Topology optimization of anisotropic structure for arbitrary objective functionals with exact free boundary representation

A new approach to performing sensitivity analysis of arbitrary objective functionals for anisotropic elasticity is proposed in this work. Three different objective functionals have been considered, and good agreement is achieved between derived topological derivatives and numerical ones. Following the verification of topological derivatives, structural topology optimizations for selected anisotropic problems are conducted. To efficiently achieve the exact free boundary representation, our Finite Element Method (FEM)-based optimization comprises two loops. In the initial loop, a fixed and coarse mesh is employed to solve the anisotropic problem and update the level-set function. Once this loop concludes, the second loop reconstructs the material domain, ensuring an exact boundary representation. The convergence of the second loop is facilitated by (1) utilizing topological derivatives instead of explicit derivatives of ϕ (similar to density derivatives) and (2) imposing the exact volume constraint on the Reaction-Diffusion Equation (RDE)-based level-set method. Moreover, we introduce a scheme to prevent structural breakdown, allowing for the standalone implementation of Loop 2 always with exact free boundary representation. The previously proposed algorithm for the exact volume constraint has been generalized to accommodate inequalities, resulting in an acceleration of the equivalent optimization process.