Finite Element Method: Physics and Solution Methods

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
	10.8 Numerical integration of element equilibrium equations
	10.9 Element assembly
	10.10 Imposing the Essential boundary conditions
		10.10.1 Symbolic representation of essential boundary conditions
		10.10.2 Numerical implementation of essential boundary conditions
	Problems
	Reference
Chapter 11 Transient thermal analysis
	11.1 Introduction
	11.2 Transient heat transfer equation
		11.2.1 Boundary/initial value problem
		11.2.2 Element equilibrium equation of one-dimensional, transient heat transfer
		11.2.3 Global equilibrium equation of one-dimensional, transient heat transfer
		11.2.4 Global boundary conditions
	11.3 Finite difference approximations to derivatives
		11.3.1 Temporal discretization of a continuous function
		11.3.2 Taylor series expansion
		11.3.3 Approximations to the first derivative of a function
	11.4 Direct time integration of the heat transfer equation
		11.4.1 Forward difference or Euler method
		11.4.2 Backward difference method
		11.4.3 Central difference or Crank-Nicholson method
		11.4.4 Generalized trapezoidal method
	11.5 Solution algorithm
		11.5.1 Explicit and implicit time integration methods
	11.6 Convergence, stability, and accuracy of time integration methods
		11.6.1 Modal expansion of the semidiscrete first-order equation
		11.6.2 Stability of the semidiscretized first-order equation
		11.6.3 Modal expansion of the generalized trapezoidal algorithm
		11.6.4 Stability of the generalized trapezoidal algorithm
		11.6.5 Fourier-von Neumann stability analysis of the generalized trapezoidal method
		11.6.6 Consistency and rate of convergence
	References
Chapter 12 Transient analysis of solids and structures
	12.1 Introduction
	12.2 Vibration of single degree of freedom systems
		12.2.1 Free vibrations: complementary solution
		12.2.2 Response to harmonic excitations: particular solution
		12.2.3 Combined response: complimentary and particular solutions
		12.2.4 Transient vibration
	12.3 Initial/boundary value problems for deformable solids
		12.3.1 Two-dimensional deformable solid
		12.3.2 One-dimensional bar
		12.3.3 Euler-Bernoulli beam
	12.4 Vibration response of an Euler-Bernoulli beam
		12.4.1 Eigenvalue problem
		12.4.2 Free vibration problem
	12.5 Semidiscrete equations of motion
		12.5.1 Two-dimensional deformable element
		12.5.2 One-dimensional elastic bar element
		12.5.3 Euler-Bernoulli beam element
	12.6 Mass matrix
		12.6.1 Consistent mass matrices
		12.6.2 Lumped mass matrix
	12.7 Damping matrix
	12.8 Global equation of motion
	12.9 Analytical analysis of vibration of semidiscrete systems
		12.9.1 Eigenvalue problem for the semidiscrete equation of motion
		12.9.2 Orthogonality of the eigenvectors
		12.9.3 Response to initial excitations by modal analysis
	12.10 Direct time integration of the equation of motion of a solid
		12.10.1 Central finite difference approximations: explicit time integration
		12.10.2 Linear and average acceleration methods: implicit time integration
		12.10.3 Newmark’s method for direct time integration
		12.10.4 α-Method for direct time integration
		12.10.5 Initial conditions
		12.10.6 Solution algorithm
	12.11 Convergence, stability, and accuracy of time integration methods
		12.11.1 Stability of the explicit method
		12.11.2 Stability and consistency of the Newmark and -methods
		Problems
	References
Appendix A
	A.1 Arithmetic
		A.1.1 Arithmetic operators
	A.2 Mathematical functions
	A.3 Matrices
		A.3.1 Subscripting and colon notation
		A.3.2 Matrix and array operations
	A.4 Relational operators and flow control
	A.5 Scripts and functions
		A.5.1 Script m-files
		A.5.2 Function m-files
	A.6 Reading and saving files
		A.6.1 Reading an input file
		A.6.2 Saving an output file by using a formatted statement
	A.7 Plotting
		A.7.1 Plot function
	References
Appendix B
	B.1 Structure of a finite element code
	B.2 Finite element program for solution of second-order ODEs
		B.2.1 Example
		B.2.2 MATLAB-based ODE solver
	B.3 Finite element program for a two-dimensional frame
		B.3.1 Data input into the finite element program frame2D
		B.3.2 Example
		B.3.3 MATLAB-based FEA code: Frame2D
Appendix C
	C.1 GUI-based analysis
	C.2 APDL-based analysis
	References
Appendix D
	D.1 Example: simply supported beam
		D.1.1 GUI-based solution for a simply supported beam under uniform pressure
		D.1.2 ANSYS APDL macro for BEAM188
		D.1.3 ANSYS APDL macro for BEAM4
	D.2 Example: suspended bridge
		D.2.1 GUI-based solution of the suspended bridge problem
		D.2.2 APDL-based solution of the suspended bridge problem
	References
Appendix E
Appendix F
	F.1 GUI-based solution of the thermomechanical deformation problem
	F.2 APDL-based solution of the thermomechanical deformation problem
Index
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