### 数学代写|拓扑学代写Topology代考|MATH6204

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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|Topology Optimization Problem

In a two-dimensional case, surface plasmon polaritons are excited by transverse magnetic (magnetic field in the $z$ direction) polarized waves, scattered by metallic nanostructures. For transverse magnetic waves propagating in the $x-y$ plane, the scattered-field formulation is used in order to reduce the dispersion error
$$\nabla \cdot\left[\varepsilon_{r}^{-1} \nabla\left(H_{z s}+H_{z i}\right)\right]+k_{0}^{2} \mu_{r}\left(H_{z s}+H_{z i}\right)=0, \text { in } \Omega$$
where $H_{z}=H_{z s}+H_{z i}$ is the total field, $H_{z s}$ and $H_{z i}$ are the scattered 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 $\omega, \varepsilon_{0}$ and $\mu_{0}$ representing the angular frequency, free space permittivity and permeability, respectively; $\Omega$ is the computational domain; the time dependence of the fields is given by the factor $e^{j \omega t}$, with $t$ representing the time. The incident field can be obtained by solving the electromagnetic equations in free space, with boundary conditions representing realistic working conditions.

The boundary conditions of Eq. $4.18$ usually include the first-order absorbing condition, periodic boundary condition and symmetric condition. The first-order absorbing condition is usually used to truncate the field distribution at infinity [46]
$$\varepsilon_{r}^{-1} \nabla H_{s z} \cdot \mathbf{n}+j k_{0} \sqrt{\varepsilon_{r}^{-1} \mu_{r}} H_{s z}=0, \text { on } \Gamma_{a b}$$
where $j$ is the imaginary unit; $\mathbf{n}$ is the unit outward normal vector at the boundary $\partial \Omega$ of the computational domain; $\Gamma_{a b}$ is the absorbing boundary included in $\partial \Omega$. Periodicity of nanostructures plays a crucial role in tuning the optical response; and single nanostructure can be approximated by the periodic case with low volume ratio of the nanostructure. Therefore, the periodic boundary condition for the scattered field, induced by the periodic incident wave, is often imposed on the piecewise pair included in $\partial \Omega$
$$\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=-e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right} \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}$$
where $\Gamma_{p d}$ and $\Gamma_{p s}$ composes one piecewise periodic boundary pair, with $\Gamma_{p d}$ and $\Gamma_{p s}$ respectively being the destination and source boundaries; $\mathbf{k}$ is the wave vector; $\mathbf{a}$ is the lattice vector of the periodic nanostructures. The symmetry of the incident wave and material distribution gives rise to the symmetrical characteristic of the scattered

field. Then the symmetric condition can be used to reduce the computational cost and ensure the computational accuracy effectively
$$\varepsilon_{r}^{-1} \nabla H_{s z} \cdot \mathbf{n}=0, \text { on } \Gamma_{s m}$$

