Efficient single-level solution of hierarchical problems in structural optimization

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Virginia Polytechnic Institute and State University


Engineering design is hierarchical in nature, and if no attempt is made to benefit from this hierarchical nature, design optimization can be very expensive. There are two alternatives to taking advantage of the hierarchical nature of structural design problems. Multi-level optimization techniques incorporate the hierarchy at the formulation stage, and result in the coordinated optimization of a hierarchy of subsystems. The use of multi-level optimization techniques often necessitates the use of equality constraints. These constraints can sometimes cause numerical difficulties during optimization. Single-level decomposition techniques take advantage of the hierarchical nature to reduce the optimization cost.

In this research the decomposition approach has been followed to reduce the computational effort in a single-level design space. A decoupling technique has been developed that retains the advantages of a partitioned system of smaller independent subsystems without an increase in the total number of design variables. A penalty function formulation using Newton's method for the solution of a sequence of unconstrained minimizations was employed. The optimization of the decoupled system is cheaper due to (i) cheaper evaluation of the hessian matrix by taking advantage of its sparsity, (ii) fewer global analyses for constraint derivative calculations, and (iii) utilizing the decoupled nature of the hessian matrix in the solution process. Further, the memory requirements of the decoupled system are much less than that of the original coupled system. These benefits increase substantially for design problems with larger and larger number of detailed design variables.

Orthotropic material properties as stiffness global variables have been shown to be effective as global variables for panels in a hierarchical wing design formulation.

The proposed decoupling technique was implemented to minimize the volume of a portal frame and a wing box. Computational savings of up to 50 percent have been obtained for medium sized problems. The savings increase as the size of the problem and the amount of decoupling is increased. The procedure is simple to implement. For truly large systems this decoupling technique provides the necessary reduction of computational effort to make the optimization process viable.