### 物理代写|光学代写Optics代考|PHS2062

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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

## 物理代写|光学代写Optics代考|Numerical Implementation

The topology optimization problems are solved by a gradient-based iterative procedure, where the gradient information is derived by the self-consistent adjoint sensitivity. The flowchart for iterative solution of the optimization problems is shown in Fig. 2.2. The iterative procedure includes the following steps: (1) initialize the design variable and optimization parameters; $(2)$ solve the wave equations with the current design variable and compute the value of the design objective; (3) solve the adjoint equations based on the solution of the wave equations; (4) compute the adjoint derivative of the design objective; (5) update the design variable using the method of moving asymptotes [20]; (6) do postprocessing, if the stopping criteria are satisfied, or else return to the step (2).

In this solution procedure, the filter radius $r$ of the PDE filter in Eq. $2.10$ is set to be $2 / 15$ of the incident wavelength; the threshold parameter $\xi$ in Eq. $2.12$ is set to be

$0.5$; the initial value of the projection parameter $\beta$ is set to be 1 and it is doubled after every fixed number of iterations until the preset maximal value of $2^{10}$ is reached. The above steps are implemented iteratively until the stopping criteria are satisfied, and the stopping criteria are specified to be the maximal iteration number and the change of the objective values in five consecutive iterations satisfying
$$\frac{1}{5} \sum_{i=1}^{4}\left|J_{k-i}-J_{k-i-1}\right| /\left|J_{k}\right| \leq \varepsilon, \beta \geq 2^{10}$$
in the $k$ th iteration, where $J_{k}$ is the objective value computed in the $k$ th iteration; $\varepsilon$ is the tolerance chosen to be $1 \times 10^{-3}$.

In the iterative procedure, the numerical solution of PDEs are implemented based on finite element method $[10,13]$. The two-dimensional wave equation, the filter equation and the corresponding adjoint equations are solved using the standard Galerkin finite element method; the three-dimensional wave equation and the corresponding adjoint equation are solved using the edge element-based finite element method with linear edge elements $[14,15]$. It is noted that the original wave equations $2.1$ and $2.2$ are solved instead of the coupled equations for the split variables, because the sole destination of splitting operation is to derive the Fréchet differentiability during the adjoint analysis.

In the optimization procedures for the two-dimensional problems, the magnetic field, design variable and filtered design variable are interpolated using linear node elements (Fig. 2.3a); and the projected design variable is interpolated using zerothorder discontinuous elements (Fig. 2.3b). In the optimization procedure for the threedimensional problems, the electric field is interpolated using linear edge elements (Fig. 2.4a); the design variable and filtered design variable is interpolated using linear nodal elements (Fig. 2.4b); the filtered design variable is converted to piecewise form by interpolating the piecewise design variable using zeroth-order discontinuous elements (Fig. $2.4 \mathrm{c}$ ), where $P_{n}$ in Eq. $2.11$ is set to be the space taken up by the finite elements.

## 物理代写|光学代写Optics代考|Optical Cloak

Optical cloak has been investigated using topology optimization by minimizing the scattering field energy around the cloak $[1,2]$. In these researches, the sensitivity analysis is implemented based on the Gâteaux differentiability of the conjugate operator. The infinite space is truncated by the first-order scattering boundary condition. As specified in [10], the first-order scattering boundary condition has reflection, and this causes the relatively lower computational accuracy compared with truncating the infinite space by PMLs.

In this section, the optical cloak is investigated using the sensitivity analysis approach with Fréchet differentiability, and the PMLs are used to achieve the scattering boundary. The cloaked object is set to be a two-dimensional circular or threedimensional spherical conductor with high conductivity $\left(\varepsilon_{r}=-1 \times 10^{4} j\right.$ ). The incident wave is set to be the uniform plane wave in free space, with frequency $1 \times 10^{9} \mathrm{~Hz}$ $(\lambda=0.3 \mathrm{~m}$ ), wave vector in positive $x$-axis, and polarization in $z$-axis.

