Kooperativer Bibliotheksverbund

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Format: 1 online resource (542 pages)

ISBN: 0-12-823200-5

Content: "Finite Element Method: Physics and Solution Methods aims to provide the reader a sound understanding of the physical systems and solution methods to enable effective use of the finite element method. This book focuses on one- and two-dimensional elasticity and heat transfer problems with detailed derivations of the governing equations. The connections between the classical variational techniques and the finite element method are carefully explained. Following the chapter addressing the classical variational methods, the finite element method is developed as a natural outcome of these methods where the governing partial differential equation is defined over a subsegment (element) of the solution domain. As well as being a guide to thorough and effective use of the finite element method, this book also functions as a reference on theory of elasticity, heat transfer, and mechanics of beams."--

Note: Includes index. , Front cover -- Half title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgments -- Chapter 1 Introduction -- 1.1 Modeling and simulation -- 1.1.1 Boundary and initial value problems -- 1.1.2 Boundary value problems -- 1.2 Solution methods -- Chapter 2 Mathematical modeling of physical systems -- 2.1 Introduction -- 2.2 Governing equations of structural mechanics -- 2.2.1 External forces, internal forces, and stress -- 2.2.2 Stress transformations -- 2.2.3 Deformation and strain -- 2.2.4 Strain compatibility conditions -- 2.2.5 Generalized Hooke's law -- 2.2.6 Two-dimensional problems -- 2.2.7 Balance laws -- 2.2.8 Boundary conditions -- 2.2.9 Total potential energy of conservative systems -- 2.3 Mechanics of a flexible beam -- 2.3.1 Equation of motion of a beam -- 2.3.2 Kinematics of the Euler-Bernoulli beam -- 2.3.3 Stresses in an Euler-Bernoulli beam -- 2.3.4 Kinematics of the Timoshenko beam -- 2.3.5 Stresses in a Timoshenko beam -- 2.3.6 Governing equations of the Euler-Bernoulli beam theory -- 2.3.7 Governing equations of the Timoshenko beam theory -- 2.4 Heat transfer -- 2.4.1 Conduction heat transfer -- 2.4.2 Convection heat transfer -- 2.4.3 Radiation heat transfer -- 2.4.4 Heat transfer equation in a one-dimensional solid -- 2.4.5 Heat transfer in a three-dimensional solid -- 2.5 Problems -- References -- Chapter 3 Integral formulations and variational methods -- 3.1 Introduction -- 3.2 Mathematical background -- 3.2.1 Divergence theorem -- 3.2.2 Green-Gauss theorem -- 3.2.3 Integration by parts -- 3.2.4 Fundamental lemma of calculus of variations -- 3.2.5 Adjoint and self-adjoint operators -- 3.3 Calculus of variations -- 3.3.1 Variation of a functional -- 3.3.2 Functional derivative -- 3.3.3 Properties of functionals -- 3.3.4 Properties of the variational derivative. , 3.3.5 Euler-Lagrange equations and boundary conditions -- 3.4 Weighted residual integral and the weak form of boundary value problems -- 3.4.1 Weighted residual integral -- 3.4.2 Boundary conditions -- 3.4.3 The weak form -- 3.4.4 Relationship between the weak form and functionals -- 3.5 Method of weighted residuals -- 3.5.1 Rayleigh-Ritz method -- 3.5.2 Galerkin method -- 3.5.3 Polynomials as basis functions for Rayleigh-Ritz and Galerkin methods -- 3.6 Problems -- References -- Chapter 4 Finite element formulation of one-dimensional boundary value problems -- 4.1 Introduction -- 4.1.1 Boundary value problem -- 4.1.2 Spatial Discretization -- 4.2 A second order, nonconstant coefficient ordinary differential equation over an element -- 4.2.1 Deflection of a one-dimensional bar -- 4.2.2 Heat transfer in a one-dimensional domain -- 4.3 One-dimensional interpolation for finite element method and shape functions -- 4.3.1 C0 continuous, linear shape functions -- 4.3.2 C0 continuous, quadratic shape functions -- 4.3.3 General form of C0 shape functions -- 4.3.4 One-dimensional, Lagrange interpolation functions -- 4.4 Equilibrium equations in finite element form -- 4.4.1 Element stiffness matrix for constant problem parameters -- 4.4.2 Element stiffness matrix for linearly varying problem parameters a, p, and q -- 4.5 Recovering specific physics from the general finite element form -- 4.6 Element assembly -- 4.7 Boundary conditions -- 4.7.1 Natural boundary conditions -- 4.7.2 Essential boundary conditions -- 4.8 Computer implementation -- 4.8.1 Main-code -- 4.8.2 Element connectivity table -- 4.8.3 Element assembly -- 4.8.4 Boundary conditions -- 4.9 Example problem -- 4.10 Problems -- Chapter 5 Finite element analysis of planar bars and trusses -- 5.1 Introduction -- 5.2 Element equilibrium equation for a planar bar -- 5.2.1 Problem definition. , 5.2.2 Weak form of the boundary value problem -- 5.2.3 Total potential energy of the system -- 5.2.4 Finite element form of the equilibrium equations of an elastic bar -- 5.3 Finite element equations for torsion of a bar -- 5.4 Coordinate transformations -- 5.4.1 Transformation of unit vectors between orthogonal coordinate systems -- 5.4.2 Transformation of equilibrium equations for the one-dimensional bar element -- 5.5 Assembly of elements -- 5.6 Boundary conditions -- 5.6.1 Formal definition -- 5.6.2 Direct assembly