## 数学代写|有限元方法代写Finite Element Method代考|EG3001

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。

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

## 数学代写|有限元方法代写Finite Element Method代考|Principal stresses and directions

Infinitely many planes can pass through a given point in a continuum, such as the plane that gives rise to the tetrahedron shown in Fig. 2.3. The normal and shear traction components $\sigma_{n n}$ and $\sigma_{n t}$ will vary according to the orientation of the cutting-plane. It is reasonable to assume that among these planes, there is one on which the shear tractions vanish, i.e., $\sigma_{n t}=0$. This plane is called the principal plane, and its orientation is called the principal orientation $\vec{n}{p}$. The normal component of the traction $\sigma{n n}$ acting on the principle plane has the magnitude $\lambda$ and it is named the principal stress.

The traction acting on the principal plane can be expressed by a simple vector argument and by using Eq. (2.17a) as follows:
\begin{aligned} &\vec{T}{n{p}}=\lambda \vec{n}{p} \ &\vec{T}{n_{p}}=[\sigma]^{T}\left{n_{p}\right} \end{aligned}
By combining these two relationships, we find,
\begin{aligned} \lambda\left(n_{p_{x}} \hat{i}+n_{p_{y}} \hat{j}+n_{p_{z}} \hat{k}\right)=& {\left[\left(\sigma_{x x} n_{p_{x}}+\tau_{y x} n_{p_{y}}+\tau_{z x} n_{p_{z}}\right) \hat{i}+\left(\tau_{x y} n_{p_{x}}+\sigma_{y y} n_{p_{y}}+\tau_{y y} n_{p_{z}}\right) \hat{j}\right.} \ &\left.+\left(\tau_{x z} n_{p_{x}}+\tau_{y z} n_{p_{y}}+\sigma_{z z} n_{p_{z}}\right) \hat{k}\right] \end{aligned}
The following system of equations can be obtained for the unknown vector components of the normal vector of the principle plane from this relationship,
\begin{aligned} \left(\sigma_{x x} n_{p_{x}}+\tau_{y x} n_{p_{y}}+\tau_{z x} n_{p_{z}}\right) &=\lambda n_{p_{x}} \ \left(\tau_{x y} n_{p_{x}}+\sigma_{y y} n_{p_{y}}+\tau_{z y} n_{p_{z}}\right.&=\lambda n_{p_{y}} \ \left(\tau_{x z} n_{p_{x}}+\tau_{y z} n_{p_{y}}+\sigma_{z z} n_{p_{z}}\right) &=\lambda n_{p_{z}} \end{aligned}
This equation can be represented in matrix form as follows:
$$\left[\begin{array}{ccc} \left(\sigma_{x x}-\lambda\right) & \tau_{y x} & \tau_{z x} \ \tau_{x y} & \left(\sigma_{y y}-\lambda\right) & \tau_{z y} \ \tau_{x z} & \tau_{y z} & \left(\sigma_{z z}-\lambda\right) \end{array}\right]\left{\begin{array}{l} n_{p_{x}} \ n_{p_{y}} \ n_{p_{z}} \end{array}\right}=0$$
or,
$$([\sigma]-\lambda[I])\left{n_{p}\right}=0$$
where $[I]$ is the identity matrix. This relationship represents the equilibrium of the principal stress $\lambda$ acting on the principal plane $\vec{n}_{p}$.

## 数学代写|有限元方法代写Finite Element Method代考|Deformation and strain

When subjected to external forces, an internal material point located, for example, at position $P$ before the loading, moves to point $P^{\prime}$ as depicted in two dimensions (a two-dimensional solid) in Fig. 2.6. The position of all material points in this solid domain are referred to a fixed Cartesian reference frame, and the position of point $P^{\prime}$ is found as follows (Fig. 2.6):
$$\vec{r}{p^{\prime}}=\vec{r}{p}+\vec{u}$$
where $\vec{u}$ is the deformation vector. For a general deformation in threedimensional space, the deformation vector is represented as follows:
$$\vec{u}=u_{x} \hat{i}+u_{y} \hat{j}+u_{z} \hat{k}$$
where $u_{x}, u_{y}$, and $u_{z}$ are the projections of $\vec{u}$ onto the $x, y$, and $z$ axes, respectively.