The computational domain is set to be square or cube with side length equal to 6 -fold of the incident wavelength respectively for the two- and three-dimensional cases; and it is enclosed by the PMLs with thickness equal to $1 / 3$-fold of the incident wavelength. The conductor is localized at the center of the computational domain, and its radius of the conductor is $3 / 4$-fold of the incident wavelength. The design domain is the two-dimensional ring or three-dimensional shell around the conductor, and the exterior radius of the design domain is $5 / 2$-fold of the incident wavelength. The material of the cloak is the dielectrics with relative permittivity $2.25$.

The objective is to minimize the normalized scattering electric field energy in the exterior of the cloak:
$$J=\frac{1}{J_{0}} \int_{\Omega \backslash\left(\Omega_{d} \cup \Omega_{c}\right)} \mathbf{E}{s} \cdot \mathbf{E}{s}^{*} \mathrm{~d} \Omega$$
where $J_{0}$ is the norm square of the scattering electric field in the exterior of the design domain fully filled by the dielectrics; $\Omega$ is the computational domain; $\Omega_{d}$ is the design domain; $\Omega_{c}$ is the cloaked conductor domain; $\mathbf{E}{s}$ is the scattering electric field, calculated to be $\frac{1}{j \varepsilon{r} \varepsilon_{0} \omega} \nabla \times\left(0,0, H_{s z}\right)$ in the two-dimensional cases.

The computational domains are discretized by $80 \times 80$ square elements and $80 \times 80 \times 80$ cube elements respectively for the two- and three-dimensional cases, where 5 layers of elements are used to discretize the PMLs. For the two-dimensional case, the topology optimization problem is sketched in Fig. 2.5a. By numerically implementing the topology optimization procedure with the derived self-consistent adjoint sensitivity, the two- and three-dimensional dielectric cloaks are derived as shown in Figs. $2.5 \mathrm{~b}$ and 2.6a. Convergent histories of the cost functions are plotted in Figs. $2.5 \mathrm{c}$ and $2.6 \mathrm{~b}$. Snapshots for the evolutionary progress of the structural topology are shown in Figs. $2.5 \mathrm{~d}$ and $2.6 \mathrm{c}$. From the convergent histories of the cost functions and snapshots for the evolutionary progress of the structural topology, one can confirm the robustness of the self-consistent adjoint sensitivity-based optimization procedure for dielectric material topology. In the initial of the numerical procedure, the design domains are fully filled by the used dielectrics; the conductor enclosed with the dielectrics scatters the incident field as shown in Fig. 2.7a with the corresponding scattering field energy shown in Fig. $2.7 \mathrm{c}$, for the two-dimensional case; and the scattering field and corresponding scattering field energy are shown in Fig. 2.8a and c, for the three-dimensional case.

