Higher-order finite elements for embedded simulation

Abstract

As demands for high-fidelity physics-based animations increase, the need for accurate methods for simulating deformable solids grows. While higherorder finite elements are commonplace in engineering due to their superior approximation properties for many problems, they have gained little traction in the computer graphics community. This may partially be explained by the need for finite element meshes to approximate the highly complex geometry of models used in graphics applications. Due to the additional perelement computational expense of higher-order elements, larger elements are needed, and the error incurred due to the geometry mismatch eradicates the benefits of higher-order discretizations. One solution to this problem is the embedding of the geometry into a coarser finite element mesh. However, to date there is no adequate, practical computational framework that permits the accurate embedding into higher-order elements. We develop a novel, robust quadrature generation method that generates theoretically guaranteed high-quality sub-cell integration rules of arbitrary polynomial accuracy. The number of quadrature points generated is bounded only by the desired degree of the polynomial, independent of the embedded geometry. Additionally, we build on recent work in the Finite Cell Method (FCM) community so as to tackle the severe ill-conditioning caused by partially filled elements by adapting an Additive-Schwarz-based preconditioner so that it is suitable for use with state-of-the-art non-linear material models from the graphics literature. Together these two contributions constitute a general-purpose framework for embedded simulation with higher-order finite elements. We finally demonstrate the benefits of our framework in several scenarios, in which second-order hexahedra and tetrahedra clearly outperform their first-order counterparts.

High-Order Incremental Potential Contact for Elastodynamic Simulation on Curved Meshes

“…Linear finite elements basis are overwhelmingly used in graphics applications, as they have the smallest number of degrees of freedom (DOF) per element and are simpler to implement. High-order basis have been shown to be beneficial to animate deformable bodies in [Bargteil and Cohen 2014], to accelerate approximate elastic deformations in [Mezger et al 2009], and to compute displacements for embedded deformations in [Longva et al 2020]. They are routinely used in engineering analysis [Jameson et al 2002] where 𝑝-refinement is often favored over ℎ-refinement as it reduces the geometric discretization error [Babuska and Guo 1988;Babuška and Guo 1992;Bassi and Rebay 1997;Luo et al 2001;Oden 1994] faster and using less degrees of freedom.…”

Enforcing essential boundary conditions on domains defined by point clouds

“…We refer to [17] for a detailed review including a collection of relevant literature. Interestingly, the FCM has recently also been picked and further developed by other communities as well (see for example contributions in the area of computer graphics [18]).…”

Simulating deformable objects for computer animation: a numerical perspective

“…A common approach for discretizing an elastodynamic system is to first apply a finite element method (FEM) in space, already at the variational level. Usually in physics-based animations linear element basis functions are used on tetrahedra (or less frequently, hexahedra), although there are works that employ higher degree elements [29]. The short course by Sifakis and Barbic [38] is a good introduction for this material and much more.…”