In this section, the variational problem for computational design is analyzed to obtain the gradient information used to iteratively evolve the design variable. According to the Refs. $[38,41,64]$, the adjoint method is an efficient approach to derive the derivative of the objective in the partial differential equation constrained variational problem. Then, the adjoint Eqs. $4.18$ and $4.21$ are obtained using the Lagrangian multiplier-based adjoint method (see Appendix $4.4$ for more details)
and
$$\left{\begin{array}{l} -\nabla \cdot\left(r^{2} \nabla \bar{\rho}{f}^{}\right)+\bar{\rho}{f}^{}=\frac{\partial \varepsilon_{r}^{-1}}{\partial \rho_{f p}} \frac{\partial \rho_{f p}}{\partial \rho_{f}} \nabla\left(H_{z s}+H_{z i}\right) \cdot \nabla \tilde{H}{z s}^{}-\frac{\partial A}{\partial \rho{f p}} \frac{\partial \rho_{f p}}{\partial \rho_{f}}, \text { in } \Omega \ r^{2} \nabla \bar{\rho}{f}^{} \cdot \mathbf{n}=j k{0} \frac{\partial \sqrt{\varepsilon_{r}^{-1}}}{\partial \rho_{f p}} \frac{\partial \rho_{f p}}{\partial \rho_{f}} \sqrt{\mu_{r}} H_{z s} \bar{H}{z s}^{}-\frac{\partial \varepsilon{r}^{-1}}{\partial \rho_{f p}} \frac{\partial \rho_{f p}}{\partial \rho_{f}} \nabla H_{z i} \cdot \mathbf{n} \bar{H}{z s}^{} \text {, on } \Gamma{a b} \ r^{2} \nabla \tilde{\rho}{f}^{} \cdot \mathbf{n}=-\frac{\partial \varepsilon{r}^{-1}}{\partial \rho_{f p}} \frac{\partial \rho_{f p}}{\partial \rho_{f}} \nabla H_{z i} \cdot \mathbf{n} \tilde{H}{z s}^{}, \text { on } \Gamma{p d} \cup \Gamma_{p s} \cup \Gamma_{s m} \end{array}\right.$$
where $\bar{H}{z s} \in \mathscr{H}^{1 *}(\Omega)$ and $\bar{\rho}{f} \in \mathscr{H}^{1 *}(\Omega)$ are the adjoint variables of the state variables $H_{z s} \in \mathscr{H}^{1}(\Omega)$ and $\rho_{f} \in \mathscr{H}^{1}(\Omega)$, respectively; $\mathscr{H}^{1}(\Omega)$ is the first-order Sobolev space, and $\mathscr{H}^{1 *}(\Omega)$ is the dual space of $\mathscr{H}^{1}(\Omega)$; for complex, * represents the conjugate operation. It is valuable to notice that $\bar{H}{z s}^{}$ and $\rho{f}^{}$ are more convenient to be solved than $\bar{H}{z s}$ and $\rho{f}$ in the adjoint Eqs. $4.11$ and $4.12$. Therefore, the adjoint Eqs. $4.11$ and $4.12$ are utilized to solve $\tilde{H}{z s}^{}$ and $\rho{f}^{}$, and $\tilde{H}{z s}$ and $\rho{f}$ can be obtained using conjugate operation. The adjoint derivative of the computational design problem is obtained as (see Appendix $4.4$ for more details)
$$\frac{\delta \hat{J}}{\delta \rho}=\operatorname{Re}\left(\frac{\partial A}{\partial \rho}-\tilde{\rho}_{f}^{*}\right), \text { in } \Omega$$

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

After adjoint analysis, the computational design problem of the nanostructures for surface plasmon polaritons can be solved using the iterative approach based on the obtained sensitivity information. The procedure for the iterative approach includes the following steps (Table 4.1): (a) the partial differential equations are solved with the current design variable; (b) the adjoint equations are solved based on the solution of the partial differential equations; (c) the adjoint derivative of the design objective is computed; (d) the design variable is updated using the method of moving asymptotes [80]. The above steps are implemented iteratively until the stopping criteria are satisfied. The stopping criterion is specified as the change of the objective values in 5 consecutive iterations satisfying
$$\frac{1}{5} \sum_{i=0}^{4}\left|J_{k-i}-J_{k-i-1}\right| /\left|J_{k}\right|<\varepsilon$$
in the $k$ th iteration, where $J_{k}$ and $\gamma_{k}$ are the objective value and distribution of the design variable in the $k$ th iteration, respectively; $\varepsilon$ is the tolerance chosen to be $1 \times 10^{-3}$. The maximal iterative number is set to be 660 .

During the solving procedure, the threshold parameter $\xi$ in Eq. $4.22$ is set to be $0.5$; the initial value of the projection parameter $\beta$ is set to be 1 and it is doubled every 60 iterations until the preset maximal value $2^{10}$ is reached; the partial differential equations and corresponding adjoint equations are solved by the finite element method. For the details on the setting of the numerical implementations, one can refer to $[21,66]$

In the computational design of nanostructures for surface plasmon polaritons, the partial differential equations are solved using the standard Galerkin finite ele-ment method. The state variables andcorresponding adjoint variables are interpolated quadratically; the Helmholtz filter and corresponding adjoint equations are solved using linear elements; and the design variable is linearly interpolated.

