## 1. MITC shell element

- 3차원 솔리드 형상으로부터 쉘형상을 표현 (유한요소 정식화가 다른 쉘요소에 비해 간단)
- 쉘 이론을 사용하지 않고 3차원 응력, 변형률을 사용하여 쉘을 표현할 수 있다.
- 임의의 형상에 대한 두꺼운 쉘과 얇은 쉘 모두 적용 가능
- Locking을 방지하기 위해 횡방향 전단 변형률에 보간법 사용

## 2. Kinematics

![Figure](images/chunk-001-fig-004.jpg)

![Figure](images/chunk-001-fig-005.jpg)

Shell의 초기 위치 벡터는 다음과 같이 shape function으로 나타낼 수 있다.

$$
{}^{0}\mathbf{X} = \sum_{i=1}^{4} \phi_{i}(\xi^{1}, \xi^{2}) {}^{0}\mathbf{X}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i}\phi_{i}(\xi^{1}, \xi^{2}) {}^{0}\mathbf{V}_{n}^{i}
$$

마찬가지로 시간이 t, $t+\Delta t$ 일 때 위치벡터는 다음과 같다.

$$
\begin{split} ^{t}\mathbf{x} &= \sum_{i=1}^{4} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t}\mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t}\mathbf{V}_{n}^{i} \\ ^{t+\Delta t}\mathbf{x} &= \sum_{i=1}^{4} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t+\Delta t}\mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t+\Delta t}\mathbf{V}_{n}^{i} \end{split}
$$

시간이 t일 때와 $t + \Delta t$ 일 때 변위 벡터는 다음과 같이 계산 할 수 있다.

$$
\begin{split} ^{t}\mathbf{u} &= ^{t}\mathbf{X}^{-0}\mathbf{X} \\ &= \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t} \mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t} \mathbf{V}_{n}^{i} - \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{X}_{i} - \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{V}_{n}^{i} \\ &= \sum_{i=1}^{4} \phi_{i} \left(^{t}\mathbf{x}_{i}^{-0}\mathbf{X}_{i}\right) + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t}\mathbf{V}_{n}^{i}^{-0}\mathbf{V}_{n}^{i}\right) \\ ^{t+\Delta t}\mathbf{u} &= ^{t+\Delta t}\mathbf{x}^{-0}\mathbf{X} \\ &= \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t+\Delta t} \mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t+\Delta t} \mathbf{V}_{n}^{i} - \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{X}_{i} - \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{V}_{n}^{i} \\ &= \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i}^{-0}\mathbf{X}_{i}\right) + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i}^{i} - \mathbf{V}_{n}^{i}\right) \end{split}
$$

따라서 시간 t와 $t+\Delta t$ 사이의 incremental displacement는 다음과 같이 나타낼 수 있다.

$$
\begin{split} & \Delta^{t}\mathbf{u} =^{t+\Delta t} \mathbf{u} - ^{t}\mathbf{u} \\ & = \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i} - ^{0}\mathbf{X}_{i}\right) + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i} - ^{0}\mathbf{V}_{n}^{i}\right) - \sum_{i=1}^{4} \phi_{i} \left(^{t}\mathbf{x}_{i} - ^{0}\mathbf{X}_{i}\right) - \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t}\mathbf{V}_{n}^{i} - ^{0}\mathbf{V}_{n}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i} - ^{t}\mathbf{x}_{i}\right) + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i} - ^{t}\mathbf{V}_{n}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \Delta^{t}\mathbf{u}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(-\alpha_{1}^{i} {}^{t}\mathbf{V}_{2}^{i} + \alpha_{2}^{i} {}^{t}\mathbf{V}_{1}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \Delta^{t}\mathbf{u}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \Delta^{t}\mathbf{V}_{n}^{i} \end{split}
$$

