# Poly-Spline Finite Element Method

arXiv: https://arxiv.org/abs/1804.03245 · HTML: https://mlsanity.com/paper/1804.03245/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2018.

## Built on by (most cited first, 1 of 1)

- [Simplicits: Mesh-Free, Geometry-Agnostic, Elastic Simulation](https://mlsanity.com/paper/2407.09497/) (2024)

## Abstract (arXiv)

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.