA1.2 Question W1 6: Answer: What can you say about the difference in the convergence behavior of this problem when a pressure loading is applied instead of a boundary condition? Displacement-control problems are more stable than loadcontrol problems. When the elastic stiffness is used instead of the true stiffness, the displacement-control analysis ran successfully but the load-control analysis failed. Table WA1 1 Displacement Control Flag Setting Number of Increments Number of Iterations Built-In Quasi-Newton 6 8 Load Control Flag Setting Number of Increments Number of Iterations Built-In * 59* Quasi-Newton 6 17 *Did not run to completion. Dassault Systèmes,

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361 Workshop 2 Reinforced Plate Under Compressive Loads Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description This problem models a reinforced plate structure subjected to in-plane compressive loading that produces localized buckling. It is a rectangular plate reinforced with beams in its two principal directions (see Figure W2 1). The plate represents part of a larger structure: the two longitudinal sides have symmetry boundary conditions, and the two transverse sides have pinned boundary conditions. In addition, springs at two major reinforcement intersections represent flexible connections to the rest of the structure. The mesh consists of S4 shell elements for both the plate and larger reinforcements and additional S3 shell and B31 beam elements for the remaining reinforcements. The entire structure is made of the same construction steel, with an initial flow stress of MPa. Gravity loads are applied followed by an in-plane load to one of the pinned sides, which compresses the plate. The plate buckles under the load. The buckling is initially localized within each of the sections bounded by the reinforcements. At higher load levels the plate experiences global buckling in a row of sections closest to the applied load. To provide stability to the numerical solution upon the anticipated buckling, the problem is solved in two different ways: once using automatic static stabilization and once using implicit dynamics (with the quasi-static application type). This workshop is based on Abaqus Example Problem As noted there, SIMULIA thanks IRCN (France) for providing this example. Dassault Systèmes,

362 W2.2 Figure W2 1 Reinforced plate model. Preliminaries Enter the working directory for this workshop:../nonconvergence/keywords/unstableplate The input file w_unstable_plate.inp contains the model geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the loading, boundary conditions, and analysis procedure. Dassault Systèmes,

363 W2.3 Static analysis (without stabilization) In this section you will complete the model and run a static analysis without stabilization. Edit w_unstable_plate.inp as described below. Constraints A beam-type multi-point constraint will be defined at the end of the plate to which the load will be applied. Sets containing the control point as well as the region to be constrained have been predefined (LOAD and FRONT, respectively, as shown in Figure W2 2). FRONT LOAD Figure W2 2 Sets used in MPC constraint. The set FRONT contains all nodes on the front of the plate with the exception of the node corresponding to the control point (contained in set LOAD). Add the option for the constraint just prior to the *END ASSEMBLY option: *ASSEMBLY : : *MPC BEAM, FRONT, LOAD *END ASSEMBLY Dassault Systèmes,

364 W2.4 Boundary conditions In the model data portion of the input file (for example, immediately after the material property definition), define the symmetry conditions described earlier as indicated in Table W2 1. The sets listed in Table W2 1 have been predefined and are shown in Figure W2 2 and Figure W2 3. Table W2 1 Set containing region DOFs to be constrained LOAD 2, 5, 6 BACK 2, 3, 5, 6 BACK-CTR 1 SIDES 1, 5, 6 SIDES BACK-CTR BACK Figure W2 3 Sets used for boundary conditions. Dassault Systèmes,

365 W2.5 The required keyword options are indicated below: *BOUNDARY LOAD, 2, 2 LOAD, 5, 6 BACK, 2, 3 BACK, 5, 6 BACK-CTR, 1, 1 SIDES, 1, 1 SIDES, 5, 6 Step definition Define a general static analysis step. Set NLGEOM=YES on the *STEP option to consider the effects of geometric nonlinearity. In this step gravity loading will be applied to the entire structure in the Y-direction (leave the first entry in the *DLOAD data line blank to invoke automatic selection of the entire model; the acceleration due to gravity is 9.81 m/s 2 ). Finally, request preselected field and history output. The required keyword options are indicated below: *STEP, NAME=STEP-1, NLGEOM=YES *STATIC 1., 1. ** *DLOAD, GRAV, 9.81, 0., 1., 0. ** *OUTPUT, FIELD, VARIABLE=PRESELECT *OUTPUT, HISTORY, VARIABLE=PRESELECT *END STEP Define a second general static analysis step. Set the maximum number of increments to 1000 and the initial time increment size to 0.1. No stabilization will be activated for the time being. In this step a compressive force of 6.46e6 N will be applied to the plate at the MPC control point. The required keyword options are indicated below: *STEP, NAME=STEP-2, NLGEOM=YES, INC=1000 *STATIC 0.1, 1. ** *CLOAD LOAD, 3, -6.46E+06 ** *OUTPUT, FIELD, VARIABLE=PRESELECT *OUTPUT, HISTORY, VARIABLE=PRESELECT *END STEP Job Save the input file as w_unstable_plate_static.inp and submit the job for analysis. Dassault Systèmes,

366 W2.6 You will find that the analysis has convergence difficulties in the second step. These coincide with the onset of local instability (localized buckling). The analysis terminates prematurely at approximately 38% of the compressive loading. Postprocessing 1. Open the output database file created by this job in Abaqus/Viewer. Contour the U2 component of displacement, as shown in Figure W2 4. Figure W2 4 Transverse displacement contours(without stabilization). The transverse displacements alternate between positive and negative near the loaded region, indicating the onset of local buckling. 2. Create a force-displacement plot for the loaded node (using field output, extract CF3 and U3 data and combine them using the techniques described earlier). The plot appears as shown in Figure W2 5. The response is linear. Some form of stabilization (viscous damping or inertia) is required to proceed past the linear regime in this problem. Figure W2 5 Force-displacement at loaded node (without stabilization). Dassault Systèmes,

367 W2.7 Static analysis (with automatic stabilization) Now, you will add automatic stabilization to the second step. Modifications to the model Edit the second step definition to add automatic stabilization. Use the default value of dissipation intensity and turn off adaptive stabilization: *STEP, NAME=STEP-2, NLGEOM=YES, INC=1000 *STATIC, STABILIZE, ALLSDTOL=0 0.1, 1. Job and postpresssing Save the input file and submit the job for analysis. In this case the job completes successfully. As before, postprocess the results in the Visualization module. Initially local out-of-plane buckling develops throughout the plate in an almost checkerboard pattern inside each one of the sections delimited by the reinforcements, as shown in Figure W2 6. Figure W2 6 Transverse displacement contours (96% of load). Later, global buckling develops along a front of sections closer to the applied load, as shown in Figure W2 7. Dassault Systèmes,

368 W2.8 Figure W2 7 Global buckling pattern. The evolution of the displacements produced by the applied load is very smooth, as shown in Figure W2 8, and does not reflect the early local instabilities in the structure. However, when the global instability develops, the curve becomes almost flat, indicating the complete loss of load carrying capacity. Figure W2 8 Force-displacement at loaded node (with stabilization). Dassault Systèmes,

