The finite element method : an introduction with partial differential equations / A.J. Davies.

Format:

Book

Author/Creator:

Davies, Alan J.

Language:

English

Subjects (All):

Finite element method.

Physical Description:

ix, 297 pages : illustrations ; 25 cm

Edition:

Second edition.

Place of Publication:

Oxford ; New York : Oxford University Press, 2011.

Summary:

The physical world around us can be described in mathematical terms. One way to do so is with partial differential equations. The finite element method is a technique for solving these equations which is particularly useful for more difficult problems such as those involving surfaces with complicated geometry.

In the first instance, Davies develops the finite element method for the solution of Poisson's equation. Time-dependent and non-linear problems are solved next. The method is then extended to a weighted residual context and the relationship with the variational approach is also explained. There are worked examples throughout and each chapter has a set of exercises with detailed solutions.

Based on many years of experience of teaching the finite element method to a varied audience, this book constitutes a major revision of the first edition. It contains new chapters on the boundary element method and computational methods, as well as a new section in the Appendix explaining the form of the partial differential equations for a variety of practical applications. The careful, relatively informal approach makes this suitable as an introductory text for undergraduate mathematicians, engineers and physical scientists. Book jacket.

Contents:

1 Historical introduction 1

2 Weighted residual and variational methods 7

2.1 Classification of differential operators 7

2.2 Self-adjoint positive definite operators 9

2.3 Weighted residual methods 12

2.4 Extremum formulation: homogeneous boundary conditions 24

2.5 Non-homogeneous boundary conditions 28

2.6 Partial differential equations: natural boundary conditions 32

2.7 The Rayleigh-Ritz method 35

2.8 The 'elastic analogy' for Poisson's equation 44

2.9 Variational methods for time-dependent problems 48

2.10 Exercises and solutions 50

3 The finite element method for elliptic problems 71

3.1 Difficulties associated with the application of weighted residual methods 71

3.2 Piecewise application of the Galerkin method 72

3.3 Terminology 73

3.4 Finite element idealization 75

3.5 Illustrative problem involving one independent variable 80

3.6 Finite element equations for Poisson's equation 91

3.7 A rectangular element for Poisson's equation 102

3.8 A triangular element for Poisson's equation 107

3.9 Exercises and solutions 114

4 Higher-order elements: the isoparametric concept 141

4.1 A two-point boundary-value problem 141

4.2 Higher-order rectangular elements 144

4.3 Higher-order triangular elements 145

4.4 Two degrees of freedom at each node 147

4.5 Condensation of internal nodal freedoms 151

4.6 Curved boundaries and higher-order elements: isoparametric elements 153

4.7 Exercises and solutions 160

5 Further topics in the finite element method 171

5.1 The variational approach 171

5.2 Collocation and least squares methods 177

5.3 Use of Galerkin's method for time-dependent and non-linear problems 179

5.4 Time-dependent problems using variational principles which are not extremal 189

5.5 The Laplace transform 192

5.6 Exercises and solutions 199

6 Convergence of the finite element method 218

6.1 A one-dimensional example 218

6.2 Two-dimensional problems involving Poisson's equation 224

6.3 Isoparametric elements: numerical integration 226

6.4 Non-conforming elements: the patch test 228

6.5 Comparison with the finite difference method: stability 229

6.6 Exercises and solutions 234

7 The boundary element method 244

7.1 Integral formulation of boundary-value problems 244

7.2 Boundary element idealization for Laplace's equation 247

7.3 A constant boundary element for Laplace's equation 251

7.4 A linear element for Laplace's equation 256

7.5 Time-dependent problems 259

7.6 Exercises and solutions 261

8 Computational aspects 270

8.1 Pre-processor 270

8.2 Solution phase 271

8.3 Post-processor 274

8.4 Finite element method (FEM) or boundary element method (BEM)? 274.

Notes:

Includes bibliographical references and index.

ISBN:

0199609136

9780199609130

OCLC:

732847901

Publisher Number:

99947894732