Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers

摘要

We propose a solution strategy for a multimaterial minimum compliancetopology optimization problem, which consists in finding the optimal allocationof a finite number of candidate (possibly anisotropic) materials inside areference domain, with the aim of maximizing the stiffness of the body. As arelevant and novel application we consider the optimization of self-assembledstructures obtained by means of diblock copolymers. Such polymers are a classof self-assembling materials which spontaneously synthesize periodicmicrostructures at the nanoscale, whose anisotropic features can be exploitedto build structures with optimal elastic response, resembling biologicaltissues exhibiting microstructures, such as bones and wood. For this purpose wepresent a new generalization of the classical Optimality Criteria algorithm toencompass a wider class of problems, where multiple candidate materials areconsidered, the orientation of the anisotropic materials is optimized, and theelastic properties of the materials are assumed to depend on a scalarparameter, which is optimized simultaneously to the material allocation andorientation. Well-posedness of the optimization problem and well-definition ofthe presented algorithm are narrowly treated and proved. The capabilities ofthe proposed method are assessed through several numerical tests.