Computational Methods in Engineering: Finite Difference, Finite Volume, Finite Element, and Dual Mesh Control Domain Methods

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
List of Symbols
About the Author
1. Introduction and Preliminaries
	1.1. Background
	1.2. Popular Numerical Methods
		1.2.1. Finite Difference Method
		1.2.2. Finite Volume Method
		1.2.3. Finite Element Method
		1.2.4. Dual Mesh Control Domain Method
	1.3. Common Features of the Numerical Methods
	1.4. Present Study
	1.5. Types of Differential Equations and Problems
		1.5.1. Preliminary Comments
		1.5.2. Order and Types of Differential Equations
		1.5.3. Types of Problems Described by Differential Equations
		1.5.4. Homogeneous and Inhomogeneous Equations
		1.5.5. Examples of IVPs and BVPs
	1.6. Taylor's Series and Elements of Matrix Theory
		1.6.1. Introduction
		1.6.2. Taylor's Series and Taylor's Formula
		1.6.3. Theory of Matrices
	1.7. Interpolation Theory
		1.7.1. Introduction
		1.7.2. Interpolating Polynomials
	1.8. Numerical Integration
		1.8.1. Preliminary Comments
		1.8.2. Trapezoidal and Simpson's Formulas
		1.8.3. Gauss Quadrature Formula
		1.8.4. Extension to Two Dimensions
	1.9. Solution of Linear Algebraic Equations
		1.9.1. Introduction
		1.9.2. Direct Methods
		1.9.3. Iterative Methods
		1.9.4. Iterative Methods for Nonlinear Equations
	1.10. Method of Manufactured Solutions
	1.11. Variational Formulations and Methods
		1.11.1. Background
		1.11.2. Integral Identities
		1.11.3. Integral Formulations and Methods of Approximation
		1.11.4. Weak (Integral) Forms
		1.11.5. The Ritz Method of Approximation
	1.12. Types of Errors
	1.13. Summary
	Problems
2. Finite Difference Method
	2.1. Finite Difference Formulas
		2.1.1. Taylor's Series
		2.1.2. Difference Formulas for First and Second Derivatives
	2.2. Solution of First-Order Ordinary Differential Equations
		2.2.1. Euler's Method
		2.2.2. Runge‒Kutta Family of Methods
		2.2.3. Coupled System of First-Order Differential Equations
	2.3. Solution of Second-Order Ordinary Differential Equations
	2.4. Solution of Partial Differential Equations
		2.4.1. One-Dimensional Problems
		2.4.2. Consistency, Stability, and Convergence
		2.4.3. Two-Dimensional Problems
	2.5. Summary
	Problems
3. Finite Volume Method
	3.1. General Idea
	3.2. One-Dimensional Problems
		3.2.1. Model Differential Equation and Domain Discretization
		3.2.2. Integral Representation of the Governing Equation
		3.2.3. Evaluation of Domain Integrals
		3.2.4. Approximation of the First Derivatives
		3.2.5. Discretized Equations for Interior Nodes
		3.2.6. Discretized Equations for Boundary Nodes
	3.3. Numerical Examples
	3.4. Two-Dimensional Problems
		3.4.1. Model Differential Equation and Domain Discretization
		3.4.2. Integral Statement over a Typical Control Volume
		3.4.3. Discretized Equations for Half-Control Volume Formulation
		3.4.4. Discretized Equations for ZFVM
		3.4.5. Numerical Examples
	3.5. Summary
	Problems
4. Finite Element Method
	4.1. Introduction
		4.1.1. Analysis Steps
		4.1.2. Remarks on the Analysis Steps
	4.2. One-Dimensional Problems
		4.2.1. Model Differential Equation
		4.2.2. Finite Element Mesh of the Geometry
		4.2.3. Approximation of the Solution over the Element
		4.2.4. Derivation of the Weak Form: The Three-Step Procedure
		4.2.5. Remarks on the Weak Form
		4.2.6. Interpolation Functions
		4.2.7. Remarks on the Interpolation Functions
		4.2.8. Finite Element Model
		4.2.9. Axisymmetric Problems
		4.2.10. Advection‒Diffusion Equation
	4.3. Two-Dimensional Problems
		4.3.1. Model Differential Equation
		4.3.2. Finite Element Approximation
		4.3.3. Weak Form
		4.3.4. Finite Element Model
		4.3.5. Axisymmetric Problems
		4.3.6. Advection‒Diffusion Equation
		4.3.7. Linear Finite Elements and Evaluation of Coefficients
		4.3.8. Higher-Order Finite Elements
		4.3.9. Assembly of Elements
		4.3.10. Numerical Examples
	4.4. Summary
	Problems
5. Dual Mesh Control Domain Method
	5.1. Introduction
	5.2. Dual Mesh Control Domain Method
	5.3. One-Dimensional Problems
		5.3.1. Model Differential Equation
		5.3.2. Primal and Dual Meshes
		5.3.3. Integral Statement over a Control Domain
		5.3.4. Discretized Equations over a Control Domain
