arXiv subjects

Dimitris Vartziotis

Publications and source records attributed to Dimitris Vartziotis.

23 records · Page 2Linked to original sources

arXiv2014

Convergence properties of a geometric mesh smoothing algorithm

We describe a simple geometric transformation of triangles which leads to an efficient and effective algorithm to smooth triangle and tetrahedral meshes. Our focus lies on the convergence properties of this algorithm: we prove the effectivity for some planar triangle meshes and further introduce dynamical methods to study the dynamics of the algorithm which may be used for any kind of algorithm based on a geometric transformation.

math.NA↗

arXiv2014

Laplacian smoothing revisited

We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The corresponding functions measuring the quality of meshes are easily seen to be convex and can be used for global optimization-based untangling and smoothing. Using a simple backtracking line-search we compare several smoothing methods with respect to the resulting mesh quality. We also discuss their effectiveness in combination with topology modification.

math.OC↗

arXiv2014

Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

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.

math.GT↗

arXiv2013

Regularizations of non-euclidean polygons

We are interested in easy geometric transformations which regularize n-polygons in the non-euclidean plane. A transformation is called easy if it can be easily implemented into an algorithm. This article is motivated by preceding work on geometric transformations on euclidean polygons and possible applications for non-euclidean meshes.

math.MG↗

arXiv2013

Efficient and Global Optimization-Based Smoothing Methods for Mixed-Volume Meshes

Some methods based on simple regularizing geometric element transformations have heuristically been shown to give runtime efficient and quality effective smoothing algorithms for meshes. We describe the mathematical framework and a systematic approach to global optimization-based versions of such methods for mixed volume meshes. In particular, we identify efficient smoothing algorithms for certain algebraic mesh quality measures. We also provide explicit constructions of potentially useful smoothing algorithms.

cs.CG↗