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

TL;DR

This paper introduces a novel topology optimization method for multiple anisotropic materials, applied to designing self-assembling nanostructures like diblock copolymers, optimizing material distribution, orientation, and properties for enhanced stiffness.

Contribution

It generalizes the Optimality Criteria algorithm to handle multiple anisotropic materials with orientation and property optimization in topology design.

Findings

Successfully optimized structures with anisotropic materials.

Demonstrated improved stiffness in self-assembled copolymer structures.

Validated the method through numerical experiments.

Abstract

We propose a solution strategy for a multimaterial minimum compliance topology optimization problem, which consists in finding the optimal allocation of a finite number of candidate (possibly anisotropic) materials inside a reference domain, with the aim of maximizing the stiffness of the body. As a relevant and novel application we consider the optimization of self-assembled structures obtained by means of diblock copolymers. Such polymers are a class of self-assembling materials which spontaneously synthesize periodic microstructures at the nanoscale, whose anisotropic features can be exploited to build structures with optimal elastic response, resembling biological tissues exhibiting microstructures, such as bones and wood. For this purpose we present a new generalization of the classical Optimality Criteria algorithm to encompass a wider class of problems, where multiple candidate materials are considered, the orientation of the anisotropic materials is optimized, and the elastic properties of the materials are assumed to depend on a scalar parameter, which is optimized simultaneously to the material allocation and orientation. Well-posedness of the optimization problem and well-definition of the presented algorithm are narrowly treated and proved. The capabilities of the proposed method are assessed through several numerical tests.