电磁场计算中的径向基函数无网格法研究
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摘要
在工程电磁场的数值计算中,基于区域剖分和局部近似技术的网格化方法常面临剖分困难以及精度与计算量的矛盾。径向基函数(Radial Basis Function,RBF)方法已在数据插值和微分方程求解中得到广泛应用,并逐步形成一类配点型无网格方法。论文将RBF方法系统地引入到电磁场数值计算中,研究RBF求解典型电磁问题的原理、方法、具体实施方案和离散模型,以期利用RBF方法无需网格剖分和近似精度较高的优点部分解决网格法应用中的局限。
论文首先给出了RBF配点插值原理和基于配点近似的微分方程求解方法。通过与传统插值方法的比较,反映出RBF插值具有精度高、稳定性好和实施简便的优势;而在微分方程求解中,RBF方法又能在较少节点下能获得连续、光滑和高精度的近似解,相对于传统数值方法体现了明显的性能优势。
接着,论文将工程电磁问题系统地划分为静态场模型、时变场模型和本征模型三大类,分别研究了每类问题的RBF求解方法,主要内容有:
①构建求解静态电磁场边值问题的RBF方法,具体包括:1)在建立一般的静态场位函数边值模型基础上,给出基于RBF配点插值的边值问题求解的基本理论,并成功应用于静电场、恒定电场和恒定磁场问题的求解;2)针对RBF插值在边界处精度相对较低现象,论文对边界条件这一关键问题开展专题研究,提出一系列边界条件处理方法,即:a.对存在突变的第一类边界条件,通过引入具有紧支撑性的拟Shannon小波函数,形成RBF与小波函数的耦合方法,实现了RBF全域逼近和小波函数局部近似的结合,有效解决了对突变边界的逼近;b.对存在导数的第二类边界条件,论文提出了虚点法、Hermite配点法和规则网格法等多种改进方案;c.对多媒质问题中的衔接边界条件,提出了分区域逼近方案,仿真表明:相对于全区域逼近,其精度更高,同时也更好体现了媒质边界的影响。
②形成求解时变电磁场扩散方程的RBF方法,论文将时变电磁场从整体上分为时谐稳态场和瞬态场,指出时谐场计算的实质是求解相量形式边值模型,进而集中研究瞬态场中时间变量处理方法,具体包括:1)应用RBF逼近理论解的空间函数,用Crank-Nilcoson差分近似时间函数,借助时间离散和空间配点建立了瞬态场时域RBF求解方法并成功应用于瞬态涡流问题的分析中;2)应用傅立叶变换实现时域和频域的转换,在时谐场RBF求解基础上建立了瞬态场频域RBF求解方法。
③形成求解电磁场本征问题的RBF方法,给出RBF方法求解本征方程的基本理论(包括RBF和多项式耦合的方法),提供了电磁参数(截止频率、截止波数和电磁场量函数)的计算方法。在规则波导和谐振腔分析中,通过与有限元方法的对比研究,反映出RBF方法计算精度高且无“伪解”现象,尤其在高阶模下仍保持较高的精度,通常达到数倍有限元节点下的计算性能。
最后,论文将RBF方法应用于微带线、电机气隙磁场、导体瞬态磁场和不规则波导等工程实例的分析中,通过与Maxwell 2D、Matlab PDE Toolbox等有限元软件包计算结果的对比,反映了RBF法求解工程电磁问题的有效性、适用性和性能特点。
论文对RBF法应用于电磁场计算的系统研究表明:RBF方法作为一类配点型无网格法,摆脱了求解中对网格剖分的依赖,解决了剖分面临的困境;同时由于全域RBF的连续性、光滑性和无限可微优点,使其在逼近中能获得较高的精度;因RBF定义为距离函数,对于维数不敏感并易于拓展到高维问题分析中;另外,RBF方法实施时基函数中心点和配点设置灵活、程序编制简便。考虑RBF方法在计算中体现的性能优势,将会成为电磁场数值计算的重要方法。
The main methods for numerical computation engineering electromagnetic field are mesh methods which are based on mesh generation and local approximation. Therefore they meet two big challenges, one is the subdivision for complex structure, the other is the contradiction between precision and computation cost. Radial basis function (RBF), widely used in data interpolation and differential equation solving, is presently applied to form a collocation type meshless method for mathematical physical equations. So this thesis introduces the RBF to solve the above problems in the numerical computation of electromagnetic field by taking advantage of its high approximation precision and non-mesh generation. Then the principles, implementation schemes and discrete model of the RBF in solving typical engineering electromagnetic problems are investigated in a systematical manner.