## 物理代写|光学代写Optics代考|Adjoint Analysis of Topology Optimization Problem

Based on the Lagrangian multiplier-based adjoint method, the augmented Lagrangian for the topology optimization problem in Eq. $2.21$ can be derived as
\begin{aligned} \hat{J}=& \int_{\Omega} A\left(H_{s z}^{R}, H_{s z}^{I}, \nabla H_{s z}^{R}, \nabla H_{s z}^{l}, \gamma_{p} ; \gamma\right)-\left(\varepsilon_{r}^{-1}\right)^{R} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right) \cdot \nabla \hat{H}{s z}^{R} \ &+\left(\varepsilon{r}^{-1}\right)^{I} \nabla\left(H_{s z}^{l}+H_{i z}^{l}\right) \cdot \nabla \hat{H}{s z}^{R}+k{0}^{2} \mu_{r}\left(H_{s z}^{R}+H_{i z}^{R}\right) \hat{H}{s z}^{R} \ &-\left(\varepsilon{r}^{-1}\right)^{I} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right) \cdot \nabla \hat{H}{s z}^{I}-\left(\varepsilon{r}^{-1}\right)^{R} \nabla\left(H_{s z}^{I}+H_{i z}^{l}\right) \cdot \nabla \hat{H}{s z}^{I} \ &+k{0}^{2} \mu_{r}\left(H_{s z}^{l}+H_{i z}^{I}\right) \hat{H}{s z}^{I} \mathrm{~d} \Omega-\int{\Omega_{p}}\left(\varepsilon_{r}^{-1}\right)^{R}\left(\mathbf{T} \nabla H_{s z}^{R}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{R}\right)|\mathbf{T}|^{-1} \ &+\left(\varepsilon{r}^{-1}\right)^{I}\left(\mathbf{T} \nabla H_{s z}^{R}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{I}\right)|\mathbf{T}|^{-1}-k{0}^{2} \mu_{r} H_{s z}^{R} \hat{H}{s z}^{R}|\mathbf{T}| \ &-\left(\varepsilon{r}^{-1}\right)^{I}\left(\mathbf{T} \nabla H_{s z}^{l}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{R}\right)|\mathbf{T}|^{-1}+\left(\varepsilon{r}^{-1}\right)^{R}\left(\mathbf{T} \nabla H_{s z}^{I}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{I}\right)|\mathbf{T}|^{-1} \ &-k{0}^{2} \mu_{r} H_{s z}^{l} \hat{H}{s z}^{l}|\mathbf{T}| \mathrm{d} \Omega+\int{\Omega_{d}} r^{2} \nabla \gamma_{f} \cdot \nabla \hat{\gamma}{f}+\gamma{f} \hat{\gamma}{f}-\gamma \hat{\gamma}{f} \mathrm{~d} \Omega \end{aligned}
where $\hat{H}{s z}^{R} \in \mathscr{H}\left(\Omega \cup \Omega{P}\right)$ with $\hat{H}{s z}^{R}=0$ on $\Gamma{D}, \hat{H}{s z}^{l} \in \mathscr{H}\left(\Omega \cup \Omega{P}\right)$ with $\hat{H}{s z}^{I}=0$ on $\Gamma{D}$, and $\hat{\gamma}{f} \in \mathscr{H}\left(\Omega{d}\right)$ are the adjoint variables of $H_{s z}^{R} \in \mathscr{H}\left(\Omega \cup \Omega_{P}\right), H_{s z}^{I} \in$ $\mathscr{H}\left(\Omega \cup \Omega_{P}\right)$, and $\gamma_{f} \in \mathscr{H}\left(\Omega_{d}\right)$ respectively; $\mathscr{H}\left(\Omega \cup \Omega_{P}\right)$ and $\mathscr{H}\left(\Omega_{d}\right)$ are the first-order Hilbert spaces for the real functions defined on $\Omega \cup \Omega_{P}$ and $\Omega_{d}$ respectively; $\mathbf{T}$ is the transformation matrix in Eq. 2.6. The first-order variational of the augmented Lagrangian to the field variables and design variable is
\begin{aligned} \delta \hat{J}=& \int_{\Omega} \frac{\partial A}{\partial H_{s z}^{R}} \delta H_{s z}^{R}+\frac{\partial A}{\partial H_{s z}^{I}} \delta H_{s z}^{I}+\frac{\partial A}{\partial \nabla H_{s z}^{R}} \cdot \nabla \delta H_{s z}^{R}+\frac{\partial A}{\partial \nabla H_{s z}^{l}} \cdot \nabla \delta H_{s z}^{I} \mathrm{~d} \Omega \ &+\sum_{n=1}^{N} \frac{1}{V_{n}} \int_{P_{n}} \frac{\partial A}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{c}} \delta \gamma_{f} \mathrm{~d} \Omega+\int_{\Omega_{d}} \frac{\partial A}{\partial \gamma} \delta \gamma \mathrm{d} \Omega+\int_{\Omega}-\left(\varepsilon_{r}^{-1}\right)^{R} \nabla \delta H_{s z}^{R} \cdot \nabla \hat{H}_{s z}^{R} \end{aligned}

