### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MAST90083

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Use of Radiation Distribution Factors When Some Surface Net Heat Fluxes Are Specified

We now address the situation where surface elements $1,2, \ldots, N$ have specified net heat fluxes and surfaces $N+1, N+2, \ldots, n$ have specified temperatures. To begin, let us consider the special case where $N=1$; that is, where only the first of $n$ surface elements has a specified net heat flux, with the remaining surfaces having specified temperatures. In this case Eq. (3.35) may be written
$$q_{1}=\varepsilon_{1}\left[\left(1-D_{11}\right) \sigma T_{1}^{4}-\sum_{j=2}^{n} \sigma T_{j}^{4} D_{i j}\right]$$
Equation (3.36) can be solved explicitly for the unknown surface temperature $T_{1}$ in terms of the known surface net heat flux $q_{1}$ and the known surface temperatures; that is,
$$\sigma T_{1}^{4}=\frac{1}{1-D_{11}}\left(\frac{q_{1}}{\varepsilon_{1}}+\sum_{j=2}^{n} \sigma T_{j}^{4} D_{i j}\right)$$
In the more general case where several surface elements have specified net heat fluxes we can rearrange Eq. (3.35) to obtain
$$q_{i}+\varepsilon_{i} \sum_{j=N+1}^{n} \sigma T_{j}^{4} D_{i j}=\varepsilon_{i} \sum_{j=1}^{N} \sigma T_{j}^{4}\left(\delta_{i j}-D_{i j}\right), \quad 1 \leq i \leq N_{\star}$$

Equation (3.38) represents $N$ equations in the $N$ unknown surface temperatures in terms of the $N$ known surface net heat fluxes and the $n-N$ known surface temperatures. It can be rewritten symbolically as
$$\Theta_{i}=\Psi_{i j} \Omega_{j},$$
where
$$\Theta_{i}=q_{i}+\varepsilon_{i} \sum_{j=N+1}^{n} \sigma T_{j}^{4} D_{i j}, \quad 1 \leq i \leq N,$$
is a known vector,
$$\Psi_{i j}=\varepsilon_{i}\left(\delta_{i j}-D_{i j}\right), \quad 1 \leq i \leq N, \quad 1 \leq j \leq N,$$
and
$$\Omega_{j}=\sigma T_{j}^{4}, \quad 1 \leq j \leq N,$$
is an unknown vector whose elements are sought. We obtain the unknown surface temperatures by inverting the matrix defined by Eq. (3.41) and then using it to operate on the vector defined by Eq. (3.40); that is,
$$\Omega_{i}=\left[\Psi_{i j}\right]^{-1} \Theta_{j}, \quad 1 \leq i \leq N .$$
The unknown surface net heat fluxes are then computed using Eq. (3.35) applied over the range $N+1 \leq i \leq n$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Bidirectional Spectral Surfaces

Experience confirms that reflection from a surface is generally neither diffuse nor specular. Rather, at a given wavelength the distribution of reflected energy depends on the mechanical and chemical preparation

of the surface and on the direction of incidence. Diffuse and specular reflections represent the two extremes of bidirectional reflectivity. Both extremes may be approached but rarely achieved in practice. Various theories have been proposed for predicting bidirectional spectral reflection and directional spectral emission and absorption for generic surfaces. The interested reader is referred to Chapter 4 in Ref. [1], where this topic is pursued in more detail. However, metrology remains the only sure path to surface models that accurately capture the directional spectral behavior of real surfaces.

A simple two-component model for directional reflectivity was introduced in Chapter 1 (Figure 1.9), where it is suggested that a directional reflection pattern can be somewhat approximated as a suitably weighted combination of spectral and diffuse reflection. Different versions of this approximation would have to be applied for each wavelength of interest, using different weight factors for each wavelength. Any success this approach might have would be due in large measure to the fact that the distribution of radiant energy within an enclosure is governed by integral equations rather than by differential equations. Integration at least partially “averages out” positive and negative excursions from reality, as illustrated in Figure 4.1. Inspection of the figure reveals that, even though

$f_{1}(x)$ is a more detailed and presumably more accurate description of local behavior than $f_{2}(x)$, it may nonetheless be true, to an acceptable approximation, that
$$\int_{0}^{1} f_{1}(x) d x \cong \int_{0}^{1} f_{2}(x) d x$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Principles Underlying a Practical Bidirectional Reflection Model

Useful bidirectional reflection models are generally based on measurements, although they are frequently informed by theory. As we learned in Sections $2.12$ and $2.13$, the optical behaviors of electrically non-conducting (dielectric) and electrically conducting (metal) surfaces are fundamentally different. In general, metal surfaces are strong specular reflectors while dielectric surfaces tend to be weak diffuse reflectors. In both cases the directional distribution of reflected radiation is known to be strongly influenced by the topography, chemical state, and degree of contamination of the surface. With the exception of certain optical components (such as mirrors, lenses, and filters), it is unlikely that a bidirectional spectral reflectivity model based entirely on theory would accurately represent the optical behavior of a surface of practical engineering interest. Therefore, in cases where high accuracy is required, a successful surface optical model must be at least semiempirical if not based entirely on measurements of the optical behavior of the surface to be modeled.

In this chapter we first demonstrate the application of semiempirical approaches for two surface coatings engineered to exhibit specific $-$ and somewhat unique – optical behaviors. In the first example we consider a highly absorptive commercial coating whose small component of reflectivity is highly directional to the point of being almost specular, and in the second example we consider another commercial coating that is highly reflective but whose reflectivity is nearly diffuse. Both of these coatings are widely used in optical applications requiring an unusual combination of both metallic and dielectric behaviors. We then follow up by presenting a completely general approach suitable for applications where a full set of experimental data is available.

We begin by recalling the bidirectional spectral reflectivity from Chapter 2,\begin{aligned} \rho_{\lambda}^{\prime \prime} &=\rho\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right) \equiv \frac{d i_{\lambda, r}\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right)}{i_{\lambda, i}\left(\lambda, \vartheta_{i}, \varphi_{i}\right) \cos \vartheta_{i} d \Omega_{i}} \ & \equiv B R D F\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right) \end{aligned}

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors When Some Surface Net Heat Fluxes Are Specified

q1=e1[(1−D11)σ吨14−∑j=2nσ吨j4D一世j]

σ吨14=11−D11(q1e1+∑j=2nσ吨j4D一世j)

q一世+e一世∑j=ñ+1nσ吨j4D一世j=e一世∑j=1ñσ吨j4(d一世j−D一世j),1≤一世≤ñ⋆

θ一世=Ψ一世jΩj,

θ一世=q一世+e一世∑j=ñ+1nσ吨j4D一世j,1≤一世≤ñ,

Ψ一世j=e一世(d一世j−D一世j),1≤一世≤ñ,1≤j≤ñ,

Ωj=σ吨j4,1≤j≤ñ,

Ω一世=[Ψ一世j]−1θj,1≤一世≤ñ.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Bidirectional Spectral Surfaces

F1(X)是对局部行为的更详细和可能更准确的描述F2(X)，但在可接受的近似值上，它可能是正确的，即

∫01F1(X)dX≅∫01F2(X)dX

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