有限元方法代写

分类：有限元方法代写

数学代写|有限元方法代写Finite Element Method代考|GENG5514

Posted on 2023年8月31日2023年8月31日 by statistics-lab

如果你也在怎样代写有限元方法finite differences method 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。有限元方法finite differences method领域中所有的物理系统都可以用边界/初值问题来表示。有限元法属于变分法的一般范畴。

有限元方法finite differences method是一类通过近似有限差分导数来求解微分方程的数值技术。空间域和时间间隔(如果适用)都被离散化，或者被分解成有限数量的步骤，并且这些离散点的解的值通过求解包含有限差分和邻近点的值的代数方程来近似。

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数学代写|有限元方法代写Finite Element Method代考|GENG5514

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

In case the material is elastically isotropic and the stress and strain tensors are symmetric the material behavior can be characterized with two material constants,
E: Elastic modulus or Young’s modulus
v: Poisson’s ratio
For a three-dimensional problem, it can be shown that the following relationships exist between the stresses and strains,
$$
\begin{aligned}
\varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \
\varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \
\varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \
\tau_{x y} & =G \gamma_{x y} \
\tau_{y z} & =G \gamma_{y z} \
\tau_{z x} & =G \gamma_{z x}
\end{aligned}
$$
where shear modulus $G=E / 2(1+v)$.

Note that Eq. (2.61a) can be inverted and expressed as follows:
$$
\begin{aligned}
\sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \
\sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \
\sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \
\tau_{x y} & =\mu \gamma_{x y} \
\tau_{y z} & =\mu \gamma_{y z} \
\tau_{z x} & =\mu \gamma_{z x}
\end{aligned}
$$
where, the Lamé constants are defined as follows:
$$
\begin{aligned}
& \lambda=\frac{v E}{(1+v)(1-2 v)} \
& \mu=G
\end{aligned}
$$

数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

Thermal stress in a one-dimensional problem: Consider a long and slender bar of length $L$ and initial temperature $T^{(0)}$. If the temperature of the bar is changed by $\Delta T$, material points in the bar would experience thermal strain proportional to the temperature change,
$$
\varepsilon^{(t h)}=\alpha \Delta T
$$
the proportionality constant $\alpha$ is a material property known as the coefficient of thermal expansion with units of $\mathrm{K}^{-1}$ or $\left({ }^{\circ} \mathrm{C}\right)^{-1}$. If the bar is not constrained on its ends, its length will change by an amount,
$$
\Delta L=\int_0^L \alpha \Delta T d x
$$
but no internal stress will develop.

On the other hand if both ends of the bar are constrained, internal forces and hence stress will develop in the bar. If such constraint conditions exist, the thermal stress in the bar can be found from Hooke’s law as follows:
$$
\sigma^{(t h)}=E \alpha \Delta T
$$
Next, consider a constrained bar subjected to external forces and change of temperature. The total strain in this bar can be found by using the superposition of the mechanical component of the strain and the thermal strain,
$$
\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T
$$
The inverse of this relation gives the corresponding total stress,
$$
\sigma=E(\varepsilon-\alpha \Delta T)
$$
Generalized stress-strain relations with thermal effects: For materials with isotropic material properties temperature change only causes normal strain in the material. The stress-strain relations for a three-dimensional isotropic material subjected to a temperature change $\Delta T$ are expressed as follows [8]:
$$
\begin{aligned}
\varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \
\varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \
\varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \
\gamma_{x y} & =\frac{\tau_{x y}}{G} \
\gamma_{y z} & =\frac{\tau_{y z}}{G} \
\gamma_{z x} & =\frac{\tau_{z x}}{G}
\end{aligned}
$$

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

如果材料是弹性各向同性的，并且应力和应变张量是对称的，则材料的行为可以用两个材料常数来表征:
E:弹性模量或杨氏模量
v:泊松比
对于三维问题，可以证明应力和应变之间存在如下关系:
$$
\begin{aligned}
\varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \
\varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \
\varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \
\tau_{x y} & =G \gamma_{x y} \
\tau_{y z} & =G \gamma_{y z} \
\tau_{z x} & =G \gamma_{z x}
\end{aligned}
$$
式中剪切模量$G=E / 2(1+v)$。

注意，Eq. (2.61a)可以反向表示为:
$$
\begin{aligned}
\sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \
\sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \
\sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \
\tau_{x y} & =\mu \gamma_{x y} \
\tau_{y z} & =\mu \gamma_{y z} \
\tau_{z x} & =\mu \gamma_{z x}
\end{aligned}
$$
其中，lam常数定义如下:
$$
\begin{aligned}
& \lambda=\frac{v E}{(1+v)(1-2 v)} \
& \mu=G
\end{aligned}
$$

数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

一维问题中的热应力:考虑一根长度为$L$，初始温度为$T^{(0)}$的细长杆。如果杆的温度变化$\Delta T$，则杆中的材料点将经历与温度变化成正比的热应变。
$$
\varepsilon^{(t h)}=\alpha \Delta T
$$
比例常数$\alpha$是一种材料性质，称为热膨胀系数，单位为$\mathrm{K}^{-1}$或$\left({ }^{\circ} \mathrm{C}\right)^{-1}$。如果棒材的两端不受约束，它的长度会发生一定的变化，
$$
\Delta L=\int_0^L \alpha \Delta T d x
$$
但不会产生内应力。

另一方面，如果杆的两端都受到约束，则内力和应力将在杆中产生。在此约束条件存在的情况下，由胡克定律可求出棒材内的热应力:
$$
\sigma^{(t h)}=E \alpha \Delta T
$$
接下来，考虑受外力和温度变化作用的约束杆。该杆的总应变可通过应变的机械分量和热应变的叠加得到。
$$
\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T
$$
这个关系的倒数给出了相应的总应力，
$$
\sigma=E(\varepsilon-\alpha \Delta T)
$$
具有热效应的广义应力-应变关系:对于具有各向同性材料性质的材料，温度变化只会引起材料的正常应变。三维各向同性材料在温度变化$\Delta T$下的应力-应变关系表示为[8]:
$$
\begin{aligned}
\varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \
\varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \
\varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \
\gamma_{x y} & =\frac{\tau_{x y}}{G} \
\gamma_{y z} & =\frac{\tau_{y z}}{G} \
\gamma_{z x} & =\frac{\tau_{z x}}{G}
\end{aligned}
$$

数学代写|有限元方法代写Finite Element Method代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|ENGR7961

Posted on 2023年8月31日2023年8月31日 by statistics-lab

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

An elastic deformation should not cause holes in a deformable body that does not have any holes before deformation. Moreover, no material overlap should be predicted by the displacement field. The strain compatibility conditions ensure that these constraints are satisfied [7].

In a planar deformation, where $u_x=u_x(x, y), u_y=u_y(x, y)$ and $u_z=0$, consider the following combination of the strains,
$$
\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}
$$

Using the definitions given in Eq. (2.47) we will find,
$$
\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0
$$
Thus we see that the relationship (a) must be equal to zero. This is the strain compatibility equation for a two-dimensional deformation in the $x, y$ plane, which imposes a specific relationship between the strains and the strain-displacement relationships.

For three-dimensional deformations where $u_x=u_x(x, y, z), u_y=u_y(x, y, z)$ and $u_z=u_z(x, y, z)$ there are a total of six strain compatibility conditions. These can be found as follows:
$$
\begin{aligned}
& \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \
& \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \
& \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \
& 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \
& 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \
& 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right)
\end{aligned}
$$

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

In previous sections it was indicated that, in general, the stress and strain tensors at a point have nine independent components each, if we do not take into account the symmetries. Therefore, the possibility exists for all of these 18 components to be interrelated. In it most general form, the linear elastic constitutive law, also known as generalized Hooke’s law, can be expressed as follows:
$$
\sigma_{i j}=c_{i j r s} \varepsilon_{r s}
$$
where the subscripts $i, j, r, s=x, y, z$ and the coefficients $c_{i j r s}$ are empirically determined. Note that the tensor notation is used in expressing Eq. (2.57) where $\sigma$ and $\varepsilon$ are second order tensors and $c_{i j r s}$ is a fourth order tensor [7]. Repeated indices imply summation, such that for $\sigma_{x x}$ the most general form of the Hooke’s law would be,
$$
\begin{aligned}
\sigma_{x x}= & c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x x z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \
& +c_{x x z y} \gamma_{z y}+c_{x x z z} \varepsilon_{z z}
\end{aligned}
$$

It can easily be deduced that 81 material properties would be required in case of an anisotropic material with no-symmetries in the strain and stress tensors. In matrix notation, Eq. (2.57) can be expressed as follows:
$$
{\sigma}=[E]{\varepsilon}
$$
where $[E]$ is an $81 \times 81$ elasticity matrix and ${\sigma}$ and ${\varepsilon}$ are $9 \times 1$ vectors.

