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1945 results in Engineering mathematics and programming

Page 60 of 98

  • 10 - Introduction to Optimization

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 374-415

  • Summary

    Introduction

    Optimization is the term often used for minimizing or maximizing a function. It is sufficient to consider the problem of minimization only; maximization of F(x) is achieved by simply minimizing —F(x). In engineering, optimization is closely related to design. The function F(x), called the merit function or objective function, is the quantity that we wish to keep as small as possible, such as the cost or weight. The components of x, known as the design variables, are the quantities that we are free to adjust. Physical dimensions (lengths, areas, angles, etc.) are common examples of design variables.

    Optimization is a large topic with many books dedicated to it. The best we can do in limited space is to introduce a few basic methods that are good enough for problems that are reasonably well behaved and do not involve too many design variables. By omitting the more sophisticated methods,we may actually not miss all that much. All optimization algorithms are unreliable to a degree — any one of them may work on one problem and fail on another. As a rule of the thumb, by going up in sophistication we gain computational efficiency, but not necessarily reliability.

    The algorithms for minimization are iterative procedures that require starting values of the design variables x. If F(x) has several local minima, the initial choice of x determines which of these will be computed. There is no guaranteed way of finding the global optimal point. One suggested procedure is to make several computer runs using different starting points and pick the best result.

  • 5 - Numerical Differentiation

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 177-192

  • Summary

    Introduction

    Numerical differentiation deals with the following problem: We are given the function y = f(x) and wish to obtain one of its derivatives at the point x = xk. The term “given” means that we either have an algorithm for computing the function or possess a set of discrete data points (xi, yi), i = 0, 1, …, n. In either case, we have access to a finite number of (x, y) data pairs from which to compute the derivative. If you suspect by now that numerical differentiation is related to interpolation, you are right — one means of finding the derivative is to approximate the function locally by a polynomial and then differentiate it. An equally effective tool is the Taylor series expansion of f(x) about the point xk, which has the advantage of providing us with information about the error involved in the approximation.

    Numerical differentiation is not a particularly accurate process. It suffers from a conflict between roundoff errors (due to limited machine precision) and errors inherent in interpolation. For this reason, a derivative of a function can never be computed with the same precision as the function itself.

  • Preface to the Second Edition

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp x-x

  • Summary

    The major change in the second edition is the replacement of NumArray (a Python extension that implements array objects) with NumPy. As a consequence, most routines listed in the text required some code changes. The reason for the changeover is the imminent discontinuance of support for NumArray and its predecessor Numeric.

    We also took the opportunity to make a few changes in the material covered:

  • Rational function interpolation was added to Chapter 3.

  • Brent's method of root finding in Chapter 4 was replaced by Ridder's method. The full-blown algorithm of Brent is a complicated procedure involving elaborate bookkeeping (a simplified version was presented in the first edition). Ridder's method is as robust and almost as efficient as Brent's method, but much easier to understand.

  • The Fletcher—Reeves method of optimization was dropped in favor of the downhill simplex method in Chapter 10. Fletcher—Reeves is a first-order method that requires knowledge of the gradients of the merit function. Because there are few practical problems where the gradients are available, the method is of limited utility. The downhill simplex algorithm is a very robust (but slow) zero-order method that often works where faster methods fail.

  • Preface to the First Edition

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp viii-ix

  • Summary

    This book is targeted primarily toward engineers and engineering students of advanced standing (juniors, seniors, and graduate students). Familiarity with a computer language is required; knowledge of engineering mechanics (statics, dynamics, and mechanics of materials) is useful, but not essential.

    The text attempts to place emphasis on numerical methods, not programming. Most engineers are not programmers, but problem solvers. They want to know what methods can be applied to a given problem, what are their strengths and pitfalls, and how to implement them. Engineers are not expected to write computer code for basic tasks from scratch; they are more likely to utilize functions and subroutines that have been already written and tested. Thus, programming by engineers is largely confined to assembling existing bits of code into a coherent package that solves the problem at hand.

    The “bit” of code is usually a function that implements a specific task. For the user the details of the code are unimportant. What matters is the interface (what goes in and what comes out) and an understanding of the method on which the algorithm is based. Since no numerical algorithm is infallible, the importance of understanding the underlying method cannot be overemphasized; it is, in fact, the rationale behind learning numerical methods.

