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.

Auteur

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.

Contenu

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 ReissnerMindlin 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 ReissnerMindlin Plates 99

5.4.1 A Single Triangular ReissnerMindlin Element 100 5.4.2 A Pa...