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The finite element method for initial value problems : mathematics and computations / Karan S. Surana, J.N. Reddy

Author
Surana, Karan S.
Published
Boca Raton, FL : CRC Press, Taylor & Francis Group, [2018]
Physical Description
xxvi, 604 pages ; 27 cm
Additional Creators
Reddy, J. N. (Junuthula Narasimha), 1945-

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Contents
Machine generated contents note: 1.Introduction -- 1.1.General overview -- 1.2.Space-time coupled methods for ωxt -- 1.3.Space-time coupled methods using space-time strip -- 1.4.Space-time decoupled or quasi methods -- 1.5.General remarks -- 1.6.Space-time coupled finite element method -- 1.7.Space-time decoupled finite element method -- 1.8.Time integration of ODEs resulting from STDFEM -- 1.9.Stability -- 1.10.Accuracy and Error -- 1.10.1.Space-time coupled FEM over space-time domain ωxt: a posteriori computation -- 1.10.2.Space-time coupled method for a space-time strip or slab with time-marching: a posteriori computation -- 1.10.3.Space-time decoupled finite element method: a posteriori computation -- 1.10.4.A priori error estimations -- 1.11.Mode superposition technique -- 1.12.Summary -- 2.Concepts from Functional Analysis and Calculus of Variations -- 2.1.General comments -- 2.2.Spaces, functions, function spaces, and operators -- 2.2.1.Space and time -- 2.2.2.Hilbert spaces Hk(ωx) of functions φ(x, y, z) -- 2.2.3.Definition of scalar product in Hk(ωx) space -- 2.2.4.Properties of scalar product in Hk(ωx) -- 2.2.5.Norm of u in Hilbert space Hk(ωx) -- 2.2.6.Seminorm of u in Hilbert space Hk(ωx) -- 2.2.7.Hilbert space H(k)(ωxt) of functions φ(x,y,z,t) -- 2.2.8.Definition of scalar product in Hk(ωxt) space -- 2.2.9.Properties of scalar product in H(k)(ωxt) -- 2.2.10.Norm of u in Hilbert space Hk(ωxt) -- 2.3.Operators -- 2.3.1.Classification of space-time differential operators -- 2.3.2.Integration by parts (IBP) -- 2.4.Elements of calculus of variations -- 2.4.1.Concept of variation of a space-time functional -- 2.4.2.Euler's equation: simplest variational problem -- 2.4.3.Variation of a space-time functional: some practical aspects -- 2.5.Riemann and Lebesgue integrals -- 2.6.Model problems -- 2.7.Summary -- 3.Space-Time Coupled Classical Methods of Approximation -- 3.1.Introduction -- 3.2.Space-time integral forms based on fundamental lemma -- 3.2.1.Classical space-time Galerkin method -- 3.2.2.Classical space-time Galerkin method with weak form -- 3.2.3.Classical space-time Petrov-Galerkin method -- 3.2.4.Classical space-time weighted residual method -- 3.2.5.Choosing No(x, t) and Ni(x, t); i = 1,2, ..., n -- 3.3.Space-time least squares process -- 3.3.1.Non-self-adjoint differential operators -- 3.3.2.Non-linear differential operators -- 3.4.STVC or STVIC of space-time integral forms -- 3.5.Model problems -- 3.5.1.Model problem 1: ID scalar wave equation -- 3.5.1.1.Space-time Galerkin method -- 3.5.1.2.Space-time Galerkin method with weak form -- 3.5.1.3.Space-time least squares method based on residual functional -- 3.5.2.Model problem 2: ID Burgers equation -- 3.5.2.1.Space-time Galerkin method -- 3.5.2.2.Space-time Galerkin method with weak form -- 3.5.2.3.Space-time least squares method based on the residual functional -- 3.6.Summary -- 4.Space-Time Finite Element Method -- 4.1.Introduction -- 4.2.Space-time domain and discretization -- 4.3.Mathematics of space-time finite element processes -- 4.3.1.Space-time finite element processes based on STGM, STPGM, and STWRM -- 4.3.2.Space-time finite element processes based on STGM/WF -- 4.3.3.Space-time finite element processes based on residual functional: STLSP -- 4.3.3.1.Non-self-adjoint space-time differential operators -- 4.3.3.2.Non-linear space-time differential operators -- 4.3.4.Summary of main steps (STLS finite element process) -- 4.4.Model problems -- 4.4.1.Model problem 1: ID scalar wave equation -- 4.4.1.1.Space-time finite element process based on STGM -- 4.4.1.2.Space-time finite element process based on STGM/WF -- 4.4.1.3.Space-time finite element process based on residual functional (STLSP) -- 4.4.1.4.Space-time finite element process based on residual functional (STLSP) using a first order system of PDEs -- 4.4.1.5.Numerical studies -- 4.4.2.Model problem 2: ID pure advection -- 4.4.2.1.Space-time finite element process based on STGM (and STGM/WF) -- 4.4.2.2.Space-time finite element process based on residual functional (STLSP) -- 4.4.2.3.Numerical studies -- 4.4.3.Model problem 3: ID convection-diffusion equation -- 4.4.3.1.Space-time finite element process based on STGM -- 4.4.3.2.Space-time finite element process based on STGM/WF -- 4.4.3.3.Space-time finite element process based on residual functional (STLSP) -- 4.4.3.4.Space-time finite element process based on STLSP using a system of first order PDEs -- 4.4.3.5.Numerical studies -- 4.4.4.Model problem 4: ID Burgers equation -- 4.4.4.1.Space-time finite element process based on STGM -- 4.4.4.2.Space-time finite element process based on STGM/WF -- 4.4.4.3.Space-time finite element process based on residual functional (STLSP) -- 4.4.4.4.Space-time finite element process based on residual functional (STLSP) using a system of first order PDEs -- 4.4.4.5.Numerical studies -- 4.4.5.Model problem 5: ID diffusion-reaction equations -- 4.4.5.1.Space-time finite element process based on residual functional (STLSP) -- 4.4.5.2.Finite element process based on residual functional (STLSP) using a system of first order PDEs -- 4.4.5.3.Numerical studies -- 4.4.6.Model problem 6: ID normal shocks -- 4.4.6.1.Space-time finite element formulation based on residual function (STLSP) -- 4.4.6.2.Numerical studies -- 4.4.7.Model problem 7: 2D phase transition -- 4.4.7.1.Mathematical model for phase transition -- 4.4.7.2.Space-time finite element formulation based on residual function (STLSP) -- 4.4.7.3.Numerical studies: liquid-solid phase transition -- 4.4.7.4.Numerical studies: solid-liquid phase transition -- 4.5.Summary -- 5.Space-Time Decoupled or Quasi Finite Element Formulation -- 5.1.Introduction and basic methodology -- 5.2.Details of space-time decoupled approach: model problems -- 5.3.Summary -- 6.Methods of Approximation for ODEs in Time -- 6.1.Introduction -- 6.2.Choice of the methods of approximation -- 6.2.1.Methods based on Taylor series -- 6.2.2.Methods based on integral forms constructed using ODEs in time -- 6.3.Basic concepts in direct integration methods -- 6.3.1.Euler's method -- 6.3.2.Runge--Kutta methods -- 6.3.2.1.Second order Runge-Kutta method (n = 2) -- 6.3.2.2.Third order Runge-Kutta method (n = 3) -- 6.3.2.3.Fourth order Runge-Kutta method (n = 4) -- 6.3.3.Numerical examples of direct integration methods -- 6.4.Basic concept in explicit methods -- 6.5.Basic concept in implicit methods -- 6.6.Time integration in structural dynamics -- 6.6.1.The central difference method (explicit method) -- 6.6.2.The Houbolt method (implicit method) -- 6.6.3.Wilson's θ method (implicit method) -- 6.6.3.1.Wilson's θ method: linear acceleration -- 6.6.3.2.Wilson's θ method: constant average acceleration -- 6.6.4.Newmark's method (implicit method) -- 6.6.4.1.Newmark's method: constant average acceleration -- 6.6.4.2.Newmark's method: linear acceleration -- 6.7.Numerical examples -- 6.7.1.ID scalar wave equation -- 6.7.1.1.Central