The Finite Element method for nonlinear structural problems
|
|
- Francine Marsh
- 5 years ago
- Views:
Transcription
1 Non Linear Computational Mechanics Athens MP06/2013 The Finite Element method for nonlinear structural problems Vincent Chiaruttini
2 Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 2
3 Non-linearities in structural problems Contact Due to the non-penetration condition Crack propagation problem under time dependant loading When crack propagates the solution becomes non-linearly time dependant Geometrical nonlinearities For large deformation, instabilities can also occur (buckling) Nonlinear constitutive relationship Non linear behaviour: elastoplasticity, damage, viscosity 3
4 Solution process Iterative algorithm Scalar example nding u j f (u) = 0 For any kind of regular function, no direct process exists Iterative algorithm building un! u j f (u) = 0 Stop when a convergence criterion is satisfied rank k j jf (uk )j < "crit Newton method Built on the linear verification of the first order Taylor development nullity f (uk+1 ) ¼ f (uk ) + (uk+1 uk ) f 0 (uk ) = 0 ) uk+1 = uk f (uk )=f 0(uk ) u0 4 u1 u2 f (u) u
5 Solution process Newton method Built on the linear verification of the first order Taylor development nullity f (uk+1 ) ¼ f (uk ) + (uk+1 uk ) f 0 (uk ) = 0 ) uk+1 = uk f (uk )=f 0(uk ) Convergence depends on u0 When converges Rank k error ek = uk u Recurrence on error relationship ek+1 ek Taylor expansion closely to the exact solution = uk+1 uk = f (uk )=f 0 (uk ) 1 00 f (uk ) = f (u)ek + f (u)e2k + o(e2k ) 2 f 0 (uk ) = f 0 (u) + f 00 (u)ek + f 000 (u)e2k + o(e2k ) 00 2f 0ek + f 00 e2k f (u) ek+1 = ek 0 + o(e ) = e + o(e ) = O(e k k k k) f + f ek + f ek 2f (u) 0 Quadratic convergence Close enough to the solution each iteration produces twice more significant new digits 5
6 Solution process Newton method Quadratic convergence Require to update the derivative at each iteration Modified Newton methods Constant direction f 0 (uk ) ¼ K = cst = f 0 (u0 ) uk+1 = uk f (uk )=K 0 Linear convergence ek+1 = (1 f (u)=k)ek + o(jek j) = O(jek j) f (u) u0 6 u1 u2 u
7 Solution process Newton method Quadratic convergence Require to update the derivative at each iteration Modified Newton methods Constant direction f 0 (uk ) ¼ K = cst = f 0 (u0 ) Linear convergence Secant update uk+1 uk uk 1 = uk + f (uk ) f (uk ) f (uk 1 ) Golden ratio convergence order f (u) u0 7 u1 u2 u ek+1 = O(jek j p )
8 Solution process Newton method for a set of equations Vectorial function F (U ) = 0 At each iteration a linear system is solved j 0 = F (U k ) + (U ) U j k where the operator constitutes the rigidity matrix at Uk Quadratic convergence when close to solution The neighbourhood where such kind of convergence is observed depends on the vectorial function properties Convergence is insured for convex functions 8
9 Examples of convergence using a Newton algorithm 9
10 Examples of convergence using a Newton algorithm 10
11 ODE integration Time dependant problem are ruled by differential equations Reduce high order differential systems to first order 2 d y dy + g(t) = r(t), 2 dt dt General formulation [Y_ ] = [F (t; [Y ])] ½ dy dt dz dt = z(t) = r(t) g(t) z(t) [Y_ (t = t0 )] = [Y0 ] Euler-type integration schemes Finite difference time discretization Y^ (tn ) = Yn ; tn+1 = tn + t Y n+1 Y n Y (t) = F (t; Y ) forward methods Y (tn+1 ) ¼ t θ-method (method-b) Y n+1 Y n = µf n+1 (Y n+1 ) + (1 µ)f n (Y n ) t Explicit Forward Euler (θ=0) Conditionally stable 1st order accurate 11 Crank-Nicholson (θ=0.5) Unconditionally stable 2nd order accurate Implicit Euler (θ=1) Unconditionally stable 1st order accurate
12 ODE integration Time dependant problem are ruled by differential equations Runge-Kutta explicit integration Minimal multiple evaluations of F (t; y(t)) on a given time increment to insure a specific 0 + ( t=2)yn00 + o( t2 ) order accuracy, based on Taylor expansion yn+1 = yn + tyn RK1 is the forward explicit Euler scheme RK2 using one mid-point sub calculation at tn+ 1 2 t yn+1=2 = yn + f (tn ; yn ) 2 yn+1 = yn + t f tn+1=2 ; yn+1=2 2nd order method RK4 popular method using 4 sub calculations yn+1 = yn + t=6 (k1 + 2k2 + 2k3 + k4 ) k1 k2 k3 k4 = f (tn ; yn ) = f (tn+1=2 ; yn + ( t=2)k1 ) = f (tn+1=2 ; yn + ( t=2)k2 ) = f (tn+1=2 ; yn + tk3 ) Initial slope on the interval Mid interval slope using k1 Mid interval slope using k2 Final slope using k3 on the interval 4th order method 12
