数学代写|有限元方法代写Finite Element Method代考|CIVE602

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数学代写|有限元方法代写Finite Element Method代考|Equations of structural dynamics

Consider matrix equations of the form
$$\mathrm{Mu}+\mathrm{Cu}+\mathrm{Ku}=\mathrm{F}$$
subjected to initial conditions
$$\mathbf{u}(0)=\mathbf{u}_0, \quad \dot{\mathbf{u}}(0)=\mathbf{v}_0$$
Such equations arise in structural dynamics, where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, and $\mathbf{K}$ is the stiffness matrix. The damping matrix $\mathbf{C}$ is often taken to be a linear combination of the mass and stiffness matrices, $\mathbf{C}=$ $c_1 \mathbf{M}+c_2 \mathbf{K}$, where $c_1$ and $c_2$ are determined from physical experiments. In the present study, we will not consider damping (i.e., $\mathbf{C}=0$ ) in the numerical examples, although the theoretical developments will account for it. Transient analysis of both bars and beams lead to equations of the type given in Eqs. (7.4.32a) and (7.4.32b). The mass and stiffness matrices for bars and beams can be found in Eqs. (7.3.40), (7.3.57), (7.3.63a), and (7.3.63b). The eigenvalue problem associated with Eq. (7.4.32a) (with $\mathbf{C}=0)$ is
$$(-\lambda \mathbf{M}+\mathbf{K}) \mathbf{u}_0=\mathbf{Q}_0, \quad \lambda=\omega^2$$
There are several numerical methods available to approximate the secondorder time derivatives and convert differential equations in time to algebraic equations (see Surana and Reddy [4] for different order RungeKutta methods, the Newmark family of methods, Wilson’s $\theta$ method, and Houbolt’s method). Recently, Kim and Reddy [5-8] have developed a number of timeapproximation schemes based on weighted-residual and leastsquares concepts [similar to the Galerkin scheme discussed in Eqs. (7.4.19)(7.4.22d)]. In the interest of simplicity and wide use, we only consider the Newmark family of time approximations and the central difference scheme.

数学代写|有限元方法代写Finite Element Method代考|Fully discretized equations

Let us consider the following $(\alpha, \gamma)$-family of approximation, where the function and its first time derivative are approximated as [following the truncated Taylor’s series notation of Eq. (7.4.15)]
\begin{aligned} & \mathbf{u}^{s+1} \approx \mathbf{u}^s+\Delta t \dot{\mathbf{u}}^s+\frac{1}{2}(\Delta t)^2\left[(1-\gamma) \ddot{\mathbf{u}}^s+\gamma \ddot{\mathbf{u}}^{s+1}\right] \ & \dot{\mathbf{u}}^{s+1} \approx \dot{\mathbf{u}}^s+a_2 \ddot{\mathbf{u}}^s+a_1 \ddot{\mathbf{u}}^{s+1} \end{aligned}
Here $\alpha$ and $\gamma$ are parameters that determine the stability and accuracy of the scheme. Equations (7.4.33) and (7.4.34) are Taylor’s series expansions of $\mathbf{u}^{s+1}$ and $\dot{\mathbf{u}}^{\mathrm{s}+1}$, respectively, about $t=t_{\mathrm{s}}$.
The fully discretized form of Eq. (7.4.32a) is obtained using the approximations introduced in Eqs. (7.4.33) and (7.4.34). First, we eliminate $\ddot{\mathbf{u}}^{\mathrm{s+1}}$ from Eqs. (7.4.33) and (7.4.34) and write the result for $\dot{\mathbf{u}}^{\text {s+1:}}$

$$\begin{gathered} \dot{\mathbf{u}}^{s+1}=a_6\left(\mathbf{u}^{s+1}-\mathbf{u}^s\right)-a_7 \dot{\mathbf{u}}^s-a_8 \ddot{\mathbf{u}}^s \ a_6=\frac{2 \alpha}{\gamma \Delta t}, \quad a_7=\frac{2 \alpha}{\gamma}-1, \quad a_8=\left(\frac{\alpha}{\gamma}-1\right) \Delta t \end{gathered}$$
Now pre-multiplying Eq. (7.4.33) with $\mathbf{M}$ and substituting for $\mathbf{M} \ddot{\mathbf{u}}^{\mathbf{s + 1}}$ from Eq. (7.4.32a), we obtain
$$\left(\mathbf{M}+\frac{\gamma(\Delta t)^2}{2} \mathbf{K}\right) \mathbf{u}^{s+1}=\mathbf{M} \mathbf{b}^s+\frac{\gamma(\Delta t)^2}{2} \mathbf{F}^{s+1}-\frac{\gamma(\Delta t)^2}{2} \mathbf{C} \dot{\mathbf{u}}^{s+1}$$
where
$$\mathbf{b}^s=\mathbf{u}^s+\Delta t \dot{\mathbf{u}}^s+\frac{1}{2}(1-\gamma)(\Delta t)^2 \ddot{\mathbf{u}}^s$$
Now, multiplying throughout with $2 /\left[\gamma(\Delta t)^2\right]$ we arrive at
$$\left(\frac{2}{\gamma(\Delta t)^2} \mathbf{M}+\mathbf{K}\right) \mathbf{u}^{s+1}=\frac{2}{\gamma(\Delta t)^2} \mathbf{M} \mathbf{b}_s+\mathbf{F}^{s+1}-\mathbf{C} \dot{\mathbf{u}}^{s+1}$$
Using Eq. (7.4.35a) for $\dot{\mathbf{u}}^{\mathrm{s}+1}$ in Eq. (7.4.36c) and collecting terms, we obtain the recursive relation:
$$\hat{\mathbf{K}} \mathbf{u}^{s+1}=\hat{\mathbf{F}}^{s, s+1}$$
where
$$\begin{gathered} \hat{\mathbf{K}}=\mathbf{K}+a_3 \mathbf{M}+a_6 \mathbf{C}, \hat{\mathbf{F}}^{s, s+1}=\mathbf{F}^{s+1}+\mathbf{M} \overline{\mathbf{u}}^s+\mathbf{C} \hat{\mathbf{u}}^s \ \overline{\mathbf{u}}^s=a_3 \mathbf{u}^s+a_4 \dot{\mathbf{u}}^s+a_5 \ddot{\mathbf{u}}^s, \hat{\mathbf{u}}^s=a_6 \mathbf{u}^s+a_7 \dot{\mathbf{u}}^s+a_8 \ddot{\mathbf{u}}^s \ a_3=\frac{2}{\gamma(\Delta t)^2}, a_4=a_3 \Delta t, \quad a_5=\frac{1}{\gamma}-1 \end{gathered}$$

数学代写|有限元方法代写Finite Element Method代考|Equations of structural dynamics

$$\mathrm{Mu}+\mathrm{Cu}+\mathrm{Ku}=\mathrm{F}$$

$$\mathbf{u}(0)=\mathbf{u}_0, \quad \dot{\mathbf{u}}(0)=\mathbf{v}_0$$

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