PDE-constrained optimization for metamaterial design under uncertainty

Abstract

The goal of this work is to explore computational strategies for solving a topology optimization problem for a metamaterial that acts as a cloak for acoustic or electromagnetic waves. The design variable is defined as a spatially varying field that represents the material density. The minimized quantity of the optimization problem is the scattered wave resulting from the reflection of the incident signal on the obstacle's surface. We extend the vanilla mathematical formulation and algorithmic implementation of this PDE-constrained optimization problem to obtain more realistic results, focusing on the definition of material surfaces, the structural integrity, and the design's manufacturability. We implement a decoupled Newton solver and an optimization continuation scheme to ensure numerical convergence of the algorithm. Both enhance the capabilities of the optimization algorithm to deal with nonlinearities and numerical instabilities, which allow for a deeper exploration of the solution space. In the formulation, incorporating the Helmholtz filter and a density projection promotes the homogeneization and definition of the material design. We fine-tune these objects and apply a coordinate descent algorithm to ensure the imposition of both techniques. Additionally, we include uncertainty in the formulation to obtain more realistic designs that contemplate real-life inaccuracies. An additive Gaussian random field is added to the parameters to represent the fluctuations of the material property or the manufacturing error. We study the mean-variance and Conditional Value at Risk risk measures under Monte Carlo simulations. We formulate and solve the optimal design under uncertainty problem for the circular obstacle, susceptible to both single-direction single-frequency incident waves and multiple-direction multiple-frequency incident waves, and encircled by a ring-shaped cloaking region, for both two and three-dimensional settings. Our numerical results show an improvement in the designs with respect to the state-of-the-art, obtaining well-defined binary materials.