Electromagnetic sensitivity analysis method and device, medium and product

Abstract

The invention discloses an electromagnetic sensitivity analysis method and device, a medium and a product, and relates to the field of electromagnetic sensitivity analysis. The method comprises the steps of obtaining a frequency domain Maxwell equation set according to a Maxwell equation set and a constitutive relation of electromagnetic isotropy in combination with expressions of dielectric constants and magnetic conductance coefficients; according to the frequency domain Maxwell equation set, an original system equation of the target two-dimensional arc-shaped structure is constructed; when the design parameters of the target two-dimensional arc-shaped structure are disturbed, determining an adjoint system equation according to an original system equation of the target two-dimensional arc-shaped structure; applying an adjoint system equation to respectively carry out one-time forward simulation and one-time adjoint simulation to obtain a forward field solution and an adjoint field solution; and calculating the sensitivity of the target function of the target two-dimensional arc-shaped structure relative to the target parameter by applying the forward field solution and the adjoint field solution. According to the invention, the efficiency and accuracy of the optimization design based on the arc boundary electromagnetic structure can be improved.

Description

Technical Field

[0001] The present invention relates to the field of electromagnetic sensitivity analysis, and in particular to an electromagnetic sensitivity analysis method, device, medium and product. Background Art

[0002] In the field of electromagnetic simulation, sensitivity analysis is a key tool for designers to evaluate system performance to changes in input parameters. By identifying parameters that have a significant impact on performance, designers can make more targeted adjustments to optimize the design to meet performance specifications. However, the computational cost of traditional finite difference methods (FDM) rises rapidly when dealing with systems with a large number of parameters or requiring intensive simulation, limiting its practicality.

[0003] To address this problem, the adjoint variable method (AVM) was proposed as an efficient sensitivity analysis tool. AVM requires only one additional simulation to obtain comprehensive first-order sensitivity data, significantly reducing the computational effort compared to FDM. This method is particularly suitable for high-frequency and complex systems, and its application in sensitivity analysis of plasma structures has demonstrated its significant advantages in computational efficiency. Despite its wide application in electromagnetic optimization, AVM still has limitations in its implementation on different structures. Summary of the invention

[0004] The purpose of the present invention is to provide an electromagnetic sensitivity analysis method, device, medium and product, which can improve the efficiency and accuracy of the optimization design based on the arc-shaped boundary electromagnetic structure.

[0005] To achieve the above object, the present invention provides the following solutions:

[0006] An electromagnetic sensitivity analysis method, the method comprising:

[0007] According to Maxwell's equations and the constitutive relation of electromagnetic isotropy, combined with the expressions of dielectric constant and permeability, the Maxwell's equations in frequency domain are obtained;

[0008] According to the frequency domain Maxwell equations, constructing the original system equations of the target two-dimensional arc structure; the original system equations of the target two-dimensional arc structure include the definition of the arc boundary, the parameterized representation of the material properties, the design parameters and the definition of the objective function related to the design parameters; the objective function is used to evaluate the response of the original system equations of the target two-dimensional arc structure;

[0009] When the design parameters of the target two-dimensional arc structure are disturbed, determining an adjoint system equation according to the original system equation of the target two-dimensional arc structure;

[0010] Applying the adjoint system equation, performing a forward simulation and an adjoint simulation respectively, and obtaining a forward field solution and an adjoint field solution;

[0011] The forward field solution and the adjoint field solution are used to calculate the sensitivity of the target function of the target two-dimensional arc-shaped structure with respect to the target parameter.

[0012] Optionally, the Maxwell equations in the frequency domain are:

[0013]

[0014] Among them, E is the vector of electric field strength; H is the vector of magnetic field strength; μ is the magnetic permeability of the medium; σ m is the magnetic permeability; ε is the dielectric constant of the medium; σ e is the conductivity; ▽× represents the rotation.

[0015] Optionally, the original system equation of the target two-dimensional arc structure is:

[0016]

[0017] C=[E nT H nT ] T ;

[0018]

[0019] Among them, B m is a function of the conductivity in the domain, B c is a matrix of constant coefficients; T is the transpose; p is a vector of design parameters; ξ is the local response integral kernel; T m is the simulation duration; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

[0020] Optionally, the objective function includes an objective function based on fixed constitutive parameters, an objective function based on frequency dependence, and an objective function based on a transient pulse function.

[0021] Optionally, the objective function based on the fixed constitutive parameters is:

[0022]

[0023] C=[E nT H nT ] T ;

[0024] Where Q is the objective function based on fixed constitutive parameters; C is the response matrix; p is the vector of design parameters; ξ is the local response integral kernel; T mis the simulation duration; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

[0025] Optionally, the frequency-dependent objective function is:

[0026]

[0027] C=[E nT H nT ] T ;

[0028] in, is the frequency-dependent objective function; C is the response matrix; ξ is the local response integral kernel; T m is the simulation duration; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

[0029] Optionally, the objective function based on the transient pulse function is:

[0030]

[0031] Among them, E(d,t 0 ) is the objective function based on the transient pulse function; δ(tt 0 ) is the selection function; E(d,t) is the objective function under abstract time; d is the position vector; t 0 is a time point; t is an abstract time point; T m is the simulation duration.

