Elements of Structural Optimization

Optimization is concerned with achieving the best outcome of a given operation 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’.

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. Still, the consideration of such problems is not a purely academic exercise, but is very helpful in the process of solving more realistic problems.

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 forms. 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 decision making processes. 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 linear and nonlinear programming problems (commonly designated as LP and NLP, respectively).

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 for 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 constantly to 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 (see Kamat and Hayduk[[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.

Most problems in structural optimization must be formulated as constrained minimization problems. In a typical structural design problem the objective function is a fairly simple function of the design variables (e.g., weight), but 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.

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, however, 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.

The first step in the analysis of a complex structure is 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.

The methods for discrete sensitivity analysis discussed in the previous chapter are very general in that they may be applied to a variety of nonstructural sensitivity analyses involving systems of linear equations, eigenvalue problems, etc. However, for structural applications they have two disadvantages. First, not all methods of structural analysis 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.

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.

Because of their superior mechanical properties compared to single phase materials, laminated fibrous 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 the complex behavior and multiple failure modes of these structures require 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 curse for the designer, in that he has more control to fine-tune his structure to meet design requirements, but only if he can figure out how to select those design variables. The possibility of achieving an efficient design that is safe against multiple failure mechanisms, 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.