The RBF method’s principles for interpolation and differential equation solution are presented firstly. Comparison study between RBF interpolation and traditional methods reveals that the RBF is characterized by precision, stability and implementation simplicity. Simultaneously, RBF can obtain continuous, smooth and precise solution only under a few nodes. On the other hand, the RBF shows obvious performance superiority in solving differential equation.
In order to form a systematic RBF method for calculating electromagnetic field, the engineering projects are divided into static field model, time varying field model and eigen model according to their characteristics. Then each type of model is separately studied and following achievement is obtained.
In chapter 3, the thesis constructed the RBF method for solving boundary value problem in static electromagnetic field. The research included: 1) On the basis of potential function boundary value model in static field, the RBF principles, implementation scheme and discrete model are presented, and then successful instances on electrostatic field, constant electric field and constant magnetic field are performed. 2) considering the RBF interpolation’s approximate precision is relatively low near the boundary, the thesis carried out special research on boundary condition in boundary value model and then proposed such boundary treatment methods as follows: i. Owing to the existence of sudden change boundary in boundary I value model, the study introduced the compact support quasi Shannon wavelet function, and formed a coupling method which consists of RBF and the wavelet basis function. This method could combine the RBF’s global approximation ability with the wavelet function’s local approximation ability. ii. Owing to the existence of derivative boundary in the boundary II value model, the paper proposed a series of improvement technology as the floating point method, the Hermite collocation and the regular grid method which reflected the performance superiority of improved method. iii. Owing to the engagement boundary in multi-medium problems, the thesis proposed the subzone approximation scheme which obtained higher precision solution than global approximation and better manifested the influence of medium boundary.
In chapter 4, the thesis formed the RBF methods for parabolic equation. Firslty, the time-varying field is classified into harmonic steady field and transient field, and then the phase form boundary model is the essence of harmonic steady field. Finally, the research focuses on the treatment of time variables in the transient fields. The studies included the following two methods --- the time-domain RBF method and the frequency domain RBF method. The former adopted the RBF approximating space function and the Crank-Nilcoson difference for time function. The latter is implemented through three stages. First, the time function is decomposed into complex exponential function by Fourier transform, then the solution in frequency domain is obtained through superposition of single frequency point time-harmonic field, and finally the solution in time domain is acquired by inverse Fourier transform. The result proved the RBF method can be successfully applied in transient eddy analysis.
In chapter 5, the thesis put forword the RBF coupled method for electromagnetic eigen model analysis which is used to calculate the electromagnetic parameters, such as cut-off frequency, cut-off wave number and eigen functions. Compared with FEM method, the RBF gained higher accuracy and non-pseudo-solution phenomenon in the waveguide and resonator analysis. Furthermore, the RBF maintained high accuracy in high order model.
In chapter 6, the RBF method was applied to the microstrip, the electrical alveolar , the transient field and irregular waveguide applications. A series of comparison between RBF and FEM were performed which were implemented with Maxwell 3D, Matlab PDE Toolbox. The results showed the RBF methods’validity in solving the electromagnetic instances.
The systematic study on RBF showed that the RBF method is independent of mesh in solution procedure, so it avoids the trouble of subdivision in complex structure. Due to the good continuity, smoothness and boundless derivative advantages of global RBF, the RBF gains higher precision than local approximation methods. In addition, RBF is defined as the distance function, therefore it is insensitive to dimension and can be easily extended to high-dimensional analysis. Finally, since RBF method is flexible in setting the centers and collocation points, it is easy in implementation and programming. Considering its feasibility, effectiveness and outstanding performance advantages in the calculation of the electromagnetic, RBF will become an important numerical method in electromagnetic calculation.