[0032] A computer device comprises: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement any one of the electromagnetic sensitivity analysis methods described above.

[0033] A computer-readable storage medium stores a computer program, which, when executed by a processor, implements any of the electromagnetic sensitivity analysis methods described above.

[0034] A computer program product comprises a computer program, wherein when the computer program is executed by a processor, the electromagnetic sensitivity analysis method described in any one of the above items is implemented.

[0035] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0036] The present invention discloses an electromagnetic sensitivity analysis method, device, medium and product, which utilizes the Adjoint Variable Method (AVM) to achieve efficient estimation of 1st to 2nd order sensitivity under the condition of only two algorithm simulations. Compared with the traditional Finite-Difference Method (FDM), the present invention significantly improves the calculation efficiency while maintaining the reliability and stability of the results. The present invention first establishes a model of the electromagnetic structure, that is, the original system equation of the target two-dimensional arc structure, and defines the objective function and the variable parameters; uses AVM to perform a forward simulation and an adjoint simulation to obtain the forward field and the adjoint field; evaluates the sensitivity of the objective function to the variable parameters through the calculation of the forward field and the adjoint field; analyzes the obtained sensitivity information to provide guidance for the optimal design of the electromagnetic structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] 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. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0038] Figure 1 It is a model diagram of the exact adjoint problem of radius parameter of the present invention.

[0039] Figure 2 It is a model diagram of the approximate adjoint problem of the radius parameter of the present invention.

[0040] Figure 3 It is the classification of different nodes of the arc-shaped structure lossy medium discontinuity surface constructed by the present invention.

[0041] Figure 4 It is a model diagram of the plane wave incident arc structure in two-dimensional TMz of the present invention.

[0042] Figure 5 It is a comparison diagram of the sensitivity of the frequency-independent constitutive parameters to the dielectric constant when scanning the relative dielectric constant of the medium in the two-dimensional example TMz mode of the present invention.

[0043] Figure 6 It is a comparison diagram of the sensitivity of the frequency-independent constitutive parameters to the conductivity when scanning the relative dielectric constant of the medium in the two-dimensional example TMz mode of the present invention.

[0044] Figure 7 It is a comparison diagram of the sensitivity of the frequency-independent constitutive parameters to the radius when scanning the relative dielectric constant of the medium in the two-dimensional example TMz mode of the present invention.

[0045] Figure 8 It is a comparison diagram of the real part of the sensitivity of the frequency-related target function to the dielectric constant in the two-dimensional example TMz mode of the present invention.

[0046] Fig. 9 It is a comparison diagram of the real part of the sensitivity of the frequency-related target function to conductivity in the two-dimensional example TMz mode of the present invention.

[0047] Fig.10 It is a comparison diagram of the real part of the sensitivity of the frequency-related target function to the radius in the two-dimensional example TMz mode of the present invention.

[0048] Fig.11 It is a comparison diagram of the imaginary part of the sensitivity of the frequency-related target function to the dielectric constant in the two-dimensional example TMz mode of the present invention.

[0049] Fig.12 It is a comparison diagram of the imaginary part of the sensitivity of the frequency-related target function to conductivity in the two-dimensional example TMz mode of the present invention.

[0050] Fig.13 It is a comparison diagram of the imaginary part of the sensitivity of the frequency-related target function to the radius in the two-dimensional example TMz mode of the present invention.

[0051] Fig.14 It is a comparison diagram of the sensitivity of the complete transient function to the dielectric constant in the two-dimensional example TMz mode of the present invention.

[0052] Fig.15 It is a comparison diagram of the sensitivity of the complete transient function to conductivity in the two-dimensional example TMz mode of the present invention.

[0053] Fig.16 It is a comparison diagram of the sensitivity of the complete transient function to the radius in the two-dimensional example TMz mode of the present invention.

[0054] Fig.17 A schematic flow chart of the electromagnetic sensitivity analysis method provided in Example 1 of the present invention.

[0055] Fig.18 This is a diagram of the internal structure of a computer device. DETAILED DESCRIPTION

[0056] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] The purpose of the present invention is to provide an electromagnetic sensitivity analysis method, device, medium and product, aiming to improve the efficiency and accuracy of the optimization design based on arc-shaped boundary electromagnetic structure.

[0058] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0059] Example 1

[0060] like Fig.17 As shown, the electromagnetic sensitivity analysis method in this embodiment includes:

[0061] Step S1: Based on the Maxwell equations and the constitutive relation of electromagnetic isotropy, combined with the expressions of dielectric constant and permeability coefficient, the Maxwell equations in the frequency domain are obtained.

[0062] Specifically, the Maxwell equations in the frequency domain are:

[0063]

[0064] Among them, E is the vector of electric field strength; H is the vector of magnetic field strength; μ is the magnetic permeability of the medium; σ m is the magnetic permeability; ε is the dielectric constant of the medium; σ e is the conductivity; ▽× represents the rotation.

