Computational Methods in Engineering

Preface to the Second Edition
Preface to the First Edition
Acknowledgments
Contents
About the Authors
1 Preliminaries
1.1 Introduction
1.1.1 Floating Point Numbers in Binary Form
1.1.2 Rounding Errors and Loss of Precision
1.1.3 Effect of Rounding on Numerical Computation
1.1.4 Taylor Series and Tuncation
1.1.5 Effect of Digital Calculations on Iteration
1.2 Mathematical and Computational Modeling
1.3 A Brief Introduction to MATLAB
1.3.1 Programming in MATLAB
1.3.2 Array and Matrices
1.3.3 Loops and Conditional Operations
1.3.4 Graphics
Part I System of Equations and Eigenvalues
2 Solution of Linear Equations
2.1 Analytical Methods of Solving a Set of Linear Equations
2.1.1 Cramer's Rule
2.1.2 Inverse of a Square Matrix
2.2 Preliminaries
2.2.1 Row operations
2.2.2 Some Useful Results
2.2.3 Condition Number of a Matrix
2.2.4 Pivoting
2.2.5 Triangular Matrices
2.3 Gauss Elimination Method
2.4 Gauss Jordan Method of Determining the Inverse Matrix
2.5 LU Decomposition or LU Factorization
2.5.1 Doolittle Decomposition
2.5.2 Crout Decomposition
2.5.3 Cholesky Decomposition
2.6 Tridiagonal Matrix Algorithm
2.6.1 Cholesky Decomposition of a Symmetric Tridiagonal Matrix
2.6.2 General Case of a Tridiagonal Matrix and the TDMA
2.7 QR Factorization
2.7.1 Gram-Schmidt Method
2.7.2 Householder Transformation and QR Factorization
2.7.3 Givens Rotation and QR Factorization
2.8 Iterative Methods of Solution
2.8.1 Jacobi and Gauss-Seidel Methods
2.8.2 Conjugate Gradient Method
3 Computation of Eigenvalues
3.1 Examples of Eigenvalues
3.1.1 Eigenvalue Problem in Geometry
3.1.2 Solution of a Set of Ordinary Differential Equations (ODE)
3.1.3 Standing Waves on a String
3.1.4 Resonance
3.1.5 Natural Frequency of a Spring Mass System
3.2 Preliminaries on Eigenvalues
3.2.1 Some Important Points
3.2.2 Similarity Transformation
3.2.3 More About the 2times2 Case
3.3 Analytical Evaluation of Eigenvalues and Eigenvectors in Simple Cases
3.4 Power Method
3.4.1 Inverse Power Method
3.4.2 Inverse Power Method with Shift
3.5 Rayleigh Quotient Iteration
3.5.1 Deflation of a Matrix
3.6 Eigenvalue Eigenvector Pair by QR Iteration
3.7 Modification of QR Iteration for Faster Convergence
3.7.1 Upper Hessenberg Form
3.7.2 QR Iteration with Shift
4 Solution of Algebraic Equations
4.1 Univariate Non-linear Equation
4.1.1 Plotting Graph: The Simplest Method
4.1.2 Bracketing Methods
4.1.3 Fixed Point Iteration Method
4.1.4 Newton-Raphson Method
4.1.5 Secant Method
4.1.6 Regula Falsi Method
4.2 Multivariate Non-linear Equations
4.2.1 Gauss-Seidel Iteration
4.2.2 Newton-Raphson Method
4.3 Root Finding and Optimization
4.3.1 Search Methods of Optimization: Univariate Case
4.4 Multidimensional Unconstrained Optimization
4.4.1 Calculus-Based Newton Method
4.4.2 Gradient Descent Search Methods
I.1 Solution of Linear Equations
I.2 Evaluation of Eigenvalues
I.3 Solution of Algebraic Equations
I.4 Optimization
Part II Interpolation, Differentiation and Integration
5 Interpolation
5.1 Polynomial Interpolation
5.2 Lagrange Interpolation Polynomial
5.2.1 Linear Interpolation
5.2.2 Quadratic Interpolation
5.2.3 Generalization
5.2.4 Lagrange Polynomials in Barycentric Form
5.2.5 Lagrange Polynomials with Equi-Spaced Data
5.3 Newton Polynomials
5.3.1 Divided Differences
5.3.2 Forward and Backward Differences
5.3.3 Newton Polynomial Using Divided, Forward or Backward Differences
5.3.4 Newton-Gregory Formulas with Equi-Spaced Data
5.4 Error Estimates of Polynomial Approximations
5.5 Polynomial approximation using Chebyshev's Nodes
5.6 Piecewise Polynomial Interpolation
5.7 Hermite Interpolation
5.7.1 Cubic Hermite Interpolating Polynomial
5.7.2 Hermite Interpolating Polynomial as Newton Polynomial
5.7.3 Generalization
5.8 Spline Interpolation and the Cubic Spline
5.8.1 General Case with Non-uniformly Spaced Data
5.8.2 Special Case with Equi-Spaced Data
5.9 Interpolation Using Rational Functions
5.9.1 Rational Functions and Their Properties
5.9.2 Comparisons of Different Rational Function Models
