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## 数学代写|拓扑学代写Topology代考|Topology Optimization Problems

Optical wave propagation is described by Maxwell’s equations. In frequency domain, Maxwell’s equations can be transformed into the wave equations. In order to reduce the dispersion error, the scattering field formulation of the wave equations are used with the time-harmonic factor $e^{j \omega t}$, where $j=\sqrt{-1}$ is the imaginary unit, $\omega$ is the angular frequency and $t$ is the time.

For the optical waves that can be reduced into two dimensions, the transverse magnetic waves with polarization perpendicular to the wave plane are the more general cases. This is because that the transverse magnetic waves can both include the description of related physics with dielectrics and noble metal, where the surface plasmonic polaritons can not be included in the transverse electric waves [16]. Therefore, this chapter focuses on the transverse magnetic waves for the two-dimensional cases. Without losing the generality, it can also be extended to the transverse electric waves with the similar procedure. For the transverse magnetic wave, the wave equation is
$$\nabla \cdot\left[\varepsilon_{r}^{-1} \nabla\left(H_{s z}+H_{i z}\right)\right]+k_{0}^{2} \mu_{r}\left(H_{s z}+H_{i z}\right)=0, \text { in } \Omega$$
where the transverse magnetic wave is polarized in the $z$-direction; $\nabla$ is the gradient operator in the Cartesian coordinate system; $H_{z}=H_{s z}+H_{i z}$ is the total field, $H_{s z}$ and $H_{i z}$ are the scattering and incident fields, respectively; $\varepsilon_{r}$ and $\mu_{r}$ are the relative permittivity and permeability, respectively; $k_{0}=\omega \sqrt{\varepsilon_{0} \mu_{0}}$ is the free space wave number with $\varepsilon_{0}$ and $\mu_{0}$ respectively representing the permittivity and permeability of the free space; $\Omega$ is a square-shaped wave propagating domain.

For the optical waves that can not be reduced into two dimensions (e.g., optical waves scattered by objects with complicated geometrical configurations), the threedimensional wave equation is used to describe the wave propagation as
$$\left{\begin{array}{l} \nabla \times\left[\mu_{r}^{-1} \nabla \times\left(\mathbf{E}{s}+\mathbf{E}{i}\right)\right]-k_{0}^{2} \varepsilon_{r}\left(\mathbf{E}{s}+\mathbf{E}{i}\right)=\mathbf{0}, \text { in } \Omega \ \nabla \cdot\left[\varepsilon_{r}\left(\mathbf{E}{s}+\mathbf{E}{i}\right)\right]=0, \text { in } \Omega \end{array}\right.$$
where the electric field $\mathbf{E}=\mathbf{E}{s}+\mathbf{E}{i}$ is the total field, $\mathbf{E}{s}$ and $\mathbf{E}{i}$ are respectively the scattering and incident fields; the second equation is the divergence-free condition of the electric displacement; $\Omega$ is a cuboid-shaped wave propagating domain.

The incident waves in Eqs. $2.1$ and $2.2$ are the optical waves propagating in the background of the computational domain. The infinite computational space is truncated by the perfectly matched layers (PMLs), which are implemented by solving the wave equations with the complex-valued coordinate transformation $[4,10,17]$ : For the two-dimensional cases,
$$\nabla_{\mathbf{x}^{\prime}} \cdot\left(\varepsilon_{r}^{-1} \nabla_{\mathbf{x}^{\prime}} H_{s z}\right)+k_{0}^{2} \mu_{r} H_{s z}=0, \text { in } \Omega_{P}$$
for the three-dimensional cases,
$$\left{\begin{array}{l} \nabla_{\mathbf{x}^{\prime}} \times\left(\mu_{r}^{-1} \nabla_{\mathbf{x}^{\prime}} \times \mathbf{E}{s}\right)-k{0}^{2} \varepsilon_{r} \mathbf{E}{s}=\mathbf{0}, \text { in } \Omega{P} \ \nabla \cdot \mathbf{E}{s}=0, \text { in } \Omega{P} \end{array}\right.$$

## 数学代写|拓扑学代写Topology代考|Split of Wave Equations

The iterative procedure is the usual approach to solve the topology optimization problems in Eqs. $2.15$ and 2.16. In the iterative approach, the design variable is evolved based on the gradient information of the cost function. The gradient information can be efficiently derived by the adjoint method [9], which is implemented based on the first-order variational of the augmented Lagrangian of the optimization problems in Eqs. $2.15$ and 2.16. However, the integral functional $A$ of the cost function contains the conjugate of the field variables, which is Gâteaux differential instead of Fréchet differential. This results in the complexity of the adjoint sensitivity, which is self-inconsistent. The self-inconsistency of the sensitivity further results in that the derived structural topology has dependence on the phase of the incident wave. The effect of the self-inconsistent adjoint sensitivity will be demonstrated by the numerical results in Sect. 2.5.1.

