A second-order measure of boundary oscillations for overhang control in topology optimization

In topology optimization, oscillating boundaries may occur when geometrical overhang angle constraints are imposed for additive manufacturability. Indeed, boundary oscillations formally satisfy the constraints, but lead to designs that are ultimately unacceptable. In order to avoid such phenomenon, in this paper we formulate a measure of boundary oscillations that can be used to define filters, penalties and constraints to suppress oscillating boundaries. In particular, first we mathematically characterize the density distribution that forms a boundary oscillation. On this basis, we formulate our measure, which is a function that is positive in correspondence of tips of oscillating boundaries and is zero everywhere else. Such analysis is first presented in a 2D framework and later extended to 3D problems. Then, we show how this measure can be employed to formulate strategies that suppress boundary oscillations. In particular, we propose an adaptive anisotropic filter and a cost penalty that fulfill this task. Numerical experiments finally show the capabilities of our measure and of the proposed oscillation control strategies. In this context, an efficient, practical implementation in a first-order finite-element space and a 3D example are provided as well. (c) 2020 Elsevier Inc. All rights reserved.