Aligning feature-mapping designs to given density fields, first and second order

Abstract

Feature-mapping methods represent structural designs by explicit geometric primitives on fixed analysis grids, an interpretable and parametric alternative to the density fields of density-based topology optimization (SIMP). In contrast to SIMP, however, feature-mapping optimization is sensitive to the initial design. We present a gradient-based approach to align a feature-mapping configuration to a given (pseudo) density field, e.g., to initialize a subsequent feature-mapping optimization. The staged approach is based on a least-squares tracking formulation, preceded by a variant that only rewards alignment and does not penalize mismatch. Without an underlying finite element simulation, iterations are cheap, but isolated features receive little sensitivity information. As a remedy, we propose an asymmetric transition function based on automatically parametrized B\'ezier curves. We demonstrate the approach for a 2D cantilever and the five-bar design; for the latter, we show optional feature minimization based on the feature scaling variable of the geometry projection method. The approach works well with first-order optimizers. The absence of a state problem and the small number of variables, however, also make a second-order formulation attractive. To the best of the authors'knowledge, we present the first Hessian formulation for feature mapping and report the behavior of first- and second-order optimizers on our benchmark problems. For completeness, we also give the exact Hessian of the state-based compliance, which requires one additional solution of the FEM system per feature variable.