To solve the problem on the self-inconsistency of the adjoint sensitivity, the wave equations in Eqs. $2.1$ and $2.2$ can be transformed by splitting the complex variables into the real and imaginary parts. By setting $H_{s z}=H_{s z}^{R}+j H_{s z}^{I}, H_{i z}=H_{i z}^{R}+j H_{i z}^{l}$, $\varepsilon_{r}^{-1}=\left(\varepsilon_{r}^{-1}\right)^{R}+j\left(\varepsilon_{r}^{-1}\right)^{I}$, and substituting these split variables into Eq. 2.1, the twodimensional wave equation is transformed into
$$\left{\begin{array}{l} \nabla \cdot\left[\left(\varepsilon_{r}^{-1}\right)^{R} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right)-\left(\varepsilon_{r}^{-1}\right)^{I} \nabla\left(H_{s z}^{l}+H_{i z}^{l}\right)\right]+k_{0}^{2} \mu_{r}\left(H_{s z}^{R}+H_{i z}^{R}\right)=0, \text { in } \Omega \ \nabla \cdot\left[\left(\varepsilon_{r}^{-1}\right)^{l} \nabla\left(H_{s z}^{R}+H_{i z}^{R}\right)+\left(\varepsilon_{r}^{-1}\right)^{R} \nabla\left(H_{s z}^{l}+H_{i z}^{l}\right)\right]+k_{0}^{2} \mu_{r}\left(H_{s z}^{l}+H_{i z}^{l}\right)=0, \text { in } \Omega \end{array}\right.$$
where the superscripts $R$ and $I$ are respectively used to mark the real and imaginary parts of the corresponding complex variable; the scattering boundary conditions for $H_{s z}^{R}$ and $H_{s z}^{I}$ are respectively implemented by solving the split wave equation in the PMLs with complex-valued coordinate transformation and zero-incident field
$$\left{\begin{array}{l} \nabla_{\mathbf{x}^{\prime}} \cdot\left[\left(\varepsilon_{r}^{-1}\right)^{R} \nabla_{\mathbf{x}^{\prime}} H_{s z}^{R}-\left(\varepsilon_{r}^{-1}\right)^{l} \nabla_{\mathbf{x}^{\prime}} H_{s z}^{l}\right]+k_{0}^{2} \mu_{r}\left(H_{s z}^{R}+H_{i z}^{R}\right)=0, \text { in } \Omega_{P} \ \nabla_{\mathbf{x}^{\prime}} \cdot\left[\left(\varepsilon_{r}^{-1}\right)^{I} \nabla_{\mathbf{x}^{\prime}} H_{s z}^{R}+\left(\varepsilon_{r}^{-1}\right)^{R} \nabla_{\mathbf{x}^{\prime}} H_{s z}^{I}\right]+k_{0}^{2} \mu_{r}\left(H_{s z}^{I}+H_{i z}^{I}\right)=0, \text { in } \Omega_{P} \end{array}\right.$$
Similarly, by setting $\mathbf{E}{s}=\mathbf{E}{s}^{R}+j \mathbf{E}{s}^{l}, \mathbf{E}{i}=\mathbf{E}{i}^{R}+j \mathbf{E}{i}^{l}, \varepsilon_{r}=\varepsilon_{r}^{R}+j \varepsilon_{r}^{I}$, and substituting these split variables into Eq. 2.2, the three-dimensional wave equation is transformed into
$$\left{\begin{array}{l} \nabla \times\left[\mu_{r}^{-1} \nabla \times\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)\right]-k_{0}^{2}\left[\varepsilon_{r}^{R}\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)-\varepsilon_{r}^{I}\left(\mathbf{E}{s}^{I}+\mathbf{E}{i}^{I}\right)\right]=\mathbf{0}, \text { in } \Omega \ \nabla \cdot \mathbf{E}{s}^{R}=0, \text { in } \Omega \ \nabla \times\left[\mu{r}^{-1} \nabla \times\left(\mathbf{E}{s}^{I}+\mathbf{E}{i}^{I}\right)\right]-k_{0}^{2}\left[\varepsilon_{r}^{I}\left(\mathbf{E}{s}^{R}+\mathbf{E}{i}^{R}\right)+\varepsilon_{r}^{R}\left(\mathbf{E}{s}^{I}+\mathbf{E}{i}^{I}\right)\right]=\mathbf{0}, \text { in } \Omega \ \nabla \cdot \mathbf{E}_{s}^{I}=0, \text { in } \Omega \end{array}\right.$$

