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

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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|Three-Dimensional Optical Waves

Based on the Lagrangian multiplier-based adjoint method, the augmented Lagrangian for the topology optimization problem in Eq. $2.22$ can be derived as
\begin{aligned} j=& \int_{\Omega} A\left(\mathbf{E}{s}^{R}, \mathbf{E}{s}^{l}, \nabla_{\mathbf{x}} \times \mathbf{E}{s}^{R}, \nabla{\mathbf{x}} \times \mathbf{E}{s}^{l}, \gamma{p} ; \gamma\right)+\mu_{r}^{-1}\left[\nabla \times\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)\right] \cdot\left(\nabla \times \hat{\mathbf{E}}{s}^{R}\right) \ &-k{0}^{2}\left[\varepsilon_{r}^{R}\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)-\varepsilon_{r}^{l}\left(\mathbf{E}{s}^{l}+\mathbf{E}{i}^{l}\right)\right] \cdot \hat{\mathbf{E}}{s}^{R}+\mu{r}^{-1}\left[\nabla \times\left(\mathbf{E}{s}^{l}+\mathbf{E}{i}^{l}\right)\right] \cdot\left(\nabla \times \hat{\mathbf{E}}{s}^{l}\right) \ &-k{0}^{2}\left[\varepsilon_{r}^{l}\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)+\varepsilon_{r}^{R}\left(\mathbf{E}{s}^{l}+\mathbf{E}{i}^{l}\right)\right] \cdot \hat{\mathbf{E}}{s}^{l} \mathrm{~d} \Omega+\int{\Omega{ }{p}} \mu{r}^{-1}\left(\mathbf{T} \nabla \times \mathbf{E}{s}^{R}\right) \ & \cdot\left(\mathbf{T} \nabla \times \hat{\mathbf{E}}{s}^{R}\right)|\mathbf{T}|^{-1}-k_{0}^{2}\left(\varepsilon_{r}^{R} \mathbf{E}{s}^{R}-\varepsilon{r}^{l} \mathbf{E}{s}^{l}\right) \cdot \hat{\mathbf{E}}{s}^{R}|\mathbf{T}|+\mu_{r}^{-1}\left(\mathbf{T} \nabla \times \mathbf{E}{s}^{l}\right) \ & \cdot\left(\mathbf{T} \nabla \times \hat{\mathbf{E}}{s}^{l}\right)|\mathbf{T}|^{-1}-k_{0}^{2}\left(\varepsilon_{r}^{l} \mathbf{E}{s}^{R}+\varepsilon{r}^{R} \mathbf{E}{s}^{l}\right) \cdot \hat{\mathbf{E}}{s}^{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 $\hat{\mathbf{E}}{s}^{R} \in \mathscr{V}{\mathbf{E}}$ with $\mathbf{n} \times \hat{\mathbf{E}}{s}^{R}=\mathbf{0}$ on $\Gamma{D}, \hat{\mathbf{E}}{s}^{I} \in \mathscr{V}{\mathbf{E}}$ with $\mathbf{n} \times \hat{\mathbf{E}}{s}^{I}=\mathbf{0}$ on $\Gamma{D}$, and $\hat{\gamma}{f} \in \mathscr{H}\left(\Omega{d}\right)$ are the adjoint variables of $\mathbf{E}{s}^{R} \in \mathscr{V}{\mathbf{E}}, \mathbf{E}{s}^{I} \in \mathscr{V}{\mathbf{E}}$, and $\gamma_{f} \in \mathscr{H}\left(\Omega_{d}\right)$ respectively; $\mathscr{V}{\mathbf{E}}$ is defined as the functional space $$\left{\mathbf{u} \in \mathscr{H}\left(\operatorname{curl} ; \Omega \cup \Omega{P}\right) \mid \nabla \cdot \mathbf{u}=0 \text { in } \Omega \cup \Omega_{P}\right}$$
with
$$\mathscr{H}\left(\operatorname{curl} ; \Omega \cup \Omega_{P}\right)=\left{\mathbf{u} \in\left(\mathscr{L}^{2}\left(\Omega \cup \Omega_{P}\right)\right)^{3} \mid \nabla \times \mathbf{u} \in\left(\mathscr{L}^{2}\left(\Omega \cup \Omega_{P}\right)\right)^{3}\right}$$
$\mathscr{L}^{2}\left(\Omega \cup \Omega_{P}\right)$ represents the second-order Lebesque space for the real functions defined on $\Omega \cup \Omega_{P}$.

