docs: reset mitc4 formulation baseline
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## 1. MITC shell element
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- 3차원 솔리드 형상으로부터 쉘형상을 표현 (유한요소 정식화가 다른 쉘요소에 비해 간단)
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- 쉘 이론을 사용하지 않고 3차원 응력, 변형률을 사용하여 쉘을 표현할 수 있다.
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- 임의의 형상에 대한 두꺼운 쉘과 얇은 쉘 모두 적용 가능
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- Locking을 방지하기 위해 횡방향 전단 변형률에 보간법 사용
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## 2. Kinematics
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Shell의 초기 위치 벡터는 다음과 같이 shape function으로 나타낼 수 있다.
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$$
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{}^{0}\mathbf{X} = \sum_{i=1}^{4} \phi_{i}(\xi^{1}, \xi^{2}) {}^{0}\mathbf{X}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i}\phi_{i}(\xi^{1}, \xi^{2}) {}^{0}\mathbf{V}_{n}^{i}
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$$
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마찬가지로 시간이 t, $t+\Delta t$ 일 때 위치벡터는 다음과 같다.
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$$
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\begin{split} ^{t}\mathbf{x} &= \sum_{i=1}^{4} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t}\mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t}\mathbf{V}_{n}^{i} \\ ^{t+\Delta t}\mathbf{x} &= \sum_{i=1}^{4} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t+\Delta t}\mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i}\left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right){}^{t+\Delta t}\mathbf{V}_{n}^{i} \end{split}
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$$
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시간이 t일 때와 $t + \Delta t$ 일 때 변위 벡터는 다음과 같이 계산 할 수 있다.
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$$
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\begin{split} ^{t}\mathbf{u} &= ^{t}\mathbf{X}^{-0}\mathbf{X} \\ &= \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t} \mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t} \mathbf{V}_{n}^{i} - \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{X}_{i} - \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{V}_{n}^{i} \\ &= \sum_{i=1}^{4} \phi_{i} \left(^{t}\mathbf{x}_{i}^{-0}\mathbf{X}_{i}\right) + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t}\mathbf{V}_{n}^{i}^{-0}\mathbf{V}_{n}^{i}\right) \\ ^{t+\Delta t}\mathbf{u} &= ^{t+\Delta t}\mathbf{x}^{-0}\mathbf{X} \\ &= \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t+\Delta t} \mathbf{x}_{i} + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{t+\Delta t} \mathbf{V}_{n}^{i} - \sum_{i=1}^{4} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{X}_{i} - \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(\boldsymbol{\xi}^{1}, \boldsymbol{\xi}^{2}\right)^{0} \mathbf{V}_{n}^{i} \\ &= \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i}^{-0}\mathbf{X}_{i}\right) + \frac{\boldsymbol{\xi}^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i}^{i} - \mathbf{V}_{n}^{i}\right) \end{split}
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$$
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따라서 시간 t와 $t+\Delta t$ 사이의 incremental displacement는 다음과 같이 나타낼 수 있다.
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$$
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\begin{split} & \Delta^{t}\mathbf{u} =^{t+\Delta t} \mathbf{u} - ^{t}\mathbf{u} \\ & = \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i} - ^{0}\mathbf{X}_{i}\right) + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i} - ^{0}\mathbf{V}_{n}^{i}\right) - \sum_{i=1}^{4} \phi_{i} \left(^{t}\mathbf{x}_{i} - ^{0}\mathbf{X}_{i}\right) - \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t}\mathbf{V}_{n}^{i} - ^{0}\mathbf{V}_{n}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \left(^{t+\Delta t}\mathbf{x}_{i} - ^{t}\mathbf{x}_{i}\right) + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(^{t+\Delta t}\mathbf{V}_{n}^{i} - ^{t}\mathbf{V}_{n}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \Delta^{t}\mathbf{u}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \left(-\alpha_{1}^{i} {}^{t}\mathbf{V}_{2}^{i} + \alpha_{2}^{i} {}^{t}\mathbf{V}_{1}^{i}\right) \\ & = \sum_{i=1}^{4} \phi_{i} \Delta^{t}\mathbf{u}_{i} + \frac{\xi^{3}}{2} \sum_{i=1}^{4} h_{i} \phi_{i} \Delta^{t}\mathbf{V}_{n}^{i} \end{split}
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$$
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위치 벡터와 변위 벡터를 matrix form으로 나타내면
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$$
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{}^{0}\mathbf{X} = {}^{0}\mathbf{N}^{0}\mathbf{X}
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$$
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$$
