A multi-physical target-oriented electromagnetic field grid adaptive regulation and optimization method in an electromagnetic system

Abstract

The present invention discloses a method for adaptively regulating and optimizing electromagnetic field grids oriented towards multi-physical target guidance in an electromagnetic system. Taking various parameters in the electromagnetic system or integrated micro-system as the target sensitivity function, after loading the grid topology, first perform a forward solution on the initial grid meshing to obtain a rough solution of the electric field; then deduce the adjoint form of the electromagnetic field wave equation to obtain the adjoint system equation; analytically obtain the multi-physical target-guided sensitivity function, extract its sensitivity coefficient, and fill this coefficient into the excitation term of the adjoint system equation; finally, solve the adjoint system equation to obtain the electric field residual weighting coefficient, calculate the error in the forward solution, and perform grid regulation and optimization through posterior error analysis. The method of the present invention has advantages such as high efficiency and high flexibility, and is used for the grid adaptive regulation and optimization of electromagnetic-thermal-mechanical integrated multi-physical problems in electromagnetic field numerical calculations, improving the numerical calculation efficiency and confidence of multi-physical fields in the electromagnetic system or integrated micro-system.

Description

Technical Field

[0001] The present invention belongs to the field of numerical calculation of electromagnetic fields, and relates to an electromagnetic field grid adaptive regulation and optimization method for multi-physical target orientation in electromagnetic systems. Background Art

[0002] In the process of numerical simulation of electromagnetic fields and multi-physical fields, especially the finite element method (FEM) based on unstructured grids, grids that meet the requirements need to be designed to obtain efficient and accurate calculations. There are the following several grid adaptive adjustment methods in engineering applications: 1. h-type: By adjusting the size of grid elements, the accuracy of numerical calculations is controlled; 2. p-type: By adjusting the order of basis functions, the accuracy of numerical calculations is controlled; 3. hp-type: Both the size of grid elements and the order of basis functions are adjusted. However, finer grids and higher polynomial orders mean an increase in the scale of calculations. Limited by the computing power of computers, workstations, servers, etc., it is necessary to minimize the scale of calculations while ensuring the calculation accuracy. For this purpose, the hp-type strategy must be flexibly adopted for adaptive grid adjustment to achieve the expected goal, and the core problem is the posterior error estimation with unknown prior probability.

[0003] For the mathematical process of posterior error estimation, there are two common strategies: residual-based estimation and recovery-based estimation. For residual-based estimation, error estimation is performed by constructing explicit or implicit residual expressions. However, due to the complexity of specific scenarios, it is difficult to establish the connection between residuals and true errors. For recovery-based estimation, the numerical gradient space is projected onto the high-order gradient space through a recovery operator to obtain a quantified error. CN110807289A discloses an integrated circuit adaptive finite element grid subdivision method based on posterior error estimation. This method solves the FEM numerical solution of grid elements, and within the superconvergent element patch of each grid node, based on the gradient approximation of the solution, a gradient of higher-order accuracy is recovered as the accurate value of the gradient of the solution. However, since this method is based on the advanced superconvergence method and depends on the quality of the original grid, it has poor robustness to low-quality grids and is difficult to handle complex boundary conditions. The existing technology cannot meet the requirements of fast, efficient, and accurate calculations.

[0004] For the physical processes in engineering applications, the error estimation method based on volume interpolation error and surface jump error (Sun D K, Cendes Z, Lee J F. Adaptive mesh refinement, h-version, for solving multiport microwave devices in three dimensions [J]. IEEE Transactions on Magnetics, 2000, 36(4): 1596 - 1599.) projects the error onto the residual charge and residual current spaces using the Green's function, providing a general method for evaluating the error of the electromagnetic wave equation. However, this method cannot solve specifically for a particular index, and due to the hyperbolic partial differential equation nature of the wave equation, error propagation will occur. The self-consistent error estimation method based on the electric and magnetic fields (Botha M M, Jin J M. Adaptive finite element-boundary integral analysis for electromagnetic fields in 3-D [J]. IEEE Transactions on Antennas and Propagation, 2005, 53(5): 1710 - 1720.) compares the errors between the solutions of the electric and magnetic fields respectively through the differential forms of Maxwell's equations to achieve the effect of error estimation. However, this method lacks flexibility and also has the problem of not being able to solve for a specific index.

