Numerical methods in finite element

Bathe K-J.,Wilson E.L.Numerical methods in finite element.Prentice-Hall civil engineering and engineering mechanics series analysis (P-H,1976)(ISBN 0136271901)(600dpi)(544p) ......Page 3
Copyright iv ......Page 4
Contents v ......Page 5
Preface xi ......Page 11
Part I MATRICES AND LINEAR ALGEBRA, 1 ......Page 16
1.1 Introduction, 3 ......Page 18
1.2 Introduction to Matrices, 4 ......Page 19
1.3 Special Matrices, 5 ......Page 20
1.4 Matrix Equality, Addition, and Multiplication by a Scalar, 8 ......Page 23
1.5 Multiplication of Matrices, 10 ......Page 25
1.6 The Inverse Matrix, 14 ......Page 29
1.7 Partitioning of Matrices, 17 ......Page 32
1.8 The Trace and Determinant of a Matrix, 19 References, 22 ......Page 37
2.1 Introduction, 23 ......Page 38
2.2 Vector Spaces, Subspaces, and the Span of a Matrix, 24 ......Page 39
2.3 Matrix Representation of Linear Transformation, 31 ......Page 46
2.4 Change of Basis, 34 ......Page 49
2.5 Matrix Representation of Variational Formulation, 37 ......Page 52
2.6 The Eigenproblem Ay = 2v, with A Being a Symmetric Matrix, 45 ......Page 60
2.7 The Rayleigh Quotient and the Minimax Characterization of Eigenvalues, 57 ......Page 72
2.8 Vector and Matrix Norms, 63 References, 68 ......Page 83
Part II THE FINITE ELEMENT METHOD, 69 ......Page 84
3.1 Introduction, 71 ......Page 86
3.2 Formulation of the Finite Element Method Using the Principle of Virtual Displacements, 82 ......Page 97
3.2.1 Displacement and Strain-Displacement Transformation Matrices for Plane Stress Analysis, 83 ......Page 98
3.2.2 General Formulation, 87 ......Page 102
3.2.3 Lumping of Structure Properties and Loads, 96 ......Page 111
3.2.4 Specialization of the General Formulation, 97 ......Page 112
3.2.5 Requirements for Monotonic Convergence, 100 ......Page 115
3.3 Derivation of Generalized Coordinate Finite Element Models, 106 ......Page 121
3.3.1 General Derivation and Specific Examples, 107 ......Page 122
3.3.2 Spatial Isotropy, 119 References, 120 ......Page 135
4.1 Introduction, 124 ......Page 139
4.2 Isoparametric Derivation of Bar Element Stiffness Matrix, 125 ......Page 140
4.3.1 Formulation of Isoparametric Finite Element Matrices in Local Coordinate Systems, 127 ......Page 142
4.3.2 Element Matrices in Global Coordinate System, 142 ......Page 157
4.4 Convergence Considerations, 144 ......Page 159
4.5 Associated Element Families, 147 ......Page 162
4.6 Numerical Integration, 151 ......Page 166
4.7 Practical Considerations in Isoparametric Element Calculations, 162 ......Page 177
4.8 Computer Program Implementation of Isoparametric Finite Elements, 166 ......Page 181
References, 169 ......Page 184
5.1 Introduction, 172 ......Page 187
5.2 Variational Formulation of Structural Mechanics Problems, 173 ......Page 188
5.3 Ritz Solution, 176 ......Page 191
5.4 Formulation of Field Problems— Example: Heat Transfer Analysis, 184 ......Page 199
5.5 Nonconforming, Mixed, and Hybrid Finite Element Models; Finite Difference Differential and Energy Methods, 188 ......Page 203
References, 199 ......Page 214
6.1 Introduction, 203 ......Page 218
6.2.1 Nodal Point and Element Information Read-In, 205 ......Page 220
6.2.2 Calculation of Element Stiffness, Mass, and Equivalent Nodal Loads, 208 ......Page 223
6.2.3 Assemblage of Structure Matrices, 209 ......Page 224
6.3 Calculation of Element Stresses, 213 ......Page 228
6.4.1 Data Input to Computer Program STAP, 214 ......Page 229
6.4.2 Listing of Program STAP, 222 References, 235 ......Page 250
Part III SOLUTION OF FINITE ELEMENT EQUILIBRIUM EQUATIONS, 237 ......Page 252
7.1 Introduction, 239 ......Page 254
7.2.1 Introduction to Gauss Elimination, 241 ......Page 256
7.2.2 The Gauss Elimination Solution, 246 ......Page 261