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= \begin{bmatrix} \phi_1 & \phi_2 & \phi_3 & \phi_4 \end{bmatrix} \begin{bmatrix} {}^{0}\mathbf{X}_1 \\ {}^{0}\mathbf{X}_2 \\ {}^{0}\mathbf{X}_3 \\ {}^{0}\mathbf{X}_4 \end{bmatrix} + \begin{bmatrix} \frac{\xi^3}{2}h_1\phi_1 & \frac{\xi^3}{2}h_2\phi_2 & \frac{\xi^3}{2}h_3\phi_3 & \frac{\xi^3}{2}h_4\phi_4 \end{bmatrix} \begin{bmatrix} {}^{0}\mathbf{V}_n^1 \\ {}^{0}\mathbf{V}_n^2 \\ {}^{0}\mathbf{V}_n^3 \\ {}^{0}\mathbf{V}_n^4 \end{bmatrix}
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$$
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$$
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= \left[ \phi_{1} \quad \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \phi_{2} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \phi_{3} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \phi_{4} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} {}^{0} \mathbf{X}_{1} \\ {}^{0} \mathbf{X}_{2} \\ {}^{0} \mathbf{X}_{2} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{3} \\ {}^{0} \mathbf{X}_{4} \\ {}^{0} \mathbf{V}_{n}^{4} \end{bmatrix}
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$$
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$$
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^{t}\mathbf{x} = ^{0}\mathbf{N}^{t}\mathbf{x}
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$$
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$$
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= \begin{bmatrix} \phi_1 & \frac{\xi^3}{2} h_1 \phi_1 & \phi_2 & \frac{\xi^3}{2} h_2 \phi_2 & \phi_3 & \frac{\xi^3}{2} h_3 \phi_3 & \phi_4 & \frac{\xi^3}{2} h_4 \phi_4 \end{bmatrix} \begin{bmatrix} {}^t \mathbf{X}_1 \\ {}^t \mathbf{V}_n^1 \\ {}^t \mathbf{X}_2 \\ {}^t \mathbf{V}_n^2 \\ {}^t \mathbf{X}_3 \\ {}^t \mathbf{V}_n^3 \\ {}^t \mathbf{X}_4 \\ {}^t \mathbf{V}_n^4 \end{bmatrix}
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$$
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$$
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^{t+\Delta t}\mathbf{x} = ^{0}\mathbf{N}^{t+\Delta t}\mathbf{x}_{n}
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$$
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$$
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= \left[ \begin{array}{cccccccccccccccccccccccccccccccccccc
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$$
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$$
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\Delta^t \mathbf{x} = {}^0 \mathbf{N} \Delta^t \mathbf{x}_n
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$$
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$$
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= \begin{bmatrix} \phi_1 & \frac{\xi^3}{2} h_1 \phi_1 & \phi_2 & \frac{\xi^3}{2} h_2 \phi_2 & \phi_3 & \frac{\xi^3}{2} h_3 \phi_3 & \phi_4 & \frac{\xi^3}{2} h_4 \phi_4 \end{bmatrix} \begin{bmatrix} \Delta \mathbf{X}_1 \\ \Delta^t \mathbf{V}_n^1 \\ \Delta^t \mathbf{X}_2 \\ \Delta^t \mathbf{V}_n^2 \\ \Delta^t \mathbf{V}_n^3 \\ \Delta^t \mathbf{V}_n^4 \\ \Delta^t \mathbf{V}_n^4 \end{bmatrix}
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$$
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변위 벡터도 마찬가지로 나타낼 수 있다.
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$$
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^{t}\mathbf{u} = ^{0}\mathbf{N}(^{t}\mathbf{x}_{n} - ^{0}\mathbf{X}_{n})
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$$
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$$
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= \left[ \phi_{1} \quad \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \phi_{2} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \phi_{3} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \phi_{4} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} {}^{t} \mathbf{X}_{1} - {}^{0} \mathbf{X}_{1} \\ {}^{t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} \\ {}^{t} \mathbf{X}_{2} - {}^{0} \mathbf{X}_{2} \\ {}^{t} \mathbf{V}_{n}^{2} - {}^{0} \mathbf{V}_{n}^{2} \\ {}^{t} \mathbf{X}_{3} - {}^{0} \mathbf{X}_{3} \\ {}^{t} \mathbf{V}_{n}^{3} - {}^{0} \mathbf{V}_{n}^{3} \\ {}^{t} \mathbf{X}_{4} - {}^{0} \mathbf{X}_{4} \\ {}^{t} \mathbf{V}_{n}^{4} - {}^{0} \mathbf{V}_{n}^{4} \right]
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$$
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$$
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\begin{aligned}
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&= \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}} \\
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&= \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} \\
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&= \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}} \\
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&= \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}} \\
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&= \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} \\
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&= \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} \\
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&= \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} \\
