Stress hessian computation in FEM
介绍
在进行软体模拟时,如果使用牛顿法计算最优的下降方向,需要计算能量密度函数\(\Psi\)关于位置\(\vec {\mathbf{x}}\) 的Hessian矩阵,即\(\frac{\partial^2 \Psi}{\partial \vec {\mathbf{x}}^2}\)。其中\(\vec {\mathbf{x}}\)是一个四面体的四个顶点的位置。因为:
\[ \begin{aligned} \frac{\partial^2 \Psi}{\partial \vec {\mathbf{x}}^2} &= \frac{\partial}{\partial \vec {\mathbf{x}}} \left(\frac{\partial \Psi}{\partial \vec {\mathbf{x}}}\right) \\ &= \frac{\partial}{\partial \vec {\mathbf{x}}} \left(\frac{\partial \Psi}{\partial F}:\frac{\partial F}{\partial \vec {\mathbf{x}}}\right) \\ &= \frac{\partial}{\partial \vec {\mathbf{x}}} \left(\frac{\partial \Psi}{\partial F}\right):\frac{\partial F}{\partial \vec {\mathbf{x}}} + \frac{\partial \Psi}{\partial F}:\frac{\partial^2 F}{\partial \vec {\mathbf{x}}^2}\\ &=\frac{\partial}{\partial \vec {\mathbf{x}}} \left(P\right):\frac{\partial F}{\partial \vec {\mathbf{x}}}\\ &=(\frac{\partial F}{\partial \vec {\mathbf{x}}}: \frac{\partial P}{\partial F}):\frac{\partial F}{\partial \vec {\mathbf{x}}} \end{aligned} \]