difference method -- 6.7.1.2.Houbolt method -- 6.7.1.3.Wilson's θ method -- 6.7.1.4.Newmark's method -- 6.8.Methods of approximation based on integral forms in time -- 6.8.1.Mathematical classification of time differential operators -- 6.8.2.Classical integral methods of approximation for ODEs in time -- 6.8.2.1.Integral form of (6.139) based on fundamental lemma -- 6.8.2.2.Classical Galerkin method in time -- 6.8.2.3.Classical Galerkin method with weak form in time -- 6.8.2.4.Classical Petrov-Galerkin method in time -- 6.8.2.5.Classical weighted residual method in time -- 6.8.2.6.Classical least squares method in time -- 6.8.2.7.When is an integral form in time for an ODE a variational formulation? -- 6.8.3.Variational consistency or variational inconsistency of time integral forms resulting from integral methods of approximation -- 6.9.Model problems -- 6.9.1.ID linear dynamics: scalar equation in modal basis -- 6.9.1.1.Classical GM, PGM, and WRM in time -- 6.9.1.2.Classical Galerkin method with weak form in time -- 6.9.1.3.Classical least squares process in time based on residual functional -- 6.9.2.ID linear dynamics: scalar equation -- 6.9.2.1.Classical GM, PGM, and WRM in time -- 6.9.2.2.Classical Galerkin method with weak form in time -- 6.9.2.3.Classical least squares process in time based on residual functional -- 6.9.3.ID non-linear dynamics: scalar equation -- 6.9.3.1.Classical GM, PGM, and WRM in time -- 6.9.3.2.Classical Galerkin method with weak form in time -- 6.9.3.3.Classical least squares process in time based on residual functional -- 6.10.Summary -- 7.Finite Element Method for ODEs in Time -- 7.1.Introduction -- 7.2.Time domain, increment of time, and time discretization -- 7.3.Finite element process in time for ODEs in time -- 7.3.1.Finite element processes based on GM, PGM, and WRM in time -- 7.3.2.Finite element processes based on GM/WF in time -- 7.3.3.Finite element processes based on residual functional: LSP in time -- 7.3.3.1.Linear time operator (non-self-adjoint) -- 7.3.3.2.Non-linear time operator -- 7.3.4.Remarks on various time finite element processes based on methods of approximation in time -- 7.4.Model problems: finite element process in time -- 7.4.1.ID linear dynamics: scalar equation in modal basis -- 7.4.1.1.Finite element processes in time based on GM376 -- 7.4.1.2.Finite element processes in time based on GM/WF -- 7.4.1.3.Finite element processes in time based on residual functional: LSP -- 7.4.1.4.Finite element processes in time based on residual functional: LSP, first order system -- 7.4.1.5.Numerical studies -- 7.4.2.ID linear dynamics: scalar equation -- 7.4.2.1.Finite element processes in time based on GM, PGM, and WRM -- 7.4.2.2.Finite element processes in time based on GM/WF -- 7.4.2.3.Finite element processes in time based on residual functional: LSP -- 7.4.2.4.Numerical studies -- and Contents note continued: 7.4.3.ID non-linear dynamics: scalar equation -- 7.4.3.1.Finite element processes in time based on GM399 -- 7.4.3.2.Finite element processes in time based on GM/WF -- 7.4.3.3.Finite element processes in time based on residual functional: LSP -- 7.4.3.4.Finite element processes in time based on residual functional: LSP, first order system -- 7.4.3.5.Numerical studies -- 7.4.4.ID scalar wave equation -- 7.4.4.1.Numerical studies -- 7.4.5.Mixing problem -- 7.4.5.1.Numerical studies -- 7.5.Summary -- 8.Stability Analysis of the Approximation Methods -- 8.1.Introduction -- 8.2.Stability of space-time coupled methods -- 8.3.Stability analysis of space-time decoupled methods -- 8.3.1.Recursive relation for time-marching solutions of ODEs -- 8.3.2.Spectral radius of [B]: boundedness