A nonlinear electromagnetic field analysis method with random effects

Abstract

The present invention belongs to the technical field of electromagnetic field calculation and analysis, and proposes a nonlinear electromagnetic field analysis method with random effects, and the steps are as follows: step S1: establish a nonlinear electromagnetic field model with random effects; step S2: split the nonlinear electromagnetic field model with random effects into two subsystems; step S3: mesh the spatial solution area and set the initial boundary values ​​corresponding to the numerical solution; step S4: generate an adaptive time step #imgabs0#; step S5: let #imgabs1#, calculate the electric field and magnetic field values ​​and record them; step S6: let #imgabs2#, judge whether #imgabs3# is less than the terminal moment #imgabs4#, if so, return to step S4; if not, obtain the electric field and magnetic field data at all times. The present invention takes into account both high computational efficiency and convergence, and only needs to solve a number of local one-dimensional subsystems, which reduces the demand for storage space and improves the calculation speed.

Description

Technical Field

[0001] The invention belongs to the technical field of electromagnetic field calculation and analysis, and in particular relates to a nonlinear electromagnetic field analysis method with random effects. Background Art

[0002] Maxwell's equations describe the propagation behavior of electromagnetic waves in a medium and play an important role in magnetic detection, optics and other fields. The propagation of electromagnetic waves in the actual environment is very complex. On the one hand, the constitutive relationship of the medium in many problems in physics and engineering is highly nonlinear. On the other hand, due to the unpredictability of the environment, the scattering source, propagation medium and electromagnetic field may not be deterministic, but random fields containing uncertainty. Therefore, it is of great significance to study the time-varying laws and statistical characteristics of nonlinear electromagnetic wave propagation with random effects.

[0003] The stochastic Maxwell equations introduce uncertainty into the problem model, giving it the essential characteristics of randomness. It is a basic model for describing electromagnetic wave propagation problems in complex random environments. However, in general, the solution to random electromagnetic problems cannot be achieved directly through the analytical form of the stochastic Maxwell equations, but relies on numerical analysis methods for approximate solutions. Therefore, the design of numerical analysis methods plays a key role in describing the time-varying laws and characteristics of random electromagnetic waves and solving random electromagnetic problems. The stochastic Maxwell equations are not only vector equations with six unknown quantities, but each component has a time dimension and multiple space dimensions. They also have random characteristics that are different from deterministic equations. In specific numerical analysis, random numerical simulations of orbits of very large magnitude are required. These characteristics greatly increase the requirements for the computational efficiency of numerical simulations of the stochastic Maxwell equations, so how to construct efficient numerical analysis methods has become the key.

[0004] There are some numerical analysis methods in the prior art when the coefficients of the random Maxwell equations have global Lipschitz continuity. For example, the time direction is based on the exponential integrator method, Euler method, Runge-Kutta method, etc.; the space direction is based on the finite difference method, discontinuous Galerkin method, wavelet method, etc. Although the above-mentioned methods have made great progress under the global Lipschitz condition, there are still great difficulties for the random Maxwell equations whose coefficients do not satisfy the Lipschitz continuity. Summary of the invention

[0005] In view of the deficiencies in the prior art, the present invention proposes a nonlinear electromagnetic field analysis method with random effects, which is suitable for solving the three-dimensional random Maxwell equations with non-global Lipschitz nonlinear terms. The adaptive explicit numerical algorithm can be used for nonlinear electromagnetic field analysis with random effects. The nonlinear electromagnetic field numerical analysis method proposed by the present invention takes into account both high computational efficiency and convergence, and only requires solving several local one-dimensional subsystems, which reduces the demand for storage space and improves the calculation speed.

