Efficient compliance-based topology optimization with many load cases scenarios

Resumo

In topology optimization setting, we can cast a variety of problems into a weighted-sum of compliances
minimization. Robust design, for example, is commonly addressed in the form of a finite sum of deterministic load
cases scenarios. Another example is the optimization of structures subjected to dynamic loads using the equivalent
static load method, where a finite set of associated loads is defined according to the displacements over time. But
when the number of loads involved is high, the solution of these problems becomes extremely expensive from a
computational point of view, due to the necessity of solving one finite element problem for each of these loading
cases along the steps of the optimization algorithm to evaluate the objective function. In this context, two methods
for dimensionality reduction of the problem are presented. First, an equivalent stochastic problem to the original
one is determined, which reduces to only a few the number of necessary load cases at each step. The second
approach uses the singular-value decomposition of the matrix that gathers the different loading scenarios to reduce
the number of linear systems to be solved. The applicability of both methods to different topology optimization
scenarios is discussed, and numerical examples are proposed to compare the final topologies obtained and quantify
the reduction in the number of necessary finite element solves.