5.9.3 Application of Rational Function Interpolation to Tabulated Data
6 Interpolation in Two and Three Dimensions
6.1 Interpolation Over a Rectangle
6.1.1 Linear Interpolation
6.1.2 Local Coordinate System for a Rectangular Element
6.1.3 Interpolating Polynomials as Products of ``Lines''
6.1.4 Lagrange Quadratic Rectangular Element
6.1.5 Quadratic Eight Noded Rectangular Element
6.2 Interpolation Over a Triangle
6.3 Interpolation in Three Dimensions
6.3.1 Hexahedral Element
6.3.2 Tetrahedral Element
7 Regression or Curve Fitting
7.1 Introduction
7.2 Method of Least Squares for Linear Regression
7.2.1 Linear Regression by Least Squares
7.2.2 Coefficient of Correlation and Goodness of Fit
7.2.3 Index of Correlation and Goodness of Fit
7.2.4 Error Estimate
7.3 Multi-linear Regression
7.3.1 Least Square Multi-Linear Fit
7.3.2 Orthogonality Method of Regression
7.4 Polynomial Regression
7.5 Nonlinear Regression
7.6 Regression Using Rational Functions
7.6.1 Application to Known Function in Tabular Form
7.6.2 Application to Experimental Data
7.7 Principal Component Analysis, Dimension Reduction and Regression
8 Numerical Differentiation
8.1 Introduction
8.2 Finite Difference Formulae Using Taylor's Series
8.3 Differentiation of Lagrange and Newton Polynomials
8.3.1 Derivatives of Lagrange Polynomials: Arbitrarily Spaced Data
8.3.2 Derivatives of Ln(x)
8.3.3 Derivatives of Lagrange Polynomials: Equi-Spaced Data
8.3.4 Higher Order Formulae Using Newton Polynomials
8.4 Numerical Partial Differentiation
8.4.1 First Derivatives in a Rectangular Element
8.4.2 Derivatives for an Arbitrary Quadrilateral
8.4.3 Second Derivative Formulas for a Rectangle
8.4.4 Linear Triangular Element
9 Numerical Integration
9.1 Introduction
9.2 Trapezoidal Rule
9.3 Simpson's Rule
9.3.1 Simpson's 1/3 Rule
9.3.2 Simpson's 3/8 Rule
9.4 Integration of Functions
9.4.1 h Fefinement: Error Estimation
9.4.2 Closed Newton Cotes Quadrature Rules
9.4.3 Romberg Method: Richardson Extrapolation
9.5 Quadrature Using Chebyshev Nodes
9.6 Gauss Quadrature
9.7 Singular Integrals
9.7.1 Open Newton Cotes Quadrature
9.8 Integrals with Infinite Range
9.8.1 Coordinate Transformation
9.9 Adaptive Quadrature
9.10 Multiple Integrals
9.10.1 Double Itegral with Fixed Limits for Both x and y
9.10.2 Double Integrals Using Newton Cotes Quadrature
9.10.3 Double Integrals Using Gauss Quadrature
9.10.4 Double Integral with Variable Limits on x or y
9.10.5 Quadrature Rules for Triangle
9.A MATLAB Routines Related to Chap. 9
9.B Further Reading
Part III Ordinary Differential Equations
10 Initial Value Problems
10.1 Introduction
10.2 Euler Method
10.2.1 Stability of Euler Method
10.3 Modified Euler Method or Heun Method
10.4 Runge Kutta (RK) Methods
10.4.1 Second Order Runge Kutta Method (RK2)
10.4.2 Fourth Order Runge Kutta Method (RK4)
10.4.3 Embedded Runge Kutta Methods
10.4.4 Adaptive Runge Kutta Methods
10.5 Predictor Corrector Methods
10.5.1 Adam-Bashforth-Moulton (ABM2) Second Order Method
10.5.2 Fourth Order Method
10.5.3 Improving Accuracy of ABM Methods
10.5.4 Adaptive ABM Method: Change of Step Size
10.6 Set of First Order ODEs
10.6.1 Euler and RK2 Applied to a Set of First Order Ordinary Differential Equations
10.6.2 Application of RK4 to Two Coupled First Order ODEs
10.7 Higher Order ODEs
10.7.1 Euler Method Applied to Second Order ODE
10.7.2 RK2 Method Applied to Second Order ODE
10.8 Stiff Equations and Backward Difference Formulae (BDF)-Based Methods
10.8.1 Implicit Euler or Backward Euler Scheme
10.8.2 Second Order Implicit Scheme
10.8.3 Higher Order Implicit Schemes Based on BDF
10.8.4 Nonlinear ODEs
11 Boundary Value Problems (ODE)
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11.1 Introduction
11.2 The ``Shooting Method''
11.2.1 Linear ODE Case
11.2.2 Nonlinear ODE Case
11.2.3 Boundary Value Problem Over Semi-infinite Domain
11.2.4 Generalization of Shooting Method for Higher Order ODEs
11.3 Finite Difference Method
11.3.1 Second Order ODE with Constant Coefficients: A Simple Example
