Finite Elements and Approximation

A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises. 1983 edition.

1. CONTINUUM BOUNDARY VALUE PROBLEMS AND THE NEED FOR NUMERICAL DISCRETIZATION. FINITE DIFFERENCE METHODS

1

(37)

1.1. Introduction,

1

(1)

1.2. Some Examples of Continuum Problems,

2

(4)

1.3. Finite Differences in One Dimension,

6

(8)

1.4. Derivative Boundary Conditions,

14

(4)

1.5. Nonlinear Problems,

18

(4)

1.6. Finite Differences in More Than One Dimension,

22

(8)

1.7. Problems Involving Irregularly Shaped Regions,

30

(2)

1.8. Nonlinear Problems in More Than One Dimension,

32

(1)

1.9. Approximation and Convergence,

33

(1)

1.10. Concluding Remarks,

34

(2)

References,

36

(1)

Suggested Further Reading,

37

(1)

2. WEIGHTED RESIDUAL METHODS: USE OF CONTINUOUS TRIAL FUNCTIONS

38

(57)

2.1. Introduction—Approximation by Trial Functions,

38

(4)

2.2. Weighted Residual Approximations,

42

(7)

2.3. Approximation to the Solutions of Differential Equations and the Use of Trial Function-Weighted Residual Forms. Boundary Conditions Satisfied by Choice of Trial Functions,

49

(8)

2.4. Simultaneous Approximation to the Solutions of Differential Equations and to the Boundary Conditions,

57

(6)

2.5. Natural Boundary Conditions,

63

(8)

2.6. Boundary Solution Methods,

71

(4)

2.7. Systems of Differential Equations,

75

(14)

2.8. Nonlinear Problems,

89

(4)

2.9. Concluding Remarks,

93

(1)

References,

93

(1)

Suggested Further Reading,

94

(1)

3. PIECEWISE DEFINED TRIAL FUNCTIONS AND THE FINITE ELEMENT METHOD

95

(66)

3.1. Introduction—The Finite Element Concept,

95

(1)

3.2. Some Typical Locally Defined Narrow-Base Shape Functions,

96

(7)

3.3. Approximation to Solutions of Differential Equations and Continuity Requirements,

103

(2)

3.4. Weak Formulation and the Galerkin Method,

105

(1)

3.5. Some One-Dimensional Problems,

106

(13)

3.6. Standard Discrete System. A Physical Analogue of the Equation Assembly Process,

119

(7)

3.7. Generalization of the Finite Element Concepts for Two- and Three-Dimensional Problems,

126

(6)

3.8. The Finite Element Method for Two-Dimensional Heat Conduction Problems,

132

(16)

3.9. Two-Dimensional Elastic Stress Analysis Using Triangular Elements,

148

(6)

3.10. Are Finite Differences a Special Case of the Finite Element Method?,

154

(3)

3.11. Concluding Remarks,

157

(3)

References,

160

(1)

Suggested Further Reading,

160

(1)

4. HIGHER ORDER FINITE ELEMENT APPROXIMATION

161

(32)

4.1. Introduction,

161

(1)

4.2. Degree of Polynomial in Trial Functions and Convergence Rates,

162

(2)

4.3. The Patch Test,

164

(1)

4.4. Standard Higher Order Shape Functions for One-Dimensional Elements with C° Continuity,

164

(7)

4.5. Hierarchical Forms of Higher Order One-Dimensional Elements with C° Continuity,

171

(7)

4.6. Two-Dimensional Rectangular Finite Element Shape Functions of Higher Order,

178

(7)

4.7. Two-Dimensional Shape Functions for Triangles,

185

(5)

4.8. Three-Dimensional Shape Functions,

190

(1)

4.9. Concluding Remarks,

190

(2)

References,

192

(1)

Suggested Further Reading,

192

(1)

5. MAPPING AND NUMERICAL INTEGRATION

193

(38)

5.1. The Concept of Mapping,

193

(13)

5.2. Numerical Integration,

206

(8)

5.3. More on Mapping,

214

(14)

5.4. Mesh Generation and Concluding Remarks,

228

(1)

References,

229

(1)

Suggested Further Reading,

230

(1)

6. VARIATIONAL METHODS

231

(35)

6.1. Introduction,

231

(1)

6.2. Variational Principles,

232

(4)

6.3. The Establishment of Natural Variational Principles,

236

(8)

6.4. Approximate Solution of Differential Equations by the Rayleigh—Ritz Method,

244

(4)

6.5. The Use of Lagrange Multipliers,

248

(6)

6.6. General Variational Principles,

254

(2)

6.7. Penalty Functions,

256

(3)

6.8. Least-Squares Method,

259

(5)

6.9. Concluding Remarks,

264

(1)

References,

265

(1)

Suggested Further Reading,

265

(1)

7. PARTIAL DISCRETIZATION AND TIME-DEPENDENT PROBLEMS

266

(43)

7.1. Introduction,

266

(1)

7.2. Partial Discretization Applied to Boundary Value Problems,

267

(3)

7.3. Time-Dependent Problems Via Partial Discretization,

270

(6)

7.4. Analytical Solution Procedures,

276

(7)

7.5. Finite Element Solution Procedures in the Time Domain,

283

(24)

References,

307

(1)

Suggested Further Reading,

308

(1)

8. GENERALIZED FINITE ELEMENTS, ERROR ESTIMATES, AND CONCLUDING REMARKS

309

(14)

8.1. The Generalized Finite Element Method,

309

(1)

8.2. The Discretization Error in a Numerical Solution,

310

(1)

8.3. A Measure of Discretization Error,

311

(2)

8.4. Estimate of Discretization Error,

313

(9)

8.5. The State of the Art,

322

(1)

References,

322

(1)

Suggested Further Reading,

322

(1)

INDEX

323

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