# 关于隐式积分方程的一些问题

${\bf F} = {\bf M}\ddot{\bf x}$

$\begin{cases} \dot{\bf v} &= {\bf FM}^{-1} \\ \dot{\bf x} &= \bf v \end{cases}$

$\begin{cases} {\bf v}^{t+h} &= {\bf v}^t + {\bf F}^{t+h}{\bf M}^{-1}h \\ {\bf x}^{t+h} &= {\bf x}^t + {\bf v}^{t+h}h \end{cases}$

\begin{align} {\bf F}^{t+h} &= {\bf F}^t + \frac{{\bf F}'}{1!}({\bf x}^{t+h}-{\bf x}^t,{\bf v}^{t+h}-{\bf v}^t) \\ &= {\bf F}^t + \frac{\partial \bf F}{\partial \bf x}\Delta {\bf x} + \frac{\partial \bf F}{\partial \bf v}\Delta {\bf v} \end{align}

$({\bf M} - \frac{\partial{\bf F}}{\partial \bf v}h - \frac{\partial{\bf F}}{\partial \bf x}h^2)\Delta {\bf v} = ({\bf F} + \frac{\partial \bf F}{\partial \bf x}{\bf v}h)h$

(配图的面的顺序，弄反了，右手系下X2与X3的位置应该换一下。。。)

\begin{aligned} F_{X_0}&= f_{\tt stiff} \cdot (X_0 - P) = f \cdot (X_0 - (\alpha_1X_1+\alpha_2X_2+\alpha_3X_3))\\ F_{X_1}&= -\alpha_1 F_{X_0} \\ F_{X_2}&= -\alpha_2 F_{X_0} \\ F_{X_3}&= -\alpha_3 F_{X_0} \end{aligned}

$$f_{\tt stiff}$$为系数。

\begin{aligned} \frac{\partial F}{\partial X_0} &= (\frac{\partial F_{X_0}}{\partial X_0}, \frac{\partial F_{X_1}}{\partial X_0},\frac{\partial F_{X_2}}{\partial X_0},\frac{\partial F_{X_3}}{\partial X_0}) \\ &= (\frac{\partial F_{X_0}}{\partial X_0}, -\alpha_1\frac{\partial F_{X_0}}{\partial X_0},-\alpha_2\frac{\partial F_{X_0}}{\partial X_0}, -\alpha_3\frac{\partial F_{X_0}}{\partial X_0}) \end{aligned}

\begin{aligned} \frac{\partial F}{\partial X_1} &= (\frac{\partial F_{X_0}}{\partial X_1}, \frac{\partial F_{X_1}}{\partial X_1},\frac{\partial F_{X_2}}{\partial X_1},\frac{\partial F_{X_3}}{\partial X_1}) \\ &= (\frac{\partial F_{X_0}}{\partial X_1}, -\alpha_1\frac{\partial F_{X_0}}{\partial X_1},-\alpha_2\frac{\partial F_{X_0}}{\partial X_1}, -\alpha_3\frac{\partial F_{X_0}}{\partial X_1}) \\ \\ \frac{\partial F}{\partial X_2} &= (\frac{\partial F_{X_0}}{\partial X_2},- \alpha_1\frac{\partial F_{X_0}}{\partial X_2},-\alpha_2\frac{\partial F_{X_0}}{\partial X_2}, -\alpha_3\frac{\partial F_{X_0}}{\partial X_2}) \\ \\ \frac{\partial F}{\partial X_3} &= (\frac{\partial F_{X_0}}{\partial X_3},-\alpha_1\frac{\partial F_{X_0}}{\partial X_3},-\alpha_2\frac{\partial F_{X_0}}{\partial X_3}, -\alpha_3\frac{\partial F_{X_0}}{\partial X_3}) \\ \end{aligned}

$\frac{\partial{\bf F}}{\partial \bf x} = \begin{bmatrix} \frac{\partial f_x}{\partial \bf x} \\ \frac{\partial f_y}{\partial \bf x} \\ \frac{\partial f_z}{\partial \bf x} \end{bmatrix} = \begin{bmatrix} \frac{\partial f_x}{\partial {\bf x}_x} \ \frac{\partial f_x}{\partial {\bf x}_y} \ \frac{\partial f_x}{\partial {\bf x}_z} \\ \frac{\partial f_y}{\partial {\bf x}_x} \ \frac{\partial f_y}{\partial {\bf x}_y} \ \frac{\partial f_y}{\partial {\bf x}_z} \\ \frac{\partial f_z}{\partial {\bf x}_x} \ \frac{\partial f_z}{\partial {\bf x}_y} \ \frac{\partial f_z}{\partial {\bf x}_z} \end{bmatrix}$

\begin{aligned} \frac{\partial F_{X_0}}{\partial X_0} &= \begin{bmatrix} &f_{\tt stiff} \ &0 \ &0 \\ &0 \ &f_{\tt stiff} \ &0 \\ &0 \ &0 \ &f_{\tt stiff} \\ \end{bmatrix} \\ \frac{\partial F_{X_0}}{\partial X_1} &= -\alpha_1 \cdot \frac{\partial F_{X_0}}{\partial X_0} \\ \frac{\partial F_{X_0}}{\partial X_2} &= -\alpha_2 \cdot \frac{\partial F_{X_0}}{\partial X_0} \\ \frac{\partial F_{X_0}}{\partial X_3} &= -\alpha_3 \cdot \frac{\partial F_{X_0}}{\partial X_0} \end{aligned}

## 参考文献

posted @ 2018-04-23 13:19  薛定谔の三味  阅读(373)  评论(0编辑  收藏  举报