[0006] The specific technical solutions of the present invention are as follows:

[0007] A nonlinear electromagnetic field analysis method with random effects comprises the following steps:

[0008] Step S1: Establish a nonlinear electromagnetic field model with random effects as follows:

[0009] (1)

[0010] Satisfy the initial value conditions and ideal conductor boundary conditions ;

[0011] in, is the unit normal vector, is the divergence operator, Time variables, is a three-dimensional space variable, is the three-dimensional space coordinate, and are the electric field intensity and magnetic field intensity respectively, and the time region is , the spatial solution area is a three-dimensional cube , They are The left and right endpoints of the interval on the axis, They are The left and right endpoints of the interval on the axis, They are The left and right endpoints of the interval on the axis, and ) are the dielectric constant and magnetic permeability, respectively. is the resistivity of the medium, express The Euclidean norm of is the random source term, represents the initial value of the electric field strength, Indicates the initial value of magnetic field strength;

[0012] Step S2: splitting the nonlinear electromagnetic field model with random effects into two subsystems;

[0013] Step S3: Meshing the spatial solution area and setting the initial boundary values ​​corresponding to the numerical solution;

[0014] Step S4: Generate adaptive time steps ;

[0015] Step S5: Set , For the Time nodes, For the At the time corresponding to each time node, the electric field data and magnetic field data are calculated and recorded;

[0016] Step S6: ,judge Is it less than the terminal time? If so, return to step S4; if not, obtain the electric field data and magnetic field data at all times.

[0017] Preferably, the two subsystems in step S2 are as follows:

[0018] (2)

[0019] (3).

[0020] Preferably, the step S3 specifically includes:

[0021] Step S3-1: Divide the spatial solution area:

[0022] ; ;

[0023] ; ;

[0024] ; ;

[0025] in, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The nodes in the axis direction, , , , They are , , The number of divisions in the axial direction, , , They are , , The step length in the axis direction, ;

[0026] Step S3-2: Let , set the initial value corresponding to the numerical solution to:

[0027] ;

[0028] ;

[0029] ;

[0030] ;

[0031] ;

[0032] ;

[0033] in, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at , , for The weight, , , for The weight, , , represents the component of the electric field strength, , , represents the component of magnetic field strength;

[0034] Step S3-3: Set the boundary value corresponding to the numerical solution to:

[0035] ;

[0036] ;

[0037] ;

[0038] in, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at .

[0039] Preferably, the step S4 specifically includes:

[0040] judge Is it less than the terminal time? If so, take the adaptive time step ; If not, take the adaptive time step ,in , intermediate variable , for The Euclidean norm of express exist The value at express exist The value at express exist The value at .

[0041] Preferably, the step S5 specifically includes: recording The electric field data at this moment is , , , the magnetic field data is , , , express exist The value at express exist The value at express exist The value at ;

[0042] Step S5-1: Calculate formula (2) The numerical solution at time is as follows:

[0043]

[0044] in, , spatial point , , and is an auxiliary variable; is the random source term The weight, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at

[0045] Step S5-2: Write formula (3) in The electric field data at this moment is , the magnetic field data is , , , the local one-dimensional method is used to calculate formula (3) in The electric field data and magnetic field data at the moment are , , and , express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at .

[0046] Compared with the prior art, the present invention has the following beneficial effects:

[0047] The present invention proposes a nonlinear electromagnetic field analysis method with random effects, which realizes efficient numerical analysis of three-dimensional random Maxwell equations with non-global Lipschitz nonlinear terms. The splitting technique is used to transform the three-dimensional spatial problem into a local one-dimensional sub-problem, so that the three-dimensional problem can be solved efficiently; for the difficulties caused by the non-global Lipschitz nonlinear term, the adaptive time step is introduced to automatically adjust the time step of each step according to the properties of the numerical solution, so that the explicit numerical algorithm has convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. The features and advantages of the present invention can be more clearly understood by referring to the drawings. The drawings are schematic and should not be understood as limiting the present invention in any way. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0049] Figure 1 The figure is a flow chart of the nonlinear electromagnetic field analysis method with random effects of the present invention.

[0050] Figure 2It is a time direction convergence order diagram of the nonlinear electromagnetic field analysis method with random effects of the present invention.