369 W2.9 Implicit dynamics (for quasi-static applications) A quasi-static solution to this problem can be obtained using the implicit dynamics procedure. The quasi-static application type for implicit dynamics provides an alternative approach to solving unstable quasi-static problems. 1. Copy the file w_unstable_plate_static.inp to w_unstable_plate_dyn.inp. 2. Modify the procedure option for the second step to invoke the implicit dynamics procedure. Set the application type to quasi-static and set the initial time increment to 0.01: *DYNAMIC, APPLICATION=QUASI-STATIC 0.01, Set the frequency at which field and history output is written to 1: *OUTPUT, FIELD, VARIABLE=PRESELECT, FREQUENCY=1 *OUTPUT, HISTORY, VARIABLE=PRESELECT, FREQUENCY=1 4. Save the file and submit the job for analysis. 5. When the analysis completes, open the file w_unstable_plate_dyn.odb in Abaqus/Viewer. 6. Postprocess the results as before. Figure W2 9 shows a comparison of the force-displacement curve obtained with the (stabilized) static and implicit dynamics procedures. The agreement between the results is excellent. Figure W2 9 Comparison of force-displacement at loaded node. Dassault Systèmes,

370 W2.10 An inspection of the model's energy content (Figure W2 10) reveals that while the load is increasing, the amount of dissipated/kinetic energy is negligible. As soon as the load flattens out, the dissipated/kinetic energy increases dramatically to absorb the work done by the applied loads. Figure W2 10 Comparison of model energies. Note: Complete input files are available for your convenience. You may consult these files if you encounter difficulties following the instructions outlined here or if you wish to check your work. The input files are named w_unstable_plate_static_complete.inp w_unstable_plate_dyn_complete.inp and are available using the Abaqus fetch utility. Dassault Systèmes,

371 Notes 371

372 372 Notes

373 Workshop 3 Crimp Forming Analysis Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description This model simulates crimp forming. Modern automobiles contain several thousand crimp joints. In a crimp joint a multi-strand wire bundle is mechanically joined to an end terminal to provide electrical continuity across the joint. The portion of the terminal that is folded over and into the wire bundle during crimping is called the grip. Figure W3 1 shows the model geometry. The grip is 0.36 mm thick and has a 60% coin at the tips. Coining is done to help the grip arms curl over the wire bundle as they are pushed against the roof of the punch during crimping. A seven-strand wire bundle is used, with each strand having a 0.31 mm diameter. A plane stress representation of the cross section is used. In reality, out-of-plane extrusion of the wire bundle and grip is a significant factor in crimp formation. To properly represent these effects, three-dimensional models are required (see Abaqus Example Problem for an example). In this workshop, this effect is not considered. The grip is formed from a half hard copper alloy; the wires are made from copper. The grip is meshed with CPS4I elements while the wires are meshed with CPS4 elements. The punch and the anvil are modeled as discrete rigid surfaces. A static analysis is difficult to achieve (even with automatic stabilization) because the model has no static stability (due to the free rigid body motion of the grip and wires) and during crimping the grip arms buckle as they are turned by the punch downward into the bundle. Thus, an implicit dynamic simulation (using the quasi-static application type) is performed. The static analysis is left as an optional exercise. Dassault Systèmes,

374 W3.2 punch wires grip anvil Figure W3 1 Crimp forming model. Preliminaries Enter the working directory for this workshop:../nonconvergence/keywords/wirecrimp The input file w_crimp.inp contains the model geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the loading, boundary conditions, and analysis procedure. Dassault Systèmes,

375 W3.3 Implicit dynamic analysis (quasi-static application) In this section you will complete the model definition and run an implicit dynamic analysis with a quasi-static application type. Note that a small amount of beta damping has been predefined to aid convergence. Edit w_crimp.inp as described below. Contact properties Coulomb friction is assumed between the individual wires ( = 0.15), between the grip and wires ( = 0.15), between the grip and anvil ( = 0.3), and between the punch and the grip ( = 0.3); frictionless contact is assumed between the two grip arms. Thus, multiple contact properties are required. The required options are: *SURFACE INTERACTION, NAME=GLOBAL *FRICTION, TAUMAX= , *SURFACE INTERACTION, NAME=GRIP-INT *FRICTION, TAUMAX= , *SURFACE INTERACTION, NAME=GRIPSELF *SURFACE INTERACTION, NAME=WIRE *FRICTION 0.15, Contact interactions General contact will be used. Choose the GLOBAL surface interaction property (defined above) as the as the global property assignment and assign individual properties according to Table W3 1. Note that the surfaces indicated in the table have been predefined: GRIP-1.GRIP is the entire exterior surface of the grip; WIRES is the union of all exterior surfaces on the wires; PUNCH-1.PUNCH is the surface of the punch; and ANVIL-1.ANVIL is the surface of the anvil. Table W3 1 First surface Second surface Property GRIP-1.GRIP PUNCH-1.PUNCH GRIP-INT GRIP-1.GRIP ANVIL-1.ANVIL GRIP-INT GRIP-1.GRIP GRIP-1.GRIP GRIPSELF WIRES WIRES WIRE GRIP-1.GRIP WIRES WIRE Dassault Systèmes,

376 W3.4 The required options are as follows: *CONTACT *CONTACT INCLUSIONS, ALL EXTERIOR *CONTACT PROPERTY ASSIGNMENT,, GLOBAL GRIP-1.GRIP, PUNCH-1.PUNCH, GRIP-INT GRIP-1.GRIP, ANVIL-1.ANVIL, GRIP-INT GRIP-1.GRIP,, GRIPSELF WIRES,, WIRE GRIP-1.GRIP, WIRES, WIRE Step definition You will define two implicit dynamic steps. In the first step the crimp operation will be performed; in the second step a springback analysis will be performed (the punch will be moved back to its original position). 1. In the first step, set NLGEOM=YES on the *STEP option to consider the effects of geometric nonlinearity and set the maximum number of increments to Choose the implicit dynamic procedure and set the application type to quasistatic, the time period to 0.2 and the initial time increment to *STEP, NAME=STEP-1, NLGEOM=YES, INC=1000 *DYNAMIC, APPLICATION=QUASI-STATIC 0.01, In this step the punch is displaced downward a distance of mm. The anvil is held motionless. Sets containing the punch and anvil reference points have been predefined (PUNCH-1.REFPT and ANVIL-1.REFPT, respectively). *BOUNDARY PUNCH-1.REFPT, 1, 1 PUNCH-1.REFPT, 2, 2, PUNCH-1.REFPT, 6, 6 ANVIL-1.REFPT, 1, 6 3. Request preselected field and history output every increment and end the step: *OUTPUT, FIELD, VARIABLE=PRESELECT, FREQUENCY=1 *OUTPUT, HISTORY, VARIABLE=PRESELECT, FREQUENCY=1 *END STEP Dassault Systèmes,

377 W In the second step, choose the implicit dynamic procedure and set the application type to quasi-static, the time period to 0.01 and the initial time increment to Move the punch back to its original position. The required options are: *STEP, NAME=STEP-2, NLGEOM=YES *DYNAMIC,APPLICATION=QUASI-STATIC 0.001,0.01 *BOUNDARY PUNCH-1.REFPT, 2, 2 *END STEP Job Save the input file and submit the job for analysis. Postprocessing When the analysis completes, open the output database file created by this job in Abaqus/Viewer. 1. Plot the Mises stress distribution on the deformed model shape at different stages of the analysis. Figure W3 2 shows the assembly after the grip arms have nearly reached the roof of the punch (126 milliseconds). The wire bundle has already been disturbed by this point. Figure W3 2 Deformed model shape at sec. Dassault Systèmes,

378 W3.6 Figure W3 3 shows the deformed shape of the crimp assembly after the grip arms have begun to curve around the roof of the punch and have partially folded over (142 milliseconds). Figure W3 3 Deformed model shape at sec. The grip arms buckle between the states indicated in Figure W3 2 and Figure W3 3. Figure W3 4 shows the final deformed shape after the forming process. The grip arms have fully folded over into the wire bundle, and the punch has made its complete downward stroke. Figure W3 4 Final deformed shape after forming process. Dassault Systèmes,