of [B] -- 8.4.Specific forms of the time approximation operator -- 8.4.1.Stability of central difference method -- 8.4.2.Stability of Houbolt method -- 8.4.3.Stability of Wilson's 6 method -- 8.4.3.1.Linear acceleration method -- 8.4.3.2.Constant average acceleration method -- 8.4.4.Stability of Newmark's method -- 8.4.4.1.Constant average acceleration method -- 8.4.4.2.Linear acceleration method -- 8.4.5.General remarks -- 8.4.6.Stability of least squares finite element method -- 8.5.Summary -- 9.Mode Superposition Technique -- 9.1.Introduction -- 9.1.1.Fundamental properties of eigenpairs -- 9.2.General remarks on free vibrations -- 9.3.Mode superposition method -- 9.3.1.Transforming initial conditions -- 9.3.2.Time response (or transient dynamic response) of undamped systems -- 9.3.3.Time response of damped systems -- 9.3.3.1.Proportional damping -- 9.3.3.2.Rayleigh damping -- 9.4.Analytical solution of undamped equations in modal basis -- 9.4.1.Constant fi -- 9.4.2.Harmonic fi -- 9.5.Analytical solution of damped equations in modal basis -- 9.5.1.Solution of homogeneous form: complementary solution -- 9.5.1.1.Critically damped system -- 9.5.1.2.Overdamped system -- 9.5.1.3.Underdamped system -- 9.5.2.Solution of nonhomogeneous form: particular solution -- 9.5.2.1.Constant fi -- 9.5.2.2.Harmonic fi -- 9.6.Analytical solutions of damped systems -- 9.6.1.Solution of homogeneous form: complementary solution -- 9.6.1.1.Critically damped system -- 9.6.1.2.Overdamped system -- 9.6.1.3.Underdamped system -- 9.6.2.Solution of nonhomogeneous form: particular solution -- 9.6.2.1.Constant [ƒ] -- 9.6.2.2.Harmonic [ƒ] -- 9.7.General remarks on modal basis and theoretical solutions -- 9.8.Model problem: 1D scalar wave equation -- 9.9.Transient response using lowest modes of vibration -- 9.10.Guyan reduction -- 9.10.1.Static condensation -- 9.10.2.Guyan reduction -- 9.11.Summary -- 10.Approximation Errors, Convergence, and Convergence Rates -- 10.1.Introduction -- 10.2.Preliminaries and some definitions -- 10.2.1.Errors -- 10.2.2.A priori error estimation and a posteriori error computation -- 10.2.3.Accuracy and time accuracy -- 10.2.4.Convergence and convergence rates -- 10.3.Space-time coupled finite element processes -- 10.3.1.A priori error estimation -- 10.3.1.1.Convergence rates -- 10.3.1.2.General remarks on a priori error estimates and use of optimal theoretical convergence rates -- 10.3.1.3.Importance and significance of higher order spaces -- 10.3.2.A posteriori error computations -- 10.3.3.Model problem: ID convection-diffusion equation -- 10.4.Space-time decoupled finite element processes -- 10.5.ODEs in time -- 10.5.1.Methods based on finite difference or finite volume techniques -- 10.5.2.Time integration methods in structural mechanics -- 10.5.2.1.Houbolt method -- 10.5.2.2.Wilson's 9 method -- 10.5.2.3.Newmark's method -- 10.5.2.4.Convergence and convergence rates of Wilson's θ and Newmark's methods -- 10.5.2.5.Model problem: ID scalar wave equation -- 10.5.2.6.General remarks -- 10.5.3.The finite element method in time -- 10.5.3.1.A priori error estimation -- 10.5.3.2.Convergence rates -- 10.5.3.3.Importance and significance of higher order spaces -- 10.5.3.4.Model problem: ID scalar wave equation -- 10.5.3.5.Model problem: mixing problem -- 10.5.3.6.A posteriori error computations -- 10.6.Summary -- 11.Mapping and Interpolation -- 11.1.Mapping in one dimension -- 11.1.1.Mapping of points -- 11.1.2.Mapping of lengths -- 11.1.3.Dependent variable behavior over ωe or ωε -- 11.2.Interpolation in one dimension -- 11.2.1.Polynomial interpolation -- 11.2.2.Lagrange interpolation in 1D -- 11.2.3.Co p-version hierarchical interpolation