Virtual Element Methods in Engineering Sciences

ISBN-13: 9783031392542 / Angielski

Peter Wriggers; Fadi Aldakheel; Blaz Hudobivnik

Spis treści

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 History and recent developments of virtual elements . . . . . . . . . . . . . 2

1.2 Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Virtual element formulation of a truss using a linear ansatz . 7

1.2.2 Quadratic ansatz for a one-dimensional virtual truss element 11

1.3 Contents of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Continuum mechanics background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Constitutive Eqations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Finite elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3 Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Potential and weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.3 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.4 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 VEM Ansatz functions and projection for solids . . . . . . . . . . . . . . . . . . . 37

3.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 General ansatz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.3 Equivalent projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1.4 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.5 Computation of the projection using symbolic software . . . . 52

3.1.6 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.7 Serendipity virtual element for a quadratic ansatz . . . . . . . . . 59

vii

viii Contents

3.1.8 Computation of the second order projection using

automatic di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.9 Higher order ansatz for virtual elements . . . . . . . . . . . . . . . . . 65

3.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.1 General ansatz space in three dimensions . . . . . . . . . . . . . . . . 67

3.2.2 Computation of the projection in three dimensions . . . . . . . . 69

3.2.3 Projection for linear ansatz in three dimensions . . . . . . . . . . . 70

4 VEM Ansatz functions and projection for the Poisson equation . . . . . . 77

4.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Computation of the projection . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.2 Projection for a linear ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.3 Projection for a quadratic ansatz . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Construction of the virtual element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1 Consistency Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1.1 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Stabilization techniques for virtual elements . . . . . . . . . . . . . . . . . . . . 90

5.2.1 Stabilization by a discrete bi-linear form . . . . . . . . . . . . . . . . . 90

5.2.2 Energy stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Assembly to the global equation system . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Numerical example for the Poisson equation . . . . . . . . . . . . . . . . . . . . 96

6 Virtual elements for elasticity problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.1 Linear elastic response of two-dimensional solids . . . . . . . . . . . . . . . . 104

6.1.1 Consistency term using Voigt notation. . . . . . . . . . . . . . . . . . . 105

6.1.2 Consistency term using tensor notation . . . . . . . . . . . . . . . . . . 108

6.1.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.4 Definition and labeling of di erent mesh types . . . . . . . . . . . . 116

6.1.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2 Finite Strain: compressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.1 Consistency term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.2 Stability term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.3 Nonlinear virtual elements for three-dimensional problems

in elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.4 General solution for nonlinear equations . . . . . . . . . . . . . . . . . 130

6.2.5 Numerical examples, compressible case . . . . . . . . . . . . . . . . . 132

6.3 Incompressible elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.3.1 Linear virtual element with constant pressure . . . . . . . . . . . . . 143

6.3.2 Quadratic serendipity virtual element with linear pressure . . 144

6.3.3 Nearly incompressible behaviour . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.4 Numerical examples, incompressible case . . . . . . . . . . . . . . . . 151

6.4 Anisotropic elastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Contents ix

6.4.1 Numerical examples, anisotropic case . . . . . . . . . . . . . . . . . . . 161

7 Virtual elements for problems in dynamics . . . . . . . . . . . . . . . . . . . . . . . . 167

7.1 Continuum formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.3 Solution algorithms for small strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3.1 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3.2 Numerical integration in time, time stepping schemes . . . . . . 174

7.4 Solution algorithms for finite strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.5.1 Transversal Beam Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.5.2 Cook’s membrane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.5.3 3D Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8 Virtual element formulation for finite plasticity . . . . . . . . . . . . . . . . . . . . 189

8.1 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.1.1 Consistency part due to projection . . . . . . . . . . . . . . . . . . . . . . 189

8.1.2 Algorithmic treatment of finite strain elasto-plasticity. . . . . . 190

8.1.3 Energy stabilization of the virtual element for finite plasticity192

8.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2.1 Necking of cylindrical bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2.2 Taylor Anvil Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9 Virtual elements for thermo-mechanical problems . . . . . . . . . . . . . . . . . 203

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.2.1 Energetic and dissipative response functions . . . . . . . . . . . . . . 205

9.2.2 Global constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.2.3 Weak form and pseudo-potential energy function . . . . . . . . . . 208

9.3 Virtual element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.4 Representative numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.4.1 Forming of a steel bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10 Virtual elements for fracture processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.1 Brittle crack-propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.1.2 Equations of brittle crack propagation . . . . . . . . . . . . . . . . . . . 220

