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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 1ý7. Problems Involving Irregularly Shaped Regions, 30 . Nonlinear Problems in More Than One Dimension, 32 .9ý. 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-Appooximation 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 2.4ý 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 2.7T Systems of Differential Equations, 75 28. Nonlinear Problems, 89 29. Concluding Remarks, 93 References, 93 Suggested Further Reading, 94 3 PIECEWIISE DEFINED TRIAL FUNCTIONS AND THE FINITE ELEMENT METHOD 95 3,. Introduction-The Finite Element Concept, 95 32. Some Typical Locally Defined Narrow-Base Shape Functions, 96 3.3. Approximation to Solutions of Differential Equations and Continuity Requirements, 103 A4. Weak Formulation and the Galerkin Method, 105 1 .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 38. The Finite Element Method for Two-Dimensional Heat Conduction Problems, 132 39. Two-Dimensional Elastic Stress Analysis Using Triangular Elements, 148 3.10. Are Finite Differences a Special Case of the Finite Eiement Method?, 154 S11i. Clonluding Remarks, 157 References, 160 Suggested Further Reading, 160 4, HIGHER ORDER FINITE ELEMENT APPROXIMATION 161 4. . 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 CO Continuity, 164 4,5. Hierarchical Forms of Higher Order One-Dimensional Elements "with C' Continuity, 171 4.6. Two-DIimensional 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 6.13 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. L 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 81. The Generalized Finite Element Method, 309 8.2 The Discretization Error in a Numerical Solution, 310 h8,3. A Measure of Discretization Error, 311 8 d4 Estimate of Discretization Error, 313 8,5 'The State of the Art, 322 References, 322 Suggested Further Reading, 322

Library of Congress subject headings for this publication: Approximation theory, Finite element method