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

∇⋅[er−1∇(H和s+H和一世)]+ķ02μr(H和s+H和一世)=0, 在 Ω

er−1∇Hs和⋅n+jķ0er−1μrHs和=0, 上 Γ一个b

\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=- e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right } \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}\left.\begin{array}{l} H_{s z}(\mathbf{x}+\mathbf{a})=H_{s z}(\mathbf{x}) e^{-j \mathbf{k} \cdot \mathbf{a}} \ \mathbf{n}(\mathbf{x}+\mathbf{a}) \cdot \nabla H_{s z}(\mathbf{x}+\mathbf{a})=- e^{-j \mathbf{k} \mathbf{a}} \mathbf{n}(\mathbf{x}) \cdot \nabla H_{s z}(\mathbf{x}) \end{array}\right } \text { for } \forall \mathbf{x} \in \Gamma_{p s}, \mathbf{x}+\mathbf{a} \in \Gamma_{p d}

er−1∇Hs和⋅n=0, 上 Γs米

$$\left{ −∇⋅(r2∇ρ¯F)+ρ¯F=∂er−1∂ρFp∂ρFp∂ρF∇(H和s+H和一世)⋅∇H~和s−∂一个∂ρFp∂ρFp∂ρF, 在 Ω r2∇ρ¯F⋅n=jķ0∂er−1∂ρFp∂ρFp∂ρFμrH和sH¯和s−∂er−1∂ρFp∂ρFp∂ρF∇H和一世⋅nH¯和s， 上 Γ一个b r2∇ρ~F⋅n=−∂er−1∂ρFp∂ρFp∂ρF∇H和一世⋅nH~和s, 上 Γpd∪Γps∪Γs米\正确的。 在H和r和H¯和s∈H1∗(Ω)一个ndρ¯F∈H1∗(Ω)一个r和吨H和一个dj○一世n吨在一个r一世一个bl和s○F吨H和s吨一个吨和在一个r一世一个bl和sH和s∈H1(Ω)一个ndρF∈H1(Ω),r和sp和C吨一世在和l是;H1(Ω)一世s吨H和F一世rs吨−○rd和r小号○b○l和在sp一个C和,一个ndH1∗(Ω)一世s吨H和d在一个lsp一个C和○FH1(Ω);F○rC○米pl和X,∗r和pr和s和n吨s吨H和C○nj在G一个吨和○p和r一个吨一世○n.我吨一世s在一个l在一个bl和吨○n○吨一世C和吨H一个吨H¯和s一个ndρF一个r和米○r和C○n在和n一世和n吨吨○b和s○l在和d吨H一个nH¯和s一个ndρF一世n吨H和一个dj○一世n吨和qs.4.11一个nd4.12.吨H和r和F○r和,吨H和一个dj○一世n吨和qs.4.11一个nd4.12一个r和在吨一世l一世和和d吨○s○l在和H~和s一个ndρF,一个ndH~和s一个ndρFC一个nb和○b吨一个一世n和d在s一世nGC○nj在G一个吨和○p和r一个吨一世○n.吨H和一个dj○一世n吨d和r一世在一个吨一世在和○F吨H和C○米p在吨一个吨一世○n一个ld和s一世Gnpr○bl和米一世s○b吨一个一世n和d一个s(s和和一个pp和nd一世X4.4F○r米○r和d和吨一个一世ls) \frac{\delta \hat{J}}{\delta \rho}=\operatorname{Re}\left(\frac{\partial A}{\partial \rho}-\tilde{\rho}_{f} ^{*}\right), \text { in } \Omega$$

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

15∑一世=04|Ĵķ−一世−Ĵķ−一世−1|/|Ĵķ|<e

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