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.

在拓扑优化中，当为附加可制造性施加几何悬垂角约束时，可能会发生振荡边界。确实，边界振荡在形式上满足了约束条件，但导致最终无法接受的设计。为了避免这种现象，在本文中，我们制定了一种边界振荡的度量，可以用来定义滤波器，惩罚和约束来抑制振荡边界。

特别地，首先我们在数学上描述形成边界振荡的密度分布。在此基础上，我们制定了测度，该测度是一个函数，与振荡边界的“尖端”相对应，为正，而在其他所有位置为零。这种分析首先在2D框架中进行，然后扩展到3D问题。

然后，我们说明如何使用此度量来制定抑制边界振荡的策略。特别是，我们提出了一种自适应各向异性滤波器和可实现此任务的成本损失。数值实验最终显示了我们的测量方法和所提出的振荡控制策略的能力。在这种情况下，还提供了在一阶有限元空间中的高效，实用的实现方式以及3D示例。