[0051] Figure 3 This is a comparison chart of the computational efficiency of the nonlinear electromagnetic field analysis method with random effects of the present invention and other numerical analysis methods. DETAILED DESCRIPTION

[0052] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

[0053] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0054] For random Maxwell equations, existing numerical analysis methods are designed for situations where the coefficients satisfy the global Lipschitz condition, and cannot handle equations where the coefficients do not satisfy the global Lipschitz condition. For random Maxwell equations with non-global Lipschitz nonlinear terms, the numerical analysis method obtained by discretizing in the time direction using an explicit method does not converge under a uniform step size, and is therefore not suitable for solving highly nonlinear random problems. For random Maxwell equations with non-global Lipschitz nonlinear terms, the numerical analysis method obtained by discretizing in the time direction using an implicit method has convergence, but its numerical simulation requires solving a high-dimensional algebraic equation at each time step, which increases the amount of calculation and reduces the computational efficiency. For three-dimensional random Maxwell equations, the electric field and magnetic field have three components respectively, and each component also has a time variable, three space variables, and a sample variable in the probability space. These characteristics require a huge storage space for numerical calculations, which greatly reduces the computational efficiency.

[0055] In view of the problems existing in the above-mentioned prior art, this embodiment proposes a nonlinear electromagnetic field analysis method with random effects. The specific process is as follows: Figure 1 As shown, the following steps are included:

[0056] Step S1: Establish a nonlinear electromagnetic field model with random effects

[0057] (1)

[0058] The equation satisfies the initial value condition , and perfect conductor (PEC) boundary conditions , ,Pick , , , . Random source term The amount Take , is the standard Brownian motion, is a natural number.

[0059] Step S2: Split the nonlinear electromagnetic field model with random effects into two subsystems, as follows:

[0060] (2)

[0061] (3)

[0062] Step S3: Mesh the spatial solution area and set the initial boundary values ​​corresponding to the numerical solution.

[0063] Step S3-1: Divide the spatial solution area:

[0064] ; ;

[0065] ; ;

[0066] ; ;

[0067] in, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The nodes in the axis direction, , , , They are , , The number of divisions in the axial direction. They are The step size in the axis direction.

[0068] Step S3-2: Set the initial value corresponding to the numerical solution to:

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074] ;

[0075] in,

[0076] ;

[0077] ;

[0078] ;

[0079] ;

[0080] ;

[0081] ;

[0082] Step S3-3: Set the boundary value corresponding to the numerical solution to:

[0083] ;

[0084] ;

[0085] ;

[0086] Step S4: Generate adaptive time steps :

[0087] judge Is it less than the terminal time? If so, take the adaptive time step ; If not, take the adaptive time step ,in , intermediate variable , for The Euclidean norm of express exist The value at express exist The value at express exist The value at .

[0088] Step S5: Calculation The numerical solution at time is , calculate the electric field data and magnetic field data

[0089] , , and record it.

[0090] Step S5-1: Calculate formula (2) The numerical solution at time is as follows:

[0091]

[0092] in, and is an auxiliary variable; is the random source term The weight, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at

[0093] Step S5-2: Write formula (3) in The electric field data at this moment is , the magnetic field data is , , , the local one-dimensional method is used to calculate formula (3) in The electric field data and magnetic field data at the moment are , , and , express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at .

[0094] Step S6: ,judge Is it less than the terminal time? If so, return to step S4; if not, obtain the electric field data and magnetic field data at all times.

[0095] Take separately The time direction convergence order of the nonlinear electromagnetic field analysis method with random effects in this embodiment is as follows: Figure 2 shown. Figure 2 This shows that the nonlinear electromagnetic field analysis method of this embodiment has convergence.

[0096] Pick , the electromagnetic field calculation efficiency is obtained by using the nonlinear electromagnetic field analysis method with random effects of this embodiment. Figure 3 shown. Figure 3 It shows that the nonlinear electromagnetic field analysis method of this embodiment can efficiently numerically analyze nonlinear electromagnetic fields with random effects.

