Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

Abstract

Abstract. The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that, the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids and octahedra, that is, the optimization process converges to regular polyhedra. We show that the (normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method (GETMe). Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally.Key words. hybrid mesh, smoothing, quality metric, quality measure, polyhedron, optimization, finite element method, GETMe, octahedron, tetrahedron, hexahedron, pyramid, prism AMS subject classifications. 52B70, 58C05, 37C101. Introduction. In the context of the finite element method, mesh quality affects numerical stability as well as solution accuracy of this method [37]. The geometric element transformation method (GETMe) as a smoothing algorithm for tetrahedral meshes is introduced in [45], which was generalized to hybrid meshes in [40,41,42,43,44]. We consider GETMe as a class of smoothing methods based on simple geometric transformations applied iteratively to all elements individually. Several numerical tests have confirmed that the geometric element transformation given in [45] and its generalizations reliably and efficiently regularizes the polyhedron types, which are relevant for the finite element method. In this paper, regularization refers to the convergence of the iteratively transformed polyhedra towards a regular polyhedron, and we ask the reader not to confuse this terminology with the methods in numerical analysis that allow you to deal with ill-conditioned problems. GETMe smoothing also significantly reduces errors in solutions of the finite element method and improves solution efficiency for meshes [44]. The search for a mathematical proof of its qualitative behavior led us to the discovery of a simple and reasonable quality measure for volume elements optimized by polyhedral generalizations of the tetrahedral GETMe algorithm in [45]. Therefore, the purpose of our work is threefold: 1. We introduce the mean volume and turn it into a scaling-invariant quality measure. 2. We consider the gradient flow of the mean volume and discuss its close relationship to the tetrahedral GETMe algorithm presented in [45]. 3. We analyze the new GETMe optimizing and untangling algorithms for volume elements induced by the gradient flow of the mean volume both theoretically and experimentally.

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Smoothing Game

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