[0065] Step S2: constructing the original system equations of the target two-dimensional arc structure according to the Maxwell equations in the frequency domain; the original system equations of the target two-dimensional arc structure include the definition of the arc boundary, the parameterized representation of the material properties, the design parameters and the definition of the objective function related to the design parameters; the objective function is used to evaluate the response of the original system equations of the target two-dimensional arc structure.

[0066] Specifically, considering the TMz case in 2D, the Maxwell curl equation is constructed by matrix, A and B are the material-related coefficients, C is the response matrix, and P is the excitation matrix. The original system equation can be obtained, that is, the forward solution matrix equation:

[0067]

[0068] The definition of arc boundaries and the parametric representation of material properties. There are many situations in which the edge changes under the arc structure, and each change needs to be weighted and stored.

[0069] Step S3: when the design parameters of the target two-dimensional arc-shaped structure are disturbed, an adjoint system equation is determined according to the original system equation of the target two-dimensional arc-shaped structure.

[0070] Specifically, when the i-th parameter p i Subject to Δp i This will cause the system matrix to be disturbed by A+ΔA i and B+ΔB i , disturbance Δp i It also causes the state variables (electric and magnetic field components in the entire domain) to be perturbed by C + ΔC i By perturbing the system and introducing the adjoint variable γ, the adjoint system is derived by the transformation:

[0071]

[0072] Among them A T With B T is the transpose of the original response A and B, I(τ) is the accompanying simulated excitation, τ is the accompanying excitation time, and the derivation of the accompanying system matrix is ​​derived using the perturbation system and mathematical methods. This equation is the basis for solving the eigenvalue.

[0073] Step S4: Apply the adjoint system equation to perform a forward simulation and an adjoint simulation respectively to obtain a forward field solution and an adjoint field solution.

[0074] Specifically, the original FDTD simulation is performed, and the difference vectors of the field components related to the design parameters are stored; the adjoint system is constructed and solved based on the adjoint variable method to obtain the adjoint field solution.

[0075] Step S5: applying the forward field solution and the adjoint field solution to calculate the sensitivity of the target function of the target two-dimensional arc-shaped structure relative to the target parameter.

[0076] Specifically, determine the forward direction before simulation and backward The perturbation Δp in the i-th parameter i There are many cases of edge changes under the arc structure, and each change needs to be weighted and stored numerically. In the time step of the forward original system simulation, the difference vector of the material parameters is stored by multiplying each affected field component by the corresponding imprint. The difference vector ΔR of the shape parameter i / Δp i In the accompanying simulation, storage ΔR i / Δp i , substitute into the sensitivity calculation expression:

[0077]

[0078] Where Q is the objective function, ξ is the kernel function of the objective function, ΔR i represents the i-th parameter residual matrix.

[0079] After calculating the sensitivity values ​​for each parameter, the traditional finite difference method is used to verify the results. First, the same 2D geometric model is constructed, the appropriate grid size is set, and then the calculation formulas of the same medium parameters and electromagnetic field components are set. The finite difference method is used to calculate and analyze the results. After obtaining the results calculated based on the adjoint sensitivity method and the finite difference method, the norm error solution formula is used to verify and analyze the two results, and the results are explained to prove the correctness of the method proposed in the present invention.

[0080] The present invention solves the problems that the existing sensitivity calculation method cannot perform boundary sensitivity analysis on the arc structure and the traditional finite difference method has a long execution time. The objective function including the objective function based on fixed constitutive parameters, the objective function based on frequency dependence and the objective function based on transient pulse function are described in detail respectively.

[0081] 1. Adjoint variable method of frequency-independent constitutive parameters:

[0082] For most structural design optimizations, the general time domain objective function can be expressed as:

[0083]

[0084] Among them, vector C is the time field solution calculated for each discrete unit in the entire computational domain; vector p is the vector of design parameters, which contains all the shape and material parameters of the considered structure; the integral kernel ξ is called the local response, which implicitly depends on p to C. Using the chain rule, the sensitivity expression of the objective function is obtained. Equation (1) is i It can be written as:

[0085]

[0086] in, The term is the relationship between Q and parameter p i The explicit derivative of , which is typically zero for most practical cases.

[0087] In the 2D FDTD case, when the field is assumed to be constant in one direction (i.e., the z direction), Maxwell's equations decompose into two uncoupled sets of equations. The first one has the component E z , H x and H y , denoted as the TMz case. The second one has the component H z 、E x and E y, denoted as the TEz case. For the TMz case, the material constituting the computational domain is assumed to be non-dispersive and isotropic. The equations are valid for any point in the computational domain, and these time domain equations are arranged in matrix form to obtain:

[0088]

[0089] in, The matrix (3) can be abbreviated as:

[0090]

[0091] This equation (4) is called the original system equation, where C = [E nT H nT ] T and

[0092]

[0093] In formula (4), the matrix B is symmetric, consisting of two symmetric matrices B m and B c Composition. m is a function of the conductivity in the domain, and B c is a matrix of constant coefficients.

