The Finite Element Method for Initial Value Problems: Mathematics and Computations

Cover......Page 1
Half Title......Page 2
Title......Page 4
Copyrights......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 20
About the Authors......Page 26
 1.1 General overview......Page 28
 1.2 Space-time coupled methods for  xt......Page 29
 1.3 Space-time coupled methods using space-time strip......Page 31
 1.4 Space-time decoupled or quasi methods......Page 33
 1.6 Space-time coupled nite element method......Page 36
 1.7 Space-time decoupled nite element method......Page 37
 1.9 Stability......Page 39
 1.10 Accuracy and Error......Page 40
  1.10.1 Space-time coupled FEM over space-time domain  xt: 愀 瀀漀猀琀攀爀椀漀爀椀 挀漀洀瀀甀琀愀琀椀漀渀......Page 41
  1.10.3 Space-time decoupled nite element method: a posteriori 挀漀洀瀀甀琀愀琀椀漀渀......Page 42
 1.11 Mode superposition technique......Page 43
 1.12 Summary......Page 44
 2.2 Spaces, functions, function spaces, and 漀瀀攀爀愀琀漀爀猀......Page 46
  2.2.1 Space and time......Page 47
  2.2.3 Denition of scalar product in Hk( x) space......Page 50
  2.2.5 Norm of u in Hilbert space Hk( x......Page 51
  2.2.8 Denition of scalar product in H(k)( xt) space......Page 52
  2.2.10 Norm of u in Hilbert space H(k)( xt......Page 53
 2.3 Operators......Page 56
  2.3.1 Classication of space-time dierential operators......Page 57
  2.3.2 Integration by parts (IBP......Page 62
 2.4 Elements of calculus of variations......Page 65
  2.4.1 Concept of variation of a space-time functional......Page 68
  2.4.2 Euler\'s equation: simplest variational problem......Page 69
  2.4.3 Variation of a space-time functional: some practical 愀猀瀀攀挀琀猀......Page 75
 2.5 Riemann and Lebesgue integrals......Page 76
 2.6 Model problems......Page 77
 2.7 Summary......Page 89
 3.1 Introduction......Page 92
  3.2.1 Classical space-time Galerkin method......Page 93
  3.2.2 Classical space-time Galerkin method with weak form......Page 95
  3.2.4 Classical space-time weighted residual method......Page 96
  3.2.5 Choosing N0(x; t) and Ni(x; t) ; i = 1; 2; : : : ; n......Page 97
  3.3.1 Non-self-adjoint dierential operators......Page 98
  3.3.2 Non-linear dierential operators......Page 100
 3.4 STVC or STVIC of space-time integral forms......Page 101
  3.5.1 Model problem 1: 1D scalar wave equation......Page 109
   3.5.1.1 Space-time Galerkin method......Page 110
   3.5.1.2 Space-time Galerkin method with weak form......Page 111
   3.5.1.3 Space-time least squares method based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀......Page 114
  3.5.2 Model problem 2: 1D Burgers equation......Page 116
   3.5.2.1 Space-time Galerkin method......Page 117
   3.5.2.2 Space-time Galerkin method with weak form......Page 119
   3.5.2.3 Space-time least squares method based on 琀栀攀 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀......Page 122
 3.6 Summary......Page 125
 4.1 Introduction......Page 130
 4.2 Space-time domain and discretization......Page 131
 4.3 Mathematics of space-time nite element processes......Page 134
  4.3.1 Space-time nite element processes based on STGM, 匀吀倀䜀䴀Ⰰ 愀渀搀 匀吀圀刀䴀......Page 135
  4.3.2 Space-time nite element processes based on STGM/WF......Page 137
  4.3.3 Space-time nite element processes based on residual  functional: STLSP......Page 139
   4.3.3.1 Non-self-adjoint space-time dierential operators ......Page 140