## 数学代写|有限元方法代写Finite Element Method代考|Principal stresses and directions

$$\lambda\left(n_{p_{x}} \hat{i}+n_{p_{y}} \hat{j}+n_{p_{z}} \hat{k}\right)=\left[\left(\sigma_{x x} n_{p_{x}}+\tau_{y x} n_{p_{y}}+\tau_{z x} n_{p_{z}}\right) \hat{i}+\left(\tau_{x y} n_{p_{x}}+\sigma_{y y} n_{p_{y}}+\tau_{y y} n_{p_{z}}\right) \hat{j} \quad+\left(\tau_{x z} r\right.\right.$$

$$\left(\sigma_{x x} n_{p_{x}}+\tau_{y x} n_{p_{y}}+\tau_{z x} n_{p_{z}}\right)=\lambda n_{p_{x}}\left(\tau_{x y} n_{p_{x}}+\sigma_{y y} n_{p_{y}}+\tau_{z y} n_{p_{z}} \quad=\lambda n_{p_{y}}\left(\tau_{x z} n_{p_{x}}+\tau_{y z} n_{p_{y}}+\sigma_{z z} n_{p_{z}}\right.\right.$$

## 数学代写|有限元方法代写Finite Element Method代考|Deformation and strain

$$\vec{r} p^{\prime}=\vec{r} p+\vec{u}$$

$$\vec{u}=u_{x} \hat{i}+u_{y} \hat{j}+u_{z} \hat{k}$$

## 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。

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

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

The state of stress of the point $P^{\prime}$ is expressed by the stress tensor in the global, Cartesian, coordinates $(x, y, z)$ by Eq. (2.5). However, the choice of this coordinate system and the small volume $(d V=d x . d y . d z)$ is arbitrary. We would like to be able to transform the stress state between different orientations.
In order to develop these transformations, we consider an oblique crosssection of the hexahedron by a plane of arbitrary, but known orientation. This results in the tetrahedral volume on one side of the plane as shown in Fig. 2.3. The traction components $\vec{T}{x}, \vec{T}{y}, \vec{T}{z}$, and $\vec{T}{n}$, acting on the small triangular areas $\Delta A_{x}, \Delta A_{y}, \Delta A_{z}$ and $\Delta A_{n}$, respectively, must be in static equilibrium in order to keep the continuum whole. Let us look into the arbitrary plane in more detail before we state this equilibrium condition.

The orientation of the plane is identified by its outward unit normal $\vec{n}$, as shown in the figure. The unit normal is defined as follows:
$$\vec{n}=n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}$$
in Cartesian coordinate system. In vector notation, the unit normal is expressed as follows:
$${n}=\left{\begin{array}{lll} n_{x} & n_{y} & n_{z} \end{array}\right}^{T}$$
The components $n_{x}, n_{y}$, and $n_{z}$ are the direction cosines of $\vec{n}$ with respect to the $(x, y, z)$ axes, and have the property,
$$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=1$$

## 数学代写|有限元方法代写Finite Element Method代考|Normal and shear components of tractions

Traction $\vec{T}{n}$ on an oblique plane $\vec{n}$ can be expressed by using the normal and shear components on the plane as follows: $$\vec{T}{n}=\sigma_{n n} \cdot \vec{n}+\sigma_{n t} \cdot \vec{t}$$
where the normal and tangential traction components are $\sigma_{n n}$ and $\sigma_{n t}$, respectively. These components can be found as follows:
$$\sigma_{n n}=\vec{T}{n} \cdot \vec{n} \quad \text { and } \quad \sigma{n t}=\vec{T}{n} \cdot \vec{t}$$ where $\vec{t}$ is a unit vector in the oblique the plane, which has the unit normal $\vec{n}$. The vector $\vec{t}$ has following components expressed in the global Cartesian system, $$\vec{t}=t{x} \hat{i}+t_{y} \hat{j}+t_{z} \hat{k} \quad \text { or } \quad{t}=\left{\begin{array}{lll} t_{x} & t_{y} & t_{z} \end{array}\right}^{T}$$
The normal component of the traction $\vec{T}{n}$ is found by using Eqs. (2.16) and (2.20) as follows: \begin{aligned} \sigma{n n}=& \vec{T}{n} \cdot \vec{n}=\left([\sigma]^{T} \cdot{n}\right) \cdot{n} \ =& {\left[\left(\sigma{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) \hat{i}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) \hat{j}\right.} \ &\left.+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) \hat{k}\right] \cdot\left(n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}\right) \end{aligned}
or, after rearranging,
\begin{aligned} \sigma_{n n}=&\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) n_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) n_{y} \ &+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) n_{z} \end{aligned}
Similarly, the tangential component of the traction $\vec{T}{n}$ is found as follows: \begin{aligned} \sigma{n t}=&\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) t_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) t_{y} \ &+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) t_{z} \end{aligned}
In fact, Eqs. (2.22) and (2.23) can be used to transform the stresses to any orientation $\vec{n}$ and $\vec{t}$.