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

在变形前没有孔洞的可变形体中，弹性变形不应造成孔洞。此外，位移场不应预测材料重叠。应变相容性条件保证了这些约束条件的满足[7]。

在平面变形中，$u_x=u_x(x, y), u_y=u_y(x, y)$和$u_z=0$，考虑以下应变组合:
$$
\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}
$$

利用式(2.47)给出的定义，我们会发现，
$$
\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0
$$
因此，我们看到关系(a)必须等于零。这是$x, y$平面上二维变形的应变兼容方程，它规定了应变与应变-位移关系之间的特定关系。

对于三维变形，$u_x=u_x(x, y, z), u_y=u_y(x, y, z)$和$u_z=u_z(x, y, z)$共有六种应变兼容条件。这些可以找到如下:
$$
\begin{aligned}
& \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \
& \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \
& \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \
& 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \
& 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \
& 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right)
\end{aligned}
$$

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

在前面的章节中指出，一般来说，如果我们不考虑对称性，一点上的应力和应变张量各有9个独立分量。因此，这18个成分存在相互关联的可能性。在最一般的形式下，线弹性本构律，也称为广义胡克定律，可以表示为:
$$
\sigma_{i j}=c_{i j r s} \varepsilon_{r s}
$$
其中下标$i, j, r, s=x, y, z$和系数$c_{i j r s}$是经验确定的。注意，在表达式Eq.(2.57)中使用了张量表示法，其中$\sigma$和$\varepsilon$是二阶张量，$c_{i j r s}$是四阶张量[7]。重复的指标意味着求和，对于$\sigma_{x x}$，胡克定律的最一般形式是，
$$
\begin{aligned}
\sigma_{x x}= & c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x x z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \
& +c_{x x z y} \gamma_{z y}+c_{x x z z} \varepsilon_{z z}
\end{aligned}
$$

可以很容易地推断出，在应变张量和应力张量不对称的各向异性材料中，需要81种材料性能。在矩阵表示法中，Eq.(2.57)可以表示为:
$$
{\sigma}=[E]{\varepsilon}
$$
其中$[E]$为$81 \times 81$弹性矩阵，${\sigma}$和${\varepsilon}$为$9 \times 1$向量。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|MECH3300

Posted on 2023年8月31日2023年8月31日 by statistics-lab

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

As stated in the introduction to Section 2.2, when a deformable body is subjected to external effects such as external forces and/or imposed displacements on its boundary, its shape will change and internal forces will develop throughout its volume. The level of deformation for given external effects depends on the material of the deformable body. Constitutive relations are empirically obtained, material specific relationships between the stress and the strain in the body. Here we are primarily interested in linear elastic relationships.

The deformation behavior of a specific material is determined experimentally. These experiments are designed such that only one of the stress components and the corresponding strain dominates the problem. This state is known as a simpleloading state.

For linear, isotropic materials tensile loading of a slender test specimen, i.e., the simple-tension test, reveals two fundamental material properties. The relationship between the normal stress and the normal strain is found by conducting a simple-tension test, as follows:
$$
\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z
$$
where $E$ is the elastic modulus of the material, also referred to as the Young’s modulus. The relationship between the longitudinal strain $\varepsilon_l$ and the transverse strain $\varepsilon_t$ represents the Poisson’s ratio, the second material property,
$$
v=-\frac{\varepsilon_t}{\varepsilon_l}
$$
The simple-shear test reveals the relationship between the shear strain and the shear stress,
$$
\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j
$$
where $G$ is the shear modulus, or modulus of rigidity. For a linear, elastic, isotropic material the following relationship holds:
$$
G=\frac{E}{2(1+v)}
$$

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

在变形前没有孔洞的可变形体中，弹性变形不应造成孔洞。此外，位移场不应预测材料重叠。应变相容性条件保证了这些约束条件的满足[7]。

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

如第2.2节介绍中所述，当一个可变形的物体受到外力和/或施加在其边界上的位移等外部影响时，其形状将发生变化，内力将在其整个体积中发展。给定外力作用的变形程度取决于可变形物体的材料。本构关系是经验得到的，是材料中应力和应变之间的具体关系。这里我们主要对线性弹性关系感兴趣。

特定材料的变形行为是通过实验确定的。这些实验是这样设计的，只有一个应力分量和相应的应变占主导地位。这种状态被称为简单加载状态。

对于线性、各向同性材料，细长试件的拉伸加载，即简单拉伸试验，揭示了材料的两个基本特性。通过简单拉伸试验得到法向应力与法向应变的关系如下:
$$
\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z
$$
其中$E$为材料的弹性模量，也称为杨氏模量。纵向应变$\varepsilon_l$与横向应变$\varepsilon_t$之间的关系表示泊松比，即材料的第二大特性;
$$
v=-\frac{\varepsilon_t}{\varepsilon_l}
$$
单剪试验揭示了剪切应变与剪应力之间的关系;
$$
\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j
$$
其中$G$为剪切模量，或刚度模量。对于线性的、弹性的、各向同性的材料，下列关系成立:
$$
G=\frac{E}{2(1+v)}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|EG55M1

Posted on 2023年7月31日2023年8月25日 by statistics-lab

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

An examination of the weak form in Eq. (9.2.10) and the finite element matrices in Eq. (9.2.19b) shows that $\psi_i^e$ should be at least linear functions of $x$ and $y$. The complete linear polynomial in $x$ and $y$ in $\Omega_e$ is of the form
$$
u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y
$$
where $c_i^e$ are constants. The set ${1, x, y}$ is linearly independent and complete. Equation (9.2.21) defines a unique plane for fixed $c_i^e$. Thus, if $u(x, y)$ is a curved surface, $u_h^e(x, y)$ approximates the surface by a plane (see Fig. 9.2.2). In particular, $u_h^e(x, y)$ is uniquely defined on a triangle by the three nodal values of $u_h^e(x, y)$; three nodes are placed at the vertices of the triangle so that the geometry of the triangle is uniquely defined, and the nodes are numbered in counterclockwise direction, as shown in Fig. 9.2.2, so that the unit normal always points upward from the domain. Let
$$
u_h^e\left(x_1, y_1\right)=u_1^e, \quad u_h^e\left(x_2, y_2\right)=u_2^e, \quad u_h^e\left(x_3, y_3\right)=u_3^e
$$
where $\left(x_i, y_i\right)$ denote the coordinates of the $i$ th vertex of the triangle. Note that the triangle is uniquely defined by the three pairs of coordinates $\left(x_i, y_i\right)$.
The three constants $c_i^e(i=1,2,3)$ in Eq. (9.2.21) can be expressed in terms of three nodal values $u_i^e(i=1,2,3)$. Thus, the polynomial in Eq. (9.2.21) is associated with a triangular element and there are three nodes identified, namely, the vertices of the triangle. Equations in (9.2.22) have the explicit form
$$
\begin{aligned}
& u_1 \equiv u_h\left(x_1, y_1\right)=c_1+c_2 x_1+c_3 y_1 \
& u_2 \equiv u_h\left(x_2, y_2\right)=c_1+c_2 x_2+c_3 y_2 \
& u_3 \equiv u_h\left(x_3, y_3\right)=c_1+c_2 x_3+c_3 y_3
\end{aligned}
$$
where the element label $e$ is omitted for simplicity. In matrix form, we have
$$
\left{\begin{array}{l}
u_1 \
u_2 \
u_3
\end{array}\right}=\left[\begin{array}{lll}
1 & x_1 & y_1 \
1 & x_2 & y_2 \
1 & x_3 & y_3
\end{array}\right]\left{\begin{array}{l}
c_1 \
c_2 \
c_3
\end{array}\right} \text { or } \mathbf{u}=\mathbf{A c}
$$

数学代写|有限元方法代写Finite Element Method代考|Linear rectangular element

Next, consider the complete polynomial
$$
u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y+c_4^e x y
$$
which contains four linearly independent terms, and is linear in $x$ and $y$, with a bilinear term in $x$ and $y$. This polynomial requires an element with four nodes. There are two possible geometric shapes: a triangle with the fourth node at the center (or centroid) of the triangle, or a rectangle with the nodes at the vertices. A triangle with a fourth node at the center does not provide a single-valued variation of $u$ at interelement boundaries, resulting in incompatible variation of $u$ at interelement boundaries, and is therefore not admissible (see Fig. 9.2.7). The linear rectangular element is a compatible element because on any side $u_h^e$ varies only linearly and there are two nodes to uniquely define it. Here we consider an approximation of the form Eq. (9.2.27) and use a rectangular element with sides $a$ and $b$ [see Fig. 9.2.8(a)]. For the sake of convenience, we choose a local coordinate system $(\bar{x}, \bar{y})$ to derive the interpolation functions. We assume that (element label is omitted)

$$
u_h(\bar{x}, \bar{y})=c_1+c_2 \bar{x}+c_3 \bar{y}+c_4 \bar{x} \bar{y}
$$
and require
$$
\begin{aligned}
& u_1=u_h(0,0)=c_1 \
& u_2=u_h(a, 0)=c_1+c_2 a \
& u_3=u_h(a, b)=c_1+c_2 a+c_3 b+c_4 a b \
& u_4=u_h(0, b)=c_1+c_3 b
\end{aligned}
$$