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&= \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} \\
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&= \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} \\
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&= \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} \\
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&= \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} \\
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&= \left[ \phi_{1} \quad \phi_{2} \quad \phi_{3} \quad \phi_{4} \right]_{\substack{t+\Delta t \\ t+\Delta t}}
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$$
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$$
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= \begin{bmatrix} \boldsymbol{\phi}_1 & \boldsymbol{\phi}_2 & \boldsymbol{\phi}_3 & \boldsymbol{\phi}_4 \end{bmatrix} \begin{bmatrix} t^{t+\Delta t} \mathbf{u}_1 \\ t^{t+\Delta t} \mathbf{u}_2 \\ t^{t+\Delta t} \mathbf{u}_3 \\ t^{t+\Delta t} \mathbf{u}_4 \end{bmatrix} + \begin{bmatrix} \underline{\xi}^3 \\ 2 \end{pmatrix} h_1 \boldsymbol{\phi}_1 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_2 \boldsymbol{\phi}_2 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_3 \boldsymbol{\phi}_3 & \underline{\xi}^3 \\ 2 \end{pmatrix} h_4 \boldsymbol{\phi}_4 \end{bmatrix} \begin{bmatrix} \Delta^t \mathbf{V}_n^1 + \Delta^0 \mathbf{V}_n^1 \\ \Delta^t \mathbf{V}_n^2 + \Delta^0 \mathbf{V}_n^2 \\ \Delta^t \mathbf{V}_n^3 + \Delta^0 \mathbf{V}_n^3 \\ \Delta^t \mathbf{V}_n^4 + \Delta^0 \mathbf{V}_n^4 \end{bmatrix}
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$$
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$$
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\left( t^{+\Delta t} \mathbf{V}_n^1 - {}^0 \mathbf{V}_n^1 = t^{+\Delta t} \mathbf{V}_n^1 - {}^t \mathbf{V}_n^1 + {}^t \mathbf{V}_n^1 - {}^0 \mathbf{V}_n^1 = \Delta^t \mathbf{V}_n^1 + \Delta^0 \mathbf{V}_n^1 \right)
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$$
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$$
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\begin{bmatrix} t^{+\Delta t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} = {}^{t+\Delta t} \mathbf{V}_{n}^{1} - {}^{t} \mathbf{V}_{n}^{1} + {}^{t} \mathbf{V}_{n}^{1} - {}^{0} \mathbf{V}_{n}^{1} = \Delta^{t} \mathbf{V}_{n}^{1} + \Delta^{0} \mathbf{V}_{n}^{1} \end{bmatrix}
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= \begin{bmatrix} \phi_{1} & \phi_{2} & \phi_{3} & \phi_{4} \end{bmatrix} \begin{bmatrix} t^{+\Delta t} \mathbf{u}_{1} \\ t^{+\Delta t} \mathbf{u}_{2} \\ t^{+\Delta t} \mathbf{u}_{3} \\ t^{+\Delta t} \mathbf{u}_{4} \end{bmatrix} + \begin{bmatrix} \underline{\xi}^{3} \\ 2 \\ h_{1} \phi_{1} & \underline{\xi}^{3} \\ 2 \\ h_{2} \phi_{2} & \underline{\xi}^{3} \\ 2 \\ h_{3} \phi_{3} & \underline{\xi}^{3} \\ 2 \\ h_{3} \phi_{3} & \underline{\xi}^{3} \\ 2 \\ h_{4} \phi_{4} \end{bmatrix} \begin{bmatrix} -\alpha_{1}^{1t} \mathbf{V}_{2}^{1} + \alpha_{2}^{1t} \mathbf{V}_{1}^{1} \\ -\alpha_{1}^{2t} \mathbf{V}_{2}^{2} + \alpha_{2}^{2t} \mathbf{V}_{1}^{2} \\ -\alpha_{1}^{3t} \mathbf{V}_{2}^{3} + \alpha_{2}^{3t} \mathbf{V}_{1}^{3} \\ -\alpha_{1}^{4t} \mathbf{V}_{2}^{4} + \alpha_{2}^{4t} \mathbf{V}_{1}^{4} \end{bmatrix}
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$$
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$$
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+ \left[ \frac{\xi^{3}}{2} h_{1} \phi_{1} \quad \frac{\xi^{3}}{2} h_{2} \phi_{2} \quad \frac{\xi^{3}}{2} h_{3} \phi_{3} \quad \frac{\xi^{3}}{2} h_{4} \phi_{4} \right] \begin{bmatrix} \Delta^{0} \mathbf{V}_{n}^{1} \\ \Delta^{0} \mathbf{V}_{n}^{2} \\ \Delta^{0} \mathbf{V}_{n}^{3} \\ \Delta^{0} \mathbf{V}_{n}^{4} \end{bmatrix}
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$$
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$$
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\Rightarrow^{t+\Delta t} \mathbf{u} = {}^{t} \mathbf{N}^{t+\Delta t} \mathbf{u}_{n} + {}^{0} \tilde{\mathbf{N}} \Delta^{0} \tilde{\mathbf{X}}^{n}
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$$
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$$
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= \begin{bmatrix} \phi_{1} & -\frac{\xi^{3}}{2} h_{1} \phi_{1} \mathbf{v}_{2}^{1} & \frac{\xi^{3}}{2} h_{1} \phi_{1} \mathbf{v}_{1}^{1} & \phi_{2} & -\frac{\xi^{3}}{2} h_{2} \phi_{2} \mathbf{v}_{2}^{2} & \frac{\xi^{3}}{2} h_{2} \phi_{2} \mathbf{v}_{1}^{2} & \phi_{3} & -\frac{\xi^{3}}{2} h_{3} \phi_{3} \mathbf{v}_{2}^{3} & \frac{\xi^{3}}{2} h_{3} \phi_{3} \mathbf{v}_{1}^{3} & \phi_{4} & -\frac{\xi^{3}}{2} h_{4} \phi_{4} \mathbf{v}_{2}^{4} & \frac{\xi^{3}}{2} h_{4} \phi_{4} \mathbf{v}_{1}^{4} \end{bmatrix} \begin{bmatrix} \mathbf{\alpha}_{1}^{1} \\ \mathbf{\alpha}_{2}^{1} \\ \mathbf{\alpha}_{1}^{2} \\ \mathbf{\alpha}_{2}^{2} \\ t + \Delta t \mathbf{u}_{3} \\ \mathbf{\alpha}_{3}^{3} \\ \mathbf{\alpha}_{4}^{3} \\ \mathbf{\alpha}_{4}^{4} \\ \mathbf{\alpha}_{4}^{4} \\ \mathbf{\alpha}_{2}^{4} \end{bmatrix}
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$$
