Introduction to finite and spectral element methods using MATLAB /

Bibliographic Details
Main Author:	Pozrikidis, C. (Constantine), 1958-
Corporate Author:	Taylor & Francis
Format:	eBook
Language:	English
Published:	Boca Raton : CRC Press, Taylor and Francis Group, 2014.
Edition:	Second edition.
Subjects:	MATLAB. MATLAB Finite element method > Data processing. Finite element method > Data processing Finite-Elemente-Methode Numerisches Verfahren Spektralmethode Electronic books.
Online Access:	Connect to the full text of this electronic book

Table of Contents:

Machine-generated contents note: 1.1. Steady diffusion with linear elements
1.1.1. Linear element interpolation
1.1.2. Element grading
1.1.3. Galerkin projection
1.1.4. Formulation of a linear algebraic system
1.1.5. Flux at the Dirichlet end
1.1.6. Galerkin finite element equations via the Dirac delta function
1.1.7. Relation to the finite difference method
1.2. Finite element assembly
1.2.1. Assembly of a linear system
1.2.2. Thomas algorithm for a tridiagonal system
1.2.3. Finite element code
1.2.4. Convection (Robin or mixed) boundary condition
1.3. Variational formulation and weighted residuals
1.3.1. Homogeneous Dirichlet boundary conditions
1.3.2. Inhomogeneous Dirichlet boundary conditions
1.3.3. Dirichlet/Neumann boundary conditions
1.3.4. Neumann/Dirichlet boundary conditions
1.4. Helmholtz equation
1.5. Steady diffusion with quadratic elements
1.5.1. Element nodes and global nodes
1.5.2. Galerkin finite element equations
1.5.3. Thomas algorithm for pentadiagonal system
1.5.4. Element matrices
1.5.5. Finite element code
1.5.6. Node condensation
1.5.7. Arbitrary interior nodes
1.6. Steady diffusion with quadratic modal expansions
2.1. Unsteady diffusion
2.1.1. Galerkin projection
2.1.2. Integrating ODEs
2.1.3. Forward Euler method
2.1.4. Numerical stability
2.1.5. Finite element code
2.1.6. Crank-Nicolson integration
2.2. Convection
2.2.1. Linear elements
2.2.2. Numerical dispersion due to spatial discretization
2.2.3. Quadratic elements
2.2.4. Integrating ODEs
2.2.5. Non-linear convection
2.3. Convection-diffusion
2.3.1. Steady linear convection-diffusion
2.3.2. Non-linear convection-diffusion
2.4. Beam bending
2.4.1. Euler-Bernoulli beam
2.5. Finite element methods for beam bending
2.5.1. Hermitian elements
2.5.2. Galerkin projection
2.5.3. Element stiffness and mass matrices
2.5.4. One-element cantilever beam
2.5.5. Cantilever beam with nodal loads
2.6. Beam buckling
2.6.1. Tip compression
2.6.2. Buckling under a compressive tip force
2.6.3. Buckling of a heavy vertical column
3.1. Element nodal sets
3.1.1. Lagrange interpolation
3.1.2. Evenly-spaced nodes
3.1.3. Element matrices
3.1.4. C° continuity and shared element nodes
3.2. Change of element nodal sets
3.3. Spectral interpolation
3.3.1. Lobatto nodal base
3.3.2. Discretization code
3.3.3. Legendre polynomials
3.3.4. Chebyshev second-kind nodal base
3.4. Lobatto interpolation and element matrices
3.4.1. Lobatto mass matrix
3.4.2. Lobatto integration quadrature
3.4.3. Computation of the Lobatto mass matrix
3.4.4. Computation of the Lobatto diffusion matrix
3.5. Spectral element code for steady diffusion
3.5.1. Spectral accuracy
3.5.2. Helmholtz equation
3.5.3. Node condensation
3.6. Modal expansion
3.6.1. Relation to the nodal expansion
3.6.2. Implementation
3.7. Lobatto modal expansion
3.7.1. Element diffusion matrix
3.7.2. Element mass matrix
3.7.3. Modal spectral element method
3.8. Arbitrary nodal sets
3.9. Unsteady diffusion
3.9.1. Crank-Nicolson discretization
3.9.2. Forward Euler discretization
4.1. Convection-diffusion in two dimensions
4.1.1. Boundary conditions
4.1.2. Galerkin projection
4.1.3. Domain discretization and interpolation
4.1.4. Galerkin finite element equations
4.1.5. Implementation of the Dirichlet boundary condition
4.1.6. Split nodes
