数学代写|拓扑学代写Topology代考|Self-consistency of Adjoint Analysis for Topology

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拓扑学是数学的一个分支,有时被称为 “橡胶板几何”,在这个分支中,如果两个物体可以通过弯曲、扭曲、拉伸和收缩等空间运动连续变形为彼此,同时不允许撕开或粘在一起的部分,则被认为是等效的。

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我们提供的拓扑学Topology及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|Self-consistency of Adjoint Analysis for Topology

数学代写|拓扑学代写Topology代考|Optimization in Frequency Domain

In frequency domain, the field variables of the optical waves are complex, comprising the amplitude and phase of the field. Cost functions of the topology optimization problems are usually included the conjugate of the field variables, e.g., the energy functionals, which are the product of the field variables and their conjugates. The energy functionals are popularly used, because of the well-posedness of their least square forms. For the sensitivity analysis of a topology optimization problem, adjoint method is popularly used, where the first-order variational is implemented for the corresponding augmented Lagrangian [10].

During the adjoint analysis procedure, it is necessary to implement the variational of the conjugate operation to the field variables. Mathematically, the conjugate operator is Gâteaux differential instead of Fréchet differential, to the field variables [130]. This was ignored in several previous researches, such as the literatures in the Refs. $[77,87]$. The Gâteaux differentiability of the conjugate operator can cause the incompleteness of the adjoint sensitivity, i.e., the adjoint sensitivity of the real-valued cost

functions is complex instead of real for the design variable, which is real-valued distribution defined on the computational domain. If this incomplete sensitivity is used directly, the design variable will be evolved to be complex during the iterative procedure, where the initial of the design variable is set to be real. This adjoint sensitivity is then self-inconsistent from the pointview of keeping the real-value property of the design variable. Therefore, the real part extraction operator is used to extract the real part of the derived adjoint sensitivity, and to artificially enforce the self-consistency of the adjoint sensitivity.

The consequence of such enforced self-inconsistency is that the derived structural topology has dependence on the phase of the incident wave. This phase-dependence is unreasonable, because the incident waves can not be inherently distinguished by only altering their phases. To solve the problem on the self-inconsistency of the adjoint sensitivity, Fréchet differentiability should be ensured for the cost function. The conjugate operator in the cost function can be removed by splitting the complex field variables into the corresponding real and imaginary parts respectively defined on real functional spaces instead of complex functional spaces. Then the Gâteaux differentiability induced by the conjugate operator is avoided. The splitting of the complex variables brings about the splitting of the wave equations, which are complex partial differential equations. The method of splitting complex partial differential equations or variational problems into their corresponding coupled systems for the real and imaginary parts of the field variables has been systematically discussed in the Refs. $[3,65]$

数学代写|拓扑学代写Topology代考|Dielectric Material Based Topology Optimization

It has been mentioned that the control of optical waves is realized by structures with complex spacial configurations using pre-selected materials and the incident waves can have complicated polarizations. Most of those situations cannot be reduced into two dimensional, except for a minority of cases involving linear polarized waves. Most of the reports on topology optimization in optics have focused on applications, including beamsplitters $[80,86]$, photonic crystals $[37,89]$, cloaks $[7,8,38]$, sensors and resonators $[104,105]$, metamaterials $[28,77,137]$, excitation of surface plasmons [10], and electromagnetic and optical antennas $[35,36,48,134]$, without presenting the systemical topology optimization methodology for optical waves propagating in three-dimensional space. Therefore, it is necessary to develop a unified and systematic topology optimization approach that sufficiently considers the physical complexity of three-dimensional optics.

It is not straightforward to develop the finite element-based topology optimization method for optical waves in three-dimensional space, because the divergence-free condition needs to be enforced. In the two-dimensional transverse electric or magnetic wave cases, the divergence-free conditions are automatically satisfied during the reducing procedure of the Maxwell’s equations with deriving the Helmholtz equations, and the node element-based Galerkin finite element method can be naturally used to directly discretize the Helmholtz equations $[53,69]$. Being different from the two-dimensional cases, the divergence-free conditions can not be automatically satisfied in solving the three-dimensional optical waves with the node element-based Galerkin finite element method, and this results in the spurious solutions.

For this problem, two dominant approaches have been developed to enforce the divergence-free conditions and eliminate the spurious solutions. The first approach is to add a penalty term with the least square form of the divergence-free condition to the weak form of the wave equation, and then discretize the weak form with node elements. However, the use of penalty term can not eliminate the divergence of the solution completely and it affects the solution accuracy. Therefore, the divergencefree condition can not be satisfied accurately by the penalty approach [53]. The second approach is the use of edge elements that assign degrees of freedom to the edges rather than to the nodes of the elements, where the vector basis with inherent satisfaction of divergence-free condition is used to implement interpolation $[53,69,72,116]$. The edge elements have also solved the problems on the inconvenience of imposing boundary conditions at material interfaces and the difficulty in treating conducting and dielectric edges and corners due to field singularities $[53,69]$. Therefore, the edge element-based finite element method is the more reasonable choice for discretizing the three-dimensional wave equations and developing the topology optimization method for three-dimensional optical waves.

