### 数学代写|有限元方法代写Finite Element Method代考|MECH ENG 4118

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled methods using space-time

In space-time coupled methods for the whole space-time domain $\bar{\Omega}{x t}=$ $[0, L] \times[0, \tau]$, the computations can be intense and sometimes prohibitive if the final time $\tau$ is large. This problem can be easily overcome by using space-time strip or slab for an increment of time $\Delta t$ and then time-marching to obtain the entire evolution. Consider the space-time domain $$\bar{\Omega}{x t}=\Omega_{x t} \cup \Gamma ; \quad \Gamma=\bigcup_{i=1}^{4} \Gamma_{i}$$
shown in Fig. 1.3. For an increment of time $\Delta t$, that is for $0 \leq t \leq \Delta t$, consider the first space-time strip $\bar{\Omega}{x t}^{(1)}=[0, L] \times[0, \Delta t]$. If we are only interested in the evolution up to time $t=\Delta t$ and not beyond $t=\Delta t$, then the evolution in the space-time domain $[0, L] \times[\Delta t, \tau]$ has not taken place yet, hence does not influence the evolution for $\bar{\Omega}{x t}^{(1)}, t \in[0, \Delta t]$. We also note that for $\bar{\Omega}{x t}^{(1)}$, the boundary at $t=\Delta t$ is open boundary that is similar to the open boundary at $t=\tau$ for the whole space-time domain. We remark that BCs and ICs for $\bar{\Omega}{x t}$ and $\bar{\Omega}{x t}^{(1)}$ are identical in the sense of those that are known and those that are not known. For $\bar{\Omega}{x t}^{(2)}$, the second space-time strip, the BCs are the same as for $\bar{\Omega}{x t}^{(1)}$ but the ICs at $t=\Delta t$ are obtained from the computed evolution for $\bar{\Omega}{x t}^{(1)}$ at $t=\Delta t$. Now, with the known ICs at $t=\Delta t$, the second space-time strip $\bar{\Omega}{x t}^{(2)}$ is exactly similar to the first space-time strip $\bar{\Omega}{x t}^{(1)}$ in terms of BCs, ICs, and open boundary. For $\bar{\Omega}{x t}^{(1)}$, $t=\Delta t$ is open boundary whereas for $\bar{\Omega}{x t}^{(2)}, t=2 \Delta t$ is open boundary. Both open boundaries are at final values of time for the corresponding space-time strips.

In this process the evolution is computed for the first space-time strip $\bar{\Omega}{x t}^{(1)}=[0, L] \times[0, \Delta t]$ and refinements are carried out (in discretization and $p$ levels in the sense of finite element processes) until the evolution for $\bar{\Omega}{x t}^{(1)}$ is a converged solution. Using this converged solution for $\bar{\Omega}{x t}^{(1)}$, ICs are extracted at $t=\Delta t$ for $\bar{\Omega}{x t}^{(2)}$ and a converged evolution is computed for the second space-time strip $\bar{\Omega}_{x t}^{(2)}$. This process is continued until $t=\tau$ is reached.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time decoupled or quasi methods

In space-time decoupled or quasi methods the solution $\phi=\phi(x, t)$ is assumed not to have simultaneous dependence on space coordinate $x$ and time $t$. Referring to the IVP (1.1) in spatial coordinate $x\left(\right.$ i.e. $\left.\mathbb{R}^{1}\right)$ and time $t$, the solution $\phi(x, t)$ is expressed as the product of two functions $g(x)$ and $h(t):$
$$\phi(x, t)=g(x) h(t)$$
where $g(x)$ is a known function that satisfies differentiability, continuity, and the completeness requirements (and others) as dictated by (1.1). We substitute (1.3) in (1.1) and obtain
$$A(g(x) h(t))-f(x, t)=0 \quad \forall x, t \in \Omega_{x t}$$
Integrating (1.4) over $\bar{\Omega}{x}=[0, L]$ while assuming $h(t)$ and its time derivatives to be constant for an instant of time, we can write $$\int{\Omega_{x}}(A(g(x) h(t))-f(x, t)) d x=0$$
Since $g(x)$ is known, the definite integral in (1.5) can be evaluated, thereby eliminating $g(x)$, its spatial derivatives (due to operator $A$ ), and more specifically spatial coordinate $x$ altogether. Hence, (1.5) reduces to
$$A h(t)-\underset{\sim}{f}(t)=0 \quad \forall t \in(0, \tau)$$
in which $A$ is a time differential operator and $f$ is only a function of time. In other words, (1.6) is an ordinary differential equation in time which can now be integrated using explicit or implicit time integration methods or finite element method in time to obtain $h(t) \forall t \in[0, \tau]$. Using this calculated $h(t)$ in (1.3), we now have the solution $\phi(x, t)$ :
$$\phi(x, t)=g(x) h(t) \quad \forall x, t \in \bar{\Omega}_{x t}=[0, L] \times[0, \tau]$$

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled methods using space-time

$$\bar{\Omega} x t=\Omega_{x t} \cup \Gamma ; \quad \Gamma=\bigcup_{i=1}^{4} \Gamma_{i}$$

## 数学代写|有限元方法代写Finite Element Method代考|Space-time decoupled or quasi methods

$(1.1)$ $x\left(\mathbb{E} \mathbb{R}^{1}\right)$ 和时间 $t$ ，解决方案 $\phi(x, t)$ 表示为两个函数的乘积 $g(x)$ 和 $h(t)$ :
$$\phi(x, t)=g(x) h(t)$$

$$A(g(x) h(t))-f(x, t)=0 \quad \forall x, t \in \Omega_{x t}$$

$$\int \Omega_{x}(A(g(x) h(t))-f(x, t)) d x=0$$

$$A h(t)-\underset{\sim}{f}(t)=0 \quad \forall t \in(0, \tau)$$

$$\phi(x, t)=g(x) h(t) \quad \forall x, t \in \bar{\Omega}_{x t}=[0, L] \times[0, \tau]$$

## 有限元方法代写

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