379 W3.7 Figure W3 5 shows the final shape of the wire bundle after springback. This figure shows that the originally round wires have been distorted during crimp formation. This distortion is essential for the correct formation of the crimp joint. The bare copper wires are actually covered by a thin layer of brittle copper oxide that forms on exposure of the copper to air. The goal of crimp forming is to break this oxide layer and expose the copper to the surface of the grip by inducing significant surface strains in each wire. Figure W3 5 Final deformed shape of the wire bundle after springback 2. Plot the internal and kinetic energy histories in the first analysis step. Figure W3 6 shows that the kinetic energy remains a small fraction of the internal energy confirming the quasi-static nature of the simulation. Figure W3 6 Model energies Dassault Systèmes,

380 W Plot the force-displacement curve for the punch (extract the data for RF2 and U2 from field output and combine the curves). Figure W3 7 shows the result. Figure W3 7 Force-displacement curve Optional: Static analysis (with stabilization) In this section you will perform the analysis using the general static analysis procedure with automatic stabilization. Step definition You will replace the implicit dynamic steps with general static ones. 1. Copy the file w_crimp.inp to w_crimp_static.inp. 2. In each step replace the *DYNAMIC option with the *STATIC, STABILIZE option. 3. Save the input file and submit the job for analysis. Monitor its progress. You will find that the analysis does not complete successfully. It terminates at approximately 40% of the forming step, corresponding to the point where the wires are about to be disturbed. This is illustrated in Figure W3 8. Recall that the wires are not restrained; thus, their free rigid body motion causes convergence difficulties for the static analysis. Dassault Systèmes,

381 W3.9 Figure W3 8 Configuration at termination of static analysis In order to circumvent the problem caused by the unrestrained wires you can try adding soft springs to hold them in place. Alternatively (or in addition to) you can try increasing the stabilization damping factor. 4. Figure W3 9 shows the result of a static analysis with the static stabilization factor equal to 5 (the default value is determined by stabilization intensity factor; in this case the default initial value for the damping factor was approximately 3). The analysis completed nearly 75% of the forming step. Figure W3 9 Configuration at termination (damping factor = 5) Dassault Systèmes,

382 W Figure W3 10 shows the result of a static analysis with the following non-default settings and controls: a. Stabilization damping factor = 100 b. Adaptive stabilization tolerance = 0.2 (default is 0.05) c. Minimum allowable time increment = 1.e 12 (default is 1.e 5 times the total time period, or 2.e 6 in this case) d. Number of attempts allowed in an increment = 10 (default is 5) e. Force residual tolerance = 0.25% (default is.5%) f. Displacement correction tolerance = 10% (default is 1%) The analysis completed nearly 90% of the forming step. Figure W3 10 Configuration at termination (damping factor = 100) Clearly it is very difficult to obtain a converged solution to this problem with a static analysis. As this problem demonstrates, the quasi-static application type for implicit dynamics represents a powerful complement to automatic static stabilization techniques and in some cases represents the only way to obtain a quasi-static solution using an implicit solution method. Note: A complete input file is available for your convenience. You may consult this file if you encounter difficulties following the instructions outlined here or if you wish to check your work. The input file is named w_crimp_complete.inp and is available using the Abaqus fetch utility. Dassault Systèmes,

383 Notes 383

384 384 Notes

385 Workshop 4 Contact: Beam Lift-Off Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to predict the final configuration of a beam as it is bent over a much stiffer body. Other results of interest are the reaction forces at the constrained nodes and the contact pressure between the beam and the body. The beam, at this stage, is modeled as a simple linear elastic material (E and 0.3). The beam is modeled in two dimensions using CPE4I elements (4-node, bilinear, plane strain quadrilateral with incompatible modes). Node-to-surface contact discretization with direct enforcement is considered (this is the default contact formulation). Question W4 1: What are the advantages of this element type over the other available plane strain solid elements? Forty elements are used along the length of the beam. Figure W4 1 Sketch of the beam. Dassault Systèmes,

386 W4.2 Desired results The desired results are shown in the following figures: Input file Figure W4 2: the deformed shape of the model, Figure W4 4: the reaction forces at nodes 1 and 101 Figure W4 5: the contact pressure between the beam and the body. The input file is../nonconvergence/keywords/contact/beam_40.inp. Figure W4 2 Deformed shape of model with loads and boundary conditions shown. Node 101 Node 1 Node 16 Figure W4 3 Deformed shape of model showing nodes 1, 16, 101. Dassault Systèmes,

387 W4.3 Figure W4 4 History of reaction forces at constrained nodes. Peak contact pressure at node 16 Figure W4 5 Contour of contact pressure on the beam. Note: Contours of CPRESS may be difficult to visualize on two-dimensional models. In Figure W4 5 the plane strain elements have been extruded for display purposes using the ODB Display Options in Abaqus/Viewer (View ODB Display Options). Question W4 2: What are some of the methods you can use to obtain the peak contact pressure in the model? Question W4 3: What is the reason that beam_40.inp does not converge? What changes must be made to make the model converge and give the desired results? Dassault Systèmes,

388 388

389 Notes 389

390 390 Notes

391 Answers 4 Contact: Beam Lift-Off Keywords Version Question W4 1: What are the advantages of this element type over the other available plane strain solid elements? Answer: The principal advantages of the CPE4I element are that it can model bending effectively (because it uses an enhanced deformation gradient formulation), it is a first-order element (and, thus, works well in contact problems), and it is fully integrated (no hourglassing problems). Question W4 2: What are some of the methods you can use to obtain the peak contact pressure in the model? Answer: You can read the result from the printed output (.dat) file. You can also look at the range of a contact pressure (CPRESS) contour plot that was created with Abaqus/Viewer. Question W4 3: What is the reason that beam_40.inp does not converge? What changes must be made to make the model converge and give the desired results? Answer: The model beam_40.inp does not converge because Abaqus cannot establish the correct contact state at the start of the analysis. Five attempts are made in the first increment of Step 2, but each fails because ultimately the solution diverges. A portion of the message file follows: ***WARNING: OVERCONSTRAINT CHECKS: NODE 1 ON THE SLAVE SURFACE AND CORRESPONDING NODE 901 ON THE MASTER SURFACE HAVE EQUAL PRESCRIBED DISPLACEMENTS NORMAL TO THE CONTACT SURFACE. SINCE THIS MAKES THE CONTACT CONSTRAINT REDUNDANT, THE CONTACT STATUS AT THE SLAVE NODE IS CHANGED FROM CLOSE TO OPEN. CONTACT PAIR (SLAVE,RIGID1) NODE 2 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (SLAVE,RIGID1) NODE 4 IS OVERCLOSED BY E-006. CONTACT PAIR (SLAVE,RIGID1) NODE 5 IS OVERCLOSED BY E-006. Dassault Systèmes,