   4.3.3.2 Non-linear space-time dierential operators......Page 142
  4.3.4 Summary of main steps (STLS nite element process......Page 144
  4.4.1 Model problem 1: 1D scalar wave equation......Page 146
   4.4.1.1 Space-time nite element process based on 匀吀䜀䴀......Page 147
   4.4.1.2 Space-time nite element process based on 匀吀䜀䴀⼀圀䘀......Page 148
   4.4.1.3 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀......Page 153
   4.4.1.4 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀⤀ 甀猀椀渀最 愀 ఀ爀猀琀 漀爀搀攀爀 system of PDEs......Page 155
   4.4.1.5 Numerical studies......Page 158
  4.4.2 Model problem 2: 1D pure advection......Page 164
   4.4.2.1 Space-time nite element process based on 匀吀䜀䴀 ⠀愀渀搀 匀吀䜀䴀⼀圀䘀......Page 169
   4.4.2.2 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀......Page 170
   4.4.2.3 Numerical studies......Page 172
  4.4.3 Model problem 3: 1D convection-diusion equation......Page 177
   4.4.3.1 Space-time nite element process based on 匀吀䜀䴀......Page 178
   4.4.3.2 Space-time nite element process based on 匀吀䜀䴀⼀圀䘀......Page 179
   4.4.3.3 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀......Page 182
   4.4.3.4 Space-time nite element process based on 匀吀䰀匀倀 甀猀椀渀最 愀 猀礀猀琀攀洀 漀昀 ఀ爀猀琀 漀爀搀攀爀 倀䐀䔀猀......Page 184
   4.4.3.5 Numerical studies......Page 187
  4.4.4 Model problem 4: 1D Burgers equation......Page 195
   4.4.4.1 Space-time nite element process based on 匀吀䜀䴀......Page 196
   4.4.4.2 Space-time nite element process based on 匀吀䜀䴀⼀圀䘀......Page 198
   4.4.4.3 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀......Page 201
   4.4.4.4 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀⤀ 甀猀椀渀最 愀 猀礀猀琀攀洀 of rst order PDEs......Page 203
   4.4.4.5 Numerical studies......Page 206
  4.4.5 Model problem 5: 1D diusion-reaction equations......Page 225
   4.4.5.1 Space-time nite element process based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀 ⠀匀吀䰀匀倀......Page 228
   4.4.5.2 Finite element process based on residual functional ⠀匀吀䰀匀倀⤀ 甀猀椀渀最 愀 猀礀猀琀攀洀 漀昀 ఀ爀猀琀 漀爀搀攀爀 PDEs......Page 231
   4.4.5.3 Numerical studies......Page 236
  4.4.6 Model problem 6: 1D normal shocks......Page 240
   4.4.6.1 Space-time nite element formulation based 漀渀 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀 ⠀匀吀䰀匀倀......Page 242
   4.4.6.2 Numerical studies......Page 244
   4.4.7.1 Mathematical model for phase transition......Page 253
   4.4.7.2 Space-time nite element formulation based  on residual function (STLSP......Page 258
   4.4.7.3 Numerical studies: liquid-solid phase transition......Page 261
   4.4.7.4 Numerical studies: solid-liquid phase transition......Page 264
 4.5 Summary......Page 269
 5.1 Introduction and basic methodology......Page 280
 5.2 Details of space-time decoupled approach: model problems......Page 282
 5.3 Summary......Page 310
 6.1 Introduction......Page 314
  6.2.1 Methods based on Taylor series......Page 315
  6.2.2 Methods based on integral forms constructed using 伀䐀䔀猀 椀渀 琀椀洀攀......Page 316
 6.3 Basic concepts in direct integration methods......Page 318
  6.3.2 Runge{Kutta methods......Page 320
   6.3.2.1 Second order Runge{Kutta method (n = 2......Page 321
   6.3.2.2 Third order Runge{Kutta method (n = 3......Page 322
  6.3.3 Numerical examples of direct integration methods......Page 323
 6.4 Basic concept in explicit methods......Page 325
 6.5 Basic concept in implicit methods......Page 326
  6.6.1 The central dierence method (explicit method......Page 328
