Hu, Liu

(School of Optoelectronic Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065)

This paper introduces the research progress and present situation of electromagnetic calculation methods, introduces several representative algorithms, and compares their advantages and disadvantages, including moment method, finite element method, finite difference time domain method and complex ray method.

Keywords: method of moments; Finite element method; Finite difference time domain method; Complex ray method

1 quotation

1864, Maxwell established a unified electromagnetic field theory on the basis of predecessors' theories (Gauss's law, Ampere's law, Faraday's law and no free magnetic pole) and experiments, and revealed the universal laws followed by all macroscopic electromagnetic phenomena in nature with mathematical models, which is the famous Maxwell equation. Solve Maxwell equations or its degenerate form in 1 1 separable variable coordinate system, and finally get the analytical solution. This method can get the exact solution of the problem with high efficiency, but its application scope is too narrow, and it can only solve simple problems with regular boundaries. For the boundary with irregular shape or arbitrary shape, high mathematical skills are needed, and even analytical solutions cannot be obtained. Since 1960s, with the development of computer technology, some numerical calculation methods of electromagnetic field have been developed and widely used. Compared with the classical electromagnetic theory, the numerical method is much less constrained by the boundary shape and can solve various complex problems. However, various numerical calculation methods have their own advantages and disadvantages. A single method is often difficult to solve a complex problem, and it is often necessary to combine various methods to learn from each other. Therefore, people pay more and more attention to the mixed method.

This paper summarizes the development of computational electromagnetics at home and abroad, and classifies the commonly used electromagnetic calculation methods.

2 Classification of numerical methods for electromagnetic fields

The numerical solutions of electromagnetic problems can be divided into two categories: time domain and frequency domain. Frequency domain technology mainly includes moment method, finite difference method and so on. Frequency domain technology developed earlier and was more mature. Time domain method is mainly time domain difference technology. The introduction of time domain method is based on the consideration of computational efficiency, and some problems need less computation when discussing in time domain. For example, when the frequency domain method is used to solve the early response of the target to the pulse, it must be sampled and calculated many times in a large bandwidth, and then the solution can be obtained by inverse Fourier transform. The calculation accuracy is affected by the number of sampling points. If the nonlinear part changes with time, the time domain method is more direct. There are also some high-frequency methods, such as GTD, UTD and ray theory.

From the form of solving equations, it can be divided into integral equation method (IE) and differential equation method (DE). Compared with DE, IE has the following characteristics: the dimension of solution region of IE method is one dimension less than that of DE method, and the error is limited to the boundary of solution region, so the accuracy is high; IE method is suitable for solving infinite domain problems, while DE method will encounter mesh truncation problem at this time. The matrix generated by IE method is full with small order, while the matrix generated by DE method is sparse with large order. IE method is difficult to deal with inhomogeneous, nonlinear and time-varying media problems, and DE method can be directly used for this kind of problems [1].

3 Introduction of several typical methods

The finite element method was put forward in 1940s, and was used in aircraft design in 1950s. Later, this method was developed and widely used in structural analysis. At present, the finite element method is very famous as a general method widely used in engineering and mathematical problems.

Finite element method is a numerical calculation method based on variational principle. The identified solution problems are:

By applying the variational principle, the boundary value problem of the required solution is transformed into the corresponding variational problem. By dividing and interpolating the region d, the variational problem is discretized into the extreme value problem of ordinary multivariate functions, and then a set of multivariate algebraic equations is obtained. By solving algebraic equations, the numerical solution of the required boundary value problem can be obtained. Generally speaking, it goes through the following steps:

① The functional corresponding to the boundary value problem of the required solution and its variational problem are given.

② Divide the D domain and select the corresponding interpolation function.

③ The variational problem is discretized into the extremum problem of multivariate function, and the following set of algebraic equations is obtained:

Where: Kij is the coefficient (stiffness) matrix; Xi is the interpolation of discrete points.

④ The numerical solution Xi (I = 1, 2, …, n) of the boundary value problem of the required solution can be obtained by choosing the appropriate algebraic solution (2).

(2) Moment method

The analysis of many electromagnetic problems comes down to an operator equation [2]:

L (f) = g (3) where: L is a linear operator, F is an unknown field or other response, and G is a known source or excitation.