392 WA4.2 CONTACT PAIR (SLAVE,RIGID1) NODE 6 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (SLAVE,RIGID1) NODE 9 OPENS. CONTACT PRESSURE/FORCE IS CONTACT PAIR (SLAVE,RIGID1) NODE 11 IS OVERCLOSED BY CONTACT PAIR (SLAVE,RIGID1) NODE 13 OPENS. CONTACT PRESSURE/FORCE IS SEVERE DISCONTINUITIES OCCURRED DURING THIS ITERATION. 3 POINTS CHANGED FROM OPEN TO CLOSED 4 POINTS CHANGED FROM CLOSED TO OPEN CONVERGENCE CHECKS FOR SEVERE DISCONTINUITY ITERATION 2 MAX. PENETRATION ERROR AT NODE 11 OF CONTACT PAIR (SLAVE,RIGID1) MAX. CONTACT FORCE ERROR AT NODE 13 OF CONTACT PAIR (SLAVE,RIGID1) THE ESTIMATED CONTACT FORCE ERROR IS LARGER THAN THE TIME-AVERAGED FORCE. AVERAGE FORCE 1.985E+03 TIME AVG. FORCE 1.985E+03 LARGEST RESIDUAL FORCE 1.480E+04 AT NODE 13 DOF 2 LARGEST INCREMENT OF DISP AT NODE 141 DOF 2 LARGEST CORRECTION TO DISP AT NODE 141 DOF 1 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE. AVERAGE MOMENT 1.466E+04 TIME AVG. MOMENT 1.466E+04 ALL MOMENT RESIDUALS ARE ZERO LARGEST INCREMENT OF ROTATION E-31 AT NODE 901 DOF 6 LARGEST CORRECTION TO ROTATION E-31 AT NODE 901 DOF 6 THE MOMENT EQUILIBRIUM EQUATIONS HAVE CONVERGED : : CONVERGENCE CHECKS FOR SEVERE DISCONTINUITY ITERATION 13 : : ***NOTE: THE SOLUTION APPEARS TO BE DIVERGING. CONVERGENCE IS JUDGED UNLIKELY. It can be difficult sometimes to sort out all the information in the message file. The job diagnostics tool available in the Abaqus/Viewer (Tools Job Diagnostics) may help you interpret some of this information. Use the job diagnostics tool to highlight the nodes that open or overclose in each SDI, as shown in Figure WA4 1. Try to identify those nodes whose contact state changes most frequently. Dassault Systèmes,

393 2. Select Contact. WA Select an SDI. 3. Select nodes. 4. Highlight nodes in the viewport. Figure WA4 1 Slave node openings in the second iteration of the second step. Dassault Systèmes,

394 WA4.4 In this case there is no single node that is involved in every SDI. However, looking at all SDIs in a given attempt shows that at least one of five nodes (2, 3, 4, 5, and 6) is involved in every SDI. These nodes are marked in Figure WA4 2. Nodes 2, 3, 4, 5, and 6 Figure WA4 2 Nodes that are chattering in the model. There are many methods that can be used to allow the model beam_40.inp to overcome its initial convergence difficulties: 1. Use the *CONTACT CONTROLS, AUTOMATIC TOLERANCES option. 2. Use penalty enforcement of the hard contact constraints. 3. Use surface-to-surface contact discretization. 4. Round the corner of the rigid body (in reality some amount of rounding is present; here a corner radius of 0.05 is sufficient to permit convergence). 5. Perturb the beam mesh near the corner of the rigid body so that the beam node does not lie exactly at the corner. The results using the automatic contact tolerances are provided below. The other methods produce results that are nearly identical to the results with automatic contact tolerances. Another method that will permit convergence is to use softened contact. However, choosing the values of the softened contact parameters can be difficult. Even if Abaqus obtains a converged solution, it will be very, very difficult to produce the desired results with softened contact alone because it fundamentally changes the contact pressures in the model. Dassault Systèmes,

395 WA4.5 Automatic contact tolerances The automatic contact tolerances are designed for exactly this type of model: one in which establishing the initial contact state is very difficult. These automatic tolerances do not affect the final results that you are interested in with this model the only results they should affect in most simulations are those in the first increment. Automatic contact tolerances are discussed in detail in Lecture 5. The reaction forces and the contact pressures for a model using the automatic tolerances are shown in Figure WA4 3 and Figure WA4 4. Figure WA4 3 Reaction forces when the automatic contact tolerances are used. Peak contact pressure at node 16 Figure WA4 4 Contact pressure contours when the automatic contact tolerances are used Dassault Systèmes,

396 396

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398 398 Notes

399 Workshop 5 Contact: Stabilization Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Introduction In many structural problems, contact is a critical factor in transferring load from one region to another. In FEA often the most challenging aspect of resolving contact-related issues is establishing stable contact conditions and hence the load path required for static equilibrium. This can be particularly difficult if the exact contact conditions are not known in advance. A loading condition is required to establish contact while a stable contact condition is required for the structure to carry load. Such a finely balanced system can be difficult to solve; however, there are a number of techniques that can be used to assist with this class of problem. The purpose of this workshop is to: 1. Introduce a particular (common) form of contact convergence problem loose/rattling contact 2. Use the job diagnostics tools to identify the cause of convergence difficulties 3. Apply modeling techniques to avoid such problems Model Description The model to be analyzed in this workshop is a bolt-up of wheel rim/hub assembly (see Figure W5 1). Symmetry has been used to reduce the model size. The wheel nuts are to be tightened to a tension of 10kN (5kN for the half nut on the symmetry plane). Dassault Systèmes,

400 W5.2 Figure W5 1 Wheel assembly Preliminaries 1. Enter the working directory for this workshop:../nonconvergence/keywords/contact_stab The input file w_wheel_disk.inp contains the model geometry, mesh, and material and section properties for the structure. You will complete the model definition by defining the loading, boundary conditions, interactions, and analysis procedure. Note that a coupling constraint is used to constrain the motion of the hub. Also, the thread engagement between the nuts and the hub is modeled with tie constraints. These have been predefined. Basic Model Setup At this stage you will complete the basic model definition. 1. Coulomb friction is assumed between all parts ( = 0.1). Define a surface property named fric. Define a friction coefficient of 0.1 and specify penalty enforcement of the normal contact constraints. The required options are as follows: *SURFACE INTERACTION, NAME=FRIC *FRICTION 0.1, *SURFACE BEHAVIOR, PENALTY Dassault Systèmes,

401 W Define the contact pairs indicated in Table W5 1. For each contact pair select the surface-to-surface discretization method and FRIC as the interaction property. Master surface RIM-HUB RIM-NUTS RIM-NUTS RIM-NUTS Slave surface HUB-RIM NUT1-RIM NUT2-RIM NUT3-RIM Table W5 1 Contact pairs The contact pairs can be defined with a single option as follows: *CONTACT PAIR, INTERACTION=FRIC, TYPE=SURFACE TO SURFACE HUB-RIM, RIM-HUB NUT1-RIM, RIM-NUTS NUT2-RIM, RIM-NUTS NUT3-RIM, RIM-NUTS 3. Define a general static step considering geometrically nonlinear effects. Set the initial time increment size to 0.05 s: *STEP, NAME=LOAD, NLGEOM=YES *STATIC 0.05, The hub reference point is completely constrained. Apply an ENCASTRE boundary condition to the set REFPOINT. 5. As indicated earlier only half the structure is modeled due to symmetry. Apply a ZSYMM boundary condition to the set ZSYMM: The required option for the boundary conditions is as follows: *BOUNDARY REFPOINT, ENCASTRE ZSYMM, ZSYMM Dassault Systèmes,