论文首先给出了RBF配点插值原理和基于配点近似的微分方程求解方法。通过与传统插值方法的比较,反映出RBF插值具有精度高、稳定性好和实施简便的优势;而在微分方程求解中,RBF方法又能在较少节点下能获得连续、光滑和高精度的近似解,相对于传统数值方法体现了明显的性能优势。
接着,论文将工程电磁问题系统地划分为静态场模型、时变场模型和本征模型三大类,分别研究了每类问题的RBF求解方法,主要内容有:
①构建求解静态电磁场边值问题的RBF方法,具体包括:1)在建立一般的静态场位函数边值模型基础上,给出基于RBF配点插值的边值问题求解的基本理论,并成功应用于静电场、恒定电场和恒定磁场问题的求解;2)针对RBF插值在边界处精度相对较低现象,论文对边界条件这一关键问题开展专题研究,提出一系列边界条件处理方法,即:a.对存在突变的第一类边界条件,通过引入具有紧支撑性的拟Shannon小波函数,形成RBF与小波函数的耦合方法,实现了RBF全域逼近和小波函数局部近似的结合,有效解决了对突变边界的逼近;b.对存在导数的第二类边界条件,论文提出了虚点法、Hermite配点法和规则网格法等多种改进方案;c.对多媒质问题中的衔接边界条件,提出了分区域逼近方案,仿真表明:相对于全区域逼近,其精度更高,同时也更好体现了媒质边界的影响。
②形成求解时变电磁场扩散方程的RBF方法,论文将时变电磁场从整体上分为时谐稳态场和瞬态场,指出时谐场计算的实质是求解相量形式边值模型,进而集中研究瞬态场中时间变量处理方法,具体包括:1)应用RBF逼近理论解的空间函数,用Crank-Nilcoson差分近似时间函数,借助时间离散和空间配点建立了瞬态场时域RBF求解方法并成功应用于瞬态涡流问题的分析中;2)应用傅立叶变换实现时域和频域的转换,在时谐场RBF求解基础上建立了瞬态场频域RBF求解方法。
③形成求解电磁场本征问题的RBF方法,给出RBF方法求解本征方程的基本理论(包括RBF和多项式耦合的方法),提供了电磁参数(截止频率、截止波数和电磁场量函数)的计算方法。在规则波导和谐振腔分析中,通过与有限元方法的对比研究,反映出RBF方法计算精度高且无“伪解”现象,尤其在高阶模下仍保持较高的精度,通常达到数倍有限元节点下的计算性能。
最后,论文将RBF方法应用于微带线、电机气隙磁场、导体瞬态磁场和不规则波导等工程实例的分析中,通过与Maxwell 2D、Matlab PDE Toolbox等有限元软件包计算结果的对比,反映了RBF法求解工程电磁问题的有效性、适用性和性能特点。
论文对RBF法应用于电磁场计算的系统研究表明:RBF方法作为一类配点型无网格法,摆脱了求解中对网格剖分的依赖,解决了剖分面临的困境;同时由于全域RBF的连续性、光滑性和无限可微优点,使其在逼近中能获得较高的精度;因RBF定义为距离函数,对于维数不敏感并易于拓展到高维问题分析中;另外,RBF方法实施时基函数中心点和配点设置灵活、程序编制简便。考虑RBF方法在计算中体现的性能优势,将会成为电磁场数值计算的重要方法。
The main methods for numerical computation engineering electromagnetic field are mesh methods which are based on mesh generation and local approximation. Therefore they meet two big challenges, one is the subdivision for complex structure, the other is the contradiction between precision and computation cost. Radial basis function (RBF), widely used in data interpolation and differential equation solving, is presently applied to form a collocation type meshless method for mathematical physical equations. So this thesis introduces the RBF to solve the above problems in the numerical computation of electromagnetic field by taking advantage of its high approximation precision and non-mesh generation. Then the principles, implementation schemes and discrete model of the RBF in solving typical engineering electromagnetic problems are investigated in a systematical manner.
