Table of contents for Finite elements and approximation / O.C. Zienkiewicz, K. Morgan.


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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
17.  Problems Involving Irregularly Shaped Regions, 30
. Nonlinear Problems in More Than One Dimension, 32
.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-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
2.4  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
2.7T  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