### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MATH 483

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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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Monte Carlo Ray-Trace Method and the Radiation Distribution Factor

The Monte Carlo ray-trace (MCRT) method is a two-step process. The first step involves estimation of the radiation distribution factor matrix

$D_{i j}$, and the second step involves multiplication of $D_{i j}$ by a vector whose components are the source strengths of the surfaces making up an enclosure. Individual elements of the distribution factor matrix may be interpreted as the sensitivity of the power absorbed by surface $j$ to the power emitted by surface $i$; that is,
$$D_{i j} \equiv \partial Q_{i j} / \partial Q_{i},$$
where $Q_{i j}$ is the total power in watts emitted from surface $i$ that is absorbed on surface $j$, and $Q_{i}$ is the total power emitted from surface $i$. If $Q_{i}$, the total power emitted from surface $i$, is known and the distribution factor matrix $D_{i j}$ is available for any combination of two surfaces $i$ and $j$, then the heat absorbed by surface $j$ is
$$Q_{j}=\sum_{\mathrm{i}=1}^{\mathrm{n}} Q_{i j}, \quad 1 \leq j \leq n$$
where $n$ is the total number of surfaces and
$$Q_{i j}=Q_{i} D_{i j} .$$
Calculation of $Q_{i}$ for a given surface condition has already been treated in Chapter 2. The current chapter deals with calculation of the distribution factor $D_{i j}$ and its subsequent use in determining the distribution of thermal radiation among surfaces of an enclosure.

If we can somehow obtain the $D_{i j}$ matrix, we already have the answer to one of the most pressing questions in optical and thermal design: “How sensitive is the heat absorbed by a specific surface $j$ to the heat emitted from a specific surface $i$ ?’ Consideration of Eqs. (3.2) and (3.3) reveals that calculation of the power distribution among the surfaces of an enclosure is a straightforward vector multiplication once the distribution factor matrix is known. Knowledge of the distribution factor matrix greatly facilitates thermal or optical design because it permits targeted analysis of heat transfer among a limited number of surfaces of particular interest.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Properties of the Total Radiation Distribution Factor

It can be demonstrated (see Problems 3.1-3.3) that, subject to the graybody assumption defined in Chapter 2 , the total radiation distribution factor has the following useful properties:

1. Conservation of energy:
$$\sum_{j=1}^{n} D_{i j}=1, \quad 1 \leq i \leq n$$
2. Reciprocity:
$$\varepsilon_{i} A_{i} D_{i j}=\varepsilon_{j} A_{j} D_{j i}, \quad 1 \leq i \leq n, \quad 1 \leq j \leq n$$
3. Combination of 1 and 2 :
$$\sum_{i=1}^{n} \varepsilon_{i} A_{i} D_{i j}=\varepsilon_{j} A_{j}, \quad 1 \leq j \leq n$$
In Eqs. (3.4)-(3.6), $n$ is the number of surface elements making up the enclosure, $\varepsilon$ is the emissivity, and $A$ is the surface area.

Equation (3.6), which is obtained by summing both sides of Eq. (3.5) over the index $i$ and then substituting Eq. (3.4) into the result, is useful for detecting and eliminating errors made during calculation of the distrihution factors for an enclosure. It can also he used to provide a statistically meaningful measure of the accuracy with which the distribution factor matrix for a given enclosure has been computed. The conservation of energy relationship, Eq. (3.4), and the reciprocity relationship, Eq. (3.5), are also useful for detecting errors or for finding unknown distribution factors from known distribution factors using distribution factor algebra. However, note that these relationships cannot be used both for error detection and for finding unknown distribution factors in the same enclosure.

Finally, we note that distribution factors can also be defined for radiation entering the enclosure through an opening $o$ with a specified directional distribution; e.g., collimated in a specific direction. In this case, the appropriate relation for defining the distribution of radiation on the surface elements making up the enclosure is
$$Q_{v j}=Q_{s} D_{v j,} \quad 1 \leq j \leq n,$$
where $Q_{o}$ is the power (W) entering the enclosure through opening $o$ and $D_{o j}$ is the fraction of this power absorbed by surface element $j$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Estimation of the Distribution Factor Matrix Using

The Monte Carlo ray-trace method is a statistical approach in which the analytical solution of a problem is avoided in favor of a numerical

simulation whose outcome is expected to be the same as that of the analytical solution but which is easier to obtain. In practice, this often means that the numerical simulation is obtainable while the equivalent analytical solution is for all practical purposes unobtainable.

In the case of a thermal radiation problem, a given quantity of energy is uniformly divided into a large number $N_{i}$ of discrete energy bundles. Here, we feel no obligation to distinguish between energy and power, as in the steady state the former is the latter multiplied by an appropriate time interval. The $N_{i}$ energy bundles are followed from their emission by surface element $i$ (or from their entry into the enclosure through opening $o$ ), through a series of reflections from other surface elements, to their absorption on surface element $j$ of the enclosure. The optical models of the enclosure walls and the laws of probability are used to determine the number of energy bundles $N_{i j}$ absorbed by a given surface element $j$, where $j=i$ is a possibility.

A consequence of the definition of the radiation distribution factor is that its value approaches the ratio of $N_{i j}$ to $N_{i}$ as $N_{i}$ increases; that is,
$$D_{i j} \approx N_{i j} / N_{i} .$$
The accuracy with which $N_{i j} / N_{i}$ estimates $D_{i j}$ depends on the number of energy bundles traced from surface $i$. Of course, as in any model-based analysis, it also depends on the accuracy with which the enclosure geometry and the surface optical models are known. Furthermore, as shown in Chapter 7, the uncertainty associated with the estimate corresponding to the value of $N_{i}$ can be stated to within a specified confidence interval. Moreover, the ultimate accuracy of the solution to a radiation heat transfer problem using the MCRT method depends in a statistically meaningful way on the product of the number of surfaces $n$ making up the enclosure and the number of rays $N$ traced per surface. The ability of the MCRT method to attribute a confidence level to the uncertainty of the results obtained is a compelling argument for its use.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Monte Carlo Ray-Trace Method and the Radiation Distribution Factor

D一世j，第二步涉及乘法D一世j由一个向量组成，其分量是构成外壳的表面的源强度。分布因子矩阵的各个元素可以解释为表面吸收功率的灵敏度j到表面发出的功率一世; 那是，

D一世j≡∂问一世j/∂问一世,

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Properties of the Total Radiation Distribution Factor

1. 能量守恒：
∑j=1nD一世j=1,1≤一世≤n
2. 互惠：
e一世一个一世D一世j=ej一个jDj一世,1≤一世≤n,1≤j≤n
3. 1 和 2 的组合：
∑一世=1ne一世一个一世D一世j=ej一个j,1≤j≤n
在方程式中。(3.4)-(3.6),n是组成外壳的表面元素的数量，e是发射率，并且一个是表面积。

D一世j≈ñ一世j/ñ一世.

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## MATLAB代写

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