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$$
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\begin{split} &\left[\mathcal{S}^{t+\Delta t}_{\phantom{t}0}\mathbf{E}_{c}\right]_{ij} = \frac{1}{2} \left(\frac{\partial^{0}\mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \xi^{j}} + \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \xi^{i}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \xi^{j}} + \frac{\partial \mathcal{S}^{t+\Delta t}\mathbf{u}}{\partial \xi^{i}} \cdot \frac{\partial^{t+\Delta t}\mathbf{u}}{\partial \xi^{j}} + \frac{\partial^{t+\Delta t}\mathbf{u}}{\partial \xi^{j}} \cdot \frac{\partial^{t}\mathbf{X}^{t+\Delta t}\mathbf{u}}{\partial \xi^{j}} \right) \\ &= \frac{1}{2} \begin{pmatrix} \frac{\partial^{0}\mathbf{X}}{\partial \xi^{i}} \cdot \frac{\partial \mathcal{S}\left({}^{t}\mathbf{N}^{t+\Delta t}\mathbf{u}_{n} + {}^{0}\tilde{\mathbf{N}}\Delta^{0}\tilde{\mathbf{X}}_{n}\right)}{\partial \xi^{j}} + \frac{\partial \mathcal{S}\left({}^{t}\mathbf{N}^{t+\Delta t}\mathbf{u}_{n} + {}^{0}\tilde{\mathbf{N}}\Delta^{0}\tilde{\mathbf{X}}_{n}\right)}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}}{\partial \xi^{j}} \cdot \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} \cdot \frac{\partial^{0}\mathbf{X}}{\partial \xi^{j}} \\ &+ \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} \cdot \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} \cdot \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}\mathbf{u}_{n}}{\partial \xi^{j}} + \frac{\partial^{t}\mathbf{X}^{t}
|
||||
$$
|
||||
|
||||
다시 가상일 항으로 돌아와서 위에 구한 Green-Lagrange strain을 대입하면
|
||||
|
||||
$$
|
||||
\begin{split} &\int_{V_{0}} \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{C} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \end{smallmatrix} \right]^{y} dV_{0} + \int_{V_{0}} \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{C} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{C} \end{smallmatrix} \right]^{y} + \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{L} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \end{smallmatrix} \right]^{y} dV_{0} \\ & \text{constant term} \end{split}
|
||||
&\left[ \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{C} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \end{smallmatrix} \right]^{ij} = \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{u}_{n} \right]^{T} \left[ \mathbf{a} \right]_{ij} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right] \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{C} \end{smallmatrix} \right]^{ijkl} \frac{1}{2} \left[ \begin{smallmatrix} t \mathbf{x}_{n} + \mathbf{0} \mathbf{X}_{n} \right]^{T} \left[ \mathbf{e} \right]_{kl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} - \mathbf{0} \mathbf{X}_{n} \right] \right] \\ \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{C} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{C} \end{smallmatrix} \right]^{ij} = \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{u}_{n} \right]^{T} \left[ \mathbf{a} \right]_{ij} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right] \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{C} \end{smallmatrix} \right]^{ijkl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right]^{T} \left[ \mathbf{a} \right]_{kl} \left[ \Delta^{t} \mathbf{u}_{n} \right] \right] \\ \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{E}_{L} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \end{smallmatrix} \right]^{ij} = \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{u}_{n} \right]^{T} \left[ \mathbf{c} \right]_{ij} \left[ \Delta^{t} \mathbf{u}_{n} \right] \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{C} \end{smallmatrix} \right]^{ijkl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} + \mathbf{0} \mathbf{X}_{n} \right]^{T} \left[ \mathbf{e} \right]_{kl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} - \mathbf{0} \mathbf{X}_{n} \right] \right] \\ = \left[ \underbrace{\delta^{t+\Delta t}}_{0} \mathbf{u}_{n} \right]^{T} \left\{ \left( \int_{V_{0}} \left[ \mathbf{a} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \right]^{ij} dV_{0} \right) \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right] \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{C} \right]^{ijkl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right]^{T} \left[ \mathbf{a} \right]_{kl} + \left[ \mathbf{c} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \right]^{ij} dV_{0} \right) \left[ \Delta^{t} \mathbf{u}_{n} \right] \right] \\ + \left( \int_{V_{0}} \left[ \mathbf{a} \right]_{ij} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right] \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{C} \right]^{ijkl} \left[ \begin{smallmatrix} t \mathbf{x}_{n} \right]^{T} \left[ \mathbf{a} \right]_{kl} + \left[ \mathbf{c} \right]_{ij} \left[ \begin{smallmatrix} t+\Delta t \\ 0 \mathbf{S}_{0} \right]^{ij} dV_{0} \right) \left[ \Delta^{t} \mathbf{u}_{n} \right] \right] \right] \end{aligned}
|
||||
$$
|
||||
|
||||
와 같다. 이제 우변의 항들을 정리해보면 아래와 같다. 여기서 body force에 대한 영향은 무시한다.