  6.6.2 The Houbolt method (implicit method......Page 330
  6.6.3 Wilson\'s  method (implicit method......Page 331
   6.6.3.1 Wilson\'s  method: linear acceleration......Page 332
   6.6.3.2 Wilson\'s  method: constant average acceleration ......Page 335
   6.6.4.1 Newmark\'s method: constant average acceleration ......Page 337
   6.6.4.2 Newmark\'s method: linear acceleration......Page 339
  6.7.1 1D scalar wave equation......Page 341
   6.7.1.1 Central dierence method......Page 343
   6.7.1.2 Houbolt method......Page 344
   6.7.1.3 Wilson\'s  method......Page 347
   6.7.1.4 Newmark\'s method......Page 348
 6.8 Methods of approximation based on integral forms in time......Page 352
  6.8.1 Mathematical classication of time dierential operators ......Page 355
   6.8.2.1 Integral form of (6.139) based on fundamental 氀攀洀洀愀......Page 356
   6.8.2.2 Classical Galerkin method in time......Page 357
   6.8.2.3 Classical Galerkin method with weak form in 琀椀洀攀......Page 358
   6.8.2.4 Classical Petrov-Galerkin method in time......Page 359
   6.8.2.5 Classical weighted residual method in time......Page 360
   6.8.2.6 Classical least squares method in time......Page 361
   6.8.2.7 When is an integral form in time for an ODE 愀 瘀愀爀椀愀琀椀漀渀愀氀 昀漀爀洀甀氀愀琀椀漀渀......Page 362
  6.8.3 Variational consistency or variational inconsistency of 琀椀洀攀 椀渀琀攀最爀愀氀 昀漀爀洀猀 爀攀猀甀氀琀椀渀最 昀爀漀洀 椀渀琀攀最爀愀氀 洀攀琀栀漀搀猀 漀昀 approximation......Page 363
 6.9 Model problems......Page 369
   6.9.1.1 Classical GM, PGM, and WRM in time......Page 370
   6.9.1.2 Classical Galerkin method with weak form in 琀椀洀攀......Page 372
   6.9.1.3 Classical least squares process in time based 漀渀 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀......Page 374
  6.9.2 1D linear dynamics: scalar equation......Page 376
   6.9.2.2 Classical Galerkin method with weak form in 琀椀洀攀......Page 377
   6.9.3.1 Classical GM, PGM, and WRM in time......Page 378
   6.9.3.2 Classical Galerkin method with weak form in 琀椀洀攀......Page 379
   6.9.3.3 Classical least squares process in time based 漀渀 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀......Page 380
 6.10 Summary......Page 382
 7.1 Introduction......Page 390
 7.2 Time domain, increment of time, and time discretization......Page 391
 7.3 Finite element process in time for ODEs in time......Page 392
  7.3.1 Finite element processes based on GM, PGM, and 圀刀䴀 椀渀 琀椀洀攀......Page 393
  7.3.2 Finite element processes based on GM/WF in time......Page 395
  7.3.3 Finite element processes based on residual functional: 䰀匀倀 椀渀 琀椀洀攀......Page 396
   7.3.3.1 Linear time operator (non-self-adjoint......Page 397
   7.3.3.2 Non-linear time operator......Page 398
  7.3.4 Remarks on various time nite element processes based 漀渀 洀攀琀栀漀搀猀 漀昀 愀瀀瀀爀漀砀椀洀愀琀椀漀渀 椀渀 琀椀洀攀......Page 401
  7.4.1 1D linear dynamics: scalar equation in modal basis......Page 402
   7.4.1.1 Finite element processes in time based on GM......Page 403
   7.4.1.2 Finite element processes in time based on 䜀䴀⼀圀䘀......Page 404
   7.4.1.3 Finite element processes in time based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀㨀 䰀匀倀......Page 408
   7.4.1.4 Finite element processes in time based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀㨀 䰀匀倀Ⰰ ఀ爀猀琀 漀爀搀攀爀 猀礀猀琀攀洀......Page 410
   7.4.1.5 Numerical studies......Page 412
   7.4.2.1 Finite element processes in time based on 䜀䴀Ⰰ 倀䜀䴀Ⰰ 愀渀搀 圀刀䴀......Page 417