위치 벡터와 변위 벡터를 matrix form으로 나타내면

$$
{}^{0}\mathbf{X} = {}^{0}\mathbf{N}^{0}\mathbf{X}
$$

$$
= \begin{bmatrix} \phi_1 & \phi_2 & \phi_3 & \phi_4 \end{bmatrix} \begin{bmatrix} {}^{0}\mathbf{X}_1 \\ {}^{0}\mathbf{X}_2 \\ {}^{0}\mathbf{X}_3 \\ {}^{0}\mathbf{X}_4 \end{bmatrix} + \begin{bmatrix} \frac{\xi^3}{2}h_1\phi_1 & \frac{\xi^3}{2}h_2\phi_2 & \frac{\xi^3}{2}h_3\phi_3 & \frac{\xi^3}{2}h_4\phi_4 \end{bmatrix} \begin{bmatrix} {}^{0}\mathbf{V}_n^1 \\ {}^{0}\mathbf{V}_n^2 \\ {}^{0}\mathbf{V}_n^3 \\ {}^{0}\mathbf{V}_n^4 \end{bmatrix}
$$

$$
= \left[ \phi_{1} \quad \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \phi_{2} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \phi_{3} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \phi_{4} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} {}^{0} \mathbf{X}_{1} \\ {}^{0} \mathbf{X}_{2} \\ {}^{0} \mathbf{X}_{2} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{4} \\ {}^{0} \mathbf{V}_{n}^{4} \end{bmatrix}
$$

$$
^{t}\mathbf{x} = ^{0}\mathbf{N}^{t}\mathbf{x}
$$

$$
= \begin{bmatrix} \phi_1 & \frac{\xi^3}{2} h_1 \phi_1 & \phi_2 & \frac{\xi^3}{2} h_2 \phi_2 & \phi_3 & \frac{\xi^3}{2} h_3 \phi_3 & \phi_4 & \frac{\xi^3}{2} h_4 \phi_4 \end{bmatrix} \begin{bmatrix} {}^t \mathbf{X}_1 \\ {}^t \mathbf{V}_n^1 \\ {}^t \mathbf{X}_2 \\ {}^t \mathbf{V}_n^2 \\ {}^t \mathbf{X}_3 \\ {}^t \mathbf{V}_n^3 \\ {}^t \mathbf{X}_4 \\ {}^t \mathbf{V}_n^4 \end{bmatrix}
$$

$$
^{t+\Delta t}\mathbf{x} = ^{0}\mathbf{N}^{t+\Delta t}\mathbf{x}_{n}
$$

$$
= \left[ \begin{array}{cccccccccccccccccccccccccccccccccccc
$$

$$
\Delta^t \mathbf{x} = {}^0 \mathbf{N} \Delta^t \mathbf{x}_n
$$

$$
= \begin{bmatrix} \phi_1 & \frac{\xi^3}{2} h_1 \phi_1 & \phi_2 & \frac{\xi^3}{2} h_2 \phi_2 & \phi_3 & \frac{\xi^3}{2} h_3 \phi_3 & \phi_4 & \frac{\xi^3}{2} h_4 \phi_4 \end{bmatrix} \begin{bmatrix} \Delta \mathbf{X}_1 \\ \Delta^t \mathbf{V}_n^1 \\ \Delta^t \mathbf{X}_2 \\ \Delta^t \mathbf{V}_n^2 \\ \Delta^t \mathbf{V}_n^3 \\ \Delta^t \mathbf{V}_n^4 \\ \Delta^t \mathbf{V}_n^4 \end{bmatrix}
$$

변위 벡터도 마찬가지로 나타낼 수 있다.

$$
^{t}\mathbf{u} = ^{0}\mathbf{N}(^{t}\mathbf{x}_{n} - ^{0}\mathbf{X}_{n})
$$

$$
= \left[ \phi_{1} \quad \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \phi_{2} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \phi_{3} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \phi_{4} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} {}^{t} \mathbf{X}_{1} - {}^{0} \mathbf{X}_{1} \\ {}^{t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} \\ {}^{t} \mathbf{X}_{2} - {}^{0} \mathbf{X}_{2} \\ {}^{t} \mathbf{V}_{n}^{2} - {}^{0} \mathbf{V}_{n}^{2} \\ {}^{t} \mathbf{X}_{3} - {}^{0} \mathbf{X}_{3} \\ {}^{t} \mathbf{V}_{n}^{3} - {}^{0} \mathbf{V}_{n}^{3} \\ {}^{t} \mathbf{X}_{4} - {}^{0} \mathbf{X}_{4} \\ {}^{t} \mathbf{V}_{n}^{4} - {}^{0} \mathbf{V}_{n}^{4} \right]
$$

