<abstract><p>We define rigorously a solution to the fourth-order total variation flow equation in $ \mathbb{R}^n $. If $ n\geq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ n\leq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ n\neq2 $. If $ n\neq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.</p></abstract>
This paper aims to develop a scheme for geometrical feature constraints in topology optimization for Additive Manufacturing (AM) without support structures based on the Partial Differential Equation (PDE) of geometrical shape features. To begin with, the basic concept of topology optimization and a level set-based topology optimization method are briefly described. Second, the PDE system for geometrical shape features is formulated. Here, aspects of the distribution of state variables are discussed using an analytical solution of the PDE. Based on the discussion, a function indicating the extended normal vector including geometrical singularity points is formulated. Third, geometrical requirements of product shape in AM without support structures – the so-called overhang constraint – are clarified in two-dimensions. A way of extending of the proposed concept to three-dimensional problems is also clarified. Additionally, geometrical singularities in the overhang constraint are discussed. Based on the PDE system and the clarified geometrical requirements, the overhang constraint including geometrical singularities is formulated. A topology optimization problem of the linear elastic problem is formulated considering the overhang constraint. A level set-based topology optimization algorithm is constructed where the Finite Element Method (FEM) is used to solve the governing equation of the linear elastic problem and the PDE, and to update the level set function. Finally, two-dimensional numerical examples are provided to confirm the validity and utility of the proposed method.