4.1.7. Variational formulation
4.2. Three-node triangles
4.2.1. Element matrices
4.2.2. Computation of the element diffusion matrix
4.2.3. Computation of the element mass matrix
4.2.4. Proof of the integration formula (4.2.36)
4.2.5. Computation of the element advection matrix
4.3. Grid generation
4.3.1. Successive subdivisions
4.3.2. Delaunay triangulation
4.3.3. Generalized connectivity matrices
4.3.4. Element and node labelling schemes
4.4. Laplace's equation with the Dirichlet boundary condition
4.5. Eigenvalues of the Laplacian operator
4.6. Convection-diffusion with the Dirichlet boundary condition
4.7. Helmholtz's equation with the Neumann boundary condition
4.8. Laplace's equation with arbitrary boundary conditions
4.9. Surface elements
4.10. Bilinear quadrilateral elements
5.1. Six-node triangular elements
5.1.1. Integral over a triangle
5.1.2. Isoparametric interpolation and element matrices
5.1.3. Element matrices and integration quadratures
5.1.4. Elements with straight edges
5.2. Grid generation
5.2.1. Circular disk
5.2.2. Square
5.2.3. L-shaped domain
5.2.4. Square with a square or circular hole
5.2.5. A rectangle with a circular hole
5.3. Laplace and Poisson equations
5.3.1. Laplace equation
5.3.2. Eigenvalues of the Laplacian operator
5.3.3. Poisson equation
5.4. Convection-diffusion with the Dirichlet boundary condition
5.5. High-order triangle expansions
5.5.1. Computation of the node interpolation functions
5.5.2. The Lebesgue constant
5.5.3. Node condensation
5.6. Appell polynomial base
5.6.1. Incomplete biorthogonality
5.6.2. Incomplete orthogonality
5.6.3. Generalized Appell polynomials
5.7. Proriol polynomial base
5.7.1. Orthogonality
5.7.2. Orthogonal expansion
5.8. High-order node distributions
5.8.1. Node distribution based on a one-dimensional master grid
5.8.2. Uniform grid
5.8.3. Lobatto grid on the triangle
5.8.4. The Fekete set
5.8.5. Further nodal distributions
5.9. Modal expansions in a triangle
5.9.1. Implementation of the modal expansion
5.9.2. Properties of the modal expansion
5.10. Surface elements
5.10.1. Surface gradient
5.10.2. Grid generation
5.11. High-order quadrilateral elements
5.11.1. Eight-node serendipity elements
5.11.2. 12-node serendipity elements
5.11.3. Grid nodes via tensor-product expansions
5.11.4. Modal expansion
6.1. Elements of elasticity theory
6.1.1. Deforma4pn and constitutive equations
6.1.2. Linear elasticity
6.2. Plane stress and plane strain analysis
6.2.1. Plane stress analysis
6.2.2. Plane strain analysis
6.2.3. Finite element formulation
6.3. Finite element plane stress analysis
6.3.1. Deformation due to an edge force
6.3.2. Deformation due to a body force
6.4. Plate bending
6.4.1. Equilibrium equations
6.4.2. Boundary conditions
6.4.3. Constitutive and governing equations
6.4.4. Circular plate
6.5. Hermite triangles
6.6. Morley's triangle
6.7. Conforming triangles
6.7.1. Six-node, 21-dof triangle
6.7.2. The Hsieh-Clough-Tocher (HCT) element
6.8. Finite element methods for plate bending
6.8.1. Formulation as a biharmonic equation
6.8.2. Formulation as a system of Poisson equations
6.9. Buckling and wrinkling
7.1. Governing equations
7.2. Finite element formulation
7.2.1. Galerkin projections
7.2.2. Discrete equations
7.3. Stokes flow
7.3.1. Governing equations
7.3.2. Galerkin finite element equations
7.3.3. Triangularization
7.4. Stokes flow in a rectangular cavity
7.5. Navier-Stokes flow
7.5.1. Steady state
7.5.2. Time integration
7.5.3. Formulation based on the pressure Poisson equation
8.1. Convection-diffusion in three dimensions
8.1.1. Boundary conditions
8.1.2. Domain discretization
8.1.3. Galerkin projection
8.1.4. Galerkin finite element equations
8.1.5. Element matrices
8.1.6. Implementation of the Dirichlet boundary condition
8.2. Tetrahedral elements
8.2.1. Parametric representation
8.2.2. Integral over the volume of a tetrahedron