13 Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 13
14 Local numerical aspects of plasticity Assumptions Small strain: linearised kinematic Quasi-static loadings: no dynamic effects Small strain elastoplastic problem Elastic domain f (¾) < 0 A usual elastic behaviour is applied Yield surface f (¾) = 0 Irreversible/dissipative phenomena can occur Example Von Mises criterion q 1 f (¾) = 3=2¾D : ¾D R with ¾ D = ¾ Tr(¾) I 3 14
15 Local numerical aspects of plasticity Small strain elastoplastic problem Plastic strain " = "e + "in = "e + "p Strain partition Rigidity ¾ = A : "e = A : (" "p ) Yield surface evolution (convex), hardening When f =0 the yield surface evolves Isotropic hardening yield surface increase Kinematic hardening yield surface translation Flow rules Plasticity (normality _ (¾) = 0 "_ = ; 0; f (¾) 0; p Cumulated plastic strain rate p(t) = Z tr 0 15 ³ 3 p "_ ( ) : "_ p ( ) d = _ for a Von Mises criterion 2
16 Local numerical aspects of plasticity Common formalism for viscoplasticity/multipotentials/large strain Strain partition Small strain e in "=" +" Elasticity e p = " +" +" vp Large strain F =EP ¾ = A : "e Flow rules Plasticity (state must stay on the yield surface at the end of the increment) p "_ = X _ s ns _ s from f_s = 0 s Viscoplasticity (state expected to be on the relevant equipotential at the end of the increment) s Àn X vp s s f s "_ = v_ m v_ = Hardening rules s K Y_ I = f ct(yi ; "p ; "vp ) 16
17 Numerical solution process Elastoplastic example Relations t 2 [0; T ] Equilibrium and strain definition r¾ + f = 0 Behaviour ¾ = A : (" "p ) 3 "_ = p_ eq ¾ D 2¾ p 1 T "= ru+r u 2 r 3 eq ¾ = k¾ D k 2 p_ 0 ¾eq R(p) 0 Boundary conditions (initial equilibrium and no plasticity) u = ud ¾ n = F ²p (t = 0) = 0 17 p_ (¾ eq R(p)) = 0
18 Numerical solution process Elastoplastic example ª p Mechanical state S(x; t) = u(x; t); "(x; t); " (x; t); ¾(x; t) Temporal discretization Incremental temporal approach using a regular grid tn+1 = tn + t The solution is search incrementaly at each time step t n+1 (all previous step states being known) Spatial discretization FE method in displacement 0 8u 2 Uad ; Z ¾ n+1 : " d = Z f n+1 :u d + Nonlinearity comes from the non linear relation between stress and strain relation Local integration of the mechanical behaviour At each point of the structure process is defined by the following process (un+1 ; Sn )! ¾ n+1 = F(un+1 ; Sn ) Global equilibrium Verified by the solution of the nonlinear variational formulation 0 R(un+1 ; u ; Sn ) = 0; 8u 2 Uad 18 F n+1 :u ds
19 Local behaviour integration process Generic interface for any constitutive equation Gauss point process (where the element variational formulation is integrated) Definition External parameters (ep) imposed as input Integrated variables (vint) Auxiliary variables (vaux) just for output Coefficients (coef) material parameters Primal and dual variables prescribed variables and associated fluxes Primal: strain increment on the current time interval (input to the local integration process) Dual: stress obtained useful for variational formulation (as output of the local integration process) F(un+1 ; Sn ) Primal variables " 19 Local GP integration ODE integration to get vint, vaux, dual variables p; "p Dual variables ¾
20 Local behaviour integration process Time integration Normality rule as an ODE 3 p "_ = p_ eq ¾ D, "pn+1 = "pn + 2¾ Z tn+1 "_ p ( ) d tn Explicit integration Runge-Kutta (RK4 usually, beware of stability condition) Implicit integration θ-methods (stable but requires a local Jacobian computation) 20
21 Local behaviour integration process Elastic prediction and correction algorithm eq Solution depends on the test ¾ R(p) 0 (is the evolution purely elastic on t?) Elastic prediction ¾ en+1 = ¾n + A : "en Convexity of the yield surface gives f (¾ en+1 ; pn ) f (¾ n+1 ; pn + pn ) 8 e < ¾ n+1 = ¾ n+1 f (¾ en+1 ; pn ) 0 purely elastic evolution, prediction is correct "pn+1 = "pn : pn+1 = pn Else a plastic evolution is observed A correction must be applied on the elastic prediction pn > 0; pn f (¾n+1 ; pn + pn ) = 0 ) f (¾n+1 ; pn + pn ) = 0 r 3 p p e ¾ n+1 = ¾n+1 A : "n "n = pn N n+1 pn > 0 2 where 21 N n+1 is the outward unit normal to the final yeld surface
