Curved Optimal Delaunay Triangulation

Leman Feng 1 Pierre Alliez 2 Laurent Busé 3 Hervé Delingette 4, 5 Mathieu Desbrun 6
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
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.
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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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