[0094] When the i-th parameter p i Subject to Δp i This will cause the system matrix to be disturbed by A+ΔA i and B+ΔB i . The matrix B c does not change because it is a constant coefficient matrix. The perturbation Δp i It also causes the state variables (electric and magnetic field components in the entire domain) to be perturbed by C + ΔC i When the excitation does not change with the i-th parameter, the difference system is:

[0095]

[0096] Among them, ΔR i is the time difference vector associated with the i-th parameter. When the disturbance Δp i Enough hours:

[0097]

[0098] Multiply both sides of equation (7) by the transpose of an undetermined adjoint vector γ and integrate by parts:

[0099]

[0100] Choose a value with zero terminal value γ(T m)=0 eliminates the first term in equation (8). Regardless of how the parameters change, it is assumed that the electric and magnetic fields have zero initial values ​​in the entire domain. When the left side of equation (8) is equal to the right side of equation (2), we have:

[0101]

[0102] This equation (9) is labeled as the adjoint system, and its excitation is given by the right-hand side vector, which is determined during the simulation of the original system (4). When solving equation (9) to obtain the value of the adjoint field γ at each time step, the sensitivity of the objective function with respect to the i-th parameter is given by:

[0103]

[0104] When the disturbance Δp i When it is small enough, the disturbance in equation (9) will not cause too much interference to the field distribution and can be ignored, that is,

[0105]

[0106] When the accompanying variable time τ = T m -t, the negative sign of the first-order derivative of γ for the time variable t is omitted. Formula (11) can be written as:

[0107]

[0108] Wherein, in equation (12), the excitation is applied backward in time. The excitation applied at time τ = 0 (with the start of the simulation) is the excitation stored at the end of the original simulation (τ = T m ). In addition, the system matrix in equation (12) is the transposed matrix of the original system equation (4). The control equation constructed in equation (3) leads to the symmetry of the system matrix, so that the adjoint system equation (12) can be expressed in an expanded form as:

[0109]

[0110] in, are the adjoint electric and magnetic fields. The derivatives in (13) are all with respect to the adjoint time variable τ. The adjoint system (12) represents an FDTD simulation with the same material properties but with different excitations. The only difference in the adjoint system is that the excitation vector depends on the objective function. The excitation used in the formula is determined during the original FDTD simulation system (4). Therefore, the same FDTD solver can be used to solve the adjoint system after making the necessary excitation changes to the original problem.

[0111] Taking a circle as an example, when the minimum unit Δp = Δr is used to perturb the length discontinuity surface, the difference vector has only one non-zero component, such as Figure 1 Medium yellow unit S r + In order to accurately solve the adjacent field, the perturbation system (9) should be solved in theory, and then the field value is stored in the yellow area. However, in practice, the perturbation adjacent system is not solved, but Figure 2 The approximate system shown is solved to optimize the solution process. r + The adjoint field in the undisturbed adjacent system region S r - The field approximation within the perturbed system is called one-to-one mapping, which has achieved good results in neighbor sensitivity estimation. This mapping is based on establishing a one-to-one correspondence between the perturbed system and the unperturbed system.

[0112] In the original simulation, the difference needs to be calculated in the unit affected by the disturbance, the i-th ΔR associated with the i-th parameter i for:

[0113]

[0114] The electric field components are multiplied by the permittivity and conductivity of the locations on the grid, for Figure 3 The internal nodes marked with dots have a dielectric constant of ε = ε 0 ε r For edge nodes marked with triangles, the effective dielectric constant is the average of the dielectric constants of air (the assumed surrounding medium) and the dielectric material ε = 0.5ε 0 ε r +0.5ε 0 For the inner and outer corner nodes marked with diamonds and pentagons, the effective dielectric constant is the weighted sum of the dielectric material and air, and its value is ε = 0.75ε respectively. 0 ε r +0.25ε 0 With ε=0.25ε 0 ε r +0.75ε 0 When the radius is disturbed, different material changes will occur at different nodes of the circle approximated by FDTD. The difference vectors generated by these changes correspond to the different non-zero vectors in equation (14).

[0115] The weights of each parameter are determined in advance as part of the parameterization step, which determines the perturbation Δp i , and the weights to be multiplied by each field component. In the original simulation, each difference is calculated by multiplying the field component by the corresponding weight using equation (14), and then storing these differences at each time step. The adjoint variable algorithm for the above two-dimensional TMz finite-difference time-domain case can be summarized as the following steps:

[0116] Step 1: Parameterization. Determine the forward and backward The perturbation Δp in the i-th parameter i The affected FDTD unit.

[0117] Second step: original simulation. At each time step, a difference vector of material parameters is stored by multiplying each affected field component by the corresponding imprint The difference vector ΔR of the shape parameter i / Δp i . Will be accompanied by incentives Stored in the observation domain of the objective function.

[0118]

[0119]

[0120] Step 3: Adjoint simulation. Store the adjoint field at all locations where the difference vector has a non-zero value. In this case, a one-to-one mapping is used. Execute Estimate sensitivity.

