Introduction to Finite Element Analysis for Engineers

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author’s Biography
Chapter 1: Introduction
	1.1. Computational Sciences and Mechanics
	1.2. Brief Mathematical Background: Linear Algebra
		1.2.1. Vectors
		1.2.2. Matrices
	1.3. Brief Mathematical Background: Partial Differential Equations
		1.3.1. Elliptic Equations: Poisson's Equation
		1.3.2. Parabolic Equations: Heat Equation
		1.3.3. Hyperbolic Equations: Wave Equation
Chapter 2: Second-Order Ordinary Differential Equations
	2.1. Model Problem
		2.1.1. Axial Deformation of a Bar
		2.1.2. One-Dimensional Heat Conduction in Solids
	2.2. A Motivational Example
	2.3. The Method of Weighted Residuals
	2.4. Weak Form
	2.5. Global Basis Functions and Matrix Formulation
	2.6. Finite Element Formulation Using Element Shape Functions
		2.6.1. Linear Element
		2.6.2. Quadratic Element
		2.6.3. Higher-Order Elements and Lagrange Interpolation Functions
	2.7. Thermoelastic Effects in One Dimension
	2.8. Numerical Evaluations of Element Matrices
	2.9. Biomedical Engineering Applications
		2.9.1. Reaction-Diffusion: Oxygen Consumption in Flat Tissues
		2.9.2. Reaction-Diffusion: Oxygen Consumption in Cylindrical Tissues and Spherical Cells
		2.9.3. Pulsatile Blood Flow in Arteries: Womersley Problem
	2.10. Problems
Chapter 3: Fourth-Order Ordinary Differential Equations
	3.1. Euler-Bernoulli Beam Theory
	3.2. Weak Form of the Beam Equation
	3.3. Finite Element Method: Beam Element
	3.4. Plane Frames
	3.5. Plane Trusses
	3.6. Principle of Virtual Displacements (Work)
	3.7. Principle of Minimum Total Potential Energy
		3.7.1. Rayleigh-Ritz Method
	3.8. Problems
Chapter 4: Elliptic Equations: Equilibrium in Two Dimensions
	4.1. Model Problem
	4.2. Weak Form
	4.3. Mesh Generation and Connectivity Matrix
	4.4. Approximations and Element Shape Functions
		4.4.1. Bilinear Triangular Element
		4.4.2. Bilinear Rectangular Element
	4.5. Element Equations and Matrices
		4.5.1. Matrices for Bilinear Triangular Element
		4.5.2. Matrices for Bilinear Rectangular Element
		4.5.3. Matrices for Natural and Mixed Boundary Conditions
	4.6. Elements Assembly and Global System
	4.7. Applications
		4.7.1. Heat Conduction in Solids
		4.7.2. Fully Developed Laminar Flow in Noncircular Ducts
		4.7.3. Torsion of Noncircular Sections
	4.8. Isoparametric Elements and Numerical Integration
		4.8.1. Shape Functions of Canonical Elements
		4.8.2. Mapping
		4.8.3. Approximations
		4.8.4. Element Matrices
	4.9. Applications
		4.9.1. Potential Flow Around 2D Airfoils, Lift
		4.9.2. Oxygen Transport and Consumption in Krogh Capillary-Tissue Cylinder
	4.10. Problems
Chapter 5: Parabolic Equations: Time-Dependent Diffusion Problems
	5.1. Model Problem
	5.2. The Weak Form
	5.3. The Weak Form for an Element
	5.4. Approximations and Element Matrices
	5.5. Temporal Approximation: Time Marching
	5.6. Transient Heat Conduction
	5.7. Biomedical Engineering Applications
		5.7.1. Bioheat: Pennes' Heat Conduction Model
		5.7.2. Unsteady Oxygen Consumption in Spherical and Cylindrical Domains
		5.7.3. Transient Oxygen Uptake in a Krogh Cylinder Tissue
	5.8. Appendix
	5.9. Problems
Chapter 6: Hyperbolic Equations: Waves and Vibrations Problems
	6.1. Model Problems
	6.2. The Weak Forms and Finite Element Models
		6.2.1. First Model Problem: Second-Order in Space and Time
		6.2.2. Second Model Problem: Transverse Vibrations of Beams
	6.3. Time Advancement Scheme: Newmark Method
	6.4. Applications: Waves and Vibrations on Strings, Bars, and Beams
	6.5. Problems
Chapter 7: Differential Eigenvalue Problems
	7.1. Natural Frequencies of Longitudinal Vibration of Bars
	7.2. Natural Frequencies of Beams and Frames
	7.3. Effects of Axial Force on Beam Deflection
	7.4. Hydrodynamic Stability: Orr-Sommerfeld Equation
		7.4.1. Orr-Sommerfeld Equation
		7.4.2. Weak Form of Orr-Sommerfeld Equation
	7.5. Problems
Chapter 8: Plane Elasticity
	8.1. Constitutive Equations for Linear Elasticity
	8.2. Principle of Virtual Displacements: Plane Elasticity
	8.3. Element Equations and Matrices
	8.4. Elements Assembly and Global System
	8.5. Uncoupled Linear Thermoelasticity
		8.5.1. Constitutive Equations
		8.5.2. Element Equations and Matrices, Thermal Loading
	8.6. Problems
Chapter 9: Kirchhoff-Love and Reissner-Mindlin Plates
	9.1. Classical Plate Theory, CPT
		9.1.1. Principle of Virtual Displacements
		9.1.2. Approximations and Element Shape Functions
		9.1.3. Element Equations and Matrices
		9.1.4. Isoparametric Hermite Elements
	9.2. Reissner-Mindlin First-Order Shear Deformation Plate Theory
		9.2.1. Principle of Virtual Displacements
		9.2.2. Approximations and Element Equations
	9.3. Dynamic Response of Plates
		9.3.1. Equation of Motion for CPT
		9.3.2. Equations of Motion for Reissner-Mindlin Plate
		9.3.3. Finite Element Model
		9.3.4. Free Vibrations: Natural Frequencies
	9.4. Fluid-Structure Interaction: Linear Analysis
		9.4.1. Coupled Fluid-Structural Model
		9.4.2. Free Vibrations of Submerged Plates: NAVMI Factors
		9.4.3. Linear Hydroelastic Stability
	9.5. Problems
Chapter 10: Nonlinear Reissner-Mindlin Plate and Applications
	10.1. Equations of Equilibrium of Finite Displacements
	10.2. Principle of Virtual Displacements
	10.3. Geometrically Nonlinear Reissner-Mindlin Plate
	10.4. Finite Element Model
	10.5. Static Deflection of Geometrically Nonlinear Plates
	10.6. Buckling and Post-Buckling of Plates
		10.6.1. Critical Loads of Plates
		10.6.2. Post-Buckling of Plates
	10.7. Thermal Buckling of Reissner-Mindlin Plate
	10.8. Nonlinear Vibrations: Element Matrices
		10.8.1. The Duffing Equation
		10.8.2. Free Vibration of Plates: Harmonic Balance
	10.9. Supersonic Panel Flutter: Limit Cycle Oscillations
	10.10. Nonlinear Hydroelastic Stability of Panels
Appendix
References
Index