Multi-Material and Multi-Constraint Topology Optimization for Structures Under Large Deformation

Topology optimization is a computational design tool for finding optimal layouts of structures and material microstructures with applications from as large as high-rise buildings to as small as material design. Within the field of topology optimization, multi-material topology optimization is an emerging area. However, most work in the field has been restricted to single material with linear elastic material behavior, limited volume constraint settings, and small deformations. To enable more design freedom and innovations, this talk will introduce a general multi-material topology optimization formulation considering both material nonlinearity and finite deformation, which is capable of optimizing both the topology and the material composition simultaneously. The proposed formulation handles an arbitrary number of candidate materials and features a generalized setting of local and global volume constraints. This feature is achieved by a novel design parametrization and a general material interpolation scheme, which can account for candidate material phases with flexible properties (such as hardening, softening, isotropic, and anisotropic behaviors).

To effectively and efficiently solve the proposed framework, we also employ several techniques. First, to efficiently handle arbitrary volume constraints, we employ and tailor the ZPR (Zhang-Paulino-Ramos) design variable update algorithm for the proposed formulation. The tailored ZPR update algorithm, which exploits the separability of the convex subproblem at each optimization step, performs robust updates of the design variables associated with each volume constraint independently and in parallel. Second, to efficiently solve nonlinear state equations resulting from both the material and geometric nonlinearities, we formulate the virtual element approximations and adopt adaptive mesh refinement and coarsening schemes to solve the finite elasticity boundary value problem, which considerably improves the computational efficiency of the framework. The novel update algorithm and solution strategies allow for high-resolution designs efficiently. Moreover, to overcome the numerical difficulty introduced by low-density element, the energy interpolate scheme is extended to account for multiple material phases. 

The proposed formulation can handle a wide range of objective functions and constraints. In this talk, we present applications of the proposed framework in several composite design problems, including compliance minimization, complaint mechanism, and more. Through those application examples using a combination of various types of nonlinear materials, we demonstrate that the proposed topology optimization formulation is robust with respect to large deformations, can effectively design soft composites by simultaneously optimizing their topology as well as material phases, and offers a promising avenue toward the systematic design of composite meta-materials and innovative engineering structures.