   7.4.2.3 Finite element processes in time based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀㨀 䰀匀倀......Page 418
   7.4.2.4 Numerical studies......Page 419
  7.4.3 1D non-linear dynamics: scalar equation......Page 424
   7.4.3.1 Finite element processes in time based on GM......Page 426
   7.4.3.2 Finite element processes in time based on  GM/WF......Page 427
   7.4.3.3 Finite element processes in time based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀㨀 䰀匀倀......Page 431
   7.4.3.4 Finite element processes in time based on 爀攀猀椀搀甀愀氀 昀甀渀挀琀椀漀渀愀氀㨀 䰀匀倀Ⰰ ఀ爀猀琀 漀爀搀攀爀 猀礀猀琀攀洀......Page 433
   7.4.3.5 Numerical studies......Page 434
  7.4.4 1D scalar wave equation......Page 439
   7.4.4.1 Numerical studies......Page 442
  7.4.5 Mixing problem......Page 443
   7.4.5.1 Numerical studies......Page 447
 7.5 Summary......Page 449
 8.1 Introduction......Page 460
 8.2 Stability of space-time coupled methods......Page 461
 8.3 Stability analysis of space-time decoupled methods......Page 462
  8.3.1 Recursive relation for time-marching solutions of ODEs......Page 464
  8.3.2 Spectral radius of [B]: boundedness of [B......Page 466
  8.4.1 Stability of central dierence method......Page 467
  8.4.2 Stability of Houbolt method......Page 469
  8.4.3 Stability of Wilson\'s  method......Page 472
   8.4.3.1 Linear acceleration method......Page 473
   8.4.3.2 Constant average acceleration method......Page 476
   8.4.4.1 Constant average acceleration method......Page 482
   8.4.4.2 Linear acceleration method......Page 485
  8.4.5 General remarks......Page 487
  8.4.6 Stability of least squares nite element method......Page 488
 8.5 Summary......Page 490
 9.1 Introduction......Page 492
  9.1.1 Fundamental properties of eigenpairs......Page 493
 9.2 General remarks on free vibrations......Page 494
 9.3 Mode superposition method......Page 495
  9.3.1 Transforming initial conditions......Page 496
  9.3.2 Time response (or transient dynamic response) of undamped 猀礀猀琀攀洀猀......Page 497
  9.3.3 Time response of damped systems......Page 499
   9.3.3.1 Proportional damping......Page 501
 9.4 Analytical solution of undamped equations in modal basis......Page 502
  9.4.1 Constant ^ fi......Page 503
 9.5 Analytical solution of damped equations in modal basis......Page 504
   9.5.1.1 Critically damped system......Page 505
  9.5.2 Solution of nonhomogeneous form: particular solution......Page 506
   9.5.2.2 Harmonic ^ fi......Page 507
  9.6.1 Solution of homogeneous form: complementary solution......Page 509
   9.6.1.1 Critically damped system......Page 510
   9.6.1.3 Underdamped system......Page 511
   9.6.2.1 Constant f......Page 512
   9.6.2.2 Harmonic f......Page 513
 9.7 General remarks on modal basis and theoretical solutions......Page 514
 9.8 Model problem: 1D scalar wave equation......Page 516
 9.9 Transient response using lowest modes of vibration......Page 517
  9.10.1 Static condensation......Page 519
  9.10.2 Guyan reduction......Page 520
 9.11 Summary......Page 521
 10.1 Introduction......Page 524
  10.2.3 Accuracy and time accuracy......Page 525
  10.2.4 Convergence and convergence rates......Page 527
  10.3.1 A priori error estimation......Page 528
   10.3.1.1 Convergence rates......Page 532
   10.3.1.2 General remarks on a priori error estimates 愀渀搀 甀猀攀 漀昀 漀瀀琀椀洀愀氀 琀栀攀漀爀攀琀椀挀愀氀 挀漀渀瘀攀爀最攀渀挀攀 rates......Page 533