10.1.3 Modeling crack propagation with virtual elements . . . . . . . . . 221

10.1.4 Computation of stress intensity factors . . . . . . . . . . . . . . . . . . 221

10.1.5 Propagation criteria: Maximum circumferential stress

criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.1.6 Stress intensity factor analysis using virtual elements . . . . . . 223

10.1.7 Cutting Technique and crack update algorithm . . . . . . . . . . . . 227

10.1.8 Crack propagation simulations based on the cutting technique231

10.2 Phase field methods for brittle fracture using virtual elements . . . . . . 235

10.2.1 Governing equations for elasticity . . . . . . . . . . . . . . . . . . . . . . 235

x Contents

10.2.2 Regularization of a sharp crack topology . . . . . . . . . . . . . . . . 235

10.2.3 Variational formulation to brittle fracture . . . . . . . . . . . . . . . . 238

10.2.4 Formulation of the virtual element method . . . . . . . . . . . . . . . 241

10.2.5 VEM for phase field brittle fracture simulations . . . . . . . . . . . 244

10.3 Phase field methods for ductile fracture using virtual elements . . . . . 246

10.3.1 Governing equations for phase field ductile fracture . . . . . . . 248

10.3.2 Formulation of the virtual element method . . . . . . . . . . . . . . . 251

10.3.3 VEM for phase field ductile fracture simulations . . . . . . . . . . 251

10.4 An adaptive scheme to follow crack paths combining phase field

and cutting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.4.2 Modeling crack propagation using VEM . . . . . . . . . . . . . . . . 258

10.4.3 A discontinuous crack propagation using phase field . . . . . . 258

10.4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10.5 Fracturing analysis using damage mechanics . . . . . . . . . . . . . . . . . . . . 268

10.5.1 Governing equations for isotropic damage model . . . . . . . . . . 268

10.5.2 Virtual element formulation for damage . . . . . . . . . . . . . . . . . 271

10.5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

11 Virtual element formulation for contact. . . . . . . . . . . . . . . . . . . . . . . . . . . 281

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

11.2 Theoretical background for contact of solids . . . . . . . . . . . . . . . . . . . . 283

11.2.1 Local contact kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

11.2.2 Constitutive relations for contact . . . . . . . . . . . . . . . . . . . . . . . 286

11.2.3 Potential form for solids in contact . . . . . . . . . . . . . . . . . . . . . . 288

11.3 Contact discretization using VEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

11.3.1 Node insertion algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

11.3.2 Inserted node and gap in the two-dimensional case . . . . . . . . 292

11.3.3 Discretization of the contact interface in 2d . . . . . . . . . . . . . . 294

11.3.4 Penalty formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

11.3.5 Augmented Lagrangian multiplier formulation . . . . . . . . . . . . 301

11.4 Stabilization of VEM in case of contact . . . . . . . . . . . . . . . . . . . . . . . . 303

11.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

11.5.1 Patch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.5.2 Hertzian contact problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.5.3 Contacting Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

11.5.4 Hertz contact for large deformations . . . . . . . . . . . . . . . . . . . . 311

11.5.5 Ironing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

11.5.6 Wall mounting of a bolt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

12 Virtual elements for homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Contents xi

13 Virtual elements for beams and plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.1 Virtual element formulations for Euler-Bernoulli beams . . . . . . . . . . 322

13.1.1 Fourth order ansatz for a one-dimensional virtual beam

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

13.2 Virtual element formulations for Kirchho -Love plates . . . . . . . . . . . 331

13.2.1 Mathematical model of the plate and constitutive relations . . 331

13.3 Formulation of the virtual element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.3.1 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.3.2 Ansatz and projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

13.3.3 Ansatz function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

13.3.4 Plate element with constant curvature . . . . . . . . . . . . . . . . . . . 339

13.3.5 Plate element with linear curvature . . . . . . . . . . . . . . . . . . . . . 342

13.3.6 Residual and sti ness matrix of the virtual plate element . . . 345

13.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

13.4.1 Notation used in the examples . . . . . . . . . . . . . . . . . . . . . . . . . . 347

13.4.2 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 348

13.4.3 Rhombic plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

13.4.4 Rectangular orthotropic plate . . . . . . . . . . . . . . . . . . . . . . . . . . 354

13.4.5 Plate with anisotropic material . . . . . . . . . . . . . . . . . . . . . . . . . 355

13.5 ⇠1 -continuous virtual elements for FEM codes . . . . . . . . . . . . . . . . . . 357

13.5.1 Clamped plate under uniform load . . . . . . . . . . . . . . . . . . . . . . 358

13.5.2 Clamped plate under point load . . . . . . . . . . . . . . . . . . . . . . . . 359

13.5.3 L-shaped plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

A Formulae in virtual element formulations . . . . . . . . . . . . . . . . . . . . . . . . . 363