    This book attempts to conform to the views outlined above. Each numerical method is explained in detail and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method, and small programs that show how the computer code is utilized in solving a problem.

Fast Multipole Boundary Element Method
  • Theory and Applications in Engineering
  • Yijun Liu
  • Published online:
    14 January 2010
    Print publication:
    24 August 2009
  • The fast multipole method is one of the most important algorithms in computing developed in the 20th century. Along with the fast multipole method, the boundary element method (BEM) has also emerged as a powerful method for modeling large-scale problems. BEM models with millions of unknowns on the boundary can now be solved on desktop computers using the fast multipole BEM. This is the first book on the fast multipole BEM, which brings together the classical theories in BEM formulations and the recent development of the fast multipole method. Two- and three-dimensional potential, elastostatic, Stokes flow, and acoustic wave problems are covered, supplemented with exercise problems and computer source codes. Applications in modeling nanocomposite materials, bio-materials, fuel cells, acoustic waves, and image-based simulations are demonstrated to show the potential of the fast multipole BEM. Enables students, researchers, and engineers to learn the BEM and fast multipole method from a single source.

Matrix Preconditioning Techniques and Applications
  • Ke Chen
  • Published online:
    06 January 2010
    Print publication:
    14 July 2005
  • Preconditioning techniques have emerged as an essential part of successful and efficient iterative solutions of matrices. Ke Chen's book offers a comprehensive introduction to these methods. A vast range of explicit and implicit sparse preconditioners are covered, including the conjugate gradient, multi-level and fast multi-pole methods, matrix and operator splitting, fast Fourier and wavelet transforms, incomplete LU and domain decomposition, Schur complements and approximate inverses. In addition, aspects of parallel realization using the MPI are discussed. Very much a users-guide, the book provides insight to the use of these techniques in areas such as acoustic wave scattering, image restoration and bifurcation problems in electrical power stations. Supporting MATLAB files are available from the Web to support and develop readers' understanding, and provide stimulus for further study. Pitched at graduate level, the book is intended to serve as a useful guide and reference for students, computational practitioners, engineers and researchers alike.

Non-Classical Problems in the Theory of Elastic Stability
  • Isaac Elishakoff, Yiwei Li, James H. Starnes, Jr
  • Published online:
    06 January 2010
    Print publication:
    29 January 2001
  • When a structure is put under an increasing compressive load, it becomes unstable and buckling occurs. Buckling is a particularly significant concern in designing shell structures such as aircraft, automobiles, ships, or bridges. This book discusses stability analysis and buckling problems and offers practical tools for dealing with uncertainties that exist in real systems. The techniques are based on two complementary theories which are developed in the text. First, the probabilistic theory of stability is presented, with particular emphasis on reliability. Both theoretical and computational issues are discussed. Secondly, the authors present the alternative to probability based on the notion of 'anti-optimization', a theory that is valid when the necessary information for probabilistic analysis is absent, that is, when only scant data are available. Design engineers, researchers, and graduate students in aerospace, mechanical, marine, and civil engineering who are concerned with issues of structural integrity will find this book a useful reference source.

Elements of the Theory of Structures
  • Jacques Heyman
  • Published online:
    18 December 2009
    Print publication:
    13 June 1996
  • A good grasp of the theory of structures - the theoretical basis by which the strength, stiffness and stability of a building can be understood - is fundamental to structural engineers and architects. Yet most modern structural analysis and design is carried out by computer, with the user isolated from the processes in action. This book provides a broad introduction to the mathematics behind a range of structural processes. The basic structural equations have been known for at least 150 years, but modern plastic theory has opened up a fundamentally new way of advancing structural theory. Paradoxically, the powerful plastic theorems can be used to examine 'classic' elastic design activity, and strong mathematical relationships exist between these two approaches. Some of the techniques used in this book may be familiar to the reader, and some may not, but each of the topics examined will give the structural engineer valuable insight into the basis of the subject. This lucid volume provides a valuable read for structural engineers and others who wish to deepen their knowledge of the structural analysis and design of buildings.