402 W5.4 First attempt Load Control At this stage you will apply the bolt loads and run the analysis. 1. For each pin region define a bolt load using a concentrated force. Specify a magnitude of for the full pin regions and 5000 for the pin region on the symmetry plane. Note that pre-tension sections have been predefined to facilitate the load definition. The forces will be defined at the pre-tension section reference nodes. Named sets (of the form LOAD-*) have been predefined. The required option is: *CLOAD LOAD-1, 1, LOAD-2, 1, LOAD-3, 1, Edit the default field output requests to include the contact state CSTATUS: *OUTPUT, FIELD, VARIABLE=PRESELECT *CONTACT OUTPUT CSTATUS, 3. Create additional history output requests for the contact normal force CFN for each of the nut-rim contact interactions: *OUTPUT, HISTORY, VARIABLE=PRESELECT *CONTACT OUTPUT, MASTER=RIM-NUTS, SLAVE=NUT1-RIM CFN1, CFN2, CFN3, CFNM *CONTACT OUTPUT, MASTER=RIM-NUTS, SLAVE=NUT2-RIM CFN1, CFN2, CFN3, CFNM *CONTACT OUTPUT, MASTER=RIM-NUTS, SLAVE=NUT3-RIM CFN1, CFN2, CFN3, CFNM 4. Save the input file as w_wheel_disk_load.inp and submit it for analysis. 5. Monitor the progress of this analysis. NOTE: This analysis is expected to fail do not panic!!! Results for first attempt The analysis should fail to complete. You will now review the information in the Abaqus output database file to see if there are any clues as to why the analysis fails. 1. Open the output database file in Abaqus/Viewer. Dassault Systèmes,

403 W5.5 It is good practice to review the contact conditions as defined in the Abaqus results file to check for potential modeling errors. 2. Use the Display Group tools to review the contact surfaces defined in the model (see Figure W5 2). Figure W5 2 Using display groups to review contact surfaces Question W5 1: Do the surfaces created seem reasonable and cover the potential/expected contact conditions? 3. Review the initial contact state (are contacting surfaces open or closed?). If contact pairs are initially overclosed, Abaqus/Standard will by default attempt to resolve the overclosure in the first increment. This often results in large unbalanced contact forces and can cause convergence problems. Question W5 2: Are any regions of the model initially overclosed? Tip: Create a contour plot of COPEN. A negative value of COPEN implies overclosure. Conversely, some surfaces will require contact to be established in order to transfer load from one region of the model to another. Dassault Systèmes,

404 W5.6 Question W5 3: Are there any regions in the model where you might expect contact but that are not initially in contact? Tip: Check CSTATUS=OPEN/COPEN=0 for the different components in the model (use the Display Group tools to help). It should be clear from this initial review of the contact state that the contact load path is not initially established. Contact has not been initiated between the nuts and the rim (see Figure W5 3). Therefore, initially there is no path defined to transfer the bolt tension loads from the nuts and hub to the wheel rim. Minimum gap is 0.005mm. No initial contact between the nuts and the rim. Figure W5 3 Contour plot of COPEN (Nuts - Rim) You will now use the Job Diagnostics tools to determine the reasons for cutbacks/convergence failure. Dassault Systèmes,

405 W Open the Job Diagnostics dialog box (Tools Job Diagnostics). 5. Check for warnings during the analysis phase using the Warnings and Errors tabbed pages (see Figure W5 4). Figure W5 4 Job Diagnostics dialog box Question W5 4: Are there any analysis warnings? Typical warnings for these problems refer to numerical singularities. Numerical singularities can often indicate that there are unconstrained degrees of freedom in the model resulting in rigid body modes. In many contact problems, the only constraint preventing rigid body modes is contact between the separate regions. If contact is not established when a load is applied then there is the potential that loading is applied to unconstrained degrees of freedom, resulting in severe instability (no structural stiffness to carry the load). This often results in numerical problems and convergence failure. Question W5 5: Are there any Numerical Singularity warnings in the model? Dassault Systèmes,

406 W Toggle on Highlight selections in viewport to show the regions of the model associated with any numerical singularity warnings. Question W5 6: Does the position of the numerical singularity warnings tie in with the initial contact state? 7. Review the information on the Residuals, Contact, and Element tabbed pages for a selection of the attempted iterations. Question W5 7: Were any increments completed? The job diagnostics provide valuable information as to exactly why an attempted increment has failed and required a cutback. Figure W5 5 shows the job diagnostics for the first attempted increment for the current model. Cutback reason given on Summary tab Residuals, Contact, and Element tabs allow identification of the offending mesh regions Figure W5 5 Job Diagnostics: First increment attempt Dassault Systèmes,

407 W For each attempted increment identify the reason given by the solver for the failure/cutback and enter the information into Table W5 2 below. Increment Attempt No. Iterations Reason for failure Severe Overclosures Table W5 2 Iteration history Question W5 8: On the evidence of the analysis history and the discussion so far do you think that resubmitting the load at a reduced load level would help the convergence? Potential Solutions It is clear that for this model, applying the pre-tension load before contact between the nut faces and the wheel rim is established will not permit a solution to be obtained without some assistance. There are a number of techniques that could be employed to avoid these convergence issues: 1) Ensure that contact is initially closed: Reposition the components in the assembly Adjust the slave nodes to lie exactly on the master surface Define contact clearance/interference This method is often effective but is not appropriate for every case; for example, it may not be possible to predict the precise contact condition of the model in advance. 2) Replace loads with prescribed displacements. In some cases it may be possible to replace the loads on the structure with conjugate displacements (for example, this would be straightforward for concentrated loads). Using displacement control instead of load control is often more stable as the application of the fixed displacement boundary conditions remove the rigid body modes and numerical singularities, making the solution inherently more stable. This method is often used in two stages: Step 1: Prescribe sufficiently large dummy displacement boundary conditions to bring the components in the model into contact. Dassault Systèmes,

408 W5.10 Step 2: Deactivate the dummy boundary conditions and replace them with the required load. This method is applicable only in certain situations. In particular, for complex loading patterns (pressures, body loads, multiple loads, etc.) it may not be possible to simply replace loads with displacement boundary conditions. 3) Apply contact stabilization to resist the rigid body modes until contact is established. This method applies viscous pressures to the contact surfaces to react the applied loads until contact is established. Contact stabilization does not suffer from the limitations of the other methods and is very easy to activate; however, it should be used with care as there is the potential to over-damp the solution. For more information refer to section Adjusting contact controls in Abaqus/Standard" in the Abaqus Analysis User's Manual. Attempt 2 Displacement Control Either of the methods discussed above could be used to attempt to resolve the convergence issues of this model. In this workshop you will try methods 2 and 3 from above (feel free to try adjusting the components in the assembly if time permits). 1. Copy the file w_wheel_disk_load.inp to w_wheel_disk_disp.inp. 2. Rename the step to DISP and specify an initial time increment size of 0.2 s: *STEP, NAME=DISP, NLGEOM=YES *STATIC 0.2, In the DISP step, delete the concentrated forces at the pre-tension section reference nodes and replace them with boundary conditions. Specify a displacement magnitude to (just large enough to close the gap). *BOUNDARY LOAD-1, 1, 1, LOAD-2, 1, 1, LOAD-3, 1, 1, Create a second general static step named LOAD. Set the initial time increment size to 0.2 s: *STEP, NAME=LOAD, NLGEOM=YES *STATIC 0.2, 1. Dassault Systèmes,

409 W In the LOAD step, delete the boundary conditions on the pre-tension section reference nodes, but retain the ones on the hub reference point and symmetry plane: *BOUNDARY, OP=NEW REFPOINT, ENCASTRE ZSYMM, ZSYMM 6. Define the bolt loads as before. Set the magnitude for the full bolts to (for the bolt on the symmetry plane set the magnitude to 5000): *CLOAD LOAD-1, 1, LOAD-2, 1, LOAD-3, 1, Save the input file and submit it for analysis. 8. Monitor the progress of this job. Question W5 9: Has loading using fixed displacements improved the convergence? Does this analysis run to completion? 9. Open the Abaqus output database and use the Job Diagnostics to examine the convergence behavior for this latest run. Question W5 10: Are there any analysis warnings produced during this attempt? Question W5 11: Has applying the displacement control method avoided any of the warnings from the previous attempt? Attempt 3 Contact stabilization 1. Copy the file w_wheel_disk_load.inp to w_wheel_disk_stab.inp. 2. Define a non-default contact controls for each contact pair to invoke contact stabilization with the default stabiliation factor: *Contact Controls, master=rim-nuts, slave=nut1-rim, stabilize=1. *Contact Controls, master=rim-nuts, slave=nut2-rim, stabilize=1. *Contact Controls, master=rim-nuts, slave=nut3-rim, stabilize=1. Dassault Systèmes,