Generally speaking, this equation is a vector equation (two-dimensional or three-dimensional). If f can be solved by an equation, it is an exact analytical solution. In most cases, we can't get the analytic form of f, so we can only use numerical method to predict it. Let f expand into a linear combination of some basic function system f 1, f2, f3, …, fn on the domain of L;

Where: an is the expansion coefficient and fn is the expansion function or basis function.

For the exact solution, the smoothness of Equation (2) is the sum of infinite terms and forms a complete set of basis functions. For the approximate solution, bring Equation (2) into Equation (1), and then apply the linearity of operator L to obtain:

m= 1，2，3，…

This set of equations can be written in matrix form F to solve F, and the method of moments is such a discrete method to transform the operator equation into a matrix equation.

In the problem of electromagnetic scattering, the ratio of characteristic scale to wavelength of scatterer is a very important parameter. He decided on the specific way to apply the method of moments. If the characteristic scale of the target can be compared with the wavelength, the general moment method can be used. If the target is large and the feature scale contains a large range, then it is necessary to choose the appropriate discrete method and discrete basis function. Affected by computer memory and computing speed, it is very difficult to solve some two-dimensional and three-dimensional problems by the method of moments, because the amount of computing memory is usually proportional to N2 or N3 (N is the number of discrete points), and the discretized ill-conditioned matrix is also difficult to solve. At this time, higher mathematical skills are needed, such as using wavelet expansion and choosing appropriate wavelet basis functions to reduce dimensions [3].

(3) Finite difference time domain method

Finite difference time domain (FDTD) method is a time domain calculation method of electromagnetic field. Traditionally, the calculation of electromagnetic field is mainly carried out in frequency domain. Over the years, time domain calculation methods have been paid more and more attention. He shows unique advantages in many aspects, especially in solving electromagnetic problems related to inhomogeneous media, scatterers with arbitrary shapes and complex structures and radiation systems. The FDTD method directly solves Maxwell's curl equation which depends on time variables, and uses the central difference approximation with second-order accuracy to directly convert the differential operator in the curl equation into a difference form, thus realizing the data sampling and compression of continuous electromagnetic fields in a certain volume and for a period of time. The electric field and magnetic field components are placed alternately in space, which ensures that the continuity condition of the tangential field component at the boundary of the medium is naturally satisfied. In Cartesian coordinate system, the position of electric field and magnetic field components in grid cells is that each magnetic field component is surrounded by four electric field components, and vice versa.

The spatial placement of this electromagnetic field conforms to the natural geometric structure of Faraday's law and Ampere's law. Therefore, FDTD algorithm is a digital simulation of continuous actual electromagnetic wave propagation process in data storage space. However, the new value of each field component at each grid point only depends on the value of that point at the same time step and the values of other fields around that point at the first half time step. This is the induction principle of electromagnetic field. These relations constitute the basic formula of FDTD method. By calculating the grid points in the simulation area step by step, the required results can be obtained after executing an appropriate number of time steps.

In the above algorithm, the time increment Δ t and the space increments Δ x, Δ y and Δ z are not independent of each other, and their values must satisfy a certain relationship to avoid numerical instability. This instability is manifested in the infinite increase of 67 with the continuation of time step when solving explicit difference equations. In order to ensure numerical stability, numerical stability conditions must be met:

Where: (for non-uniform areas, the maximum value of c should be selected) [4].

The numerical calculation of Maxwell's equation by difference method will also cause the dispersion of analog wave modes in the grid, that is, the propagation speed of digital wave modes in FDTD grid will change with wavelength, propagation direction and discretization in the grid. This dispersion will lead to pulse waveform distortion, artificial anisotropy and virtual diffraction caused by non-physical reasons, so numerical dispersion must be considered. If grids with different sizes or different dielectric regions are used in the simulation space, the ratio of grid size to wavelength will be a function of position, and non-physical diffraction and reflection will occur at the interface of different grids or media, which should also be quantitatively studied to ensure the accuracy of FDTD algorithm. In the open problem, the electromagnetic field will occupy an infinite space, and because the computer memory is always limited, it can only simulate a limited space, so the difference grid will be truncated somewhere, which requires that the wave will not have obvious reflection at the grid truncation, so the wave propagating outward will be like propagating in infinite space. This is to set the absorption boundary condition at the truncation, so that the wave propagating to the truncation is absorbed by the boundary without reflection. Of course, it is impossible to be completely reflection-free. At present, some absorption boundary conditions have been created to meet the requirements of accuracy, such as the absorption boundary conditions derived by Mur.

