Norrie / De Vries The Finite Element Method

Fundamentals and Applications
1. Auflage 2014
ISBN: 978-1-4832-1891-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 336 Seiten, Web PDF

The Finite Element Method: Fundamentals and Applications demonstrates the generality of the finite element method by providing a unified treatment of fundamentals and a broad coverage of applications. Topics covered include field problems and their approximate solutions; the variational method based on the Hilbert space; and the Ritz finite element method. Finite element applications in solid and structural mechanics are also discussed. Comprised of 16 chapters, this book begins with an introduction to the formulation and classification of physical problems, followed by a review of field or continuum problems and their approximate solutions by the method of trial functions. It is shown that the finite element method is a subclass of the method of trial functions and that a finite element formulation can, in principle, be developed for most trial function procedures. Variational and residual trial function methods are considered in some detail and their convergence is examined. After discussing the calculus of variations, both in classical and Hilbert space form, the fundamentals of the finite element method are analyzed. The variational approach is illustrated by outlining the Ritz finite element method. The application of the finite element method to solid and structural mechanics is also considered. This monograph will appeal to undergraduate and graduate students, engineers, scientists, and applied mathematicians.

Autoren/Hrsg.

Norrie, Douglas H.

De Vries, Gerard

Inhaltsverzeichnis

1;Front Cover;1
2;The Finite Element Method;4
3;Copyright Page;5
4;Table of Contents;6
5;Dedication;5
6;PREFACE;12
7;ACKNOWLEDGMENTS;14
8;Chapter 1. The Formulation of Physical Problems;16
8.1;1.1 Introduction;16
8.2;1.2 Classification of Physical Problems;17
8.3;1.3 Classification of the Equations of a System;18
8.4;References;21
9;Chapter 2. Field Problems and Their Approximate Solutions;22
9.1;2.1 Formulation of Field (Continuous) Problems;22
9.2;2.2 Classification of Field Problems;26
9.3;2.3 Equilibrium Field Problems;28
9.4;2.4 Eigenvalue Field Problems;31
9.5;2.5 Propagation Field Problems;32
9.6;2.6 Summary of Governing Equations;32
9.7;2.7 Approximate Solution of Field
Problems;33
9.8;2.8 Trial Function Methods in Equilibrium Problems;36
9.9;2.9 Trial Function Methods in Eigenvalue Problems;44
9.10;2.10 Trial Function Methods in Propagation Problems;48
9.11;2.11 Accuracy, Stability, and Convergence;51
9.12;2.12 Approximate Solutions for Nonlinear Problems;56
9.13;2.13 The Extension to Vector Problems;56
9.14;2.14 The Finite Element Method;56
9.15;References;60
10;Chapter 3. The Variational Calculus and Its Application;63
10.1;3.1 Maxima and Minima of Functions;63
10.2;3.2 The Lagrange Multipliers;65
10.3;3.3 Maxima and Minima of Functionals;68
10.4;3.4 Variational Principles in Physical Phenomena;73
10.5;References;74
11;Chapter 4. The Variational Method Based on the Hilbert Space;76
11.1;4.1 The Hilbert Function Space;76
11.2;4.2 Equilibrium and Eigenvalue Problems;78
11.3;4.3 The Variational Solution of the Equilibrium Problem;79
11.4;4.4 Inhomogeneous Boundary Conditions;82
11.5;4.5 Natural Boundary Conditions;83
11.6;4.6 The Variational Solution of the Eigenvalue Problem;84
11.7;References;86
12;Chapter 5. Fundamentals of the Finite Element Approach;87
12.1;5.1 Classification of Finite Element Methods;87
12.2;5.2 The Finite Element Approximation;88
12.3;5.3 Elements and Their Shape Functions;94
12.4;5.4 Variational Finite Element Methods;111
12.5;5.5 Residual Finite Element Methods;117
12.6;5.6 The Direct Finite Element Method;123
12.7;5.7 Significant Features of a Finite Element Method;125
12.8;5.8 The Coefficient Finite Element Method;127
12.9;5.9 The Cell Finite Element Method;129
12.10;5.10 Convergence in the Finite Element Method;129
12.11;References;141
13;Chapter 6. The Ritz Finite Element Method (Classical);144
13.1;6.1 Statement of the Problem;144
13.2;6.2 The Equivalent Variational Problem;145
13.3;6.3 The Subdivision of the Region;146
13.4;6.4 The Element Shape Function;148
13.5;6.5 The Subdivision of the Functional;149
