XPBI: Position-Based Dynamics with Smoothing Kernels Handles Continuum Inelasticity

Abstract

PBD and its extension, XPBD, have been predominantly applied to compliant constrained elastodynamics, with their potential in finite strain (visco-) elastoplasticity remaining underexplored. XPBD is often perceived to stand in contrast to other meshless methods, such as the MPM. MPM is based on discretizing the weak form of governing partial differential equations within a continuum domain, coupled with a hybrid Lagrangian-Eulerian method for tracking deformation gradients. In contrast, XPBD formulates specific constraints, whether hard or compliant, to positional degrees of freedom. We revisit this perception by investigating the potential of XPBD in handling inelastic materials that are described with classical continuum mechanics-based yield surfaces and elastoplastic flow rules. Our inspiration is that a robust estimation of the velocity gradient is a sufficiently useful key to effectively tracking deformation gradients in XPBD simulations. By further incorporating implicit inelastic constitutive relationships, we introduce a plasticity in-the-loop updated Lagrangian augmentation to XPBD. This enhancement enables the simulation of elastoplastic, viscoplastic, and granular substances following their standard constitutive laws. We demonstrate the effectiveness of our method through high-resolution and real-time simulations of diverse materials such as snow, sand, and plasticine, and its integration with standard XPBD simulations of cloth and water.

Figures (20)

• Figure 1: Noodles. We simulate noodles modeled using Von Mises plasticity as it is pressed through a cylindrical mold.
• Figure 2: Cloth. XPBI fits into traditional XPBD pipeline and naturally couples updated Lagrangian materials (viscoplastic paint) and mesh-based geometry (cloth).
• Figure 3: XPBI simulates Hershel-Bulkley shear thinning ($h=0.3$), viscoplastic ($h=1.0$), and shear thickening materials ($h=3.0$), where $h$ controls a power law flow rate detailed in yue2015continuum.
• Figure 4: Candy Camponotus. We simulate the brittle fracture of a candy shaped like a camponotus falling onto the ground.
• Figure 5: Deformation Gradient Evolution. The dotted line (bottom) illustrates the evolution of the deformation gradient in the updated Lagrangian view, transitioning from $\bm{F}_p^0$ (initial configuration) to $\bm{F}_p^{n+1}$ (updated configuration), with $\bm{F}_p^{n}$ (current configuration) serving as the reference state. This facilitates tracking large deformations. The solid line loop (top) depicts iterations of our XPBI algorithm to simulate $t^n \rightarrow t^{n+1}$, alternating between an xpbd iteration and a fixed point iteration. During iteration $k$, $\bm{F}_p^{E, tr}$ is first estimated based on the current gradient of $\bm{v}^{(k)}$ (§ \ref{['sec:gradient']}). Then, plasticity is applied through projection to obtain $\bm{F}_p^{(k+1)}$ (§ \ref{['sec:implicit_plasticity']}), and finally, $\bm{v}^{(k+1)}$ is updated by solving constraints (§ \ref{['sec:algorithm_overview']}).