Content: Cover
Half title
Title
Copyright
Dedication
Contents
Preface
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 Di erential 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 Di erential 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 Residuals2.4 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 E ects in One Dimension
2.8 Numerical Evaluations of Element Matrices
2.9 Biomedical Engineering Applications
2.9.1 Reaction-Di usion: Oxygen Consumption in Flat Tissues
2.9.2 Reaction-Di usion: Oxygen Consumption in Cylindrical Tissues and Spherical Cells 2.9.3 Pulsatile Blood Flow in Arteries: Womersley Problem2.10 Problems
Chapter 3 Fourth-Order Ordinary Di erential 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 Element4.4.2 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 Matrices4.9 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 Di usion 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 Cylin-drical Domains