数学代写|拓扑学代写Topology代考|Metal Material-Based Topology Optimization

A metal surface with a negative real part of the permittivity can trap optical waves with achieving surface plasmon polaritons $[31,81]$. The metals used for surface plasmon polaritons are usually noble metals, e.g., silver (Ag), gold (Au) and Aluminum (Al). At optical frequencies, the metal’s free electrons can sustain, under certain conditions, oscillations with distinct resonance frequencies $[66,127,140]$. The existence of surface plasmons is characteristic for the interaction between metal and light, where the Kretschmann-Raether and Otto configurations are commonly used for plasmon excitation.

Many innovative concepts of surface plasmon polaritons have been developed over the past few years, e.g., localized surface plasmon resonances [100], extraordinary optical transmission [30] and transformational plasmon optics $[52,60]$. Correspondingly, many related applications have also been proposed for surface plasmon polaritons, e.g., biomolecular manipulation and labeling [23], surface enhanced Raman spectroscopy [70], chemical and biological sensors [33], photo-voltaics [22], nearfield lithography and imaging [84], optical trapping [67, 76], nano optic circuits [1], opto-electronic devices, wavelength-tunable filters, optical modulators [16,34,41, $42]$, plasmonic Luneburg lens and surface plasmonic cloaking $[52,60]$.

All those applications utilize surface plasmonic polaritons by reasonably designing the nanoscale structures. For the design of surface plasmonic devices, transformational plasmon optics has been developed to determine the refractive index distribution and guide the propagation of surface plasmon polaritons $[52,60]$. However, the derived distribution has extreme electromagnetic properties, i.e., permittivity and permeability, which are achieved based on the use of metamaterials. The tailored microstructure of such metamaterials has to be much smaller than the wavelength, and this makes it very challenging to realize the desired magnetic properties for optical frequencies. To overcome the similar problems in bulk optical waves and use the conventional simple isotropic dielectric readily available in nature, topology optimization-based computational design method has been used to implement the inverse design of cloak for bulk optical wave $[9,12,38]$. On surface plasmons, most related researches were focused on designing nanostructures by tuning a handful of structural parameters $[73,78,114]$, although the topology optimization method has been applied for the grating couplers to enhance the excitation efficiency of surface plasmons [11]. Therefore, Chap. 4 focuses on the systematic research of fullparameter computational design of nanostructures for surface plasmon polaritons.
Chapter 4 is organized as follows. The topology optimization-based computational design methodology is stated in Sect. 4.1. By specifying the desired functionality and performance of the nanostructures for surface plasmon polaritons, several results are obtained in Sect. $4.2$ for localized surface plasmon resonances, extraordinary optical transmission and surface plasmonic cloaking, respectively. And this chapter is concluded in Sect. $4.3$.

数学代写|拓扑学代写Topology代考|Self-consistency of Adjoint Analysis for Topology

拓扑学代考

数学代写|拓扑学代写Topology代考|Optimization in Frequency Domain

在频域中,光波的场变量是复杂的,包括场的幅度和相位。拓扑优化问题的成本函数通常包括场变量的共轭,例如能量泛函,它是场变量及其共轭的乘积。能量泛函被广泛使用,因为它们的最小二乘形式具有适定性。对于拓扑优化问题的敏感性分析,通常使用伴随方法,其中对相应的增广拉格朗日[10]实施一阶变分。

在伴随分析过程中,需要对场变量进行共轭运算的变分。在数学上,共轭算子是场变量的 Gâteaux 微分而不是 Fréchet 微分 [130]。这在之前的一些研究中被忽略了,例如参考文献中的文献。[77,87]. 共轭算子的Gâteaux可微性会导致伴随敏感性的不完全性,即实值成本的伴随敏感性

对于设计变量,函数是复数而不是实数,这是在计算域上定义的实值分布。如果直接使用这种不完全敏感性,则在迭代过程中设计变量将演变为复杂的,其中设计变量的初始设置为实数。从保持设计变量的实值属性的角度来看,这种伴随敏感性是自不一致的。因此,使用实部提取算子来提取导出的伴随灵敏度的实部,人为地增强伴随灵敏度的自洽性。

这种强制自不一致的结果是导出的结构拓扑依赖于入射波的相位。这种相位依赖性是不合理的,因为仅通过改变它们的相位不能固有地区分入射波。为了解决伴随灵敏度的自不一致性问题,代价函数需要保证 Fréchet 可微性。成本函数中的共轭算子可以通过将复场变量拆分为分别定义在实函数空间而不是复函数空间上的相应实部和虚部来去除。然后避免了由共轭算子引起的 Gâteaux 可微性。复变量的分裂导致波动方程的分裂,是复杂的偏微分方程。将复杂的偏微分方程或变分问题分解为相应的场变量实部和虚部耦合系统的方法已在参考文献中进行了系统讨论。[3,65]

