Introduction Rigid Body with Controllable Friction

Acceleration: AsyncMPM and IMEX-MPM

Improving Particle Distribution

s

The Material Point Method (MPM) is a numerical method capable of simulating

the behavior of liquids, deformable solids, and granular materials. In this work, we

propose novel techniques to make it even more versatile and efficient:

1. A surface tension model to MPM liquid.

2. Improved particle distribution using an anisotropic position correction model.

3. Rigid body simulation in MPM, where friction is modeled as plasticity.

4. Two new ways to speed up MPM: Asynchronization, and implicit-explicit mixed

time integration.

5. Investigation on divergence-free particle-grid transfer on APIC and PolyPIC.

Figure 3: A 3D water surface scene simulated using the CSS model. The total particle number is

88374. (Simulated with JIXIE, rendered by Houdini.)

Enriching and accelerating the Material Point Method Yuanming Hu1 Yu Fang1 Ziheng Ge2 Ziyin Qu3 Andre Pradahana3 Chenfanfu Jiang3

1Tsinghua University 2University of Science and Technology of China 3University of Pennsylvania

The Continuum Surface Force Model (CSF):

The Continuum Surface Stress Model (CSS):

Normal vector: 𝑛 =𝛻𝐶

𝛻𝐶

Curvature: 𝜅 = 𝛻 ∙ 𝑛

Surface force: 𝑓 = 𝛾𝜅𝑛

Calculate

on grid

Surface stress: 𝜎 = 𝛾(1

𝑑𝛻𝐶 𝐼𝑛 −

(𝛻𝐶)𝑇𝛻𝐶

𝛻𝐶)

The Smoothed Particle Hydrodynamics Model (SPH):

Cohesion force:

𝑓 𝑖𝑗 = −𝛾𝑚𝑖𝑚𝑗𝐶( 𝑥𝑖 − 𝑥𝑗 )𝑥𝑖−𝑥𝑗

𝑥𝑖−𝑥𝑗

There are some choices of kernel C(r),

and we use the one shown in blue plot.

𝛻𝐶 𝑝 = Σ𝑖𝛻𝑤𝑖𝑝𝑚𝑖

𝛻𝐶 𝑖 = Σ𝑝𝛻𝑤𝑖𝑝𝑚𝑝

𝜅𝑖 = Σ𝑗 𝛻𝑤𝑗𝑖𝑇𝑛𝑗

Calculate on particles

Figure 2: A 2D dam break simulation.

Figure 1: An overview of the new Material Point Method. Our contributions are highlighted in red.

AsyncMPM: a novel asynchronous time integration scheme for MPM which offers

temporal adaptivity when objects of different stiffness or velocity coexist. We show

via multiple test scenes that our asynchronous MPM leads to 6x speed up in Table 1.

IMEX-MPM: We propose an effective approach to use implicit and explicit time

integragtion scheme in different regions of MPM. The regions are automatically

detected based on material stiffness.

Simulating rigid bodies in MPM is an open problem. To this end, we Soft

Rigidification, which preserves energy and coupled naturally with other parts of

MPM. In short, this is achieved by penalizing non-rigid position of rigid body

particles before particle position update.

In order to model rigid surface friction, we use a tailored representation of

deformation for rigid bodies (left), as well as a plasticity model for objects with

different surface friction, by changing the amount of shearing allowed (right).

Constitutive Model: Constitutive model relating material stress to material state

(such as deformation gradient F) is needed for governing the material responses

under deformations. Commonly used hyper-elastic constitutive models include Neo-

Hookean Model and Fixed Corotated Model.

For liquid simulation, J-based constitutive model is used, where 𝐽 is the determinant

of deformation gradient 𝐹. Water is modeled to be weakly incompressible with

partial stress

𝜎𝑤 = −𝑃𝑤𝐼, 𝑃𝑤 = 𝑘1

𝐽𝑤𝛾− 1

The pressure 𝑃𝑤 is designed to stiffly penalize liquid volume change, which is

characterized in terms of the determinant of the liquid deformation gradient

𝐽𝑤 = det (𝐹𝑤). Here 𝑘 is the bulk modulus of water, and 𝛾 is a term that more stiffly

penalizes large deviations from incompressibility.

Weakly Compressible Liquids with Surface Tension

Surface tension arises as a result of attractive forces between molecules in a fluid.

As a force acting only on the surface, it balances radially inward inter-molecular

attractive force with radially outward pressure gradient force across the surface. Figure 4: Comparison of dam break without position correction(left) and with position

correction(right).

An inherent problem of MPM is uneven spatial particle distributions during

simulation, which in turn can cause “holes” due to missed fluid cells and lead to

artifacts such as bumpy surfaces. To prevent this, we use anisotropic position

correction to improve particle distribution. The anisotropic displacement vector ∆𝑃

is computed by

∆𝑃𝑖= −∆𝑡𝛾𝑠𝑑∑𝑟

𝑟𝑊𝑠𝑚𝑜𝑜𝑡ℎ(𝑟, 𝑑)

𝑟 =1

2𝑘𝑠𝐶𝑖

−1 + 𝐶𝑗−1 𝑝𝑗 − 𝑝𝑖 ,

After we obtain ∆𝑃, we recompute our new velocity by

𝑢 𝑥 =∑𝑖𝑚𝑖𝑢𝑖𝑊𝑠ℎ𝑎𝑟𝑝(𝑝𝑖 − 𝑥, 𝛼𝑢𝑑𝑖)

∑𝑖𝑚𝑖𝑊𝑠ℎ𝑎𝑟𝑝(𝑝𝑖 − 𝑥, 𝛼𝑢𝑑𝑖)

Divergence-Free Particle-Grid Transfer

The particle-grid transfer in MPM usually leads to velocity mode loss, since there are more

degrees of freedom (DoFs) on the grid than those on the particles. Such loss is commonly

considered to be harmful and dissipative. However, thinking positively, if we intentionally

drop the divergence part of the velocity field, such “loss” can actually be considered as

pressure projection, making fluid more incompressible.

We tried this idea with APIC while it turns out APIC itself is too dissipative, making the

fluid viscous. Therefore, we investigate PolyPIC, which is indeed lossless and thereby

promising for future research on divergence-free transfer.

Initial PPIC 20 iter. APIC 20 iter. Initial PPIC 20 iter. APIC 20 iter.

Platforms with

increasing

friction from

left to right

We focus on the following three models for surface tension: (𝛾 is a constant

coefficient for surface tension strength.)