11.3.2 Second Order ODE with Constant Coefficients: A Variant
11.3.3 Application of FDM Using Non-uniform Grids
11.3.4 Solution of Nonlinear Case by FDM
11.3.5 Application of FDM to Second Order ODE with Variable Coefficients
11.4 Collocation Method
11.5 Method of Weighted Residuals
11.6 Finite Eement Method
11.6.1 Elements
11.6.2 Weighting Function
11.6.3 Second Order ODE with Linear Element
11.6.4 Finite Element Method Applied to Structural Problems
11.7 Finite Volume Method
11.7.1 Background
11.7.2 Discretization
11.7.3 Simple Example with Discretization Scheme 1
11.7.4 Simple Example with Disctretization Scheme 2
11.7.5 Example Using Piecewise Linear Function
11.A MATLAB Routines Related to Chap. 11
11.B Further Reading
III.1 Initial Value Problems
III.2 Boundary Value Problems
Part IV Partial Differential Equations
12 Introduction to PDEs
12.1 Preliminaries
12.2 Second Order PDE with Constant Coefficients
12.3 Numerical Solution Methods for PDEs
12.4 MATLAB Functions Related to PDE
13 Laplace and Poisson Equations
13.1 Introduction
13.2 Finite Difference Solution
13.2.1 Discretization of Computational Domain
13.2.2 Different Types of Boundary Conditions
13.2.3 Alternate Direction Implicit or ADI Method
13.3 Elliptic Equations in Other Coordinate Systems
13.4 Elliptic Equation Over Irregular Domain
13.5 FEM and FVM Applied to Elliptic Problems
14 Advection and Diffusion Equations
14.1 Introduction
14.2 The Advection Equation
14.2.1 Finite Difference Schemes for Advection Equation
14.3 Nonlinear Advection
14.3.1 Advection Equation with Varying a
14.3.2 Nonlinear Advection Equation
14.4 Parabolic PDE: Transient Diffusion Equation
14.4.1 Explicit Formulation
14.4.2 Implicit Method
14.4.3 Crank-Nicolson or Semi-implicit Method
14.5 FEM Analysis of Heat Equation
14.5.1 Galerkin FEM
14.6 Advection with Diffusion
14.6.1 FTCS
14.6.2 Upwind for Advection and Central Difference for Diffusion
14.6.3 Crank-Nicolson Scheme
14.7 Multidimensional Advection
14.7.1 Upwind Scheme
14.7.2 Operator Splitting and Upwind Scheme
14.8 Diffusion Equation in Multiple Dimensions
14.8.1 Explicit Formulation
14.8.2 Implicit and Crank-Nicolson Schemes
14.8.3 ADI Method
15 Wave Equation
15.1 Introduction
15.2 General Solution of the Wave Equation
15.2.1 d'Alembert's Solution to the One-Dimensional Wave Equation
15.2.2 Boundary Conditions
15.2.3 A Useful Theorem
15.3 Numerical Solution of the One-Dimensional Wave Equation
15.3.1 Explicit scheme
15.3.2 Stability Analysis of the Explicit Scheme
15.3.3 Implicit Scheme
15.3.4 Stability of Implicit Scheme
15.4 Waves in a Diaphragm
15.4.1 Explicit Scheme for One-Dimensional Wave in a Circular Diaphragm
15.4.2 Waves in Two Dimensions—Waves in a Rectangular Diaphragm or Plate
15.4.3 Waves in Two Dimensions—Waves in a Circular Diaphragm or Plate
IV.1 Types of Partial Differential Equations
IV.2 Laplace and Poisson Equations
IV.3 Advection and Diffusion Equations
IV.4 Wave Equation
16 Application Examples
16.1 Introduction
16.2 Modeling and Simulation
16.2.1 Operating Point of a Fan-Duct System
16.2.2 Pumps Operating in Parallel
16.3 Operating Point of a Heat Exchanger
16.3.1 Automobile Problem
16.4 Analysis of Data
16.4.1 Fit to Hygrometer Data
16.4.2 Practical Application of Regression
16.5 Moment of Inertia Calculations
16.5.1 Moment of Inertia of Triangles
16.5.2 Moment of Inertia of Area with Curved Boundary
16.6 Second Order System Response without and with Feedback Control
16.7 Circuit Analysis—Electrical or Thermal
16.8 Solution of Boundary Value Problems
16.8.1 Radial Fin of Uniform Thickness
16.8.2 Post-processing of Data and Error Analysis
16.8.3 Solution of Boundary Value Problem Subject to Third Kind Boundary Conditions
16.9 Examples in Two Dimensions
16.9.1 Steady Heat Transfer in an L-shaped Region
16.9.2 2D Heat Transfer with Heat Generation: Use of Non-uniform Grids
A Epilogue
Beyond the Book—The Way Ahead
Researchers Developing Their Own Code
Users of Commercial Code
Where Does One Look for Help?
Free Resources
Licensed Resources
Index