数学代写|拓扑学代写Topology代考|Dielectric Material Based Topology Optimization

已经提到,光波的控制是通过使用预选材料的具有复杂空间配置的结构来实现的,并且入射波可以具有复杂的偏振。除了少数涉及线偏振波的情况外,大多数情况都不能简化为二维。大多数关于光学拓扑优化的报告都集中在应用上,包括分束器[80,86], 光子晶体[37,89], 斗篷[7,8,38]、传感器和谐振器[104,105], 超材料[28,77,137],表面等离子体激发[10],以及电磁和光学天线[35,36,48,134],没有提出光波在三维空间中传播的系统拓扑优化方法。因此,有必要开发一种统一、系统的拓扑优化方法,充分考虑三维光学的物理复杂性。

在三维空间中开发基于有限元的光波拓扑优化方法并不简单,因为需要强制执行无散度条件。在二维横向电波或磁波情况下,麦克斯韦方程组的归约过程自动满足无散度条件,推导亥姆霍兹方程组,自然可以使用基于节点元的伽辽金有限元法直接求解离散亥姆霍兹方程[53,69]. 与二维情况不同,基于节点元的伽辽金有限元法求解三维光波不能自动满足无散度条件,从而产生伪解。

对于这个问题,已经开发了两种主要的方法来强制执行无散度条件并消除虚假解决方案。第一种方法是在波动方程的弱形式上加入一个无散条件的最小二乘形式的惩罚项,然后用节点元素对弱形式进行离散化。但是,惩罚项的使用并不能完全消除解的发散,影响了解的准确性。因此,惩罚方法[53]不能准确地满足无散条件。第二种方法是使用边缘元素,将自由度分配给边缘而不是元素的节点,其中使用固有满足无散条件的向量基来实现插值[53,69,72,116]. 边缘单元还解决了在材料界面处施加边界条件的不便,以及由于场奇异性而难以处理导电和介电边角的问题。[53,69]. 因此,基于边缘元的有限元法是三维波动方程离散化和三维光波拓扑优化方法的较合理选择。

数学代写|拓扑学代写Topology代考|Metal Material-Based Topology Optimization

介电常数为负实部的金属表面可以捕获光波并实现表面等离子体激元[31,81]. 用于表面等离子体激元的金属通常是贵金属,例如银(Ag)、金(Au)和铝(Al)。在光学频率下,金属的自由电子可以在某些条件下维持具有不同共振频率的振荡[66,127,140]. 表面等离子体的存在是金属和光之间相互作用的特征,其中 Kretschmann-Raether 和 Otto 配置通常用于等离子体激发。

在过去几年中,已经开发了许多表面等离子体激元的创新概念,例如,局部表面等离子体共振 [100]、非凡的光学传输 [30] 和转换等离子体光学[52,60]. 相应地,表面等离子体激元也被提出了许多相关应用,例如,生物分子操纵和标记[23]、表面增强拉曼光谱[70]、化学和生物传感器[33]、光伏[22]、近场光刻和成像 [84],光捕获 [67, 76],纳米光学电路 [1],光电器件,波长可调滤波器,光调制器 [16,34,41,42], 等离子体 Luneburg 透镜和表面等离子体隐形[52,60].

所有这些应用都通过合理设计纳米级结构来利用表面等离子体激元。对于表面等离子体装置的设计,已经开发了转换等离子体光学来确定折射率分布并引导表面等离子体激元的传播[52,60]. 然而,导出的分布具有极端的电磁特性,即介电常数和磁导率,这是基于超材料的使用而实现的。这种超材料的定制微结构必须比波长小得多,这使得实现光学频率所需的磁性非常具有挑战性。为了克服体光波中的类似问题,利用自然界中容易获得的常规简单各向同性电介质,采用基于拓扑优化的计算设计方法实现了体光波斗篷的逆设计。[9,12,38]. 在表面等离激元上,大多数相关研究都集中在通过调整少数结构参数来设计纳米结构[73,78,114],尽管拓扑优化方法已应用于光栅耦合器以提高表面等离子体的激发效率[11]。因此,章。4专注于表面等离子体激元纳米结构全参数计算设计的系统研究。
第 4 章组织如下。基于拓扑优化的计算设计方法在第 3 节中说明。4.1。通过指定表面等离子激元纳米结构所需的功能和性能,在 Sect 中获得了几个结果。4.2分别用于局部表面等离子体共振、非凡的光学传输和表面等离子体隐形。本章结束于Sect。4.3.

数学代写|拓扑学代写Topology代考 请认准statistics-lab™

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金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题,以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法,其中不假设数据来自于由少数参数决定的规定模型;这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型(GLM)归属统计学领域,是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。

有限元方法代写

有限元方法(FEM)是一种流行的方法,用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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随机分析代写


随机微积分是数学的一个分支,对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。

回归分析代写

多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

R语言代写问卷设计与分析代写
PYTHON代写回归分析与线性模型代写
MATLAB代写方差分析与试验设计代写
STATA代写机器学习/统计学习代写
SPSS代写计量经济学代写
EVIEWS代写时间序列分析代写
EXCEL代写深度学习代写
SQL代写各种数据建模与可视化代写

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