Finite element method physics and solution methods /

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Muftu, Sinan
Formato: Electrónico eBook
Idioma:Inglés
Publicado: London : Academic Press, 2022.
Temas:
Finite element method.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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