### 数学代写|计算方法代写computational method代考|Introduction to the finite element method

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
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• (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|计算方法代写computational method代考|Introduction to the finite element method

This book covers the fundamentals of the finite element method in the context of numerical simulation with specific reference to the simulation of the response of structural and mechanical components to mechanical and thermal loads.

We begin with the question: what is the meaning of the term “simulation”? By its dictionary definition, simulation is the imitative representation of the functioning of one system or process by means of the functioning of another. For instance, the membrane analogy introduced by Prandtl ${ }^{1}$ in 1903 made it possible to find the shearing stresses in bars of arbitrary cross-section, loaded by a twisting moment, through mapping the deflected shape of a thin elastic membrane. In other words, the distribution and magnitude of shearing stress in a twisted bar can be simulated by the deflected shape of an elastic membrane.

The membrane analogy exists because two unrelated phenomena can be modeled by the same partial differential equation. The physical meaning associated with the coefficients of the differential equation depends on which problem is being solved. However, the solution of one is proportional to the solution of the other: At corresponding points the shearing stress in a bar, subjected to a twisting moment, is oriented in the direction of the tangent to the contour lines of a deflected thin membrane and its magnitude is proportional to the slope of the membrane. Furthermore, the volume enclosed by the deflected membrane is proportional to the twisting moment.

In the pre-computer years the membrane analogy provided practical means for estimating shearing stresses in prismatic bars. This involved cutting the shape of the cross-section out of sheet metal or a wood panel, covering the hole with a thin elastic membrane, applying pressure to the membrane and mapping the contours of the deflected membrane. In present-day practice both problems would be formulated as mathematical problems which would then be solved by a numerical method, most likely by the finite element method.

There are many other useful analogies. For example, the same differential equations simulate the response of assemblies of mechanical components, such as linear spring-mass-viscous damper systems and assemblies of electrical components, such as capacitors, inductors and resistors. This has been exploited by the use of analogue computers. Obviously, it is much easier to build and manipulate electrical circuitry than mechanical assemblies. In present-day practice both simulation problems would be formulated as mathematical problems which would be solved by a numerical method.

At the heart of simulation of aspects of physical reality is a mathematical problem cast in a generalized form ${ }^{2}$. The solution of the mathematical problem is approximated by a numerical method,

such as the finite element method, which is the subject of this book. The quantities of interest (QoI) are extracted from the approximate solution. The errors of approximation in the QoI depend on how the mathematical problem was discretized ${ }^{3}$ and how the QoI were extracted from the numerical solution. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst.
Estimation and control of numerical errors are fundamentally important in numerical simulation. Consider, for example, the problem of design certification. Design rules are typically stated in the form
$$F_{\max } \leq F_{\text {all }}$$
where $F_{\max }>0$ (resp. $F_{\text {all }}>0$ ) is the maximum (resp. allowable) value of a quantity of interest, for example the first principal stress. Since in numerical simulation only an approximation to $F_{\max }$ is available, denoted by $F_{\text {num }}$, it is necessary to know the size of the numerical error $\tau$ :
$$\left|F_{\max }-F_{\text {num }}\right| \leq \tau F_{\max }{ }^{\circ}$$
In design and design certification the worst case scenario has to be considered, which is underestimation of $F_{\max }$, that is,
$$F_{\text {num }}=(1-t) F_{\max } .$$
Therefore it has to be shown that
$$F_{\text {num }} \leq(1-\tau) F_{\text {all. }} .$$

## 数学代写|计算方法代写computational method代考|An introductory problem

We introduce the finite element method through approximating the exact solution of the following second order ordinary differential equation
$$-\left(\kappa u^{\prime}\right)^{\prime}+c u=f \quad \text { on the closed interval } \bar{I}=[0 \leq x \leq \ell]$$
with the boundary conditions
$$u(0)=u(\ell)=0$$
where the prime indicates differentiation with respect to $x$. It is assumed that $0<\alpha \leq \kappa(x) \leq \beta<\infty$ where $\alpha$ and $\beta$ are real numbers, $\kappa^{\prime}<\infty$ on $\bar{I}, c \geq 0$ and $f=f(x)$ are defined such that the indicated operations are meaningful on $I$. For example, the indicated operations would not be meaningful if $\left(\kappa u^{\prime}\right)^{\prime}, c$ or $f$ would not be finite in one or more points on the interval $0 \leq x \leq \ell$. The function $f$ is called a forcing function.
We seek an approximation to $u$ in the form:
$$u_{n}=\sum_{j=1}^{n} a_{j} \varphi_{j}(x), \quad \varphi_{j}(0)=\varphi_{j}(\ell)=0 \text { for all } j$$
where $\varphi_{j}(x)$ are fixed functions, called basis functions, and $a_{j}$ are the coefficients of the basis functions to be determined. Note that the basis functions satisfy the zero boundary conditions.

