Finite Element Analysis with Error Estimators

Finite Element Analysis with Error Estimators
An Introduction to the FEM and Adaptive Error Analysis for Engineering Students
J. E. Akin
Contents
Preface . xi
Notation xiii
1. Introduction . 1
1.1 Finite element methods . 1
1.2 Capabilities of FEA . 3
1.3 Outline of finite element procedures . 8
1.4 Assembly into the system equations . 12
1.5 Error concepts . 21
1.6 Exercises . 22
1.7 Bibliography . 24
2. Mathematical preliminaries . 26
2.1 Introduction . 26
2.2 Linear spaces and norms . 28
2.3 Sobolev norms* 29
2.4 Dual problems, self-adjointness 29
2.5 Weighted residuals 31
2.6 Boundary condition terms . 35
2.7 Adding more unknowns 39
2.8 Numerical integration . 39
2.9 Integration by parts . 41
2.10 Finite element model problem 41
2.11 Continuous nodal flux recovery 56
2.12 A one-dimensional example error analysis . 59
2.13 General boundary condition choices . 67
2.14 General matrix partitions 69
2.15 Elliptic boundary value problems . 70
2.16 Initial value problems . 77
2.17 Eigen-problems . 80vi Contents
2.18 Equivalent forms* . 83
2.19 Exercises . 86
2.20 Bibliography . 90
3. Element interpolation and local coordinates 92
3.1 Introduction . 92
3.2 Linear interpolation . 92
3.3 Quadratic interpolation . 96
3.4 Lagrange interpolation 97
3.5 Hermitian interpolation . 98
3.6 Hierarchial interpolation . 101
3.7 Space-time interpolation* . 106
3.8 Nodally exact interpolations* . 106
3.9 Interpolation error* . 107
3.10 Gradient estimates* 110
3.11 Exercises . 113
3.12 Bibliography . 115
4. One-dimensional integration . 116
4.1 Introduction . 116
4.2 Local coordinate Jacobian . 116
4.3 Exact polynomial integration* 117
4.4 Numerical integration . 119
4.5 Variable Jacobians . 123
4.6 Exercises . 126
4.7 Bibliography . 126
5. Error estimates for elliptic problems 127
5.1 Introduction . 127
5.2 Error estimates 131
5.3 Hierarchical error indicator . 132
5.4 Flux balancing error estimates 136
5.5 Element adaptivity 138
5.6 H adaptivity . 139
5.7 P adaptivity . 139
5.8 HP adaptivity 140
5.9 Exercises . 141
5.10 Bibliography . 143Contents vii
6. Super-convergent patch recovery 146
6.1 Patch implementation database . 146
6.2 SCP nodal flux averaging 158
6.3 Computing the SCP element error estimate 164
6.4 Hessian matrix* 166
6.5 Exercises . 176
6.6 Bibliography . 176
7. Variational methods 178
7.1 Introduction . 178
7.2 Structural mechanics . 179
7.3 Finite element analysis 180
7.4 Continuous elastic bar . 185
7.5 Thermal loads on a bar* . 192
7.6 Reaction flux recovery for an element* 196
7.7 Heat transfer in a rod . 199
7.8 Element validation* 202
7.9 Euler’s equations of variational calculus* 208
7.10 Exercises . 210
7.11 Bibliography . 213
8. Cylindrical analysis problems 215
8.1 Introduction . 215
8.2 Heat conduction in a cylinder . 215
8.3 Cylindrical stress analysis . 225
8.4 Exercises . 229
8.4 Bibliography . 229
9. General interpolation 231
9.1 Introduction . 231
9.2 Unit coordinate interpolation 231
9.3 Natural coordinates . 238
9.4 Isoparametric and subparametric elements . 239
9.5 Hierarchical interpolation . 247
9.6 Differential geometry* 252
9.7 Mass properties* . 256
9.9 Interpolation error* . 257
9.9 Element distortion . 258
9.10 Space-time interpolation* . 260
9.11 Exercises . 262
9.12 Bibliography . 263viii Contents
10. Integration methods 265
10.1 Introduction . 265
10.2 Unit coordinate integration . 265
10.3 Simplex coordinate integration . 267
10.4 Numerical integration . 270
10.5 Typical source distribution integrals* . 273
10.6 Minimal, optimal, reduced and selected integration* . 276
10.7 Exercises . 279
10.8 Bibliography . 280
11. Scalar fields 281
11.1 Introduction . 281
11.2 Variational formulation . 281
11.3 Element and boundary matrices 284
11.4 Linear triangle element . 289
11.5 Linear triangle applications . 291
11.6 Bilinear rectangles* 316
11.7 General 2-d elements 318
11.8 Numerically integrated arrays . 319
11.9 Strong diagonal gradient SCP test case 322
11.10 Orthtropic conduction . 337
11.11 Axisymmetric conduction . 344
11.12 Torsion 350
11.13 Introduction to linear flows . 358
11.14 Potential flow 358
11.15 Axisymmetric plasma equilibria* . 365
11.16 Slider bearing lubrication 370
11.17 Transient scalar fields 377
11.18 Exercises . 381
11.19 Bibliography . 382
12. Vector fields . 384
12.1 Introduction . 384
12.2 Displacement based stress analysis . 384
12.3 Planar models 389
12.4 Matrices for the constant strain triangle 395
12.5 Stress and strain transformations* 407
12.6 Axisymmetric solid stress* . 412
12.7 General solid stress* . 413
12.8 Anisotropic materials* 413Contents ix
12.9 Circular hole in an infinite plate 416
12.10 Dynamics of solids 428
12.11 Exercises . 435
12.11 Bibliography . 435
Index .
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