有限元方法代考

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

对式(9.2.10)中的弱形式和式(9.2.19b)中的有限元矩阵的检验表明，$\psi_i^e$至少应该是$x$和$y$的线性函数。$\Omega_e$中$x$和$y$的完全线性多项式的形式为
$$
u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y
$$
其中$c_i^e$是常数。集合${1, x, y}$是线性无关且完备的。式(9.2.21)定义了固定$c_i^e$的唯一平面。因此，如果$u(x, y)$是一个曲面，则$u_h^e(x, y)$近似为一个平面(见图9.2.2)。特别地，$u_h^e(x, y)$在三角形上由$u_h^e(x, y)$的三个节点值唯一地定义;在三角形的顶点处放置三个节点，以唯一定义三角形的几何形状，并按逆时针方向编号，如图9.2.2所示，使单位法线始终指向域的上方。让
$$
u_h^e\left(x_1, y_1\right)=u_1^e, \quad u_h^e\left(x_2, y_2\right)=u_2^e, \quad u_h^e\left(x_3, y_3\right)=u_3^e
$$
其中$\left(x_i, y_i\right)$表示三角形的第$i$顶点的坐标。注意，三角形是由三对坐标$\left(x_i, y_i\right)$唯一定义的。
式(9.2.21)中的三个常数$c_i^e(i=1,2,3)$可以用三个节点值$u_i^e(i=1,2,3)$表示。因此，将Eq.(9.2.21)中的多项式与一个三角形元素联系起来，并确定了三个节点，即三角形的顶点。式(9.2.22)中的方程有显式
$$
\begin{aligned}
& u_1 \equiv u_h\left(x_1, y_1\right)=c_1+c_2 x_1+c_3 y_1 \
& u_2 \equiv u_h\left(x_2, y_2\right)=c_1+c_2 x_2+c_3 y_2 \
& u_3 \equiv u_h\left(x_3, y_3\right)=c_1+c_2 x_3+c_3 y_3
\end{aligned}
$$
其中为简单起见省略了元素标签$e$。在矩阵形式中，我们有
$$
\left{\begin{array}{l}
u_1 \
u_2 \
u_3
\end{array}\right}=\left[\begin{array}{lll}
1 & x_1 & y_1 \
1 & x_2 & y_2 \
1 & x_3 & y_3
\end{array}\right]\left{\begin{array}{l}
c_1 \
c_2 \
c_3
\end{array}\right} \text { or } \mathbf{u}=\mathbf{A c}
$$

数学代写|有限元方法代写Finite Element Method代考|Linear rectangular element

接下来，考虑完全多项式
$$
u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y+c_4^e x y
$$
它包含四个线性无关的项，在$x$和$y$中是线性的，在$x$和$y$中有一个双线性项。这个多项式需要一个有四个节点的元素。有两种可能的几何形状:第四个节点位于三角形中心(或质心)的三角形，或节点位于顶点的矩形。中心有第四个节点的三角形在元素间边界处不能提供$u$的单值变化，导致元素间边界处$u$的变化不相容，因此不允许(见图9.2.7)。线性矩形元素是一个兼容的元素，因为在任何一边$u_h^e$都是线性变化的，并且有两个节点可以唯一地定义它。这里我们考虑形式为Eq.(9.2.27)的近似，并使用边长为$a$和$b$的矩形单元[见图9.2.8(a)]。为方便起见，我们选择一个局部坐标系$(\bar{x}, \bar{y})$来推导插值函数。我们假设(元素标签被省略)

$$
u_h(\bar{x}, \bar{y})=c_1+c_2 \bar{x}+c_3 \bar{y}+c_4 \bar{x} \bar{y}
$$
并且要求
$$
\begin{aligned}
& u_1=u_h(0,0)=c_1 \
& u_2=u_h(a, 0)=c_1+c_2 a \
& u_3=u_h(a, b)=c_1+c_2 a+c_3 b+c_4 a b \
& u_4=u_h(0, b)=c_1+c_3 b
\end{aligned}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|AMCS329

Posted on 2023年7月31日2023年8月25日 by statistics-lab

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

In two dimensions there is more than one simple geometric shape that can be used as a finite element (see Fig. 9.2.1). As we shall see shortly, the interpolation functions depend not only on the number of nodes in the element and the number of unknowns per node, but also on the shape of the element. The shape of the element must be such that its geometry is uniquely defined by a set of points, which serve as the element nodes in the development of the interpolation functions. As will be discussed later in this section, a triangle is the simplest geometric shape, followed by a rectangle.

The representation of a given region by a set of elements (i.e., discretization or mesh generation) is an important step in finite element analysis. The choice of element type, number of elements, and density of elements depends on the geometry of the domain, the problem to be analyzed, and the degree of accuracy desired. Of course, there are no specific formulae to obtain this information. In general, the analyst is guided by his or her technical background, insight into the physics of the problem being modeled (e.g., a qualitative understanding of the solution), and experience with finite element modeling. The general rules of mesh generation for finite element formulations include:

Select elements that characterize the governing equations of the problem.
The number, shape, and type (i.e., linear or quadratic) of elements should be such that the geometry of the domain is represented as accurately as desired.
The density of elements should be such that regions of large gradients of the solution are adequately modeled (i.e., use more elements or higher-order elements in regions of large gradients).
Mesh refinements should vary gradually from high-density regions to low-density regions. If transition elements are used, they should be used away from critical regions (i.e., regions of large gradients). Transition elements are those which connect lower-order elements to higher-order elements (e.g., linear to quadratic).

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

In the development of the weak form we need only consider a typical element. We assume that $\Omega_e$ is a typical element, whether triangular or quadrilateral, of the finite element mesh, and we develop the finite element model of Eq. (9.2.1) over $\Omega_e$. Various two-dimensional elements will be discussed in the sequel.

Following the three-step procedure presented in Chapters 2 and 3, we develop the weak form of Eq. (9.2.1) over the typical element $\Omega_e$. The first step is to multiply Eq. (9.2.1) with a weight function $w$, which is assumed to be differentiable once with respect to $x$ and $y$, and then integrate the equation over the element domain $\Omega_e$ :
$$
0=\int_{\Omega_{\varepsilon}} w\left[-\frac{\partial}{\partial x}\left(F_1\right)-\frac{\partial}{\partial y}\left(F_2\right)+a_{00} u-f\right] d x d y
$$
where
$$
F_1=a_{11} \frac{\partial u}{\partial x}+a_{12} \frac{\partial u}{\partial y}, \quad F_2=a_{21} \frac{\partial u}{\partial x}+a_{22} \frac{\partial u}{\partial y}
$$
In the second step we distribute the differentiation among $u$ and $w$ equally. To achieve this we integrate the first two terms in (9.2.4a) by parts. First we note the identities
$$
\begin{aligned}
& \frac{\partial}{\partial x}\left(w F_1\right)=\frac{\partial w}{\partial x} F_1+w \frac{\partial F_1}{\partial x} \quad \text { or } \quad-w \frac{\partial F_1}{\partial x}=\frac{\partial w}{\partial x} F_1-\frac{\partial}{\partial x}\left(w F_1\right) \
& \frac{\partial}{\partial y}\left(w F_2\right)=\frac{\partial w}{\partial y} F_2+w \frac{\partial F_2}{\partial y} \quad \text { or } \quad-w \frac{\partial F_2}{\partial y}=\frac{\partial w}{\partial y} F_2-\frac{\partial}{\partial y}\left(w F_2\right)
\end{aligned}
$$

有限元方法代考

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

在二维空间中，不止一种简单的几何形状可以用作有限元(见图9.2.1)。我们很快就会看到，插值函数不仅取决于元素中的节点数和每个节点的未知数数，还取决于元素的形状。元素的形状必须由一组点唯一地定义，这些点在插值函数的开发中充当元素节点。正如本节后面将要讨论的，三角形是最简单的几何形状，其次是矩形。

用一组单元表示给定区域(即离散化或网格生成)是有限元分析中的重要步骤。元素类型、元素数量和元素密度的选择取决于域的几何形状、要分析的问题和所需的精度程度。当然，没有特定的公式来获得这些信息。一般来说，分析人员是由他或她的技术背景、对正在建模的问题的物理特性的洞察(例如，对解决方案的定性理解)以及有限元素建模的经验来指导的。有限元公式网格生成的一般规则包括:

选择表征问题控制方程的元素。

元素的数量、形状和类型(即线性或二次型)应该使域的几何形状像期望的那样精确地表示出来。

元素的密度应该使溶液的大梯度区域得到充分的建模(即，在大梯度区域使用更多的元素或高阶元素)。

网格细化应该从高密度区域逐渐变化到低密度区域。如果使用过渡元素，它们应该远离关键区域(即大梯度区域)。过渡元素是那些连接低阶元素到高阶元素的元素(例如，线性到二次元)。