Based on the Lagrangian multiplier-based adjoint method and the adjoint analysis of the transformed topology optimization problem in Eq. 2.21, the weak forms of the adjoint equations for the topology optimization problem in Eq. $2.15$ are derived as (the details are presented in Appendix 2.7.1):
Find $\hat{H}{s z}$ with $\operatorname{Re}\left(\hat{H}{s z}\right) \in \mathscr{H}\left(\Omega \cup \Omega_{P}\right), \operatorname{Im}\left(\hat{H}{s z}\right) \in \mathscr{H}\left(\Omega \cup \Omega{P}\right)$ and $\hat{H}{s z}=0$ on $\Gamma{D}$, such that
\begin{aligned} &\int_{\Omega}\left(\frac{\partial A}{\partial \operatorname{Re}\left(H_{s z}\right)}-j \frac{\partial A}{\partial I m\left(H_{s z}\right)}\right) \phi+\left(\frac{\partial A}{\partial \nabla \operatorname{Re}\left(H_{s z}\right)}-j \frac{\partial A}{\partial \nabla I m\left(H_{s z}\right)}\right) \cdot \nabla \phi \ &-\varepsilon_{r}^{-1} \nabla \hat{H}{s z}^{} \cdot \nabla \phi+k{0}^{2} \mu_{r} \hat{H}{s z}^{} \phi \mathrm{d} \Omega+\int{\Omega_{p}}-\varepsilon_{r}^{-1}\left(\mathbf{T} \nabla \hat{H}{s z}^{}\right) \cdot(\mathbf{T} \nabla \phi)|\mathbf{T}|^{-1} \ &+k{0}^{2} \mu_{r} \hat{H}{s z}^{} \phi|\mathbf{T}| \mathrm{d} \Omega=0, \forall \phi \in \mathscr{H}\left(\Omega \cup \Omega{P}\right) \end{aligned}
and
Find $\hat{\gamma}{f} \in \mathscr{H}\left(\Omega{d}\right)$ such that
\begin{aligned} &\int_{\Omega_{d}} r^{2} \nabla \hat{\gamma}{f} \cdot \nabla \varphi+\hat{\gamma}{f} \varphi+\left[\sum_{n=1}^{N} \frac{1}{V_{n}} \int_{P_{n}} \frac{\partial A}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{c}}-\operatorname{Re}\left(\frac{\partial \varepsilon_{r}^{-1}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}+H_{i z}\right)\right)\right. \ &\left.\cdot \operatorname{Re}\left(\nabla \hat{H}{s z}^{}\right)+\operatorname{Im}\left(\frac{\partial \varepsilon{r}^{-1}}{\partial \gamma_{p}} \frac{\partial \gamma_{p}}{\partial \gamma_{e}} \nabla\left(H_{s z}+H_{i z}\right)\right) \cdot \operatorname{Im}\left(\nabla \hat{H}{s z}^{}\right) \mathrm{d} \Omega\right] \varphi \mathrm{d} \Omega=0 \ &\forall \varphi \in \mathscr{H}\left(\Omega{d}\right), \end{aligned}
where $\hat{H}{s z}$ and $\hat{\gamma}{f}$ are the adjoint variables of $H_{s z}$ and $\gamma_{f}$ respectively; ${ }^{*}$ is the operator used to implement the conjugate of a complex variable; $\Gamma_{D}$ is the perfect magnetic conductor boundary; $\mathscr{H}\left(\Omega \cup \Omega_{P}\right)$ 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; $R e$ and $I m$ are operators used to extract the real and imaginary parts of a complex. The adjoint sensitivity is derived as
$$\delta \hat{J}=\int_{\Omega_{d}}\left(\frac{\partial A}{\partial \gamma}-\hat{\gamma}{f}\right) \delta \gamma \mathrm{d} \Omega, \forall \delta \gamma \in \mathscr{L}^{2}\left(\Omega{d}\right),$$