The RBF method’s principles for interpolation and differential equation solution are presented firstly. Comparison study between RBF interpolation and traditional methods reveals that the RBF is characterized by precision, stability and implementation simplicity. Simultaneously, RBF can obtain continuous, smooth and precise solution only under a few nodes. On the other hand, the RBF shows obvious performance superiority in solving differential equation.
In order to form a systematic RBF method for calculating electromagnetic field, the engineering projects are divided into static field model, time varying field model and eigen model according to their characteristics. Then each type of model is separately studied and following achievement is obtained.
In chapter 3, the thesis constructed the RBF method for solving boundary value problem in static electromagnetic field. The research included: 1) On the basis of potential function boundary value model in static field, the RBF principles, implementation scheme and discrete model are presented, and then successful instances on electrostatic field, constant electric field and constant magnetic field are performed. 2) considering the RBF interpolation’s approximate precision is relatively low near the boundary, the thesis carried out special research on boundary condition in boundary value model and then proposed such boundary treatment methods as follows: i. Owing to the existence of sudden change boundary in boundary I value model, the study introduced the compact support quasi Shannon wavelet function, and formed a coupling method which consists of RBF and the wavelet basis function. This method could combine the RBF’s global approximation ability with the wavelet function’s local approximation ability. ii. Owing to the existence of derivative boundary in the boundary II value model, the paper proposed a series of improvement technology as the floating point method, the Hermite collocation and the regular grid method which reflected the performance superiority of improved method. iii. Owing to the engagement boundary in multi-medium problems, the thesis proposed the subzone approximation scheme which obtained higher precision solution than global approximation and better manifested the influence of medium boundary.
In chapter 4, the thesis formed the RBF methods for parabolic equation. Firslty, the time-varying field is classified into harmonic steady field and transient field, and then the phase form boundary model is the essence of harmonic steady field. Finally, the research focuses on the treatment of time variables in the transient fields. The studies included the following two methods --- the time-domain RBF method and the frequency domain RBF method. The former adopted the RBF approximating space function and the Crank-Nilcoson difference for time function. The latter is implemented through three stages. First, the time function is decomposed into complex exponential function by Fourier transform, then the solution in frequency domain is obtained through superposition of single frequency point time-harmonic field, and finally the solution in time domain is acquired by inverse Fourier transform. The result proved the RBF method can be successfully applied in transient eddy analysis.
In chapter 5, the thesis put forword the RBF coupled method for electromagnetic eigen model analysis which is used to calculate the electromagnetic parameters, such as cut-off frequency, cut-off wave number and eigen functions. Compared with FEM method, the RBF gained higher accuracy and non-pseudo-solution phenomenon in the waveguide and resonator analysis. Furthermore, the RBF maintained high accuracy in high order model.
In chapter 6, the RBF method was applied to the microstrip, the electrical alveolar , the transient field and irregular waveguide applications. A series of comparison between RBF and FEM were performed which were implemented with Maxwell 3D, Matlab PDE Toolbox. The results showed the RBF methods’validity in solving the electromagnetic instances.
The systematic study on RBF showed that the RBF method is independent of mesh in solution procedure, so it avoids the trouble of subdivision in complex structure. Due to the good continuity, smoothness and boundless derivative advantages of global RBF, the RBF gains higher precision than local approximation methods. In addition, RBF is defined as the distance function, therefore it is insensitive to dimension and can be easily extended to high-dimensional analysis. Finally, since RBF method is flexible in setting the centers and collocation points, it is easy in implementation and programming. Considering its feasibility, effectiveness and outstanding performance advantages in the calculation of the electromagnetic, RBF will become an important numerical method in electromagnetic calculation.
引文
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