### 数学代写|拓扑学代写Topology代考|Numerical Examples

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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

## 数学代写|拓扑学代写Topology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:
$$J=\frac{1}{J_{0}} \int_{\Omega 2 \backslash\left(\Omega_{d} \cup \Omega_{c}\right)} \mathbf{E}{s} \cdot \mathbf{E}{s}^{*} \mathrm{~d} \Omega$$
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 $\frac{1}{j \varepsilon{r} \varepsilon_{0}} \nabla \times\left(0,0, H_{s z}\right)$ 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. $2.8 \mathrm{a}$ and $\mathrm{c}$, for the three-dimensional case.

## 数学代写|拓扑学代写Topology代考|Nanostructures for Localized Surface Plasmon

Localized surface plasmon resonances are the strong interaction between nanostructures and visible light through the resonant excitations of collective oscillations of conduction electrons. In localized surface plasmon resonances, the local optical field

near the nanostructure can be many orders of magnitude higher than the incident field, and the incident field around the resonant peak wavelength is scattered strongly; the enhanced electric field is confined within only a tiny region of the nanometer length scale near the surface of the nanostructures and decays significantly thereafter [19].
In this section, the computational design is carried out for the metal nanostructures of localized surface plasmon resonances using the self-consistent adjoint sensitivitybased topology optimization procedure. The material for the nanostructure is set to be the noble metal Au with complex relative permittivity referred to [12]. The computational domain is set to be two-dimensional square and three-dimensional cube with side length equal to $1 / 2$-fold of the incident wavelength, and it is enclosed by the PMLs with thickness equal to $1 / 20$-fold of the incident wavelength. The

incident wave is set to be the uniform plane wave in free space, with frequency $3.75 \times 10^{14} \mathrm{~Hz}(\lambda=800 \mathrm{~nm})$, wave vector in positive $x$-axis and polarization in $z$ axis. The design domain is set to be two-dimensional circle and three-dimensional sphere localized at the center of the design domain, and the radius of the design domain is $1 / 5$-fold of the incident wavelength. The desired localized surface plasmon resonance is at the center of the design domain.

The strength of localized surface plasmon resonances is measured by the enhancement factor defined at the specified position. Therefore, the cost function can be set to maximize the normalized enhancement factor expressed as
$$J=\frac{1}{J_{0}} \int_{\Omega} \mathbf{E} \cdot \mathbf{E}^{} \delta\left(\mathbf{x}-\mathbf{x}{c}\right) \mathrm{d} \Omega$$ where $J{0}=\int_{\Omega} \mathbf{E}{i} \cdot \mathbf{E}{i}^{} \delta\left(\mathbf{x}-\mathbf{x}{c}\right) \mathrm{d} \Omega$ is the 2 -norm of the electric field at the localized surface plasmon resonance position; $\Omega$ is the computational domain; $\mathbf{x}{c}$ is the desired localized surface plasmon resonance position, set to be the center of the design domain; $\mathbf{E}$ is the electric field, calculated to be $\frac{1}{j \varepsilon_{r} \varepsilon_{0 \omega}} \nabla \times\left(0,0, H_{z}\right)$ in the twodimensional case.

## 数学代写|拓扑学代写Topology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}^{I}, \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}^{I}\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}^{I}\right) \cdot \nabla \hat{H}{s z}^{I} \ &+k{0}^{2} \mu_{r}\left(H_{s z}^{I}+H_{i z}^{l}\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}^{I} \hat{H}{s z}^{I}|\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 $\hat{H}{s z}^{l}=0$ on $\Gamma{D}$, and $\hat{\gamma}{f} \in \mathscr{H}\left(\Omega{d}\right)$ are the adjoint variables of $H_{s z}^{R} \in \mathscr{H}\left(\Omega \cup \Omega_{P}\right), H_{s z}^{l} \in$ $\mathscr{H}\left(\Omega \cup \Omega_{P}\right)$, 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}^{I}} \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_{e}} \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}

## 数学代写|拓扑学代写Topology代考|Optical Cloak

Ĵ=1Ĵ0∫Ω2∖(Ωd∪ΩC)和s⋅和s∗ dΩ

## 数学代写|拓扑学代写Topology代考|Nanostructures for Localized Surface Plasmon

Ĵ=1Ĵ0∫Ω和⋅和d(X−XC)dΩ在哪里Ĵ0=∫Ω和一世⋅和一世d(X−XC)dΩ是局部表面等离子体共振位置处电场的 2 范数；Ω是计算域；XC是所需的局部表面等离子体共振位置，设置为设计域的中心；和是电场，计算为1jere0ω∇×(0,0,H和)在二维情况下。

## 数学代写|拓扑学代写Topology代考|Adjoint Analysis of Topology Optimization Problem

Ĵ^=∫Ω一个(Hs和R,Hs和我,∇Hs和R,∇Hs和我,Cp;C)−(er−1)R∇(Hs和R+H一世和R)⋅∇H^s和R +(er−1)我∇(Hs和l+H一世和我)⋅∇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一世和我)⋅∇H^s和我 +ķ02μr(Hs和我+H一世和l)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和我H^s和我|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和我⋅∇dHs和我 dΩ +∑n=1ñ1在n∫磷n∂一个∂Cp∂Cp∂C和dCF dΩ+∫Ωd∂一个∂CdCdΩ+∫Ω−(er−1)R∇dHs和R⋅∇H^s和R

## 有限元方法代写

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## MATLAB代写

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