|
||||
|
||||
$$
|
||||
\begin{split} &\int_{\partial V_{0m}} \mathcal{S}^{t+\Delta t} \mathbf{u}^{t+\Delta t}_{0} \mathbf{F}^{t+\Delta t}_{0} \mathbf{S}^{t+\Delta t} \tilde{\mathbf{n}} dA_{0} = \int_{\partial V_{0m}} \mathcal{S} \left( {}^{t} \mathbf{N}^{t+\Delta t} \mathbf{u}_{n} + {}^{0} \tilde{\mathbf{N}} \Delta^{0} \tilde{\mathbf{X}}_{n} \right)^{t+\Delta t}_{0} \mathbf{F}^{t+\Delta t} \tilde{\mathbf{n}} dA_{0} \\ &= \int_{\partial V_{0m}} \mathcal{S} \left( {}^{t} \mathbf{N}^{t+\Delta t} \mathbf{u}_{n} \right)^{t+\Delta t}_{0} \mathbf{F}^{t+\Delta t}_{0} \mathbf{S}^{t+\Delta t} \tilde{\mathbf{n}} dA_{0} \\ &= \left[ \mathcal{S}^{t+\Delta t} \mathbf{u}_{n} \right]^{T} \int_{\partial V_{0m}} \left[ {}^{t} \mathbf{N} \right]^{Tt+\Delta t}_{0} \mathbf{F}^{t+\Delta t}_{0} \tilde{\mathbf{N}} dA_{0} \\ &= \left[ \mathcal{S}^{t+\Delta t} \mathbf{u}_{n} \right]^{T} \int_{\partial V_{0}} \left[ {}^{t} \mathbf{N} \right]^{T} [\mathbf{t}] dA_{0} \end{split}
|
||||
$$
|
||||
|
||||
따라서 가상변위를 지워 모든 식을 정리하면
|
||||
|
||||
$$
|
||||
\begin{split} &\int_{V_{0}} \delta^{t+\Delta t} \mathbf{u} \cdot \boldsymbol{\rho}_{0}^{t+\Delta t} \ddot{\mathbf{u}} dV_{0} + \int_{V_{0}} \delta^{t+\Delta t}_{0} \mathbf{E} :^{t+\Delta t}_{0} \mathbf{S} dV_{0} = \int_{\partial V_{0m}} \delta^{t+\Delta t} \mathbf{u}^{t+\Delta t}_{0} \mathbf{F}^{t+\Delta t}_{0} \mathbf{S}^{t+\Delta t} \tilde{\mathbf{n}} dA_{0} + \int_{V_{0}} \delta^{t+\Delta t} \mathbf{u} \boldsymbol{\rho}_{0}^{t+\Delta t} \mathbf{f} dV_{0} \\ &\Rightarrow \underbrace{\int_{V_{0}} \boldsymbol{\rho} \begin{bmatrix} {}^{t} \mathbf{N} \end{bmatrix}^{T} \begin{bmatrix} {}^{t} \mathbf{N} \end{bmatrix} J dV_{0}}_{\mathbf{M}} {}^{t+\Delta t} \ddot{\mathbf{u}} + \underbrace{\int_{V_{0}} \left[ \mathbf{a} \right]_{ij} \begin{bmatrix} {}^{t+\Delta t}_{0} \mathbf{S}_{0} \end{bmatrix}^{ij} dV_{0}^{t} \mathbf{x}}_{\mathbf{f}_{int}} \\ &+ \underbrace{\int_{V_{0}} \left[ \mathbf{a} \right]_{ij} \begin{bmatrix} {}^{t} \mathbf{x}_{n} \end{bmatrix}^{T} [{}^{t} \mathbf{x}_{n} \end{bmatrix}^{T} [\mathbf{a}]_{kl} + \left[ \mathbf{c} \right]_{ij} \begin{bmatrix} {}^{t+\Delta t}_{0} \mathbf{S}_{0} \end{bmatrix}^{ij} dV_{0}^{t} \Delta^{t} \mathbf{u} = \underbrace{\int_{\partial V_{0m}} \left[ {}^{t} \mathbf{N} \right]^{T} [\mathbf{t}] dA_{0}}_{\mathbf{P}_{dist}} + \mathbf{P}_{con} \\ &\Rightarrow \mathbf{M}^{t+\Delta t} \ddot{\mathbf{u}} + \mathbf{K}_{t} \Delta^{t} \mathbf{u} = \mathbf{P}_{dist} + \mathbf{P}_{con} - \mathbf{f}_{int} \end{split}
|
||||
$$
|
||||
|
||||
와 같이 정리할 수 있다. 여기서 $\mathbf{M}$ 은 mass matrix, $\mathbf{K}_{\iota}$ 는 tangent stiffness matrix, $\mathbf{P}_{dist}$ 는 분포하중에 의한 힘, $\mathbf{P}_{con}$ 는 집중하중, $\mathbf{f}_{int}$ 는 변형에 의한 힘을 나타낸다.
