Advanced Numerical Methods with Matlab 2

Abdelkhalak El Hami is Professeur des Universités at the Institut National des Sciences Appliquées (INSA-Rouen) in France and is in charge of the Normandy Conservatoire National des Arts et Metiers (CNAM) Chair of Mechanics, as well as several European pedagogical projects. He is an expert in fluid-structure interaction studies and reliability. Bouchaib Radi is Professor of Higher Education in the Faculty of Science and Technology of Settat Hassan First University in Morocco. His work specializes in material forming processes, optimization and reliability.

• Preface ixPart 1. Solving Equations 1Chapter 1. Solving Nonlinear Equations 31.1 Introduction 31.2 Separating the roots 31.3 Approximating a separated root 41.3.1 Bisection method (or dichotomy method) 41.3.2 Fixed-point method 61.3.3 First convergence criterion 71.3.4 Iterative stopping criteria.81.3.5 Second convergence criterion (local criterion) 91.3.6 Newton’s method (or the method of tangents) 101.3.7 Secant method 121.3.8 Regula falsi method (or false position method) 171.4 Order of an iterative process.191.5 Using Matlab 191.5.1 Finding the roots of polynomials 191.5.2 Bisection method 211.5.3 Newton’s method 22Chapter 2. Numerically Solving Differential Equations 252.1 Introduction 252.2 Cauchy problem and discretization 272.3 Euler’s method 302.3.1 Interpretation 302.3.2 Convergence 302.4 One-step Runge–Kutta method 312.4.1 Second-order Runge–Kutta method 322.4.2 Fourth-order Runge–Kutta method 332.5 Multi-step Adams methods 362.5.1 Open Adams methods 362.5.2 Closed Adams formulas 392.6 Predictor–Corrector method.412.7 Using Matlab 43Part 2. Solving PDEs 47Chapter 3. Finite Difference Methods 493.1 Introduction 493.2 Presentation of the finite difference method 513.2.1 Convergence, consistency and stability 533.2.2 Courant–Friedrichs–Lewy condition 563.2.3 Von Neumann stability analysis 573.3 Hyperbolic equations 583.3.1 Key results 593.3.2 Numerical schemes for solving the transport equation 633.3.3 Wave equation 663.3.4 Burgers equation 683.4 Elliptic equations 723.4.1 Poisson equation 723.5 Parabolic equations 743.5.1 Heat equation 743.6 Using Matlab 76Chapter 4. Finite Element Method 834.1 Introduction 834.2 One-dimensional finite element methods 834.3 Two-dimensional finite element methods 884.4 General procedure of the method 934.5 Finite element method for computing elastic structures 934.5.1 Linear elasticity 934.5.2 Variational formulation of the linear elasticity problem 974.5.3 Planar linear elasticity problems 994.5.4 Applying the finite element method to planar problems 1014.5.5 Axisymmetric problems.1054.5.6 Three-dimensional problems 1074.6 Using Matlab 1074.6.1 Solving Poisson’s equation 1084.6.2 Solving the heat equation.1114.6.3 Computing structures 112Chapter 5. Finite Volume Methods 1175.1 Introduction 1175.2 Finite volume method (FVM) 1185.2.1 Conservation properties of the method 1185.2.2 The stages of the method.1195.2.3 Convergence 1205.2.4 Consistency 1205.2.5 Stability 1205.3 Advection schemes 1215.3.1 Two-dimensional FVM. 1265.3.2 Convection-diffusion equation 1295.3.3 Central differencing scheme 1315.3.4 Upwind (decentered) scheme 1335.3.5 Hybrid scheme 1365.3.6 Power-law scheme 1365.3.7 QUICK scheme 1375.3.8 Higher-order schemes 1395.3.9 Unsteady one-dimensional convection-diffusion Equation 1405.3.10 Explicit scheme 1425.3.11 Crank–Nicolson scheme.1425.3.12 Implicit scheme 1435.4 Using Matlab 144Chapter 6. Meshless Methods. 1476.1 Introduction 1476.2 Limitations of the FEM and motivation of meshless methods 1486.3 Examples of meshless methods1486.3.1 Advantages of meshless methods 1496.3.2 Disadvantages of meshless methods1506.3.3 Comparison of the finite element method and meshless methods 1516.4 Basis of meshless methods 1516.4.1 Approximations 1516.4.2 Kernel (weight) functions.1526.4.3 Completeness 1526.4.4 Partition of unity 1526.5 Meshless method (EFG) 1536.5.1 Theory 1536.5.2 Moving Least-Squares Approximation 1536.6 Application of the meshless method to elasticity 1636.6.1 Formulation of static linear elasticity 1636.6.2 Imposing essential boundary conditions 1656.7 Numerical examples 1706.7.1 Fixed-free beam 1706.7.2 Compressed block 1716.8 Using Matlab 173Part 3. Appendices 179Appendix 1181Appendix 2189Bibliography 195Index 199