arXiv subjects

Lenka Ptackova

Publications and source records attributed to Lenka Ptackova.

4 recordsLinked to original sources

arXiv2026

Domain Decomposition for Mean Curvature Flow of Surface Polygonal Meshes

We examine the use of domain decomposition for potentially more efficient mean curvature flow of surface meshes, whose faces are arbitrary simple polygons. We first test traditional domain decomposition methods with and without overlap of deconstructed domains. And we present adapted Robin transmission conditions of optimized Schwarz method. We then analyze the resulting smoothing from the point of view of shape quality and texture deformation. By decomposing the initial mesh into two sub-meshes, we solve two smaller boundary value problems instead of one big problem, and we can process these two tasks almost entirely in parallel.

math.NA↗

arXiv2025

A discrete wedge product on general polygonal meshes

Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete exterior derivative, has been a challenging task. The cup product of cochains is traditionally considered to be an appropriate discrete wedge product. However, only the case of pure triangle or pure quadrilateral surface meshes has been studied thoroughly. In this work, we extend this tradition to general polygonal meshes. Specifically, we present explicit formulas for calculation of a cup/discrete wedge product on surface meshes that correspond to 2--dimensional pseudomanifolds, whose 2--dimensional faces are any simple polygons. We rigorously prove that the proposed product satisfies the definition of an abstract cup product; notably, we show that the product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Furthermore, the product is associative on the cohomology level, but not on the cochain level in general. We analyze the lack of associativity on the cochain level and prove that the error tends to zero under refinement of the mesh. We thus argue that the proposed product is an appropriate discretization of the wedge product of differential forms on general polygonal meshes.

math.AT↗

arXiv2024

Bézier curves and the Takagi function

We consider Bézier curves with complex parameters, and we determine explicitly the affine iterated function system (IFS) corresponding to the de Casteljau subdivision algorithm, together with the complex parametric domain over which such an IFS has a unique global connected attractor. For a specific family of complex parameters having vanishing imaginary part, we prove that the Takagi fractal curve is the attractor, under suitable scaling.

math.NA↗

arXiv2024

A simple and complete discrete exterior calculus on general polygonal meshes

Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal-to-primal Hodge star operator. Combining these three `basic operators' we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz-Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

math.NA↗