[0005] For target-oriented posterior error analysis, adjoint sensitivity analysis is a practical method mainly applied to the solution of structural, material, and topology optimization problems in numerical calculations. It analyzes the sensitivity of the objective function to the design variables by introducing adjoint variables and adjoint system equations. This method requires solving both the wave equation system of the forward problem and the adjoint equation system of the reverse problem in optimization problems, and it has not been fully applied in the adaptive regulation of electromagnetic field meshes. Currently, the literature (Harmon J J, Key C, Estep D, et al. Adjoint-Based Accelerated Adaptive Refinement in Frequency Domain 3-D Finite Element Method Scattering Problems [J]. IEEE Transactions on Antennas and Propagation, 2020, 69(2): 940 - 949.) only performs p-type adaptive regulation on the meshes of scattering problems, with a narrow scope of application and inflexible regulation. Summary of the Invention

[0006] Aiming at the defects in the prior art, the object of the present invention is to provide a multi-physical target-oriented electromagnetic field grid adaptive regulation and optimization method for electromagnetic field numerical calculation.

[0007] The technical solution adopted by the present invention is as follows:

[0008] A multi-physical target-oriented electromagnetic field grid adaptive regulation and optimization method for an electromagnetic system takes various physical parameters in the electromagnetic system or integrated microsystem as the target sensitivity function. After loading the grid topology, first perform a forward solution for the initial grid division to obtain a rough solution of the electric field of the electromagnetic field wave equation; then, derive the adjoint form of the electromagnetic field wave equation to obtain the adjoint system equation; further, analytically obtain the multi-physical target-oriented sensitivity function, extract its sensitivity coefficient, and fill this coefficient into the excitation term of the adjoint system equation; finally, solve the adjoint system equation to obtain the electric field residual weighting coefficient, calculate the error in the forward solution, and perform grid regulation and optimization through posterior error analysis.

[0009] Furthermore, the method specifically may include the following steps:

[0010] In the first step, establish a geometric model of the electromagnetic system, perform an initial grid division on it to obtain a computational electromagnetic grid topology, that is, shape function information, element (point, edge, face, body) adjacency relationship, and boundary conditions.

[0011] In the second step: Load the obtained computational electromagnetic grid topology, perform weak formulation for the electromagnetic field wave equation using the Galerkin method, and then perform discretization using the vector finite element method.

[0012] In the third step: Solve the discretized electromagnetic field wave equation in the second step, and expand the solved electric field in the entire computational region to obtain a rough solution of the electric field.

[0013] In the fourth step: For the electromagnetic field wave equation, derive its adjoint form to obtain the adjoint system equation; use the Galerkin method and the vector finite element method to perform weak formulation and discretization on it respectively.

[0014] In the fifth step: Analytically obtain the multi-physical target-oriented sensitivity function, extract its sensitivity coefficient, and fill this coefficient into the excitation term of the adjoint system equation.

[0015] In the sixth step: Solve the discretized adjoint system equation obtained from the fourth step to the fifth step, and expand the solved adjoint variable in the entire computational region using a high-order approximation method to obtain the electric field residual weighting coefficient.

[0016] Step 7: Take the inner product of the residual of the rough solution of the electric field in Step 3 and the weighting coefficient of the electric field residual in Step 6 in the L2 space to obtain a quantifiable posterior error;

[0017] Step 8: Use the quantifiable posterior error obtained in Step 7 to perform hp-type adjustment on the grid topology;

[0018] Step 9: Repeat Steps 2 to 8 until the posterior error meets the convergence requirement.

[0019] In the above technical solution, further, in Steps 2 and 4, the vector finite element method is used for discretization, and the discretization space is the tangentially continuous Nédélec space. The order of the basis function of the Nédélec space is k, where 0 ≤ k ≤ p max , p max is the highest order of the basis function of the Nédélec space; in the high-order approximation method described in Step 6, the high-order approximation space is the tangentially continuous Nédélec space, and the order of the basis function of the Nédélec space is k + 1.