(4) complex ray method

Complex ray is a high-frequency approximation method to solve the wave field propagation and scattering problems. According to the analytical method and calculation formula of geometrical optics theory and geometrical diffraction theory, he solved the amplitude and phase of complex ray trajectory and field in the complex space of analytic continuation, thus directly obtaining the propagation and scattering law of local inhomogeneous wave (falling wave) [5]. Complex ray method is the general name of a series of processing methods such as complex ray tracing, complex ray paraxial approximation, complex ray expansion and complex ray diffraction. Its * * * features are: by extending the coordinates of ray reference points to complex space, a simple and unified ray beam analysis model in real space is established; Through Fermat's principle and its extension, an effective description method of saddle point field under the framework of ray optics is constructed by using the processing technology based on complex ray tracing or complex ray paraxial approximation. For example, the complex ray tracing method directly extends the ray tracing method and field strength calculation formula used in ray optics to complex space, and uses the extended complex Fermat principle to search for complex rays, so as to find complex ray trajectories and complex ray fields. The characteristic of this method is that it can effectively describe the beam propagation in space based on the ray optics method, and provides a simple method for analyzing the beam propagation. Its disadvantage is that it is necessary to search a two-dimensional or four-dimensional complex ray trajectory for each given observation point, which is a very time-consuming computer iterative process.

4 Comparison and progress of several methods

The transplantation of finite element method into the field of electromagnetic engineering was in the sixties and seventies of last century, which is relatively novel. The advantage of finite element method is that it is suitable for definite solution problems with complex boundary shape or boundary conditions and complex medium. Each link of this method can be standardized, and a general calculation program can be obtained with high calculation accuracy. However, the calculation program of this method is complicated and lengthy. Because it is a domain solution, there are many elements and nodes, which leads to the complexity of initial data, and the final equations have a large number of elements and long calculation time, which requires the storage of the computer itself. For many problems in electromagnetism, finite element method produces strips (if the node numbers are correct) and sparse matrices (many matrix elements are 0). But the finite element method itself can only solve the open domain problem. The first step of finite element numerical analysis is to discretize the target. People have been studying this problem for many years, trying to find an effective and convenient discretization method, but because of the particularity of electromagnetic field, this problem has not been solved well. The crux of the problem is that, on the one hand, the general subdivision method is difficult to apply to complex structures; On the other hand, because the density of subdivision is closely related to the storage capacity of the final coefficient matrix, people have adopted many methods to reduce the storage capacity, such as multi-grid method, but these methods are difficult to realize [6].

Mesh generation and encryption is one of the bottlenecks in the development of finite element method, and adaptive mesh generation and encryption technology can solve this problem well. Adaptive mesh generation adjusts the mesh density according to the result of solving the field distribution, adopts high-order interpolation function in the dense area of the mesh to further improve the accuracy, and encrypts many times in the area where the field distribution changes sharply.

In recent years, the development of finite element method is accelerating day by day, and new progress has been made in the combination with other theories, and considerable application results have been achieved, such as adaptive mesh generation, three-dimensional field modeling, coupling problem, open domain problem, dealing with high magnetic materials and media with hysteresis saturation nonlinear characteristics. And some work that is still in the exploratory stage, such as quasi-problem, application of artificial intelligence and expert system in optimal design of electromagnetic devices, edge-based finite element method, etc. All these make the finite element method.

The method of moments discretizes a continuous equation into an algebraic equation, which is suitable for solving both differential equations and integral equations. His solution process is simple, the solution steps are unified and the application is convenient. But he needs some mathematical skills, such as the degree of discretization, the choice of basis function and weight function, and the process of matrix solution. In addition, it must be pointed out that the method of moments can achieve the required accuracy, the analytical part is simple, and the calculation amount is very large, even if a high-speed and large-capacity computer is used, the calculation task is very heavy. The method of moments has been widely used in antenna analysis and electromagnetic scattering, and has been successfully applied in antenna and antenna array radiation, scattering, microstrip and lossy structure analysis, propagation on uneven ground and electromagnetic absorption in human body.

