Stochastic Topology Design Optimization for Continuous Elastic Materials

In this paper, we develop a stochastic model for topology optimization. We find robust structures that minimize the compliance for a given main load having a stochastic behavior. We propose a model that takes into account the expected value of the compliance and its variance. We show that, similarly to the case of truss structures, these values can be computed with an equivalent deterministic approach and the stochastic model can be transformed into a nonlinear programming problem, reducing the complexity of this kind of problems. First, we obtain an explicit expression (at the continuous level) of the expected compliance and its variance, then we consider a numerical discretization (by using a finite element method) of this expression and finally we use an optimization algorithm. This approach allows to solve design problems which include point, surface or volume loads with dependent or independent perturbations. We check the capacity of our formulation to generate structures that are robust to main loads and their perturbations by considering several 2D and 3D numerical examples. To this end, we analyze the behavior of our model by studying the impact on the optimized solutions of the expected-compliance and variance weight coefficients, the laws used to describe the random loads, the variance of the perturbations and the dependence/independence of the perturbations. Then, the results are compared with similar ones found in the literature for a different modeling approach