Let us find $a_{j}$ such that the integral $I$ defined by
$$I=\frac{1}{2} \int_{0}^{\ell}\left(\kappa\left(u^{\prime}-u_{n}^{\prime}\right)^{2}+c\left(u-u_{n}\right)^{2}\right) d x$$
is minimum. While there are other plausible criteria for selecting $a_{j}$, we will see that this criterion is fundamentally important in the finite element method. Differentiating $I$ with respect to $a_{i}$ and letting the derivative equal to zero, we have:
$$\frac{d I}{d a_{i}}=\int_{0}^{\ell}\left(\kappa\left(u^{\prime}-u_{n}^{\prime}\right) \varphi_{i}^{\prime}+c\left(u-u_{n}\right) \varphi_{i}\right) d x=0, \quad i=1,2, \ldots, n .$$
Using the product rule: $\left(\kappa u^{\prime} \varphi_{i}\right)^{\prime}=\left(\kappa u^{\prime}\right)^{\prime} \varphi_{i}+\kappa u^{\prime} \varphi_{i}^{\prime}$ we write
\begin{aligned} \int_{0}^{\ell} \kappa u^{\prime} \varphi_{i}^{\prime} d x &=\int_{0}^{\ell}\left(\left(\kappa u^{\prime} \varphi_{i}\right)^{\prime}-\left(\kappa u^{\prime}\right)^{\prime} \varphi_{i}\right) d x \ &=\underbrace{\left(\kappa u^{\prime} \varphi_{i}\right){x=\ell}}{=0}-\underbrace{\left(\kappa u^{\prime} \varphi_{i}\right){x=0}}{=0}-\int_{0}^{\ell}\left(\kappa u^{\prime}\right)^{\prime} \varphi_{i} d x \end{aligned}
The underbraced terms vanish on account of the boundary conditions, see eq. (1.7). On substituting this expression into eq. (1.9), we get
$$\int_{0}^{t} \underbrace{\left(-\left(\kappa u^{\prime}\right)^{\prime}+c u\right)}{=f(x)} \varphi{i} d x-\int_{0}^{\ell}\left(\kappa u_{n}^{\prime} \varphi_{i}^{\prime}+c u_{n} \varphi_{i}\right) d x=0$$
which will be written as
$$\int_{0}^{t}\left(\kappa u_{n}^{\prime} \varphi_{i}^{\prime}+c u_{n} \varphi_{i}\right) d x=\int_{0}^{t} f \varphi_{i} d x, \quad i=1,2, \ldots, n$$
We define
$$k_{i j}=\int_{0}^{\ell} \kappa \varphi_{i}^{\prime} \varphi_{j}^{\prime} d x, \quad m_{i j}=\int_{0}^{\ell} c \varphi_{i} \varphi_{j} d x, \quad r_{i}=\int_{0}^{\ell} f \varphi_{i} d x$$
and write eq. (1.11) in the following form
$$\sum_{j=1}^{n}\left(k_{i j}+m_{i j}\right) a_{j}=r_{i}, \quad i=1,2, \ldots, n$$
which represents $n$ simultaneous equations in $n$ unknowns. It is usually written in matrix form:
$$([K]+[M]){a}={r} .$$
On solving these equations we find an approximation $u_{n}$ to the exact solution $u$ in the sense that $u_{n}$ minimizes the integral $I$.

## 数学代写|计算方法代写computational method代考|The exact solution

If eq. (1.5) holds then for an arbitrary function $v=v(x)$, subject only to the restriction that all of the operations indicated in the following are properly defined, we have
$$\int_{0}^{t}\left(\left(-\kappa u^{\prime}\right)^{\prime}+c u-f\right) v d x=0 .$$
Using the product rule; $\left(\kappa u^{\prime} v\right)^{\prime}=\left(\kappa u^{\prime}\right)^{\prime} v+\kappa u^{\prime} v^{\prime}$ we get
$$\int_{0}^{\ell}\left(-\kappa u^{\prime}\right)^{\prime} v d x=-\left(\kappa u^{\prime} v\right){x=\ell}+\left(\kappa u^{\prime} v\right){x=0}+\int_{0}^{\ell} \kappa u^{\prime} v^{\prime} d x$$
therefore eq. (1.17) is transformed to:
$$\int_{0}^{\ell}\left(\kappa u^{\prime} v^{\prime}+c u v\right) d x=\int_{0}^{\ell} f v d x+\left(\kappa u^{\prime} v\right){x=\ell}-\left(\kappa u^{\prime} v\right){x=0} .$$

We introduce the following notation:
$$B(u, v) \stackrel{\operatorname{def}}{=} \int_{0}^{\ell}\left(\kappa u^{\prime} v^{\prime}+c u v\right) d x$$
where $B(u, v)$ is a bilinear form. A bilinear form has the property that it is linear with respect to each of its two arguments. The properties of bilinear forms are listed Section A.1.3 of Appendix A. We define the linear form:
$$F(v) \stackrel{\text { def }}{=} \int_{0}^{t} f v d x+\left(\kappa u^{\prime} v\right){x=\ell}-\left(\kappa u^{\prime} v\right){x=0^{-}}$$
The forcing function $f(x)$ may be a sum of forcing functions: $f(x)=f_{1}(x)+f_{2}(x)+\ldots$, some or all of which may be the Dirac delta function ${ }^{4}$ multiplied by a constant. For example if $f_{k}(x)=F_{0} \delta\left(x_{0}\right)$ then
$$\int_{0}^{\ell} f_{k}(x) v d x=\int_{0}^{\ell} F_{0} \delta\left(x_{0}\right) v d x=F_{0} v\left(x_{0}\right) .$$
The properties of linear forms are listed in Section A.1.2. Note that $F_{0} v\left(x_{0}\right)$ in eq. (1.21) is a linear form only if $v$ is continuous and bounded.