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

在弱形式的发展中，我们只需要考虑一个典型的元素。我们假设$\Omega_e$是有限元网格的典型单元，无论是三角形还是四边形，并且我们在$\Omega_e$上开发了Eq.(9.2.1)的有限元模型。各种二维元素将在续集中讨论。

按照第2章和第3章中提出的三步步骤，我们在典型元素$\Omega_e$上推导出方程(9.2.1)的弱形式。第一步是将Eq.(9.2.1)与权函数$w$相乘，假设权函数对$x$和$y$可微一次，然后在元素域$\Omega_e$上对方程积分:
$$
0=\int_{\Omega_{\varepsilon}} w\left[-\frac{\partial}{\partial x}\left(F_1\right)-\frac{\partial}{\partial y}\left(F_2\right)+a_{00} u-f\right] d x d y
$$
在哪里
$$
F_1=a_{11} \frac{\partial u}{\partial x}+a_{12} \frac{\partial u}{\partial y}, \quad F_2=a_{21} \frac{\partial u}{\partial x}+a_{22} \frac{\partial u}{\partial y}
$$
在第二步中，我们将微分均匀地分布在$u$和$w$之间。为了达到这个目的，我们将(9.2.4a)中的前两项按部分积分。首先我们注意到恒等式
$$
\begin{aligned}
& \frac{\partial}{\partial x}\left(w F_1\right)=\frac{\partial w}{\partial x} F_1+w \frac{\partial F_1}{\partial x} \quad \text { or } \quad-w \frac{\partial F_1}{\partial x}=\frac{\partial w}{\partial x} F_1-\frac{\partial}{\partial x}\left(w F_1\right) \
& \frac{\partial}{\partial y}\left(w F_2\right)=\frac{\partial w}{\partial y} F_2+w \frac{\partial F_2}{\partial y} \quad \text { or } \quad-w \frac{\partial F_2}{\partial y}=\frac{\partial w}{\partial y} F_2-\frac{\partial}{\partial y}\left(w F_2\right)
\end{aligned}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|ME672

Posted on 2023年7月31日2023年8月25日 by statistics-lab

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

A typical finite element program consists of three basic units (see Fig. 8.3.1):

Preprocessor
Processor
Postprocessor
In the preprocessor part of the program, the input data of the problem are read in and/or generated. This includes the geometry (e.g., length of the domain and boundary conditions), the data of the problem (e.g., coefficients in the differential equation), finite element mesh information (e.g., element type, number of elements, element length, coordinates of the nodes, and connectivity matrix), and indicators for various options (e.g., print, no print, type of field problem analyzed, static analysis, eigenvalue analysis, transient analysis, and degree of interpolation).

In the processor part, all steps of the finite element analysis discussed in the preceding chapters, except for postprocessing, are performed. The major steps of the processor are:

Generation of the element matrices using numerical integration.
Assembly of element equations.
Imposition of the boundary conditions.
Solution of the algebraic equations for the nodal values of the primary variables.

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

The preprocessor unit consists of reading input data and generating finite element mesh, and printing the data and mesh information. The input data to a finite element program consist of element type, IELEM (i.e., Lagrange element or Hermite element), number of elements in the mesh (NEM), specified boundary conditions on primary and secondary variables (number of boundary conditions, global node number and degree of freedom, and specified values of the degrees of freedom), the global coordinates of global nodes, and element properties [e.g., coefficients $a(x), b(x), c(x), f(x)$, etc.] If a uniform mesh is used, the length of the domain should be read in, and global coordinates of the nodes can be generated in the program.

The preprocessor portion that deals with the generation of finite element mesh information (when not supplied by the user) can be separated into a subroutine (MESH1D), depending on the convenience and complexity of the program. Mesh generation includes computation of the global coordinates $X_I$ and the connectivity array NOD $\left(=B_{i j}\right)$. Recall that the connectivity matrix describes the relationship between element nodes to global nodes:
$\operatorname{NOD}(n, j)=$ Global node number corresponding to the $j$ th (local) node of element $n$
This array is used in the assembly procedure as well as to transfer information from element to the global system and vice versa. For example, to extract the vector ELX of global coordinates of element nodes from the vector GLX of global coordinates of global nodes, we can use the matrix NOD as follows. The global coordinate $x_i^{(n)}$ of the $i$ th node of the $n$th element is the same as the global coordinate $X_I$ of the global node $I$, where $I=$ $\operatorname{NOD}(n, i)$ :
$$
\left{x_i^{(n)}\right}=\left{X_l\right}, \quad I=\operatorname{NOD}(n, i) \rightarrow \operatorname{ELX}(i)=\operatorname{GLX}(\operatorname{NOD}(n, i))
$$

有限元方法代考

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

典型的有限元程序由三个基本单元组成(如图8.3.1所示):

预处理机

处理机

后处理程序
在程序的预处理器部分，问题的输入数据被读入和/或生成。这包括几何(例如，域的长度和边界条件)，问题的数据(例如，微分方程中的系数)，有限元网格信息(例如，元素类型，元素数量，元素长度，节点坐标和连通性矩阵)，以及各种选项的指标(例如，打印，不打印，分析的现场问题类型，静态分析，特征值分析，瞬态分析和插值程度)。

在处理器部分，除后处理外，执行前面章节中讨论的有限元分析的所有步骤。处理器的主要步骤是:

用数值积分法生成元素矩阵。

单元方程的装配。

施加边界条件。

求解主变量节点值的代数方程。

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

预处理器单元包括读取输入数据、生成有限元网格、打印数据和网格信息。有限元程序的输入数据包括单元类型、IELEM(即拉格朗日单元或Hermite单元)、网格中单元数(NEM)、主、次变量的指定边界条件(边界条件个数、全局节点数和自由度、自由度的指定值)、全局节点的全局坐标和单元属性[如系数$a(x), b(x), c(x), f(x)$等]。读入域的长度，在程序中生成节点的全局坐标。

根据程序的便利性和复杂性，处理生成有限元网格信息(当用户不提供时)的预处理器部分可以分离成子程序(MESH1D)。网格生成包括全局坐标$X_I$和连通性数组NOD $\left(=B_{i j}\right)$的计算。回想一下，连接性矩阵描述了元素节点与全局节点之间的关系:
$\operatorname{NOD}(n, j)=$元素$n$的第$j$个(本地)节点对应的全局节点号
该数组用于组装过程以及将信息从元素传递到全局系统，反之亦然。例如，要从全局节点的全局坐标的向量GLX中提取元素节点的全局坐标的向量ELX，我们可以使用如下的矩阵NOD。$n$元素的$i$第th节点的全局坐标$x_i^{(n)}$与全局节点$I$的全局坐标$X_I$相同，其中$I=$$\operatorname{NOD}(n, i)$:
$$
\left{x_i^{(n)}\right}=\left{X_l\right}, \quad I=\operatorname{NOD}(n, i) \rightarrow \operatorname{ELX}(i)=\operatorname{GLX}(\operatorname{NOD}(n, i))
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|AMCS329

Posted on 2023年7月17日2023年8月24日 by statistics-lab

如果你也在怎样代写有限元方法finite differences method 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。有限元方法finite differences method领域中所有的物理系统都可以用边界/初值问题来表示。在这本书中，我们主要集中在解决一维和二维线性弹性和传热问题。将这些解决方案技术扩展到三维分析是直接的，因此为了保持表示的清晰性并避免重复，这里不进行讨论。

有限元方法finite differences method提供了一种系统的方法来推导简单子区域的近似函数，通过这些近似函数可以表示几何上复杂的区域。在有限元法中，近似函数是分段多项式(即只在子区域上定义的多项式，称为单元)。

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

Numerical evaluation of integrals, called numerical integration or numerical quadrature, involves approximation of the integrand by a polynomial of sufficient degree, because the integral of a polynomial can be evaluated exactly. The integrals are generally expressed in terms of the coordinate appearing in the problem description (like $x$ or $r$ ). We shall call $x$ and $r$ as the problem coordinates or global coordinates. Since the finite element approximation (or interpolation) functions are derived over an element, it is convenient to use a local (i.e., element) coordinate, such as $\bar{x}$ used earlier.
For example, consider the integral,
$$
I_e=\int_{x_a^e}^{x_b^e} F_e(x) d x=\int_0^{h_e} F^e(\bar{x}) d \bar{x}
$$
where $F^e$ is a function of $x=x_a^e+\bar{x}$, and $h_e=x_b^e-x_a^e$. We approximate the function $F^e(\bar{x})$ in $\Omega^e$ by a polynomial
$$
F^e(\bar{x}) \approx \sum_{I=1}^N F_I^e \psi_I^e(\bar{x})
$$
where $F_I^e$ denotes the value of $F^e(\bar{x})$ at the $I$ th point $\bar{x}_I$ of the interval $\left[0, h_e\right]$ and $\psi_I^e(\bar{x})$ are polynomials of degree $N-1$ in the interval. The representation can be viewed as the finite element interpolation of $F^e(\bar{x})$, where $F_I^e$ is the value of the function at the Ith node. The interpolation functions $\psi_I^e(\bar{x})$ can be of the Lagrange type or the Hermite type.

