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# 数学代写|有限元方法代写finite differences method代考|One-Dimensional Heat Transfer

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## 数学代写|有限元代写Finite Element Method代考|One-Dimensional Heat Transfer

The direct approach can also be used to develop finite element models of one-dimensional heat transfer. The relations between temperatures and heats at the ends of a surface insulated solid bar or two surfaces of a plane wall can be developed using the basic principles of heat transfer. We have
\begin{aligned} \text { temperature gradient } & =\text { difference in temperature } / \text { length } \ \text { heat flux, } q & =\text { conductivity } \times(- \text { temperature gradient }) \ \text { heat, } Q & =\text { heat flux } \times \text { area of cross section } \end{aligned}
Then, if there is no internal heat generation and the temperature is assumed to vary linearly between the ends of the bar of length (or plane wall of thickness) $h_e$, cross-sectional area $A_e$, and conductivity $k_e$, the heats at the left and right ends of the bar are
\begin{aligned} & Q_1^e=A_e q_1^e=-A_e k_e \frac{T_2^e-T_1^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_1^e-T_2^e\right) \ & Q_2^e=A_e q_2^e=-A_e k_e \frac{T_1^e-T_2^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_2^e-T_1^e\right) \end{aligned}
Technically, both $Q_1^e$ and $Q_2^e$ are heat inputs. In matrix form, we have
$$\frac{A_e k_e}{h_e}\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} T_1^e \ T_2^e \end{array}\right}=\left{\begin{array}{l} Q_1^e \ Q_2^e \end{array}\right}$$

## 数学代写|有限元代写Finite Element Method代考|Model Boundary Value Problem

Consider the problem of finding the function $u(x)$ that satisfies the differential equation
$$-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u-f=0 \text { for } 0<x<L$$
and the boundary conditions
$$u(0)=u_0,\left.\quad\left(a \frac{d u}{d x}\right)\right|_{x=L}=Q_L$$
where $a=a(x), c=c(x), f=f(x), u_0$, and $Q_L$ are known quantities, called the data of the problem. Equation (3.4.1) arises in connection with the analytical description of many physical processes. For example, conduction and convection heat transfer in the bar shown in Fig. 3.4.1(a), flow through channels and pipes, transverse deflection of cables, axial deformation of bars shown in Fig. 3.4.1(b), and many other physical processes are described by Eq. (3.4.1). A sample list of field problems described by Eq. (3.4.1) when $c(x)=0$ is presented in Table 3.4.1. Thus, if we can develop a numerical procedure by which Eq. (3.4.1) can be solved for all possible boundary conditions, the procedure can be used to solve all field problems listed in Table 3.4.1. This fact provides us with the motivation to use Eq. (3.4.1) as the model second-order equation in one dimension.

## 数学代写|有限元代写Finite Element Method代考|One-Dimensional Heat Transfer

\begin{aligned} \text { temperature gradient } & =\text { difference in temperature } / \text { length } \ \text { heat flux, } q & =\text { conductivity } \times(- \text { temperature gradient }) \ \text { heat, } Q & =\text { heat flux } \times \text { area of cross section } \end{aligned}

\begin{aligned} & Q_1^e=A_e q_1^e=-A_e k_e \frac{T_2^e-T_1^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_1^e-T_2^e\right) \ & Q_2^e=A_e q_2^e=-A_e k_e \frac{T_1^e-T_2^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_2^e-T_1^e\right) \end{aligned}

$$\frac{A_e k_e}{h_e}\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} T_1^e \ T_2^e \end{array}\right}=\left{\begin{array}{l} Q_1^e \ Q_2^e \end{array}\right}$$

## 数学代写|有限元代写Finite Element Method代考|Model Boundary Value Problem

$$-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u-f=0 \text { for } 0<x<L$$

$$u(0)=u_0,\left.\quad\left(a \frac{d u}{d x}\right)\right|_{x=L}=Q_L$$

## MATLAB代写

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