FESADev/docs/Paper/mitc공부/mitc공부_003.md

$$
\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.

$$
\mathbf{u}_{n+1} = \mathbf{u}_n + h\dot{\mathbf{u}}_n + h^2\left(\frac{1}{2} - \beta\right)\ddot{\mathbf{u}}_n + h^2\beta\ddot{\mathbf{u}}_{n+1} = \mathbf{u}_n + h\dot{\mathbf{u}}_n + \frac{h^2}{2}\ddot{\mathbf{u}}_n - h^2\beta\ddot{\mathbf{u}}_n + h^2\beta\ddot{\mathbf{u}}_{n+1}
\dot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_n + h(1 - \gamma)\ddot{\mathbf{u}}_n + h\gamma\ddot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_n + h\ddot{\mathbf{u}}_n + \gamma h\ddot{\mathbf{u}}_{n+1} - \gamma h\ddot{\mathbf{u}}_n
$$

위의 식을 가속도와 속도로 나타내면

$$
h^{2}\beta\ddot{\mathbf{u}}_{n+1} = \mathbf{u}_{n+1} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
\dot{\mathbf{u}}_{n+1} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1} - \gamma h\ddot{\mathbf{u}}_{n}
$$

여기서 마찬가지로 k+1 반복에서 평형을 이룬다면

$$
h^{2}\beta\ddot{\mathbf{u}}_{n+1}^{k+1} = \mathbf{u}_{n+1}^{k+1} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
\dot{\mathbf{u}}_{n+1}^{k+1} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k+1} - \gamma h\ddot{\mathbf{u}}_{n}
$$

와 같고 반복에 대한 항을 선형화 시키면

$$
h^{2}\beta\ddot{\mathbf{u}}_{n+1}^{k} + h^{2}\beta\Delta\ddot{\mathbf{u}}_{n+1}^{k} = \mathbf{u}_{n+1}^{k} + \Delta\mathbf{u}_{n+1}^{k} - \mathbf{u}_{n} - h\dot{\mathbf{u}}_{n} - \frac{h^{2}}{2}\ddot{\mathbf{u}}_{n} + h^{2}\beta\ddot{\mathbf{u}}_{n}
\dot{\mathbf{u}}_{n+1}^{k} + \Delta\dot{\mathbf{u}}_{n+1}^{k} = \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k} + \gamma h\Delta\ddot{\mathbf{u}}_{n+1}^{k} - \gamma h\ddot{\mathbf{u}}_{n}
$$

위 식을 다음과 같이 k 번째 반복의 가속도와 속도, k 번째 반복의 미소 가속도와 미소 속도 항으로 분리 할수 있다.

$$
\begin{split} \ddot{\mathbf{u}}_{n+1}^{k} &= \frac{1}{h^{2}\beta}\mathbf{u}_{n+1}^{k} - \frac{1}{h^{2}\beta}\mathbf{u}_{n} - \frac{1}{h\beta}\dot{\mathbf{u}}_{n} - \frac{1}{2\beta}\ddot{\mathbf{u}}_{n} + \ddot{\mathbf{u}}_{n} \\ \dot{\mathbf{u}}_{n+1}^{k} &= \dot{\mathbf{u}}_{n} + h\ddot{\mathbf{u}}_{n} + \gamma h\ddot{\mathbf{u}}_{n+1}^{k} - \gamma h\ddot{\mathbf{u}}_{n} = \frac{\gamma}{h\beta}\mathbf{u}_{n+1}^{k} - \frac{\gamma}{h\beta}\mathbf{u}_{n} + \left(1 - \frac{\gamma}{\beta}\right)\dot{\mathbf{u}}_{n} + h\left(1 - \frac{\gamma}{2\beta}\right)\ddot{\mathbf{u}}_{n} \\ \Delta \ddot{\mathbf{u}}_{n+1}^{k} &= \frac{1}{h^{2}\beta}\Delta\mathbf{u}_{n+1}^{k} \\ \Delta \dot{\mathbf{u}}_{n+1}^{k} &= \gamma h\Delta \ddot{\mathbf{u}}_{n+1}^{k} = \frac{\gamma}{h\beta}\Delta\mathbf{u}_{n+1}^{k} \end{split}
$$

위의 미소 가속도, 미소 속도를 대입하면