Substitution of Eq. (8.2.2) into Eq. (8.2.1) and evaluation of the integral give an approximate value of the integral $I_e$. For example, suppose that we choose linear interpolation of $F^e(\bar{x})$, as shown in Fig. 8.2.1. For $N=2$, we have
$$
\psi_1^e=1-\frac{\bar{x}}{h_e}, \quad \psi_2^e=\frac{\bar{x}}{h_e}, \quad \int_0^{h_e} \psi_I^e(\bar{x}) d \bar{x}=\frac{h_e}{2} \quad(I=1,2)
$$
and
$$
I_e=\frac{h_e}{2}\left(F_1^e+F_2^e\right), \quad F_1^e=F^e(0), \quad F_2^e=F^e\left(h_e\right)
$$
Thus, the value of the integral is given by the area of a trapezoid used to approximate the area under the function $F^e(\bar{x})$ (see Fig. 8.2.1). Equation (8.2.3b) is known as the trapezoidal rule of numerical integration.

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

Of all the quadrature formulae, the Gauss-Legendre quadrature is the most commonly used. The details of the method itself will be discussed shortly. The quadrature formula requires the integral to be cast as one to be evaluated over the interval (element) $\hat{\bar{\Omega}}^e \equiv[-1,1}$ This requires the transformation of the problem coordinate $x$ (i.e., the coordinate used to describe the governing equation) to the element coordinate $\xi$ such that the integrals are posed on [-1, 1] (see Fig. 8.2.2). Thus, the relationship between $x$ and $\xi$ is linear and it can be established with the help of the conditions
when $x=x_a^e, \quad \xi=-1 ;$ and when $x=x_b^e, \quad \xi=1$
This transformation between $x$ and $\xi$ can be represented by the linear “stretch” mapping
$$
x=a^e+b^e \xi \text { for } x \text { in } \bar{\Omega}^e=\left[x_a^e, x_b^e\right]
$$
where $a^e$ and $b^e$ are constants to be determined such that conditions in Eq. (8.2.6a) hold:
$$
x_a^e=a^e+b^e \cdot(-1), \quad x_b^e=a^e+b^e .
$$
Solving for $a^e$ and $b^e$, we obtain
$$
b^e=\frac{1}{2}\left(x_b^e-x_a^e\right)=\frac{1}{2} h_e, \quad a^e=\frac{1}{2}\left(x_b^e+x_a^e\right)=x_a^e+\frac{1}{2} h_e
$$
Hence the transformation between $x$ and $\xi$ becomes
$$
x(\xi)=\frac{1}{2}\left(x_b^e+x_a^e\right)+\frac{1}{2}\left(x_b^e-x_a^e\right) \xi=x_a^e \frac{1}{2}(1-\xi) x_b^e \frac{1}{2}(1+\xi)=x_a^e+\frac{1}{2} h_e(1+\xi)
$$
where $x_a^e$ and $x_b^e$ denote the global coordinates of the left end and the right end, respectively, of the element $\bar{\Omega}^e$ and $h_e$ is the element length (see Fig. 8.2.2).
The local coordinate $\xi$ is called the normal coordinate or natural coordinate, and its values always lie between -1 and 1 , with its origin at the center of the 1-D element. The local coordinate $\xi$ is useful in two ways: it is (1) convenient in constructing the interpolation functions and (2) required in the numerical integration based on the Gauss-Legendre quadrature.
The derivation of the Lagrange family of interpolation functions in terms of the natural coordinate $\xi$ is made easy by the following interpolation property of the approximation functions:
$$
\psi_i^e\left(\xi_j\right)= \begin{cases}1 & \text { if } i=j \ 0 & \text { if } i \neq j\end{cases}
$$
where $\xi_j$ is the $\xi$-coordinate of the $j$ th node in the element. For an element with $n$ nodes, the Lagrange interpolation functions $\psi_i^e(i=1,2, \ldots, n)$ are polynomials of degree $n-1$. To construct $\psi_i^e$ satisfying the properties in Eq. (8.2.8), we proceed as follows. For each $\psi_i^e$, we form the product of $n-1$ linear functions $\xi-\xi_j(j=1,2, \ldots, i-1, i+1, \ldots, n$ and $j \neq i)$ :
$$
\psi_i^e=c_i\left(\xi-\xi_1\right)\left(\xi-\xi_2\right) \cdots\left(\xi-\xi_{i-1}\right)\left(\xi-\xi_{i+1}\right) \cdots\left(\xi-\xi_n\right)
$$
Note that $\psi_i^e$ is zero at all nodes except the $i$ th. Next we determine the constant $c_i$ such that $\psi_i^e=1$ at $\xi-\xi_i$ :
$$
c_i=\left[\left(\xi_i-\xi_1\right)\left(\xi_i-\xi_2\right) \cdots\left(\xi_i-\xi_{i-1}\right)\left(\xi_i-\xi_{i+1}\right) \cdots\left(\xi_i-\xi_n\right)\right]^{-1}
$$
Thus, the interpolation function associated with node $i$ is
$$
\psi_i^e(\xi)=\frac{\left(\xi-\xi_1\right)\left(\xi-\xi_2\right) \cdots\left(\xi-\xi_{i-1}\right)\left(\xi-\xi_{i+1}\right) \cdots\left(\xi-\xi_n\right)}{\left(\xi_i-\xi_1\right)\left(\xi_i-\xi_2\right) \cdots\left(\xi_i-\xi_{i-1}\right)\left(\xi_i-\xi_{i+1}\right) \cdots\left(\xi_i-\xi_n\right)}
$$

有限元方法代考

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

积分的数值计算，称为数值积分或数值正交，涉及到用一个充分次多项式逼近被积函数，因为多项式的积分可以精确地求值。积分通常用问题描述(如$x$或$r$)中出现的坐标来表示。我们称$x$和$r$为问题坐标或全局坐标。由于有限元逼近(或插值)函数是在一个元素上导出的，因此使用局部(即元素)坐标很方便，例如前面使用的$\bar{x}$。
例如，考虑积分，
$$
I_e=\int_{x_a^e}^{x_b^e} F_e(x) d x=\int_0^{h_e} F^e(\bar{x}) d \bar{x}
$$
其中$F^e$是$x=x_a^e+\bar{x}$的函数，$h_e=x_b^e-x_a^e$。我们用多项式近似$\Omega^e$中的$F^e(\bar{x})$函数
$$
F^e(\bar{x}) \approx \sum_{I=1}^N F_I^e \psi_I^e(\bar{x})
$$
式中$F_I^e$为区间$I$第th点$\bar{x}_I$处$F^e(\bar{x})$的值$\left[0, h_e\right]$, $\psi_I^e(\bar{x})$为区间内$N-1$次多项式。表示可以看作是$F^e(\bar{x})$的有限元插值，其中$F_I^e$是函数在第i个节点处的值。插值函数$\psi_I^e(\bar{x})$可以是拉格朗日类型或埃尔米特类型。

将Eq.(8.2.2)代入Eq.(8.2.1)并对积分进行计算，得到积分$I_e$的近似值。例如，假设我们选择$F^e(\bar{x})$线性插值，如图8.2.1所示。对于$N=2$，我们有
$$
\psi_1^e=1-\frac{\bar{x}}{h_e}, \quad \psi_2^e=\frac{\bar{x}}{h_e}, \quad \int_0^{h_e} \psi_I^e(\bar{x}) d \bar{x}=\frac{h_e}{2} \quad(I=1,2)
$$
和
$$
I_e=\frac{h_e}{2}\left(F_1^e+F_2^e\right), \quad F_1^e=F^e(0), \quad F_2^e=F^e\left(h_e\right)
$$
因此，积分的值由用于近似函数$F^e(\bar{x})$下的面积的梯形面积给出(见图8.2.1)。式(8.2.3b)称为数值积分的梯形法则。