[0097] Through the nonlinear electromagnetic field analysis method described in this embodiment, efficient numerical simulation of nonlinear electromagnetic fields with random effects is achieved, which minimizes the calculation scale and improves the calculation efficiency. It does not need to solve ultra-large-scale algebraic equations at each time step, and can ensure the convergence of the numerical analysis method.

[0098] In the present invention, unless otherwise clearly specified and limited, a first feature being "above" or "below" a second feature may include that the first and second features are in direct contact, or may include that the first and second features are not in direct contact but are in contact through another feature between them. Moreover, a first feature being "above", "above" and "above" a second feature includes that the first feature is directly above and obliquely above the second feature, or simply indicates that the first feature is higher in level than the second feature. A first feature being "below", "below" and "below" a second feature includes that the first feature is directly below and obliquely below the second feature, or simply indicates that the first feature is lower in level than the second feature.

[0099] In the present invention, the terms "first", "second", "third", "fourth" are used for descriptive purposes only and cannot be understood as indicating or implying relative importance. The term "plurality" refers to two or more, unless otherwise clearly defined.

[0100] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A nonlinear electromagnetic field analysis method with random effects, It is characterized in that The following steps are involved: Step S1: Establish a nonlinear electromagnetic field model with random effects as follows: (1) Satisfy the initial value conditions and ideal conductor boundary conditions ; in, is the unit normal vector, is the divergence operator, Time variables, is a three-dimensional space variable, is the three-dimensional space coordinate, and are the electric field intensity and magnetic field intensity respectively, and the time region is , the spatial solution area is a three-dimensional cube , They are The left and right endpoints of the interval on the axis, They are The left and right endpoints of the interval on the axis, They are The left and right endpoints of the interval on the axis, and ) are the dielectric constant and magnetic permeability, respectively. is the resistivity of the medium, express The Euclidean norm of is the random source term, represents the initial value of the electric field strength, Indicates the initial value of magnetic field strength; Step S2: splitting the nonlinear electromagnetic field model with random effects into two subsystems; Step S3: Meshing the spatial solution area and setting the initial boundary values ​​corresponding to the numerical solution; Step S4: Generate adaptive time steps ; Step S5: Set , For the Time nodes, For the At the time corresponding to each time node, the electric field data and magnetic field data are calculated and recorded; Step S6: ,judge Is it less than the terminal time? If yes, then return to step S4; if no, then obtain the electric field data and magnetic field data at all times; The step S4 specifically includes: judge Is it less than the terminal time? If so, take the adaptive time step ; If not, take the adaptive time step ,in , intermediate variable , for The Euclidean norm of express exist The value at express exist The value at express exist The value at .

2. The nonlinear electromagnetic field analysis method according to claim 1, It is characterized in that The two subsystems in step S2 are as follows: (2) (3)。 3. The nonlinear electromagnetic field analysis method according to claim 2, It is characterized in that The step S3 specifically includes: Step S3-1: Divide the spatial solution area: ； ； ； ； ； ； in, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The midpoint of the mesh node in the axial direction, for The nodes in the axis direction, for The nodes in the axis direction, , , , They are , , The number of divisions in the axial direction, , , They are , , The step length in the axis direction, ; Step S3-2: Let , set the initial value corresponding to the numerical solution to: ； ； ； ； ； ； in, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at , , for The weight, , , for The weight, , , represents the component of the electric field strength, , , represents the component of magnetic field strength; Step S3-3: Set the boundary value corresponding to the numerical solution to: ； ； ； in, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at .

4. The nonlinear electromagnetic field analysis method according to claim 3, It is characterized in that The step S5 specifically includes: The electric field data at this moment is , , , the magnetic field data is , , , express exist The value at express exist The value at express exist The value at ; Step S5-1: Calculate formula (2) The numerical solution at time is as follows: in, , spatial point , , and is an auxiliary variable; is the random source term The weight, express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at Step S5-2: Write formula (3) in The electric field data at this moment is , the magnetic field data is , , , the local one-dimensional method is used to calculate formula (3) in The electric field data and magnetic field data at the moment are , , and , express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at express exist The value at .

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