FDTD replaces the differential expression of Maxwell curl equation in time domain with the finite difference expression, and obtains the finite difference expression about the field component. It can be modeled in different coordinate systems for different research objects, so it has these advantages and is easy to model complex media. Through a time domain analysis and calculation, the frequency response in the whole same frequency band can be obtained by Fourier transform. It can be distributed in real time on site to accurately simulate the radiation characteristics and scattering characteristics of various radiation sources and scatterers; The calculation time is short. However, due to the limitation of computer storage capacity, the grid space of FDTD analysis method cannot be infinitely increased, which makes FDTD method not suitable for large-size and thin-structured media. Because the minimum size of this thin structure is much smaller than the FDTD grid size, if the thin structure is fitted with a grid, the grid size can only be reduced, which will inevitably lead to the increase of computer storage capacity. Therefore, it is necessary to combine FDTD with other technologies that are developing vigorously at present, such as time domain integral equation /FDTD method, FDTD/ moment method and so on. FDTD has a wide range of applications, such as mobile phone radiation, antennas, electromagnetic interference characteristics of different building structures, microstrip lines and so on.

Complex ray technology has the characteristics of simple physical model, convenient mathematical processing and high calculation efficiency, and has important research value in application fields such as scattering characteristics analysis of complex targets. The typical processing method is to disperse the incident plane wave into a set of complex source point fields with parallel beams. Through ray tracing, field strength calculation and the superposition of the contributions of each ray field in the case of a specific target, the high-frequency asymptotic solution of the scattering field at a specific observation position can be obtained. At present, the complex ray analysis method has been successfully used to analyze the scattering characteristics of typical targets such as aircraft antenna and radome (radar cabin), wing-body joint and inlet, coated metal plate, corner reflector and so on. Although the calculation error of complex ray technique can be controlled by parameter adjustment, it is a high-frequency approximate calculation method, which brings inevitable calculation error due to the dispersion of incident wave field and the introduction of saddle point contribution. Generally speaking, the complex ray method still has unique advantages in the field of electromagnetic scattering of targets, especially for complex

Handling of miscellaneous objects.

5 conclusion

The numerical calculation methods of electromagnetism are far more than those mentioned above, and there are boundary element method, Green's function method and so on. In specific problems, we should adopt different methods instead of sticking to these methods, and we can also use them comprehensively to achieve the best results.

The numerical calculation of electromagnetism is a computational art, which spans many disciplines and is an organic combination of mathematical theory, electromagnetic theory and computer. In principle, the broadband range from DC to light belongs to his research scope. In order to keep up with the development of science and technology in the world, we should vigorously study the parallel calculation method of electromagnetic field and constantly expand its application fields, such as bioelectronics, electromagnetic positive and negative problems in complex media, medical applications, microwave remote sensing applications, chaos and bifurcation in nonlinear electromagnetics, microelectronics and nanoelectronics.

refer to

[1] Govin i. Progress and prospect of computational electromagnetics [J]. Acta Electronica Sinica,1995,23 (10): 62-69.

Liu. Numerical method of electromagnetic field [M]. Wuhan: Huazhong University of Science and Technology Press, 199 1.

Zhang Cheng, Zheng Hongxing. Solving electromagnetic integral equation by wavelet moment method [J]. Journal of Ningxia University (Natural Science Edition), 2000,21(1): 76-79.

Wang Changqing. Finite difference time domain method [J].Acta microwave, 1989, (4):8- 18.

Ruan Qianying. Complex ray theory and its application [M]. Chengdu: Electronic Industry Press, 1995438+0.

Fang Jing, Wang Wenbing. Analysis of radiation characteristics of back-cavity antenna by combining finite element method and moment method [J].Acta Microwave, 2000,16 (2):139-143.

Yang, man. FDTD analysis method of electromagnetic field [J]. Modern Electronic Technology, 200 1, (1 1): 73-74.

Hong Wei. Research progress of computational electromagnetics [J]. Southeast University RB (Natural Science Edition), 2002,32 (3): 335-339.

Wang Changqing, Zhu Xili. Finite difference time domain method in electromagnetic field calculation [M]. Beijing: Peking University Publishing House, 1994.

Lou Renhai, Fu Guoxing, Yuan Jinghong. Electromagnetic theory [M]. Chengdu: University of Electronic Science and Technology of China Press, 1996.

Modern electronic technology