$$
\begin{aligned}
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{x}_{1} - {}^{0} \mathbf{X}_{1}}{\mathbf{x}_{2} - {}^{0} \mathbf{X}_{2}} + \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{V}_{n}^{2} - {}^{0} \mathbf{V}_{n}^{2}}{\mathbf{V}_{n}^{2} - {}^{0} \mathbf{V}_{n}^{3}} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{u}_{1}}{\mathbf{u}_{2}} + \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right]_{\substack{\Delta^{t} \mathbf{V}_{n}^{1} + \Delta^{0} \mathbf{V}_{n}^{1} \\ \Delta^{t} \mathbf{V}_{n}^{2} + \Delta^{0} \mathbf{V}_{n}^{3}}}^{t+\Delta t} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{u}_{1}}{\mathbf{u}_{2}} + \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right]_{\substack{\Delta^{t} \mathbf{V}_{n}^{1} + \Delta^{0} \mathbf{V}_{n}^{1} \\ \Delta^{t} \mathbf{V}_{n}^{3} + \Delta^{0} \mathbf{V}_{n}^{3} \\ \Delta^{t} \mathbf{V}_{n}^{3} + \Delta^{0} \mathbf{V}_{n}^{3}} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{u}_{1}}{\mathbf{u}_{2}} + \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right]_{\substack{\Delta^{t} \mathbf{V}_{n}^{3} + \Delta^{0} \mathbf{V}_{n}^{3} \\ \Delta^{t} \mathbf{V}_{n}^{3} + \Delta^{0} \mathbf{V}_{n}^{3}} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \frac{\mathbf{u}_{1}}{\mathbf{u}_{2}} + \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right]_{\substack{t+\Delta t \\ \Delta^{t} \mathbf{V}_{n}^{3} + \Delta^{0} \mathbf{V}_{n}^{3}}}^{t+\Delta t} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{2} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{2} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{3} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{3} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{3} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}^{t+\Delta t} \mathbf{u}_{3} \\
&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}
$$

$$
= \begin{bmatrix} \boldsymbol{\phi}_1 & \boldsymbol{\phi}_2 & \boldsymbol{\phi}_3 & \boldsymbol{\phi}_4 \end{bmatrix} \begin{bmatrix} t^{t+\Delta t} \mathbf{u}_1 \\ t^{t+\Delta t} \mathbf{u}_2 \\ t^{t+\Delta t} \mathbf{u}_3 \\ t^{t+\Delta t} \mathbf{u}_4 \end{bmatrix} + \begin{bmatrix} \underline{\xi}^3 \\ 2 \end{pmatrix} h_1 \boldsymbol{\phi}_1 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_2 \boldsymbol{\phi}_2 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_3 \boldsymbol{\phi}_3 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_4 \boldsymbol{\phi}_4 \end{bmatrix} \begin{bmatrix} \Delta^t \mathbf{V}_n^1 + \Delta^0 \mathbf{V}_n^1 \\ \Delta^t \mathbf{V}_n^2 + \Delta^0 \mathbf{V}_n^2 \\ \Delta^t \mathbf{V}_n^3 + \Delta^0 \mathbf{V}_n^3 \\ \Delta^t \mathbf{V}_n^4 + \Delta^0 \mathbf{V}_n^4 \end{bmatrix}
$$

$$
\left( t^{+\Delta t} \mathbf{V}_n^1 - {}^0 \mathbf{V}_n^1 = t^{+\Delta t} \mathbf{V}_n^1 - {}^t \mathbf{V}_n^1 + {}^t \mathbf{V}_n^1 - {}^0 \mathbf{V}_n^1 = \Delta^t \mathbf{V}_n^1 + \Delta^0 \mathbf{V}_n^1 \right)
$$

