Nonlinear analysis in ADINA Structures

Transcription

1 Nonlinear analysis in ADINA Structures Theodore Sussman, Ph.D. ADINA R&D, Inc,

2 Topics presented Types of nonlinearities Materially nonlinear only Geometrically nonlinear analysis Deformation-dependent loads Theory of nonlinear iterations full Newton iterations line searches limiting displacements ATS low-speed dynamics Stiffness stabilization TLA and TLA-S methods Birth-death Prescribed displacements Geometric imperfections ADINA R&D, Inc,

3 Nonlinear analysis in ADINA Structures, theory Much of the theory used by ADINA Structures can be found in the following reference: K.J. Bathe, Finite Element Procedures, 2 nd ed, Cambridge, MA, Klaus-Jürgen Bathe, 2014 and also in the papers by Bathe and co-workers. Much of the material presented in this lecture is taken from Chapter 6 of the textbook. ADINA R&D, Inc,

4 Materially nonlinear only Materially nonlinear only (MNO) Elastic-plastic material, viscoelastic material, etc. u nodal point displacements e strains e E elastic strains e P plastic strains P e change in plastic strains τ Cauchy stresses F nodal point forces Symbolically τ f E ( e ) ADINA R&D, Inc,

5 Geometric nonlinearities Geometric nonlinearities include Euler buckling P effect stress stiffening large rotations large strains 3 HL 3EI All of these effects are included by the use of a Lagrangian formulation (for example, total Lagrangian), with appropriate stress and strain measures, and with an appropriate material model. The details depend upon the element type and material model. We will discuss the nonlinearities due to deformation-dependent loading later. ADINA R&D, Inc,

6 Geometric nonlinearities - large disp, small strains Large displacements / small strains with an elastic orthotropic material: ε Green-Lagrange strains S 2nd Piola-Kirchhoff stresses S11 C11 C12 C S C C S 33 C S12 C S 23 symmetric C S C Symbolically τ f () e is replaced by S f () ε same stress-strain matrix as in linear analysis ADINA R&D, Inc,

7 Geometric nonlinearities - large disp, small strains This procedure includes all of the geometric nonlinearities mentioned above, with the exception of nonlinearities due to large strains. This procedure gives correct results provided that the strains are small, because, for small strains ε e and S τ. In fact, the definition of small strains is based on these conditions. Numerically, small strains are less than about 2%. If the strains are larger, the results will start to deviate from the results that you might expect. For example, the response in compression will not be the same as the response in tension. There is no real material for which S is linearly related to ε, when the strains are large. ADINA R&D, Inc,

8 Geometric nonlinearities - large disp, small strains Large displacements / small strains with an elastic-plastic material: ε E elastic Green-Lagrange strains ε P plastic Green-Lagrange strains Symbolically E τ f ( e ) is replaced by E S f ( ε ) As in the elastic-orthotropic case, this procedure gives correct results provided that the strains are small. We will give more details in the lecture on plasticity. ADINA R&D, Inc,

9 Geometric nonlinearities - large disp, large strains Large displacements / large strains with an elastic-plastic material, ULH formulation: X deformation gradient X P plastic deformation gradient X E elastic deformation gradient R E elastic rotation tensor U E elastic stretch tensor E ER elastic Hencky strains E Symbolically τ f ( e ) ER is replaced by τ f ( E ) τ rotated Cauchy stresses ADINA R&D, Inc,

10 Geometric nonlinearities - large disp, large strains The Hencky strains are a three-dimensional extension of the logarithmic (true) strains, and the Cauchy stresses are the true stresses (force per unit deformed area). So the material relationship is a good approximation for real materials. Therefore the ULH procedure can be used in both small and large strain situations. In addition, large rotations are accounted for. For some materials, there is also the possibility of using the ULJ formulation (updated Lagrangian Jaumann), but we will not discuss this further. ADINA R&D, Inc,

11 Geometric nonlinearities - large disp, large strains Large displacements / large strains with a rubberlike material. W S ε W strain energy density Because the material model is directly applicable to large strains, the procedure also is directly applicable to large strains. ADINA R&D, Inc,

12 Deformation-dependent loads When the model undergoes large displacements, the loads can also be geometrically nonlinear. If the model deforms significantly, there are two options: ADINA R&D, Inc,

13 Deformation-dependent loads By default, a pressure load is deformation-dependent if it is applied to a geometrically nonlinear element (large displacement kinematics). Deformation-dependent pressure loads can slow down the rate of convergence of the equilibrium iterations. Concentrated loads can also be deformation-dependent (follower loads). Centrifugal loads can also be deformation-dependent (load softening). ADINA R&D, Inc,

14 Equations of motion, statics The finite element equations of motion for ADINA Structures are written in terms of nodal point forces. The left superscript t+t means evaluated at time t+t. tt tt 0 R F Vector of external nodal point forces (from applied loads) Vector of internal nodal point forces (from element stresses) ADINA R&D, Inc,

15 Equations of motion The unknowns in ADINA Structures are the nodal point displacements and rotations. t t U F F U tt tt Lots of theory contained in this equation! R R U tt tt In many cases, the external forces are independent of the nodal point displacements (deformation-independent loads). ADINA R&D, Inc,

16 Equations of motion, linear special case If R is deformation-independent and F is a linear function of U, then the analysis is linear, and we can then write F K U tt tt and tt tt K U R But frequently F is a nonlinear function of U, or R is deformationdependent, so the equations of motion are nonlinear and we must solve them iteratively. tt tt 0 R F ADINA R&D, Inc,

17 Equations of motion, nonlinear case We suppose that the solution at time t is known, and we seek the solution at time t+t. The first approximation to the the solution at time t+t is U tt (0) t where (i) is an iteration counter (starting from 0). U t t (0) U Using we compute t t (0) F, t t (0) R and also t t (0) K where tt K (0) tt F U (0) ideally. ADINA R&D, Inc,

18 Equations of motion, nonlinear case Next, we compute the out-of-balance loads vector and a correction to the displacement vector K ΔU R F t t (0) t t (0) t t (0) to obtain a new trial displacement vector U = U +ΔU tt (1) tt (0) R F tt (0) tt (0) U tt ( i1) In general, given, ADINA Structures iterates as follows: Full Newton iterations without line searches Compute F tt ( i1), R tt ( i1) K U R F tt ( i1) tt ( i1) tt ( i1) U U U tt () i tt ( i1), K tt ( i1) ADINA R&D, Inc,

19 One DOF visualization of iterations The solution process can be visualized for one DOF in graphical form: Force t t R Slope t K Slope t t (1) K t t (1) F Iteration 0 Iteration 1 t R t U t t (1) U t t U Displacement ADINA R&D, Inc,

20 Convergence norms Norm of the out-of-balance force vector: R F tt ( i1) tt ( i1) 2 Norm of the incremental displacement vector: t t () i U 2 (for each of these quantities, there are two norms, one for the translational degrees of freedom and one for the rotational degrees of freedom) Out-of-balance energy norm U R F () it tt ( i1) tt ( i1) This norm has units of energy, but is not related to the total strain energy of the model or any other physical energy. ADINA R&D, Inc,

21 Convergence tolerances Force/moment Displacement/rotation R tt ( i1) tt ( i1) RNORM F 2 RTOL Energy t t () i U DNORM 2 DTOL U R F U R F () it tt ( i1) tt ( i1) (1) T tt (0) tt (0) ETOL ADINA R&D, Inc,

22 Line searches Instead of we use U U U tt () i tt ( i1) U U U tt () i tt ( i1) where is iteratively chosen to satisfy U R F U R F T tt () i tt () i T tt ( i1) tt ( i1) STOL The default is no line searches. ADINA R&D, Inc,

23 Limiting displacements Instead of U U U tt () i tt ( i1) FAC U we use in which FAC is chosen to make the largest incremental displacement component smaller than a certain value (MAXDISP in the TOLERANCES ITERATION command). The default value of MAXDISP (selected by the choice MAXDISP=0.0) depends on the model: - If the model is static and includes contact, MAXDISP is 1% of the largest model dimension. - Otherwise MAXDISP is set very large so the limiting displacement feature is not used. ADINA R&D, Inc,