   10.3.1.3 Importance and signicance of higher order 猀瀀愀挀攀猀......Page 534
  10.3.3 Model problem: 1D convection-diusion equation......Page 535
 10.5 ODEs in time......Page 538
   10.5.2.1 Houbolt method......Page 539
   10.5.2.2 Wilson\'s  method......Page 540
   10.5.2.3 Newmark\'s method......Page 542
   10.5.2.5 Model problem: 1D scalar wave equation......Page 544
   10.5.2.6 General remarks......Page 547
  10.5.3 The nite element method in time......Page 548
   10.5.3.1 A priori error estimation......Page 549
   10.5.3.2 Convergence rates......Page 553
   10.5.3.3 Importance and signicance of higher order 猀瀀愀挀攀猀......Page 554
   10.5.3.4 Model problem: 1D scalar wave equation......Page 555
   10.5.3.5 Model problem: mixing problem......Page 557
   10.5.3.6 A posteriori error computations......Page 558
 10.6 Summary......Page 559
  11.1.2 Mapping of lengths......Page 564
  11.2.2 Lagrange interpolation in 1D......Page 565
  11.2.3 C0 p-version hierarchical interpolation functions in 1D......Page 566
  11.2.4 p-version interpolations of class C1( e) in 1D......Page 567
  11.3.1 Mapping of points......Page 568
  11.3.3 Mapping of areas......Page 569
 11.4 Interpolation in two dimensions: quadrilateral elements......Page 570
  11.4.1 Obtaining derivatives of (; ) with respect to x and y......Page 571
  11.4.2 C00 local approximations over    or  m: quadrilateral  elements......Page 572
  11.4.3 C00 p-version hierarchical local approximations based 漀渀 䰀愀最爀愀渀最攀 瀀漀氀礀渀漀洀椀愀氀猀......Page 574
  11.4.4 Cij p-version hierarchical local approximations: rectangular 昀愀洀椀氀礀 漀昀 攀氀攀洀攀渀琀猀......Page 577
  11.4.5 2D Cij( e) approximations for quadrilateral elements......Page 578
  11.5.2 1D p-version C0 hierarchical approximation functions ⠀䰀攀最攀渀搀爀攀 瀀漀氀礀渀漀洀椀愀氀猀......Page 580
  11.6.1 Chebyshev polynomials......Page 581
  11.6.3 2D p-version C00 hierarchical interpolation functions 昀漀爀 焀甀愀搀爀椀氀愀琀攀爀愀氀 攀氀攀洀攀渀琀猀 ⠀䌀栀攀戀礀猀栀攀瘀 瀀漀氀礀渀漀洀椀愀氀猀......Page 582
  11.7.1 Area coordinates......Page 583
 11.8 Serendipity family of C00 interpolations......Page 585
  11.9.1 Mapping of points......Page 587
  11.9.2 Mapping of lengths......Page 588
  11.10.1 Obtaining derivatives of e 栀⠀᠀㬀 ᄀ㬀 က⤀ 眀椀琀栀 爀攀猀瀀攀挀琀 琀漀 x; y; z......Page 589
   11.10.2.2 Tensor product: C000( e) and Cijk( e......Page 590
  11.11.2 Lagrange family C000 tetrahedron elements based on 瘀漀氀甀洀攀 挀漀漀爀搀椀渀愀琀攀猀......Page 591
  11.11.3 Higher degree C000 basis functions using volume coordinates ......Page 593
   11.11.3.1 Four-node linear tetrahedron element (p-level 漀昀 漀渀攀......Page 594
 11.12 Summary......Page 595
 A.1 Introduction......Page 600
  A.2.1 1D pure advection......Page 601
  A.2.2 1D convection-diusion equation......Page 603
  A.2.3 1D Burgers equation......Page 604
  A.2.4 1D wave propagation in elastic medium (structural 搀礀渀愀洀椀挀猀......Page 606
  A.2.6 1D diusion-reaction equations......Page 608
  A.2.7 1D compressible 漀眀 ⠀䔀甀氀攀爀椀愀渀 搀攀猀挀爀椀瀀琀椀漀渀......Page 610
   A.2.7.2 Momentum equation......Page 611
   A.2.7.3 Energy equation......Page 612
   A.2.7.5 Equation of state......Page 613
   A.2.7.6 Summary of the dimensionless form of the 䜀䐀䔀猀......Page 614
  B.1.1 Line integrals over  m =   = [..1; 1......Page 616
  B.1.2 Area integrals over  m =   = [..1; 1]  [..1; 1......Page 617
  B.1.3 Volume integrals over  m =   = [..1; 1][..1; 1] 嬀⸀⸀㄀㬀 ㄀......Page 618
 B.2 Gauss quadrature over triangular domains......Page 620
Index......Page 624