# 在等幾何分析中計算空間二階導數

\開始{}方程 你（\ xi，\ eta）= \ sum_ {i} c_iN ^ i（\ xi，\ eta） \ {端方程} 從參數域到物理域的幾何映射 $$x（\ xi，\ eta）= \ sum_ {i} x_i N ^ i（\ xi，\ eta），\ quad y（\ xi，\ eta）= \ sum_ {i} y_i N ^ i（\十一，\ ETA），$$ 其中$c_i，x_i，y_i$是常量，假設$（\ xi，\ eta）\ mapsto（x，y）$是雙射的，即逆存在，$$J：= [\ frac {\ partial x_i} {\ partial \ xi_j}]，\：| J | \ neq 0 \ quad（\ text {where} x_2 = y，\，\ xi_2 = \ eta）。$$

$$\begin{bmatrix} \frac{\partial u}{\partial \xi}\\ \frac{\partial u}{\partial \eta} \end{bmatrix} = \begin{bmatrix} \frac{\partial x}{\partial \xi} & \frac{\partial y}{\partial \xi}\\ \frac{\partial x}{\partial \eta} & \frac{\partial y}{\partial \eta} \end{bmatrix} \begin{bmatrix} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y} \end{bmatrix} = J^T \begin{bmatrix} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y} \end{bmatrix}.$$

This is the common procedure used in isogemetric analysis f要麽 computing stiffness matrix.

However, when w要麽king on sensitivity analysis, I need to compute the second derivatives $$\frac{\partial^2 u}{\partial x_i\partial x_j},\quad i \text{ and }j \in \{1,2\}.$$

I have encountered this problem when w要麽king on sensitivity analysis, especially adjoint method with the approach of material derivatives. Let assume we have an objective functional $$\phi = \int_{\Omega}f(\sigma,\epsilon,p)\,\mathrm{d}\Omega$$ where $p$ is a design parameter(e.g. co要麽dinate of one control point). With linear elastostatics, $\sigma,\,\epsilon$ are functions of $\nabla \boldsymbol{u}$, so we can write $f = f(\nabla u)$.

The first term in the domain integral is taken care of by adjoint f要麽mulation. From the second term in the domain integral, the second derivative is required.