Computational Methods in Engineering

The book is designed to serve as a textbook for courses offered to graduate and upper-undergraduate students enrolled in mechanical engineering. The book attempts to make students with mathematical backgrounds comfortable with numerical methods. The book also serves as a handy reference for practicing engineers who are interested in applications. The book is written in an easy-to-understand manner, with the essence of each numerical method clearly stated. This makes it easy for professional engineers, students, and early career researchers to follow the material presented in the book. The structure of the book has been modeled accordingly. It is divided into four modules: i) solution of a system of equations and eigenvalues which includes linear equations, determining eigenvalues, and solution of nonlinear equations; ii) function approximations: interpolation, data fit, numerical differentiation, and numerical integration; iii) solution of ordinary differential equations―initial value problems and boundary value problems; and iv) solution of partial differential equations―parabolic, elliptic, and hyperbolic PDEs. Each section of the book includes exercises to reinforce the concepts, and problems have been added at the end of each chapter. Exercise problems may be solved by using computational tools such as scientific calculators, spreadsheet programs, and MATLAB codes. The detailed coverage and pedagogical tools make this an ideal textbook for students, early career researchers, and professionals. 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) [DELETE] 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? 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