Poly-Spline Finite Element Method

TL;DR

This paper presents a novel finite element method pipeline that automatically generates hybrid meshes with high-order basis functions, achieving cubic convergence and reducing degrees of freedom for solving PDEs in complex volumes.

Contribution

It introduces a hybrid hexahedral-dominant mesh construction with high-order basis functions, enabling efficient and accurate PDE solutions with minimal user intervention.

Findings

Converges cubically under refinement.

Uses around 50% fewer degrees of freedom than traditional methods.

Successfully solves Poisson's equation on diverse models.

Abstract

We introduce an integrated meshing and finite element method pipeline enabling black-box solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order basis on its elements, combining triquadratic B-splines, triquadratic hexahedra (27 degrees of freedom), and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson's equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface.