# Particle-In-Cell 旋转动量减小

P2G前所有粒子的angular momentum是
\mathbf{L}_{tot}^{P,n} = \sum_p \mathbf{x}_p^n \times m_p \mathbf{v}_p^n

P2G后所有网格顶点的angular momentum是
\mathbf{L}_{tot}^{G,n} = \sum_i \mathbf{x}_i^n \times m_i^n \mathbf{v}_i^n

\mathbf{L}_{tot}^{G,n+1} = \sum_i \mathbf{x}_i^n \times m_i^n \tilde{\mathbf{v}}_i^{n+1}

G2P后所有网格顶点的angular momentum是
\mathbf{L}_{tot}^{P,n+1} = \sum_p \mathbf{x}_p^n \times m_p \mathbf{v}_p^{n+1}

\mathbf{L}_{tot}^{G,n} = \sum_i \mathbf{x}_i^n \times \sum_{p} w_{ip}^n m_p\mathbf{v}_p^n =- \sum_p m_p \mathbf{v}_p^n \times \sum_i w_{ip}^n \mathbf{x}_i^n

\mathbf{L}_{tot}^{G,n} = - \sum_p m_p \mathbf{v}_p^n \times \mathbf{x}_p^n = \mathbf{L}_{tot}^{P,n}

\mathbf{L}_{tot}^{P,n+1} = \sum_p \mathbf{x}_p^n \times m_p \sum_i w_{ip}^n\mathbf{\tilde{v}}_i^{n+1}

\mathbf{L}_{tot}^{P,n+1} = \sum_p \sum_i w_{ip}^n \mathbf{x}_i^n \times m_p \sum_i w_{ip}^n\mathbf{\tilde{v}}_i^{n+1} = \sum_p \sum_i (w_{ip}^n)^2 \mathbf{x}_i^n \times m_p \mathbf{\tilde{v}}_i^{n+1}

\mathbf{L}_{tot}^{G,n+1} = \sum_i \mathbf{x}_i^n \times m_i^n \tilde{\mathbf{v}}_i^{n+1} = \sum_i \mathbf{x}_i^n \times \sum_p w_{ip}^n m_p \tilde{\mathbf{v}}_i^{n+1} = \sum_p \sum_i w_{ip}^n \mathbf{x}_i^n \times m_p \tilde{\mathbf{v}}_i^{n+1}

09.22 更新

\mathbf{L}_{tot}^{P,n+1} = \sum_p \mathbf{x}_p^n \times m_p \sum_i w_{ip}^n\mathbf{\tilde{v}}_i^{n+1} = \sum_p \sum_i \mathbf{x}_p^n \times m_p w_{ip}^n\mathbf{\tilde{v}}_i^{n+1}

\mathbf{L}_{tot}^{G,n+1} = \sum_p \sum_i w_{ip}^n \mathbf{x}_i^n \times m_p \tilde{\mathbf{v}}_i^{n+1} = \sum_p \sum_i \mathbf{x}_i^n \times m_p w_{ip}^n \tilde{\mathbf{v}}_i^{n+1}

\mathbf{L}_p^{n+1} = \sum_i (\mathbf{x}_p^n-\mathbf{x}_i^n) \times m_p w_{ip}^n \tilde{\mathbf{v}}_i^{n+1}

3 Likes

1. 如果grid角动量定义在moved grid nodes上，那相应的particle角动量也应该定义在moved particle上？
2. 好像具体实现的时候也没有move grid nodes这一步，particle的物理量也是从fixed grid nodes gather过来的。