|
||||
|
||||
## 4. Constitutive matrix
|
||||
|
||||
Plane stress 가정을 사용하는 구성행렬은 다음과 같다.
|
||||
|
||||
$$
|
||||
\begin{bmatrix} {}^{t+\Delta t} \mathbf{C} \end{bmatrix}_{x^1 x^2 x^3} = \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 & 0 & 0 & 0 \\ n & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0
|
||||
$$
|
||||
|
||||
위의 구성 행렬은 local Cartesian coordinate에서 정의되었기 때문에 transformation matrix를 이용하여 natural coordinate로 바꾸어 줄 수 있다.
|
||||
|
||||
$$
|
||||
\begin{bmatrix} t + \Delta t \\ 0 \end{bmatrix}_{\mathcal{E}^1 \mathcal{E}^2 \mathcal{E}^3} = \begin{bmatrix} t + \Delta t \\ 0 \end{bmatrix}^T \begin{bmatrix} t + \Delta t \\ 0 \end{bmatrix}_{x^1 x^2 x^3} \begin{bmatrix} t + \Delta t \\ 0 \end{bmatrix}
|
||||
$$
|
||||
|
||||
이 때 전단보정계수 $\kappa$ 는 $\frac{5}{6}$ 를 사용하였다.
|
||||
|
||||
$$
|
||||
E_{kl} = \tilde{E}_{mn} \underbrace{\left(\mathbf{E}_{k} \cdot \mathbf{G}^{m}\right) \left(\mathbf{E}_{l} \cdot \mathbf{G}^{n}\right)}_{=\mathbf{T}} = \tilde{E}_{mn} \left(\mathbf{E}_{k} \cdot \frac{\partial X^{a}}{\partial \xi^{m}} \mathbf{E}_{a}\right) \left(\mathbf{E}_{l} \cdot \frac{\partial X^{b}}{\partial \xi^{n}} \mathbf{E}_{b}\right) = \frac{\partial X^{k}}{\partial \xi^{m}} \frac{\partial X^{l}}{\partial \xi^{n}}
|
||||
$$
|
||||
|
||||
$$
|
||||
[\mathbf{T}] = \begin{bmatrix} \frac{\partial X^{1}}{\partial \xi^{1}} \frac{\partial X^{1}}{\partial \xi^{1}} & \frac{\partial X^{1}}{\partial \xi^{2}} \frac{\partial X^{1}}{\partial \xi^{2}} & \frac{\partial X^{1}}{\partial \xi^{3}} \frac{\partial X^{1}}{\partial \xi^{3}} & \frac{\partial X^{1}}{\partial \xi^{2}} \frac{\partial X^{1}}{\partial \xi^{3}} & \frac{\partial X^{1}}{\partial \xi^{2}} \frac{\partial X^{1}}{\partial \xi^{3}} & \frac{\partial X^{1}}{\partial \xi^{2}} \frac{\partial X^{1}}{\partial \xi^{2}} \\ \frac{\partial X^{2}}{\partial \xi^{1}} \frac{\partial X^{2}}{\partial \xi^{1}} & \frac{\partial X^{2}}{\partial \xi^{2}} \frac{\partial X^{2}}{\partial \xi^{2}} & \frac{\partial X^{2}}{\partial \xi^{3}} \frac{\partial X^{2}}{\partial \xi^{3}} & \frac{\partial X^{2}}{\partial \xi^{2}} \frac{\partial X^{2}}{\partial \xi^{3}} & \frac{\partial X^{1}}{\partial \xi^{3}} \frac{\partial X^{2}}{\partial \xi^{2}} \\ \frac{\partial X^{3}}{\partial \xi^{1}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \\ \frac{\partial X^{3}}{\partial \xi^{1}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \\ \frac{\partial X^{3}}{\partial \xi^{1}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \\ \frac{\partial X^{3}}{\partial \xi^{1}} \frac{\partial X^{3}}{\partial \xi^{3}} & \frac{\partial X^{3}}{\partial \xi^{2}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial X^{3}}{\partial \xi^{3}} \frac{\partial
|
||||
$$
|
||||
|
||||
## 5. Nonlinear Newmark- \beta integration method
|
||||
|
||||
먼저 물체의 비선형 운동방정식은 다음과 같다.
|
||||
|
||||
$$
|
||||
M\ddot{\mathbf{u}} + \mathbf{C}(\dot{\mathbf{u}})\dot{\mathbf{u}} + \mathbf{K}(\mathbf{u})\mathbf{u} = \mathbf{P}
|
||||
$$
|
||||
|
||||
여기서 밀도는 시간이나 변위에 따라 변화하지 않는다고 가정하면 $\mathbf{M}$ 은 항상 일정하다. 또한 구조물의 동적 문제이기 때문에 $\mathbf{C}$ 는 없다고 생각 할 수 있다. n+1시간에서 평형방정식을 생각하면
|
||||
|
||||
$$
|
||||
\mathbf{M}\ddot{\mathbf{u}}_{n+1} + \mathbf{K}(\mathbf{u}_{n+1})\mathbf{u}_{n+1} = \mathbf{P}_{n+1}
|
||||
$$
|
||||
|
||||
와 같다. 운동방정식이 비선형이기 때문에 n+1시간에서 평형을 만족하는 변위와 가속도를 계산하기 위해서 반복 계산이 필요하다. 따라서 Newton-Raphson method를 사용하여 반복계산을 수행하였다. k+1번째 반복에서 평형이 이루어졌다면 식은 다음과 같다.