A.1 Integration over polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

A.2 Computation of volume by surface integrals . . . . . . . . . . . . . . . . . . . . 365

B I-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

O autorze

Professor Dr.-Ing. habil. P. Wriggers studied Civil Engineering at the University Hannover; he obtained his Dr.-Ing degree at the University of Hannover in 1980 on “Contact-impact problems.” Since April 2022, he is Emeritus Professor at Leibniz Universität Hannover. Peter Wriggers is Member of the “Braunschweigische Wissenschaftliche Gesellschaft,” the Academy of Science and Literature in Mainz, the German National Academy of Engineering “acatech” and the National Academy of Croatia. He was President of GAMM, President of GACM and Vice-President of IACM. Furthermore, he acts as Editor-in-Chief for the International Journal “Computational Mechanics” and “Computational Particle Mechanics.” He was awarded the Fellowship of IACM and received the “Computational Mechanics Award” and the “IACM Award” of IACM, the “Euler Medal” of ECCOMAS as well as three honorary degrees from the Universities of Poznan, ENS Cachan and TU Darmstadt.

Professor Dr.-Ing. habil. F. Aldakheel is since April 2023 professor for high performance computing at Leibniz Universität Hannover. After studying engineering in Aleppo, he initially worked at Alfurat University in Syria before moving to the Institute of Applied Mechanics at the University of Stuttgart for the master and Ph.D. studies and then the postdoc period. There he was course director for the international master's programme "Computational Mechanics of Materials and Structures" (COMMAS) as well as local director for the excellence programme "Erasmus Mundus Master of Science in Computational Mechanics". Most recently, he was Chief-Engineer/Group-Leader at the Institute for Continuum Mechanics at Leibniz Universität Hannover and Associate Professor (Honorary) at the Zienkiewicz Centre for Computational Engineering at Swansea University, UK. He has been awarded numerous awards, among them the Richard-von-Mises Prize of GAMM (Association of Applied Mathematics and Mechanics). His research interests are related to the modeling of material behaviors, variational principles, computational solid mechanics, structural mechanics, finite and virtual element methods, multiphysics and multi-scales problems, machine learning, energy transition and experimental validation.

Dr. Blaž Hudobivnik studied Civil Engineering at the University of Ljubljana. He was awarded his Doctoral degree in 2016 under the supervision of Prof. Jože Korelc. He worked as Young researcher/Researcher between 2011 and October 2016 at the University of Ljubljana and after that he was employed as Postdoctoral researcher until April 2023 at the Institute of Continuum Mechanics at the Leibniz Universität Hannover. Since April 2023 he is employed in industry as simulation expert in mechanical design of batteries. His primary research fields are efficient implementation of nonlinear coupled problems, the development of the virtual element method and its application to a wide range of engineering problems. This includes 2D and 3D applications for linear and nonlinear materials, for static and dynamic solids, plate and contact problems, coupled problems (thermo-hydro-mechanics), phase field methods, multi-scale and optimization problems. Alongside research, he advises other institute members in numerical implementations due to his expert knowledge of the Software-Tool AceGen/AceFEM, developed by his doctoral advisor Prof. Korelc.

Blurb

This book provides a comprehensive treatment of the virtual element method (VEM) for engineering applications, focusing on its application in solid mechanics. Starting with a continuum mechanics background, the book establishes the necessary foundation for understanding the subsequent chapters. It then delves into the VEM's Ansatz functions and projection techniques, both for solids and the Poisson equation, which are fundamental to the method. The book explores the virtual element formulation for elasticity problems, offering insights into its advantages and capabilities. Moving beyond elasticity, the VEM is extended to problems in dynamics, enabling the analysis of dynamic systems with accuracy and efficiency. The book also covers the virtual element formulation for finite plasticity, providing a framework for simulating the behavior of materials undergoing plastic deformation. Furthermore, the VEM is applied to thermo-mechanical problems, where it allows for the investigation of coupled thermal and mechanical effects. The book dedicates a significant portion to the virtual elements for fracture processes, presenting techniques to model and analyze fractures in engineering structures. It also addresses contact problems, showcasing the VEM's effectiveness in dealing with contact phenomena. The virtual element method's versatility is further demonstrated through its application in homogenization, offering a means to understand the effective behavior of composite materials and heterogeneous structures. Finally, the book concludes with the virtual elements for beams and plates, exploring their application in these specific structural elements. Throughout the book, the authors emphasize the advantages of the virtual element method over traditional finite element discretization schemes, highlighting its accuracy, flexibility, and computational efficiency in various engineering contexts.