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

Unlike most finite element books that cover time dependent processes (IVPs) in a cursory manner, **The Finite Element Method for Initial Value Problems: Mathematics and Computations** focuses on the mathematical details as well as applications of space-time coupled and space-time decoupled finite element methods for IVPs. Space-time operator classification, space-time methods of approximation, and space-time calculus of variations are used to establish unconditional stability of space-time methods during the evolution. Space-time decoupled methods are also presented with the same rigor. Stability of space-time decoupled methods, time integration of ODEs including the finite element method in time are presented in detail with applications. Modal basis, normal mode synthesis techniques, error estimation, and a posteriori error computations for space-time coupled as well as space-time decoupled methods are presented. This book is aimed at a second-semester graduate level course in FEM. 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: a posteriori computation......Page 41 1.10.3 Space-time decoupled nite element method: a posteriori computation......Page 42 1.11 Mode superposition technique......Page 43 1.12 Summary......Page 44 2.2 Spaces, functions, function spaces, and operators......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 aspects......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 residual functional......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 the residual functional......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, STPGM, and STWRM......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 STGM......Page 147 4.4.1.2 Space-time nite element process based on STGM/WF......Page 148 4.4.1.3 Space-time nite element process based on residual functional (STLSP......Page 153 4.4.1.4 Space-time nite element process based on residual functional (STLSP) using a rst order 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 STGM (and STGM/WF......Page 169 4.4.2.2 Space-time nite element process based on residual functional (STLSP......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 STGM......Page 178 4.4.3.2 Space-time nite element process based on STGM/WF......Page 179 4.4.3.3 Space-time nite element process based on residual functional (STLSP......Page 182 4.4.3.4 Space-time nite element process based on STLSP using a system of rst order PDEs......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 STGM......Page 196 4.4.4.2 Space-time nite element process based on STGM/WF......Page 198 4.4.4.3 Space-time nite element process based on residual functional (STLSP......Page 201 4.4.4.4 Space-time nite element process based on residual functional (STLSP) using a system 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 residual functional (STLSP......Page 228 4.4.5.2 Finite element process based on residual functional (STLSP) using a system of rst order 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 on residual function (STLSP......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 ODEs in time......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 lemma......Page 356 6.8.2.2 Classical Galerkin method in time......Page 357 6.8.2.3 Classical Galerkin method with weak form in time......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 a variational formulation......Page 362 6.8.3 Variational consistency or variational inconsistency of time integral forms resulting from integral methods of 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 time......Page 372 6.9.1.3 Classical least squares process in time based on residual functional......Page 374 6.9.2 1D linear dynamics: scalar equation......Page 376 6.9.2.2 Classical Galerkin method with weak form in time......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 time......Page 379 6.9.3.3 Classical least squares process in time based on residual functional......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 WRM in time......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: LSP in time......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 on methods of approximation in time......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 GM/WF......Page 404 7.4.1.3 Finite element processes in time based on residual functional: LSP......Page 408 7.4.1.4 Finite element processes in time based on residual functional: LSP, rst order system......Page 410 7.4.1.5 Numerical studies......Page 412 7.4.2.1 Finite element processes in time based on GM, PGM, and WRM......Page 417 7.4.2.3 Finite element processes in time based on residual functional: LSP......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 residual functional: LSP......Page 431 7.4.3.4 Finite element processes in time based on residual functional: LSP, rst order system......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 systems......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 and use of optimal theoretical convergence rates......Page 533 10.3.1.3 Importance and signi cance of higher order spaces......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 spaces......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 on Lagrange polynomials......Page 574 11.4.4 Cij p-version hierarchical local approximations: rectangular family of elements......Page 577 11.4.5 2D Cij( e) approximations for quadrilateral elements......Page 578 11.5.2 1D p-version C0 hierarchical approximation functions (Legendre polynomials......Page 580 11.6.1 Chebyshev polynomials......Page 581 11.6.3 2D p-version C00 hierarchical interpolation functions for quadrilateral elements (Chebyshev polynomials......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 h(; ; ) with respect to 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 volume coordinates......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 of one......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 dynamics......Page 606 A.2.6 1D diusion-reaction equations......Page 608 A.2.7 1D compressible ow (Eulerian description......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 GDEs......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] [..1; 1......Page 618 B.2 Gauss quadrature over triangular domains......Page 620 Index......Page 624