Material Point Method for Snow Simulation (SIGGRAPH 2013)

mpm........

1. Introduction

2. Related Work

3. Paper Overview

4. Material Point Method

4-1. Full Method

1. Rasterize particle data to the grid

• transfer mass and velocity to the grid
• mass
  $m_i^n = \sum_p m_pw_{ip}^n$
• velocity (운동량 보존을 위한 식)
  $v_i^n = \sum_p v_p^nm_pw_{ip}^n/m_i^n$

2. Compute Particle volumes and densities

• 첫 time step 에서만 실행
• particle의 부피 설정
• Cell의 밀도
  $m_i^0/h^3$
• particle의 가중치와 Volume
  $\rho_p^0=\sum_i m_i^0w_{ip}^0/h^3$
  $V_p^0=m_p/\rho_p^0$

3. Compute grid forces

• Eq.6

4. Update velocities on grid

• Section 10 using $v_i^*$

5. Grid-based body collisions

• Section 8 (Body Collision) using $v_i^*$

6. Solve the linear system

• implicit) Eq.9 for semi-implicit integration
• explicit) $v_i^{n+1} = v_i^*$ 로 계산

7. Update deformation gradient

• $F_p^{n+1} = (I+\Delta t\nabla v_p^{n+1})F_p^n$
• $\nabla v_p^{n+1}=\sum_i v_i^{n+1}(\nabla w_{ip}^n)^T$
• Section 7

8. Update particle velocities

• new particle velocities
  $v_p^{n+1}=(1-\alpha)v_{PICp}^{n+1}+\alpha v_{FLIPp}^{n+1}$
• PIC part
  $v_{PICp}^{n+1} = \sum_i v_i^{n+1}w_{ip}^n$
• FLIP part
  $v_{FLIPp}^{n+1}=v_p^n+\sum_i(v_i^{n+1}-v_i^n)w_{ip}^n$
• $\alpha$
  $\alpha = 0.95$

9. Particle-based body collisions

• Section 8 using $v_p^{n+1}$

10. Update particle positions

• $x_p^{n+1}=x_p^n+\Delta t v_p^{n+1}$

5. Comstitutive Model

• hyperelasticity 에 있어서 finite-strain multiplicative plasticity 를 간소화함
• plastic yield criteria : using principal stretches
• hyperelasticity energy density : using Lame parameters
• Elasto-plastic energy density function
  $\Psi(F_E, F_P) = \mu(F_P)||F_E- R_E||^2_F + \frac{\lambda(F_P}{2}(J_E-1)^2$ - Eq(1)
  $F$: Deformation Gradient ($F=F_EF_P)$
  $F_E$ : elastic part , $F_P$ : plastic part
• plastic deformation gradient
  $\mu(F_P) = \mu_0e^{\xi(1-J_P)}$ and $\lambda(F_P) = \lambda_0e^{\xi(1-J_P)}$ - Eq(2)
  $J_E = def(F_E)$ and $J_P = det(F_P)$, $F_E = R_ES_E$
  $\lambda_0, \mu_0$ : initial Lame coefficient
  $\xi$ : dimensionless plastic hardening parameter
  $\theta_c$ and $\theta_s$ : compression and stretch thresholds to start plastic deformation(fracture)
  -> $F_E$의 값을 $[1-\theta_c, \ 1+\theta_s]$로 제한
• deforming plastically
  • small deformation 영역에서는 elastic ($F_E$ dependency)
  • deformation gradient 가 threshold를 넘은 경우, deforming plastically (Section.7)
  • Eq.2에 나온 material property 를 변화시킴
    • 압력(packing)에 있어서는 크게
    • stretch(farcture)에 있어서는 작게
• Threshold
  • $\theta_c$ and $\theta_s$ : material이 breaking 되는 시간 결정
• Parameters
  