|
||||
|
||||
$$
|
||||
\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k+1} + \mathbf{K}(\mathbf{u}_{n+1}^{k+1})\mathbf{u}_{n+1}^{k+1} = \mathbf{P}_{n+1}^{k+1}
|
||||
$$
|
||||
|
||||
위 식을 정리하면
|
||||
|
||||
$$
|
||||
\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k+1} + \mathbf{K}(\mathbf{u}_{n+1}^{k+1})\mathbf{u}_{n+1}^{k+1} - \mathbf{P}_{n+1}^{k+1} = 0 = \mathbf{R}_{n+1}^{k+1}
|
||||
$$
|
||||
|
||||
와 같고 Taylor series expansion을 통해 선형화 시키면
|
||||
|
||||
$$
|
||||
\mathbf{R}_{n+1}^{k+1} = \mathbf{R}_{n+1}^{k} + \frac{\partial \mathbf{R}_{n+1}^{k}}{\partial \mathbf{u}_{n+1}^{k}} \Delta \mathbf{u}_{n+1}^{k} + \frac{\partial \mathbf{R}_{n+1}^{k}}{\partial \dot{\mathbf{u}}_{n+1}^{k}} \Delta \dot{\mathbf{u}}_{n+1}^{k} + \frac{\partial \mathbf{R}_{n+1}^{k}}{\partial \ddot{\mathbf{u}}_{n+1}^{k}} \Delta \ddot{\mathbf{u}}_{n+1}^{k}
|
||||
$$
|
||||
|
||||
와 같다. 이를 풀어 쓰면
|
||||
|
||||
$$
|
||||
0 = \mathbf{P}_{n+1}^{k} + \frac{\partial \mathbf{P}_{n+1}^{k}}{\partial \mathbf{u}_{n+1}^{k}} \Delta \mathbf{u}_{n+1}^{k} - \left\{ \mathbf{M} \ddot{\mathbf{u}}_{n+1}^{k} + \underbrace{\mathbf{K} \left( \mathbf{u}_{n+1}^{k} \right) \mathbf{u}_{n+1}^{k}}_{\mathbf{f}_{int} \left( \mathbf{u}_{n+1}^{k} \right)} \right\} - \left\{ \mathbf{M} \Delta \ddot{\mathbf{u}}_{n+1}^{k} + \underbrace{\frac{\partial \left( \mathbf{K} \left( \mathbf{u}_{n+1}^{k} \right) \mathbf{u}_{n+1}^{k} \right)}{\partial \mathbf{u}_{n+1}^{k}}}_{\mathbf{K}_{t}} \Delta \mathbf{u}_{n+1}^{k} \right\}
|
||||
\Rightarrow \mathbf{M} \Delta \ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{K}_{t} \Delta \mathbf{u}_{n+1}^{k} - \mathbf{P}_{t} \Delta \mathbf{u}_{n+1}^{k} = \mathbf{P}_{n+1}^{k} - \left\{ \mathbf{M} \ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{f}_{int} \left( \mathbf{u}_{n+1}^{k} \right) \right\}
|
||||
$$
|
||||
|
||||
와 같다. Newmark- $\beta$ method를 적용하면 n+1시간에서 변위와 속도를 구할 수 있다.
|
||||
|
||||
$$
|
||||
\mathbf{u}_{n+1} = \mathbf{u}_n + h\dot{\mathbf{u}}_n + h^2\left(\frac{1}{2} - \beta\right)\ddot{\mathbf{u}}_n + h^2\beta\ddot{\mathbf{u}}_{n+1} = \mathbf{u}_n + h\dot{\mathbf{u}}_n + \frac{h^2}{2}\ddot{\mathbf{u}}_n - h^2\beta\ddot{\mathbf{u}}_n + h^2\beta\ddot{\mathbf{u}}_{n+1}
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\dot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_n + h(1 - \gamma)\ddot{\mathbf{u}}_n + h\gamma\ddot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_n + h\ddot{\mathbf{u}}_n + \gamma h\ddot{\mathbf{u}}_{n+1} - \gamma h\ddot{\mathbf{u}}_n
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$$
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위의 식을 가속도와 속도로 나타내면
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$$
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h^{2}\beta\ddot{\mathbf{u}}_{n+1} = \mathbf{u}_{n+1} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
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\dot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1} - \gamma h\ddot{\mathbf{u}}_{n}
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$$
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여기서 마찬가지로 k+1 반복에서 평형을 이룬다면
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$$
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h^{2}\beta\ddot{\mathbf{u}}_{n+1}^{k+1} = \mathbf{u}_{n+1}^{k+1} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
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\dot{\mathbf{u}}_{n+1}^{k+1} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k+1} - \gamma h\ddot{\mathbf{u}}_{n}
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$$
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와 같고 반복에 대한 항을 선형화 시키면
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$$
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h^{2}\beta\ddot{\mathbf{u}}_{n+1}^{k} + h^{2}\beta\Delta\ddot{\mathbf{u}}_{n+1}^{k} = \mathbf{u}_{n+1}^{k} + \Delta\mathbf{u}_{n+1}^{k} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
|
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\dot{\mathbf{u}}_{n+1}^{k} + \Delta\dot{\mathbf{u}}_{n+1}^{k} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k} + \gamma h\Delta\ddot{\mathbf{u}}_{n+1}^{k} - \gamma h\ddot{\mathbf{u}}_{n}
|
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$$
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위 식을 다음과 같이 k 번째 반복의 가속도와 속도, k 번째 반복의 미소 가속도와 미소 속도 항으로 분리 할수 있다.
