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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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}

## 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

E:弹性模量或杨氏模量
v:泊松比

\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}

\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}

\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

$$\varepsilon^{(t h)}=\alpha \Delta T$$

$$\Delta L=\int_0^L \alpha \Delta T d x$$

$$\sigma^{(t h)}=E \alpha \Delta T$$

$$\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T$$

$$\sigma=E(\varepsilon-\alpha \Delta T)$$

\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}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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.

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

$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$

\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

$$\sigma_{i j}=c_{i j r s} \varepsilon_{r s}$$

\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}

$${\sigma}=[E]{\varepsilon}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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

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:
$$G=\frac{E}{2(1+v)}$$

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

$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$

\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

$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$

$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$

$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$

$$G=\frac{E}{2(1+v)}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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}

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

$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y$$

$$u_h^e\left(x_1, y_1\right)=u_1^e, \quad u_h^e\left(x_2, y_2\right)=u_2^e, \quad u_h^e\left(x_3, y_3\right)=u_3^e$$

\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}

$$\left{\begin{array}{l} u_1 \ u_2 \ u_3 \end{array}\right}=\left[\begin{array}{lll} 1 & x_1 & y_1 \ 1 & x_2 & y_2 \ 1 & x_3 & y_3 \end{array}\right]\left{\begin{array}{l} c_1 \ c_2 \ c_3 \end{array}\right} \text { or } \mathbf{u}=\mathbf{A c}$$

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

$$u_h^e(x, y)=c_1^e+c_2^e x+c_3^e y+c_4^e x y$$

$$u_h(\bar{x}, \bar{y})=c_1+c_2 \bar{x}+c_3 \bar{y}+c_4 \bar{x} \bar{y}$$

\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}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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:

1. Select elements that characterize the governing equations of the problem.
2. 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.
3. 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).
4. 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}

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

$$0=\int_{\Omega_{\varepsilon}} w\left[-\frac{\partial}{\partial x}\left(F_1\right)-\frac{\partial}{\partial y}\left(F_2\right)+a_{00} u-f\right] d x d y$$

$$F_1=a_{11} \frac{\partial u}{\partial x}+a_{12} \frac{\partial u}{\partial y}, \quad F_2=a_{21} \frac{\partial u}{\partial x}+a_{22} \frac{\partial u}{\partial y}$$

\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}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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):

1. Preprocessor
2. Processor
3. 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:

1. Generation of the element matrices using numerical integration.
2. Assembly of element equations.
3. Imposition of the boundary conditions.
4. 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 $I=$ $\operatorname{NOD}(n, i)$ :
$$\left{x_i^{(n)}\right}=\left{X_l\right}, \quad I=\operatorname{NOD}(n, i) \rightarrow \operatorname{ELX}(i)=\operatorname{GLX}(\operatorname{NOD}(n, i))$$

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

$\operatorname{NOD}(n, j)=$元素$n$的第$j$个(本地)节点对应的全局节点号

$$(-\lambda \mathbf{M}+\mathbf{K}) \mathbf{u}_0=\mathbf{Q}_0, \quad \lambda=\omega^2$$

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

\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}

$$\begin{gathered} \dot{\mathbf{u}}^{s+1}=a_6\left(\mathbf{u}^{s+1}-\mathbf{u}^s\right)-a_7 \dot{\mathbf{u}}^s-a_8 \ddot{\mathbf{u}}^s \ a_6=\frac{2 \alpha}{\gamma \Delta t}, \quad a_7=\frac{2 \alpha}{\gamma}-1, \quad a_8=\left(\frac{\alpha}{\gamma}-1\right) \Delta t \end{gathered}$$

$$\left(\mathbf{M}+\frac{\gamma(\Delta t)^2}{2} \mathbf{K}\right) \mathbf{u}^{s+1}=\mathbf{M} \mathbf{b}^s+\frac{\gamma(\Delta t)^2}{2} \mathbf{F}^{s+1}-\frac{\gamma(\Delta t)^2}{2} \mathbf{C} \dot{\mathbf{u}}^{s+1}$$

$$\mathbf{b}^s=\mathbf{u}^s+\Delta t \dot{\mathbf{u}}^s+\frac{1}{2}(1-\gamma)(\Delta t)^2 \ddot{\mathbf{u}}^s$$

$$\left(\frac{2}{\gamma(\Delta t)^2} \mathbf{M}+\mathbf{K}\right) \mathbf{u}^{s+1}=\frac{2}{\gamma(\Delta t)^2} \mathbf{M} \mathbf{b}_s+\mathbf{F}^{s+1}-\mathbf{C} \dot{\mathbf{u}}^{s+1}$$

$$\hat{\mathbf{K}} \mathbf{u}^{s+1}=\hat{\mathbf{F}}^{s, s+1}$$

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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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|Euler-Bernoulli beam theory

$$\frac{d^2}{d x^2}\left(E I \frac{d^2 W}{d x^2}\right)+N^0 \frac{d^2 W}{d x^2}=0$$

7.3.4.2 Timoshenko梁理论

\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}

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

7.3.5.1传热和棒状问题(二阶方程)

$$-\frac{d}{d x}\left(a \frac{d U}{d x}\right)+c_0 U-c \lambda U=0, \quad 0<x<L$$

$$0=\int_{x_a^e}^{x_b^{\prime}}\left(a \frac{d w_i}{d x} \frac{d U_h}{d x}+c_0 w_i U_h-\lambda c w_i U_h\right) d x-Q_1^e w_i\left(x_a^e\right)-Q_n^e w_i\left(x_b^e\right)$$

$$Q_1^e=-\left[a \frac{d U_h}{d x}\right]{x_a^e}, \quad Q_n^e=\left[a \frac{d U_h}{d x}\right]{x_b^e}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 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.

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

$$c_1 \frac{\partial u}{\partial t}-\frac{\partial}{\partial x}\left(k A \frac{\partial u}{\partial x}\right)=f(x, t), \quad 00$$

$$c_2 \frac{\partial^2 u}{\partial t^2}-\frac{\partial}{\partial x}\left(E A \frac{\partial u}{\partial x}\right)=f(x, t), 00$$

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

$$c_2 \frac{\partial^2 w}{\partial t^2}-c_3 \frac{\partial^4 w}{\partial t^2 \partial x^2}+\frac{\partial^2}{\partial x^2}\left(E I \frac{\partial^2 w}{\partial x^2}\right)=q(x, t), 00$$

$$A_t u+A_{x t} u+A_x u=f(\mathbf{x}, t) \text { in } \Omega$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。