Applied Optimization
  • Formulation and Algorithms for Engineering Systems
  • Ross Baldick
  • Published online:
    03 December 2009
    Print publication:
    04 May 2006
  • The starting point in the formulation of any numerical problem is to take an intuitive idea about the problem in question and to translate it into precise mathematical language. This book provides step-by-step descriptions of how to formulate numerical problems and develops techniques for solving them. A number of engineering case studies motivate the development of efficient algorithms that involve, in some cases, transformation of the problem from its initial formulation into a more tractable form. Five general problem classes are considered: linear systems of equations, non-linear systems of equations, unconstrained optimization, equality-constrained optimization and inequality-constrained optimization. The book contains many worked examples and homework exercises and is suitable for students of engineering or operations research taking courses in optimization. Supplementary material including solutions, lecture slides and appendices are available online at www.cambridge.org/9780521855648.

Chaotic Behaviour of Deterministic Dissipative Systems
  • Milos Marek, Igor Schreiber
  • Published online:
    24 November 2009
    Print publication:
    16 May 1991
  • This is a graduate text surveying both the theoretical and experimental aspects of chaotic behaviour. Over the course of the past two decades it has been discovered that relatively simple, deterministic, nonlinear mathematical models that describe dynamic phenomena in various physical, chemical, biological and other systems yield solutions which are aperiodic and depend very sensitively on the initial conditions. This phenomenon is known as deterministic chaos. The authors present chaos as a model of many seemingly random processes in nature. Basic notions from the theory of dynamical systems and bifurcation theory, together with the properties of chaotic solutions, are then described and are illustrated by examples. A review of the numerical methods used both in studies of mathematical models and in the interpretation of experimental data is also provided.

Numerical Methods in Engineering with MATLAB®
  • Jaan Kiusalaas
  • Published online:
    21 November 2009
    Print publication:
    01 August 2005
  • Numerical Methods in Engineering with MATLAB® is a text for engineering students and a reference for practicing engineers, especially those who wish to explore the power and efficiency of MATLAB®. Examples and applications were chosen for their relevance to real world problems, and where numerical solutions are most efficient. Numerical methods are discussed thoroughly and illustrated with problems involving both hand computation and programming. MATLAB® mfiles accompany each method and are available on the book web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method. MATLAB® was chosen as the example language because of its ubiquitous use in engineering studies and practice. Moreover, it is widely available to students on school networks and through inexpensive educational versions. Explore numerical methods with MATLAB®, a great tool for teaching scientific computation.

A Compendium of Partial Differential Equation Models
  • Method of Lines Analysis with Matlab
  • William E. Schiesser, Graham W. Griffiths
  • Published online:
    08 October 2009
    Print publication:
    16 March 2009
  • Mathematical modelling of physical and chemical systems is used extensively throughout science, engineering, and applied mathematics. To use mathematical models, one needs solutions to the model equations; this generally requires numerical methods. This book presents numerical methods and associated computer code in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs). The authors focus on the method of lines (MOL), a well-established procedure for all major classes of PDEs, where the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code related to the associated PDE model.

  • 10 - Introduction to Optimization

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with MATLAB®
    Published online:
    05 June 2012
    Print publication:
    30 September 2009, pp 380-414

  • Summary

    Find x that minimizes F(x) subject to g(x) = 0, h(x) ≤ 0

    Introduction

    Optimization is the term often used for minimizing or maximizing a function. It is sufficient to consider the problem of minimization only; maximization of F(x) is achieved by simply minimizing − F(x). In engineering, optimization is closely related to design. The function F(x), called the merit function or objective function, is the quantity that we wish to keep as small as possible, such as the cost or weight. The components of x, known as the design variables, are the quantities that we are free to adjust. Physical dimensions (lengths, areas, angles, etc.) are common examples of design variables.

    Optimization is a large topic with many books dedicated to it. The best we can do in limited space is to introduce a few basic methods that are good enough for problems that are reasonably well behaved and do not involve too many design variables.

    The algorithms for minimization are iterative procedures that require starting values of the design variables x. If F(x) has several local minima, the initial choice of x determines which of these will be computed. There is no guaranteed way of finding the global optimal point. One suggested procedure is to make several computer runs using different starting points and pick the best result.

    More often than not, the design is also subjected to restrictions, or constraints, which may have the form of equalities or inequalities. As an example, take the minimum weight design of a roof truss that has to carry a certain loading. Assume that the layout of the members is given, so that the design variables are the cross-sectional areas of the members.

Page 60 of 98