410 W Edit the field output requests to include CDSTRESS (contact damping stresses): *OUTPUT, FIELD, VARIABLE=PRESELECT *CONTACT OUTPUT CSTATUS, CDSTRESS 4. Save the input file and submit it for analysis. 5. Monitor the progress of this job. Question W5 12: Has the application of contact stabilization allowed the analysis to complete? 6. Open the Abaqus output database and use the Job Diagnostics to examine the convergence behavior for this latest run. Question W5 13: Are there any analysis warnings produced during this attempt? Question W5 14: Has applying the stabilization method avoided any of the warnings from the first attempt? As mentioned earlier, applying viscous damping/stabilization forces should be used with care as over-damping the solution may lead to inaccuracies in the results. Therefore, you will check to see if applying viscous damping forces in this model have adversely affected the results. 7. Create a contour plot of the contact damping pressure CDPRESS. Question W5 15: What is the peak value of CDPRESS? TIP: You will need to review the complete time history to determine the peak value. The Contact Stabilization Factor is ramped down to zero by the end of the step so the peak CDPRESS value will not be at the end of the step. Question W5 16: How does the peak value of CDPRESS compare to the peak contact pressure (CPRESS)? The output variable ALLSD represents the energy dissipated by viscous forces such as contact stabilization. This can be used to evaluate the level of impact of the artificial stabilization forces on the solution. As a general rule of thumb, the stabilization energy should be a small percentage (say < 5%) of the model s internal energy (ALLSE if purely elastic; otherwise, ALLIE). 8. Using history output, plot the time histories of the static dissipation energy ALLSD and the total elastic strain energy, ALLSE. The plot appears as shown in Figure W5 6. Dassault Systèmes,

411 W5.13 Figure W5 6 Energy histories Question W5 17: Is ALLSD small compared to ALLSD? 9. Edit the contact controls to apply a scale factor of 0.1 to the default value: *Contact Controls, master=rim-nuts, slave=nut1-rim, stabilize=0.1 *Contact Controls, master=rim-nuts, slave=nut2-rim, stabilize=0.1 *Contact Controls, master=rim-nuts, slave=nut3-rim, stabilize= Resubmit the analysis. 11. Finally, review the results (stresses, strains, contact forces, etc.) from the w_wheel_disk_disp and w_wheel_disk_stab jobs. Question W5 18: How do the results from the two different approaches compare? Note: Complete input files are available for your convenience. You may consult these files if you encounter difficulties following the instructions outlined here or if you wish to check your work. The input files are named w_wheel_disk_load_complete.inp w_wheel_disk_disp_complete.inp w_wheel_disk_stab_complete.inp w_wheel_disk_stab2_complete.inp and are available using the Abaqus fetch utility. Dassault Systèmes,

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415 Answers 5 Contact: Stabilization Keywords Version Question W5 1: Do the surfaces created seem reasonable and cover the potential/expected contact conditions? Answer: The main contact areas are all covered. Although in the absence of friction, an extra set of contact pairs between the hub pins and the wheel rims may be required (see Figure WA5 1) Figure WA5 1 Additional contacts Dassault Systèmes,

416 WA5.2 Question W5 2: Are any regions of the model initially overclosed? Answer: No. Question W5 3: Are there any regions in the model where you might expect contact that are not initially in contact? Answer: Yes. You would expect that the underside of the nuts will eventually contact the wheel rim. Question W5 4: Are there any analysis warnings? Answer: Yes. There are three analysis warnings during each increment attempt. Question W5 5: Are there any Numerical Singularity warnings in the model? Answer: Yes. All of the warnings relate to numerical singularities. Question W5 6: Does the position of the numerical singularity warnings tie in with the initial contact state? Answer: Yes. The highlighted nodes refer to regions of the hub where the bolt pre-tension is applied. In order to carry the bolt preload, contact must be established between the underside of the nuts and the wheel rim. Recall that this contact interface is initially open; hence, there is no way to react the pre-tension force, resulting in the numerical singularity warnings. Question W5 7: Were any increments completed? Answer: No. Each attempt fails after a single iteration. Question W5 8: On the evidence of the analysis history and the discussion so far do you think that resubmitting the load at a reduced load level would help the convergence? Dassault Systèmes,

417 WA5.3 Answer: There is no evidence to suggest that reducing the load level will help with convergence. Note that automatic time incrementation has already reduced the load level five times, with the same convergence behavior for each attempt, before completely giving up. Question W5 9: Has loading using fixed displacements improved the convergence? Does this analysis run to completion? Answer: Yes. The analysis completes with no cutbacks. Question W5 10: Are there any analysis warnings produced during this attempt? Answer: No. Question W5 11: Has applying the displacement control method avoided any of the warnings from the previous attempt? Answer: Yes. Establishing contact via displacement control has avoided the numerical singularity warnings. Question W5 12: Has the application of contact stabilization allowed the analysis to complete? Answer: Yes. The analysis completes with no cutbacks. Question W5 13: Are there any analysis warnings produced during this attempt? Answer: No. Question W5 14: Has applying the stabilization method avoided any of the warnings from the first attempt? Answer: Yes. Establishing contact using contact stabilization has avoided the numerical singularity warnings. Question W5 15: What is the peak value of CDPRESS? Answer: MPa Dassault Systèmes,

418 WA5.4 Tip: Create a maximum envelope plot of CDPRESS over all frames (Tools Create Field Output From Frames). Question W5 16: How does the peak value of CDPRESS compare to the peak contact pressure (CPRESS)? Answer: Even with the default contact stabilization the peak CDPRESS is small compared to the CPRESS for this analysis (peak CDPRESS=2.616MPa vs. peak CPRESS=125.4MPa). Question W5 17: Is ALLSD small compared to ALLSD? Answer: ALLSD is a little high (~6% of ALLSE); thus, reduction of the stabilization damping factor is recommended. Question W5 18: How do the results from the two different approaches compare? Answer: The results of both techniques match very well, with very little differences in the stress, strain, CPRESS, etc. Dassault Systèmes,

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421 Workshop 6 Element Selection Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to study the performance of different element types. We consider two problems: the bending of a long, slender cantilever beam and the uniaxial tension of a nearly incompressible elastic solid. Each problem is solved with displacement formulation elements as well as hybrid elements. The difference in the performance of each element formulation is assessed. Desired results We compare the convergence history of each problem when solved with the displacement and hybrid elements. Input file Enter the working directory for this workshop:../nonconvergence/keywords/element/ The base input files are slender_beam1.inp and solid1.inp. Dassault Systèmes,

