Decoupling Simulation Accuracy from Mesh Quality

Abstract

For a given PDE problem, three main factors affect the accuracy of FEM solutions: basis order, mesh resolution, and mesh element quality. The first two factors are easy to control, while controlling element shape quality is a challenge, with fundamental limitations on what can be achieved.

We propose to use p-refinement (increasing element degree) to decouple the approximation error of the finite element method from the domain mesh quality for elliptic PDEs.

Our technique produces an accurate solution even on meshes with badly shaped elements, with a slightly higher running time due to the higher cost of high-order elements. We demonstrate that it is able to automatically adapt the basis to badly shaped elements, ensuring an error consistent with high-quality meshing, without any per-mesh parameter tuning. Our construction reduces to traditional fixed-degree FEM methods on high-quality meshes with identical performance.

Our construction decreases the burden on meshing algorithms, reducing the need for often expensive mesh optimization and automatically compensates for badly shaped elements, which are present due to boundary constraints or limitations of current meshing methods. By tackling mesh generation and finite element simulation jointly, we obtain a pipeline that is both more efficient and more robust than combinations of existing state of the art meshing and FEM algorithms.

Publication
In ACM Transactions on Graphics (TOG) — Proceedings of ACM SIGGRAPH Asia
Date

Citation

@article{Martinez:2018:DSA,
    author = {Teseo Schneider and Yixin Hu and Jérémie Dumas and Xifeng Gao and Daniele Panozzo and Denis Zorin},
    journal = {ACM Transactions on Graphics},
    link = {},
    month = oct,
    number = {6},
    publisher = {Association for Computing Machinery (ACM)},
    title = {Decoupling Simulation Accuracy from Mesh Quality},
    volume = {37},
    year = {2018}
}

Acknowledgments

We are grateful to Wenzel Jakob for the Mitsuba renderer, and to NYU HPC staff for providing computing cluster service. This work was partially supported by the NSF CAREER award 1652515, the NSF grant IIS-1320635, the NSF grant DMS-1436591, the NSF grant 1835712, the SNSF grant P2TIP2-175859, a gift from Adobe Research, and a gift from nTopology.