Design Sensitivity Analysis of Structural Systems







Design Sensitivity Analysis

of Structural Systems

This is Volume 177 in

MATHEMATICS IN

SCIENCE AND ENGINEERING

A Series of Monographs and Textbooks

Edited by WILLIAM F. AMES,

Georgia Institute of Technology

The complete listing of books in this series is available from the Publisher

upon request.



Preface

Structural design sensitivity analysis concerns the relationship between

design variables available to the engineer and structural response or state

variables that are determined by the laws of mechanics. The dependence of

structural response measures such as displacement, stress, natural frequency,

and buckling load on design variables such as truss member crosssectional

area, plate thickness, and component shape is implicitly defined through the state equations

ofstructural mechanics. Attention is restricted in

this text to linear structural mechanics; i.e., to structures whose governing

equations (matrix, ordinary differential, or partial differential) are linear in

the state variables, once the design variable is fixed. Since design variables

appear in the coefficients oflinear operators, however, the state equations

are nonlinear as functions of state anddesign. The mathematical challenge is

to treat the nonlinear problem ofdesign sensitivity analysis with methods

that take advantage of mathematical properties ofthe linear (for fixed design)

state operators.A substantial literature on the technical aspects

ofstructural design sensitivity analysis exists. Some is devoted directly to the subject,

but most is imbedded in papers devoted to structural optimization. The premise

ofthis text is that a comprehensive theory ofstructural design sensitivity analysis

for linear elastic structures can be treated in a unified way. The objective of

the text is to provide a complete treatment of the theory and numerical

methods of structural design sensitivity analysis. The theory supports optimality

criteria methods ofstructural optimization and serves as the foundation for iterative methods

ofstructural optimization. One ofthe most common

methods ofstructural design involves decisions made by the designer,

based on experience and intuition. This conventional mode of structural



design can be substantially enhanced ifthe designer is provided with design

sensitivity information that explains what the influence ofdesign changes

will be, without requiring trial and error.

The advanced state ofthe art offinite element structural analysis provides

a reliable tool for evaluation ofstructural designs. In its present form, however,

it is used to identify technical problems, but it gives the designer little

help in identifying ways to modify the design to avoid problems or improve

desired qualities. Using design-sensitivity information that can be generated

by methods that exploit the finite-element formulation, the designer can

carry out systematic trade-offanalysis and improve the design. The numerical

efficiency of finite-element-based design sensitivity analysis and the

emergence ofinteractive graphics technology and CAD systems are factors

that suggest the time is right for interactive computer-aided design ofstructures,

using design sensitivity information and any of a variety of modem

optimization methods. It is encouraging to note that as ofSeptember 1983, the

MacNeal-Schwendler Corporation introduced one ofthe design sensitivity

analysis methods presented in Chapter 1in NASTRANfinite element

code. It is hoped that this represents a trend that will be followed by other

codes. In addition to developing design sensitivity formulas and numerical

methods, the authors have attempted to present a unified and relatively

complete mathematical theory ofstructural design sensitivity analysis. Recent

developments in functional analysis and linear operator theory provide

a foundation for rigorous mathematical analysis ofthe problem. Inaddition

to fulfilling one's instincts to be accurate, complete, and pure of heart, the

mathematical theory provides valuable insights and some surprisingly practical

results. The mathematical theory shows that positive definiteness (actually

strong ellipticity) of the operators of structural mechanics for stable

elastic structures is the property that provides most ofthe theoretical results

and makes numerical methods work. In the case of repeated eigenvalues,

which are now known to arise systematically in optimized designs,

the theory shows that repeated eigenvalues are not generally differentiable with respect

to design, but are only directionally differentiable. Erroneous results that

have appeared in the literature under the assumption ofdifferentiability can

now be corrected. Since such pathological problems and dangers lurk in

broad classes ofoptimum structures, it is hoped that the mathematical tools

presented in this text will help in constructing truly optimum structures that

do not have mathematically induced flaws.

The authors have attempted to write this text to meet the needs ofboth the

engineer who is interested in applications and the mathematician and theoretically

inclined engineer who are interested in the mathematical subtleties

ofthe subject. Toaccomplish this objective, each chapterhas been written to





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