Finite Elements

Notation xi

1 COMPUTATIONAL ENGINEERING SCIENCE 1

1.1 Engineering simulation 1

1.2 A problem solving environment 2

1.3 Problem statements in engineering 4

1.4 Decisions on forming WS^N 6

1.5 Discrete approximate WS^h implementation 8

1.6 Chapter summary 9

1.7 Chapter references 10

2 PROBLEM STATEMENTS 11

2.1 Engineering simulation 11

2.2 Continuum mechanics viewpoint 12

2.3 Continuum conservation law forms 12

2.4 Constitutive closure for conservation law PDEs 14

2.5 Engineering science continuum mechanics 18

2.6 Chapter references 20

3 SOME INTRODUCTORY MATERIAL 21

3.1 Introduction 21

3.2 Multi-dimensional PDEs, separation of variables 22

3.3 Theoretical foundations, GWS^h 27

3.4 A legacy FD construction 28

3.5 An FD approximate solution 30

3.6 Lagrange interpolation polynomials 31

3.7 Chapter summary 32

3.8 Exercises 34

3.9 Chapter references 34

4 HEAT CONDUCTION35

4.1 A steady heat conduction example 35

4.2 Weak form approximation, error minimization 37

4.3 GWS^N discrete implementation, FE basis38

4.4 Finite element matrix statement 41

4.5 Assembly of {WS}_e to form algebraic GWS^h 43

4.6 Solution accuracy, error distribution 45

4.7 Convergence, boundary heat flux 47

4.8 Chapter summary 47

4.9 Exercises 48

4.10 Chapter reference 48

5 STEADY HEAT TRANSFER, n =149

5.1 Introduction 49

5.2 Steady heat transfer, n = 1 50

5.3 FE k = 1 trial space basis matrix library 52

5.4 Object-oriented GWS^h programming 55

5.5 Higher completeness degree trial space bases58

5.6 Global theory, asymptotic error estimate 62

5.7 Non-smooth data, theory generalization 66

5.8 Temperature dependent conductivity, non-linearity 69

5.9 Static condensation, p-elements 72

5.10 Chapter summary 75

5.11 Exercises 76

5.12 Computer labs 77

5.13 Chapter references 78

6 ENGINEERING SCIENCES, n =1 79

6.1 Introduction 79

6.2 The Euler-Bernoulli beam equation 80

6.3 Euler-Bernoulli beam theory GWS^h reformulation 85

6.4 The Timoshenko beam theory 92

6.5 Mechanical vibrations of a beam 99

6.6 Fluid mechanics, potential flow 106

6.7 Electromagnetic plane wave propagation110

6.8 Convective-radiative finned cylinder heat transfer 112

6.9 Chapter summary 120

6.10 Exercises122

6.10 Computer labs 123

6.11 Chapter references 124

> 1 125

7.1 Introduction 125

7.2 Multi-dimensional FE bases and DOF 126

7.3 Multi-dimensional FE operations 129

7.4 The NC k = 1,2 basis FE matrix library 132

7.5 NC basis {WS}_e template, accuracy, convergence 136

7.6 The tensor product basis element family 139

7.7 Gauss numerical quadrature, k = 1 TP basis library 141

7.8 Convection-radiation BC GWS^h implementation 146

7.9 Linear basis GWS^h template unification 150

7.10 Accuracy, convergence revisited 152

7.11 Chapter summary 153

7.12 Exercises155

7.13 Computer labs 155

7.14 Chapter references 156

8 FINITE DIFFERENCES OF OPINION 159

8.1 The FD-FE correlation159

8.2 The FV-FE correlation162

8.3 Chapter summary 167

8.4 Exercises168

9 CONVECTION-DIFFUSION, n = 1 169

9.1 Introduction169

9.2 The Galerkin weak statement 170

9.3 GWS^h completion for time dependence172

9.4 GWS^h + qTS algorithm templates 173

9.5 GWS^h + qTS algorithm asymptotic error estimates 175

9.6 Performance verification test cases 177

9.7 Dispersive error characterization 180

9.8 A modified Galerkin weak statement 184

9.9 Verification problem statements revisited 187

9.10 Unsteady heat conduction 190

9.11 Chapter summary 193

9.12 Exercises 193

9.13 Computer labs 194

9.14 Chapter references 195

> 1 197

10.1 The problem statement 197

10.2 GWS^h + qTS formulation reprise 198

10.3 Matrix library additions, templates 200

10.4 mPDE Galerkin weak forms, theoretical analyses 202

10.5 Verification, benchmarking and validation 207

10.6 Mass transport, the rotating cone verification 208

10.7 The gaussian plume benchmark 211

10.8 The steady n-D Peclet problem verification 213

10.9 Mass transport, a validated n = 3 experiment 215

10.10 Numerical linear algebra, matrix iteration 222

10.11 Newton and AF TP jacobian templates 227

10.12 Chapter summary 229

10.13 Exercises231

10.14 Computer labs 231

10.15 Chapter references232

> 1 235

11.1 Introduction 235

11.2 Structural mechanics236

11.3 Structural mechanics, virtual work FE form 240

11.4 Plane stress/strain, GWS^h implementation 242

11.5 Elasticity computer lab 246

11.6 Fluid mechanics, incompressible-thermal flow 251

11.7 Vorticity-streamfunction GWS^h + qTS algorithm 254

11.8 An isothermal INS validation experiment 258

11.9 Multi-mode convection heat transfer262

11.10 Mechanical vibrations, normal mode GWS^h 267

11.11 Normal modes of a vibrating membrane270

11.12 Multi-physics solid-fluid mass transport 276

11.13 Chapter summary 280

11.14 Exercises 282

11.15 Computer labs283

11.14 Chapter references 284

12 CONCLUSION 287

Index 289