[0121] 2. Adjoint variable method of frequency-dependent objective function:

[0122] When the objective function is to evaluate the spectrum of the field in the observation domain, this spectrum is summarized as:

[0123]

[0124] When evaluating the sensitivity of equation (15) to different parameters at different frequencies, it is necessary to take the accompanying analog excitation as q(t). The discrete approximate sum of the equation is:

[0125]

[0126] Using any frequency f in equation (16) m To tune the frequency, due to the use of N T time samples, the spectrum becomes periodic. Usually the frequency is evenly distributed, with: m =mΔf,Δf=1 / (N T Δt),τ n =nΔτ=nΔt, under this frequency selection, the spectrum form is modified as follows:

[0127]

[0128] From the above formula, only a small part of the frequency in formula (17) is needed to cover the frequency band of the excitation source. Using the inverse discrete Fourier transform to evaluate the time domain signal value using the frequency domain spectrum is:

[0129]

[0130] Combining the conjugation property with equation (16), we have:

[0131]

[0132] It can be obtained that the general broadband adjoint excitation q(t) is written as the sum of cosines of different amplitudes and phases. The data provided by the mth frequency expansion (18) is changed to:

[0133]

[0134] From equation (19), we can see that the steady-state phase of each excitation frequency, the result of broadband excitation q(t) is the broadband adjoint response γ of the entire computational domain q (τ). At this point, the broadband adjoint response γ at all time steps is calculated q (τ). Due to the different frequency components in the signal q(t), these phasors determine the amplitude and phase of the steady-state sinusoidal response, resulting in a sinusoidal excitation Spectrum The jth component of The spectral response of the real adjoint excitation is then calculated using the formula:

[0135]

[0136] All terms in equation (21) are known, so the above equation can be used to accurately calculate the corresponding steady-state response at any frequency, thereby evaluating the real and imaginary sensitivity of the spectrum at all frequencies. At the same time, the imaginary response under this excitation is obtained:

[0137]

[0138] Formula (21) and Formula (22) are expressed by 2πf m T m =2π(mΔf)(N T Δτ)=2mπ,ΔfΔτ=(1 / N T ), which is determined by the frequency selection, and the time domain response is:

[0139]

[0140] The phase at all frequencies is determined by equations (23) and (24), and the corresponding steady-state sine function is calculated:

[0141]

[0142] This data is used to calculate the sensitivity of the relevant objective function corresponding to different parameters at the required frequency through formula (10).

[0143] The difference between the frequency-dependent objective function and the frequency-independent constitutive parameter algorithm is mainly reflected in: (1) there is no need to store the adjoint excitation source in the original simulation, because the adjoint excitation is independent of the original simulation; (2) the adjoint simulation broadband excitation q(t) is selected, the adjoint simulation is performed, the adjoint response of the disturbed domain is stored, the above equations (20) to (26) are calculated, and finally equation (10) is executed to obtain the sensitivity.

[0144] 3. Adjoint variable method of complete transient function:

[0145] The sensitivity analysis of the transient function is the sensitivity of the field value at a point in the observation space to all parameter changes at all time steps of the entire simulation. To evaluate the sensitivity of the complete transient function, we first start from the transient function of a single time step. Suppose we want to evaluate the electric field in a certain direction and determine the field value E(d,t 0 ), the objective function is expressed as:

[0146]

[0147] δ(tt in formula (27) 0 ) is a selection function that is intended to determine the specific time step and select the time t according to the requirements. 0 In order to determine the adjoint simulation excitation corresponding to the objective function of equation (27), similar to the reasoning of the first method, equation (12) is used to obtain:

[0148]

[0149] In formula (28), a corresponds to a vector with a value of 1, where the δ function is independent of the response.

[0150] Because of the existence of the delta function, the adjoint excitation is a pulse function, and the excitation of the adjoint simulation is the current density source at the observation point d. Under this excitation source, the adjoint simulation formula (12) is executed, and the corresponding adjoint simulation formula is obtained as follows:

[0151]

[0152] At the same time, the original difference vector is stored in the original simulation, and the field value E(d,t 0 ) for all target parameters.

[0153] When the target time is any other step t 2 When , the adjoint simulation formula under the time objective function is obtained in the same way:

[0154]

[0155] Comparing the objective functions of two different time steps, we find that the system is the same, the excitation is time-shifted, and when simulating regional linear and time-invariant materials, we get:

[0156] γ 2 (τ) = γ 1 (τ+τ 1 -τ 2 ),τ≥τ 2 (31)

[0157] That is, with the response γ 2 (τ) is γ 1 The time-shift result of (τ) can be obtained without the need to simulate equation (30), and its accompanying response can be obtained. The result can be derived from the result of equation (29).

[0158] Through the above derivation, we can get the sensitivity of the target position field value to the parameters in a more complete time. Using the simulation of formula (29), we can derive the accompanying response field in the entire time step, and then calculate the objective function E(d,t 0 ) to the parameters.