The definitions of $B(u, v)$ and $F(v)$ are modified depending on the boundary conditions. Before proceeding further we need the following definitions.

## 数学代写|计算方法代写computational method代考|Introduction to the finite element method

F最大限度≤F全部

|F最大限度−F在一个 |≤τF最大限度∘

F在一个 =(1−吨)F最大限度.

F在一个 ≤(1−τ)F全部。 .

## 数学代写|计算方法代写computational method代考|An introductory problem

−(ķ在′)′+C在=F 在闭区间 一世¯=[0≤X≤ℓ]

d一世d一种一世=∫0ℓ(ķ(在′−在n′)披一世′+C(在−在n)披一世)dX=0,一世=1,2,…,n.

\begin{aligned} \int_{0}^{\ell} \kappa u^{\prime} \varphi_{i}^{\prime} dx &=\int_{0}^{\ell} \left(\left(\kappa u^{\prime} \varphi_{i}\right)^{\prime}-\left(\kappa u^{\prime}\right)^{\prime} \varphi_{ i}\right) dx \ &=\underbrace{\left(\kappa u^{\prime} \varphi_{i}\right) {x=\ell}} {=0}-\underbrace{\left(\ kappa u^{\prime} \varphi_{i}\right) {x=0}} {=0}-\int_{0}^{\ell}\left(\kappa u^{\prime}\right) ^{\prime} \varphi_{i} dx \end{对齐} 吨H和在nd和rbr一种C和d吨和r米s在一种n一世sH这n一种CC这在n吨这F吨H和b这在nd一种r是C这nd一世吨一世这ns,s和和和q.(1.7).这ns在bs吨一世吨在吨一世nG吨H一世s和Xpr和ss一世这n一世n吨这和q.(1.9),在和G和吨 \int_{0}^{t} \underbrace{\left(-\left(\kappa u^{\prime}\right)^{\prime}+cu\right)} {=f(x)} \varphi {i} d x-\int_{0}^{\ell}\left(\kappa u_{n}^{\prime} \varphi_{i}^{\prime}+c u_{n} \varphi_{i }\right) dx=0 在H一世CH在一世llb和在r一世吨吨和n一种s \int_{0}^{t}\left(\kappa u_{n}^{\prime} \varphi_{i}^{\prime}+c u_{n} \varphi_{i}\right) dx=\ int_{0}^{t} f \varphi_{i} dx, \quad i=1,2, \ldots, n 在和d和F一世n和 k_{ij}=\int_{0}^{\ell} \kappa \varphi_{i}^{\prime} \varphi_{j}^{\prime} dx, \quad m_{ij}=\int_{0 }^{\ell} c \varphi_{i} \varphi_{j} dx, \quad r_{i}=\int_{0}^{\ell} f \varphi_{i} dx 一种nd在r一世吨和和q.(1.11)一世n吨H和F这ll这在一世nGF这r米 \sum_{j=1}^{n}\left(k_{ij}+m_{ij}\right) a_{j}=r_{i}, \quad i=1,2, \ldots, n 在H一世CHr和pr和s和n吨sns一世米在l吨一种n和这在s和q在一种吨一世这ns一世nn在nķn这在ns.一世吨一世s在s在一种ll是在r一世吨吨和n一世n米一种吨r一世XF这r米: ([K]+[M]){a}={r} 。

## 数学代写|计算方法代写computational method代考|The exact solution

∫0吨((−ķ在′)′+C在−F)在dX=0.

∫0ℓ(−ķ在′)′在dX=−(ķ在′在)X=ℓ+(ķ在′在)X=0+∫0ℓķ在′在′dX

∫0ℓ(ķ在′在′+C在在)dX=∫0ℓF在dX+(ķ在′在)X=ℓ−(ķ在′在)X=0.

$$F(v) \stackrel{\text { def }}{=} \int_{0}^{t} fvd x+ \left(\kappa u^{\prime} v\right) {x=\ell}-\left(\kappa u^{\prime} v\right) {x=0^{-}} 吨H和F这rC一世nGF在nC吨一世这nF(X)米一种是b和一种s在米这FF这rC一世nGF在nC吨一世这ns:F(X)=F1(X)+F2(X)+…,s这米和这r一种ll这F在H一世CH米一种是b和吨H和D一世r一种Cd和l吨一种F在nC吨一世这n4米在l吨一世pl一世和db是一种C这ns吨一种n吨.F这r和X一种米pl和一世FFķ(X)=F0d(X0)吨H和n \int_{0}^{\ell} f_{k}(x) vdx=\int_{0}^{\ell} F_{0} \delta\left(x_{0}\right) vdx=F_{0 } v\left(x_{0}\right) 。$$

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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。