422 W6.2 Cantilever beam In this section you will model the bending of a long, slender cantilever beam. First, study the performance of displacement formulation beam elements for beams of different length. 1. Copy slender_beam1.inp to a file named slender_beam1a.inp and submit the job. 2. Enter the number of increments and the total number of iterations required to complete the analysis in Table W6 1. You can find this information at the end of the analysis message file slender_beam1a.msg. 3. Copy slender_beam1a.inp to slender_beam1b.inp. Modify slender_beam1b.inp so that the length of the beam is 5000 units. Submit the input file. 4. Repeat Step Copy slender_beam1b.inp to slender_beam1c.inp. Modify slender_beam1c.inp so that the length of the beam is units. Submit the input file. 6. Repeat Step 2. Next, you will study the performance of hybrid formulation beam elements. 1. Copy slender_beam1a.inp to slender_beam2a.inp. 2. Modify slender_beam2a.inp so that hybrid beam elements are used (B21H). Submit the job. 3. Enter the number of increments and the total number of iterations required to complete the analysis in Table W Copy slender_beam2a.inp to slender_beam2b.inp. Modify slender_beam2b.inp so that the length of the beam is 5000 units. Submit the input file. 5. Repeat Step Copy slender_beam2b.inp to slender_beam2c.inp. Modify slender_beam2c.inp so that the length of the beam is units. Submit the input file. 7. Repeat Step 3. Question W6 1: Do the results (displacements, stresses, etc.) differ between the displacement element model and the hybrid element model? Question W6 2: Why does the hybrid element model converge more easily than the displacement element model as the beam length increases? Dassault Systèmes,

423 W6.3 Table W6 1 B21 Elements Length Number of Increments Number of Iterations B21H Elements Length Number of Increments Number of Iterations Next, you will study the iteration history for the long beam models (i.e., length =12000). 1. Open the message file slender_beam1c.msg in a text editor or open slender_beam1c.odb in Abaqus/Viewer and access the job diagnostic tool (Tools Job Diagnostics). Locate the iteration information printed for increment Enter the largest residual force for each iteration of increment 4 in Table W Repeat 2 for increment 5. Question W6 3: What do you notice about the iteration history for the model with displacement elements? Table W6 2 B21 Elements (Length = 12000) Increment Iteration Largest Residual Force Dassault Systèmes,

424 W Open the message file slender_beam2c.msg in a text editor or open slender_beam1c.odb in Abaqus/Viewer and access the job diagnostic tool. Locate the iteration information printed for increment Enter the largest residual force for each iteration of increment 4 in Table W Repeat Step 5 for increment 5. Question W6 4: How does the iteration history for the hybrid element model compare with that for the displacement element model? Table W6 3 B21H Elements (Length = 12000) Increment Iteration Largest Residual Force Dassault Systèmes,

425 W6.5 Incompressibility in an elastic solid This task investigates the convergence behavior of an analysis of a nearly incompressible elastic solid under uniaxial tension. You will analyze one-element models consisting of reduced-integration plane strain elements. A parametric study will show the change in convergence behavior as Poisson s ratio approaches a value of Submit the input file solid1.inp for analysis. 2. Enter the number of increments and the total number of iterations required to complete the analysis in Table W Modify solid1.inp so that Poisson s ratio is set closer to 0.5. Use values of 0.499, , and Enter the number of increments and the total number of iterations required to complete each analysis in Table W Copy solid1.inp to solid2.inp. Modify solid2.inp so that hybrid elements are used (CPE4RH). Submit the job. 6. Repeat Step 2 Step 4 for solid2.inp. Question W6 5: How does the iteration history for the hybrid element model compare with that for the displacement element model? Table W6 4 CPE4R Elements ν Number of Increments Number of Iterations CPE4RH Elements ν Number of Increments Number of Iterations Dassault Systèmes,

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429 Answers 6 Element Selection Keywords Version Question W6 1: Do the results (displacements, stresses, etc.) differ between the displacement element model and the hybrid element model? Answer: The displacements and stresses are essentially same for the two element formulations (differences, if any, are less than 0.1%). Question W6 2: Why does the hybrid element model converge more easily than the displacement element model as the beam length increases? Answer: As the beam length increases, the ratio between its axial stiffness and bending stiffness becomes greater and greater. Hybrid elements are specially designed for use in such cases. As a result, the models with hybrid elements converge more rapidly than the models using displacement-based elements. Table WA6 1 B21 Elements Length Number of Increments Number of Iterations B21H Elements Length Number of Increments Number of Iterations Dassault Systèmes,

430 WA6.2 Question W6 3: What do you notice about the iteration history for the model with displacement elements? Answer: The force residual does not decrease monotonically. In fact, the largest force residual fluctuates between very large and very small numbers. The large axial forces produce large displacement corrections. Table WA6 2 B21 Elements (Length = 12000) Increment Iteration Largest Residual Force E E E E E E 03 Dassault Systèmes,

431 WA6.3 Question W6 4: How does the iteration history for the hybrid element model compare with that for the displacement element model? Answer: The model with the hybrid elements requires fewer iterations to complete the analysis than the model with the displacementbased elements. This is particularly noticeable as the beam gets longer and more inextensible. In addition, the peak residuals are more stable from iteration to iteration, which usually results in fewer cutbacks. Table WA6 3 B21H Elements (Length = 12000) Increment Iteration Largest Residual Force E E E E 03 Dassault Systèmes,

432 WA6.4 Question W6 5: How does the iteration history for the hybrid element model compare with that for the displacement element model? Answer: As the material becomes more incompressible (i.e., Poisson s ratio approaches 0.5), the models with the displacement-based elements require more iterations to complete the analysis than the models with hybrid elements. The models with the hybrid elements require the same number of iterations regardless of the value chosen for ν. Table WA6 4 CPE4R Elements ν Number of Increments Number of Iterations CPE4RH Elements ν Number of Increments Number of Iterations Dassault Systèmes,

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435 Workshop 7 Limit Load Analysis Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description The goal of this analysis is to study the response of a pressurized fluid channel when different forms of nonlinearity are present, including geometric and material nonlinearity. The loading consists of an internally applied pressure of 2000 psi. NHIST Desired results Figure W7 1 Fluid channel model. The desired results include load-deflection curves that compare the internal pressure against the vertical displacement (bulging) of the structure for each of the runs described below. Dassault Systèmes,

436 W7.2 Input file Enter the working directory for this workshop:../nonconvergence/keywords/riks/ The base input file is channel1.inp. Linear analysis In this section you will analyze the linear elastic response of the structure. The results serve as a baseline for the subsequent nonlinear analyses. 1. To run the job channel1.inp and create the load-deflection curve using Abaqus/Viewer: a. Open channel1.inp in a text editor. b. Request history output of the displacements (U) at a node that belongs to node set NHIST. The appropriate output request is *OUTPUT, HISTORY, FREQUENCY=1 *NODE OUTPUT, NSET=NHIST U c. Add the following option if you want to obtain the preselected field output: *OUTPUT, FIELD, FREQUENCY=10, VARIABLE=PRESELECT d. Run the analysis. When the job completes, open the.odb file in Abaqus/Viewer. e. In the Results Tree, expand the History Output container for the output database file named channel1.inp. f. Click mouse button 3 on the data named Spatial Displacement: U2 at Node 3919 in NSET NHIST. From the menu that appears, select Save As. Name the X Y data linear-dsp and click OK. g. In the Results Tree, double-click the XYData container. h. In the Create XY Data dialog box, choose Operate on XY data and click Continue. i. Exchange the X- and Y-values of linear-dsp using the swap operator and multiply the swapped curve with the magnitude of the pressure load (i.e., 2000 in this case). j. Click Save As. Name the X Y data linear-loaddsp and click OK. k. Click Cancel to close the Operate on XY Data dialog box. l. Expand the XYData container, and double-click the curve named linear- LoadDsp. m. Double-click the Y-axis to open the Axis Options dialog box; change the Y-axis title to Load. Dassault Systèmes,