6. Stress-based forces and linearization

• Total elastic potential energy -> energy density $\Psi$
  $\int_{\Omega_0}\Psi(F_E(X),\ F_P(X))dX$ -Eq (3)
• stress-based forces에 대한 MPM 공간 이산화 = Eulerian grid node material positions에 대한 에너지의 이산 근사값을 미분한 것
  -> BUT Eulerian 그리드는 변형되지 않으므로, 그리드 노드의 위치 변화는 그리드 노드의 속도에 의해 결정된다.
• 수식
  • 변경된 그리드 노드의 위치
    $x_i$ : 그리드 노드 $i$의 위치
    $\hat{x}_i=x_i+\Delta tv_i$ : 현재 그리드 속도 $v_i$에 대한 그리드 노드의 deformation location
  • total elastic potential
    $\Phi(\hat{x}) = \sum_pV_p^0\Psi(\hat{F}_{E_p}(\hat{x}),F_{P_p}^n)$ : total elastic potential
    (vector of all nodes $\hat{x}_i$ denote to $\hat{x}$)
    $V_P^0$ : 처음에 particle p가 차지한 volume
    $F_{P_p}^n$ : time $t^n$에서의 particle $p$의 plastic part of $F$
  • $\hat{F}_{E_p}$
    $\hat{F}_{E_p}$ : $\hat{x}$의 elastic part
    -> $\hat{F}_{E_p}(\hat{x})=(I+\sum_i(\hat{x_i}-x_i)(\nabla w_{ip}^n)^T)F_{E_p}^n$ - Eq(4)
  • stress-based forces에 대한 MPM 공간 이산화
    $-f_i(\hat{x})=\frac{\partial \Phi}{\partial \hat{x_i}}=\sum_pV_p^0\frac{\partial \Psi}{\partial F_E}(\hat{F}_{E_P}(\hat{x}),F_{P_P}^n)(F_{E_P}^n)^T\nabla w_{ip}^n$ - Eq(5)
    $f_i(\hat{x})$ : 그리드 노드 $i$에서 elastic stress에 대한 힘
    -> Cauchy stress로 표현
    $\sigma_p=\frac{1}{J_p^n}\frac{\partial \Psi}{\partial F_E}(\hat{F}_{E_P},F_{P_P}^n)(F_{E_P}^n)^T$ 사용해서
    $f_i(\hat{x})=-\sum_pV_p^n\sigma_p\nabla w_{ip}^n$ - Eq(6)
    $V_p^n=J_p^nV_p^0$ : 시간 $t^n$ 에 particle $p$에 의해 차지된 물질의 부피
• Grid velocity update가 목표
  • 앞에서 말한 것들은 MPM 공간 이산화와 elastic potential 간의 관계
  • Full method 3에서 말한 compute grid forces 단계에 사용
  • potential에 대한 Hessian 사용해서 elastic part의 implicit step update표현
    $-\delta f_i=\sum_j\frac{\partial^2\Phi}{\partial\hat{x_i}\partial\hat{x_j}}(\hat{x})\delta u_j = \sum_pV_p^0A_p(F_{E_P}^n)^T\nabla w_{ip}^n$ -Eq (7)
    $A_P=\frac{\partial^2\Psi}{\partial F_E\partial F_E}(F_E(\hat{x}),F_{P_P}^n) : (\sum_j\delta u_j(\nabla w_{jp}^n)^TF_{E_P}^n)$ -Eq (8)
    -> $A=C:D$ 뜻은 $A_{ij}=C_{ijkl}D_{kl}$

6.1 Semi-implicit update

7. Deformation gradient update

• temporary defining
  $\hat{F}_{P_P}^{n+1} = (I+\Delta t\nabla v_p^{n+1})F_{E_P}^n$ and $\hat{F}_{P_P}^{n+1} = F_{P_P}^n$
  -> 초기의 변화는 deformation gradient 의 elastic part에 영향을 미친다.
  $F_P^{n+1}=(I+\Delta t\nabla v_p^{n+1})F_{E_P}^nF_{P_P}^n=\hat{F}_{E_P}^{n+1}\hat{F}_{P_P}^{n+1}$
• Next Step
  • $\hat{F}_{E_P}^{n+1}$중에서 critical deformation threshold를 넘는 것을 $\hat{F}_{P_P}^{n+1}$에 넣기
  • SVD 써서 $\hat{F}_{E_P}^{n+1}=U_P\hat{\Sigma}_pV_p^T$
    $\Sigma_p = clamp(\hat{\Sigma_p}, [1-\theta_c, 1+\theta_s])$
• Final
  • elastic and plastic component of deformation gradient
    $F_{E_P}^{n+1}=U_p\Sigma_pV_p^T$ and $F_{P_P}^{n+1}=V_p\Sigma_p^{-1}U_p^TF_p^{n+1}$
    따라서 $F_p^{n+1}=F_{E_p}^{n+1}F_{P_p}^{n+1}$

8. Body Collisions

• collision check twice in each time step
  • 1st : grid velocity에 힘이 가해진 직후의 grid velocity $v_i^*$
  • 2nd : particle velocity $v_p^{n+1}$의 위치를 업데이트 하기 전 보간에 의한 소실을 막기 위해서
• in case of collision
  $n=\nabla \phi$ : local normal
  $v_{co}$ : object velocity
  • $v$ : particle/grid velocity
    -> transform into reference frame of the collision object
    -> $v_{rel}=v-v_{co}$
    if) bodied separate ($v_n = v_{rel}\cdot n\ge 0)$ -> no coliision
    let) $v_t=v_{rel}-nv_n$ : tangential portion of the relative velocity
    -> if) sticking impluse required ($||v_t|| \le -\mu v_n)$ -> $v'_{rel}=0$
    -> if) dynamic friction $v'_{rel}=v_t+\mu v_nv_t/||v_t||$ ($\mu$ : coefficient of friction)
• Finally
  $v'=v'_{rel}+v_{co}$
  -> relative velocity back into world coordinates
• Types of Collision objects
  • Rigid Case
  • Deforming Case
• Finally
  • sticky collision
    • $v'_{rel}=0$