Introduction to Finite and Spectral Element Methods Using MATLAB Second Edition C. Pozrikidis University of Massachusetts Amherst, USA (g) CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor St Francis Croup, an informa business A CHAPMAN & HALL BOOK
Contents Preface FSELIB software library Frequently asked questions xiii xvii xxiii 1 The finite element method in one dimension 1 1.1 Steady diffusion with linear elements 1 1.1.1 Linear element interpolation 3 1.1.2 Element grading 4 1.1.3 Galerkin projection 7 1.1.4 Formulation of a linear algebraic system 11 1.1.5 Flux at the Dirichlet end 14 1.1.6 Galerkin finite element equations via the Dirac delta function 16 1.1.7 Relation to the finite difference method 20 1.2 Finite element assembly 21 1.2.1 Assembly of a linear system 25 1.2.2 Thomas algorithm for a tridiagonal system 25 1.2.3 Finite element code 27 1.2.4 Convection (Robin or mixed) boundary condition 34 1.3 Variational formulation and weighted residuals 35 1.3.1 Homogeneous Dirichlet boundary conditions 36 1.3.2 Inhomogeneous Dirichlet boundary conditions 39 1.3.3 Dirichlet/Neumann boundary conditions 41 1.3.4 Neumann/Dirichlet boundary conditions 43 1.4 Helmholtz equation 45 1.5 Steady diffusion with quadratic elements 51 1.5.1 Element nodes and global nodes 51 1.5.2 Galerkin finite element equations 53 1.5.3 Thomas algorithm for pentadiagonal system 57 1.5.4 Element matrices 58 1.5.5 Finite element code 62 1.5.6 Node condensation 67 1.5.7 Arbitrary interior nodes 73 1.6 Steady diffusion with quadratic modal expansions 78 v
Finite and Spectral Element Methods Using MATLAB Further applications in one dimension 87 2.1 Unsteady diffusion 87 2.1.1 Galerkin projection 88 2.1.2 Integrating ODEs 90 2.1.3 Forward Euler method 90 2.1.4 Numerical stability 91 2.1.5 Finite element code 96 2.1.6 Crank-Nicolson integration 101 2.2 Convection 105 2.2.1 Linear elements 106 2.2.2 Numerical dispersion due to spatial discretization 109 2.2.3 Quadratic elements 110 2.2.4 Integrating ODEs 110 2.2.5 Nonlinear convection 113 2.3 Convection-diffusion 114 2.3.1 Steady linear convection-diffusion 114 2.3.2 Nonlinear convection-diffusion 119 2.4 Beam bending 120 2.4.1 Euler-Bernoulli beam 120 2.5 Finite element methods for beam bending 125 2.5.1 Hermitian elements 126 2.5.2 Galerkin projection 128 2.5.3 Element stiffness and mass matrices 131 2.5.4 One-element cantilever beam 132 2.5.5 Cantilever beam with nodal loads 136 2.6 Beam buckling 140 2.6.1 Tip compression 142 2.6.2 Buckling under a compressive tip force 143 2.6.3 Buckling of a heavy vertical column 145 High-order and spectral elements in one dimension 153 3.1 Element nodal sets 154 3.1.1 Lagrange interpolation 155 3.1.2 Evenly spaced nodes 156 3.1.3 Element matrices 158 3.1.4 C continuity and shared element nodes 159 3.2 Change of element nodal sets 161 3.3 Spectral interpolation 165 3.3.1 Lobatto nodal base 166 3.3.2 Discretization code 174 3.3.3 Legendre polynomials 174 3.3.4 Chebyshev second-kind nodal base 178 3.4 Lobatto interpolation and element matrices 180 3.4.1 Lobatto mass matrix 181 3.4.2 Lobatto integration quadrature 182
Contents vii 3.4.3 Computation of the Lobatto mass matrix 183 3.4.4 Computation of the Lobatto diffusion matrix 191 3.5 Spectral element code for steady diffusion 196 3.5.1 Spectral accuracy 201 3.5.2 Helmholtz equation 206 3.5.3 Node condensation 206 3.6 Modal expansion 212 3.6.1 Relation to the nodal expansion 213 3.6.2 Implementation 215 3.7 Lobatto modal expansion 216 3.7.1 Element diffusion matrix 216 3.7.2 Element mass matrix 220 3.7.3 Modal spectral element method 224 3.8 Arbitrary nodal sets 224 3.9 Unsteady diffusion 231 3.9.1 Crank-Nicolson discretization 233 3.9.2 Forward Euler discretization 240 4 The finite element method in two dimensions 243 4.1 Convection-diffusion in two dimensions 244 4.1.1 Boundary conditions 246 4.1.2 Galerkin projection 246 4.1.3 Domain discretization and interpolation 248 4.1.4 Galerkin finite element equations 250 4.1.5 Implementation of the Dirichlet boundary condition 251 4.1.6 Split nodes 253 4.1.7 Variational formulation 253 4.2 Three-node triangles 257 4.2.1 Element matrices 261 4.2.2 Computation of the element diffusion matrix 263 4.2.3 Computation of the element mass matrix 263 4.2.4 Proof of the integration formula (4.2.36) 265 4.2.5 Computation of the element advection matrix 267 4.3 Grid generation 270 4.3.1 Successive subdivisions 270 4.3.2 Delaunay triangulation 274 4.3.3 Generalized connectivity matrices 279 4.3.4 Element and node labeling schemes 282 4.4 Laplace's equation with the Dirichlet boundary condition 285 4.5 Eigenvalues of the Laplacian operator 293 4.6 Convection-diffusion with the Dirichlet boundary condition 297 4.7 Helmholtz's equation with the Neumann boundary condition.... 