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Since this book is concerned with the finite element solutions of differential equations arising mostly in engineering, although the basic developments are also valid for problems in applied sciences, it is useful to review the governing equations, including the boundary conditions (but not the initial conditions) of a continuum occupying a closed bounded region $\Omega$ with boundary $\Gamma$ which serve as references in the coming chapters. Although sufficient background is given every time we consider a differential equation
or a problem for analysis, the summary of equations included in this section serves as a quick reference. The equations are summarized under three subject areas of engineering that will receive considerable attention in this book: heat transfer, fluid mechanics, and solid mechanics. Since the present exposure is only to summarize the equations, the readers may wish to consult books that contain detailed treatment of the subjects.

数学代写|有限元代写Finite Element Method代考|Heat Transfer

The principle of conservation of energy applied to a solid medium yields
$$\rho c_v \frac{\partial T}{\partial t}-\boldsymbol{\nabla} \cdot(k \nabla T)=g$$
where $\nabla$ is the del operator (see Section 2.2.1.3), $T$ is the temperature, $g$ is the rate of internal heat generation per unit volume, $k$ is the conductivity of the (isotropic) solid, $\rho$ is the density, and $c v$ is the specific heat at constant volume. The expanded form of Eq. (2.6.1) (for constant $\mathrm{k}$ ) in rectangular Cartesian system $(x, y, z)$ is given by
$$\rho c_v \frac{\partial T}{\partial t}-k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)=g$$
In cylindrical coordinate system $(r, \theta, z)$, Eq.(2.6.1) takes the form
$$\rho c_v \frac{\partial T}{\partial t}-k\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}+\frac{\partial^2 T}{\partial z^2}\right]=g$$
The second-order equations in Eqs. (2.6.1)-(2.6.3) are to be solved subjected to suitable boundary conditions. The boundary conditions involve specifying either the value of the temperature $T$ or balancing the heat flux normal to the boundary $q_n=\hat{\mathbf{n}} \cdot \mathbf{q}$ at a boundary point:
$$T=\hat{T} \quad \text { or } \quad q_n+\beta\left(T-T_{\infty}\right)=\hat{q}$$
where $\hat{T}$ and $\hat{q}$ denote the specified temperature and heat flux, respectively. Heat flux vector $\mathbf{q}$ is related to the gradient of temperature by Fourier’s heat conduction law (for the isotropic case)

数学代写|有限元代写Finite Element Method代考|Heat Transfer

$$\rho c_v \frac{\partial T}{\partial t}-\boldsymbol{\nabla} \cdot(k \nabla T)=g$$

$$\rho c_v \frac{\partial T}{\partial t}-k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)=g$$

$$\rho c_v \frac{\partial T}{\partial t}-k\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}+\frac{\partial^2 T}{\partial z^2}\right]=g$$

$$T=\hat{T} \quad \text { or } \quad q_n+\beta\left(T-T_{\infty}\right)=\hat{q}$$

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。