$$
\begin{bmatrix} t^{+\Delta t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} = {}^{t+\Delta t} \mathbf{V}_{n}^{1} - {}^{t} \mathbf{V}_{n}^{1} + {}^{t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} = \Delta^{t} \mathbf{V}_{n}^{1} + \Delta^{0} \mathbf{V}_{n}^{1} \end{bmatrix}
= \begin{bmatrix} \phi_{1} & \phi_{2} & \phi_{3} & \phi_{4} \end{bmatrix} \begin{bmatrix} t^{+\Delta t} \mathbf{u}_{1} \\ t^{+\Delta t} \mathbf{u}_{2} \\ t^{+\Delta t} \mathbf{u}_{3} \\ t^{+\Delta t} \mathbf{u}_{4} \end{bmatrix} + \begin{bmatrix} \underline{\xi}^{3} \\ 2 \\ h_{1} \phi_{1} & \underline{\xi}^{3} \\ 2 \\ h_{2} \phi_{2} & \underline{\xi}^{3} \\ 2 \\ h_{3} \phi_{3} & \underline{\xi}^{3} \\ 2 \\ h_{3} \phi_{3} & \underline{\xi}^{3} \\ 2 \\ h_{4} \phi_{4} \end{bmatrix} \begin{bmatrix} -\alpha_{1}^{1t} \mathbf{V}_{2}^{1} + \alpha_{2}^{1t} \mathbf{V}_{1}^{1} \\ -\alpha_{1}^{2t} \mathbf{V}_{2}^{2} + \alpha_{2}^{2t} \mathbf{V}_{1}^{2} \\ -\alpha_{1}^{3t} \mathbf{V}_{2}^{3} + \alpha_{2}^{3t} \mathbf{V}_{1}^{3} \\ -\alpha_{1}^{4t} \mathbf{V}_{2}^{4} + \alpha_{2}^{4t} \mathbf{V}_{1}^{4} \end{bmatrix}
$$

$$
+ \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} \Delta^{0} \mathbf{V}_{n}^{1} \\ \Delta^{0} \mathbf{V}_{n}^{2} \\ \Delta^{0} \mathbf{V}_{n}^{3} \\ \Delta^{0} \mathbf{V}_{n}^{4} \end{bmatrix}
$$

$$
\Rightarrow^{t+\Delta t} \mathbf{u} = {}^{t} \mathbf{N}^{t+\Delta t} \mathbf{u}_{n} + {}^{0} \tilde{\mathbf{N}} \Delta^{0} \tilde{\mathbf{X}}^{n}
$$

$$
= \begin{bmatrix} \phi_{1} & -\frac{\xi^{3}}{2} h_{1} \phi_{1} \mathbf{v}_{2}^{1} & \frac{\xi^{3}}{2} h_{1} \phi_{1} \mathbf{v}_{1}^{1} & \phi_{2} & -\frac{\xi^{3}}{2} h_{2} \phi_{2} \mathbf{v}_{2}^{2} & \frac{\xi^{3}}{2} h_{2} \phi_{2} \mathbf{v}_{1}^{2} & \phi_{3} & -\frac{\xi^{3}}{2} h_{3} \phi_{3} \mathbf{v}_{2}^{3} & \frac{\xi^{3}}{2} h_{3} \phi_{3} \mathbf{v}_{1}^{3} & \phi_{4} & -\frac{\xi^{3}}{2} h_{4} \phi_{4} \mathbf{v}_{2}^{4} & \frac{\xi^{3}}{2} h_{4} \phi_{4} \mathbf{v}_{1}^{4} \end{bmatrix} \begin{bmatrix} \mathbf{\alpha}_{1}^{1} \\ \mathbf{\alpha}_{2}^{1} \\ \mathbf{\alpha}_{1}^{2} \\ \mathbf{\alpha}_{2}^{2} \\ t + \Delta t \mathbf{u}_{3} \\ \mathbf{\alpha}_{3}^{3} \\ \mathbf{\alpha}_{4}^{3} \\ \mathbf{\alpha}_{4}^{4} \\ \mathbf{\alpha}_{4}^{4} \\ \mathbf{\alpha}_{2}^{4} \end{bmatrix}
$$