ISOCOMP: Unified geometric and material composition for optimal topology design

In this paper, a unified strategy is developed to simultaneously insert inclusions or holes of regular shape as well as redistribute the material to effect optimal topologies of solids. We demonstrate the unified optimal design strategy through three possible choices of design variables: (1) purely geometrical, (2) purely material, and (3) geometrical-material. We couple the geometrical approach with the topological derivative of the objective function and a condition derived for optimally inserting an infinitesimal ellipsoidal heterogeneity (hole or inclusion) into the structure. The approximations of the geometry, material and behavioral fields are isoparametric (or isogeometric) and are composed consistent with the Hierarchical Partition of Unity Field Compositions (HPFC) theory (Rayasam et al., Int J Numer Methods Eng 72(12):1452-1489, 2007). Specifically, analogous to the constructive solid geometry procedure of CAD, the complex material as well as the behavioral field is modeled hierarchically through a series of pair-wise compositions of primitive fields defined on the primitive geometrical domains. The geometrical, material and behavioral approximations are made using Non-Uniform Rational B-Splines (NURBS) basis functions. Thus, the proposed approach seamlessly unifies the explicit representation of boundary shapes with the implicit representations of boundaries arising out of material redistribution, and is termed ISOCOMP, or isoparametric compositions for topology optimization. The methodology is demonstrated first on a set of example problems that increase in complexity of design variable choice culminating in simultaneous optimization of hole location, hole shape and material distribution within the domain. This is followed by a detailed case study involving topology optimization of a bicycle dropout..