Fast Corotated Elastic SPH Solids with Implicit Zero-Energy Mode Control

Abstract

We develop a new operator splitting formulation for the simulation of corotated linearly elastic solids with Smoothed Particle Hydrodynamics (SPH). Based on the technique of Kugelstadt et al. [2018] originally developed for the Finite Element Method (FEM), we split the elastic energy into two separate terms corresponding to stretching and volume conservation, and based on this principle, we design a splitting scheme compatible with SPH. The operator splitting scheme enables us to treat the two terms separately, and because the stretching forces lead to a stiffness matrix that is constant in time, we are able to prefactor the system matrix for the implicit integration step. Solid-solid contact and fluid-solid interaction is achieved through a unified pressure solve. We demonstrate more than an order of magnitude improvement in computation time compared to a state-of-the-art SPH simulator for elastic solids. We further improve the stability and reliability of the simulation through several additional contributions. We introduce a new implicit penalty mechanism that suppresses zero-energy modes inherent in the SPH formulation for elastic solids, and present a new, physics-inspired sampling algorithm for generating high-quality particle distributions for the rest shape of an elastic solid. We finally also devise an efficient method for interpolating vertex positions of a high-resolution surface mesh based on the SPH particle positions for use in high-fidelity visualization.

A Survey on SPH Methods in Computer Graphics

“…However, the force minimizes only in the direction of the vector x ij . In contrast to that, Kugelstadt et al [KBFF*21] propose an implicit zero‐energy mode control by minimizing the quadratic energy function…”

“…A meshless representation of a solid can be obtained by sampling its geometry. Different sampling techniques are discussed in [KBFF*21].…”

“…Kugelstadt et al [KKB18, KBFF*21] split the corotated linear constitutive model (see Eq. (79)) in a stretching term and a volume term and keep the rotation constant during the step.…”

A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling

“…Peer et al [2018] proposed an implicit scheme and applied kernel gradient correction [Bonet and Lok 1999] to obtain a first-order consistent SPH formulation for the deformation gradient. Incorporating solid particles into the preexisting fluid pressure solver can resolve contact handling, but SPH still faces numerical issues such as the zero-mode [Ganzenmüller 2015; Kugelstadt et al 2021] when simulating elastic objects. Additionally, the pressure solver will treat solid objects as incompressible under compression, which may not be applicable in all cases.…”

“…On the other hand, our method (0.45 min/frame) is slower than IISPH (0.31 min/frame) and DFSPH (0.15 min/frame) due to the more sophisticated boundary handling strategy. However, our proposed approach can couple SPH fluids and elastic solids with arbitrary constitutive models, while most existing SPH methods [Kugelstadt et al 2021;Peer et al 2018] treat elastic solids as incompressible, which is not generally applicable. 5.2.2 Solid-Fluid Coupling.…”