[0159] So far, the sensitivity analysis method of the fixed constitutive parameters, frequency-dependent objective function and transient pulse function to the response of the arc structure has been obtained. The above formula can be used to calculate the sensitivity analysis of the arc structure under 2D conditions.

[0160] After the formula is derived, the present invention performs code verification, and the specific method includes the following steps:

[0161] (1) Starting from the differential form of Maxwell's equations, according to the constitutive relation of electromagnetic isotropy and combining the expressions of dielectric constant and permeability, the frequency domain form of Maxwell's equations is given.

[0162] (2) For the 2D case, the Maxwell equations in the frequency domain are simplified and expressed in matrix form, forming an equation system containing key electromagnetic field parameters.

[0163] (3) According to the derived adjoint simulation system, the adjoint excitation domain and excitation source are reselected in the forward simulation model according to different objective functions, the simulation of the adjoint system is completed, and the unit response affected by the disturbance parameters is recorded.

[0164] (4) According to the derived adjoint simulation system, the adjoint excitation domain and adjoint excitation source are reselected in the forward simulation model according to the different objective functions to complete the adjoint system simulation. The response changes of the units affected by the disturbance parameters are stored in the adjoint system.

[0165] (5) The sensitivity estimation formula was applied to complete the sensitivity calculation. This step significantly improved the analysis efficiency and reduced the demand for computing resources.

[0166] (6) By using the forward simulation multiple times, changing the corresponding parameters each time, and storing the measured values ​​of the objective function, the number of simulations required for the finite difference method is completed and comprehensive data is obtained.

[0167] (7) Use the finite difference formula to estimate the sensitivity of the corresponding parameters.

[0168] (8) Compare and verify the finite difference results with the results calculated by the electromagnetic sensitivity analysis method based on the arc-shaped boundary electromagnetic structure, and make a relative error data table to verify the correctness of the results of this method.

[0169] The following is a specific application example to illustrate the two-dimensional plane wave excitation model-frequency-independent constitutive parameters, the two-dimensional plane wave excitation model-frequency-dependent objective function and the two-dimensional plane wave excitation model-complete transient function.

[0170] For the 2D plane wave excitation model - frequency independent constitutive parameters, such as Figure 4 The figure shows a rectangle with a spatial size of 98 mm × 96 mm in the x and y directions. The spatial step is set to Δx = Δy = 1 mm, and the time step is Δt = 2 ps. The plane wave propagates in the positive y direction at y = 0, and the Gaussian pulse time domain form is: p(n) = exp[-(tt 0 ) 2 / τ 2 ], where τ = 0.1c -1 , t 0 ≈4.5τ, t=nΔt.

[0171] The entire simulation space is truncated by 8 layers of CPML (Convolutional Perfectly Matched Layer), and a circular non-dispersive isotropic medium with a radius of 20 mm is placed in the middle. The objective function is the integral of the square of the electric field value on the line y=60Δy in the observation domain, as shown in formula (32):

[0172] where Ω is the line y=60Δy from x=-30Δx to x=30Δx. The goal is to evaluate the function of equation (31) for the parameter p=[ε r σ e r] T The sensitivity of the disturbance Δp = [0.10.01S / m 1mm] T .

[0173] Figure 5-Figure 7The results of using the adjoint variable method and the finite difference method to calculate the objective function Q for the parameter vector p = [ε r σ e r] T The sensitivity curve below. Figure 5-Figure 7 As shown in the figure, the sensitivity calculated by the adjoint variable method proposed in the present invention is consistent with the result of the finite difference method, and is even close to the sensitivity of the second-order finite difference calculation in terms of the sensitivity to the material parameters. The sensitivity calculated for the radius of the arc structure parameter is also well consistent with the FDM result. At the same time, in order to obtain the numerical error, the norm ratio is applied to the unified error range (0,1), which is expressed as Error = ||AVM-FDM|| 2 / ||FDM|| 2 , the results are placed in Table 1, and it is found that the error is extremely small, indicating the accuracy of the AVM calculation results.

[0174] Table 1 Norm error between AVM and FDM under constitutive independent objective function

[0175] Error type <![CDATA[ε r ]]> <![CDATA[σ e ]]> r AVM and CFD <![CDATA[2.0255×10 -3 ]]> <![CDATA[5.1667×10 -4 ]]> <![CDATA[6.2553×10 -2 ]]> AVM and FFD <![CDATA[2.7232×10 -3 ]]> <![CDATA[5.5774×10 -4 ]]> <![CDATA[7.3643×10 -3 ]]> AVM and BFD <![CDATA[1.3269×10 -3 ]]> <![CDATA[4.7567×10 -4 ]]> <![CDATA[1.4296×10 -1 ]]>

[0176] It is important to note that the number of simulations used in this example using forward and backward finite differences is 4, and the central difference is even 7, but the adjoint variable method only uses two simulations, which greatly reduces the computational efficiency. At the same time, as the number of evaluation parameters increases, the number of simulations required for finite differences will continue to increase, but the adjoint variable method only needs two simulations to complete.