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

$$\vec{n}=n_{x} \hat{i}+n_{y} \hat{j}+n_{z} \hat{k}$$

${n}=|$ left $\left{\right.$ begin ${a r r a y}{| l}_{-}{x} \& n_{-}{y} \& n_{-}{z} \backslash$ lend ${a r r a y} \backslash r_{i g h t} \wedge \wedge{T}$

$$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=1$$

## 数学代写|有限元方法代写Finite Element Method代考|Normal and shear components of tractions

$$\vec{T} n=\sigma_{n n} \cdot \vec{n}+\sigma_{n t} \cdot \vec{t}$$

$$\sigma_{n n}=\vec{T} n \cdot \vec{n} \quad \text { and } \quad \sigma n t=\vec{T} n \cdot \vec{t}$$

㸻引的法向分量 $\vec{T} n$ 是通过使用方程式找到的。(2.16) 和 (2.20) 如下:
$$\sigma n n=\vec{T} n \cdot \vec{n}=\left([\sigma]^{T} \cdot n\right) \cdot n=\quad\left[\left(\sigma x x n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) \hat{i}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) \hat{j}+\left(\tau_{x z}\right.\right.$$

$$\sigma_{n n}=\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) n_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) n_{y} \quad+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) n_{z}$$

$$\sigma n t=\left(\sigma_{x x} n_{x}+\tau_{y x} n_{y}+\tau_{z x} n_{z}\right) t_{x}+\left(\tau_{x y} n_{x}+\sigma_{y y} n_{y}+\tau_{z y} n_{z}\right) t_{y} \quad+\left(\tau_{x z} n_{x}+\tau_{y z} n_{y}+\sigma_{z z} n_{z}\right) t_{z}$$

## 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|JEE350

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。

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

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

In this work, modeling refers to mathematical formulation of a physical process. This requires background in the related subjects, certain mathematical tools, and experimental observations. In Chapter 2, we present the formulation of models for deformation of elastic solids and transfer and storage of thermal energy in solids and fluids. Solution of the mathematical model can be a challenging task and forms the general background of this work. Analytical solutions which can be expressed as relatively straight forward relationships between the dependent and independent variables exist only for a relatively small number of situations where the geometry and the physical nature of the problem can be simplified. Numerical methods are used otherwise. Among the numerical solution methods for solving PDEs are the finite difference, variational, and finite element methods.

The finite difference method (FDM) is implemented on the differential form of the BVP. The derivative operators of the PDE are approximated by finite difference operators. The solution domain is discretized in to a grid, and the unknowns are the values of the dependent variable at the nodes. The discretized version of the PDE is evaluated at each grid point. This results in a set of algebraic equations which can be represented in matrix form,
$$[K]{D}={R}$$
where $[K]$ is the stiffness matrix representing the discretized form of the partial derivatives, ${D}$ is the vector of unknown nodal values of the dependent variable, and ${R}$ is the loading vector representing the external effects. The boundary conditions often require specialized treatment of the finite difference operators and modify the $[K]$ matrix. The FDM is effective over relatively simple shapes such as rectangular and cylindrical domains in two-dimensional problems and parallelepiped or spherical domains in three-dimensional problems.

## 数学代写|有限元方法代写Finite Element Method代考|Mathematical modeling of physical systems

The goal of this chapter is to give brief descriptions to modeling of deformation of linear elastic solids and thermal energy transfer and storage in physical systems. More detailed discussion of these topics can be found in the specialized references provided at the end of this chapter. Our goal is to demonstrate how to obtain mathematical models (representations) of physical systems by using the fundamental laws of physics. Thus, we will show that deformation of elastic solids can be describéd by using Newton’s laws of motion. This will reesult in equations of motion represented as partial differential equations. Vibration of a long and slender bar (Section 2.1), deflection of a general deformable body (Section 2.2), and deflection of beams (Section 2.3) constitute examples of such systems. The principle of conservation of energy will be used to describe effects of heat transfer in a continuum (Section 2.4).