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

在所有的正交公式中，高斯-勒让德正交是最常用的。稍后将讨论该方法本身的细节。积分公式要求将积分转换为一个在区间(元素)$\hat{\bar{\Omega}}^e \equiv[-1,1}$上求值的积分，这需要将问题坐标$x$(即，用于描述控制方程的坐标)转换为元素坐标$\xi$，以便将积分置于[- 1,1]上(见图8.2.2)。因此，$x$与$\xi$之间的关系是线性的，可以借助条件来建立
当$x=x_a^e, \quad \xi=-1 ;$和当$x=x_b^e, \quad \xi=1$
$x$和$\xi$之间的转换可以用线性“拉伸”映射表示
$$
x=a^e+b^e \xi \text { for } x \text { in } \bar{\Omega}^e=\left[x_a^e, x_b^e\right]
$$
其中$a^e$和$b^e$为待确定的常数，使式(8.2.6a)中的条件成立:
$$
x_a^e=a^e+b^e \cdot(-1), \quad x_b^e=a^e+b^e .
$$
求解$a^e$和$b^e$，得到
$$
b^e=\frac{1}{2}\left(x_b^e-x_a^e\right)=\frac{1}{2} h_e, \quad a^e=\frac{1}{2}\left(x_b^e+x_a^e\right)=x_a^e+\frac{1}{2} h_e
$$
因此$x$和$\xi$之间的转换变成
$$
x(\xi)=\frac{1}{2}\left(x_b^e+x_a^e\right)+\frac{1}{2}\left(x_b^e-x_a^e\right) \xi=x_a^e \frac{1}{2}(1-\xi) x_b^e \frac{1}{2}(1+\xi)=x_a^e+\frac{1}{2} h_e(1+\xi)
$$
其中$x_a^e$和$x_b^e$分别为元素左端和右端的全局坐标，$\bar{\Omega}^e$和$h_e$为元素长度(见图8.2.2)。
局部坐标$\xi$称为法坐标或自然坐标，其值始终在-1到1之间，原点位于一维元素的中心。局部坐标$\xi$在两个方面是有用的:(1)在构造插值函数时是方便的;(2)在基于高斯-勒让德正交的数值积分中是必需的。
用自然坐标$\xi$来推导拉格朗日插值函数族是很容易的，因为近似函数的以下插值性质:
$$
\psi_i^e\left(\xi_j\right)= \begin{cases}1 & \text { if } i=j \ 0 & \text { if } i \neq j\end{cases}
$$
其中$\xi_j$是元素中$j$节点的$\xi$ -坐标。对于具有$n$节点的元素，拉格朗日插值函数$\psi_i^e(i=1,2, \ldots, n)$是次多项式$n-1$。为了构造满足式(8.2.8)中的性质的$\psi_i^e$，我们进行如下操作。对于每个$\psi_i^e$，我们形成$n-1$线性函数$\xi-\xi_j(j=1,2, \ldots, i-1, i+1, \ldots, n$与$j \neq i)$的乘积:
$$
\psi_i^e=c_i\left(\xi-\xi_1\right)\left(\xi-\xi_2\right) \cdots\left(\xi-\xi_{i-1}\right)\left(\xi-\xi_{i+1}\right) \cdots\left(\xi-\xi_n\right)
$$
注意，除了第$i$个节点外，$\psi_i^e$在所有节点上都为零。接下来，我们确定常数$c_i$，使$\psi_i^e=1$ at $\xi-\xi_i$:
$$
c_i=\left[\left(\xi_i-\xi_1\right)\left(\xi_i-\xi_2\right) \cdots\left(\xi_i-\xi_{i-1}\right)\left(\xi_i-\xi_{i+1}\right) \cdots\left(\xi_i-\xi_n\right)\right]^{-1}
$$
因此，节点$i$对应的插值函数为
$$
\psi_i^e(\xi)=\frac{\left(\xi-\xi_1\right)\left(\xi-\xi_2\right) \cdots\left(\xi-\xi_{i-1}\right)\left(\xi-\xi_{i+1}\right) \cdots\left(\xi-\xi_n\right)}{\left(\xi_i-\xi_1\right)\left(\xi_i-\xi_2\right) \cdots\left(\xi_i-\xi_{i-1}\right)\left(\xi_i-\xi_{i+1}\right) \cdots\left(\xi_i-\xi_n\right)}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年7月17日2023年8月23日 by statistics-lab

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
$$
\begin{gathered}
\hat{\mathbf{K}}=\mathbf{K}+a_3 \mathbf{M}+a_6 \mathbf{C}, \hat{\mathbf{F}}^{s, s+1}=\mathbf{F}^{s+1}+\mathbf{M} \overline{\mathbf{u}}^s+\mathbf{C} \hat{\mathbf{u}}^s \
\overline{\mathbf{u}}^s=a_3 \mathbf{u}^s+a_4 \dot{\mathbf{u}}^s+a_5 \ddot{\mathbf{u}}^s, \hat{\mathbf{u}}^s=a_6 \mathbf{u}^s+a_7 \dot{\mathbf{u}}^s+a_8 \ddot{\mathbf{u}}^s \
a_3=\frac{2}{\gamma(\Delta t)^2}, a_4=a_3 \Delta t, \quad a_5=\frac{1}{\gamma}-1
\end{gathered}
$$

有限元方法代考

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

考虑这样的矩阵方程
$$
\mathrm{Mu}+\mathrm{Cu}+\mathrm{Ku}=\mathrm{F}
$$
受制于初始条件
$$
\mathbf{u}(0)=\mathbf{u}_0, \quad \dot{\mathbf{u}}(0)=\mathbf{v}_0
$$
这种方程出现在结构动力学中，其中$\mathbf{M}$为质量矩阵，$\mathbf{C}$为阻尼矩阵，$\mathbf{K}$为刚度矩阵。阻尼矩阵$\mathbf{C}$通常被认为是质量和刚度矩阵的线性组合$\mathbf{C}=$$c_1 \mathbf{M}+c_2 \mathbf{K}$，其中$c_1$和$c_2$是由物理实验确定的。在本研究中，我们将不考虑数值例子中的阻尼(即$\mathbf{C}=0$)，尽管理论发展将解释它。杆和梁的瞬态分析得到式所示的方程。(7.4.32a)和(7.4.32b)。杆和梁的质量和刚度矩阵见式。(7.3.40)、(7.3.57)、(7.3.63a)和(7.3.63b)。与Eq. (7.4.32a) (with $\mathbf{C}=0)$)相关的特征值问题是
$$
(-\lambda \mathbf{M}+\mathbf{K}) \mathbf{u}_0=\mathbf{Q}_0, \quad \lambda=\omega^2
$$
有几种数值方法可用于近似二阶时间导数并将微分方程在时间上转换为代数方程(参见Surana和Reddy[4]，了解不同阶的RungeKutta方法、Newmark系列方法、Wilson的$\theta$方法和Houbolt的方法)。最近，Kim和Reddy[5-8]开发了一些基于加权残差和最小二乘概念的时间逼近方案[类似于式中讨论的Galerkin方案]。[7.4.19][7.4.22]。为了简便和广泛使用，我们只考虑纽马克时间近似族和中心差分格式。

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

让我们考虑以下$(\alpha, \gamma)$ -近似族，其中函数及其一阶导数近似为[遵循截断的泰勒级数符号式(7.4.15)]
$$
\begin{aligned}
& \mathbf{u}^{s+1} \approx \mathbf{u}^s+\Delta t \dot{\mathbf{u}}^s+\frac{1}{2}(\Delta t)^2\left[(1-\gamma) \ddot{\mathbf{u}}^s+\gamma \ddot{\mathbf{u}}^{s+1}\right] \
& \dot{\mathbf{u}}^{s+1} \approx \dot{\mathbf{u}}^s+a_2 \ddot{\mathbf{u}}^s+a_1 \ddot{\mathbf{u}}^{s+1}
\end{aligned}
$$
其中$\alpha$和$\gamma$是决定方案稳定性和准确性的参数。式(7.4.33)和式(7.4.34)分别是$\mathbf{u}^{s+1}$和$\dot{\mathbf{u}}^{\mathrm{s}+1}$关于$t=t_{\mathrm{s}}$的泰勒级数展开式。
利用式中引入的近似得到式(7.4.32a)的完全离散形式。(7.4.33)和(7.4.34)。首先，我们从等式中消去$\ddot{\mathbf{u}}^{\mathrm{s+1}}$。(7.4.33)和式(7.4.34)，并写出结果 $\dot{\mathbf{u}}^{\text {s+1:}}$