[0177] For the two-dimensional plane wave excitation model-frequency-dependent objective function, the same model is Figure 4 As shown in the example, the objective function is changed to a frequency-dependent function:

[0178]

[0179] Where Ω is y = 60Δy, and the output port is from x = -30Δx to x = 30Δx, E m (Ω) is the normalized result at the output port and is a constant in this problem. The plane wave excitation used is adjusted to a Gaussian pulse modulated by a cosine function: p(n) = cos[ω c (tt 0 )]exp[-(tt 0 ) 2 / τ 2 ]. Where τ = 0.1c -1 , t 0 ≈4.5τ,ω c =3π×10 9 , t=nΔt.

[0180] Adjoint simulation requires specifying the time distribution and frequency distribution of the adjoint excitation. Given the adjoint simulation excitation q(t) = sin[ω c (n-0.5)Δt]exp[-((n-0.5)-500Δt) 2 / 2(125Δt) 2 ]. Where τ = 0.1c -1 , t 0 ≈4.5τ,ω c =3π×10 9 , n is the time step. At the same time, according to f m =mΔf,Δf=1 / (N T Δt),τ n =nΔτ=nΔt set the uniform frequency distribution, and 12 frequency points are selected as the objective function analysis.

[0181] Table 2 Norm error between AVM and FDM under frequency-dependent objective function

[0182]

[0183] Figure 8-Figure 13 The sensitivity of the real and imaginary parts of the spectrum to the target parameters at the considered frequencies is shown in Fig. 2. The sensitivity estimates are in good agreement with the forward finite difference, backward finite difference and central finite difference approximations, further verifying the correctness of the method. Figures 8 to 13 As shown. At the same time, Table 2 shows the norm error between AVM and FDM under the frequency-dependent objective function. As expected, the result is between (0, 1), which reflects the accuracy of AVM. The sensitivity of the spectrum to the dielectric constant and conductivity is very small and can be ignored because the material of the discontinuity is a good conductor. Among them, the objective function is less sensitive to the radius, which is attributed to the fact that the perturbation Δr used is only one unit. Even under such a small perturbation, the results still show a high degree of consistency with the results of the finite difference method (FDM), which verifies the accuracy of the adjoint variable method (AVM) in the analysis of arc structures.

[0184] For the 2D Plane Wave Excitation Model - Full Transient Function, select Figure 3 In the model, a point at x = 25Δx, y = 55Δy above the prototype ring is selected as the field value detection point, and the field value E at this point is evaluated. Z During the whole simulation time, the parameter p = [ε r σ e r] T The sensitivity of the plane wave excitation is a Gaussian pulse modulated by a cosine function: p(n) = cos[ω c (tt 0 )]exp[-(tt0 ) 2 / τ 2 ]. Where τ = 0.1c -1 , t 0 ≈4.5τ,ω c =3π×10 9 , t=nΔt.

[0185] Table 3 Norm error between AVM and FDM under transient function

[0186] Error type <![CDATA[ε r ]]> <![CDATA[σ e ]]> r AVM and CFD <![CDATA[3.6592×10 -3 ]]> <![CDATA[4.0714×10 -3 ]]> <![CDATA[1.1738×10 -1 ]]> AVM and FFD <![CDATA[1.3879×10 -2 ]]> <![CDATA[4.9260×10 -3 ]]> <![CDATA[1.3187×10 -1 ]]> AVM and BFD <![CDATA[1.9305×10 -2 ]]> <![CDATA[4.1873×10 -3 ]]> <![CDATA[1.3244×10 -1 ]]>

[0187] The comparison results are as follows Figure 14-16 As shown in Figure 3, the sensitivity of the dielectric constant, conductivity and radius calculated by the adjoint variable method shows a high consistency with the complete transient sensitivity estimated by the central finite difference method. At the same time, the norm errors of AVM and FDM are given in Table 3. This result fully proves the comparability of the two methods in terms of accuracy and provides reliable data support for the further application of the adjoint variable method in the field of electromagnetic simulation optimization.

[0188] In order to broaden the application scope of AVM and solve the efficiency and accuracy problems of traditional methods in complex structure analysis, the present invention innovatively applies it to the sensitivity analysis of arc structures. It can not only accurately evaluate the impact of design parameter changes on the performance of arc structures, but also provide important guidance for the design of arc structures in high-frequency communications and radar systems at different frequencies.

[0189] In addition, the present invention has completed the sensitivity analysis of the arc structure response under fixed constitutive parameters, frequency-dependent objective functions, and transient pulse functions, providing a new perspective for evaluating the dynamic performance of structures in complex electromagnetic environments. It not only provides a new and efficient tool for the sensitivity analysis of arc structures, but also lays the foundation for the application of more complex structural analysis in the future. Through the introduction of this method, the sensitivity analysis of any complex structure can be realized, thereby completing the structural optimization.