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 \mathbf{E}{s}^{R}} \cdot \delta \mathbf{E}{s}^{R}+\frac{\partial A}{\partial \mathbf{E}{s}^{l}} \cdot \delta \mathbf{E}{s}^{l}+\frac{\partial A}{\partial \nabla \times \mathbf{E}{s}^{R}} \cdot\left(\nabla \times \delta \mathbf{E}{s}^{R}\right)+\frac{\partial A}{\partial \nabla \times \mathbf{E}{s}^{l}} \ &\left(\nabla \times \delta \mathbf{E}{s}^{I}\right) \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} \mu_{r}^{-1}\left(\nabla \times \delta \mathbf{E}{s}^{R}\right) \cdot\left(\nabla \times \hat{\mathbf{E}}{s}^{R}\right)-k_{0}^{2}\left(\varepsilon_{r}^{R} \delta \mathbf{E}{s}^{R}-\varepsilon{r}^{l} \delta \mathbf{E}{s}^{l}\right) \cdot \hat{\mathbf{E}}{s}^{R} \mathrm{~d} \Omega \end{aligned}

## 数学代写|拓扑学代写Topology代考|Magnetic Field Formulation

Maxwell’s equations are widely used to describe the propagation optical waves. Under the time-harmonic assumption, the following magnetic field wave equation can be derived by setting the time-dependent factor to be $e^{\text {jot }}$ :
$$\left{\begin{array}{l} \nabla \times\left[\varepsilon_{r}^{-1} \nabla \times\left(\mathbf{H}{s}+\mathbf{H}{i}\right)\right]-k_{0}^{2} \mu_{r}\left(\mathbf{H}{s}+\mathbf{H}{i}\right)=\mathbf{0}, \text { in } \Omega \ \nabla \cdot \mathbf{H}{s}=0, \text { in } \Omega \end{array}\right.$$ where the scattering field formulation is used with the magnetic field $\mathbf{H}$ split into two parts, i.e., the incident wave $\mathbf{H}{i}$ and scattering field $\mathbf{H}{s}$; the second equation is the divergence-free condition of the scattering field; the incident wave is the wave propagating in free space, and it satisfies the divergence-free condition $\nabla \cdot \mathbf{H}{i}=0 ; \varepsilon_{r}$ and $\mu_{r}$ are respectively the relative permittivity and permeability of the propagation medium; $\omega$ is the angular frequency; $t$ is the time; $j=\sqrt{-1}$ is the imaginary unit; $k_{0}=\omega \sqrt{\varepsilon_{0} \mu_{0}}$ is the free space wave number, with $\varepsilon_{0}$ and $\mu_{0}$ respectively representing the free space permittivity and permeability; $\Omega \subset \mathbb{R}^{3}$ is the computational domain. To truncate the wave field towards infinite space and investigate the field in a given space without artefacts, boundary conditions need to be imposed on the border $\partial \Omega \subset \mathbb{R}^{2}$ of the computational domain $\Omega$.

The boundary conditions for Eq.3.1 usually include a first-order adsorbing condition, as well as perfect magnetic and electric conditions. The first-order absorbing condition can be used to truncate the field distribution at infinity [28]
$$\mathbf{n} \times\left(\varepsilon_{r}^{-1} \nabla \times \mathbf{H}{s}\right)-j k{0} \sqrt{\varepsilon_{r}^{-1} \mu_{r}} \mathbf{n} \times\left(\mathbf{H}{s} \times \mathbf{n}\right)=\mathbf{0}, \text { on } \Gamma{a}$$
where $\mathbf{n}$ is the unit outward normal vector at the trace $\partial \Omega ; \Gamma_{a} \subset \partial \Omega$ is the absorbing boundary. The perfect magnetic and electric conditions are used to describe the truncation of the field at perfect magnetic and electric conductors, where the tangential continuity of the field is ensured
$$\left{\begin{array}{l} \mathbf{n} \times\left(\mathbf{H}{s}+\mathbf{H}{i}\right)=\mathbf{0}, \text { on } \Gamma_{P M C} \ \mathbf{n} \times\left[\varepsilon_{r}^{-1} \nabla \times\left(\mathbf{H}{s}+\mathbf{H}{i}\right)\right]=\mathbf{0}, \text { on } \Gamma_{P E C} \end{array}\right.$$ where $\Gamma_{P M C}$ and $\Gamma_{P E C}$ are the perfect magnetic and electric boundaries respectively. The perfect magnetic boundary condition can also be used to express the symmetry of the field.

