High-Order Continuous Geometrical Validity

TL;DR

This paper introduces a robust, efficient algorithm for testing the geometrical validity of high-order finite elements during deformation, ensuring simulation stability without manual tuning.

Contribution

The authors present a novel conservative algorithm combining adaptive Bézier refinement and bisection search, which maintains robustness and accuracy in validity testing using floating point arithmetic.

Findings

Prevents invalid configurations in high-order elastodynamic simulations.

Operates efficiently with minimal runtime overhead.

Outperforms existing inexact methods in robustness and generality.

Abstract

We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform over a piecewise-linear trajectory. Our algorithm uses a combination of adaptive B\'ezier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element's polynomial geometric map becomes negative in the transition from one configuration to another. Unlike previous approaches, our method preserves its properties also when implemented using floating point arithmetic: This feature comes at a small additional runtime cost compared to existing inexact methods, making it a drop-in replacement for current validity tests, while providing superior robustness and generality. To prove the practical effectiveness of our algorithm, we demonstrate its use in a high-order Incremental Potential Contact (IPC) elastodynamic simulator, and we experimentally show that it prevents invalid, simulation-breaking configurations that would otherwise occur using inexact methods, without the need for manual parameter tuning.