High order unfitted finite element discretizations for explicit boundary representations

TL;DR

This paper introduces a new computational pipeline for solving PDEs on complex geometries defined by explicit boundary representations, using novel algorithms for geometric discretization and quadrature generation, enabling high-order accuracy.

Contribution

It presents a novel automatic method for discretizing nonlinear boundary geometries and generating quadratures, extending unfitted finite element methods to explicit CAD-based domains.

Findings

Successfully applied to PDE simulations on CAD-defined nonlinear domains

Demonstrated robustness and high-order accuracy of the geometric algorithms

Proved the correctness of the proposed quadrature generation algorithm

Abstract

When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer a significant advantage in dealing with complex geometries, eliminating the need for generating unstructured body-fitted meshes. However, current unfitted finite elements on nonlinear geometries are restricted to implicit (possibly high-order) level set geometries. In this work, we introduce a novel automatic computational pipeline to approximate solutions of partial differential equations on domains defined by explicit nonlinear boundary representations. For the geometrical discretization, we propose a novel algorithm to generate quadratures for the bulk and surface integration on nonlinear polytopes required to compute all the terms in unfitted finite element methods. The algorithm relies on a nonlinear triangulation of the boundary, a kd-tree refinement of the surface cells that simplify the nonlinear intersections of surface and background cells to simple cases that are diffeomorphically equivalent to linear intersections, robust polynomial root-finding algorithms and surface parameterization techniques. We prove the correctness of the proposed algorithm. We have successfully applied this algorithm to simulate partial differential equations with unfitted finite elements on nonlinear domains described by computer-aided design models, demonstrating the robustness of the geometric algorithm and showing high-order accuracy of the overall method.