22 Local behaviour integration process Elastic prediction and correction algorithm eq Solution depends on the test ¾ R(p) 0 (is the evolution purely elastic on t?) Elastic prediction ¾ en+1 = ¾n + A : "en Convexity of the yield surface gives f (¾ en+1 ; pn ) f (¾ n+1 ; pn + pn ) 8 e < ¾ n+1 = ¾ n+1 f (¾ en+1 ; pn ) 0 purely elastic evolution, prediction is correct "pn+1 = "pn : pn+1 = pn Else a plastic evolution is observed A correction must be applied on the elastic prediction pn > 0; pn f (¾n+1 ; pn + pn ) = 0 ) f (¾n+1 ; pn + pn ) = 0 r 3 p p e ¾ n+1 = ¾n+1 A : "n "n = pn N n+1 pn > 0 2 where 22 N n+1 is the outward unit normal to the final yeld surface
23 Radial return algorithm For von Mises based criteria or isotropic or linear kinematic hardening p The corrective term A : " is oriented by the final normal that can be computed a priori n For von Mises criterion the final normal is collinear to the elastic predictor r deviator N n+1 = ¾ edn+1 k¾edn+1 k eq e;eq ¾n+1 = ¾n+1 3¹ pn ¾ n+1 = A : pn e ¾Dn+1 f (¾n+1 ; pn+1 ) = 0 ¾Dn+1 3¹ pn R(pn + pn ) = 0 that allows to find f (¾n ; pn ) = 0 pn ¾Dn 23 3 N n+1 2 ¹ Lamé coefficient Final state on the yeld surface gives to verify e;eq ¾n+1 ¾en+1
24 Generalized radial return algorithm Closest point projection technique For a generalized normality " = f (¾n+1 ; pn+1 ) = I = p e ¾Dn+1 ¾Dn+1 f (¾n ; pn ) = 0 Fluxes P ¾ = A : " " ¾Dn YI = MI I = 24 ¾ YI = " A : " 0 # " A 0 0 # p
25 Generic formulation of the local integration process 25
26 Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 26
27 Global aspects Solving the global discrete problem Global problem 0 with R(un+1 ; u ; Sn ) = R(un+1 ; u ; Sn ) = 0; 8u 2 Uad Z Z Z F (un+1 ; Sn ) : " d f n+1 :u F n+1 :u ds Global Newton algorithm At each time step the solution is searched iteratively starting from the last converged solution leading to the following linear system at iteration i i 0 R(uin+1 ; u ; Sn ) + R0 (uin+1 ; u ; Sn )[ui+1 u ] = 0; 8u 2 U n+1 ad n+1 To obtain the residual evaluation, it is necessary to perform the local integration to get F(uin+1 ; Sn ) To calculate the global consistent tangent operator, it is necessary to obtain a local consistent tangent operator (that can be kept constant Zif a modified Newton algorithm with constant operator is R = (un+1 ; Sn ) : " i+1 When the residual is small enough the solution is reached R(un+1 ; u ; Sn ) < N L 0 27 (uin+1 ; u ; Sn )
28 Local consistent tangent operator In the local integration process 28
29 Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 29
30 Viscoelastic with isotropic hardening 30
31 Viscoelastic with isotropic hardening Implementation 31
32 Viscoelastic with isotropic hardening Implementation 32
33 Viscoelastic with isotropic hardening Implementation 33
34 Large strain viscoelastic with isotropic hardening 34
35 System of residuals 35
36 Algorithm 36
37 Tangent matrix 37
38 Outline Generalities on computational strategies for nonlinear problems Examples (contact, crack propagation, non-linear behaviour, geometrical non-linearities) Classical algorithm for nonlinear or time dependant problems Local numerical aspects of plasticity Elastic-plastic behaviour Local integration of non-linear models Global numerical aspects of plasticity Solution process Consistent tangent matrix Examples of solution process Presentation of Z-mat 38
39 Presentation of the Z-mat library 39
40 Inside Z-mat 40
41 Object oriented design 41
42 Isotropic and non-linear kinematic model 42
43 Datafile examples 43
44 ZebFront interface layer for efficient behaviour development 44
45 Explicit model implementation - ZebFront 45
46 Implicit model implementation - ZebFront 46
47 Multimat capabilities 47
48 Multimat capabilities 48
49 References Bonnet M., Frangi A. (2006) Analyse des solides déformables par la méthode des éléments finis. Editions Ecole Polytechnique. Belytschko, T., Liu, W., and Moran, B. (2000). Nonlinear Finite Elements for Continua and Structures. Besson, J., Cailletaud, G., Chaboche, J.-L., and Forest, S. (2001). Mecanique non linéaire des matériaux. Hermes. Ciarlet, P. and Lions, J. (1995). Handbook of Numerical Analysis : Finite Element Methods (P.1), Numerical Methods for Solids (P.2). North Holland. Dhatt, G. and Touzot, G. (1981). Une présentation de la méthode des élements finis. Maloine. Hughes, T. (1987). The finite element method: Linear static and dynamic finite element analysis. Prentice Hall Inc. Simo, J. and Hughes, T. (1997). Computational Inelasticity. Springer Verlag. Zienkiewicz, O. and Taylor, R. (2000). The finite element method, Vol. I-III (Vol.1: The Basis, Vol.2: Solid Mechanics, Vol. 3: Fluid dynamics). Butterworth Heinemann.