24 Example convergence history Here is sample printout from the.out file (.log file for FSI). OUT-OF- NORM OF BALANCE OUT-OF-BALANCE NORM OF INCREMENTAL ENERGY FORCE MOMENT DISP. ROTN. CFORCE NODE-DOF NODE-DOF NODE-DOF NODE-DOF CFNORM MAX VALUE MAX VALUE MAX VALUE MAX VALUE ITE= E E E E E E Y 25-X 250-Z 121-Y 0.00E E E E E-05 ITE= E E E E E E X 45-Z 1-X 24-Y 0.00E E E E E-05 ITE= E E E E E E X 45-Y 56-Z 111-X 0.00E E E E E-05 ITE= E E E E E E Y 47-Z 1-X 114-Z 0.00E E E E E-06 ITE= E E E E E E Y 47-Z 1-X 114-Z 0.00E E E E E-06 ITE= E E E E E E Y 47-Z 1-X 114-Z 0.00E E E E E-06 CONVERGENCE RATIOS CONVERGENCE RATIOS OUT-OF-BALANCE LOAD FOR OUT-OF-BALANCE FOR INCREMENTAL VECTOR CALCULATION ENERGY FORCE DISP. CFORCE BETA RATIO MOMENT ROTN. (ITERNS) COMPARE WITH COMPARE WITH ETOL RTOL DTOL RCTOL (NOT USED)(NOT USED) 1.00E E E E E E E E E E E E E E+00 ( 1) 1.70E E E E E E E E+00 ( 1) 7.60E E E E E E E E+00 ( 3) 7.20E E E E E E E E+00 ( 3) 2.20E E E E E E E E+00 ( 1) ADINA R&D, Inc,

25 Convergence history view (not the same data as shown in the previous slide) ADINA R&D, Inc,

26 Iteration patterns, quadratic convergence Example: ITE= E E E E E E X Y E E E E E+00 ITE= E E E E E E Y 0 21-Z E E E E E+00 ITE= E E E E E E Z 0 21-Z E E E E E+00 ITE= E E E E E E X Y E E E E E+00 ITE= E E E E E E X Z E E E E E+00 This is the fastest convergence. Near convergence, the current force norm is roughly the square of the previous force norm. Quadratic convergence is only observed if the stiffness matrix is truly tangent. Observing quadratic convergence gives some assurance that the solution has truly converged. ADINA R&D, Inc,

27 Iteration patterns, linear convergence Example: ITE= E E E E E E X 22-Y 34-X 16-Z 4.60E E E E E-05 Near convergence, the current force norm is a factor times the previous force norm. ITE= E E E E E E Z 55-Y 66-X 23-X 4.60E E E E E-06 ITE= E E E E E E X 56-Y 17-X 45-Z 4.60E E E E E-07 ITE= E E E E E E X 56-Y 17-X 45-Z 4.60E+01) 2.50E E E E-08 Linear convergence is observed if the stiffness matrix is not truly tangent. Deformationdependent loads Certain material models ADINA R&D, Inc,

28 Iteration patterns, divergence Example: ITE= E E E E E E X 2314-Y 1232-X 3421-X 0.00E E E E E-03 ITE= E E E E E E X 3238-Y 1523-X 1962-Y 1.20E E E E E-02 ITE= E E E E E E X 456-X 132-Y 429-Z 3.60E E E E E-02 ITE= E E E E E E X 962-Y 434-Z 347-X 6.60E E E E E-02 ITE= E E E E E E X 562-Y 983-X 4713-Z 1.20E E E E E-03 ADINA R&D, Inc,

29 What can we do when the solution doesn t converge? Use more iterations (only if the solution was converging already) Use line searches Use smaller load increments Manually reduce the time step Use the ATS method (with or without low-speed dynamics) Use physical intuition to determine and resolve the problem. ADINA R&D, Inc,