302 4.8 Laplace's equation with arbitrary boundary conditions 308 4.9 Surface elements 314 4.10 Bilinear quadrilateral elements 317
Vlll Finite and Spectral Element Methods Using MATLAB 5 Quadratic and spectral elements in two dimensions 321 5.1 Six-node triangular elements 321 5.1.1 Integral over a triangle 326 5.1.2 Isoparametric interpolation and element matrices 326 5.1.3 Element matrices and integration quadratures 328 5.1.4 Elements with straight edges 335 5.2 Grid generation 340 5.2.1 Circular disk 340 5.2.2 Square 347 5.2.3 L-shaped domain 347 5.2.4 Square with a square or circular hole 347 5.2.5 A rectangle with a circular hole 350 5.3 Laplace and Poisson equations 359 5.3.1 Laplace equation 359 5.3.2 Eigenvalues of the Laplacian operator 365 5.3.3 Poisson equation 365 5.4 Convection-diffusion with the Dirichlet boundary condition 374 5.5 High-order triangle expansions 380 5.5.1 Computation of the node interpolation functions 383 5.5.2 The Lebesgue constant 387 5.5.3 Node condensation 387 5.6 Appell polynomial base 387 5.6.1 Incomplete biorthogonality 390 5.6.2 Incomplete orthogonality 391 5.6.3 Generalized Appell polynomials 391 5.7 Proriol polynomial base 392 5.7.1 Orthogonality 394 5.7.2 Orthogonal expansion 395 5.8 High-order node distributions 396 5.8.1 Node distribution based on a one-dimensional master grid.. 396 5.8.2 Uniform grid 398 5.8.3 Lobatto grid on the triangle 402 5.8.4 The Fekete set 405 5.8.5 Further nodal distributions 406 5.9 Modal expansions in a triangle 407 5.9.1 Implementation of the modal expansion 412 5.9.2 Properties of the modal expansion 412 5.10 Surface elements 413 5.10.1 Surface gradient 414 5.10.2 Grid generation 415 5.11 High-order quadrilateral elements 416 5.11.1 Eight-node serendipity elements 417 5.11.2 12-node serendipity elements 419 5.11.3 Grid nodes via tensor-product expansions 422 5.11.4 Modal expansion 424
Contents ix 6 Applications in mechanics 429 6.1 Elements of elasticity theory 429 6.1.1 Deformation and constitutive equations 431 6.1.2 Linear elasticity 432 6.2 Plane stress and plane strain analysis 434 6.2.1 Plane stress analysis 434 6.2.2 Plane strain analysis 439 6.2.3 Finite element formulation 440 6.3 Finite element plane stress analysis 443 6.3.1 Deformation due to an edge force 445 6.3.2 Deformation due to a body force 460 6.4 Plate bending 470 6.4.1 Equilibrium equations 474 6.4.2 Boundary conditions 475 6.4.3 Constitutive and governing equations 477 6.4.4 Circular plate 480 6.5 Hermite triangles 482 6.6 Morley's triangle 493 6.7 Conforming triangles 496 6.7.1 Six-node, 21-dof triangle 497 6.7.2 The Hsieh-Clough-Tbcher (HCT) element 498 6.8 Finite element methods for plate bending 500 6.8.1 Formulation as a biharmonic equation 506 6.8.2 Formulation as a system of Poisson equations 518 6.9 Buckling and wrinkling 523 7 Viscous flow 541 7.1 Governing equations 541 7.2 Finite element formulation 544 7.2.1 Galerkin projections 544 7.2.2 Discrete equations 546 7.3 Stokes flow 547 7.3.1 Governing equations 547 7.3.2 Galerkin finite element equations 548 7.3.3 Triangularization 552 7.4 Stokes flow in a rectangular cavity 553 7.5 Navier-Stokes flow 562 7.5.1 Steady state 564 7.5.2 Time integration 564 7.5.3 Formulation based on the pressure Poisson equation 566 8 Finite and spectral element methods in three dimensions 569 8.1 Convection-diffusion in three dimensions 569 8.1.1 Boundary conditions 570 8.1.2 Domain discretization 571