Introduction to Finite Element computations
Non Linear Computational Mechanics Athens MP06/2012 Introduction to Finite Element computations Vincent Chiaruttini, Georges Cailletaud vincent.chiaruttini@onera.fr Outline Continuous to discrete problem
More informationMODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD
MODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD Andrzej Skrzat, Rzeszow University of Technology, Powst. Warszawy 8, Rzeszow, Poland Abstract: User-defined material models which can be used
More informationNon-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 17, 2017, Lesson 5
Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 17, 2017, Lesson 5 1 Politecnico di Milano, February 17, 2017, Lesson 5 2 Outline
More informationDEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS
DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS Mohsen Safaei, Wim De Waele Ghent University, Laboratory Soete, Belgium Abstract The present work relates to the
More informationConstitutive models: Incremental plasticity Drücker s postulate
Constitutive models: Incremental plasticity Drücker s postulate if consistency condition associated plastic law, associated plasticity - plastic flow law associated with the limit (loading) surface Prager
More informationCourse in. Geometric nonlinearity. Nonlinear FEM. Computational Mechanics, AAU, Esbjerg
Course in Nonlinear FEM Geometric nonlinearity Nonlinear FEM Outline Lecture 1 Introduction Lecture 2 Geometric nonlinearity Lecture 3 Material nonlinearity Lecture 4 Material nonlinearity it continued
More informationMODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Class Meeting #5: Integration of Constitutive Equations
MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #5: Integration of Constitutive Equations Structural Equilibrium: Incremental Tangent Stiffness
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More informationChapter 2 Finite Element Formulations
Chapter 2 Finite Element Formulations The governing equations for problems solved by the finite element method are typically formulated by partial differential equations in their original form. These are
More informationLecture V: The game-engine loop & Time Integration
Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change
More informationMODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Uniaxial Model: Strain-Driven Format of Elastoplasticity
MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #3: Elastoplastic Concrete Models Uniaxial Model: Strain-Driven Format of Elastoplasticity Triaxial
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationNon-linear and time-dependent material models in Mentat & MARC. Tutorial with Background and Exercises
Non-linear and time-dependent material models in Mentat & MARC Tutorial with Background and Exercises Eindhoven University of Technology Department of Mechanical Engineering Piet Schreurs July 7, 2009
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationThe Finite Element Method II
[ 1 The Finite Element Method II Non-Linear finite element Use of Constitutive Relations Xinghong LIU Phd student 02.11.2007 [ 2 Finite element equilibrium equations: kinematic variables Displacement Strain-displacement
More informationImplicit integration and clarifies nonlinear relations of behaviors
Titre : Intégration implicite et explicite des relations d[...] Date : 25/09/2013 Page : 1/14 Implicit integration and clarifies nonlinear relations of behaviors Summary This document describes two methods
More informationCode_Aster. Reaction of the steel subjected to corrosion
Titre : Comportement de l acier soumis à la corrosion Date : 16/0/01 Page : 1/1 Reaction of the steel subjected to corrosion Summarized: This documentation the model presents steel reaction of subjected
More informationNumerical modelling of elastic-plastic deformation at crack tips in composite material under stress wave loading
JOURNAL DE PHYSIQUE IV Colloque C8, supplément au Journal de Physique III, Volume 4, septembre 1994 C8-53 Numerical modelling of elastic-plastic deformation at crack tips in composite material under stress
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems. Prof. Dr. Eleni Chatzi Lecture ST1-19 November, 2015
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture ST1-19 November, 2015 Institute of Structural Engineering Method of Finite Elements II 1 Constitutive
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationAn Evaluation of Simplified Methods to Compute the Mechanical Steady State
An Evaluation of Simplified Methods to Compute the Mechanical Steady State T. Herbland a,b, G. Cailletaud a, S. Quilici a, H. Jaffal b, M. Afzali b a Mines Paris Paris Tech, CNRS UMR 7633, BP 87, 91003
More informationComputational Inelasticity FHLN05. Assignment A non-linear elasto-plastic problem
Computational Inelasticity FHLN05 Assignment 2017 A non-linear elasto-plastic problem General instructions A written report should be submitted to the Division of Solid Mechanics no later than October
More informationSensitivity and Reliability Analysis of Nonlinear Frame Structures