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

The variational problem in Eq. $3.7$ is analyzed to derive the gradient information used to evolve the design variable. It has been clarified that the adjoint method is an efficient approach with which to derive the gradient expressions of a PDE constrained optimization problem [22]. Being different from the conventional case, the functional space for the wave equation in Eq. $3.1$ needs to be chosen to satisfy the divergence-free condition [37]
$$\mathscr{V}{\mathbf{H}} \doteq\left{\mathbf{u} \in \mathscr{H}\left(\text { curl; } \Omega \text { ) } \mid \nabla \cdot \mathbf{u}=0, \text { in } \Omega ; \mathbf{n} \times \mathbf{u}=\mathbf{0}, \text { on } \Gamma{P M C}\right}\right.$$
where
$$\mathscr{H}(\text { curl; } \Omega)=\left{\mathbf{u} \in\left(L^{2}(\Omega)\right)^{3} \mid \nabla \times \mathbf{u} \in\left(L^{2}(\Omega)\right)^{3}\right}$$
and $L^{2}(\Omega)$ is the second-order Lebesgue integrable functional space. Then, according to the Kurash-Kuhn-Tucker condition of the PDE-constrained optimization problem [22], the adjoint equations of the wave equation and PDE filter can be obtained as
Find $\mathbf{H}{s a} \in \mathcal{V}{\mathbf{H}}$ such that
\begin{aligned} &\int_{\Omega} \frac{\partial A}{\partial \mathbf{H}{s}} \cdot \phi+\frac{\partial A}{\partial \nabla \times \mathbf{H}{s}} \cdot(\nabla \times \phi)+\varepsilon_{r}^{-1}\left(\nabla \times \overline{\mathbf{H}}{s a}\right) \cdot(\nabla \times \phi)-k{0}^{2} \mu_{r} \overline{\mathbf{H}}{s a} \cdot \boldsymbol{\phi} \mathrm{d} \Omega \ &+\int{\Gamma_{a}} j k_{0} \sqrt{\varepsilon_{r}^{-1} \mu_{r}}\left(\mathbf{n} \times \overline{\mathbf{H}}{s a} \times \mathbf{n}\right) \cdot(\mathbf{n} \times \boldsymbol{\phi} \times \mathbf{n})+\frac{\partial B}{\partial \mathbf{H}{s}} \cdot \phi \mathrm{d} \Gamma \ &+\int_{\Gamma_{P E c}} \frac{\partial B}{\partial \mathbf{H}{s}} \cdot \phi \mathrm{d} \Gamma=0, \forall \phi \in \mathscr{V}{\mathbf{H}} \end{aligned}

## 数学代写|拓扑学代写Topology代考|Three-Dimensional Optical Waves

j=∫Ω一个(和sR,和sl,∇X×和sR,∇X×和sl,Cp;C)+μr−1[∇×(和sR+和一世R)]⋅(∇×和^sR) −ķ02[erR(和sR+和一世R)−erl(和sl+和一世l)]⋅和^sR+μr−1[∇×(和sl+和一世l)]⋅(∇×和^sl) −ķ02[erl(和sR+和一世R)+erR(和sl+和一世l)]⋅和^sl dΩ+∫Ωpμr−1(吨∇×和sR) ⋅(吨∇×和^sR)|吨|−1−ķ02(erR和sR−erl和sl)⋅和^sR|吨|+μr−1(吨∇×和sl) ⋅(吨∇×和^sl)|吨|−1−ķ02(erl和sR+erR和sl)⋅和^sl|吨|dΩ +∫Ωdr2∇CF⋅∇C^F+CFC^F−CC^F dΩ

