Kirsch Design-Oriented Analysis of Structures

A Unified Approach
1. Auflage 2006
ISBN: 978-0-306-48631-9
Verlag: Springer Netherland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 95, 255 Seiten, eBook

Reihe: Solid Mechanics and Its Applications

This book was developed while I was teaching graduate courses on analysis, design and optimization of structures, in the United States, Europe and Israel. Structural analysis is a main part of any design problem, and the analysis often must be repeated many times during the design process. Much work has been done on design-oriented analysis of structures recently and many studies have been published. The purpose of the book is to collect together selected topics of this literature and to present them in a unified approach. It meets the need for a general text covering the basic concepts and methods as well as recent developments in this area. This should prove useful to students, researchers, consultants and practicing engineers involved in analysis and design of structures. Previous books on structural analysis do not cover most of the material presented in the book. The book deals with the problem of multiple repeated analyses (reanalysis) of structures that is common to numerous analysis and design tasks. Reanalysis is needed in many areas such as structural optimization, analysis of damaged structures, nonlinear analysis, probabilistic analysis, controlled structures, smart structures and adaptive structures. It is related to a wide range of applications in such fields as Aerospace Engineering, Civil Engineering, Mechanical Engineering and Naval Architecture.

Zielgruppe

Research

Autoren/Hrsg.

Kirsch, Uri

Inhaltsverzeichnis

Concepts and Methods.- Structural Analysis.- Reanalsis of Structures.- Direct Methods.- Local Approximations.- Global Approximations.- A Unified Approach.- Combined Approximations (CA).- Simplified Solution Procedures.- Topological and Geometrical Changes.- Design Sensitivity Analysis.- Nonlinear Reanalysis.- Vibration Reanalysis.

Leseproben

6 Global Approximations (p. 85-86)

Global (multipoint) approximations are obtained by analyzing the structure at a number of design points, and they are valid for the whole design space (or, at least, large regions of it). This type of approximation may require much computational effort, particularly in problems with large numbers of design variables. This difficulty can be alleviated by the approach presented in Chapter 7.

Polynomial fitting and response surface methods are introduced in Section
6.1. In response surface methods, the response functions are replaced by simple functions (polynomials), which are fitted to data computed at a set of selected design points. So far in practice, the use of these methods has been limited to problems with a few design variables.

Reduced-basis methods are presented in Section 6.2. Using this approach, we approximate the response of a large system, which is originally described by many degrees of freedom, by a linear combination of a few pre-selected basis vectors. The problem is then stated in terms of a small number of unknown coefficients of the basis vectors. This approach is most effective in cases where highly accurate approximations can be achieved by the reduced system of equations. A basic question in using reduced basis methods relates to the choice of an appropriate set of the basis vectors. Response vectors of previously analyzed designs could be used, but an ad hoc or intuitive choice of these vectors may not lead to satisfactory approximations.

In addition, calculation of the basis vectors requires several exact analyses of the structure for the basis design points, which might involve extensive computational effort. A method for selecting the basis vectors that provides efficient and accurate results is presented in Section 7.1.1.

The conjugate gradient method described in Section 6.3 is an iterative method for solving a set of linear equations. The problem can be stated equivalently as the minimization of a quadratic function. The method generates a set of conjugate vectors such that the solution requires little storage and computation. If the quadratic function is minimized sequentially in an n dimensional space, once along each of a set of n conjugate directions, the minimum will be found at or before the nth step, regardless of the starting point.

A preconditioned conjugate gradient method, intended to accelerate convergence of ill-conditioned problems by transformation of the set of linear equations, is then developed.

6.1 POLYNOMIAL FITTING AND RESPONSE SURFACE

When the number of design variables is small it might be practical to analyze the structure at a number of design points, and use the response at those points to construct a polynomial approximation to the response at other points. Polynomial approximations obtained by analyzing the structure at a number of design points are global approximations. Obtaining such approximations can be quite expensive for problems with large numbers of design variables. For example, if the object is to fit the structural response by a quadratic polynomial, it is necessary to analyze the structure for at least n(n+1)/2 design points (typically, many more are required to ensure a robust approximation), where n is the number of design variables [1]. The most common global approximation is the response surface approach. Using this approach, we compute the response functions at a number of points, and then fit an analytical response surface, such as a polynomial, to the data.