Finite elements and approximation

목차 (Table of Contents)

• CONTENTS
• 1. CONTINUUM BOUNDARY VALUE PROBLEMS AND THE NEED FOR NUMERICAL DISCRETIZATION. FINITE DIFFERENCE METHODS = 1
• 1.1. Introduction = 1
• 1.2. Some Examples of Continuum Problems = 2
• 1.3. Finite Differences in One Dimension = 6
• 1.4. Derivative Boundary Conditions = 14
• 1.5. Nonlinear Problems = 18
• 1.6. Finite Differences in More Than One Dimension = 22
• 1.7. Problems Involving Irregularly Shaped Regions = 30
• 1.8. Nonlinear Problems in More Than One Dimension = 32
• 1.9. Approximation and Convergence = 33
• 1.10. Concluding Remarks = 34
• References = 36
• Suggested Further Reading = 37
• 2. WEIGHTED RESIDUAL METHODS: USE OF CONTINUOUS TRIAL FUNCTIONS = 38
• 2.1. Introduction Approximation by Trial Functions = 38
• 2.2. Weighted Residual Approximations = 42
• 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
• 2.4. Simultaneous Approximation to the Solutions of Differential Equations and to the Boundary Conditions = 57
• 2.5. Natural Boundary Conditions = 63
• 2.6. Boundary Solution Methods = 71
• 2.7. Systems of Differential Equations = 75
• 2.8. Nonlinear Problems = 89
• 2.9. Concluding Remarks = 93
• References = 93
• Suggested Further Reading = 94
• 3. PIECEWISE DEFINED TRIAL FUNCTIONS AND THE FINITE ELEMENT METHOD = 95
• 3.1. Introduction the Finite Element Concept = 95
• 3.2. Some Typical Locally Defined Narrow-Base Shape Functions = 96
• 3.3. Approximation to Solutions of Differential Equations and Continuity Requirements = 103
• 3.4. Weak Formulation and the Galerkin Method = 105
• 3.5. Some One-Dimensional Problems = 106
• 3.6. Standard Discrete System. A Physical Analogue of the Equation Assembly Process = 119
• 3.7. Generalization of the Finite Element Concepts for Two- and Three-Dimensional Problems = 126
• 3.8. The Finite Element Method for Two-Dimensional Heat Conduction Problems = 132
• 3.9. Two-Dimensional Elastic Stress Analysis Using Triangular Elements = 148
• 3.10. Are Finite Differences a Special Case of the Finite Element Method? = 154
• 3.11. Concluding Remarks = 157
• References = 160
• Suggested Further Reading = 160
• 4. HIGHER ORDER FINITE ELEMENT APPROXIMATION = 161
• 4.1. Introduction = 161
• 4.2. Degree of Polynomial in Trial Functions and Convergence Rates = 162
• 4.3. The Patch Test = 164
• 4.4. Standard Higher Order Shape Functions for One-Dimensional Elements with $$C^o$$ Continuity = 164
• 4.5. Hierarchical Forms of Higher Order One-Dimensional Elements with $$C^o$$ Continuity = 171
• 4.6. Two-Dimensional Rectangular Finite Element Shape Functions of Higher Order = 178
• 4.7. Two-Dimensional Shape Functions for Triangles = 185
• 4.8. Three-Dimensional Shape Functions = 190
• 4.9. Concluding Remarks = 190
• References = 192
• Suggested Further Reading = 192
• 5. MAPPING AND NUMERICAL INTEGRATION = 193
• 5.1. The Concept of Mapping = 193
• 5.2. Numerical Integration = 206
• 5.3. More on Mapping = 214
• 5.4. Mesh Generation and Concluding Remarks = 228
• References = 229
• Suggested Further Reading = 230
• 6. VARIATIONAL METHODS = 231
• 6.1. Introduction = 231
• 6.2. Variational Principles = 232
• 6.3. The Establishment of Natural Variational Principles = 236
• 6.4. Approximate Solution of Differential Equations by the Rayleigh-Ritz Method = 244
• 6.5. The Use of Lagrange Multipliers = 248 ^
• 6.6. General Variational Principles = 254
• 6.7. Penalty Functions = 256
• 6.8. Least-Squares Method = 259
• 6.9. Concluding Remarks = 264
• References = 265
• Suggested Further Reading = 265
• 7. PARTIAL DISCRETIZATION AND TIME-DEPENDENT PROBLEMS = 266
• 7.1. Introduction = 266
• 7.2. Partial Discretization Applied to Boundary Value Problems = 267
• 7.3. Time-Dependent Problems Via Partial Discretization = 270
• 7.4. Analytical Solution Procedures = 276
• 7.5. Finite Element Solution Procedures in the Time Domain = 283
• References = 307
• Suggested Further Reading = 308
• 8. GENERALIZED FINITE ELEMENTS, ERROR ESTIMATES, AND CONCLUDING REMARKS = 309
• 8.1. The Generalized Finite Element Method = 309
• 8.2. The Discretization Hrror in a Numerical Solution = 310
• 8.3. A Measure of Discretization Error = 311
• 8.4. Estimate of Discretization Error = 313
• 8.5. The State of the Art = 322
• References = 322
• Suggested Further Reading = 322
• INDEX = 323