7.2.3 Computer Implementation of Gauss Elimination, 249 ......Page 264
7.2.4 Cholesky Factorization, Static Condensation, Substructures, and Frontal Solution, 258 ......Page 273
7.2.5 Solution of Equations with Symmetric Nonpositive Definite Coefficient Matrices, 268 ......Page 283
7.3 Direct Solutions Using Orthogonal Matrices, 282 ......Page 297
7.3.1 The Givens Factorization, 283 ......Page 298
7.3.2 The Householder Factorization, 287 ......Page 302
7.4 The Gauss-Seidel Iterative Solution, 291 ......Page 306
7.5 Solution Errors, 294 References, 303 ......Page 318
8.1 Introduction, 308 ......Page 323
8.2 Direct Integration Methods, 309 ......Page 324
8.2.1 The Central Difference Method, 310 ......Page 325
8.2.2 The Houbolt Method, 316 ......Page 331
8.2.3 The Wilson 0 Method, 319 ......Page 334
8.2.4 The Newmark Method, 322 ......Page 337
8.3 Mode Superposition, 326 ......Page 341
8.3.1 Change of Basis to Modal Generalized Displacements, 327 ......Page 342
8.3.2 Analysis with Damping Neglected, 330 ......Page 345
8.3.3 Analysis with Damping Included, 337 ......Page 352
References, 343 ......Page 358
9.1 Introduction, 345 ......Page 360
9.2 Direct Integration Approximation and Load Operators, 347 ......Page 362
9.2.2 The Houbolt Method, 348 ......Page 363
9.2.3 The Wilson 9 Method, 349 ......Page 364
9.2.4 The Newmark Method, 350 ......Page 365
9.3 Stability Analysis, 351 ......Page 366
9.4 Accuracy Analysis, 355 References, 361 ......Page 376
10.1 Introduction, 363 ......Page 378
10.2.1 Properties of the Eigenvectors, 366 ......Page 381
10.2.2 The Characteristic Polynomials of the Eigenproblem Kf = aMf and of Its Associated Constraint Problems, 371 ......Page 386
10.2.3 Shifting, 378 ......Page 393
10.2.4 Effect of Zero Mass, 379 ......Page 394
10.2.5 Transformation of the Generalized Eigenproblem Kf = aMf  to a Standard Form, 381 ......Page 396
10.3 Approximate Solution Techniques, 387 ......Page 402
10.3.1 Static Condensation, 388 ......Page 403
10.3.2 Rayleigh-Ritz Analysis, 394 ......Page 409
10.3.3 Component Mode Synthesis, 404 ......Page 419
10.4 Solution Errors, 408 References, 414 ......Page 429
11.1 Introduction, 417 ......Page 432
11.2 Vector Iteration Methods, 419 ......Page 434
11.2.1 Inverse Iteration, 420 ......Page 435
11.2.2 Forward Iteration, 428 ......Page 443
11.2.3 Shifting in Vector Iteration, 431 ......Page 446
11.2.4 Rayleigh Quotient Iteration, 436 ......Page 451
11.2.5 Matrix Deflation and Gram-Schmidt Orthogonalization, 439 ......Page 454
11.2.6 Some Practical Considerations Concerning Vector Iterations, 442 ......Page 457
11.3 Transformation Methods, 443 ......Page 458
11.3.1 The Jacobi Method, 445 ......Page 460
11.3.2 The Generalized Jacobi Method, 452 ......Page 467
11.3.3 The Householder-QR-Inverse Iteration Solution, 460 ......Page 475
11.4.1 Explicit Polynomial Iteration, 473 ......Page 488
11.4.2 Implicit Polynomial Iteration, 475 ......Page 490
11.5 Methods Based on the Sturm Sequence Property, 478 References, 482 ......Page 497
12.1 Introduction, 485 ......Page 500
12.2 The Determinant Search Method, 486 ......Page 501
12.2.1 Preliminary Considerations, 487 ......Page 502
12.2.2 The Solution Algorithm, 488 ......Page 503
12.2.3 Final Remarks Concerning the Determinant Search Solution, 490 ......Page 505
12.3 The Subspace Iteration Method, 494 ......Page 509
12.3.1 Preliminary Considerations, 495 ......Page 510
12.3.2 Subspace Iteration, 499 ......Page 514
12.3.3 Starting Iteration Vectors, 501 ......Page 516
12.3.4 Convergence, 504 ......Page 519
12.3.5 Final Remarks Concerning the Subspace Iteration Method, 506 ......Page 521
12.4 Selection of Solution Technique, 517 References, 520 ......Page 535
Index, 523 ......Page 538
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