Sensitivity and Reliability Analysis of Nonlinear Frame Structures Michael H. Scott Associate Professor School of Civil and Construction Engineering Applied Mathematics and Computation Seminar April 8,
More informationNew implicit method for analysis of problems in nonlinear structural dynamics
Applied and Computational Mechanics 5 (2011) 15 20 New implicit method for analysis of problems in nonlinear structural dynamics A. A. Gholampour a,, M. Ghassemieh a a School of Civil Engineering, University
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationA return-map algorithm for general associative isotropic elasto-plastic materials in large deformation regimes
International Journal of Plasticity 15 (1999) 1359±1378 return-map algorithm for general associative isotropic elasto-plastic materials in large deformation regimes F. uricchio a, *, R.L. Taylor b a Dipartimento
More informationTransactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN
A computational method for the analysis of viscoelastic structures containing defects G. Ghazlan," C. Petit," S. Caperaa* " Civil Engineering Laboratory, University of Limoges, 19300 Egletons, France &
More informationENGN 2290: Plasticity Computational plasticity in Abaqus
ENGN 229: Plasticity Computational plasticity in Abaqus The purpose of these exercises is to build a familiarity with using user-material subroutines (UMATs) in Abaqus/Standard. Abaqus/Standard is a finite-element
More information6 th Pipeline Technology Conference 2011
6 th Pipeline Technology Conference 2011 Mechanical behavior of metal loss in heat affected zone of welded joints in low carbon steel pipes. M. J. Fernández C. 1, J. L. González V. 2, G. Jarvio C. 3, J.
More informationA Krylov Subspace Accelerated Newton Algorithm
A Krylov Subspace Accelerated Newton Algorithm Michael H. Scott 1 and Gregory L. Fenves 2 ABSTRACT In the nonlinear analysis of structural systems, the Newton-Raphson (NR) method is the most common method
More informationReturn Mapping Algorithms and Stress Predictors for Failure Analysis in Geomechanics
Return Mapping Algorithms and Stress Predictors for Failure Analysis in Geomechanics Jinsong Huang 1 and D. V. Griffiths 2 Abstract: Two well-known return mapping algorithms, the closest point projection
More informationAlternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE. University of Liège Aerospace & Mechanical Engineering
University of Liège Aerospace & Mechanical Engineering Alternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE Van Dung NGUYEN Innocent NIYONZIMA Aerospace & Mechanical engineering
More informationINTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS
INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS SHEN R. WU and LEI GU WILEY A JOHN WILEY & SONS, INC., PUBLICATION ! PREFACE xv PARTI FUNDAMENTALS 1 1 INTRODUCTION 3
More informationMECHANICS OF MATERIALS. EQUATIONS AND THEOREMS
1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal
More informationMathematical Modeling. of Large Elastic-Plastic Deformations
Applied Mathematical Sciences, Vol. 8, 04, no. 60, 99 996 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.04.447 Mathematical Modeling of Large Elastic-Plastic Deformations L. U. Sultanov Kazan
More informationELASTOPLASTICITY THEORY by V. A. Lubarda
ELASTOPLASTICITY THEORY by V. A. Lubarda Contents Preface xiii Part 1. ELEMENTS OF CONTINUUM MECHANICS 1 Chapter 1. TENSOR PRELIMINARIES 3 1.1. Vectors 3 1.2. Second-Order Tensors 4 1.3. Eigenvalues and
More informationThe Finite Element Method for Solid and Structural Mechanics
The Finite Element Method for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in
More informationelastoplastic contact problems D. Martin and M.H. Aliabadi Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK
Non-conforming BEM elastoplastic contact problems D. Martin and M.H. Aliabadi discretisation in Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Abstract In this paper,
More informationComputational Engineering
Coordinating unit: 205 - ESEIAAT - Terrassa School of Industrial, Aerospace and Audiovisual Engineering Teaching unit: 220 - ETSEIAT - Terrassa School of Industrial and Aeronautical Engineering Academic
More informationOn the Numerical Modelling of Orthotropic Large Strain Elastoplasticity
63 Advances in 63 On the Numerical Modelling of Orthotropic Large Strain Elastoplasticity I. Karsaj, C. Sansour and J. Soric Summary A constitutive model for orthotropic yield function at large strain
More informationA Three-Dimensional Anisotropic Viscoelastic Generalized Maxwell Model for Ageing Wood Composites
A Three-Dimensional Anisotropic Viscoelastic Generalized for Ageing Wood Composites J. Deteix, G. Djoumna, A. Fortin, A. Cloutier and P. Blanchet Society of Wood Science and Technology, 51st Annual Convention
More informationMHA042 - Material mechanics: Duggafrågor
MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of
More informationCM - Computational Mechanics
Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2018 200 - FME - School of Mathematics and Statistics 751 - DECA - Department of Civil and Environmental Engineering 749 - MAT - Department
More informationMathematical Background
CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous
More informationTangent Modulus in Numerical Integration of Constitutive Relations and its Influence on Convergence of N-R Method
Applied and Computational Mechanics 3 (2009) 27 38 Tangent Modulus in Numerical Integration of Constitutive Relations and its Influence on Convergence of N-R Method R. Halama a,, Z. Poruba a a Faculty
More informationFEM for elastic-plastic problems
FEM for elastic-plastic problems Jerzy Pamin e-mail: JPamin@L5.pk.edu.pl With thanks to: P. Mika, A. Winnicki, A. Wosatko TNO DIANA http://www.tnodiana.com FEAP http://www.ce.berkeley.edu/feap Lecture
More informationUsing Thermal Boundary Conditions in SOLIDWORKS Simulation to Simulate a Press Fit Connection
Using Thermal Boundary Conditions in SOLIDWORKS Simulation to Simulate a Press Fit Connection Simulating a press fit condition in SOLIDWORKS Simulation can be very challenging when there is a large amount
More informationPROGRAMMING THE TRANSIENT EXPLICIT FINITE ELEMENT ANALYSIS WITH MATLAB
U.P.B. Sci. Bull., Series D, Vol. 75, Iss. 2, 2013 ISSN 1454-2358 PROGRAMMING THE TRANSIENT EXPLICIT FINITE ELEMENT ANALYSIS WITH MATLAB Andrei Dragoş Mircea SÎRBU 1, László FARKAS 2 Modern research in
More informationA unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation
A unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation S. Bordère a and J.-P. Caltagirone b a. CNRS, Univ. Bordeaux, ICMCB,
More information1 Introduction IPICSE-2016
(06) DOI: 0.05/ matecconf/06860006 IPICSE-06 Numerical algorithm for solving of nonlinear problems of structural mechanics based on the continuation method in combination with the dynamic relaxation method
More informationComputational Inelasticity FHLN05. Assignment A non-linear elasto-plastic problem
Computational Inelasticity FHLN05 Assignment 2018 A non-linear elasto-plastic problem General instructions A written report should be submitted to the Division of Solid Mechanics no later than November
More informationA Comparative Analysis of Linear and Nonlinear Kinematic Hardening Rules in Computational Elastoplasticity
TECHNISCHE MECHANIK, 32, 2-5, (212), 164 173 submitted: October 2, 211 A Comparative Analysis of Linear and Nonlinear Kinematic Hardening Rules in Computational Elastoplasticity Fabio De Angelis In this
More informationChap. 20: Initial-Value Problems
Chap. 20: Initial-Value Problems Ordinary Differential Equations Goal: to solve differential equations of the form: dy dt f t, y The methods in this chapter are all one-step methods and have the general
More informationAn example of panel solution in the elastic-plastic regime
An example of panel solution in the elastic-plastic regime Piotr Mika May, 2014 2013-05-08 1. Example solution of the panel with ABAQUS program The purpose is to analyze the elastic-plastic panel. The
More informationAn example solution of a panel in the elastic-plastic regime
An example solution of a panel in the elastic-plastic regime Piotr Mika May, 2013 1. Example solution of the panel with ABAQUS program The purpose is to analyze an elastic-plastic panel. The elastic solution
More informationBifurcation Analysis in Geomechanics
Bifurcation Analysis in Geomechanics I. VARDOULAKIS Department of Engineering Science National Technical University of Athens Greece and J. SULEM Centre d'enseignement et de Recherche en Mecanique des
More informationFVM for Fluid-Structure Interaction with Large Structural Displacements
FVM for Fluid-Structure Interaction with Large Structural Displacements Željko Tuković and Hrvoje Jasak Zeljko.Tukovic@fsb.hr, h.jasak@wikki.co.uk Faculty of Mechanical Engineering and Naval Architecture
More informationCoupled Thermomechanical Contact Problems
Coupled Thermomechanical Contact Problems Computational Modeling of Solidification Processes C. Agelet de Saracibar, M. Chiumenti, M. Cervera ETS Ingenieros de Caminos, Canales y Puertos, Barcelona, UPC
More informationEXTENDED ABSTRACT. Dynamic analysis of elastic solids by the finite element method. Vítor Hugo Amaral Carreiro
EXTENDED ABSTRACT Dynamic analysis of elastic solids by the finite element method Vítor Hugo Amaral Carreiro Supervisor: Professor Fernando Manuel Fernandes Simões June 2009 Summary The finite element
More informationHERCULES-2 Project. Deliverable: D4.4. TMF model for new cylinder head. <Final> 28 February March 2018
HERCULES-2 Project Fuel Flexible, Near Zero Emissions, Adaptive Performance Marine Engine Deliverable: D4.4 TMF model for new cylinder head Nature of the Deliverable: Due date of the Deliverable:
More informationSSNV221 Hydrostatic test with a behavior DRUCK_PRAGER linear and parabolic
Titre : SSNV221 - Essai hydrostatique avec un comportement[...] Date : 08/08/2011 Page : 1/10 SSNV221 Hydrostatic test with a behavior DRUCK_PRAGER linear and parabolic Summary: The case test proposes
More informationMulti-scale digital image correlation of strain localization
