On Isogeometric Subdivision Methods for PDEs on Surfaces

Bert Juettler (1) , Angelos Mantzaflaris (2, 3, 4) , Ricardo Perl (5) , Rumpf M. (5)

1 Institute of Applied Geometry [Linz]
2 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
3 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
4 RICAM - Johann Radon Institute for Computational and Applied Mathematics
5 Hausdorff Center for Mathematics and Institute for Numerical Simulation - University of Bonn

Subdivision surfaces are proven to be a powerful tool in geometric modeling and computer graphics, due to the great flexibility they offer in capturing irregular topologies. This paper discusses the robust and efficient implementation of an isogeometric discretization approach to partial differential equations on surfaces using subdivision methodology. Elliptic equations with the Laplace-Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problems are considered. Thereby, efficiency relies on the proper choice of a numerical quadrature scheme which preserves the expected higher order consistency. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. In this paper, the focus is on Loop's subdivision scheme on triangular meshes. Based on a series of numerical experiments, different quadrature schemes are compared and a mid-edge quadrature, which is easy-to-implement via lookup tables, turns out to be a preferable choice due to its robustness and efficiency.

Mots clés: subdivision methods; isogeometric analysis; PDEs on surfaces; numerical quadrature

Domaines: Analyse numérique [math.NA]; Modélisation et simulation; Calcul formel [cs.SC]

Bert Juettler, Angelos Mantzaflaris, Ricardo Perl, Rumpf M.. On Isogeometric Subdivision Methods for PDEs on Surfaces. Computer Methods in Applied Mechanics and Engineering, 2016, 302, pp.131-146. ⟨10.1016/j.cma.2016.01.005⟩. ⟨hal-02272118⟩