An implicit updated lagrangian formulation for liquids with large surface energy

Abstract

We present an updated Lagrangian discretization of surface tension forces for the simulation of liquids with moderate to extreme surface tension effects. The potential energy associated with surface tension is proportional to the surface area of the liquid. We design discrete forces as gradients of this energy with respect to the motion of the fluid over a time step. We show that this naturally allows for inversion of the Hessian of the potential energy required with the use of Newton's method to solve the systems of nonlinear equations associated with implicit time stepping. The rotational invariance of the surface tension energy makes it non-convex and we define a definiteness fix procedure as in [Teran et al. 2005]. We design a novel level-set-based boundary quadrature technique to discretize the surface area calculation in our energy based formulation. Our approach works most naturally with Particle-In-Cell [Harlow 1964] techniques and we demonstrate our approach with a weakly incompressible model for liquid discretized with the Material Point Method [Sulsky et al. 1994]. We show that our approach is essential for allowing efficient implicit numerical integration in the limit of high surface tension materials like liquid metals.

A momentum-conserving implicit material point method for surface tension with contact angles and spatial gradients

“…We call these additional temporary particles balance particles. We show that our novel resampling provides improved behavior over the original approach of Hyde et al [2020], even in the case of standard, non-Marangoni surface tension effects. Furthermore, although other resampling techniques exist for PIC methods [Edwards and Bridson 2012;Gao et al 2017b;Yue et al 2015], we note that ours is the first to guarantee perfect conservation when using generalized particle velocities associated with the Affine Particle-in-Cell (APIC) method [Fu et al 2017;Jiang et al 2015.…”

“…We build on the work of Hyde et al [2020] and show that efficient implicit time stepping with Marangoni effects is achievable with PIC. As in [Hyde et al 2020] we observe that similarities with hyperelasticity suggest that the Material Point Method (MPM) [Sulsky et al 1994] is the appropriate version of PIC.…”

“…We build on the work of Hyde et al [2020] and show that efficient implicit time stepping with Marangoni effects is achievable with PIC. As in [Hyde et al 2020] we observe that similarities with hyperelasticity suggest that the Material Point Method (MPM) [Sulsky et al 1994] is the appropriate version of PIC. We show that by building our discretization from the energy in Equation (1) rather than the more commonly adopted traction condition in Equation (2), we can naturally compute the first and second variations of the potential needed when setting up and solving the nonlinear systems of equations associated with fully implicit temporal discretization.…”

“…While our method is a generalization of the MPM technique in [Hyde et al 2020], we also improve on its core functionality. The Hyde et al [2020] approach is characterized by the introduction of additional surface tension particles at each time step which is designed to represent the liquid interface Γ and its area weighted boundary normals.…”

“…While our method is a generalization of the MPM technique in [Hyde et al 2020], we also improve on its core functionality. The Hyde et al [2020] approach is characterized by the introduction of additional surface tension particles at each time step which is designed to represent the liquid interface Γ and its area weighted boundary normals. These surface tension particles are temporary and are deleted at the end of the time step to prevent excessive growth in particle count or macroscopic particle resampling.…”