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1990 | Buch | 2. Auflage

Introduction Kapitel lesen Erstes Kapitel lesen

Elements of Structural Optimization

verfasst von: Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat

Verlag: Springer Netherlands

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

The field of structural optimization is still a relatively new field undergoing rapid changes in methods and focus. Until recently there was a severe imbalance between the enormous amount of literature on the subject, and the paucity of applications to practical design problems. This imbalance is being gradually redressed now. There is still no shortage of new publications, but there are also exciting applications of the methods of structural optimizations in the automotive, aerospace, civil engineering, machine design and other engineering fields. As a result of the growing pace of applications, research into structural optimization methods is increasingly driven by real-life problems. Most engineers who design structures employ complex general-purpose software packages for structural analysis. Often they do not have any access to the source the details of program, and even more frequently they have only scant knowledge of the structural analysis algorithms used in this software packages. Therefore the major challenge faced by researchers in structural optimization is to develop methods that are suitable for use with such software packages. Another major challenge is the high computational cost associated with the analysis of many complex real-life problems. In many cases the engineer who has the task of designing a structure cannot afford to analyze it more than a handful of times.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Optimization is concerned with achieving the best outcome of a given objective while satisfying certain restrictions. Human beings, guided and influenced by their natural surroundings, almost instinctively perform all functions in a manner that economizes energy or minimizes discomfort and pain. The motivation is to exploit the available limited resources in a manner that maximizes output or profit. The early inventions of the lever or the pulley mechanisms are clear manifestations of man’s desire to maximize mechanical efficiency. Innumerable other such examples abound in the saga of human history. Douglas Wilde [1] provides an interesting account of the origin of the word optimum and the definition of an optimal design. We will paraphrase Wilde and offer the definition of an optimal design as being ‘the best feasible design according to a preselected quantitative measure of effectiveness’.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 2. Classical Tools in Structural Optimization
Abstract
Classical optimization tools used for finding the maxima and minima of functions and functionals have direct applications in the field of structural optimization. The words classical tools are implied here to encompass the classical techniques of ordinary differential calculus and the calculus of variations. Exact solutions to a few relatively simple unconstrained or equality constrained problems have been obtained in the literature using these two techniques. It must be pointed out, however, that such problems are often the result of simplifying assumptions which at times lack realism, and result in unreasonable configurations. One would hasten to question if the consideration of such problems is purely an academic exercise; it is quite the contrary.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 3. Linear Mathematical Programming
Abstract
Mathematical programming is concerned with the extremization of a function f defined over an n-dimensional design space R n and bounded by a set S in the design space. The set S may be defined by equality or inequality constraints, and these constraints may assume linear or nonlinear form. The function f together with the set S in the domain of f is called a mathematical program or a mathematical programming problem. This terminology is in common usage in the context of problems which arise in planning and scheduling which are generally studied under operations research; the branch of mathematics concerned with the decision making process. Mathematical programming problems may be classified into several different categories depending on the nature and form of the design variables, constraint functions, and the objective function. However, only two of these categories are of interest to us, namely the linear and the nonlinear programming (commonly designated as LP and NLP, respectively).
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 4. Unconstrained Optimization
Abstract
In this chapter we study mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important because of several reasons. First of all, if the design is at a stage where no constraints are active then the process of determining a search direction and travel distance for minimizing the objective function involves an unconstrained function minimization algorithm. Of course in such a case one has to constantly watch for constraint violations during the move in design space. Secondly, a constrained optimization problem can be cast as an unconstrained minimization problem even if the constraints are active. The penalty function and multiplier methods discussed in Chapter 5 are examples of such indirect methods that transform the constrained minimization problem into an equivalent unconstrained problem. Finally, unconstrained minimization strategies are becoming increasingly popular as techniques suitable for linear and nonlinear structural analysis problems [1] which involve solution of a system of linear or nonlinear equations. The solution of such systems may be posed as finding the minimum of the potential energy of the system or the minimum of the residuals of the equations in a least squared sense.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 5. Constrained Optimization
Abstract
