TOPOLOGICAL SENSITIVITY-BASED ANALYSIS AND OPTIMIZATION OF MICROSTRUCTURED INTERFACES

This paper concerns the optimization of microstructures within a surface when considering the propagation of scalar waves across a periodic row of inclusions embedded within a homogeneous matrix. The approach relies on the low-frequency homogenized model, which consists, in the present case, of some effective jump conditions through a discontinuity within the ambient medium. The topological derivatives of the effective parameters defining these jump conditions are computed from an asymptotic analysis. Their expressions are validated numerically and then used to study the sensitivity of the homogenized model to the geometry in the case of elliptic inclusions. Finally, a topological optimization algorithm is used to minimize a given cost functional. This relies on the expression of the topological derivatives to iteratively perform phase changes in the unit cell characterizing the material, and on FFT-accelerated solvers previously adapted to solve the band cell problems underlying the homogenized model. To illustrate this approach, the resulting procedure is applied to the design of a microstructure that minimizes transmitted fields along a given direction.