Optimal Fine-Scale Structures in Compliance Minimization for a Shear Load

We return to a classic problem of structural optimization whose solution requires microstructure. It is well-known that perimeter penalization assures the existence of an optimal design. We are interested in the regime where the perimeter penalization is weak; i.e., in the effect of perimeter as a selection mechanism in structural optimization. To explore this topic in a simple yet challenging example, we focus on a two-dimensional elastic shape optimization problem involving the optimal removal of material from a rectangular region loaded in shear. We consider the minimization of a weighted sum of volume, perimeter, and compliance (i.e., the work done by the load), focusing on the behavior as the weight ɛ of the perimeter term tends to 0. Our main result concerns the scaling of the optimal value with respect to ɛ. Our analysis combines an upper bound and a lower bound. The upper bound is proved by finding a near-optimal structure, which resembles a rank-2 laminate except that the approximate interfaces are replaced by branching constructions. The lower bound, which shows that no other microstructure can be much better, uses arguments based on the Hashin-Shtrikman variational principle. The regime being considered here is particularly difficult to explore numerically due to the intrinsic nonconvexity of structural optimization and the spatial complexity of the optimal structures. While perimeter has been considered as a selection mechanism in other problems involving microstructure, the example considered here is novel because optimality seems to require the use of two well-separated length scales.