Curved Optimal Delaunay Triangulation

Leman Feng 1 Pierre Alliez 2 Laurent Busé 3 Hervé Delingette 4, 5 Mathieu Desbrun 6
1 ENPC - École des Ponts ParisTech
2 TITANE - Geometric Modeling of 3D Environments
CRISAM - Inria Sophia Antipolis - Méditerranée
3 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , National and Kapodistrian University of Athens
4 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
5 EPIONE - E-Patient : Images, données & mOdèles pour la médeciNe numériquE
CRISAM - Inria Sophia Antipolis - Méditerranée
6 CS CALTECH - Computer Science Department
Abstract : Meshes with curvilinear elements hold the appealing promise of enhanced geometric flexibility and higher-order numerical accuracy compared to their commonly-used straight-edge counterparts. However, the generation of curved meshes remains a computationally expensive endeavor with current meshing approaches: high-order parametric elements are notoriously difficult to conform to a given boundary geometry, and enforcing a smooth and non-degenerate Jacobian everywhere brings additional numerical difficulties to the meshing of complex domains. In this paper, we propose an extension of Optimal Delaunay Triangulations (ODT) to curved and graded isotropic meshes. By exploiting a continuum mechanics interpretation of ODT instead of the usual approximation theoretical foundations, we formulate a very robust geometry and topology optimization of Bézier meshes based on a new simple functional promoting isotropic and uniform Jacobians throughout the domain. We demonstrate that our resulting curved meshes can adapt to complex domains with high precision even for a small count of elements thanks to the added flexibility afforded by more control points and higher order basis functions.
Keywords : Optimal Delaunay Triangulations Optimal Delau-nay Triangulations Additional Key Words and Phrases: Higher-order meshing CCS Concepts: • Mathematics of computing → Mesh generation Bézier elements ACM Reference Format: Mesh generation Bézier elements Higher-order meshing higher order finite elements
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Leman Feng, Pierre Alliez, Laurent Busé, Hervé Delingette, Mathieu Desbrun. Curved Optimal Delaunay Triangulation. ACM Transactions on Graphics, Association for Computing Machinery, 2018, Proceedings of SIGGRAPH 2018, 37 (4), pp.16. ⟨10.1145/3197517.3201358⟩. ⟨hal-01826055⟩

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