Approximate reanalysis in topology optimization

Despite rapid improvements in computer performance there is still room for advances in computational efficiency for handling structural optimization applications due to the high computational cost involved in the optimization process. In general, the total computational cost of the optimization procedure is governed by the complexity of three aspects: the model, the analysis and the optimization [1]. Solving problems that consider high complexities in all three aspects is presently somewhat limited in problem size, thus motivating the search for methods and procedures that require reduced computational resources but yield high quality results. In this paper, the implementation of an approximate reanalysis method in topology optimization is investigated. We apply the so-called nested approach where optimization is performed in the design variables only and where the equilibrium equations are treated as function calls. The goal is to reduce the computation involved in repeated solutions of the equilibrium equations, which for large problems will dominate the computational cost of the whole process. The approximate reanalysis performed here is based on the Combined Approximations (CA) approach, proposed by Kirsch [2] for linear static reanalysis and later used successfully also in vibration, dynamic and nonlinear reanalysis. The main feature of CA is the integration of a local series expansion in a global reduced basis solution. When using CA for repeated structural analysis, one can substantially reduce the number of required factorizations of the stiffness matrix, and thus removing a significant portion of the computational effort. In this study, CA has been implemented for topology optimization of linear structures, and the accuracy of the solutions obtained has been benchmarked against solutions presented in the literature ([3] and [4]). The relative efficiency of the method has been evaluated by comparing with the case in which all analysis equations are solved by a direct method. There is a clear trade-off between the accuracy and the efficiency of the approximate procedure. Both are mainly influenced by two parameters: the frequency of full matrix decompositions and the number of vectors generated for the reduced basis. More frequent decompositions will result in smaller changes in stiffness and therefore a more accurate approximation, but at the cost of matrix factorizations; similarly, using more basis vectors will give a better representation of the actual solution but will increase the computational effort. For all the trial cases considered, the optimal designs and the objective values obtained by the approximate procedure were practically