Comparison of various move limit strategies in structural optimization

Many recent improvements have been made in the field of optimization, among which is the introduction of the theory of approximations to improve problem convergence and reduce the computational burden of repeated full analyses. This theory states that an approximation can be made in the neighborhood of the current design point, and the optimizer can make repeated calls to this approximation in lieu of the analysis program (usually finite elements). However, when such approximations are introduced, a strategy must be formulated which determines the n-dimensional domain around the current design point where the approximation is still applicable. The bounds of this domain are called move limits and typically, they are uniformly applied to all the design variables according to the experience of the user. The objectives of this research are to compare two recently proposed strategies for the calculation of move limits, and propose either new or hybrid methods that allow the design problem to converge to the correct answer in the least number of iterations. This paper treats the move limit as a function of two parameters: the curvature of the objectivelconstraint functions at the current design point, and the effect of a particular design variable on the constraints. The curvature of the function is indicated by the magnitude of the exponent obtained by the Two-Point Exponential approximation1. The effect of the design variable on the constraint can be determined by effectiveness coefficients2. The Two-Point Exponential approximation uses previous design histories to find a better fitting approximation. This approximation uses an exponent which is obtained by matching slopes at the previous design point. This expo*l Mechanical Engineering student. Research Experience for Undergraduates participant at Clemson University '2 Assistant Professor, Mechanical Engineering Department, Member AlAA and ASME. nent provides a measure of curvature3 for eachconstraint and with respect to each design variable. Next, depending on the magnitude of the exponents specific to a single design variable, move limits can be assigned based on some absolute minimum and maximum allowable limits that the user prescribes. Using the Kreisselmeier-Steinhauser 6.S . ) function2, an overall effectiveness coefficient which relates the effect of a particular design variable on all of the constraints simultaneously can be derived. Then, based on the variation between these effectiveness coefficients, move limits for each variable can be computed. These move limits are also based on some absolute minimum and maximum that the user prescribes. Both strategies are implemented and compared. Speed of convergence is tabulated for various minimum and maximum allowable move limits. In particular, the variations of the individual move-limits computed at each iteration for both methods are compared to identify differences between the two strategies and devise new methods. Five variations and combinations of the two strategies are tested on a structural design problem. The comparison of these move limit strategies shows that the method based on the curvature of active constraints provides the best balance between accuracy and speed of convergence. Introduction In many modern engineering design problems, optimizationmust be used. Typically, theengineer mustfirstestimate the initial design parameters, analyze this design through some analysis process (finite elements in the case of structural design), and then check to see if the design is optimal; i.e. no improvement in the objective function can be Copyright 01994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved 1 made without violating one or more constraint functions. If requires a minimal amount of effort. the optimum design isnot achieved, then the analysis is connected to an optimizer. This optimizer attempts to minimize the objective function while satisfying the constraints typically by using gradient information. During the process, the design parameters are changed, and the optimizer makes repeated calls to the analysis program to evaluate subsequent designs. This iterative process demands an extremely large amount of computer time for real-world problems, which in turn raises the cost of the design. A major improvement to this classical design approach is to use the approximation theory. This theory states that one can approximate the given problem, optimize this approximation, and then check for global convergence. However, since the approximations are only valid in a reduced space, some mechanism to avoid misleading the optimizer is needed. A flowchart of this process is shown in Figure 1. Many approximation schemes can be used, but the most common is the simple Taylor series expansion of the functions around the design point. This is given by the following equation: Because the first derivatives at the design point are already known either directly from the finite element analysis, or from some mereda t ion scheme, this approximation Many other approximations were developed over the last two decades. Some are more appropriate to certain types of problems. others try to over-constrain the problem to ensure a feasible solution. Notable approximations are the reciprocal appr~ximation~>*~, the modified approximation7, the Conservative Convex or Hybrid approximations. By introducing any of these approximations, however, the problem mentioned earlier arises: how can we determine the distance from the current design point which limits the applicability of the approximation? Until recent research in the subject, engineers have simply chosen these move limits, usually constant throughout the optimization and uniform over the design variables, based on their experience with the particular problem they are dealing with. This paper summarizes past research in variable move limit strategies and proposes several new combinations or modifiiations of these strategies. By choosing appropriate move limits, the speed of problem convergence can be improved drastically. Also, current research in multi-disciplinary optimization demands an effkient method for assigning move limits -by setting these correctly, one problem may be optimized from the viewpoint of different engineering fields. In effect, the mwe limit would define an "allowable change" in a particular design variable for multi-disciplinary optimization. Initialize Variables I