8.2.3. Element subdivision into eight tetrahedra
8.2.4. Element subdivision into 12 tetrahedra
8.2.5. Isoparametric interpolation
8.2.6. Element diffusion matrix
8.2.7. Element mass matrix
8.2.8. Proof of the integration formula (8.2.36)
8.2.9. Element advection
matrix
8.3. Domain discretization into four-node tetrahedra
8.3.1. Delaunay tessellation
8.4. Finite element codes with four-node tetrahedra
8.4.1. Laplace's equation
8.4.2. Eigenvalues of the Laplacian operator
8.5. Orthogonal polynomials over a tetrahedron
8.5.1. Karniadakis and Sherwin polynomials
8.5.2. Orthogonal expansion
8.6. High-order and spectral tetrahedral elements
8.6.1. Uniform node distributions
8.6.2. Arbitrary node distributions
8.6.3. Spectral node distributions
8.6.4. Gradient of the element node interpolation functions
8.6.5. Numerical integration
8.7. 10-node quadratic tetrahedra
8.7.1. Node interpolation functions
8.7.2. Element diffusion and mass matrices
8.7.3. Domain discretization
8.7.4. Laplace's equation
8.7.5. Eigenvalues of the Laplacian operator
8.8. Modal expansions in a tetrahedron
8.9. Hexahedral elements
8.9.1. Parametric representation
8.9.2. Integral over the volume of the hexahedron
8.9.3. High-order and spectral hexahedral elements
8.9.4. Modal expansion
A. Mathematical supplement
A.1. Index notation
A.2. Kronecker's delta
A.3. Alternating tensor
Note continued: A.4. Two- and three-dimensional vectors
A.5. Del or nabla operator
A.6. Gradient and divergence
A.7. Vector identities
A.8. Gauss divergence theorem
A.9. Gauss divergence theorem in the plane
A.10. Stokes's theorem
B. Orthogonal polynomials
B.1. Definitions and basic properties
B.1.1. Orthogonality against lower-degree polynomials
B.1.2. Roots of orthogonal polynomials
B.1.3. Discrete orthogonality
B.1.4. Gram polynomials
B.1.5. Recursion relation
B.1.6. Evaluation as the determinant of a tridiagonal matrix
B.1.7. Clenshaw's algorithm
B.1.8. Gram-Schmidt orthogonalization
B.1.9. Orthonormal polynomials
B.1.10. Christoffel-Darboux formula
B.2. Gaussian integration quadratures
B.2.1. Evaluation of the integration weights
B.2.2. Standard Gaussian quadratures
B.3. Lobatto integration quadrature
B.4. Chebyshev integration quadrature
B.5. Legendre polynomials
B.6. Lobatto polynomials
B.7. Chebyshev polynomials
B.8. Jacobi polynomials
C. Linear solvers
C.1. Gauss elimination
C.1.1. Pivoting
C.1.2. Implementation
C.1.3. Symmetric matrices
C.1.4. Computational cost
C.1.5. Gauss elimination code
C.1.6. Multiple right-hand sides
C.1.7. Computation of the inverse
C.1.8. Gauss-Jordan reduction
C.2. Iterative methods based on matrix splitting
C.2.1. Jacobi's method
C.2.2. Gauss-Seidel method
C.2.3. Successive over-relaxation (SOR)
C.2.4. Operator- and grid-based splitting
C.3. Iterative methods based on path search
C.3.1. Symmetric and positive-definite matrices
C.3.2. General methods
C.4. Finite element system-solvers
D. Function interpolation
D.1. The interpolating polynomial
D.1.1. Vandermonde matrix
D.1.2. Generalized Vandermonde matrix
D.1.3. Newton interpolation
D.2. Lagrange interpolation
D.2.1. Cauchy relations
D.2.2. Representation in terms of a generating polynomial
D.2.3. First derivative and the node differentiation matrix
D.2.4. Representation in terms of the Vandermonde matrix
D.2.5. Lagrange polynomials corresponding to polynomial roots
D.2.6. Lagrange polynomials for Hermite interpolation
D.3. Error in polynomial interpolation
D.3.1. Convergence and the Lebesgue constant
D.4. Chebyshev interpolation
D.5. Lobatto interpolation
D.6. Interpolation in two and higher dimensions
E. Element grid generation
F. Glossary
G. MATLAB primer
G.1. Programming in MATLAB
G.1.1. Grammar and syntax
G.1.2. Precision
G.1.3. MATLAB commands
G.1.4. Elementary examples
G.2. MATLAB functions
G.3. Numerical methods
G.4. MATLAB graphics.