### 数学代写|拓扑学代写Topology代考| Dielectric Material-Based Topology Optimization for Nano-Optics

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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|Adjoint Analysis for Electric Field-Based Topology

The Lagrangian multiplier-based adjoint sensitivity analysis of variational problem in Eq. $3.23$ is implemented as follows. The functional space and trace operators of Eq. 3.14 are similarly defined as that in Sect. 3.1.2, except that
$$\mathscr{V}{\mathbf{E}} \doteq\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u}=0, \text { in } \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0} \text {, on } \Gamma{P E C}\right}$$
According to the Kurash-Kuhn-Tucker condition of the PDE constrained optimization problem [22], the adjoint equations can be obtained as

Find $\mathbf{E}{s a} \in \mathscr{V}{\mathbf{E}}$ such that
\begin{aligned} &\int_{\Omega} \frac{\partial A}{\partial \mathbf{E}{s}} \cdot \phi+\frac{\partial A}{\partial \nabla \times \mathbf{E}{s}} \cdot(\nabla \times \phi)+\mu_{r}^{-1}\left(\nabla \times \overline{\mathbf{E}}{s a}\right) \cdot(\nabla \times \phi)-k{0}^{2} \varepsilon_{r} \overline{\mathbf{E}}{s a} \cdot \phi \mathrm{d} \Omega \ &+\int{\Gamma_{a}} j k_{0} \sqrt{\varepsilon_{r} \mu_{r}^{-1}}\left(\mathbf{n} \times \overline{\mathbf{E}}{s a} \times \mathbf{n}\right) \cdot(\mathbf{n} \times \phi \times \mathbf{n})+\frac{\partial B}{\partial \mathbf{E}{s}} \cdot \phi \mathrm{d} \Gamma \ &+\int_{\Gamma_{P M c}} \frac{\partial B}{\partial \mathbf{E}{s}} \cdot \phi \mathrm{d} \Gamma=0, \forall \phi \in \mathscr{V}{\mathbf{E}} \ &\text { Find } \gamma_{f a} \in \mathscr{H}(\Omega) \text { such that } \ &\int_{\Omega} r^{2} \nabla_{\gamma_{f a}} \cdot \nabla \phi+\gamma_{f a} \phi+A_{\gamma_{e}} \phi-S_{\gamma_{c}} \phi \mathrm{d} \Omega=0, \forall \phi \in \mathscr{H}(\Omega) \end{aligned}
where $A_{\gamma_{c}}(\Omega)$ is defined as
$$A_{\gamma_{e}}=\sum_{n=1}^{N} A_{\gamma_{n e}}\left(\Omega_{n}\right)$$
with
$$A_{\gamma_{n e s}}\left(\Omega_{n}\right)=\left{\begin{array}{l} \frac{1}{V_{\Omega_{n}}} \int_{\Omega_{n}} \frac{\partial A}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \mathrm{~d} \Omega, \forall \mathbf{x} \in \Omega_{n} \ 0, \forall \mathbf{x} \in \Omega \backslash \Omega_{n} \end{array}\right.$$
and $S_{\gamma_{c}}(\Omega)$ is defined to be
$$S_{\gamma_{e}}=\sum_{n=1}^{N} S_{\gamma_{a, e}}\left(\Omega_{n}\right)$$
with
$$S_{\gamma_{n, c}}\left(\Omega_{n}\right)=\left{\begin{array}{l} \frac{1}{V_{\Omega_{n}}} \int_{\Omega_{n}} k_{0}^{2} \frac{\partial \varepsilon_{r}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}}\left(\mathbf{E}{s}+\mathbf{E}{i}\right) \cdot \overline{\mathbf{E}}{s a} \mathrm{~d} \Omega, \forall \mathbf{x} \in \Omega{n} \ 0, \forall \mathbf{x} \in \Omega \backslash \Omega_{n} \end{array}\right.$$
The adjoint derivative of the cost functional can be derived as
$$\delta J=\int_{\Omega} \operatorname{Re}\left(\frac{\partial A}{\partial \gamma}-\bar{\gamma}_{f a}\right) \delta \gamma \mathrm{d} \Omega$$

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

In the wave equations and corresponding adjoint equations, a divergence-free condition needs to be satisfied for both the state variable and the adjoint variable. Therefore,

the edge element-based finite element method is utilized to solve the wave equations and adjoint equations, where brick elements are used to discretize the computational domain and simultaneously ensure the divergence-free condition [28]. For the Helmholtz filter, the filter Eq. $3.4$ and its adjoint equation are solved using the standard Galerkin finite element method.

The topology optimization method for three-dimensional optical waves is implemented by a gradient-based iterative procedure, where the gradient information is derived by sensitivity analysis as demonstrated in Sects. 3.1.2 and 3.1.4 respectively corresponding to the variational problems in Eqs.3.7 and 3.23. The flowcharts for iteratively solving the variational problems (Eqs. $3.7$ and $3.23$ ) respectively corresponding to the magnetic field formulation and electric field formulation are respectively shown in Fig. 3.1a and b. The iterative procedure includes the following steps: (a) solve the wave equations with the current design variable; (b) solve the adjoint equations based on the solution of the wave equations; (c) compute the adjoint derivative of the design objective; and (d) update the design variable using the method of moving asymptotes [47].