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. In this section, the equations of motion for small deflections of linear, elastic materials are presented. In particular, we are interested in small deformations of linear, elastic solids. To this end, following are discussed: $i$ ) concepts of external and internal forces and the concept of stress, ii) elastic deformations and the concept of small strain, iii) linear elastic constitutive relations, iv) balance laws, and $v$ ) total potential energy of a deformable body.

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

$$[K] D=R$$

## 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|Find2022

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。

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

## 数学代写|有限元方法代写Finite Element Method代考|Boundary and initial value problems

Consider a system with dependent variables $u, v$, and $w$ defined over a domain $\Omega$, which itself occupies a subsection of space (Fig. 1.1). In general, each variable can take different values at different points in the domain and these values can also vary in time. Spatial position of a point $P$ in the domain $\Omega$ can be identified with respect to a spatial reference system (e.g., $(x, y, z)$ ). If the position of point $P$ also varies in time, the position of point $P$ is said to be time dependent. Thus, for example, if $u$ is a function of space and time $u=u(x, y, z, t)$. In these notes, we will consider boundary value problems (BVPs) and initial value problems that are formulated by using PDEs. A very general representation of such a problem can be given as follows:
$$\mathcal{L}(u, v, w)=f \text { in } \Omega, \text { for } 0 \leq t \leq \tau$$
where $\mathcal{L}(\cdot)$ is a differential operator of independent spatial variables $x, y, z$ and time $t, f=f(x, y, z, t)$ is typically a function that represents the internal effects that act on the system, and $\tau$ is the duration of interest.

The dependent variables interact with the outside of the domain $\Omega$ through the boundary $\Gamma$ of the domain, and typically experience changes as a result of the external effects that are imposed on the boundary. These external effects are known as the boundary conditions which depend on the physics of the problem.

The Dirichlet boundary condition represents a prescribed value for a dependent variable,
$$u=u_{b}(t) \text { on } \Gamma_{E}$$
Here the variable $u$ of the solution domain is prescribed to $u_{b}$ on a segment of the boundary $\Gamma_{E}$. In general, this prescribed variable can be a function of time $t$. The Dirichlet boundary condition is also known as the essential boundary condition.
The von Neumann boundary condition typically describes the external effects that cause a change in the system. Such effects include external forces, heat flow, etc. As we will demonstrate later in the notes, the von Neumann boundary conditions are typically represented as follows:
$$\mathcal{B}(u, v, w)=g(t) \text { on } \Gamma_{N}$$
where $\mathcal{B}(\cdot)$ is another differential operator, $g$ is a given function, and $\Gamma_{N}$ represents the segment of the boundary over which the von Neumann boundary condition is applied. The von Neumann boundary condition, also known as the natural boundary condition or the nonessential boundary condition, can also vary in time.

## 数学代写|有限元方法代写Finite Element Method代考|Boundary value problems

In some problems, only the steady state of the dependent variables is of interest and the temporal variation is neglected (or negligible). Thus, for example, $u$ becomes only a function of the spatial dimensions $u=u(x, y, z)$. A steady state boundary value problem can be formulated by dropping the time dependence as follows:
$$\mathcal{L}(u, v, w)=f \text { in } \Omega$$
where for a boundary value problem $\mathcal{L}(\cdot)$ is a differential operator of the independent spatial variables $(x, y, z)$ and $f=f(x, y, z)$. A steady state boundary value problem is also subject to the Dirichlet and/or von Neumann conditions on the boundary of the domain.
Example 1.1 Equation of motion of a solid bar
a) Derive the equation of motion of an elastic bar in terms of its deflection $u(x, t)$. Initially, assume that the bar has a variable cross-sectional area $A(x)$ and that it is subjected to distributed axial load $q(x, t)$ and a concentrated force $F$ at its free end as shown in Fig. 1.2. Also assume small deflections, linear elastic material behavior with constant elastic modulus $E$, and constant mass density $\rho$.
b) Obtain the steady state solution for the case of constant cross-section and zero distributed force.