30 ATS (automatic time-stepping method) When ADINA Structures uses the ATS method, ADINA Structures cuts the step size increment and retries the step, when there is no convergence. For a 1 DOF system Force t t R 1 2 t t R t R t t U 1 t 2 U tt U Displacement ADINA R&D, Inc,

31 ATS example, graphical representation t t Original time step, t NO CONVERGENCE ADINA R&D, Inc,

32 ATS example, graphical representation t t/2 t First subdivided time step NO CONVERGENCE Original time step, t NO CONVERGENCE ADINA R&D, Inc,

33 ATS example, graphical representation t t /4 t/2 t First subdivided time step NO CONVERGENCE Second subdivided time step CONVERGED Original time step, t NO CONVERGENCE ADINA R&D, Inc,

34 ATS example, graphical representation t t/2 t t /4 CONVERGED t Same as the Time Step Prior to Subdivision ADINA R&D, Inc,

35 ATS example, graphical representation t t/2 t t /4 CONVERGED t /4 t Same as the Time Step that Gave Convergence Same as the Time Step Prior to Subdivision ADINA R&D, Inc,

36 ATS example, graphical representation t t/2 t t /4 CONVERGED t /4 t Same as the Time Step that Gave Convergence (t t /4) Same as the Time Step Prior to Subdivision Return to Original Time Step Specified ADINA R&D, Inc,

37 ATS notes The ATS method only saves the solution for the user-specified time steps, and for the last converged solution if this solution does not correspond to a user-specified time step. After the solution is obtained for a user-specified time step, the program chooses the next time step according to one of the following options: Same as time step prior to subdivision (RESTORE=YES) Same as time step that gave convergence (RESTORE=NO) Solution time matches original next solution time (always used for iterative FSI) (RESTORE=ORIGINAL) Automatic, based on problem characteristics (RESTORE=AUTOMATIC) If the ATS method does not converge, the program can optionally turn on low-speed dynamics. An additional option specifies the time duration during which lowspeed dynamics is activated; after this period of time, the program deactivates low-speed dynamics. ADINA R&D, Inc,

38 Low-speed dynamics The ATS method includes a special low-speed dynamics option in static analysis. When low-speed dynamics is active, ADINA Structures includes mass and damping effects in the (otherwise static) analysis. M U C U K U R F tt () i tt () i tt ( i1) () i tt tt ( i1) where C M K,, user-specified Reasons to use low-speed dynamics: Allow for solution of models with rigid-body modes. Allow for solutions of models with local snap-through or buckling instabilities Dynamics smooths the response, hence convergence is easier. ADINA R&D, Inc,

39 Low-speed dynamics, continued The mass matrix is evaluated from the density and the low-speed inertia factor (default = 1). The damping matrix is evaluated using Rayleigh damping: C M K where and are user-specified parameters (defaults 0 and 10-4 ). Two basic options for damping specification: Damping factors applied to entire model (in which case = 0). Damping factors applied to individual element groups, using the Rayleigh damping options. ADINA R&D, Inc,

40 Low-speed dynamics, continued When low-speed dynamics is used, the time step size will influence the results. can be interpreted as a decay time (time constant of exponential decay). Increasing increases the decay time. It is recommended that either The time step size be at least 10, or After the solution is obtained, the load be held constant for a period of time at least 10 to allow the dynamic solution to decay away. ADINA R&D, Inc,

41 Low-speed dynamics, continued The solution indicators can be used to assess the magnitudes of the inertia and damping forces: External force indicator Damping force indicator Inertia force indicator I I R U EF I R U DF I R I D U ==================================================================================== S O L U T I O N A C C U R A C Y I N D I C A T O R S FOR LOW SPEED DYNAMICS, CONTACT DAMPING, SHELL DRILLING AND STIFFNESS STABILIZATION EXTERNAL DRILLING DAMPING INERTIA CONTACT STIFFNESS FORCES FORCES FORCES FORCES DAMP.FORCES STABIL E E E E E % of ext.forces ====================================================================================== ADINA R&D, Inc,