\begin{aligned} &+\left(\varepsilon_{r}^{-1}\right)^{l} \nabla \delta H_{s z}^{I} \cdot \nabla \hat{H}{s z}^{R}+k{0}^{2} \mu_{r} \delta H_{s z}^{R} \hat{H}{s z}^{R}-\left(\varepsilon{r}^{-1}\right)^{l} \nabla \delta H_{s z}^{R} \cdot \nabla \hat{H}{s z}^{I}-\left(\varepsilon{r}^{-1}\right)^{R} \nabla \delta H_{s z}^{I} \ &\cdot \nabla \hat{H}{s z}^{l}+k{0}^{2} \mu_{r} \delta H_{s z}^{l} \hat{H}{s z}^{l} \mathrm{~d} \Omega-\int{\Omega_{p}}\left(\varepsilon_{r}^{-1}\right)^{R}\left[\mathbf{T}\left(\nabla \delta H_{s z}^{R}\right)\right] \cdot\left[\mathbf{T}\left(\nabla \hat{H}{s z}^{R}\right)\right]|\mathbf{T}|^{-1} \ &+\left(\varepsilon{r}^{-1}\right)^{l}\left[\mathbf{T}\left(\nabla \delta H_{s z}^{R}\right)\right] \cdot\left[\mathbf{T}\left(\nabla \hat{H}{s z}^{l}\right)\right]|\mathbf{T}|^{-1}-k{0}^{2} \mu_{r} \delta H_{s z}^{R} \hat{H}{s z}^{R}|\mathbf{T}| \ &-\left(\varepsilon{r}^{-1}\right)^{l}\left(\mathbf{T} \nabla \delta H_{s z}^{l}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{R}\right)|\mathbf{T}|^{-1}+\left(\varepsilon{r}^{-1}\right)^{R}\left(\mathbf{T} \nabla \delta H_{s z}^{l}\right) \cdot\left(\mathbf{T} \nabla \hat{H}{s z}^{l}\right)|\mathbf{T}|^{-1} \ &-k{0}^{2} \mu_{r} \delta H_{s z}^{l} \hat{H}{s z}^{l}|\mathbf{T}| \mathrm{d} \Omega+\int{\Omega_{d}} r^{2} \nabla \delta \gamma_{f} \cdot \nabla \hat{\gamma}{f}+\delta \gamma{f} \hat{\gamma}{f}-\delta \gamma \hat{\gamma}{f} \mathrm{~d} \Omega \ &+\sum_{n=1}^{N} \frac{1}{V_{n}} \int_{P_{n}}-\frac{\partial\left(\varepsilon_{r}^{-1}\right)^{R}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right) \cdot \nabla \hat{H}{s z}^{R} \delta \gamma{f} \ &+\frac{\partial\left(\varepsilon_{r}^{-1}\right)^{I}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}^{l}+H_{i z}^{l}\right) \cdot \nabla \hat{H}{s z}^{R} \delta \gamma{f}-\frac{\partial\left(\varepsilon_{r}^{-1}\right)^{l}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right) \ &\cdot \nabla \hat{H}{s z}^{l} \delta \gamma{f}-\frac{\partial\left(\varepsilon_{r}^{-1}\right)^{R}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}^{l}+H_{i z}^{l}\right) \cdot \nabla \hat{H}{s z}^{l} \delta \gamma{f} \mathrm{~d} \Omega \end{aligned}
where $\delta H_{s z}^{R}, \delta H_{s z}^{1}, \delta \gamma_{f}$ and $\delta \gamma$ are the first-order variational of $H_{s z}^{R}, H_{s z}^{l}, \gamma_{f}$ and $\gamma$ respectively.

