NINI
11 lines
1.5 KiB
Markdown
11 lines
1.5 KiB
Markdown
$$
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\mathbf{M}\Delta\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{K}_{t}\Delta\mathbf{u}_{n+1}^{k} - \mathbf{P}_{t}\Delta\mathbf{u}_{n+1}^{k} = \mathbf{P}_{n+1}^{k} - \left\{\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{f}_{\text{int}}\left(\mathbf{u}_{n+1}^{k}\right)\right\}
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\Rightarrow \left[\frac{1}{h^{2}\beta}\mathbf{M} + \mathbf{K}_{t}\left(\mathbf{u}_{n+1}^{k}\right) - \mathbf{P}_{t}\left(\mathbf{u}_{n+1}^{k}\right)\right]\Delta\mathbf{u}_{n+1}^{k} = \underbrace{\mathbf{P}_{n+1}^{k} - \left\{\mathbf{M}\ddot{\mathbf{u}}_{n+1}^{k} + \mathbf{f}_{\text{int}}\left(\mathbf{u}_{n+1}^{k}\right)\right\}}_{\mathbf{R}\left(\mathbf{u}_{n+1}^{k}\right)}
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$$
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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.
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$$
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\begin{split} &\mathbf{u}_{n+1}^{k+1} = \mathbf{u}_{n+1}^k + \Delta \mathbf{u}_{n+1}^k \\ &\dot{\mathbf{u}}_{n+1}^{k+1} = \frac{\gamma}{h\beta} \mathbf{u}_{n+1}^{k+1} - \left(\frac{\gamma}{h\beta} \mathbf{u}_n - \left(1 - \frac{\gamma}{\beta}\right) \dot{\mathbf{u}}_n - h \left(1 - \frac{\gamma}{2\beta}\right) \ddot{\mathbf{u}}_n\right) \\ &\ddot{\mathbf{u}}_{n+1}^{k+1} = \frac{1}{h^2\beta} \mathbf{u}_{n+1}^{k+1} - \left(\frac{1}{h^2\beta} \mathbf{u}_n + \frac{1}{h\beta} \dot{\mathbf{u}}_n + \frac{1}{2\beta} \ddot{\mathbf{u}}_n - \ddot{\mathbf{u}}_n\right) \end{split}
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$$
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