42 Stiffness stabilization Sometimes the structure is unstable (has rigid body modes) at the start of the analysis. The equation solver cannot give a solution when there are rigid-body modes. Rigid-body modes cause zero pivots in the factorized stiffness matrix. ADINA R&D, Inc,

43 Zero pivots can be removed using stiffness stabilization Scale all diagonal stiffness terms (excluding contact diagonals) without modifying the right-hand-side load vector K 1 K ii stab ii Physically, attaches weak springs to all degrees of freedom, but in such a way to not affect the solution in nonlinear analysis. ADINA R&D, Inc,

44 Stiffness stabilization does not affect the solution in nonlinear analysis Stiffness stabilization only modifies K, element nodal forces F are not modified i1 i i1 i1 K U R F tt tt tt converges when out-of-balance load is sufficiently small Hence, the converged solution is same as without stabilization. However, as K is modified, rate of convergence might be worsened, so that more iterations are required. ADINA R&D, Inc,

45 TLA, TLA-S methods Goal: apply a load in nonlinear static analysis without excessive user input. User applies total load level (TLA = Total Load Application). ADINA Structures automatically applies the load using a ramp time function. ADINA Structures chooses the number of time steps. ADINA Structures increases/decreases the time step size depending upon the number of equilibrium iterations in the previous time step. By default: 1 st time step has size 1/50 th of the total time. Max number of equilibrium iterations = 30. Max number of time step subdivisions used in ATS method = 64 Max limiting incremental displacement = 5% model dimension ADINA R&D, Inc,

46 TLA-S = TLA with stabilization low-speed dynamics contact damping stiffness stabilization TLA, TLA-S methods At the end of the equilibrium iterations for each step, ADINA Structures prints the solution indicators: ================================================================================ S O L U T I O N A C C U R A C Y I N D I C A T O R S FOR LOW SPEED DYNAMICS (LSD), CONTACT DAMPING, AND SHELL DRILLING STIFFNESS EXTERNAL DRILLING LSD DAMPING LSD INERTIA CONTACT FORCES FORCES FORCES FORCES DAMP.FORCES E E E E % of ext.forces ================================================================================ If indicators are < 1%, solution is accurate ADINA R&D, Inc,

47 Element birth/death Elements can be born, or can die, during the solution. You can specify the birth and death times as part of the model definition: Birth Death Birth/death Elements can die as a result of material rupture. ADINA R&D, Inc,

48 Element birth/death You should not specify the birth and death times to be exactly equal to solution times. Solution times 1, 2, etc. If you want the element to be born at time 2, birth time can be between and If you want the element to die at time 3, death time can be between and The element is assumed to be stress-free when it is born. Birth time = to 1.999, element is stress free at time 1. ADINA R&D, Inc,

49 The dying element mass and stiffness can be removed from the model over a period of time. The element starts to die at the userspecified time, and is completely removed from the model after an additional period of time. Element death decay time Element death decay time ADINA R&D, Inc,

50 Prescribed displacements Displacements can be prescribed as a function of time (similar to other load types). When the arrival time option is used, the prescribed displacement becomes active only after the arrival time. Arrival time is 10 Relative prescribed displacement option. Total prescribed displacement = preexisting displacement + relative prescribed displacement. Arrival time is 10 ADINA R&D, Inc,

51 Prescribed displacements - time unloading Time unloading with zero user-specified unloading force: Unloading time is time 2.5 Time unloading with nonzero user-specified unloading force R u : Unloading time is time 2.5 ADINA R&D, Inc,

52 Prescribed displacements - force unloading Unloading starts when the specified force is exceeded: User-specified unloading force = 1000 ADINA R&D, Inc,

53 Geometric imperfections Before the analysis begins, the nodal coordinates can be updated by imperfection displacements. The "imperfect" nodal coordinates are used during the analysis, and the computed displacements are based on the imperfect nodal coordinates. The imperfect mesh is stress-free at the start of the analysis. ADINA R&D, Inc,

54 Geometric imperfections in collapse analysis Buckling modes come from a linearized buckling analysis. Each buckling mode can be independently scaled. ADINA R&D, Inc,