## 物理代写|光学代写Optics代考|Numerical Implementation

0.5; 投影参数的初始值b设置为1，每固定迭代次数加倍，直到达到预设的最大值210到达了。上述步骤迭代执行直到满足停止条件，停止条件指定为最大迭代次数和连续五次迭代中目标值的变化满足

15∑一世=14|Ĵķ−一世−Ĵķ−一世−1|/|Ĵķ|≤e,b≥210

## 物理代写|光学代写Optics代考|Optical Cloak

Ĵ=1Ĵ0∫Ω∖(Ωd∪ΩC)和s⋅和s∗ dΩ

## 物理代写|光学代写Optics代考|Adjoint Analysis of Topology Optimization Problem

Ĵ^=∫Ω一个(Hs和R,Hs和我,∇Hs和R,∇Hs和l,Cp;C)−(er−1)R∇(Hs和R+H一世和R)⋅∇H^s和R +(er−1)我∇(Hs和l+H一世和l)⋅∇H^s和R+ķ02μr(Hs和R+H一世和R)H^s和R −(er−1)我∇(Hs和R+H一世和R)⋅∇H^s和我−(er−1)R∇(Hs和我+H一世和l)⋅∇H^s和我 +ķ02μr(Hs和l+H一世和我)H^s和我 dΩ−∫Ωp(er−1)R(吨∇Hs和R)⋅(吨∇H^s和R)|吨|−1 +(er−1)我(吨∇Hs和R)⋅(吨∇H^s和我)|吨|−1−ķ02μrHs和RH^s和R|吨| −(er−1)我(吨∇Hs和l)⋅(吨∇H^s和R)|吨|−1+(er−1)R(吨∇Hs和我)⋅(吨∇H^s和我)|吨|−1 −ķ02μrHs和lH^s和l|吨|dΩ+∫Ωdr2∇CF⋅∇C^F+CFC^F−CC^F dΩ

dĴ^=∫Ω∂一个∂Hs和RdHs和R+∂一个∂Hs和我dHs和我+∂一个∂∇Hs和R⋅∇dHs和R+∂一个∂∇Hs和l⋅∇dHs和我 dΩ +∑n=1ñ1在n∫磷n∂一个∂Cp∂Cp∂CCdCF dΩ+∫Ωd∂一个∂CdCdΩ+∫Ω−(er−1)R∇dHs和R⋅∇H^s和R

+(er−1)l∇dHs和我⋅∇H^s和R+ķ02μrdHs和RH^s和R−(er−1)l∇dHs和R⋅∇H^s和我−(er−1)R∇dHs和我 ⋅∇H^s和l+ķ02μrdHs和lH^s和l dΩ−∫Ωp(er−1)R[吨(∇dHs和R)]⋅[吨(∇H^s和R)]|吨|−1 +(er−1)l[吨(∇dHs和R)]⋅[吨(∇H^s和l)]|吨|−1−ķ02μrdHs和RH^s和R|吨| −(er−1)l(吨∇dHs和l)⋅(吨∇H^s和R)|吨|−1+(er−1)R(吨∇dHs和l)⋅(吨∇H^s和l)|吨|−1 −ķ02μrdHs和lH^s和l|吨|dΩ+∫Ωdr2∇dCF⋅∇C^F+dCFC^F−dCC^F dΩ +∑n=1ñ1在n∫磷n−∂(er−1)R∂Cp∂Cp∂C和∇(Hs和R+H一世和R)⋅∇H^s和RdCF +∂(er−1)我∂Cp∂Cp∂C和∇(Hs和l+H一世和l)⋅∇H^s和RdCF−∂(er−1)l∂Cp∂Cp∂C和∇(Hs和R+H一世和R) ⋅∇H^s和ldCF−∂(er−1)R∂Cp∂Cp∂C和∇(Hs和l+H一世和l)⋅∇H^s和ldCF dΩ

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