x Finite and Spectral Element Methods Using MATLAB 8.1.3 Galerkin projection 571 8.1.4 Galerkin finite element equations 572 8.1.5 Element matrices 573 8.1.6 Implementation of the Dirichlet boundary condition 574 8.2 Tetrahedral elements 574 8.2.1 Parametric representation. 576 8.2.2 Integral over the volume of a tetrahedron 578 8.2.3 Element subdivision into eight tetrahedra 579 8.2.4 Element subdivision into 12 tetrahedra 583 8.2.5 Isoparametric interpolation 583 8.2.6 Element diffusion matrix 583 8.2.7 Element mass matrix 594 8.2.8 Proof of the integration formula (8.2.36) 595 8.2.9 Element advection matrix 597 8.3 Domain discretization into four-node tetrahedra 598 8.3.1 Delaunay tessellation 598 8.4 Finite element codes with four-node tetrahedra 605 8.4.1 Laplace's equation 605 8.4.2 Eigenvalues of the Laplacian operator 612 8.5 Orthogonal polynomials over a tetrahedron 615 8.5.1 Karniadakis and Sherwin polynomials 616 8.5.2 Orthogonal expansion 618 8.6 High-order and spectral tetrahedral elements 620 8.6.1 Uniform node distributions 621 8.6.2 Arbitrary node distributions 625 8.6.3 Spectral node distributions 625 8.6.4 Gradient of the element node interpolation functions 628 8.6.5 Numerical integration 628 8.7 10-node quadratic tetrahedra 629 8.7.1 Node interpolation functions 631 8.7.2 Element diffusion and mass matrices 632 8.7.3 Domain discretization 632 8.7.4 Laplace's equation 651 8.7.5 Eigenvalues of the Laplacian operator 651 8.8 Modal expansions in a tetrahedron 661 8.9 Hexahedral elements 664 8.9.1 Parametric representation 665 8.9.2 Integral over the volume of the hexahedron 666 8.9.3 High-order and spectral hexahedral elements 667 8.9.4 Modal expansion 668 Appendices 673 A Mathematical supplement 675 A.l Index notation 675
Contents xi A.2 Kronecker's delta 675 A.3 Alternating tensor 676 A.4 Two- and three-dimensional vectors 676 A.5 Del or nabla operator 677 A.6 Gradient and divergence 677 A.7 Vector identities 678 A.8 Gauss divergence theorem 679 A.9 Gauss divergence theorem in the plane 680 A.10 Stokes's theorem 681 B Orthogonal polynomials 683 B.l Definitions and basic properties 683 B.1.1 Orthogonality against lower-degree polynomials 684 B.l.2 Roots of orthogonal polynomials 686 B.l.3 Discrete orthogonality 686 B.l.4 Gram polynomials 687 B.1.5 Recursion relation 687 B.1.6 Evaluation as the determinant of a tridiagonal matrix... 689 B.l.7 Clenshaw's algorithm 690 B.l.8 Gram-Schmidt orthogonalization 690 B.1.9 Orthonormal polynomials 692 B.l. 10 Christoffel-Darboux formula 692 B.2 Gaussian integration quadratures 694 B.2.1 Evaluation of the integration weights 695 B.2.2 Standard Gaussian quadratures 696 B.3 Lobatto integration quadrature 697 B.4 Chebyshev integration quadrature 699 B.5 Legendre polynomials 700 B.6 Lobatto polynomials 702 B.7 Chebyshev polynomials 703 B.8 Jacobi polynomials 706 C Linear solvers 709 C.l Gauss elimination 709 C.l.l Pivoting 711 C.l.2 Implementation 712 C.l.3 Symmetric matrices 715 C.l.4 Computational cost 715 C.l.5 Gauss elimination code 715 C.1.6 Multiple right-hand sides 715 C.l.7 Computation of the inverse 721 C.l.8 Gauss-Jordan reduction 721 C.2 Iterative methods based on matrix splitting 722 C.2.1 Jacobi's method 723 C.2.2 Gauss-Seidel method 723
xii Finite and Spectral Element Methods Using MATLAB C.2.3 Successive over-relaxation (SOR) 724 C.2.4 Operator- and grid-based splitting 724 C.3 Iterative methods based on path search 725 C.3.1 Symmetric and positive-definite matrices 725 C.3.2 General methods 733 C.4 Finite element system solvers 734 D Function interpolation 735 D.l The interpolating polynomial 736 D.l.l Vandermonde matrix 736 D.I.2 Generalized Vandermonde matrix 738 D.I.3 Newton interpolation 739 D.2 Lagrange interpolation 740 D.2.1 Cauchy relations 740 D.2.2 Representation in terms of a generating polynomial 741 D.2.3 First derivative and the node differentiation matrix 742 D.2.4 Representation in terms of the Vandermonde matrix 745 D.2.5 Lagrange polynomials corresponding to polynomial roots.. 748 D.2.6 Lagrange polynomials for Hermite interpolation 749 D.3 Error in polynomial interpolation 751 D.3.1 Convergence and the Lebesgue constant 752 D.4 Chebyshev interpolation 755 D.5 Lobatto interpolation 756 D.6 Interpolation in two and higher dimensions 760 E Element grid generation 763 F Glossary 765 G MATLAB primer 767 G.l Programming in MATLAB 768 G.l.l Grammar and syntax 768 G.l.2 Precision 769 G.l.3 MATLAB commands 771 G.l.4 Elementary examples 773 G.2 MATLAB functions 777 G.3 Numerical methods 780 G.4 MATLAB graphics 781 References 789 Index 795