Lagrangian evolution approach to surface-patch quadrangulation

• Published in: Applications of mathematics (Prague) Vol. 66; no. 4; pp. 509 - 551
• Main Authors: Húska, Martin, Medl’a, Matej, Mikula, Karol, Morigi, Serena
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2021; Springer Nature B.V
• Subjects: Analysis; Apexes; Applications of Mathematics; Boundaries; Classical and Continuum Physics; Evolution; Layouts; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Mesh generation; Minimal surfaces; Optimization; Theoretical; Topology; 35K93; 65M08; non-conforming mesh; mesh partitioning; patch layout; Lagrangian evolution; 35K55
• ISSN: 0862-7940; 1572-9109
• DOI: 10.21136/AM.2021.0366-19

We present a method for the generation of a pure quad mesh approximating a discrete manifold of arbitrary topology that preserves the patch layout characterizing the intrinsic object structure. A three-step procedure constitutes the core of our approach which first extracts the patch layout of the object by a topological partitioning of the digital shape, then computes the minimal surface given by the boundaries of the patch layout (basic quad layout) and then evolves it towards the object boundaries. The Lagrangian evolution of the initial surface (basic quad layout) in the direction of the gradient of the signed distance function is smoothed by a mean curvature term. The direct control over the global quality of the generated quad mesh is provided by two types of tangential redistributions: area-based, to equally distribute the size of the quads, and angle-based, to preserve quad corner angles. Experimental results showed that the proposed method generates pure quad meshes of arbitrary topology objects, composed of well-shaped evenly distributed elements with few extraordinary vertices.