Projective Dynamics: Fusing Constraint Projections for Fast Simulation

最新推荐文章于 2024-09-25 17:16:50 发布

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最新推荐文章于 2024-09-25 17:16:50 发布

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(3)->(4)

q n + 1 - q n - h v n - h 2 M f e x t = h 2 M f i n t (q n + 1)

$q_{n+1}- q_n-hv_n-\frac{h^2}{M}f_{ext}= \frac{h^2}{M}f_{int}(q_{n+1})$

1 h 2 M (q n + 1 - s n) = f i n t (q n + 1)

$\frac{1}{h^2}M(q_{n+1}- s_n)= f_{int}(q_{n+1})$

右边项左移然后对 $q_{n+1}$ 积分, 注意 $q_{n+1}$ 为矩阵, 矩阵的积分请参见http://blog.youkuaiyun.com/seamanj/article/details/53300058

\int 1 h 2 M (q n + 1 - s n) d q n + 1 - \int f i n t (q n + 1) d q n + 1 = 1 h 2 \int M 1 2 (q n + 1 - s n) d (M 1 2 (q n + 1 - s n)) + \sum i W i (q n + 1) = 1 2 h 2 (M 1 2 (q n + 1 - s n) : M 1 2 (q n + 1 - s n)) + \sum i W i (q n + 1) = 1 2 h 2 ∥ M 1 2 (q n + 1 - s n) ∥ 2 F + \sum i W i (q n + 1)

$\begin{align} & \int\frac{1}{h^2}M(q_{n+1}- s_n)dq_{n+1} - \int f_{int}(q_{n+1})dq_{n+1} \\ &=\frac{1}{h^2}\int M^{\frac{1}{2}}(q_{n+1}- s_n)d(M^{\frac{1}{2}}(q_{n+1}- s_n)) + \sum_i W_i(q_{n+1})\\ & =\frac{1}{2h^2}(M^{\frac{1}{2}}(q_{n+1}- s_n):M^{\frac{1}{2}}(q_{n+1}- s_n)) + \sum_i W_i(q_{n+1})\\ &=\frac{1}{2h^2}\|M^{\frac{1}{2}}(q_{n+1}- s_n)\|_F^2 + \sum_i W_i(q_{n+1}) \end{align}$

得证.

(8)式中的第二、三项是(4)式中的W项

(8)->(10)
对(8)式相对 $q$ 求导,然后令其为0
这里写图片描述

4.Gauss-Seidel 和 Jacobi 算法的比较

这里写图片描述

5.令 $\delta_C(q+\Delta q)=0$
则 $C(q+\Delta q) = C(q) + \nabla C(q):\Delta q=C(q) +tr(\nabla C(q)^T\Delta q) = 0$

对(12)式求导

\partial ( 1 2 ( M 1 2 Δ q : M 1 2 Δ q ) + λ ( C ( q ) + \nabla C ( q ) : Δ q ) ) \partial Δ q = M Δ q + λ \nabla C (q) = 0

$\frac{\partial (\frac{1}{2}(M^{\frac{1}{2}}\Delta q:M^{\frac{1}{2}}\Delta q)+\lambda( C(q) + \nabla C(q):\Delta q) )}{\partial \Delta q} = M\Delta q + \lambda \nabla C(q) = 0$