# 数学代写|有限元方法代写Finite Element Method代考|MEE356

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## 数学代写|有限元方法代写Finite Element Method代考|Properties of the variational derivative

The following properties of the variational derivative are analogous to differentiation and they are stated without proof. For $F_1=F_1(u)$ and $F_2=F_2(u)$ it can be shown that the following properties of the derivative operator hold for the variational derivative operator as well [3]:
Variation of the sums: $\quad \delta\left(F_1 \pm F_2\right)=\delta F_1 \pm \delta F_2$
Variation of the products: $\quad \delta\left(F_1 F_2\right)=F_2 \delta F_1+F_1 \delta F_2$
Variation of the ratios: $\quad \delta\left(\frac{F_1}{F_2}\right)=\frac{F_2 \delta F_1-F_1 \delta F_2}{F_2^2}$
Variation of the powers: $\quad \delta\left[\left(F_1\right)^n\right]=n\left(F_1\right)^{n-1} \delta F_1$
Commutation with differential operator: $\quad \delta\left(\frac{d u}{d x}\right)=\frac{d}{d x}(\delta u)$ Commutation with integral operator: $\delta \int_0^b u(x) d x=\int_a^b \delta u(x) d x \quad$ (3.40f)
Example $3.3$ :
Linear functional: $l(v)=\int_0^L v f d x+\frac{d v}{d x}(L) M_0$
Bilinear functional: $B(v, w)=\int_0^L a \frac{d v}{d x} \frac{d w}{d x} d x$ where $f(x), M_0, a=a(x)$ are given.

## 数学代写|有限元方法代写Finite Element Method代考|Euler–Lagrange equations and boundary conditions

We indicated that the value of $I(u)$ depends on the chosen varied path $\tilde{u}$. It can be stated that as the chosen path approaches the extremal path, $\tilde{u} \rightarrow u$, the variation of the functional will approach zero, $\delta I \rightarrow 0$. Thus, for the extremal path we have the condition, $\delta I=0$ stated in Eq. (3.31). Let us evaluate the condition $\delta I=0$ by using the definition of the variational derivative given in Eq. (3.33). Note that we will be using many of the properties of the variational derivative operator stated in Eqs. (3.40a)-(3.40f) without explicitly pointing to them.
\begin{aligned} \delta I(u) &=\delta \int_{x_0}^{x_L} F\left(x, u, u_{, x}\right) d x=0 \ &=\int_{x_0}^{x_L} \delta F\left(x, u, u_{, x}\right) d x=\int_{x_0}^{x_L}\left(\frac{\partial F}{\partial u} \delta u+\frac{\partial F}{\partial u_{, x}} \delta u_{, x}\right) d x=0 \end{aligned}
Note that the term $\delta u_{, x}=\delta(\partial u / \partial x)$ or $\delta u_{, x}=\partial(\delta u) / \partial x$. Thus, Eq. (3.41) can be stated as,
$$\delta I(u)=\int_{x_0}^{x_L}\left(\frac{\partial F}{\partial u} \delta u+\frac{\partial F}{\partial u_{, x}} \frac{\partial(\delta u)}{\partial x}\right) d x=0$$
By using integration by parts on the second term of this equation we can obtain an equivalent expression that only has $\delta u$ in the integral,
\begin{aligned} \delta I(u) &=\int_{x_0}^{x_L} \frac{\partial F}{\partial u} \delta u d x+\int_{x_0}^{x_L} \frac{\partial F}{\partial u_x} \frac{\partial(\delta u)}{\partial x} d x \ &=\int_{x_0}^{x_L} \frac{\partial F}{\partial u} \delta u d x+\left[\delta u\left(\frac{\partial F}{\partial u_x}\right)\right]{x_0}^{x_L}-\int{x_0}^{x_L} \delta u \frac{\partial}{\partial x}\left(\frac{\partial F}{\partial u_x}\right) d x=0 \end{aligned}
After rearranging we find the following final form:
$$\delta I(u)=\int_{x_0}^{x_L}\left[\frac{\partial F}{\partial u}-\frac{\partial}{\partial x}\left(\frac{\partial F}{\partial u_u x}\right)\right] \delta u d x+\delta u\left(x_L\right)\left(\frac{\partial F}{\partial u_x}\right){x_L}-\delta u\left(x_0\right)\left(\frac{\partial F}{\partial u{, x}}\right)_{x_0}=0$$

# 有限元方法代考

## 数学代写|有限元方法代写有限元法代考|欧拉-拉格朗日方程和边界条件

\begin{aligned} \delta I(u) &=\delta \int_{x_0}^{x_L} F\left(x, u, u_{, x}\right) d x=0 \ &=\int_{x_0}^{x_L} \delta F\left(x, u, u_{, x}\right) d x=\int_{x_0}^{x_L}\left(\frac{\partial F}{\partial u} \delta u+\frac{\partial F}{\partial u_{, x}} \delta u_{, x}\right) d x=0 \end{aligned}

$$\delta I(u)=\int_{x_0}^{x_L}\left(\frac{\partial F}{\partial u} \delta u+\frac{\partial F}{\partial u_{, x}} \frac{\partial(\delta u)}{\partial x}\right) d x=0$$

\begin{aligned} \delta I(u) &=\int_{x_0}^{x_L} \frac{\partial F}{\partial u} \delta u d x+\int_{x_0}^{x_L} \frac{\partial F}{\partial u_x} \frac{\partial(\delta u)}{\partial x} d x \ &=\int_{x_0}^{x_L} \frac{\partial F}{\partial u} \delta u d x+\left[\delta u\left(\frac{\partial F}{\partial u_x}\right)\right]{x_0}^{x_L}-\int{x_0}^{x_L} \delta u \frac{\partial}{\partial x}\left(\frac{\partial F}{\partial u_x}\right) d x=0 \end{aligned}

$$\delta I(u)=\int_{x_0}^{x_L}\left[\frac{\partial F}{\partial u}-\frac{\partial}{\partial x}\left(\frac{\partial F}{\partial u_u x}\right)\right] \delta u d x+\delta u\left(x_L\right)\left(\frac{\partial F}{\partial u_x}\right){x_L}-\delta u\left(x_0\right)\left(\frac{\partial F}{\partial u{, x}}\right)_{x_0}=0$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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