$$
\begin{gathered}
\dot{\mathbf{u}}^{s+1}=a_6\left(\mathbf{u}^{s+1}-\mathbf{u}^s\right)-a_7 \dot{\mathbf{u}}^s-a_8 \ddot{\mathbf{u}}^s \
a_6=\frac{2 \alpha}{\gamma \Delta t}, \quad a_7=\frac{2 \alpha}{\gamma}-1, \quad a_8=\left(\frac{\alpha}{\gamma}-1\right) \Delta t
\end{gathered}
$$
现在将式(7.4.33)与$\mathbf{M}$预乘，并将式(7.4.32a)中的$\mathbf{M} \ddot{\mathbf{u}}^{\mathbf{s + 1}}$代入，我们得到
$$
\left(\mathbf{M}+\frac{\gamma(\Delta t)^2}{2} \mathbf{K}\right) \mathbf{u}^{s+1}=\mathbf{M} \mathbf{b}^s+\frac{\gamma(\Delta t)^2}{2} \mathbf{F}^{s+1}-\frac{\gamma(\Delta t)^2}{2} \mathbf{C} \dot{\mathbf{u}}^{s+1}
$$
在哪里
$$
\mathbf{b}^s=\mathbf{u}^s+\Delta t \dot{\mathbf{u}}^s+\frac{1}{2}(1-\gamma)(\Delta t)^2 \ddot{\mathbf{u}}^s
$$
现在，乘以$2 /\left[\gamma(\Delta t)^2\right]$我们得到
$$
\left(\frac{2}{\gamma(\Delta t)^2} \mathbf{M}+\mathbf{K}\right) \mathbf{u}^{s+1}=\frac{2}{\gamma(\Delta t)^2} \mathbf{M} \mathbf{b}_s+\mathbf{F}^{s+1}-\mathbf{C} \dot{\mathbf{u}}^{s+1}
$$
利用式(7.4.36c)中$\dot{\mathbf{u}}^{\mathrm{s}+1}$的式(7.4.35a)和收集项，可以得到递归关系:
$$
\hat{\mathbf{K}} \mathbf{u}^{s+1}=\hat{\mathbf{F}}^{s, s+1}
$$
在哪里
$$
\begin{gathered}
\hat{\mathbf{K}}=\mathbf{K}+a_3 \mathbf{M}+a_6 \mathbf{C}, \hat{\mathbf{F}}^{s, s+1}=\mathbf{F}^{s+1}+\mathbf{M} \overline{\mathbf{u}}^s+\mathbf{C} \hat{\mathbf{u}}^s \
\overline{\mathbf{u}}^s=a_3 \mathbf{u}^s+a_4 \dot{\mathbf{u}}^s+a_5 \ddot{\mathbf{u}}^s, \hat{\mathbf{u}}^s=a_6 \mathbf{u}^s+a_7 \dot{\mathbf{u}}^s+a_8 \ddot{\mathbf{u}}^s \
a_3=\frac{2}{\gamma(\Delta t)^2}, a_4=a_3 \Delta t, \quad a_5=\frac{1}{\gamma}-1
\end{gathered}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|MEE721

Posted on 2023年7月17日2023年8月23日 by statistics-lab

数学代写|有限元方法代写Finite Element Method代考|Euler-Bernoulli beam theory

The study of buckling (also called stability) of beam-columns also leads to an eigenvalue problem. For example, equation governing the onset of buckling of a column subjected to an axial compressive force $N^0$ (see Fig. 7.3.1) according to the Euler-Bernoulli beam theory is (see Reddy [1,2])
$$
\frac{d^2}{d x^2}\left(E I \frac{d^2 W}{d x^2}\right)+N^0 \frac{d^2 W}{d x^2}=0
$$
where $W(x)$ is the lateral deflection at the onset of buckling. Equation (7.3.31) describes an eigenvalue problem with $\lambda=N^0$. The smallest value of $N^0$ is called the critical buckling load.
7.3.4.2 Timoshenko beam theory
For the Timoshenko beam theory, the equations governing buckling of beams are given by
$$
\begin{aligned}
&- \frac{d}{d x}\left[G A K_s\left(\frac{d W}{d x}+S\right)\right]+N^0 \frac{d^2 W}{d x^2}=0 \
&-\frac{d}{d x}\left(E I \frac{d S}{d x}\right)+G A K_s\left(\frac{d W}{d x}+S\right)=0
\end{aligned}
$$
Here $W(x)$ and $S(x)$ denote the transverse deflection and rotation,respectively, at the onset of buckling. Equations (7.3.32a) and (7.3.32b) together define an eigenvalue problem of finding the buckling load $N^0$ (eigenvalue) and the associated mode shape defined by $W(x)$ and $S(x)$ (eigenvector).
This completes the descriptions of the types of eigenvalue problems that will be treated in this chapter. The task of determining natural frequencies and mode shapes of a structure undergoing free (or natural) vibration is termed modal analysis. In addition, we also study buckling of beam-columns. In the following sections, we develop weak forms and finite element models of the eigenvalue problems described here. Numerical examples will be presented to illustrate the procedure of determining eigenvalues and eigenvectors, although this exercise may be familiar to the readers from other courses (e.g., a course on vibrations).

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

In this section, we develop finite element models of eigenvalue problems described by differential equations of heat transfer, bars, and beams. In view of the close similarity between the differential equations governing eigenvalue and boundary value problems, the steps involved in the construction of their finite element models are entirely analogous. The eigenvalue problems described by differential equations are reduced to algebraic eigenvalue problems by means of finite element approximations. The methods of solution of algebraic eigenvalue problems are then used to solve for the eigenvalues $\lambda$ and eigenvectors.

We note that a continuous system has an infinite number of eigenvalues while a discrete system has only a finite number of eigenvalues. The number of eigenvalues is equal to the number of unconstrained primary degrees of freedom in the mesh. The number of eigenvalues will increase with a mesh refinement, and the newly added eigenvalues will be larger in magnitude.
7.3.5.1 Heat transfer and bar-like problems (second-order equations)
Governing equation. The eigenvalue problem associated with onedimensional heat flow and straight bars both share the same type of governing equation:

$$
-\frac{d}{d x}\left(a \frac{d U}{d x}\right)+c_0 U-c \lambda U=0, \quad 0<x<L
$$
Here $a, c_0$ and $c$ are known parameters (data) that depend on the physical problem, $\lambda$ is the eigenvalue, and $U$ is the eigenfunction. Special cases of Eq. (7.3.33) are given below.
Heat transfer: $a=k A, c_0=P \beta, c=\rho c_v A$
Bars: $a=E A, c_0=0, c=c_2=\rho A$
Weak form. In view of the discussion of Chapters 3 and 4 , the weak form of Eq. (7.3.33) over $\Omega^e=\left(x_a^e, x_b^e\right)$ can be readily obtained as
$$
0=\int_{x_a^e}^{x_b^{\prime}}\left(a \frac{d w_i}{d x} \frac{d U_h}{d x}+c_0 w_i U_h-\lambda c w_i U_h\right) d x-Q_1^e w_i\left(x_a^e\right)-Q_n^e w_i\left(x_b^e\right)
$$
where $U_h$ is an approximation of $U, w_i$ is the ith weight function (which will be replaced with $\psi_i^e$ in deriving the finite element model), and $Q_1^e$ and $Q_n^e$ are the secondary variables at the first and last nodes, respectively, of a finite element with $n$ nodes (for eigenvalue problems, we take $Q_i^e=0$ for $1<i<n$ ):
$$
Q_1^e=-\left[a \frac{d U_h}{d x}\right]{x_a^e}, \quad Q_n^e=\left[a \frac{d U_h}{d x}\right]{x_b^e}
$$

数学代写|有限元方法代写Finite Element Method代考|One-Dimensional Heat Flow

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Euler-Bernoulli beam theory

研究梁柱的屈曲(也称为稳定性)也会导致一个特征值问题。例如，根据欧拉-伯努利梁理论，柱在轴向压缩力$N^0$(见图7.3.1)作用下的屈曲起始方程为(见Reddy[1,2])。
$$
\frac{d^2}{d x^2}\left(E I \frac{d^2 W}{d x^2}\right)+N^0 \frac{d^2 W}{d x^2}=0
$$
其中$W(x)$为屈曲开始时的侧向挠度。式(7.3.31)描述了一个带有$\lambda=N^0$的特征值问题。$N^0$的最小值称为临界屈曲载荷。
7.3.4.2 Timoshenko梁理论
对于Timoshenko梁理论，控制梁屈曲的方程由
$$
\begin{aligned}
&- \frac{d}{d x}\left[G A K_s\left(\frac{d W}{d x}+S\right)\right]+N^0 \frac{d^2 W}{d x^2}=0 \
&-\frac{d}{d x}\left(E I \frac{d S}{d x}\right)+G A K_s\left(\frac{d W}{d x}+S\right)=0
\end{aligned}
$$
这里$W(x)$和$S(x)$分别表示屈曲开始时的横向挠度和旋转。式(7.3.32a)和式(7.3.32b)共同定义了一个求屈曲载荷$N^0$ (eigenvalue)和相关模态振型$W(x)$和$S(x)$ (eigenvector)的特征值问题。
这就完成了本章将要讨论的特征值问题类型的描述。确定经受自由(或固有)振动的结构的固有频率和模态振型的任务称为模态分析。此外，我们还研究了梁柱的屈曲。在接下来的章节中，我们将开发这里描述的特征值问题的弱形式和有限元模型。本文将给出数值例子来说明确定特征值和特征向量的过程，尽管读者可能对其他课程(例如振动课程)的练习很熟悉。

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

在本节中，我们开发了由传热、杆和梁的微分方程描述的特征值问题的有限元模型。由于特征值问题和边值问题的微分方程非常相似，它们的有限元模型的建立步骤是完全相似的。利用有限元逼近的方法，将微分方程所描述的特征值问题化为代数特征值问题。然后用代数特征值问题的解法求解特征值$\lambda$和特征向量。