of the active degrees of freedom -- 5.6.3 Numerical implementation of the boundary conditions -- 5.7 Effects of initial stress or initial strain -- 5.7.1 Thermal stresses -- 5.7.2 Initial stresses -- 5.8 Postprocessing: Computation of stresses and reaction forces -- 5.8.1 Computation of stresses in members -- 5.8.2 Reaction forces -- 5.9 Error and convergence in finite element analysis -- Problems -- Reference -- Chapter 6 Euler-Bernoulli beam element -- 6.1 Introduction -- 6.2 C1-Continuous interpolation function -- 6.3 Element equilibrium equation -- 6.3.1 Problem definition -- 6.3.2 Weak form of the boundary value problem -- 6.3.3 Total potential energy of a beam element -- 6.3.4 Finite element form of the equilibrium equations of an Euler-Bernoulli beam -- 6.4 General beam element with membrane and bending capabilities -- 6.5 Coordinate transformations -- 6.5.1 Vector transformation between orthogonal coordinate systems in a two-dimensional plane -- 6.5.2 Transformation of equilibrium equations for the Euler-Bernoulli beam element with axial deformation -- 6.6 Assembly, boundary conditions, and reaction forces -- 6.7 Postprocessing and computation of stresses in members -- Example 6.1 -- Problems -- Reference -- Chapter 7 Isoparametric elements for two-dimensional elastic solids -- 7.1 Introduction. , 7.2 Solution domain and its boundary -- 7.2.1 Outward unit normal and tangent vectors along the boundary -- 7.3 Equations of equilibrium for two-dimensional elastic solids -- 7.4 General finite element form of equilibrium equations for a two-dimensional element -- 7.4.1 Variational form of the equation of equilibrium -- 7.4.2 Finite element form of the equation of equilibrium -- 7.5 Interpolation across a two-dimensional domain -- 7.5.1 Two-dimensional polynomials -- 7.5.2 Two-dimensional shape functions -- 7.6 Mapping between general quadrilateral and rectangular domains -- 7.6.1 Jacobian matrix and Jacobian determinant -- 7.6.2 Differential area in curvilinear coordinates -- 7.7 Mapped isoparametric elements -- 7.7.1 Strain-displacement operator matrix, [B] -- 7.7.2 Finite element form of the element equilibrium equations for a Q4-element -- 7.8 Numerical integration using Gauss quadrature -- 7.8.1 Coordinate transformation -- 7.8.2 Derivation of second-order Gauss quadrature -- 7.8.3 Integration of two-dimensional functions by Gauss quadrature -- 7.9 Numerical evaluation of the element equilibrium equations -- 7.10 Global equilibrium equations and boundary conditions -- 7.10.1 Assembly of global equilibrium equation -- 7.10.2 General treatment of the boundary conditions -- 7.10.3 Numerical implementation of the boundary conditions -- 7.11 Postprocessing of the solution -- References -- Chapter 8 Rectangular and triangular elements for two-dimensional elastic solids -- 8.1 Introduction -- 8.1.1 Total potential energy of an element for a two-dimensional elasticity problem -- 8.1.2 High-level derivation of the element equilibrium equations -- 8.2 Two-dimensional interpolation functions -- 8.2.1 Interpolation and shape functions in plane quadrilateral elements -- 8.2.2 Interpolation and shape functions in plane triangular elements. , 8.3 Bilinear rectangular element (Q4) -- 8.3.1 Element stiffness matrix -- 8.3.2 Consistent nodal force vector -- 8.4 Constant strain triangle (CST) element -- 8.5 Element defects -- 8.5.1 Constant strain triangle element -- 8.5.2 Bilinear rectangle (Q4) -- 8.6 Higher order elements -- 8.6.1 Quadratic triangle (linear strain triangle) -- 8.6.2 Q8 quadratic rectangle -- 8.6.3 Q9 quadratic rectangle -- 8.6.4 Q6 quadratic rectangle -- 8.7 Assembly, boundary conditions, solution, and postprocessing -- References -- Chapter 9 Finite element analysis of one-dimensional heat transfer problems -- 9.1 Introduction -- 9.2 One-dimensional heat transfer -- 9.2.1 Boundary conditions for one-dimensional heat transfer -- 9.3 Finite element formulation of the one-dimensional, steady state, heat transfer problem -- 9.3.1 Element equilibrium equations for a generic one-dimensional element -- 9.3.2 Finite element form with linear interpolation -- 9.4 Element equilibrium equations: general ordinary differential equation -- 9.5 Element assembly -- 9.6 Boundary conditions -- 9.6.1 Natural boundary conditions -- 9.6.2 Essential boundary conditions -- 9.7 Computer implementation -- Problems -- Chapter 10 Heat transfer problems in two-dimensions -- 10.1 Introduction -- 10.2 Solution domain and its boundary -- 10.3 The heat equation and its boundary conditions -- 10.3.1 Boundary conditions for heat transfer in two-dimensional domain -- 10.4 The weak form of heat transfer equation in two dimensions -- 10.5 The finite element form of the two-dimensional heat transfer problem -- 10.5.1 Finite element form with linear, quadrilateral (Q4) element -- 10.6 Natural boundary conditions -- 10.6.1 Internal edges -- 10.6.2 External edges subjected to prescribed heat flux -- 10.6.3 External edges subjected to convection -- 10.6.4 External edges subjected to radiation. , 10.7 Summary of finite element form of the heat equation and natural boundary conditions.

Additional Edition: Print version: Muftu, Sinan Finite Element Method San Diego : Elsevier Science & Technology,c2022 ISBN 9780128211274

Language: English