### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT5003

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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|Binning of Rays on a Surface Element; Illustrative Example

Meshing and indexing are treated in Section 1.6. A closely related idea is binning, in which indexing allows us to keep track of the spatial distribution of absorbed energy bundles. In the case of the instrument in Figure 3.2, it might be desirable to know how the power incident to the entrance aperture is distributed across the detector face. In this situation, after executing Step 11, control is transferred to a binning step not shown in Figure 3.1. The detector surface, subdivided into a number of bins, is illustrated in Figure 3.5. In a given application the actual number of bins and their spatial distribution would be dictated by the shape of the surface and the desired spatial resolution.

No attempt has been made in creating Figure $3.5$ to suggest a particular scheme for determining the spacing between consecutive radial rings. However, two schemes come to mind: (i) either the ring boundaries could be equally spaced or (ii) they could be spaced to achieve equal-area rings. The former scheme is, of course, easy to achieve but is probably

not as useful as the latter. A plot of power absorbed as a function of radial position would be easier to interpret if the rings were divided into equal areas. This may be achieved using the formula
$$r_{m}=(d / 2) \sqrt{m / M}$$
where $d$ is the diameter of the detector, $M$ is the total number of radial bands to be created, and $r_{m}$ is the radial position of the outer radius of the $m$ th band. It is both natural and convenient to base the circumferential divisions on equal angles,
$$\varphi_{n}=2 \pi n / N,$$
where $N$ is the total number of angular sectors, $n$ is the $n$th such sector, and $\varphi_{n}$ is the angle corresponding to the upper angular limit of the sector.

We arrive at the binning step with knowledge of $x_{1}, y_{1}, z_{1}$, and $P_{1}$, where $P_{1}$ is the power carried by the current ray. The Matlab function for determining the values of $m$ and $n$ and updating the power accumulated in bin $(m, n)$ appears in Figure 3.6. The “floor” operator on lines 8 and 12 of Figure $3.6$ returns the least integer in the argument.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Thermal and Optical Analysis

Figure $3.7$ is a detailed data sheet for the radiometric instrument concept illustrated in Figure 3.2. Our task is to provide a preliminary evaluation of its optical and thermal performance. The approach is to model the

optical behavior of the instrument under the assumption of specularly reflecting mirrors and diffuse-specular non-mirror surfaces in the visible wavelength range $(0.4-0.7 \mu \mathrm{m})$, and to model the thermal behavior of the instrument in the infrared. Optical characterization will require assessment of baffle and telescope performance, while thermal characterization will require estimation of the distribution factor matrix. The listing for the Matlab code developed to accomplish this task is available at the companion website listed on p. xix. Study of the code listing reveals that the component geometries and dimensions have been entered “by hand.” However, it is possible to import system geometries directly into Matlab from the CAD environment where they were created.

The first step is to determine the distance from the entrance aperture to the focal plane. This task is accomplished by performing a numerical experiment in which an on-axis collimated beam consisting of 100000 rays is introduced through the instrument aperture. The detector is then displaced back and forth along the optical axis until the waist diameter of the twice-reflected beam is minimized. The result of this process, obtained by iteration, is shown in Figure 3.8, in which the rays have converged to a circle of minimum diameter at $z=58.5823 \mathrm{~mm}$. The notation “TBD” in Figure $3.7$ can now be replaced with this value, and the preliminary design of the instrument is complete.

In Figures $3.9$ and 3.10, the intersections of rays with the primary and secondary mirrors are shown as individual dots. These and similar plots for the other surfaces of the instrument are invaluable for confirming the basic design assumptions, such as symmetry in this example.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors for the Case of Specified Surface Temperatures

Consider the special case where all of the surface elements making up the generic enclosure shown schematically in Figure $3.17$ have specified temperatures, and where all of the surface net heat fluxes are unknown. Note that the view of some surface elements may be either partially or fully blocked when viewed from other surface elements. This does not present a problem in the following analysis because partial or full blockage of direct radiation has already been accounted for in the definition and estimation of the distribution factors, as has the effect of surface curvature.

Consistent with the definition of the diffuse-specular graybody radiation distribution factor, we can express the radiation heat flux $\left(\mathrm{W} \mathrm{m}^{-2}\right.$ )

absorbed by surface element $i$ due to emission from all of the surface elements making up the enclosure
$$q_{i, a}=\frac{Q_{i, a}}{A_{i}}=\frac{1}{A_{i}} \sum_{j=1}^{n} \varepsilon_{j} A_{j} \sigma T_{j}^{4} D_{j i}, \quad 1 \leq i \leq n,$$
which, with the introduction of reciprocity, Eq. (3.5), becomes
$$q_{i, a}=\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4} D_{i j} . \quad 1 \leq i \leq n .$$
The radiation heat flux emitted from surface element $i$ is
$$q_{i, e}=\frac{Q_{i, e}}{A_{i}}=\frac{\varepsilon_{i} A_{i} \sigma T_{i}^{4}}{A_{i}}=\varepsilon_{i} \sigma T_{i}^{4}, \quad 1 \leq i \leq n$$
Then the net heat flux from surface element $i$ is
$$q_{i}=q_{i, e}-q_{i, a}=\varepsilon_{i} \sigma T_{i}^{4}-\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4} D_{i j}, \quad 1 \leq i \leq n$$
or, with the introduction of the Kronecker delta
$$\delta_{i j} \equiv \begin{cases}1, & i=j \ 0, & i \neq j\end{cases}$$
Eq. (3.33) can be written
$$q_{i}=\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4}\left(\delta_{i j}-D_{i j}\right) \quad 1 \leq i \leq n$$

r米=(d/2)米/米

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors for the Case of Specified Surface Temperatures

q一世,一个=问一世,一个一个一世=1一个一世∑j=1nej一个jσ吨j4Dj一世,1≤一世≤n,

q一世,一个=e一世∑j=1nσ吨j4D一世j.1≤一世≤n.

q一世,和=问一世,和一个一世=e一世一个一世σ吨一世4一个一世=e一世σ吨一世4,1≤一世≤n

q一世=q一世,和−q一世,一个=e一世σ吨一世4−e一世∑j=1nσ吨j4D一世j,1≤一世≤n

d一世j≡{1,一世=j 0,一世≠j

q一世=e一世∑j=1nσ吨j4(d一世j−D一世j)1≤一世≤n

## 有限元方法代写

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## MATLAB代写

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