Structural Optimization Methods for Large Scale Problems: Status and Limitations

Abstract

This paper presents results from recent numerical experiments supported by theoretical arguments which indicate where are the limits of current optimization methods when applied to problems involving a large number of design variables as well as a large number of constraints, many of them being active. This is typical of optimal sizing problems with local stress constraints especially when composite materials are employed. It is shown that in both primal and dual methods the CPU time spent in the optimizer is related to the numerical effort needed to invert a symmetric positive definite matrix of size jact, jact being the effective number of active constraints, i.e. constraints associated with positive Lagrange multipliers. This CPU time varies with jact3. When the number m of constraints increases, jact has a tendency to grow, but there is a limit. Indeed another well known theoretical property is that the number of active constraints jact should not exceed the number of free primal variables iact, i.e. the number of variables that do not reach a lower or upper bound. This number iact is itself of course smaller than the real number of design variables n. This leads to the conclusion that for problems with many active constraints the CPU time could grow as fast as n3. With respect to m the increase in CPU time remains approximately linear. Some practical applications to real life industrial problems will be briefly shown: design optimisation of an aircraft composite wing with local buckling constraints and topology optimization of an engine pylon.