[0020] Further, the adjoint form is derived from the electromagnetic field wave equation through the adjoint operator, and the expression is:

[0021]

[0022] where is the linear operator corresponding to the electromagnetic field wave equation, * represents the adjoint operator. For a scalar, it represents the conjugate, and for a vector and a tensor, it represents the conjugate transpose. v is the adjoint variable, μ r is the relative magnetic permeability, k0 is the wave number of the electromagnetic wave in free space, and ε r is the relative permittivity.

[0023] Further, the sensitivity function of the electric field is determined by the equivalent error current at the specified position and is expressed as:

[0024]

[0025] where j is the imaginary unit, ω is the angular frequency, μ0 is the vacuum magnetic permeability, w is the unit vector in the direction to be concerned, J is the current density, Ω i is the i-th element to be concerned, H is the magnetic field strength, ε is the permittivity, E is the vector electric field, and μ r is the relative magnetic permeability, and k0 is the wave number;

[0026] According to the Riesz representation theorem, we have:

[0027]

[0028] where p is the extracted sensitivity coefficient.

[0029] Furthermore, the multi-physical objectives may be various parameters such as S-parameters, far-field radiation, power, etc.

[0030] The beneficial effects of the present invention are as follows: The present invention utilizes the adjoint system equation and is not affected by the generation, propagation, and accumulation of errors; it is target-oriented, with a high degree of grid customization, low grid calculation cost, high accuracy, and good convergence; based on the weighting of the sensitivity function and the residual, the influence of errors on accuracy can be directly quantified; error elimination is performed through hp-type adjustment, and different strategies can be used for regulation according to different indicators; the high-order approximation method is used and does not depend on the quality of the initial grid. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a flowchart of the adaptive regulation analysis method of the present invention.

[0032] Figure 2 It is a schematic diagram of the first-order vector basis function.

[0033] Figure 3 It is a two-dimensional schematic diagram of h-type regulation and p-type regulation, (a) h-type regulation, (b) p-type regulation.

[0034] Figure 4 It is the customized grid design result with the electric field amplitude at the specified receiving point as the receiving point. And the comparison with the non-customized grid, (a) initial grid, (b) non-customized grid, (c) customized grid (receiving above), (d) customized grid (receiving below), and the present invention greatly reduces the scale of grid generation.

[0035] Figure 5 It is the geometric structure of the coupled-line bandpass filter to be solved in the example of the present invention.

[0036] Figure 6 is Figure 5 The optimized result of the customized grid of the coupled-line bandpass filter to be solved in the example, (a) initial grid, (b) adaptive optimized grid.

[0037] Figure 7 is Figure 5 The comparison of the number of degrees of freedom required by the present invention and the traditional regulation method in the example, the x-axis is the number of degrees of freedom, and the y-axis is the calculated S-parameter. Within the error tolerance range, the number of degrees of freedom is reduced by one order of magnitude.

[0038] Figure 8 It is a discontinuous medium with the electric field power as the sensitivity function.

[0039] Figure 9 It is the electric field distribution diagram before and after the optimization of the discontinuous medium. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0040] The technical solution of the present invention will be further described in detail below in conjunction with the attached drawings and specific examples.

[0041] The initial mesh of the electromagnetic system or integrated microsystem can be generated using various finite element mesh generation software, and its shape function information, element (point, edge, face, volume) adjacency relationship, and boundary conditions are clarified and read in.

[0042] The electromagnetic field is described by the wave equation:

[0043]

[0044] The calculation boundary is truncated by a first-order absorbing boundary, that is

[0045]

[0046] The descriptions of each symbol are as follows:

[0047] μ r Relative magnetic permeability

[0048] ε r Relative permittivity

[0049] k0 Wave number of electromagnetic wave in free space

[0050] Z0 Characteristic impedance of the medium

[0051] J Current density

[0052] n Outer normal of the boundary

[0053] E Vector electric field

[0054] Formulas (1) and (2) together describe the forward problem of the electromagnetic field wave equation. Using the initial mesh topology obtained in the first step for Galerkin weak formulation and vector finite element discretization, the discrete form of the wave equation is obtained:

[0055] Among them, the stiffness matrix is:

[0056]

[0057] The boundary condition is:

[0058]

[0059] Among them, N i 、N j are basis functions in the Nédélec vector basis function space.