Solution 1.1a: The solution domain $\Omega$ for this problem spans $0<x<L$. The boundaries $\Gamma$ of the solution domain are located at $x=0$ and $x=L$. Internal forces develop in the bar in response to external loading. The internal normal force $N(x)$ at the cross-section $x$ can be defined as follows:
$$N(x)=\bar{\sigma}(x) A(x)$$
where the average normal stress $\bar{\sigma}$ is defined as follows:
$$\bar{\sigma}(x)=\frac{1}{A(x)} \int_{A(x)} \sigma d A$$
and where $\sigma$ is the internal normal stress, $A$ is the cross-sectional area of the bar.

## 数学代写|有限元方法代写Finite Element Method代考|Boundary and initial value problems

$$\mathcal{L}(u, v, w)=f \text { in } \Omega, \text { for } 0 \leq t \leq \tau$$

Dirichlet 边界条件表示因变量的规定值，
$$u=u_{b}(t) \text { on } \Gamma_{E}$$

$$\mathcal{B}(u, v, w)=g(t) \text { on } \Gamma_{N}$$

## 数学代写|有限元方法代写Finite Element Method代考|Boundary value problems

$$\mathcal{L}(u, v, w)=f \text { in } \Omega$$

b) 获得恒定截面和零分布力情况下的稳态解。

$$N(x)=\bar{\sigma}(x) A(x)$$

$$\bar{\sigma}(x)=\frac{1}{A(x)} \int_{A(x)} \sigma d A$$

## 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。

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

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

From the material presented in Sections $1.2-1.4$ it is clear that one could entertain any of the methods of approximation listed in Section 1.2, spacetime coupled, or space-time decoupled approaches for obtaining numerical solutions of the IVPs.

In this book we only consider finite element method in conjunction with space-time coupled and space-time decoupled approaches for obtaining numerical solutions of the IVPs. The finite element method for both approaches has rigorous mathematical foundation, hence in this approach it is always possible to ascertain feasibility, stability, and accuracy of the resulting computational processes. Error estimation, error computation, convergence, and convergence rates are additional meritorious features of the finite element processes for IVPs compared to all other methods listed in Section 1.2.
In the following sections we present a brief description of space-time coupled and space-time decoupled finite element processes, their merits and shortcomings, time integration techniques for ODEs in time resulting from decoupling space and time, stability of computational processes, error estimation, error computation, and convergence.

Some additional topics related to linear structural and linear solid mechanics such as mode superposition techniques of obtaining time evolution are also discussed.

## 数学代写|有限元方法代写Finite Element Method代考|Space-time coupled finite element method

In the initial value problem (1.1), the operator $A$ is a space-time differential operator. Thus, in order to address STFEM for totality of all IVPs in a problem- and application-independent fashion we must mathematically classify space-time differential operators appearing in all IVPs into groups. For these groups of space-time operators we can consider space-time methods of approximation such as space-time Galerkin method (STGM), space-time Petrov-Galerkin method (STPGM), space-time weighted residual method (STWRM), space-time Galerkin method with weak form (STGM/WF), spacetime least squares method or process (STLSM or STLSP), etc., thereby addressing totality of all IVPs. The space-time integral forms resulting from these space-time methods of approximation are necessary conditions.

By making a correspondence of these integral forms to the space-time calculus of variations we can determine which integral forms lead to unconditionally stable computational processes. The space-time integral forms that satisfy all elements of the space-time calculus of variations are termed space-time variationally consistent (STVC) integral forms. These integral forms result in unconditionally stable computational processes during the entire evolution. The integral forms in which one or more aspects of the space-time calculus of variations is not satisfied are termed space-time variationally inconsistent (STVIC) integral forms. In STVIC integral forms, unconditional stability of the computations is not always ensured.

## 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|MECH ENG 4118

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。

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## 有限元方法代写

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|Find 2022

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。

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

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

The physical processes encountered in all branches of sciences and engineering can be classified into two major categories: time-dependent processes and stationary processes. Time-dependent processes describe evolutions in which quantities of interest change with time. If the quantities of interest cease to change in an evolution then the evolution is said to have reached a stationary state. Not all evolutions have stationary states. The evolutions without a stationary state are often referred to as unsteady processes. Stationary processes are those in which the quantities of interest do not depend upon time. For a stationary process to be valid or viable, it must correspond to the stationary state of an evolution. Every process in nature is an evolution. Nonetheless it is sometimes convenient to consider their stationary state. In this book we only consider non-stationary processes, i.e. evolutions that may have a stationary state or may be unsteady.