functions in 1D -- 11.2.4.p-version interpolations of class C1(ωe) in 1D -- 11.2.5.Higher order global differentiability p-version interpolations in 1D -- 11.3.Mapping in two dimensions: quadrilateral elements -- 11.3.1.Mapping of points -- 11.3.2.Mapping of lengths -- 11.3.3.Mapping of areas -- 11.3.4.Mapping in 2D using boundary nodes -- 11.4.Interpolation in two dimensions: quadrilateral elements -- 11.4.1.Obtaining derivatives of φ(ε, η) with respect to x and y -- 11.4.2.C00 local approximations over ωεη or ωm: quadrilateral elements -- 11.4.3.C00 p-version hierarchical local approximations based on Lagrange polynomials -- 11.4.4.C00 p-version hierarchical local approximations: rectangular family of elements -- 11.4.5.2D Cij(ωe) approximations for quadrilateral elements -- 11.5.1D and 2D approximations based on Legendre polynomials -- 11.5.1.Legendre polynomials -- 11.5.2.1D p-version C00 hierarchical approximation functions (Legendre polynomials) -- 11.5.3.2D p-version C00 hierarchical interpolation functions for quadrilateral elements (Legendre polynomials) -- 11.5.4.2D Cij p-version interpolations functions for quadrilateral elements (Legendre polynomials) -- 11.6.1D and 2D interpolations based on Chebyshev polynomials -- 11.6.1.Chebyshev polynomials -- 11.6.2.ID C0 p-version hierarchical interpolations based on Chebyshev polynomials -- 11.6.3.2D p-version C00 hierarchical interpolation functions for quadrilateral elements (Chebyshev polynomials) -- 11.6.4.2D Cij p-version interpolation functions for quadrilateral elements (Chebyshev polynomials) -- 11.7.Interpolation in two dimensions: triangular elements -- 11.7.1.Area coordinates -- 11.8.Serendipity family of C00 interpolations -- 11.9.Mapping in three dimensions: hexahedron elements -- 11.9.1.Mapping of points -- 11.9.2.Mapping of lengths -- 11.9.3.Mapping of volumes -- 11.10.Interpolation in three dimensions: hexahedron elements -- 11.10.1.Obtaining derivatives of φeh(ε, η, ζ) with respect to x,y,z -- 11.10.2.Local approximation for a dependent variable φ over ωm -- 11.10.2.1.Polynomial approximation: C000 (ωe) -- 11.10.2.2.Tensor product: C000(ωe) and Cijk(ωe) -- 11.11.Interpolation in three dimensions: tetrahedron elements -- 11.11.1.Basis functions of class C000(ωe) based on Lagrange interpolations -- 11.11.2.Lagrange family C000 tetrahedron elements based on volume coordinates -- 11.11.3.Higher degree C000 basis functions using volume coordinates -- 11.11.3.1.Four-node linear tetrahedron element (p-level of one) -- 11.11.3.2.A ten-node tetrahedron element (p-level of 2) -- 11.12.Summary -- Appendix A: Nondimensionalizing Mathematical Models -- A.1.Introduction -- A.2.Model problems -- A.2.1.1D pure advection -- A.2.2.1D convection-diffusion equation -- A.2.3.1D Burgers equation -- A.2.4.1D wave propagation in elastic medium (structural dynamics) -- A.2.5.1D scalar wave equation -- A.2.6.1D diffusion-reaction equations -- A.2.7.1D compressible flow (Eulerian description) -- A.2.7.1.Continuity equation -- A.2.7.2.Momentum equation -- A.2.7.3.Energy equation -- A.2.7.4.Constitutive equations -- A.2.7.5.Equation of state -- A.2.7.6.Summary of the dimensionless form of the GDEs -- Appendix B Numerical Integration using Gauss Quadrature -- B.1.Gauss quadrature in R1, R2, and R3 -- B.1.1.Line integrals over ωm = ωε = [-1,1] -- B.1.2.Area integrals over ωm = ωεηζ = [-1,1] x [-1,1] -- B.1.3.Volume integrals over ωm = εηζ = [-1,1] x [-1,1] x [-1,1] -- B.2.Gauss quadrature over triangular domains.
Subject(s)
ISBN
9781138576377 hardcover
1138576379 hardcover
Bibliography Note
Includes bibliographical references and index.
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