Finite element analysis with error estimators: an introduction to the FEM and adaptive error analysis for engineering students

J. E. Akin0750667222, 9780750667227, 9780080472751

This key text is written for senior undergraduate and graduate engineering students. It delivers a complete introduction to finite element methods and to automatic adaptation (error estimation) that will enable students to understand and use FEA as a true engineering tool. It has been specifically developed to be accessible to non-mathematics students and provides the only complete text for FEA with error estimators for non-mathematicians. Error estimation is taught on nearly half of all FEM courses for engineers at senior undergraduate and postgraduate level; no other existing textbook for this market covers this topic. * The only introductory FEA text with error estimation for students of engineering, scientific computing and applied mathematics * Includes source code for creating and proving FEA error estimators * Complete with homework exercises and supporting website with instructor’s solutions manual

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Table of contents :
Cover……Page 1
Half Title Page……Page 2
Title Page……Page 3
Copyright……Page 4
Contents……Page 5
Preface……Page 11
Features of the text and accompanying resources……Page 12
Notation……Page 13
1.1 Finite element methods……Page 19
1.2 Capabilities of FEA……Page 21
1.3 Outline of finite element procedures……Page 26
1.4 Assembly into the system equations……Page 30
1.5 Error concepts……Page 39
1.6 Exercises……Page 40
1.7 Bibliography……Page 42
2.1 Introduction……Page 44
2.2 Linear spaces and norms……Page 46
2.4 Dual problem, self-adjointness……Page 47
2.5 Weighted residuals……Page 49
2.6 Boundary condition terms……Page 53
2.8 Numerical integration……Page 57
2.10 Finite element model problem……Page 59
2.11 Continuous nodal flux recovery……Page 74
2.12 A one-dimensional example error analysis……Page 77
2.13 General boundary condition choices……Page 85
2.14 General matrix partitions……Page 87
2.15 Elliptic boundary value problems……Page 88
2.16 Initial value problems……Page 95
2.17 Eigen-problems……Page 98
2.18 Equivalent forms……Page 101
2.19 Exercises……Page 104
2.20 Bibliography……Page 108
3.2 Linear interpolation……Page 110
3.3 Quadratic interpolation……Page 114
3.4 Lagrange interpolation……Page 115
3.5 Hermitian interpolation……Page 116
3.6 Hierarchical interpolation……Page 119
3.8 Nodally exact interpolations……Page 124
3.9 Interpolation error……Page 125
3.10 Gradient estimates……Page 128
3.11 Exercises……Page 131
3.12 Bibliography……Page 133
4.2 Local coordinate Jacobian……Page 134
4.3 Exact polynomial integration……Page 135
4.4 Numerical integration……Page 137
4.5 Variable Jacobians……Page 141
4.7 Bibliography……Page 144
5.1 Introduction……Page 145
5.2 Error estimates……Page 149
5.3 Hierarchical error indicator……Page 150
5.4 Flux balancing error estimates……Page 154
5.5 Element adaptivity……Page 156
5.7 P-adaptivity……Page 157
5.8 HP-adaptivity……Page 158
5.9 Exercises……Page 159
5.10 Bibliography……Page 161
6.1 Patch implementation database……Page 164
6.2 SCP nodal flux averaging……Page 176
6.3 Computing the SCP element error estimates……Page 182
6.4 Hessian matrix……Page 184
6.6 Bibliography……Page 194
7.1 Introduction……Page 196
7.2 Structural mechanics……Page 197
7.3 Finite element analysis……Page 198
7.4 Continuous elastic bar……Page 203
7.5 Thermal loads on a bar……Page 210
7.6 Reaction flux recovery for an element……Page 214
7.7 Heat transfer in a rod……Page 217
7.8 Element validation……Page 220
7.9 Euler’s equations of variational calculus……Page 226
7.10 Exercises……Page 228
7.11 Bibliography……Page 231
8.2 Heat conduction in a cylinder……Page 233
8.3 Cylindrical stress analysis……Page 243
8.4 Exercises……Page 247
8.5 Bibliography……Page 248
9.2 Unit coordinate interpolation……Page 249
9.3 Natural coordinates……Page 256
9.4 Isoparametric and subparametric elements……Page 257
9.5 Hierarchical interpolation……Page 265
9.6 Differential geometry……Page 270
9.7 Mass properties……Page 274
9.8 Interpolation error……Page 275
9.9 Element distortion……Page 276
9.10 Space-time interpolation……Page 278
9.11 Exercises……Page 280
9.12 Bibliography……Page 281
10.2 Unit coordinate integration……Page 283
10.3 Simplex coordinate integration……Page 285
10.4 Numerical integration……Page 288
10.5 Typical source distribution integrals……Page 291
10.6 Minimal, optimal, reduced and selected integration……Page 294
10.7 Exercises……Page 297
10.8 Bibliography……Page 298
11.2 Variational formulation……Page 299
11.3 Element and boundary matrices……Page 303
11.4 Linear triangular element……Page 307
11.5 Linear triangle applications……Page 309
11.6 Bilinear rectangles……Page 334
11.7 General 2-d elements……Page 336
11.8 Numerically integrated arrays……Page 337
11.9 Strong diagonal gradient SCP test case……Page 340
11.10 Orthotropic conduction……Page 355
11.11 Axisymmetric conductions……Page 362
11.12 Torsion……Page 368
11.14 Potential flow……Page 376
11.15 Axisymmetric plasma equilibria……Page 383
11.16 Slider bearing lubrication……Page 388
11.17 Transient scalar fields……Page 395
11.18 Exercises……Page 399
11.19 Bibliography……Page 400
12.2 Displacement based stress analysis summary……Page 402
12.3 Planar models……Page 407
12.4 Matrices for the constant strain triangle (CST)……Page 413
12.5 Stress and strain transformations……Page 425
12.6 Axisymmetric solid stress……Page 430
12.8 Anisotropic materials……Page 431
12.9 Circular hole in an infinite plate……Page 434
12.10 Dynamics of solids……Page 446
12.12 Bibliography……Page 453
Index……Page 455