我们注意到一个连续系统有无限个特征值，而一个离散系统只有有限个特征值。特征值的个数等于网格中无约束的初级自由度的个数。随着网格的细化，特征值的数量会增加，新增加的特征值的大小也会增大。
7.3.5.1传热和棒状问题(二阶方程)
控制方程。一维热流和直杆的特征值问题具有相同类型的控制方程:

$$
-\frac{d}{d x}\left(a \frac{d U}{d x}\right)+c_0 U-c \lambda U=0, \quad 0<x<L
$$
这里$a, c_0$和$c$是依赖于物理问题的已知参数(数据)，$\lambda$是特征值，$U$是特征函数。(7.3.33)式的特殊情况如下。
传热:$a=k A, c_0=P \beta, c=\rho c_v A$
酒吧:$a=E A, c_0=0, c=c_2=\rho A$
弱形式。考虑到第3章和第4章的讨论，(7.3.33)式除以$\Omega^e=\left(x_a^e, x_b^e\right)$的弱形式可以很容易地得到
$$
0=\int_{x_a^e}^{x_b^{\prime}}\left(a \frac{d w_i}{d x} \frac{d U_h}{d x}+c_0 w_i U_h-\lambda c w_i U_h\right) d x-Q_1^e w_i\left(x_a^e\right)-Q_n^e w_i\left(x_b^e\right)
$$
其中$U_h$是近似的$U, w_i$是第i个权重函数(在推导有限元模型时将用$\psi_i^e$代替)，$Q_1^e$和$Q_n^e$分别是具有$n$节点的有限元的第一个和最后一个节点上的次要变量(对于特征值问题，我们将$Q_i^e=0$作为$1<i<n$):
$$
Q_1^e=-\left[a \frac{d U_h}{d x}\right]{x_a^e}, \quad Q_n^e=\left[a \frac{d U_h}{d x}\right]{x_b^e}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|有限元方法代写Finite Element Method代考|One-Dimensional Heat Flow

Posted on 2023年7月1日2023年7月1日 by statistics-lab

如果你也在怎样代写有限元方法finite differences method 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。有限元方法finite differences method在数值分析中，是一类通过用有限差分逼近导数解决微分方程的数值技术。空间域和时间间隔（如果适用）都被离散化，或被分成有限的步骤，通过解决包含有限差分和附近点的数值的代数方程来逼近这些离散点的解的数值。

有限元方法finite differences method有限差分法将可能是非线性的常微分方程（ODE）或偏微分方程（PDE）转换成可以用矩阵代数技术解决的线性方程系统。现代计算机可以有效地进行这些线性代数计算，再加上其相对容易实现，使得FDM在现代数值分析中得到了广泛的应用。今天，FDM与有限元方法一样，是数值解决PDE的最常用方法之一。

数学代写|有限元方法代写Finite Element Method代考|One-Dimensional Heat Flow

The principle of balance of energy, which can be stated as the time-rate of change of internal energy of a system is equal to the heat input to the system, for a one-dimensional heat flow (e.g., in a plane wall or a fin) results in the equation
$$
c_1 \frac{\partial u}{\partial t}-\frac{\partial}{\partial x}\left(k A \frac{\partial u}{\partial x}\right)=f(x, t), \quad 00
$$
where $u$ denotes the temperature above a reference temperature $\left(u=T-T_0\right.$ ), $c_1=c_v \rho A, k$ denotes the thermal conductivity, $\rho$ is the mass density, $A$ is the cross-sectional area, $c_v$ is the specific heat at constant volume, and $f$ is the internal heat generation per unit length, all of which can be, in general, known functions of position $x$ and time $t$.

The following equation of motion arises in connection with the axial motion
of a bar:
$$
c_2 \frac{\partial^2 u}{\partial t^2}-\frac{\partial}{\partial x}\left(E A \frac{\partial u}{\partial x}\right)=f(x, t), 00
$$
where $u$ denotes the axial displacement, $c_2=\rho A, E$ is the modulus of elasticity, $A$ is the area of cross section, $\rho$ is the mass density, and $f$ is the axial force per unit length.

数学代写|有限元方法代写Finite Element Method代考|Bending of Beams: The Euler–Bernoulli Beam Theory

The equation of motion of bending of beams using the Euler-Bernoulli beam theory is given by (see Examples 2.3.5 and 2.3.6 and the textbook by Reddy [1], pp. 73-76 for the development of the EBT)
$$
c_2 \frac{\partial^2 w}{\partial t^2}-c_3 \frac{\partial^4 w}{\partial t^2 \partial x^2}+\frac{\partial^2}{\partial x^2}\left(E I \frac{\partial^2 w}{\partial x^2}\right)=q(x, t), 00
$$
where $c_2=\rho A$ and $c_3=\rho I ; \rho$ denotes the mass density per unit length, $A$ the area of cross section, $E$ the modulus, and $I$ the moment of inertia.

$$
A_t u+A_{x t} u+A_x u=f(\mathbf{x}, t) \text { in } \Omega
$$
where $A_t$ is a linear differential operator in time $t, A_x$ is a linear differential operator in the spatial coordinates $\mathbf{x}, A_{x t}$ is a linear differential operator in both $t$ and $\mathbf{x}, f$ is a “forcing” function of position $\mathbf{x}$ and time $t$. Examples of the operator equation Eq. (7.2.5) are provided by Eqs. (7.2.1)-(7.2.4b), where operators $A_t, A_{x t}$, and $A_x$ can be readily identified $\left[A_{x t} \neq 0\right.$ only in Eq. (7.2.3)].
Equations containing first-order time derivatives are called parabolic equations while those containing second-order time derivatives are termed hyperbolic equations. The operator equations that describe the steady-state response can be obtained by setting the time-dependent parts to zero. Analysis of the time-dependent problems to determine their time-dependent solution $u(\mathbf{x}, t)$ is known as the transient analysis and $u(\mathbf{x}, t)$ is called the transient response, and it is presented in Section 7.4. The eigenvalue problem associated with a time-dependent problem can be derived from the governing equations of motion by assuming a suitable (i.e., decaying or periodic type) solution form. Details are presented in the next section.

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|One-Dimensional Heat Flow

对于一维热流(如平面壁面或翅片内的热流)，能量平衡原理可以表示为系统内能的时间变化率等于系统输入的热量
$$
c_1 \frac{\partial u}{\partial t}-\frac{\partial}{\partial x}\left(k A \frac{\partial u}{\partial x}\right)=f(x, t), \quad 00
$$
其中$u$为参考温度以上的温度($\left(u=T-T_0\right.$)， $c_1=c_v \rho A, k$为导热系数，$\rho$为质量密度，$A$为截面积，$c_v$为定容比热，$f$为单位长度内部产生的热量，一般来说，这些都可以是位置$x$和时间$t$的已知函数。

下面的运动方程与轴向运动有关
酒吧的:
$$
c_2 \frac{\partial^2 u}{\partial t^2}-\frac{\partial}{\partial x}\left(E A \frac{\partial u}{\partial x}\right)=f(x, t), 00
$$
式中$u$为轴向位移，$c_2=\rho A, E$为弹性模量，$A$为截面面积，$\rho$为质量密度，$f$为单位长度轴向力。

数学代写|有限元方法代写Finite Element Method代考|Bending of Beams: The Euler–Bernoulli Beam Theory

采用欧拉-伯努利梁理论的梁的弯曲运动方程由(例2.3.5和2.3.6以及Reddy[1]的教材73-76页)给出。
$$
c_2 \frac{\partial^2 w}{\partial t^2}-c_3 \frac{\partial^4 w}{\partial t^2 \partial x^2}+\frac{\partial^2}{\partial x^2}\left(E I \frac{\partial^2 w}{\partial x^2}\right)=q(x, t), 00
$$
式中$c_2=\rho A$和$c_3=\rho I ; \rho$为单位长度的质量密度，$A$为截面面积，$E$为模量，$I$为转动惯量。

$$
A_t u+A_{x t} u+A_x u=f(\mathbf{x}, t) \text { in } \Omega
$$
其中$A_t$是时间上的线性微分算子$t, A_x$是空间坐标上的线性微分算子$\mathbf{x}, A_{x t}$是$t$和$\mathbf{x}, f$的线性微分算子是位置$\mathbf{x}$和时间$t$的“强制”函数。方程提供了算子方程Eq.(7.2.5)的例子。(7.2.1)-(7.2.4b)，其中运算符$A_t, A_{x t}$和$A_x$可以很容易地识别$\left[A_{x t} \neq 0\right.$只在式(7.2.3)中]。
含有一阶时间导数的方程称为抛物方程，而含有二阶时间导数的方程称为双曲方程。描述稳态响应的算子方程可以通过将与时间相关的部分设置为零来获得。分析时变问题以确定其时变解$u(\mathbf{x}, t)$称为暂态分析，$u(\mathbf{x}, t)$称为暂态响应，在第7.4节中给出。与时间相关问题相关的特征值问题可以通过假设合适的(即衰减型或周期型)解形式从运动控制方程中导出。详细信息将在下一节中介绍。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写