437 W7.3 Nonlinear analysis I (NLGEOM) Now, you will include the effects of geometric nonlinearity in this problem. 1. Copy channel1.inp to nlgeom.inp. 2. Modify nlgeom.inp to account for geometric nonlinearity effects. 3. Submit the job for analysis. 4. When the analysis completes, open nlgeom.odb in Abaqus/Viewer and create the load-deflection curve using the method described for the linear analysis. Name the curve nlgeom-loaddsp. Compare the load-deflection curves for each analysis run thus far. Tip: Use [Ctrl]+Click in the XYData container to select more than one data object for plotting. Click mouse button 3 and select Plot from the menu that appears. Question W7 1: Do geometric nonlinearity effects play an important role in this analysis when the material response is linear? Nonlinear analysis II ( PLASTIC) Next, you will include the effects of combined geometric and material nonlinearity. 1. Copy nlgeom.inp to plastic-1.inp. 2. Modify plastic-1.inp to include the effects of classical Mises plasticity. The true stress plastic strain data pairs are given by 3700., , , , , , , , Submit the job, and create the load-deflection curve. Compare the load-deflection curves for each analysis run thus far. Question W7 2: Why does this analysis terminate before the total load is applied? Nonlinear analysis III ( PLASTIC) In this analysis we artificially extend the stress plastic strain curve so that the slope of the curve at psi is maintained until the plastic strain is 1.0. Question W7 3: What is the effect of the additional hardening? 1. Copy plastic-1.inp to plastic-2.inp. Dassault Systèmes,

438 2. Modify the data under the PLASTIC option so that the slope of the curve at psi is maintained until the plastic strain is 1.0. W7.4 Tip: You can use the command line interface (CLI) in Abaqus/CAE as a calculator. For example, to determine slope of the curve at psi and then the stress when the plastic strain is equal to 1.0 enter the commands below: 3. Run the job, and postprocess as before. 4. Compare the load-deflection curve with those obtained previously. Question W7 4: Why was this analysis able to run to completion when the previous one didn t? Nonlinear analysis IV (Riks) Usually, the response of the structure in the vicinity of its limit load is of interest. In such instances we use an alternative solution scheme known as the Riks method. In this section you will consider the problem with the original plastic hardening curve (plastic-1.inp). 1. Copy plastic-1.inp to riks-1.inp. Modify riks-1.inp so that the Riks method is used. 2. Set the parameters for the Riks analysis so that the initial arc length increment is 0.1 and the total arc length scale factor for the step is Specify that the analysis will terminate when the vertical displacement of node 3919 is in. 4. Run the analysis job. 5. In Abaqus/Viewer, create the load-deflection curve named riks1-loaddsp using the Riks analysis results. In the previous analysis jobs the load magnitude was ramped linearly during the analysis. In a Riks analysis the variation of the load magnitude is found as part of the solution. The load proportionality factor (LPF), which multiplies the load defined in the step, is provided as history data in the output database file. To create the load-deflection curve for this analysis, combine the displacement history data with the load proportionality factor data multiplied by the applied pressure magnitude. Tip: Edit the curve riks1-loaddsp (click mouse button 3 on it in the XYData container and select Edit from the menu that appears) and set the Y-axis type to Time (to be consistent with the previously created curves). Otherwise, multiple Y- axes will be displayed. Dassault Systèmes,

439 W7.5 Question W7 5: Why happens when the limit load is reached? Look at the message file for clues. For example, what does the drastic reduction in arc length increment imply? Nonlinear analysis V (Riks) The previous analysis reversed direction at the point where the limit load is reached. The message file indicates that the incremental arc length in the 25th increment of riks-1.inp is reduced drastically. This implies that the incremental arc length was too large to continue the analysis successfully past the limit point. Thus, we restrict the maximum arc length increment. 1. Copy riks-1.inp to riks-2.inp. Modify the Riks parameters so that the maximum arc length increment is Run the job, and postprocess as before. Compare the load-deflection curve with those obtained previously. The load-deflection curves for all the workshop analyses are shown in Figure W7 2. Saved X Y data objects are available only for the duration of your Abaqus/Viewer session. However, you can copy X Y data to output database files (Tools XY Data Copy to ODB) as long as you opened the output database files with write privileges. Alternatively, you can report X Y data to a text file (Report XY). Figure W7 2 Load-deflection curves for fluid channel analyses. Dassault Systèmes,

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443 Answers 7 Limit Load Analysis Keywords Version Question W7 1: Do geometric nonlinearity effects play an important role in this analysis when the material response is linear? Answer: No. The load-deflection curve is linear even when nonlinear geometric effects are included in the analysis. Question W7 2: Why does this analysis terminate before the total load is applied? Answer: The analysis terminates when the limit load of the structure is reached. Special solution techniques are required to advance the solution past this point. Question W7 3: What is the effect of the additional hardening? Answer: Abaqus assumes perfect plasticity beyond the last data pair specified under the PLASTIC option. Extending the yield curve until a plastic strain of 1.0 delays the onset of perfect plasticity (zero stiffness). Question W7 4: Why was this analysis able to run to completion when the previous one didn t? Answer: The additional (artificial) hardening gives the material some stiffness beyond the previous point of perfect plasticity, thus allowing a greater pressure load to be applied. Dassault Systèmes,

444 WA7.2 Question W7 5: Why happens when the limit load is reached? Look at the message file for clues. For example, what does the drastic reduction in arc length increment imply? Answer: Even though a special solution technique is used, the analysis reverses direction when the limit point is reached. The message file indicates the incremental arc length in the 25th increment is reduced significantly (4 orders of magnitude). Instability problems that exhibit a sharp transition often require a limit on the maximum incremental arc length to get past the transition point. Dassault Systèmes,

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447 Workshop 8 Ball Impact Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the Interactive version of these instructions. Please complete either the Keywords or Interactive version of this workshop. Problem description A rigid sphere representing the surface of a sports ball impacts a plate as shown in Figure W8 1. The diameter of the rigid sphere is 28 mm. The ball consists of three components: a solid core, a shell cover, and an outer surface. In an effort to simplify the modeling process, the properties of the shell cover and the solid core are reduced to a lumped mass, rotary inertia, stiffness, and damping. Rigid bodies with mass and rotary inertia properties are used to model the solid core and shell cover. The rigid body reference points for these parts are linked with a Cartesian-Cardan connector. The connector represents the translational and torsional stiffness and damping between the core and the shell cover. The outer surface of the ball (also modeled as a rigid body) will come into contact with the plate. A Cartesian-Cardan connector links the shell cover reference point to the ball (outer surface) reference point. It represents the translational and torsional stiffness and damping between the shell cover and the outer surface of the ball. This discrete modeling technique is adopted because meshing the shell cover and core makes the simulation too expensive. The deformable plate is modeled as a linear elastic material. Incompatible mode elements (C3D8I) are used to model the plate because they perform well in bending. The dimensions of the plate are 120 mm by 50 mm. The material properties are: Elastic modulus: MPa Poisson s ratio: 0.29 Density: tonne/mm 3 The consistent set of units used in the model is mm, tonne, N, MPa. The edges of the plate are pinned. The initial velocity of the ball is V z = 30 V y = mm/s mm/s; Dassault Systèmes,

448 W Figure W8 1 Ball impact model. Desired results The goal of this analysis is to develop and test this discrete technique for modeling the core and the cover of the ball. If a valid and accurate model can be developed, it will simplify the design of future balls. The data that will be needed in the future are the velocity histories of the core, cover, and reference node, such as those shown in Figure W8 2. Figure W8 2 Velocity history of core and shell cover of the ball when = Dassault Systèmes,