## 数学代写|拓扑学代写Topology代考|Topology Optimization Problems

∇⋅[er−1∇(Hs和+H一世和)]+ķ02μr(Hs和+H一世和)=0, 在 Ω

$$\left{ ∇×[μr−1∇×(和s+和一世)]−ķ02er(和s+和一世)=0, 在 Ω ∇⋅[er(和s+和一世)]=0, 在 Ω\正确的。$$

∇X′⋅(er−1∇X′Hs和)+ķ02μrHs和=0, 在 Ω磷

$$\left{ ∇X′×(μr−1∇X′×和s)−ķ02er和s=0, 在 Ω磷 ∇⋅和s=0, 在 Ω磷\正确的。$$

## 数学代写|拓扑学代写Topology代考|Split of Wave Equations

$$\left{ ∇⋅[(er−1)R∇(Hs和R+H一世和R)−(er−1)我∇(Hs和l+H一世和l)]+ķ02μr(Hs和R+H一世和R)=0, 在 Ω ∇⋅[(er−1)l∇(Hs和R+H一世和R)+(er−1)R∇(Hs和l+H一世和l)]+ķ02μr(Hs和l+H一世和l)=0, 在 Ω\正确的。 在H和r和吨H和s在p和rsCr一世p吨sR一个nd我一个r和r和sp和C吨一世在和l是在s和d吨○米一个rķ吨H和r和一个l一个nd一世米一个G一世n一个r是p一个r吨s○F吨H和C○rr和sp○nd一世nGC○米pl和X在一个r一世一个bl和;吨H和sC一个吨吨和r一世nGb○在nd一个r是C○nd一世吨一世○nsF○rHs和R一个ndHs和我一个r和r和sp和C吨一世在和l是一世米pl和米和n吨和db是s○l在一世nG吨H和spl一世吨在一个在和和q在一个吨一世○n一世n吨H和磷米大号s在一世吨HC○米pl和X−在一个l在和dC○○rd一世n一个吨和吨r一个nsF○r米一个吨一世○n一个nd和和r○−一世nC一世d和n吨F一世和ld \剩下{ ∇X′⋅[(er−1)R∇X′Hs和R−(er−1)l∇X′Hs和l]+ķ02μr(Hs和R+H一世和R)=0, 在 Ω磷 ∇X′⋅[(er−1)我∇X′Hs和R+(er−1)R∇X′Hs和我]+ķ02μr(Hs和我+H一世和我)=0, 在 Ω磷\正确的。 小号一世米一世l一个rl是,b是s和吨吨一世nG和s=和sR+j和sl,和一世=和一世R+j和一世l,er=erR+jer我,一个nds在bs吨一世吨在吨一世nG吨H和s和spl一世吨在一个r一世一个bl和s一世n吨○和q.2.2,吨H和吨Hr和和−d一世米和ns一世○n一个l在一个在和和q在一个吨一世○n一世s吨r一个nsF○r米和d一世n吨○ \剩下{ ∇×[μr−1∇×(和sR+和一世R)]−ķ02[erR(和sR+和一世R)−er我(和s我+和一世我)]=0, 在 Ω ∇⋅和sR=0, 在 Ω ∇×[μr−1∇×(和s我+和一世我)]−ķ02[er我(和sR+和一世R)+erR(和s我+和一世我)]=0, 在 Ω ∇⋅和s我=0, 在 Ω\正确的。$$

∫Ω(∂一个∂回覆⁡(Hs和)−j∂一个∂我米(Hs和))φ+(∂一个∂∇回覆⁡(Hs和)−j∂一个∂∇我米(Hs和))⋅∇φ −er−1∇H^s和⋅∇φ+ķ02μrH^s和φdΩ+∫Ωp−er−1(吨∇H^s和)⋅(吨∇φ)|吨|−1 +ķ02μrH^s和φ|吨|dΩ=0,∀φ∈H(Ω∪Ω磷)

∫Ωdr2∇C^F⋅∇披+C^F披+[∑n=1ñ1在n∫磷n∂一个∂Cp∂Cp∂CC−回覆⁡(∂er−1∂Cp∂Cp∂C和∇(Hs和+H一世和)) ⋅回覆⁡(∇H^s和)+在里面⁡(∂er−1∂Cp∂Cp∂C和∇(Hs和+H一世和))⋅在里面⁡(∇H^s和)dΩ]披dΩ=0 ∀披∈H(Ωd),

dĴ^=∫Ωd(∂一个∂C−C^F)dCdΩ,∀dC∈大号2(Ωd),

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