Equilibrium Finite Element Formulations

A comprehensive treatment of the theory and practice of equilibrium finite element analysis in the context of solid and structural mechanics

Equilibrium Finite Element Formulations is an up to date exposition on hybrid equilibrium finite elements, which are based on the direct approximation of the stress fields. The focus is on their derivation and on the advantages that strong forms of equilibrium can have, either when used independently or together with the more conventional displacement based elements. These elements solve two important problems of concern to computational structural mechanics: a rational basis for error estimation, which leads to bounds on quantities of interest that are vital for verification of the output and provision of outputs immediately useful to the engineer for structural design and assessment.

Key features:

• Unique in its coverage of equilibrium – an essential reference work for those seeking solutions that are strongly equilibrated. The approach is not widely known, and should be of benefit to structural design and assessment.
• Thorough explanations of the formulations for: 2D and 3D continua, thick and thin bending of plates and potential problems; covering mainly linear aspects of behaviour, but also with some excursions into non-linearity.
• Highly relevant to the verification of numerical solutions, the basis for obtaining bounds of the errors is explained in detail.
• Simple illustrative examples are given, together with their physical interpretations.
• The most relevant issues regarding the computational implementation of this approach are presented.

When strong equilibrium and finite elements are to be combined, the book is a must-have reference for postgraduate students, researchers in software development or numerical analysis, and industrial practitioners who want to keep up to date with progress in simulation tools.

J.P. Moitinho de Almeida is an Associate Professor in the Department of Civil Engineering, Architecture and Georesources in Instituto Superior Técnico, at the University of Lisbon. His research interests include non-conventional finite element formulations, and procedures for graphic processing of results in computational mechanics.

Edward A. W. Maunder is an Honorary Fellow at the College of Engineering, Mathematics and Physical Sciences at the University of Exeter. His research interests are in the areas of computational structural mechanics and the development of stress based equilibrium finite element models for the design and assessment of structures.

