Introduction. Finite and Spectral Element Methods Using MATLAB. Second Edition. C. Pozrikidis. University of Massachusetts Amherst, USA

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1 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.

2 Contents Preface FSELIB software library Frequently asked questions xiii xvii xxiii 1 The finite element method in one dimension Steady diffusion with linear elements Linear element interpolation Element grading Galerkin projection Formulation of a linear algebraic system Flux at the Dirichlet end Galerkin finite element equations via the Dirac delta function Relation to the finite difference method Finite element assembly Assembly of a linear system Thomas algorithm for a tridiagonal system Finite element code Convection (Robin or mixed) boundary condition Variational formulation and weighted residuals Homogeneous Dirichlet boundary conditions Inhomogeneous Dirichlet boundary conditions Dirichlet/Neumann boundary conditions Neumann/Dirichlet boundary conditions Helmholtz equation Steady diffusion with quadratic elements Element nodes and global nodes Galerkin finite element equations Thomas algorithm for pentadiagonal system Element matrices Finite element code Node condensation Arbitrary interior nodes 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.

3 Finite and Spectral Element Methods Using MATLAB Further applications in one dimension Unsteady diffusion Galerkin projection Integrating ODEs Forward Euler method Numerical stability Finite element code Crank-Nicolson integration Convection Linear elements Numerical dispersion due to spatial discretization Quadratic elements Integrating ODEs Nonlinear convection Convection-diffusion Steady linear convection-diffusion Nonlinear convection-diffusion Beam bending Euler-Bernoulli beam Finite element methods for beam bending Hermitian elements Galerkin projection Element stiffness and mass matrices One-element cantilever beam Cantilever beam with nodal loads Beam buckling Tip compression Buckling under a compressive tip force Buckling of a heavy vertical column 145 High-order and spectral elements in one dimension Element nodal sets Lagrange interpolation Evenly spaced nodes Element matrices C continuity and shared element nodes Change of element nodal sets Spectral interpolation Lobatto nodal base Discretization code Legendre polynomials Chebyshev second-kind nodal base Lobatto interpolation and element matrices Lobatto mass matrix 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.

4 Contents vii Computation of the Lobatto mass matrix Computation of the Lobatto diffusion matrix Spectral element code for steady diffusion Spectral accuracy Helmholtz equation Node condensation Modal expansion Relation to the nodal expansion Implementation Lobatto modal expansion Element diffusion matrix Element mass matrix Modal spectral element method Arbitrary nodal sets Unsteady diffusion Crank-Nicolson discretization Forward Euler discretization The finite element method in two dimensions Convection-diffusion in two dimensions Boundary conditions Galerkin projection Domain discretization and interpolation Galerkin finite element equations Implementation of the Dirichlet boundary condition Split nodes Variational formulation Three-node triangles Element matrices Computation of the element diffusion matrix Computation of the element mass matrix Proof of the integration formula (4.2.36) Computation of the element advection matrix Grid generation Successive subdivisions Delaunay triangulation Generalized connectivity matrices Element and node labeling schemes Laplace's equation with the Dirichlet boundary condition Eigenvalues of the Laplacian operator Convection-diffusion with the Dirichlet boundary condition Helmholtz's equation with the Neumann boundary condition Laplace's equation with arbitrary boundary conditions Surface elements 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.

5 Vlll Finite and Spectral Element Methods Using MATLAB 5 Quadratic and spectral elements in two dimensions Six-node triangular elements Integral over a triangle Isoparametric interpolation and element matrices Element matrices and integration quadratures Elements with straight edges Grid generation Circular disk Square L-shaped domain Square with a square or circular hole A rectangle with a circular hole Laplace and Poisson equations Laplace equation Eigenvalues of the Laplacian operator Poisson equation Convection-diffusion with the Dirichlet boundary condition High-order triangle expansions Computation of the node interpolation functions The Lebesgue constant Node condensation Appell polynomial base Incomplete biorthogonality Incomplete orthogonality Generalized Appell polynomials Proriol polynomial base Orthogonality Orthogonal expansion High-order node distributions Node distribution based on a one-dimensional master grid Uniform grid Lobatto grid on the triangle The Fekete set Further nodal distributions Modal expansions in a triangle Implementation of the modal expansion Properties of the modal expansion Surface elements Surface gradient Grid generation High-order quadrilateral elements Eight-node serendipity elements node serendipity elements Grid nodes via tensor-product expansions 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.

6 Contents ix 6 Applications in mechanics Elements of elasticity theory Deformation and constitutive equations Linear elasticity Plane stress and plane strain analysis Plane stress analysis Plane strain analysis Finite element formulation Finite element plane stress analysis Deformation due to an edge force Deformation due to a body force Plate bending Equilibrium equations Boundary conditions Constitutive and governing equations Circular plate Hermite triangles Morley's triangle Conforming triangles Six-node, 21-dof triangle The Hsieh-Clough-Tbcher (HCT) element Finite element methods for plate bending Formulation as a biharmonic equation Formulation as a system of Poisson equations Buckling and wrinkling Viscous flow Governing equations Finite element formulation Galerkin projections Discrete equations Stokes flow Governing equations Galerkin finite element equations Triangularization Stokes flow in a rectangular cavity Navier-Stokes flow Steady state Time integration Formulation based on the pressure Poisson equation Finite and spectral element methods in three dimensions Convection-diffusion in three dimensions Boundary conditions 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.

7 x Finite and Spectral Element Methods Using MATLAB Galerkin projection Galerkin finite element equations Element matrices Implementation of the Dirichlet boundary condition Tetrahedral elements Parametric representation Integral over the volume of a tetrahedron Element subdivision into eight tetrahedra Element subdivision into 12 tetrahedra Isoparametric interpolation Element diffusion matrix Element mass matrix Proof of the integration formula (8.2.36) Element advection matrix Domain discretization into four-node tetrahedra Delaunay tessellation Finite element codes with four-node tetrahedra Laplace's equation Eigenvalues of the Laplacian operator Orthogonal polynomials over a tetrahedron Karniadakis and Sherwin polynomials Orthogonal expansion High-order and spectral tetrahedral elements Uniform node distributions Arbitrary node distributions Spectral node distributions Gradient of the element node interpolation functions Numerical integration node quadratic tetrahedra Node interpolation functions Element diffusion and mass matrices Domain discretization Laplace's equation Eigenvalues of the Laplacian operator Modal expansions in a tetrahedron Hexahedral elements Parametric representation Integral over the volume of the hexahedron High-order and spectral hexahedral elements 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 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 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.

9 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 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

Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,

Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup, Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business