Most problems in structural optimization cannot be formulated as unconstrained minimization problems. In fact, in a typical structural design problem the objective function is a fairly simple function of the design variables (e.g. weight). However, the design has to satisfy a host of stress, displacement, buckling, and frequency constraints. These constraints are usually complex functions of the design variables available only from an analysis of a finite element model of the structure. This chapter offers a review of methods that are commonly used to solve such constrained problems.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 6. Aspects of The Optimization Process in Practice
Abstract
Occasionally, a structural analyst will write a design program that includes the calculation of structural response as well as an implementation of a constrained optimization algorithm, such as those discussed in Chapter 5. More often the analyst will have a structural analysis package, such as a finite-element program, as well as an optimization software package available to him. The task of the analyst is to combine the two so as to bring them to bear on the structural design problem that he wishes to solve.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 7. Sensitivity of Discrete Systems
Abstract
The first step in the analysis of a complex structure is typically a spatial discretization of the continuum equations into a finite element, finite difference or a similar model. The analysis problem then requires the solution of algebraic equations (static response), algebraic eigenvalue problems (buckling or vibration) or ordinary differential equations (transient response). The sensitivity calculation is then equivalent to the mathematical problem of obtaining the derivatives of the solutions of those equations with respect to their coefficients. This is the main subject of the present chapter.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 8. Introduction to Variational Sensitivity Analysis
Abstract
The methods for discrete sensitivity analysis discussed in the previous chapter are very general in that they are equally applicable to nonstructural sensitivity analysis involving systems of linear equations, eigenvalue problems, etc. However, for structural applications they also have two disadvantages. First, not all structural analysis solution methods lead to the type of discretized equations that are discussed in Chapter 7. For example, shell-of-revolution codes such as FASOR [1] directly integrate the equations of equilibrium without first converting them to systems of algebraic equations. Second, operating on the discretized equations often requires access to the source code of the structural analysis program which implements these equations. Unfortunately, many of the popular structural analysis programs do not provide such access to most users. It is desirable, therefore, to have sensitivity analysis methods that are more generally applicable and can be implemented without extensive access to and knowledge of the insides of structural analysis programs. Variational methods of sensitivity analysis achieve this goal by differentiating the equations governing the structure before they are discretized. The resulting sensitivity equations can then be solved with the aid of a structural analysis program. It is not even essential that the same program be used for the analysis and the sensitivity calculations.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 9. Dual and Optimality Criteria Methods
Abstract
During the seventies people who worked in the field of structural optimization were divided into two distinct and somewhat belligerent camps. On the one side the mathematical programming camp believed in employing the general methods of linear and nonlinear programming such as the simplex method, penalty function techniques, gradient projection techniques or methods of feasible directions (see Chapter 5). On the other side the optimality criteria camp subscribed to the use of methods described in the present chapter. The mathematical programming camp decried the lack of generality of optimality criteria methods, and the optimality criteria camp sneered at the inefficiency of the general mathematical programming approach.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 10. Decomposition and Multilevel Optimization
Abstract
The resources required for the solution of an optimization problem typically increase with the dimensionality of the problem at a rate which is more than linear. That is, if we double the number of design variables in a problem, the cost of solution will typically more than double. Large problems may also require excessive computer memory allocations. For these reasons we often seek ways of breaking a large optimization problem into a series of smaller problems.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Chapter 11. Optimum Design of Laminated Composite Structures
Abstract
Because of their superior mechanical properties, laminated composite materials are finding a wide range of applications in structural design, especially for lightweight structures that have stringent stiffness and strength requirements. Designing with laminated composites, on the other hand, has become a challenge for the designer because of a wide range of parameters that can be varied, and because of the complex behavior of these structures that mandates sophisticated analysis techniques. Finding an efficient composite structural design that meets the requirements of a certain application can be achieved not only by sizing the cross-sectional areas and member thicknesses, but also by global or local tailoring of the material properties through selective use of orientation, number, and stacking sequence of laminae that make up the composite laminate. The increased number of design variables is both a blessing and a nightmare for the designer, in that he has more control to fine-tune his structure to meet the requirements of a design situation, but only if he can figure out how to select all those design variables. The possibility of achieving a design that meets multiple requirements efficiently, coupled with the difficulty in selecting the values of a large set of design variables makes structural optimization an obvious tool for the design of laminated composite structures.
Raphael T. Haftka, Zafer Gürdal, Manohar P. Kamat
Backmatter
Metadaten
Titel
Elements of Structural Optimization
verfasst von
Raphael T. Haftka
Zafer Gürdal
Manohar P. Kamat
Copyright-Jahr
1990
Verlag
Springer Netherlands
Electronic ISBN
978-94-015-7862-2
Print ISBN
978-94-015-7864-6
DOI
https://doi.org/10.1007/978-94-015-7862-2