Approximation in non-hierarchic system optimization

This paper reports on the effectiveness of a nonhierarchic systems optimization algorithm in application to complex coupled systems problems. A second order nonhierarchic systems optimization algorithm developed in earlier studies is modified in this study to provide for individual constraintlstate modelling. A cumulative constraint formulation was used in previous implementation studies. The test problems in this study are each complex coupled systems. Complex coupled systems require an iterative solution strategy to evaluate system states. Nonhierarchic algorithm development is driven by these types of problems and their study is imperative. The algorithm successfully optimizes each of the complex coupled systems. A significant reduction in the number of system analyses required for optimization is observed as compared to conventional optimization using the generalized reduced gradient method. Overview The non-hierarchic systems optimization algorithm used in this research was initially proposed in RenaudlGabriele9. The strategy provides for concurrent nonlinear subspace optimizations followed by a coordination procedure of system approximation. The system approximations are built using design information generated during the subspace optimizations. The original algorithm formulation (RenaudlGabriele9) used an accumulated approximation strategy (Rasmussen8) for system approximation. The accumulated approximations were built about a first order basis function of approximation. Implementation of the algorithm as reported in RenaudlGabriele9 provided robust coordination in nonhierarchic system optimization studies. The algorithm was able to locate the well documented optimums for the welded beam problem of ~agsdell/Phillips7 and the speed reducer problem of Golinski4. The study also identified areas of the algorithm that could be improved. The f is t order based *Assistant Professor, Member, AIAA, ASME Wssociate Professor, Member, ASME Copyright O American Institute of Aeronautics and Astronautics. Inc., 1992. All rights reserved accumulated approximation of the objective function in the coordination procedure of system approximation was very accurate. However the accumulated approximation of cumulative constraints in the coordination procedure of system approximation was observed to be less predictive. Changes in the active set within a cumulative constraint were not handled well by the first order based accumulated approximation. This condition produced cycling of the objective function value (i.e., oscillation) during nonhierarchic optimization. To improve algorithm performance, the second order based coordination procedure of system approximation was developed in Renaud/GabrielelO. The second order modification was aimed at improving the accuracy of cumulative constraint approximation in the coordination procedure. The second order basis is obtained numerically using data already available in the design data basc. The design data base stores design site information generated during the subspace optimizations. A quadratic polynomial approximation to the design is formed using ~anderplaatsl7 strategy. A weighted least squares solution strategy is employed to solve for the second order terms in the Vanderplaats strategy. Exact data in the design data basc is more heavily weighted in the least squares solution procedure. The resulting quadratic polynomial forms the basis function of accumulated approximation replacing the linear basis used in the original formulation. In Renaud/GabrielelO improved convergence is observed for the welded beam test problem. The improved convergence (is., less cycling) is attributed to the improved accuracy of cumulative constraint approximations when using sccond order based approximating functions. Additional studies using the second order based coordination procedure of system approximation indicated that replacing the cumulative constraints with their component constraints may improve algorithm performance. Implementation of the second order bascd coordination procedure of system approximation was less effective in reducing cycling when applied to the Golinski speed reducer problem. The speed reducer cumulative constraints were composed of a large number of individual constraints as compared to the cumulative constraints in the welded beam test problem. With a larger number of individual constraints assigned to a cumulative constraint, it will more likely undergo a change in its active set during the coordination procedure. It is difficult to approximate these changes in the cumulative constraints. Poor cumulative constraint approximations lead to cyclic convergence. Approximating individual constraints/states in the coordination procedure of system approximation should more accurately predict active set changes. The subspace optimization of Renaud/GabrielelO is modified to reflect individual constraint/state modelling in this study. One drawback to the modelling of individual constraints is the increased size of the design data base. The use of individual constraint modelling or cumulative constraint modelling is ultimately a user choice in algorithm implementation. The proposed algorithm with individual constraint modelling is applied to complex coupled test problems in this study. In previous implementation studies the problem coupling was non-complex (i.e., diagonal coefficient matrix in the GSE). In this study the algorithm is seen to be effective in non-hierarchic optimization of the alkylation process problem of Sauer/Colvillel2 and the electronic packaging problem of Renaudll and Korngold, et a1.15. 11. Algorithm Overview This section presents an overview of the nonhierarchic system optimization algorithm proposed in RenaudIGabrielelO and modified in this research. The reader is referred to Figure 1 which depicts the algorithm flowchart. The method optimizes decomposed subspaces concurrently, followed by a coordination procedure of system approximation. In this study, optimization of a second order system approximation problem is used as a coordination procedure for directing system convergence and resolving subspace conflicts. The second order system approximation problem is formed using information obtained during concurrent subspace optimizations. This corresponds to existing design practice where individual design groups optimize their component designs, and tradeoffs or compromises are made based on the "whole" of the designers' previous experience. In the algorithm, information obtained during the subspace optimizations is used to represent designers' knowledge and data for respective design subspaces. The subspace optimizations serve to improve the component designs of each design group, while simultaneously building a design data base used in the coordination procedure of system approximation. The design data base stores state information for design vectors investigated during subspace optimizations. This information is "free" as it is generated by the local optimizerts) during the subspace optimizations. The cost of storing the design data base represents additional overhead intrinsic to approximation methods. System coupling is maintained and updated using the GSE approach as introduced by Sobieskil3.14. The algorithm, notation and each module in Figure 1 are detailed in Renaud/GabrielelO.