		5.3.5. Numerical Examples
	5.4. Two-Dimensional Problems
		5.4.1. Preliminary Comments
		5.4.2. Model Equation
		5.4.3. Discretized Equations
		5.4.4. Numerical Examples
		5.4.5. Advection‒Diffusion Equation
	5.5. Summary
	Problems
6. Nonlinear Problems with a Single Unknown
	6.1. Introduction
	6.2. One-Dimensional Problems
		6.2.1. Model Differential Equation
		6.2.2. Finite Element Method
		6.2.3. Dual Mesh Control Domain Method
		6.2.4. Numerical Examples
	6.3. Two-Dimensional Problems
		6.3.1. Model Differential Equation
		6.3.2. Finite Element Method
		6.3.3. Dual Mesh Control Domain Formulation
		6.3.4. Numerical Examples
	6.4. Summary
	Problems
7. Bending of Straight Beams
	7.1. Introduction
		7.1.1. Background
		7.1.2. Functionally Graded Structures
		7.1.3. Present Study
	7.2. Linear Theories of FGM Beams
		7.2.1. Euler‒Bernoulli Beam Theory
		7.2.2. Timoshenko Beam Theory
	7.3. Linear Finite Element Models
		7.3.1. Euler‒Bernoulli Beam Theory
		7.3.2. Timoshenko Beam Theory
	7.4. Linear Dual Mesh Control Domain Model
		7.4.1. Euler‒Bernoulli Beam Theory
		7.4.2. Timoshenko Beams
	7.5. Numerical Results for Linear Problems
	7.6. Nonlinear Analysis of Beams
		7.6.1. Euler‒Bernoulli Beam Theory
		7.6.2. Timoshenko Beam Theory
		7.6.3. Dual Mesh Control Domain Models
		7.6.4. Linearization of Equations
		7.6.5. Numerical Results
	7.7. Summary
	Problems
8. Bending of Axisymmetric Circular Plates
	8.1. General Kinematic and Constitutive Relations
		8.1.1. Geometry and Coordinate System
		8.1.2. Kinematic Relations
		8.1.3. Constitutive Equations
	8.2. Classical Theory of Plates
		8.2.1. Displacements and Strains
		8.2.2. Equilibrium Equations
		8.2.3. Governing Equations in Terms of Displacements
		8.2.4. Equations in Terms of Displacements and Bending Moment
	8.3. First-Order Shear Deformation Plate Theory
		8.3.1. Displacements and Strains
		8.3.2. Equations of Equilibrium
		8.3.3. Equations of Equilibrium in Terms of Displacements
	8.4. Finite Element Models
		8.4.1. Introduction
		8.4.2. Displacement Model of the CPT
		8.4.3. Mixed Model of the CPT
		8.4.4. Displacement Model of the FST
	8.5. Dual Mesh Control Domain Models
		8.5.1. Preliminary Comments
		8.5.2. Mixed Model of the Classical Plate Theory
		8.5.3. Displacement Model of the FST
	8.6. Numerical Results
		8.6.1. Preliminary Comments
		8.6.2. Linear Analysis
		8.6.3. Nonlinear Analysis
	8.7. Summary
	Problems
9. Plane Elasticity and Viscous Incompressible Flows
	9.1. Introduction
		9.1.1. Two-Dimensional Elasticity
		9.1.2. Flows of Viscous Fluids
	9.2. Governing Equations
		9.2.1. Plane Elasticity
		9.2.2. Two-Dimensional Flows of Viscous Incompressible Fluids
	9.3. Finite Element Model of Plane Elasticity
		9.3.1. Weak Forms
		9.3.2. Finite Element Model
	9.4. Dual Mesh Control Domain Model of Plane Elasticity
		9.4.1. Governing Equations
		9.4.2. Control Domain Statements
		9.4.3. Discretized Equations
	9.5. Finite Element Model of Creeping Flows
		9.5.1. Penalty Function Formulation
		9.5.2. Finite Element Model
	9.6. Dual Mesh Control Domain Model of Creeping Flows
		9.6.1. Governing Equations
		9.6.2. Control Domain Statements
		9.6.3. Discretized Equations
	9.7. Discrete Models of the Navier‒Stokes Equations
		9.7.1. Finite Element Model
		9.7.2. Dual Mesh Control Domain Model
	9.8. Numerical Examples
	9.9. DMCDM with Arbitrary Meshes: 2D Elasticity
		9.9.1. Preliminary Comments
		9.9.2. Discretized Equations over an Arbitrary Control Domain
		9.9.3. Control Domains at the Boundary
		9.9.4. Numerical Examples
	9.10. Summary
	Problems
10. Bending of Flat Plates
	10.1. Introduction
	10.2. Governing Equations
		10.2.1. Displacement Field
		10.2.2. Principle of Virtual Displacements
		10.2.3. Governing Equations of Equilibrium
		10.2.4. Relations between Stress Resultants and Displacements
	10.3. Finite Element Model Development
		10.3.1. Weak Forms
		10.3.2. Finite Element Model
		10.3.3. Tangent Stiffness Coefficients
		10.3.4. Shear and Membrane Locking
	10.4. Dual Mesh Control Domain Model
		10.4.1. Primal and Dual Meshes
		10.4.2. Discretized Equations
		10.4.3. Shear Locking
	10.5. Numerical Examples
		10.5.1. Linear Analysis
		10.5.2. Results of Nonlinear Analysis
	10.6. Summary
References
Index