\left{\mathbf{u}\in\mathscr{H}\left(\operatorname{curl};\Omega\cup\Omega{P}\right)\mid\nabla\cdot\mathbf{u}=0\文本 { in } \Omega\cup\Omega_{P}\right}\left{\mathbf{u}\in\mathscr{H}\left(\operatorname{curl};\Omega\cup\Omega{P}\right)\mid\nabla\cdot\mathbf{u}=0\文本 { in } \Omega\cup\Omega_{P}\right}

\mathscr{H}\left(\operatorname{curl} ; \Omega \cup \Omega_{P}\right)=\left{\mathbf{u} \in\left(\mathscr{L}^{2}\左(\Omega \cup \Omega_{P}\right)\right)^{3} \mid \nabla \times \mathbf{u} \in\left(\mathscr{L}^{2}\left(\欧米茄 \cup \Omega_{P}\right)\right)^{3}\right}\mathscr{H}\left(\operatorname{curl} ; \Omega \cup \Omega_{P}\right)=\left{\mathbf{u} \in\left(\mathscr{L}^{2}\左(\Omega \cup \Omega_{P}\right)\right)^{3} \mid \nabla \times \mathbf{u} \in\left(\mathscr{L}^{2}\left(\欧米茄 \cup \Omega_{P}\right)\right)^{3}\right}

dĴ^=∫Ω∂一个∂和sR⋅d和sR+∂一个∂和sl⋅d和sl+∂一个∂∇×和sR⋅(∇×d和sR)+∂一个∂∇×和sl (∇×d和s我)dΩ+∑n=1ñ1在n∫磷n∂一个∂Cp∂Cp∂C和dCF dΩ+∫Ωd∂一个∂CdCdΩ +∫Ωμr−1(∇×d和sR)⋅(∇×和^sR)−ķ02(erRd和sR−erld和sl)⋅和^sR dΩ

## 数学代写|拓扑学代写Topology代考|Magnetic Field Formulation

$$\左{ ∇×[er−1∇×(Hs+H一世)]−ķ02μr(Hs+H一世)=0, 在 Ω ∇⋅Hs=0, 在 Ω\正确的。$$ 其中散射场公式与磁场一起使用H分为两部分，即入射波H一世和散射场Hs; 第二个方程是散射场的无散度条件；入射波是在自由空间中传播的波，它满足无散度条件∇⋅H一世=0;er和μr分别是传播介质的相对介电常数和磁导率；ω是角频率；吨是时间；j=−1是虚数单位；ķ0=ωe0μ0是自由空间波数，其中e0和μ0分别代表自由空间介电常数和磁导率；Ω⊂R3是计算域。为了将波场向无限空间截断并研究给定空间中没有伪影的场，需要在边界上施加边界条件∂Ω⊂R2计算域的Ω.

Eq.3.1 的边界条件通常包括一阶吸附条件，以及完美的磁电条件。一阶吸收条件可用于截断无穷远处的场分布 [28]

n×(er−1∇×Hs)−jķ0er−1μrn×(Hs×n)=0, 上 Γ一个

$$\left{ n×(Hs+H一世)=0, 上 Γ磷米C n×[er−1∇×(Hs+H一世)]=0, 上 Γ磷和C\正确的。$$ 在哪里Γ磷米C和Γ磷和C分别是完美的磁边界和电边界。完美的磁边界条件也可以用来表示场的对称性。

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

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

\mathscr{H}(\text{curl;}\Omega)=\left{\mathbf{u}\in\left(L^{2}(\Omega)\right)^{3}\mid\nabla\次 \mathbf{u}\in\left(L^{2}(\Omega)\right)^{3}\right}\mathscr{H}(\text{curl;}\Omega)=\left{\mathbf{u}\in\left(L^{2}(\Omega)\right)^{3}\mid\nabla\次 \mathbf{u}\in\left(L^{2}(\Omega)\right)^{3}\right}

FindHs一个∈在H这样

∫Ω∂一个∂Hs⋅φ+∂一个∂∇×Hs⋅(∇×φ)+er−1(∇×H¯s一个)⋅(∇×φ)−ķ02μrH¯s一个⋅φdΩ +∫Γ一个jķ0er−1μr(n×H¯s一个×n)⋅(n×φ×n)+∂乙∂Hs⋅φdΓ +∫Γ磷和C∂乙∂Hs⋅φdΓ=0,∀φ∈在H

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