Preface xiii

List of Symbols xvii

1 Introduction 1

1.1 Prerequisites 1

1.2 What Is Meant by Equilibrium? Weak to Strong Forms 2

1.3 What Do We Gain From Strong Forms of Equilibrium? 3

1.4 What Paths Have Been Followed to Achieve Strong Forms of Equilibrium? 5

1.5 Industrial Perspectives 6

1.5.1 Simulation Governance 7

1.5.2 Equilibrium in Structural Design and Assessment 7

1.6 The Structure of the Book 8

References 9

2 Basic Concepts Illustrated by Simple Examples 11

2.1 Symmetric Bi-Material Strip 12

2.2 Kirchhoff Plate With a Line Load 16

2.2.1 Kinematically Admissible Solutions 16

2.2.2 Statically Admissible Solutions 19

2.2.3 Assessment of the Solutions Obtained 20

References 21

3 Equilibrium in Other Finite Element Formulations 22

3.1 Conforming Formulations and Nodal Equilibrium 22

3.2 Pian’s Hybrid Formulation 25

3.3 Mixed Stress Formulations 27

3.4 Variants of the Displacement Based Formulations With Stronger Forms of Equilibrium 28

3.4.1 Fraeijs de Veubeke’s Equilibrated Triangle 29

3.4.2 Triangular Equilibrium Elements for Plate Bending 30

3.4.3 Other Variants 31

3.5 Trefftz Formulations 32

3.6 Formulations Based on the Approximation of a Stress Potential 33

3.7 The Symmetric Bi-Material Strip Revisited 33

References 40

4 Formulation of Hybrid Equilibrium Elements 43

4.1 Approximation of the Stresses 43

4.2 Approximation of the Boundary Displacements 45

4.3 Assembling the Approximations 48

4.4 Enforcement of Equilibrium at the Boundaries of the Elements 48

4.5 Enforcement of Compatibility 51

4.6 Governing System 53

4.7 Existence and Uniqueness of the Solution 54

4.8 Elements for Specific Types of Problem 57

4.8.1 Continua in 2D 57

4.8.1.1 Exemplification of the Assembly Process 58

4.8.1.2 A Simple Numerical Example 60

4.8.2 Continua in 3D 62

4.8.3 Plate Bending 63

4.8.3.1 Reissner–Mindlin Theory 64

4.8.3.2 Kirchhoff Theory 65

4.8.3.3 Example 66

4.8.4 Potential Problems of Lower Order 66

4.9 The Case of Geometries With a Non-Linear Mapping 68

4.10 Compatibility Defaults 69

4.11 The Dimension of the System of Equations 70

References 71

5 Analysis of the Kinematic Stability of Hybrid Equilibrium Elements 73

5.1 Algebraic and Duality Concepts Related to Spurious Kinematic Modes 73

5.2 Spurious Kinematic Modes in Models of 2D Continua 76

5.2.1 Single Triangular Elements 77

5.2.2 A Pair of Triangular Elements With a Common Interface 80

5.2.3 Star Patches of 2D Elements 82

5.2.3.1 Open Stars of Degree 0 84

5.2.3.2 Closed Stars of Degree 0 84

5.2.3.3 Open Stars of Degree 1 84

5.2.3.4 Closed Stars of Degree 1 85

5.2.3.5 Open Stars of Degree 2 85

5.2.3.6 Closed Stars of Degree 2 85

5.2.3.7 Examples of Unstable Closed Star Patches of Degree 2 86

5.2.3.8 Stars of Degree 3 or Higher 87

5.2.4 Observations for General 2D Meshes 87

5.3 Spurious Kinematic Modes in Models of 3D Continua 90

5.3.1 Single Tetrahedral Elements 90

5.3.1.1 Spurious Modes Associated With a Single Edge 92

5.3.1.2 Spurious Modes Associated With a Single Face 94

5.3.2 A Pair of Tetrahedral Elements 94

5.3.2.1 Primary Interface Spurious Modes 95

5.3.2.2 Pairs of Tetrahedral Elements With Coplanar Faces 96

5.3.3 Star Patches of Tetrahedral Elements 97

5.3.3.1 Edge-Centred Patches 98

5.3.3.2 Vertex-Centred Patches 98

5.4 Spurious Kinematic Modes in Models of Reissner–Mindlin Plates 99

5.4.1 A Single Triangular Reissner–Mindlin Element 100

5.4.2 A Pair of Reissner–Mindlin Elements 102

5.4.3 Star Patches of Reissner–Mindlin Elements 103

5.4.3.1 Open Stars of Degree 1 103

5.4.3.2 Closed Star Patches of Degree 1 103

5.4.3.3 Open Stars of Degree 2 103

5.4.3.4 Closed Star Patches of Degree 2 103

5.4.4 Observations for General Meshes of Reissner–Mindlin Elements 104

5.5 The Stability of Plates Modelled With Kirchhoff Elements 105

5.6 The Stability of Models for Potential Problems 106

5.7 How Do We Obtain a Stable Mesh for General Structural Models? 108

5.7.1 General Procedures 108

5.7.2 Macro-Elements 108

References 109

6 Practical Aspects of the Kinematic Stability of Hybrid Equilibrium Elements 111

6.1 Identification of Rigid Body and Spurious Kinematic Modes 111

6.1.1 Spurious Kinematic and Rigid Body Modes of an Element 112

6.1.2 Spurious Kinematic and Rigid Body Modes of a Mesh 113

6.2 Blocking the Spurious Modes 115

6.3 An Illustration of the Procedures to Remove Spurious Modes 116

6.4 How Do We Recognize Admissible Loads? 117

6.5 Quasi-Simplicial Hybrid Elements Created by Hierarchical Mesh Refinement 118

6.6 Non-Simplicial Hybrid Elements 120

6.7 A Cautionary Tale of ‘Near Misses’ 120

References 125

7 A Variational Basis of the Hybrid Equilibrium Formulation 126

7.1 Potential Energy and Complementary Potential Energy 126

7.1.1 Existence and Uniqueness of Solutions 129

7.1.2 Properties of the Exact Solution 129

7.1.3 The Formal Relation Between Both Energies 130

7.2 Hybrid Complementary Potential Energy 131

7.3 Properties of the Generalized Complementary Energy 132

