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Unformatted text preview: Constrained Optimization 5 Most problems in structural optimization must be formulated as constrained min- imization 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. The methods described in this chapter are for use when the computational cost of evaluating the objective function and constraints is small or moderate. In these meth- ods the objective function or constraints these are calculated exactly (e.g., by a finite element program) whenever they are required by the optimization algorithm. This approach can require hundreds of evaluations of objective function and constraints, and is not practical for problems where a single evaluation is computationally ex- pensive. For these more expensive problems we go through an intermediate stage of constructing approximations for the objective function and constraints, or at least for the more expensive functions. The optimization is then performed on the approx- imate problem. This approximation process is described in the next chapter. The basic problem that we consider in this chapter is the minimization of a function subject to equality and inequality constraints minimize f ( x ) such that h i ( x ) = 0 , i = 1 , . . . , n e , g j ( x ) , j = 1 , . . . , n g . (5 . 1) The constraints divide the design space into two domains, the feasible domain where the constraints are satisfied, and the infeasible domain where at least one of the constraints is violated. In most practical problems the minimum is found on the boundary between the feasible and infeasible domains, that is at a point where g j ( x ) = 0 for at least one j . Otherwise, the inequality constraints may be removed without altering the solution. In most structural optimization problems the inequality constraints prescribe limits on sizes, stresses, displacements, etc. These limits have 159 Chapter 5: Constrained Optimization great impact on the design, so that typically several of the inequality constraints are active at the minimum. While the methods described in this section are powerful, they can often per- form poorly when design variables and constraints are scaled improperly. To prevent ill-conditioning, all the design variables should have similar magnitudes, and all con- straints should have similar values when they are at similar levels of criticality. A common practice is to normalize constraints such that g ( x ) = 0 . 1 correspond to a ten percent margin in a response quantity. For example, if the constraint is an upper limit a on a stress measure , then the constraint may be written as g = 1...
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