[0060] For vector basis functions, the high-order derivative components are usually jointly composed of the curl field (irrotational) and the divergence field (solenoidal), each contributing a 0.5-order component to each order of the basis function. Taking the first-order basis function as an example:

[0061]

[0062] and and represent the Lagrange basis functions corresponding to the nodes. The node positions are at both ends of the edges corresponding to the vector basis functions.

[0063] Figure 2 Their vector schematic diagrams are given. It is observed that they are mutually orthogonal and independent, and each has first-order curl completeness and divergence completeness. Each order of basis functions stacks to form hierarchical basis functions, which are characterized in that the high-order basis functions explicitly contain the information of the low-order basis functions. Therefore, within the same element, multiple basis functions of different orders can be adopted and maintain a certain linear independence. Therefore, for the basis function order elevation of a single edge, face, or volume, it does not affect other edges, faces, and volumes. This method enables the mesh topology to have p-type adaptive adjustment capabilities.

[0064] The rough solution of the electric field obtained by solving the equation is expanded in the entire computational domain as:

[0065]

[0066] where N k represents the total number of basis functions when the order of the spatial basis function is k; E i represents the solution of the electric field obtained by solving the equation.

[0067] The adjoint form of the wave equation is derived from the adjoint operator, and its expression is:

[0068]

[0069] where * represents the adjoint operator. For a scalar, it represents the conjugate, and for a vector and a tensor, it represents the conjugate transpose.

[0070] By performing Galerkin weak formulation and vector finite element discretization on the adjoint form, the discrete form of the adjoint system equation can be obtained.

[0071] The multi-physics goal-oriented sensitivity function generally has a linear relationship with the electric field. The sensitivity function of the electric field is determined by the equivalent error current at the specified position:

[0072]

[0073] where is the sensitivity function of the equivalent error current at a specified position. After linear transformation of this equation, a multi-physics target-oriented sensitivity function can be customized, and parameters such as S-parameters, far-field radiation, and power can all be expressed by this equation. According to the Riesz representation theorem.

[0074]

[0075] where p is the extracted sensitivity coefficient.

[0076] The excitation term of the adjoint system equation is filled with the sensitivity coefficient. Therefore, the complete adjoint equation can be expressed as:

[0077]

[0078] The adjoint variable v is expanded in the computational domain using the high-order approximation method. The principle is as follows:

[0079] Assume that q(φ) is the sensitivity function in scalar form (the dependent variable in this invention is the electric field solution), and the expression of the h-type approximation method is

[0080]

[0081] The expression of the p-type approximation method is:

[0082]

[0083] Their meanings are respectively to obtain the derivative information on the left side of the equation using the encrypted grid (the size changes from h to h / 2) or the increased order of the basis function (the order changes from p to p + 1). This invention adopts the latter method for high-order approximation.

[0084] For the electric field equation mentioned in this invention, assume that according to the left side terms of (11) and (12), the adjoint variable v is expanded, and the obtained weighted coefficient of the electric field residual is u, which is substituted into the inner product (9) expressed by the Riesz representation theorem

[0085]

[0086] Formula (13) can be used as a quantifiable posterior error, and hp-type regulation can be flexibly carried out according to its magnitude and discontinuity; Figure 3 Two-dimensional schematic diagrams of h-type regulation and p-type regulation are given, where Figure 3 -a is the schematic diagram of h-type regulation, Figure 3 -b is the schematic diagram of p-type regulation.

[0087] Figure 4The electric field amplitude at the specified receiving point is given as the customized grid design result of the receiving point, as well as the comparison with the non-customized grid. (a) Initial grid, (b) Non-customized grid, (c) Customized grid (upper receiving), (d) Customized grid (lower receiving). It can be seen that the method of the present invention greatly reduces the scale of grid generation.

[0088] At the same time, we provide a case for solving the S-parameters of a coupled-line bandpass filter. Figure 5 is the geometric structure to be solved. The customized grid optimization results of the coupled-line bandpass filter to be solved by using the method of the present invention are as Figure 6 shown, where (a) is the initial grid and (b) is the adaptive optimization grid. Figure 7 is the comparison of the number of degrees of freedom required by using the method of the present invention and the traditional regulation method in this case. The x-axis is the number of degrees of freedom, and the y-axis is the calculated S-parameters. It can be seen that within the error tolerance range, the number of degrees of freedom is reduced by one order of magnitude. Figure 8 is a schematic diagram of a discontinuous medium with the electric field power as the sensitivity function. Figure 9 are the electric field distribution diagrams before and after the optimization of the discontinuous medium.