A mathematical description of most stationary processes in sciences and engineering often leads to a system of ordinary or partial differential equations. These mathematical descriptions of the stationary processes are referred to as boundary value problems (BVPs). Since stationary processes are independent of time, the partial differential equations describing their behavior only involve dependent variables and space coordinates as independent variables. On the other hand, mathematical descriptions of evolutions lead to partial differential equations in dependent variables, space coordinates, and time and are referred to as initial value problems (IVPs).

In case of simple physical systems, the mathematical descriptions of IVPs may be simple enough to permit analytical solutions. However, most physical systems of interest may be quite complicated and their mathematical description (IVPs) may be complex enough not to permit analytical solutions. In such cases, two alternatives are possible. In the first case, one could undertake simplifications of the mathematical descriptions to a point that analytical solutions are possible. In this approach, the simplified forms may not be descriptive of the actual behavior and sometimes this simplification may not be possible at all. In the second alternative, we abandon the possibility of theoretical solutions altogether as viable means of solving complex practical problems involving IVPs and instead we resort to numerical meth-ods for obtaining numerical solutions of IVPs. The finite element method (FEM) is one such method of solving IVPs numerically and constitutes the subject matter of this book. Before we delve deeper into the FEM for IVPs, it is perhaps fitting to discuss a little about the broader classes of available methods for obtaining numerical solutions of IVPs.

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

We note that since $\phi=\phi(x, t)$, the solution exhibits simultaneous dependence on spatial coordinates $x$ and time $t$. This feature is intrinsic in the mathematical description (1.1) of the physics.

Thus, the most rational approach to undertake for the solution of (1.1) (approximate or otherwise) is to preserve simultaneous dependence of $\phi$ on $x$ and $t$. Such methods are known as space-time coupled methods. Broadly speaking, in such methods time $t$ is treated as another independent variable in addition to spatial coordinates. Fig. $1.1$ shows space-time domain $\bar{\Omega}{x t}=$ $\Omega{x t} \cup \Gamma ; \Gamma=\cup_{i=1}^{4} \Gamma_{i}$ with closed boundary $\Gamma$. For the sake of discussion, as an example we could have a boundary condition $(\mathrm{BC})$ at $x=0 \forall t \in[0, \tau]$, boundary $\Gamma_{1}$, as well as at $x=L \forall t \in[0, \tau]$, boundary $\Gamma_{2}$, and an initial condition $(\mathrm{IC})$ at $t=0 \forall x \in[0, L]$, boundary $\Gamma_{3}$. Boundary $\Gamma_{4}$ at final value of time $t=\tau$ is open, i.e. at this boundary only the evolution (the solution of (1.1) subjected to these $\mathrm{BCs}$ and $\mathrm{IC})$, will yield the function $\phi(x, \tau)$ and its spatial and time derivatives.

When the initial value problem contains two spatial coordinates, we have space-time slab $\bar{\Omega}{x t}$ shown in Fig. $1.2$ in which $$\Omega{x t}=\left(0, L_{1}\right) \times\left(0, L_{2}\right) \times(0, \tau)$$
is a prism. In this case $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}$, and $\Gamma_{4}$ are faces of the prism (surfaces). For illustration, the possible choices of BCs and ICs could be: BCs on $\Gamma_{1}=$ $A D D_{1} A_{1}$ and $\Gamma_{2}=B C C_{1} B_{1}$, IC on $\Gamma_{3}=A B C D$, and $\Gamma_{4}=A_{1} B_{1} C_{1} D_{1}$ is the open boundary. This concept of space-time slab can be extended for three spatial dimensions and time. Using space-time domain shown in Fig. $1.1$ or $1.2$ and treating time as another independent variable, we could consider the following methods of approximation.

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

$x=0 \forall t \in[0, \tau]$ ，边界 $\Gamma_{1}$ ，以及在 $x=L \forall t \in[0, \tau]$, 边界 $\Gamma_{2}$ ，和一个初始条件 (IC)在 $t=0 \forall x \in[0, L]$ ，边 界 $\Gamma_{3}$. 边界 $\Gamma_{4}$ 在最终时间值 $t=\tau$ 是开放的，即在这个边界上只有演化 ( (1.1) 的解受到这些BCs和IC)，将产 生函数 $\phi(x, \tau)$ 及其空间和时间导数。

$$\Omega x t=\left(0, L_{1}\right) \times\left(0, L_{2}\right) \times(0, \tau)$$

## 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。