The Finite Element Method: An Introduction with Partial Differential Equations

Cover......Page 1
Contents......Page 8
1 Historical introduction......Page 12
2.1 Classification of differential operators......Page 18
2.2 Self-adjoint positive definite operators......Page 20
2.3 Weighted residual methods......Page 23
2.4 Extremum formulation: homogeneous boundary conditions......Page 35
2.5 Non-homogeneous boundary conditions......Page 39
2.6 Partial differential equations: natural boundary conditions......Page 43
2.7 The Rayleigh–Ritz method......Page 46
2.8 The ‘elastic analogy’ for Poisson’s equation......Page 55
2.9 Variational methods for time-dependent problems......Page 59
2.10 Exercises and solutions......Page 61
3.1 Difficulties associated with the application of weighted residual methods......Page 82
3.2 Piecewise application of the Galerkin method......Page 83
3.3 Terminology......Page 84
3.4 Finite element idealization......Page 86
3.5 Illustrative problem involving one independent variable......Page 91
3.6 Finite element equations for Poisson’s equation......Page 102
3.7 A rectangular element for Poisson’s equation......Page 113
3.8 A triangular element for Poisson’s equation......Page 118
3.9 Exercises and solutions......Page 125
4.1 A two-point boundary-value problem......Page 152
4.2 Higher-order rectangular elements......Page 155
4.3 Higher-order triangular elements......Page 156
4.4 Two degrees of freedom at each node......Page 158
4.5 Condensation of internal nodal freedoms......Page 162
4.6 Curved boundaries and higher-order elements: isoparametric elements......Page 164
4.7 Exercises and solutions......Page 171
5.1 The variational approach......Page 182
5.2 Collocation and least squares methods......Page 188
5.3 Use of Galerkin’s method for time-dependent and non-linear problems......Page 190
5.4 Time-dependent problems using variational principles which are not extremal......Page 200
5.5 The Laplace transform......Page 203
5.6 Exercises and solutions......Page 210
6.1 A one-dimensional example......Page 229
6.2 Two-dimensional problems involving Poisson’s equation......Page 235
6.3 Isoparametric elements: numerical integration......Page 237
6.4 Non-conforming elements: the patch test......Page 239
6.5 Comparison with the finite difference method: stability......Page 240
6.6 Exercises and solutions......Page 245
7.1 Integral formulation of boundary-value problems......Page 255
7.2 Boundary element idealization for Laplace’s equation......Page 258
7.3 A constant boundary element for Laplace’s equation......Page 262
7.4 A linear element for Laplace’s equation......Page 267
7.5 Time-dependent problems......Page 270
7.6 Exercises and solutions......Page 272
8.1 Pre-processor......Page 281
8.2 Solution phase......Page 282
8.4 Finite element method (FEM) or boundary element method (BEM)?......Page 285
A.1 Parabolic problems......Page 287
A.2 Elliptic problems......Page 288
A.3 Hyperbolic problems......Page 289
A.4 Initial and boundary conditions......Page 290
Appendix B: Some integral theorems of the vector calculus......Page 291
Appendix C: A formula for integrating products of area coordinates over a triangle......Page 293
D.2 Two-dimensional Gauss quadrature......Page 295
D.3 Logarithmic Gauss quadrature......Page 296
Appendix E: Stehfest’s formula and weights for numerical Laplace transform inversion......Page 298
References......Page 299
F......Page 306
P......Page 307
Y......Page 308