Multi-scale digital image correlation of strain localization J. Marty a, J. Réthoré a, A. Combescure a a. Laboratoire de Mécanique des Contacts et des Strcutures, INSA Lyon / UMR CNRS 5259 2 Avenue des
More informationTheory of Plasticity. Lecture Notes
Theory of Plasticity Lecture Notes Spring 2012 Contents I Theory of Plasticity 1 1 Mechanical Theory of Plasticity 2 1.1 Field Equations for A Mechanical Theory.................... 2 1.1.1 Strain-displacement
More informationHSNV140 - Thermoplasticity with restoration of work hardening: test of blocked dilatometry
Titre : HSNV140 - Thermo-plasticité avec restauration d'éc[...] Date : 29/06/2015 Page : 1/11 HSNV140 - Thermoplasticity with restoration of work hardening: test of blocked dilatometry Summary: This test
More informationPseudo-Force Incremental Methods
. 19 Pseudo-Force Incremental Methods 19 1 Chapter 19: PSEUDO-FORCE INCREMENTAL METHODS 19 2 TABLE OF CONTENTS Page 19.1. Pseudo Force Formulation 19 3 19.2. Computing the Reference Stiffness and Internal
More informationEsben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer
Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 1-21 September, 2017 Institute of Structural Engineering
More informationThe Finite Element Method for Computational Structural Mechanics
The Finite Element Method for Computational Structural Mechanics Martin Kronbichler Applied Scientific Computing (Tillämpad beräkningsvetenskap) January 29, 2010 Martin Kronbichler (TDB) FEM for CSM January
More informationSTRESS UPDATE ALGORITHM FOR NON-ASSOCIATED FLOW METAL PLASTICITY
STRESS UPDATE ALGORITHM FOR NON-ASSOCIATED FLOW METAL PLASTICITY Mohsen Safaei 1, a, Wim De Waele 1,b 1 Laboratorium Soete, Department of Mechanical Construction and Production, Ghent University, Technologiepark
More informationnumerical implementation and application for life prediction of rocket combustors Tel: +49 (0)
2nd Workshop on Structural Analsysis of Lightweight Structures. 30 th May 2012, Natters, Austria Continuum damage mechanics with ANSYS USERMAT: numerical implementation and application for life prediction
More informationComparison of Models for Finite Plasticity
Comparison of Models for Finite Plasticity A numerical study Patrizio Neff and Christian Wieners California Institute of Technology (Universität Darmstadt) Universität Augsburg (Universität Heidelberg)
More informationStudy of a Nuclear Power Plant Containment Damage Caused by Impact of a Plane
International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 1, Issue 4(December 2012), PP.48-53 Study of a Nuclear Power Plant Containment Damage
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationMaria Cameron. f(x) = 1 n
Maria Cameron 1. Local algorithms for solving nonlinear equations Here we discuss local methods for nonlinear equations r(x) =. These methods are Newton, inexact Newton and quasi-newton. We will show that
More informationSSNV231 Sphere digs under internal pressure in great deformations
Titre : SSNV231 Sphère creuse sous pression interne en g[...] Date : 25/02/2014 Page : 1/18 SSNV231 Sphere digs under internal pressure in great deformations Summary: The objective of this test is to validate
More informationA Simple and Accurate Elastoplastic Model Dependent on the Third Invariant and Applied to a Wide Range of Stress Triaxiality
A Simple and Accurate Elastoplastic Model Dependent on the Third Invariant and Applied to a Wide Range of Stress Triaxiality Lucival Malcher Department of Mechanical Engineering Faculty of Tecnology, University
More informationOrdinary differential equations - Initial value problems
Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations - Initial value problems In this chapter
More informationNumerical Methods I Solving Nonlinear Equations
Numerical Methods I Solving Nonlinear Equations Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 16th, 2014 A. Donev (Courant Institute)
More informationPSEUDO ELASTIC ANALYSIS OF MATERIAL NON-LINEAR PROBLEMS USING ELEMENT FREE GALERKIN METHOD
Journal of the Chinese Institute of Engineers, Vol. 27, No. 4, pp. 505-516 (2004) 505 PSEUDO ELASTIC ANALYSIS OF MATERIAL NON-LINEAR PROBLEMS USING ELEMENT FREE GALERKIN METHOD Raju Sethuraman* and Cherku
More informationAnalytical formulation of Modified Upper Bound theorem
CHAPTER 3 Analytical formulation of Modified Upper Bound theorem 3.1 Introduction In the mathematical theory of elasticity, the principles of minimum potential energy and minimum complimentary energy are
More informationOverview of Solution Methods
20 Overview of Solution Methods NFEM Ch 20 Slide 1 Nonlinear Structural Analysis is a Multilevel Continuation Process Stages Increments Iterations Individual stage Increments Iterations NFEM Ch 20 Slide
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20
2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20 REVIEW Lecture 19: Finite Volume Methods Review: Basic elements of a FV scheme and steps to step-up a FV scheme One Dimensional examples d x j x j 1/2
More informationEngineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.
Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 21 Numerical Differentiation PowerPoints organized by Dr. Michael R. Gustafson II, Duke University 1 All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationLecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity
Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling
More informationNUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMIC FRACTURE
NUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMIC FRACTURE Kevin G. Wang (1), Patrick Lea (2), and Charbel Farhat (3) (1) Department of Aerospace, California Institute of Technology
More informationFatigue Damage Development in a Steel Based MMC
Fatigue Damage Development in a Steel Based MMC V. Tvergaard 1,T.O/ rts Pedersen 1 Abstract: The development of fatigue damage in a toolsteel metal matrix discontinuously reinforced with TiC particulates
More informationStresses Analysis of Petroleum Pipe Finite Element under Internal Pressure
ISSN : 48-96, Vol. 6, Issue 8, ( Part -4 August 06, pp.3-38 RESEARCH ARTICLE Stresses Analysis of Petroleum Pipe Finite Element under Internal Pressure Dr.Ragbe.M.Abdusslam Eng. Khaled.S.Bagar ABSTRACT
More informationScientific Computing II
Scientific Computing II Molecular Dynamics Numerics Michael Bader SCCS Technical University of Munich Summer 018 Recall: Molecular Dynamics System of ODEs resulting force acting on a molecule: F i = j
More informationLinearized theory of elasticity
Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark
More informationNonlinear analysis in ADINA Structures
Nonlinear analysis in ADINA Structures Theodore Sussman, Ph.D. ADINA R&D, Inc, 2016 1 Topics presented Types of nonlinearities Materially nonlinear only Geometrically nonlinear analysis Deformation-dependent
More informationCombined Isotropic-Kinematic Hardening Laws with Anisotropic Back-stress Evolution for Orthotropic Fiber-Reinforced Composites
Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution for Orthotropic Fiber- Reinforced Composites Combined Isotropic-Kinematic Hardening Laws with Antropic Back-stress Evolution
More informationSolving Ordinary Differential Equations
Solving Ordinary Differential Equations Sanzheng Qiao Department of Computing and Software McMaster University March, 2014 Outline 1 Initial Value Problem Euler s Method Runge-Kutta Methods Multistep Methods
More informationA return mapping algorithm for unified strength theory model
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2015; 104:749 766 Published online 10 June 2015 in Wiley Online Library (wileyonlinelibrary.com)..4937 A return mapping
More informationGEOMETRIC NONLINEAR ANALYSIS
GEOMETRIC NONLINEAR ANALYSIS The approach for solving problems with geometric nonlinearity is presented. The ESAComp solution relies on Elmer open-source computational tool [1] for multiphysics problems.
More informationContinuation methods for non-linear analysis
Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Outline Definition of
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS NUMERICAL FLUID MECHANICS FALL 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS FALL 2011 QUIZ 2 The goals of this quiz 2 are to: (i) ask some general
More informationANSYS Mechanical Basic Structural Nonlinearities
Lecture 4 Rate Independent Plasticity ANSYS Mechanical Basic Structural Nonlinearities 1 Chapter Overview The following will be covered in this Chapter: A. Background Elasticity/Plasticity B. Yield Criteria
More informationINTRODUCTION TO COMPUTER METHODS FOR O.D.E.
INTRODUCTION TO COMPUTER METHODS FOR O.D.E. 0. Introduction. The goal of this handout is to introduce some of the ideas behind the basic computer algorithms to approximate solutions to differential equations.
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More information