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."--Provided by publisher Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface to the second edition -- Preface -- Acknowledgments -- About the author -- I: Data approximations -- 1 Classical interpolation methods -- 1.1 Newton interpolation -- 1.1.1 Equidistant Newton interpolation -- 1.1.2 Computational example -- 1.2 Lagrange interpolation -- 1.2.1 Equidistant Lagrange interpolation -- 1.2.2 Computational example -- 1.2.3 Parametric 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 -- 2.1.1 Equidistant natural spline approximation -- 2.1.2 Computational example -- 2.2 Bezier splines -- 2.2.1 Rational Bezier splines -- 2.2.2 Computational example -- 2.3 Approximation with B-splines -- 2.3.1 Computational example -- 2.3.2 Nonuniform B-splines -- 2.3.3 Nonuniform rational B-splines -- 2.4 Surface spline approximation -- 2.4.1 Coons surfaces -- 2.4.2 Computational example -- 2.4.3 Bezier surfaces -- 2.4.4 Triangular surface patches -- 2.4.5 Computational example -- 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 -- 3.6.1 Computational example -- 3.7 Trigonometric least squares approximation -- 3.7.1 Computational example -- 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 -- 4.2.1 Gram-Schmidt orthogonalization -- 4.2.2 Computational example 4.3 Chebyshev approximation -- 4.3.1 Collapsing a power series -- 4.3.2 Computational example -- 4.4 Fourier approximation -- 4.4.1 Computational example -- 4.4.2 Complex Fourier approximation -- 4.5 Padé approximation -- 4.5.1 Computational example -- 4.6 Approximating matrix functions -- 4.6.1 Taylor series approximation of matrix functions -- References -- 5 Numerical differentiation -- 5.1 Finite difference formulae -- 5.1.1 Three-point finite difference formulae -- 5.1.2 Computational example -- 5.2 Higher order derivatives -- 5.3 Richardson extrapolation -- 5.3.1 Computational example -- 5.4 Multipoint finite difference formulae -- References -- 6 Numerical integration -- 6.1 The Newton-Cotes class -- 6.1.1 The trapezoid rule -- 6.1.2 Simpson's rule -- 6.1.3 Computational example -- 6.1.4 Open Newton-Cotes formulae -- 6.2 Advanced Newton-Cotes methods -- 6.2.1 Composite methods -- 6.2.2 Romberg's method -- 6.2.3 Computational example -- 6.3 Gaussian quadrature -- 6.3.1 Computational example -- 6.4 Integration of functions of multiple variables -- 6.4.1 Gaussian cubature -- 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 -- 7.1.1 The method of bisection -- 7.1.2 The secant method -- 7.1.3 Fixed point iteration -- 7.1.4 Computational example -- 7.2 Newton's method -- 7.3 Solution of algebraic equations -- 7.3.1 Sturm sequence -- 7.3.2 Horner's scheme of evaluating polynomials -- 7.3.3 Computational example -- 7.4 Aitken's acceleration -- 7.4.1 Computational example -- References -- 8 Systems of nonlinear equations -- 8.1 The generalized fixed point method -- 8.2 The method of steepest descent -- 8.2.1 Computational example -- 8.3 The generalization of Newton's method -- 8.3.1 Computational example 11.3.6 Fehlberg's method of step size adjustment -- 11.3.7 Stability of multistep techniques -- 11.4 Initial value problems of systems of ordinary differential equations -- 11.4.1 Picard's successive approximation -- 11.4.2 Runge-Kutta solution -- 11.4.3 Computational example -- 11.5 Initial value problems of higher order ordinary differential equations -- 11.5.1 Computational example -- 11.6 Linearization of second order initial value problems -- 11.7 Transient response analysis application -- 11.7.1 Model order reduction -- References -- 12 Boundary value problems -- 12.1 Boundary value problems of ordinary differential equations -- 12.1.1 Nonlinear boundary value problems -- 12.1.2 Linear boundary value problems -- 12.1.3 Approximate solution of boundary value problems -- 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 -- 12.4.1 Computational example -- 12.5 The finite element method -- 12.5.1 Finite element shape functions -- 12.5.2 Finite element matrix generation and assembly -- 12.5.3 Computational example -- 12.6 Finite element analysis of three-dimensional continuum -- 12.6.1 Tetrahedral finite element -- 12.6.2 Finite element matrix in parametric coordinates -- 12.6.3 Local to global coordinate transformation -- 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 -- 13.4.1 Computational example -- 13.5 Neumann approximation -- 13.5.1 Computational example -- 13.6 Nystrom method 8.4 Quasi-Newton method -- 8.4.1 Computational example -- 8.5 Nonlinear static analysis application -- References -- 9 Iterative solution of linear systems -- 9.1 Iterative solution of linear systems -- 9.2 Splitting methods -- 9.2.1 Jacobi method -- 9.2.2 Gauss-Seidel method -- 9.2.3 Successive overrelaxation method -- 9.2.4 Computational formulation of splitting methods -- 9.3 Ritz-Galerkin method -- 9.4 Conjugate gradient method -- 9.4.1 Computational example -- 9.5 Preconditioning techniques -- 9.5.1 Approximate inverse -- 9.6 Biconjugate gradient method -- 9.7 Least squares systems -- 9.7.1 QR factorization -- 9.7.2 Computational example -- 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 -- 10.1.1 The technique of deflation -- 10.1.2 Computational example -- 10.1.3 Improving the eigenvalue accuracy -- 10.1.4 Computational example -- 10.2 The Rayleigh-Ritz procedure -- 10.3 The Lanczos method -- 10.3.1 Truncated Lanczos process accuracy -- 10.4 The solution of the tridiagonal eigenvalue problem -- 10.5 The biorthogonal Lanczos method -- 10.5.1 Computational example -- 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 -- 11.2.1 Euler's method -- 11.2.2 Taylor methods -- 11.2.3 Computational example -- 11.2.4 Runge-Kutta methods -- 11.2.5 Computational example -- 11.3 Multistep methods -- 11.3.1 Explicit methods -- 11.3.2 Implicit methods -- 11.3.3 Predictor-corrector technique -- 11.3.4 Gragg's method of extrapolation -- 11.3.5 Computational example 13.6.1 Computational example -- 13.7 Nonlinear integral equations -- 13.7.1 Computational example -- 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 -- 14.2.1 Computational example -- 14.3 Quadratic optimization -- 14.3.1 Computational example -- 14.4 Gradient-based methods -- 14.4.1 Computational example -- 14.5 Global optimization -- 14.6 Topology optimization -- 14.6.1 Deformation constraint -- 14.6.2 Natural frequency constraint -- 14.7 Structural compliance application -- References -- Closing remarks -- List of figures -- List of tables -- Annotation -- Index "Presenting numerous examples, algorithms, and industrial applications, Approximation Techniques for Engineers is your complete guide to the major techniques used in modern engineering practice. Whether you need approximations for discrete data of continuous functions, or you're looking for approximate solutions to engineering problems, everything you need is nestled between the covers of this book. Now you can benefit from Louis Komzsik's years of industrial experience to gain a working knowledge of a vast array of approximation techniques through this complete and self-contained resource"--Publisher's description