[0190] The present invention is particularly suitable for sensitivity analysis of arc boundary parameters, and for the first time expands the application scope of AVM to such boundaries. Through the method of the present invention, the time domain field, frequency domain field and far-field information distribution of electromagnetic scattering can be quickly obtained under limited computing resources, which is suitable for optimization problems of electromagnetic structures such as plasma models and complex antenna models. Compared with the prior art, the beneficial effects of the present invention are: significantly reducing the computing time and resources required for the optimization design of electromagnetic structures; providing a stable and efficient electromagnetic sensitivity analysis method; and broadening the application scope of AVM in the optimization of arc-boundary electromagnetic structures.

[0191] Example 2

[0192] A computer device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the electromagnetic sensitivity analysis method in Example 1.

[0193] Example 3

[0194] A computer-readable storage medium stores a computer program, which implements the electromagnetic sensitivity analysis method in embodiment 1 when executed by a processor.

[0195] Example 4

[0196] A computer program product includes a computer program, which implements the electromagnetic sensitivity analysis method in embodiment 1 when executed by a processor.

[0197] Example 5

[0198] A computer device, which may be a database, may have an internal structure as shown in Fig.18 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, referred to as I / O) and a communication interface. Among them, the processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store pending transactions. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, the electromagnetic sensitivity analysis method in Example 1 is implemented.

[0199] It should be noted that the object information (including but not limited to object device information, object personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in the present invention are all information and data authorized by the object or fully authorized by all parties, and the collection, use and processing of relevant data need to comply with relevant laws, regulations and standards of relevant countries and regions.

[0200] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiments can be completed by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to the memory, database or other medium used in the embodiments provided by the present invention can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. As an illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM). The database involved in each embodiment provided by the present invention may include at least one of a relational database and a non-relational database. Non-relational databases may include distributed databases based on blockchains, etc., but are not limited thereto. The processor involved in each embodiment provided by the present invention may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but are not limited thereto.

[0201] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0202] The principles and implementation methods of the present invention are described in this article using specific examples. The description of the above embodiments is only used to help understand the method and core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. An electromagnetic sensitivity analysis method, characterized in that: The method comprises: According to Maxwell's equations and the constitutive relation of electromagnetic isotropy, combined with the expressions of dielectric constant and permeability, the Maxwell's equations in frequency domain are obtained; According to the frequency domain Maxwell equations, constructing the original system equations of the target two-dimensional arc structure; the original system equations of the target two-dimensional arc structure include the definition of the arc boundary, the parameterized representation of the material properties, the design parameters and the definition of the objective function related to the design parameters; the objective function is used to evaluate the response of the original system equations of the target two-dimensional arc structure; When the design parameters of the target two-dimensional arc structure are disturbed, determining an adjoint system equation according to the original system equation of the target two-dimensional arc structure; Applying the adjoint system equation, performing a forward simulation and an adjoint simulation respectively, and obtaining a forward field solution and an adjoint field solution; The forward field solution and the adjoint field solution are used to calculate the sensitivity of the target function of the target two-dimensional arc-shaped structure with respect to the target parameter.

2. The electromagnetic sensitivity analysis method according to claim 1, characterized in that: The Maxwell equations in the frequency domain are: Among them, E is the vector of electric field strength; H is the vector of magnetic field strength; μ is the magnetic permeability of the medium; σ m is the magnetic permeability; ε is the dielectric constant of the medium; σ e is the conductivity; × represents the rotation.

3. The electromagnetic sensitivity analysis method according to claim 1, characterized in that: The original system equation of the target two-dimensional arc structure is: C=[E nT H nT ] T ; Among them, B m is a function of the conductivity in the domain, B c is a matrix of constant coefficients; T is the transpose; P is the excitation matrix; μ is the magnetic permeability of the medium; σ m is the magnetic permeability; ε is the dielectric constant of the medium; σ e is the electrical conductivity; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

4. The electromagnetic sensitivity analysis method according to claim 1, characterized in that: The objective function includes an objective function based on fixed constitutive parameters, an objective function based on frequency dependence, and an objective function based on a transient pulse function.

5. The electromagnetic sensitivity analysis method according to claim 4, characterized in that: The objective function based on the fixed constitutive parameters is: C=[E nT H nT ] T ; Where Q is the objective function based on fixed constitutive parameters; C is the response matrix; p is the vector of design parameters; ξ is the local response integral kernel; T m is the simulation duration; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

6. The electromagnetic sensitivity analysis method according to claim 4, characterized in that: The objective function based on frequency dependence is: C=[E nT H nT ] T ; in, is the frequency-dependent objective function; C is the response matrix; ξ is the local response integral kernel; T m is the simulation duration; E nT is the electric field strength vector; H nT is the magnetic field strength vector.

7. The electromagnetic sensitivity analysis method according to claim 4, characterized in that: The objective function based on the transient pulse function is: Where E(d, t0) is the objective function based on the transient pulse function; δ(t-t0) is the selection function; E(d, t) is the objective function under abstract time; d is the position vector; t0 is the time point; t is the abstract time point; T m is the simulation duration.

8. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the electromagnetic sensitivity analysis method according to any one of claims 1 to 7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the electromagnetic sensitivity analysis method described in any one of claims 1 to 7 is implemented.

10. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the electromagnetic sensitivity analysis method described in any one of claims 1 to 7 is implemented.

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