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• 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 conditions for 1D heat transfer

The physics of heat transfer dictate that the temperature $T$, or the heat flux $f$ to take on specified values on the boundary of the body. In Chapter 3 , it is shown that only $T$ or $f$ can be specified at a given position on the boundary; but, they cannot be defined simultaneously at the same position. The boundary conditions of a 1D heat transfer problem are defined as follows:
at $x=x_0$ either $T=T_b$ or $\left(-k_x \frac{\partial T}{\partial x}\right) n=f_B$ at $x=x_L$ either $T=T_b$ or $\left(-k_x \frac{\partial T}{\partial x}\right) n=f_B$
Note that $n$ represents the unit outward normal of the boundary, $n=-1$, $+1$ at $x=x_0, x_L$ respectively, $T_b$ is the specified (prescribed) value of the temperature on the boundary and $f_B$ is the prescribed heat flux. Depending on the conditions the prescribed heat flux is dictated by one (or combination) of the following physical mechanisms ${ }^3$,
Insulated Boundary: $f_B=f_B(t)$
Convected Boundary: $f_B=h\left(T-T_{\infty}\right)$
Radiated Boundary: $f_B=\sigma \psi\left(T^4-T_s^4\right)$
In addition, to the boundary conditions, the time dependent nature of the problem requires the initial condition of the body temperature to be specified,
$$T(x, 0)=T^{(0)}(x)$$

数学代写|有限元方法代写Finite Element Method代考|Heat transfer in a three-dimensional solid

Consider an arbitrary domain $\Omega$ enclosed by a surface $\Gamma$ as shown in Fig. 2.22. Conservation of (heat) energy in such a domain requires the following balance statement to be satisfied [4]: rate of change of heat energy in this volume should be equal to the sum of the heat energy flowing across the boundary of the volume per unit time and the heat energy generated inside the volume per unit time.
Flow of thermal energy per unit area and per unit time in a continuous medium is characterized by the heat flux vector $\vec{f}$. The heat flux that contributes to the energy balance is the normal component of $\vec{f}$ with respect to the boundary it is crossing. This is expressed by $\vec{f} \cdot \vec{n}$ where $\vec{n}$ is the unit outward normal of the boundary. Thus a positive value for $\vec{f} \cdot \vec{n}$ indicates an outward flow of energy from the boundary.
The thermal energy stored within the volume is expressed by,
thermal energy $=\int_{\Omega} c \rho T d \Omega$
where $c$ is the heat capacity, $\rho$ is the mass density and $V$ is the volume of the material. The conservation of thermal energy can then be expressed mathematically as follows:
$$\frac{\partial}{\partial t} \int_{\Omega} c \rho T d \Omega=-\int_{\Gamma} \vec{f} . \vec{n} d \Gamma+\int_{\Omega} Q d \Omega$$
where $Q$ is the internal energy generated per unit volume and unit time. The $-$ sign in this equation indicates that thermal energy is lost from the volume.
Recall that for any vectorial variable $\vec{v}$, the divergence theorem [9] states the integral of the gradient of the vector field inside the volume is equal to the surface integral of the normal component of the vector field as follows:
$$\int_{\Omega} \nabla \cdot \vec{v} d \Omega=\int_{\Gamma} \vec{v} \cdot \vec{n} d \Gamma$$
where $\nabla$ is the gradient operator, defined in Cartesian coordinates, as follows:
$$\nabla(\cdot)=\frac{\partial(\cdot)}{\partial x} \hat{i}+\frac{\partial(\cdot)}{\partial y} \hat{j}+\frac{\partial(\cdot)}{\partial z} \hat{k}$$

有限元方法代考

数学代写|有限元方法代写Finite – Element Method代考|一维传热的边界条件

at $x=x_0$ 要么 $T=T_b$ 或 $\left(-k_x \frac{\partial T}{\partial x}\right) n=f_B$ 在 $x=x_L$ 要么 $T=T_b$ 或 $\left(-k_x \frac{\partial T}{\partial x}\right) n=f_B$

$$T(x, 0)=T^{(0)}(x)$$

数学代写|有限元方法代写Finite – Element Method代考|三维固体中的传热

$$\frac{\partial}{\partial t} \int_{\Omega} c \rho T d \Omega=-\int_{\Gamma} \vec{f} . \vec{n} d \Gamma+\int_{\Omega} Q d \Omega$$

$$\int_{\Omega} \nabla \cdot \vec{v} d \Omega=\int_{\Gamma} \vec{v} \cdot \vec{n} d \Gamma$$

$$\nabla(\cdot)=\frac{\partial(\cdot)}{\partial x} \hat{i}+\frac{\partial(\cdot)}{\partial y} \hat{j}+\frac{\partial(\cdot)}{\partial z} \hat{k}$$

有限元方法代写

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MATLAB代写

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