I have $u\in\mathbb{R}^3$ and the term $\epsilon = (\nabla u)^\top + \nabla u$.
Since $\nabla u = \begin{pmatrix} \frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} & \frac{\partial u_1}{\partial z} \\ \frac{\partial u_2}{\partial x} & \frac{\partial u_2}{\partial y} & \frac{\partial u_2}{\partial z} \\ \frac{\partial u_3}{\partial x} & \frac{\partial u_3}{\partial y} & \frac{\partial u_3}{\partial z}\end{pmatrix}$, I have summarized $\epsilon$ to be $\epsilon=\begin{pmatrix} \frac{\partial u_1}{\partial x} + \frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} + \frac{\partial u_2}{\partial x} & \frac{\partial u_1}{\partial z} + \frac{\partial u_3}{\partial x} \\ \frac{\partial u_2}{\partial x} + \frac{\partial u_1}{\partial y} & \frac{\partial u_2}{\partial y} + \frac{\partial u_2}{\partial y} & \frac{\partial u_2}{\partial z} + \frac{\partial u_3}{\partial y} \\ \frac{\partial u_3}{\partial x} + \frac{\partial u_1}{\partial z} & \frac{\partial u_3}{\partial y} + \frac{\partial u_2}{\partial z} & \frac{\partial u_3}{\partial z} + \frac{\partial u_3}{\partial z} \end{pmatrix}$.
With some boundary conditions of the problem I'm trying to solve I can reduce $\epsilon$ to $U = \begin{pmatrix} \frac{\partial u_1}{\partial x} & \frac{\partial u_2}{\partial y} & \frac{\partial u_3}{\partial z} & \frac{\partial u_2}{\partial z} + \frac{\partial u_3}{\partial y} & \frac{\partial u_1}{\partial z} + \frac{\partial u_3}{\partial x} & \frac{\partial u_1}{\partial y} + \frac{\partial u_2}{\partial x} \end{pmatrix}^\mathbf{T}$.
Now I have the term $\int_\Omega \nabla (eU) \cdot v \text{ d}x$ with $e\in\mathbb{R}^{3\times 6}, v\in\mathbb{R}^{3}$. To eliminate the second order derivatives of $u$ I want to integrate it by parts.
My solution to this looks like this: $eU\cdot v - \int_\Omega e U \cdot \nabla v \text{ d}x$.
Now I have two questions:
- Is the way of summarizing the terms in $\epsilon$ correct in the first place?
- Is my integration by parts correct? I feel unsure about it because I've somehow expected both terms to stay in integrals but maybe changing the integration boundaries to $\partial\Omega$.
I appreciate any help. Thank you.