### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT31511

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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|Scattering and Refraction

The power incident to a surface that is neither absorbed locally nor reflected is then either scattered in a process analogous to reflection, or it is refracted. In the MCRT description of radiation heat transfer, scattering is modeled by subdividing the incident ray into many equal-power rays, with each scattered ray continuing in a direction determined by an appropriate scattering model. The complex phenomenon of scattering is treated in detail in Chapter 5 , which deals with radiation propagating through a participating medium. The simplest and most basic model for scattering, which is used in the early chapters of this book, is the assumption that scattering can be neglected, as is often the case in radiation heat transfer.

In the ray-trace description of geometrical optics, refraction refers to the abrupt change in direction of the ransmitted ray as it passes through an interface. The Snell-Descartes law, illustrated in Figure 1.11, represents reality very well, especially for interfaces between air and common materials used in the fabrication of lenses, filters, retarder plates, and windows. According to the Snell-Descartes law
$$\sin \left(\vartheta_{1}\right) / \sin \left(\vartheta_{2}\right)=n_{2} / n_{1},$$
where $n_{1}$ and $n_{2}$ are the refractive indices of the two materials whose interface provokes refraction. Problem $1.14$ is an important application of this principle in applied optics.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Meshing and Indexing

The MCRT method used throughout this book requires that the modeling space be subdivided into surface and volume elements, i.e., that it be appropriately meshed. While entire books have been written on this topic alone, the limited treatment offered here is adequate for the needs of most MCRT analyses. The meshes used in the MCRT method must be amenable to indexing. Indexing refers to the process of systematically numbering the surface and volume elements in such a way that the numbers, called indices, can be determined algorithmically from the coordinates of a point lying on a surface element or within a volume element.
Pedagogical considerations favor limitation of the discussion presented here to rectilinear spaces, i.e., to spaces that can be represented by rectangular solid blocks. As used here, the word “solid”‘ implies only that the spaces are three-dimensional. Many, if not most, enclosures of practical engineering interest can be accurately represented using a rectilinear mesh if care is taken to ensure that the surface element unit normal vectors represent the actual local curvature. The methods presented in this section can be extended to spaces consisting of trapezoidal, cylindrical (both circular and noncircular), and spheroidal solids.

Consider the hollow, three-dimensional rectilinear space illustrated in Figure 1.12. Use of the MCRT method often requires that the space be divided into $N$ surface elements and $n-N$ volume elements, with a unique number, or “index”, algorithmically assigned to each element. Furthermore, square surface elements and cubic volume elements are highly desirable. Finally, the resulting mesh must be sufficiently dense to assure adequate spatial resolution of the results obtained using an MCRT analysis. How do we go about satisfying all of these requirements? Consider the following numerical examples.

Electromagnetic (EM) waves, whose properties are explored in this chapter, carry energy from one location to another, even – indeed, especially – in a vacuum. The mathematical form of the magnitude of the electric field component of an EM wave propagating along the $x$-axis is
$$E(x, t)=E_{0} e^{i(k x-\omega t)},$$
where $E\left(\mathrm{~V} \mathrm{~m}^{-1}\right)$ is the instantaneous electric field strength at position $x(\mathrm{~m})$ and time $t(\mathrm{~s}), E_{0}$ is the amplitude of its oscillation, $k\left(\mathrm{~m}^{-1}\right)$ is

the wave number, and $\omega\left(\mathrm{r} \mathrm{s}^{-1}\right)$ is the angular frequency. An analogous equation can be written for the magnetic field component, $H\left(\mathrm{Am}^{-1}\right)$. Figure $2.1$ illustrates a $y$-polarized EM wave propagating along the $x$-axis. The power carried by this EM wave is given by the Poynting vector,
$$\boldsymbol{P}=\boldsymbol{E} \times \boldsymbol{H}=E_{y} \boldsymbol{j} \times H_{z} \boldsymbol{k}=E_{y} H_{z} \boldsymbol{i}\left(\mathrm{W} / \mathrm{m}^{2}\right) .$$
This is the basic mechanism of radiation heat transfer.
The frequency $v=\omega / 2 \pi\left(\mathrm{s}^{-1}\right)$ of an electromagnetic wave is determined at its origin and does not change as it propagates. However, its wavelength $\lambda=2 \pi / k(\mathrm{~m})$ varies according to the speed of light $c\left(\mathrm{~m} \mathrm{~s}^{-1}\right)$ in the medium through which it propagates according to
$$\lambda=c / v=c_{0} / n v,$$
where $n \equiv c_{0} / c$ is the index of refraction and $c_{0}$ is the speed of light in a vacuum $\left(\sim 2.9979 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right)$.

λ=C/在=C0/n在,

## 有限元方法代写

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## MATLAB代写

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