Claims

1. A multi-physics target-oriented electromagnetic field grid adaptive regulation and optimization method in an electromagnetic system, characterized in that Taking various physical parameters in an electromagnetic system or an integrated micro-system as the target sensitivity function, after loading the grid topology, first perform a forward solution for the initial grid discretization to obtain a rough solution of the electric field of the electromagnetic field wave equation; then, derive the adjoint form of the electromagnetic field wave equation to obtain the adjoint system equation; further, analytically obtain the multi-physical target-oriented sensitivity function, extract its sensitivity coefficient, and fill this coefficient into the excitation term of the adjoint system equation; finally, solve the adjoint system equation to obtain the electric field residual weighting coefficient, calculate the error in the forward solution, and perform grid control optimization through posterior error analysis; specifically, it includes the following steps: First step, establish the geometric model of the electromagnetic system, perform initial grid discretization on it to obtain the computational electromagnetic grid topology, that is, shape function information, element adjacency relationship, and boundary conditions; Second step: Load the obtained computational electromagnetic grid topology, perform weak formulation on the wave equation of the electromagnetic field using the Galerkin method, and then perform discretization using the vector finite element method; Third step: Solve the discretized electromagnetic field wave equation in the second step, and expand the solved electric field in the entire computational domain to obtain a rough solution of the electric field; Fourth step: For the electromagnetic field wave equation, derive its adjoint form to obtain the adjoint system equation; use the Galerkin method and the vector finite element method to perform weak formulation and discretization on it respectively; Fifth step: Analytically obtain the multi-physical target-oriented sensitivity function, extract its sensitivity coefficient, and fill this coefficient into the excitation term of the adjoint system equation; Sixth step: Solve the discretized adjoint system equation obtained from the fourth step to the fifth step, and expand the solved adjoint variable in the entire computational domain using the high-order approximation method to obtain the electric field residual weighting coefficient; Seventh step: Take the inner product of the residual of the rough solution of the electric field in the third step and the electric field residual weighting coefficient in the sixth step in the L2 space to obtain a quantifiable posterior error; Eighth step: Use the quantifiable posterior error obtained in the seventh step to perform hp-type adjustment on the grid topology; Ninth step: Repeat the second step to the eighth step until the posterior error meets the convergence requirement.

2. The electromagnetic field grid adaptive regulation and optimization method for multi-physical target orientation in an electromagnetic system according to claim 1, wherein: In Step 2 and Step 4, discretization is performed using the vector finite element method, and the discretized space is the Nédélec space with tangential continuity. The order of the basis functions of the Nédélec space is , where , is the highest order of the basis functions of the Nédélec space; in the high-order approximation method described in Step 6, the space for high-order approximation is the Nédélec space with tangential continuity, and the order of the basis functions of the Nédélec space is .

3. The electromagnetic field grid adaptive regulation and optimization method for multi-physical target orientation in an electromagnetic system according to claim 1, wherein: The adjoint form is derived and generated from the electromagnetic field wave equation through the adjoint operator, and the expression is: Among them is the linear operator corresponding to the electromagnetic field wave equation, * represents the adjoint operator. For a scalar, it represents the conjugate, and for a vector and a tensor, it represents the conjugate transpose. is the adjoint variable. is the relative magnetic permeability. is the wavenumber of electromagnetic waves in free space. is the relative permittivity.

4. The electromagnetic field grid adaptive regulation and optimization method for multi-physical target orientation in an electromagnetic system according to claim 1, wherein: Electric field sensitivity function Determined by the equivalent error current at a specified location and expressed as: Wherein, is the imaginary unit, is the angular frequency, is the magnetic permeability of vacuum, is the unit vector in the direction to be concerned, is the current density, is the i-th unit to be concerned, H is the magnetic field strength, is the permittivity, E is the vector electric field, is the relative magnetic permeability, is the wave number; According to the Riesz representation theorem, there is: Among them is the extracted sensitivity coefficient.

5. The electromagnetic field grid adaptive regulation and optimization method for multi - physical target - oriented in an electromagnetic system according to claim 1, characterized in that: The multi-physical targets are various parameters such as S-parameters, far-field radiation, and power.

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