Approximation Techniques for Engineers: Second Edition

This second edition includes eleven new sections based on the approximation of matrix functions, deflating the solution space and improving the accuracy of approximate solutions, iterative solution of initial value problems of systems of ordinary differential equations, and the method of trial functions for boundary value problems. The topics of the two new chapters are integral equations and mathematical optimization. The book provides alternative solutions to software tools amenable to hand computations to validate the results obtained by "black box" solvers. It also offers an insight into the mathematics behind many CAD, CAE tools of the industry. The book aims to provide a working knowledge of the various approximation techniques for engineering practice.

Table of Contents:
I Data approximations 1 Classical interpolation methods 1.1 Newton interpolation 1.2 Lagrange interpolation 1.3 Hermite interpolation 1.3.1 Computational example 1.4 Interpolation of functions of two variables with polynomials References 2 Approximation with splines 2.1 Natural cubic splines 2.2 Bezier splines 2.3 Approximation with B-splines 2.4 Surface spline approximation References 3 Least squares approximation 3.1 The least squares principle 3.2 Linear least squares approximation 3.3 Polynomial least squares approximation 3.4 Computational example 3.5 Exponential and logarithmic least squares approximations 3.6 Nonlinear least squares approximation 3.7 Trigonometric least squares approximation 3.8 Directional least squares approximation 3.9 Weighted least squares approximation References 4 Approximation of functions 4.1 Least squares approximation of functions 4.2 Approximation with Legendre polynomials 4.3 Chebyshev approximation 4.4 Fourier approximation 4.5 Pad'e approximation 4.6 Approximating matrix functions References 5 Numerical differentiation 5.1 Finite difference formulae 5.2 Higher order derivatives 5.3 Richardson’s extrapolation 5.4 Multipoint finite difference formulae References 6 Numerical integration 6.1 The Newton-Cotes class 6.2 Advanced Newton-Cotes methods 6.3 Gaussian quadrature 6.4 Integration of functions of multiple variables 6.5 Chebyshev quadrature 6.6 Numerical integration of periodic functions References II Approximate solutions 7 Nonlinear equations in one variable 7.1 General equations 7.2 Newton’s method 7.3 Solution of algebraic equations 7.4 Aitken’s acceleration References 8 Systems of nonlinear equations 8.1 The generalized fixed point method 8.2 The method of steepest descent 8.3 The generalization of Newton’s method 8.4 Quasi-Newton method 8.5 Nonlinear static analysis application References 9 Iterative solution of linear systems 9.1 Iterative solution of linear systems 9.2 Splitting methods 9.3 Ritz-Galerkin method 9.4 Conjugate gradient method 9.5 Preconditioning techniques 9.6 Biconjugate gradient method 9.7 Least squares systems 9.8 The minimum residual approach 9.9 Algebraic multigrid method 9.10 Linear static analysis application References 10 Approximate solution of eigenvalue problems 10.1 Classical iterations 10.2 The Rayleigh-Ritz procedure 10.3 The Lanczos method 10.4 The solution of the tridiagonal eigenvalue problem 10.5 The biorthogonal Lanczos method 10.6 The Arnoldi method 10.7 The block Lanczos method 10.7.1 Preconditioned block Lanczos method 10.8 Normal modes analysis application References 11 Initial value problems 11.1 Solution of initial value problems 11.2 Single-step methods 11.3 Multistep methods 11.4 Initial value problems of systems of ordinary differential equations 11.5 Initial value problems of higher order ordinary differential equations 11.6 Linearization of second order initial value problems 11.7 Transient response analysis application References 12 Boundary value problems 12.1 Boundary value problems of ordinary differential equations 12.2 The finite difference method for boundary value problems of ordinary differential equations 12.3 Boundary value problems of partial differential equations 12.4 The finite difference method for boundary value problems of partial differential equations 12.5 The finite element method 12.6 Finite element analysis of three-dimensional continuum 12.7 Fluid-structure interaction application References 13 Integral equations 13.1 Converting initial value problems to integral equations 13.2 Converting boundary value problems to integral equations 13.3 Classification of integral equations 13.4 Fredholm solution 13.5 Neumann approximation 13.6 Nystrom method 13.7 Nonlinear integral equations 13.8 Integro-differential equations 13.8.1 Computational example 13.9 Boundary integral method application References 14 Mathematical optimization 14.1 Existence of solution 14.2 Penalty method 14.3 Quadratic optimization 14.4 Gradient based methods 14.5 Global optimization 14.6 Topology optimization 14.7 Structural compliance application References List of figures List of tables Annotation Index Closing remarks