7.4 The Babuška–Brezzi Condition and Hybrid Equilibrium Elements 133

References 134

8 Recovery of Complementary Solutions 135

8.1 General Features of Partition of Unity Functions 136

8.2 Recovery of Compatibility From an Equilibrated Solution 138

8.2.1 Derivation of uÞ E 140

8.2.2 An Illustration of the Technique 141

8.3 Recovery of Equilibrium From a Compatible Solution 143

8.3.1 Recovery From Star Patches: The General Case 144

8.3.2 Recovery From Star Patches: The Case of Linear Displacements 146

8.3.3 Element by Element Recovery of Equilibrium 150

8.3.3.1 Resolution of the Vertex Forces 150

8.3.3.2 Derivation of Statically Equivalent Codiffusive Tractions 153

8.3.3.3 Admissibility of the Derived Tractions 155

8.3.3.4 Derivation of the Element Stress Fields 156

8.4 Numerical Examples 157

8.4.1 Recovery of Compatibility From an Equilibrated Solution 157

8.4.2 Recovery of Equilibrium From a Compatible Solution 160

8.5 Extensions of the Recovery Procedures 163

8.5.1 Reissner–Mindlin Theory 163

8.5.2 Kirchhoff Theory 163

8.5.2.1 Recovery of Compatibility 163

8.5.2.2 Recovery of Equilibrium 164

8.5.3 Non-Simplicial Elements 164

References 164

9 Dual Analyses for Error Estimation & Adaptivity 166

9.1 Global Error Bounds 167

9.1.1 Revisiting the simple example 170

9.2 Estimation of the Error Distribution and Global Mesh Adaptation 177

9.2.2 The Convergence of the Simple Example 180

9.3 Obtaining Local Quantities of Interest 184

9.4 Bounding the Error of Local Outputs 187

9.4.1 Background 187

9.4.2 Bounds of the Error of Outputs Obtained From Complementary Solutions 187

9.5 Local Outputs for the Kirchhoff Plate With a Line Load 189

9.5.1 The Displacement at the Corner 190

9.5.2 The Average Displacement on the Loaded Side 192

9.5.3 The Average Displacement on the Free Side 193

9.6 Estimation of the Error Distribution and Mesh Adaptation for Local Quantities 194

9.7 Adaptivity for Multiple Loads and Multiple Outputs 195

References 196

10 Dynamic Analyses 199

10.1 Toupin’s Principle for Elastodynamics 200

10.2 Derivation of the Equilibrium Finite Element Equations 201

10.3 Analysis in the Frequency Domain 203

10.4 Analysis in the Time Domain 205

10.5 No Direct Bounds of the Eigenfrequencies? 206

10.6 Example 207

10.6.1 Eigenfrequencies 207

10.6.2 Forced Vibrations 209

References 211

11 Non-Linear Analyses 212

11.1 Elastic Contact 212

11.2 Material Non-Linearity 214

11.2.1 Non-Linear Elasticity 214

11.2.2 Elastoplastic Constitutive Relations 215

11.2.2.1 Direct Implementation 216

11.2.2.2 A Standard Return Mapping Implementation 217

11.2.2.3 A Return Mapping Implementation for Plasticity Defined in the Strain Space 218

11.2.2.4 Imposing the Yield Condition in a Weak Form 219

11.3 Limit Analysis 220

11.3.1 Introduction 220

11.3.2 General Statement of the Problem as a Mathematical Programme 220

11.3.2.1 Formulation (1) 221

11.3.2.2 Formulation (2) 221

11.3.2.3 Yield Constraints 222

11.3.2.4 Application of the Complementary (Dual) Programme 222

11.3.3 Implementation for Plate Bending Problems 222

11.3.4 Numerical Example 223

11.4 Geometric Non-Linearity 224

11.4.1 Weak Compatibility for Large Displacements With Small Strains 225

11.4.2 Equilibrium 227

11.4.3 Transformation of Boundary Displacement Parameters and Generalized Tractions 228

11.4.4 Governing System 229

11.4.5 Determination of the Rigid Body Displacements 229

11.4.6 Tangent Form of the Governing System 230

11.4.6.1 Variation of the Rigid Body Displacements 230

11.4.6.2 The Effect of a Variation in the Boundary Displacement Parameters on the Associated Transformations 231

11.4.6.3 Tangent Form of the Governing System for an Element 233

11.4.7 Large Displacements and Spurious Kinematic Modes 233

11.4.7.1 Numerical Example 234

References 235

A Fundamental Equations of Structural Mechanics 237

A.1 The General Elastostatic Problem 237

A.1.1 Two Dimensional Elasticity 237

A.1.2 Three Dimensional Elasticity 238

A.1.3 Shear Stresses and Warping of a Beam Section 240

A.1.4 Plate Bending 245

A.1.4.1 Reissner–Mindlin Theory 247

A.1.4.2 Kirchhoff Theory 248

A.2 Compatibility of Strains 250

A.2.1 Integrability Conditions 250

A.2.2 Enforcement of the Kinematic Boundary Conditions 251

A.3 General Elastodynamic Problem 252

References 252

B Computer Programs for Equilibrium Finite Element Formulations 254

B.1 Auxiliary Programs 255

B.1.1 gmsh 255

B.1.2 The mche and mchf Classes 258

B.1.3 mtimesx 258

B.2 Structure of the Programs 259

B.2.1 Definition of the Mesh 259

B.2.2 Definition of the Material Properties and Boundary Conditions 260

B.2.3 Definition of the Approximation Functions 261

B.2.4 Enforcement of Boundary Conditions 263

B.2.5 Processing the Solutions 265

B.2.6 Code Snippets 265

B.2.6.1 Computing the Flexibility Matrix of an Element 265

B.2.6.2 The Equilibrium Matrix of a Side of a Plane Element 267

References 269

Subject Index 271

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.