13.6;6.6 The Minimization Condition;151
13.7;6.7 The Element Matrix Equation;151
13.8;6.8 The System Matrix Equation;157
13.9;6.9 Insertion of the Dirichlet Boundary Condition;159
13.10;6.10 The Finite Element Approximation;160
13.11;6.11 The Two-Dimensional Region;161
13.12;6.12 Structural Formulations of the Finite Element Method;161
13.13;References;167
14;Chapter 7. The Ritz Finite Element Method (Hilbert Space);169
14.1;7.1 The Ritz Finite Element Method for the Equilibrium Problem;169
14.2;7.2 Rayleigh-Ritz Finite Element Solution for the Eigenvalue Problem;173
14.3;References;177
15;Chapter 8. Finite Element Applications in Solid and Structural Mechanics;178
15.1;8.1 The Solid Mechanics Formulation of the Finite Element Method;178
15.2;8.2 The Structural Formulation of the Finite Element Method;194
15.3;References;198
16;Chapter 9. The Laplace or Potential Field;200
16.1;9.1 The Laplace Equation;200
16.2;9.2 The Variational Formulation for the Laplace Field;207
16.3;9.3 The Ritz Finite Element Solution of the Laplace Field;208
16.4;9.4 Summary;221
16.5;References;222
17;Chapter 10. Laplace and Associated Boundary-Value Problems;223
17.1;10.1 The Potential Flow Field;223
17.2;10.2 The Electrostatic Field;234
17.3;10.3 The Thermal Conduction Field;234
17.4;10.4 Porous Media Flows;235
17.5;10.5 The Quasi-Harmonic Equation;236
17.6;10.6 The Poisson Equation;237
17.7;10.7 Unsteady Potential Fields (Moving Boundaries);237
17.8;10.8 Lifting Bodies with Appreciable Boundary Displacement;239
17.9;References;240
18;Chapter 11. The Helmholtz and Wave Equations;242
18.1;11.1 Physical Phenomena and the Helmholtz Equation;242
18.2;11.2 Physical Phenomena and the Wave Equation;245
18.3;References;247
19;Chapter 12. The Diffusion Equation;248
19.1;12.1 Forms of the Diffusion Equation;248
19.2;12.2 The Finite Element Solution of the Diffusion Equation;249
19.3;References;253
20;Chapter 13. Finite Element Applications to Viscous Flow;255
20.1;13.1 Oden and Somogyi;255
20.2;13.2 Tong;256
20.3;13.3 Baker;258
20.4;13.4 Leonard;259
20.5;13.5 Atkinson et al.;259
20.6;13.6 Reddi;259
20.7;13.7 Argyris and Scharpf;259
20.8;13.8 Other Formulations;260
20.9;References;260
21;Chapter 14. Finite Element Applications to Compressible Flow;261
21.1;14.1 Leonard;261
21.2;14.2 Gelder;262
21.3;14.3 De Vries, Berard, and Norrie;263
21.4;14.4 Reddi and Chu;265
21.5;14.5 Other Formulations;265
21.6;References;266
22;Chapter 15. Finite Element Applications to More General Fluid Flows;267
22.1;15.1 Skiba;267
22.2;15.2 Oden—I;268
22.3;15.3 Oden—II;269
22.4;15.4 Oden—III;269
22.5;15.5 De Vries and Norrie;270
22.6;15.6 Baker;270
22.7;15.7 Bramlette and Mallett;271
22.8;15.8 Other Formulations;271
22.9;References;271
23;Chapter 16. Other Finite Element Applications;273
23.1;16.1 Solid-Fluid Coupled Vibrations;273
23.2;16.2 Further Finite Element Applications;275
23.3;16.3 Further Developments;275
23.4;References;275
24;Appendix A. Matrix Algebra;278
24.1;A.1 Matrix Definitions;279
24.2;A.2 Matrix Algebra;282
24.3;A.3 Quadratic and Linear Forms;287
24.4;References;288
25;Appendix B. The Differential and Integral Calculus of Matrices;289
25.1;B.1 Definition of Differentiation and Integration of Matrices;289
25.2;B.2 Differentiation of a Function of a Matrix with Respect to the Matrix;291
25.3;B.3 Partial Differentiation of Matrices;292
25.4;B.4 Differentiation of Functions of Several Variables;294
25.5;References;295
26;Appendix C. The Transformation Matrix;296
26.1;C.1 The Functional Relationship between Coordinate Systems;297
26.2;C.2 The Local Transformation Matrix;297
26.3;C.3 The Linear Transformation;299
26.4;C.4 The Translation Matrix;300
26.5;C.5 The Rotation Matrix;305
26.6;C.6 Successive Transformations;307
26.7;C.7 Transformation of Matrices;309
26.8;C.8 Principal Axes, Diagonalization, and Eigenvalues;310
26.9;References;311
27;Additional References;312
27.1;A. Mathematical Methods;312
27.2;B. Variational Principles and Formulations;313
27.3;C. Finite Element Analysis;316
27.4;D. Finite Element Solutions of Physical Problems;318
27.5;E. Finite Element Computational Procedures;322
27.6;F. Input/Output Data Procedures;323
27.7;G. Large-Scale Finite Element Systems;323
28;AUTHOR INDEX;324
29;SUBJECT INDEX;331