During the solving procedure, the filter radius $r$ of the Helmholtz filter in Eq.3.4 is set to be the size of the finite elements used to discretize the computational domain; the threshold parameter $\xi$ in Eqs. $3.5$ and $3.21$ 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 $2^{10}$ is reached (eleven cycles). The above steps are implemented iteratively until the stopping criterion is satisfied, specified to be 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 $1 \times 10^{-3}$. Because the iteration number is set to be 40 before doubling the projection parameter, the maximal iterative number is set to be $440 .$

In the optimization procedure for magnetic field described optical waves, the magnetic field is interpolated using linear edge elements (Fig. 3.2a); the design variable and filtered design variable is interpolated using linear nodal element (Fig. 3.2b). In the optimization procedure for electric field described optical waves, the electric field is interpolated using linear edge elements (Fig. 3.2a); the design variable and filtered design variable is interpolated using linear nodal elements (Fig. 3.2b); the filtered design variable is converted to piecewise form by interpolating the piecewise design variable using zeroth-order discontinuous elements (Fig. $3.2 \mathrm{c}$ ), where $\Omega_{n}$ in Eq. $3.20$ is set to be the space taken up by the brick elements.

## 数学代写|拓扑学代写Topology代考|Cloak for Perfect Conductor

The cloaks for perfect conductor are inversely designed using the developed method. This is a typical min-type optimization problem. Topology optimization-based inverse design of two-dimensional optical cloaks have been investigated for transverse magnetic and transverse electric incident waves, where two-dimensional is the reduced case with an infinite extension assumed in the third dimension $[3,4$, 16]. Three-dimensional design is more flexible and practical for the consideration of realistic situations.

In the following, optical cloaks are designed for a spherical perfect conductor. To cloak the sphere, the scattering field should be minimized to achieve phase matching of the total field around the conductor. The inverse domain of the cloak is set to be a cube with side length equal to 7 times the incident wavelength, as shown in Fig. 3.3, where the cloak domain is set to be a spherical shell with external and internal radii equal to $2.5$ and $0.75$ times the incident wavelength, and the cloaked conductor is enclosed in a central spherical domain with a radius equal to $0.75$ times the incident wavelength. The computational domain is discretized by $63 \times 63 \times 63$ brick elements.

For a magnetic field described optical cloak, the objective in Eq. $3.8$ is set to minimize the normalized square norm of the scattered magnetic field
$$J=\frac{1}{J_{0}} \int_{\Omega_{a}} \mathbf{H}{s} \cdot \overline{\mathbf{H}}{s} \mathrm{~d} \Omega$$
where $\Omega_{o}$ is the domain outside the spherical shell-shaped design domain; $J_{0}$ is the square norm of the uncloaked scattered magnetic field in the outside domain of the cloak. The obtained cloak topology, found by solving the corresponding topology optimization problem, is shown in Fig. 3.4a, with incident wave, uncloaked field, and cloaked field shown in Fig. $3.4 \mathrm{c}$, d, and e, where the incident wave is set to be the uniform plane wave $\mathbf{H}{i}=\left(0,0, e^{-j k{0} x}\right)$ with $k_{0}=20 \pi \mathrm{rad} / \mathrm{m}$. For an electric field described optical cloak, the objective in Eq. $3.24$ is set to minimize the normalized square norm of the scattered electric field
$$J=\frac{1}{J_{0}} \int_{\Omega_{o}} \mathbf{E}{s} \cdot \overline{\mathbf{E}}{s} \mathrm{~d} \Omega$$

## 数学代写|拓扑学代写Topology代考|Adjoint Analysis for Electric Field-Based Topology

\mathscr{V}{\mathbf{E}} \doteq\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u} =0, \text { } \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0} \text {, on } \Gamma{PEC}\right}\mathscr{V}{\mathbf{E}} \doteq\left{\mathbf{u} \in \mathscr{H}(\operatorname{curl} ; \Omega) \mid \nabla \cdot \mathbf{u} =0, \text { } \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0} \text {, on } \Gamma{PEC}\right}

∫Ω∂一个∂和s⋅φ+∂一个∂∇×和s⋅(∇×φ)+μr−1(∇×和¯s一个)⋅(∇×φ)−ķ02er和¯s一个⋅φdΩ +∫Γ一个jķ0erμr−1(n×和¯s一个×n)⋅(n×φ×n)+∂乙∂和s⋅φdΓ +∫Γ磷米C∂乙∂和s⋅φdΓ=0,∀φ∈在和  寻找 CF一个∈H(Ω) 这样  ∫Ωr2∇CF一个⋅∇φ+CF一个φ+一个C和φ−小号CCφdΩ=0,∀φ∈H(Ω)

$$A_{\ gamma_ {nes}}\left(\Omega_{n}\right)=\left{ 1在Ωn∫Ωn∂一个∂Cp∂Cp∂C和 dΩ,∀X∈Ωn 0,∀X∈Ω∖Ωn\正确的。 一个nd小号CC(Ω)一世sd和F一世n和d吨○b和 S_{\gamma_{e}}=\sum_{n=1}^{N} S_{\gamma_{a, e}}\left(\Omega_{n}\right) 在一世吨H S_{\gamma_{n, c}}\left(\Omega_{n}\right)=\left{ 1在Ωn∫Ωnķ02∂er∂Cp∂Cp∂C和(和s+和一世)⋅和¯s一个 dΩ,∀X∈Ωn 0,∀X∈Ω∖Ωn\正确的。 吨H和一个dj○一世n吨d和r一世在一个吨一世在和○F吨H和C○s吨F在nC吨一世○n一个lC一个nb和d和r一世在和d一个s \delta J=\int_{\Omega} \operatorname{Re}\left(\frac{\partial A}{\partial \gamma}-\bar{\gamma}_{fa}\right) \delta \gamma\数学{d}\欧米茄$$

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

15∑一世=14|Ĵķ−一世−Ĵķ−一世−1|/|Ĵķ|≤e,b≥210

## 数学代写|拓扑学代写Topology代考|Cloak for Perfect Conductor

Ĵ=1Ĵ0∫Ω一个Hs⋅H¯s dΩ

Ĵ=1Ĵ0∫Ω○和s⋅和¯s dΩ

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## MATLAB代写

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