  This paper considers the design of structures made of engineered materials,
accounting for uncertainty in material properties. We present a topology
optimization approach that optimizes the structural shape and topology at the
macroscale assuming design-independent uncertain microstructures. The
structural geometry at the macroscale is described by an explicit level set
approach, and the macroscopic structural response is predicted by the eXtended
Finite Element Method (XFEM). We describe the microscopic layout by either an
analytic geometric model with uncertain parameters or a level cut from a
Gaussian random field. The macroscale properties of the microstructured
material are predicted by homogenization. Considering the large number of
possible microscale configurations, one of the main challenges of solving such
topology optimization problems is the computational cost of estimating the
statistical moments of the cost and constraint functions and their gradients
with respect to the design variables. Methods for predicting these moments,
such as Monte Carlo sampling, and Taylor series and polynomial chaos expansions
often require many random samples resulting in an impractical computation. To
reduce this cost, we propose an approach wherein, at every design iteration, we
only use a small number of microstructure configurations to generate an
independent, stochastic approximation of the gradients. These gradients are
then used either with a gradient descent algorithm, namely Adam, or the
globally convergent method of moving asymptotes (GCMMA). Three numerical
examples from structural mechanics are used to show that the proposed approach
provides a computationally efficient way for macroscale topology optimization
in the presence of microstructural uncertainty and enables the designers to
consider a new class of problems that are out of reach today with conventional
tools.