|
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|
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$$
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\begin{split} \ddot{\mathbf{u}}_{n+1}^{k} &= \frac{1}{h^{2}\beta}\mathbf{u}_{n+1}^{k} - \frac{1}{h^{2}\beta}\mathbf{u}_{n} - \frac{1}{h\beta}\dot{\mathbf{u}}_{n} - \frac{1}{2\beta}\ddot{\mathbf{u}}_{n} + \ddot{\mathbf{u}}_{n} \\ \dot{\mathbf{u}}_{n+1}^{k} &= \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k} - \gamma h\ddot{\mathbf{u}}_{n} = \frac{\gamma}{h\beta}\mathbf{u}_{n+1}^{k} - \frac{\gamma}{h\beta}\mathbf{u}_{n} + \left(1 - \frac{\gamma}{\beta}\right)\dot{\mathbf{u}}_{n} + h\left(1 - \frac{\gamma}{2\beta}\right)\ddot{\mathbf{u}}_{n} \\ \Delta \ddot{\mathbf{u}}_{n+1}^{k} &= \frac{1}{h^{2}\beta}\Delta\mathbf{u}_{n+1}^{k} \\ \Delta \dot{\mathbf{u}}_{n+1}^{k} &= \gamma h\Delta \ddot{\mathbf{u}}_{n+1}^{k} = \frac{\gamma}{h\beta}\Delta\mathbf{u}_{n+1}^{k} \end{split}
|
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$$
|
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|
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위의 미소 가속도, 미소 속도를 대입하면
|
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@@ -0,0 +1,10 @@
|
||||
$$
|
||||
\mathbf{M}\Delta\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{K}_{t}\Delta\mathbf{u}_{n+1}^{k} - \mathbf{P}_{t}\Delta\mathbf{u}_{n+1}^{k} = \mathbf{P}_{n+1}^{k} - \left\{\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{f}_{\text{int}}\left(\mathbf{u}_{n+1}^{k}\right)\right\}
|
||||
\Rightarrow \left[\frac{1}{h^{2}\beta}\mathbf{M} + \mathbf{K}_{t}\left(\mathbf{u}_{n+1}^{k}\right) - \mathbf{P}_{t}\left(\mathbf{u}_{n+1}^{k}\right)\right]\Delta\mathbf{u}_{n+1}^{k} = \underbrace{\mathbf{P}_{n+1}^{k} - \left\{\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{f}_{\text{int}}\left(\mathbf{u}_{n+1}^{k}\right)\right\}}_{\mathbf{R}\left(\mathbf{u}_{n+1}^{k}\right)}
|
||||
$$
|
||||
|
||||
여기서 $\mathbf{f}_{\mathrm{int}}\left(\mathbf{u}_{n+1}^{k}\right)$ 와 $\mathbf{K}_{t}\left(\mathbf{u}_{n+1}^{k}\right)$ 는 $\mathbf{u}_{n+1}^{k}$ 의 함수이기 때문에 반복이 수행될 때마다 다시 계산해 주어야 한다. 이후 다음 반복에 대한 변위, 속도, 가속도는 다음과 같다.
|
||||
|
||||
$$
|
||||
\begin{split} &\mathbf{u}_{n+1}^{k+1} = \mathbf{u}_{n+1}^k + \Delta \mathbf{u}_{n+1}^k \\ &\dot{\mathbf{u}}_{n+1}^{k+1} = \frac{\gamma}{h\beta} \mathbf{u}_{n+1}^{k+1} - \left(\frac{\gamma}{h\beta} \mathbf{u}_n - \left(1 - \frac{\gamma}{\beta}\right) \dot{\mathbf{u}}_n - h \left(1 - \frac{\gamma}{2\beta}\right) \ddot{\mathbf{u}}_n\right) \\ &\ddot{\mathbf{u}}_{n+1}^{k+1} = \frac{1}{h^2\beta} \mathbf{u}_{n+1}^{k+1} - \left(\frac{1}{h^2\beta} \mathbf{u}_n + \frac{1}{h\beta} \dot{\mathbf{u}}_n + \frac{1}{2\beta} \ddot{\mathbf{u}}_n - \ddot{\mathbf{u}}_n\right) \end{split}
|
||||
$$
|
||||
Reference in New Issue
Block a user