A mesh adaptive method based on recovery-type a posteriori error estimate

Abstract

This invention belongs to the field of numerical electromagnetic field solution, specifically a mesh adaptive method based on reconstructive posterior error estimation. The invention first performs finite element modeling of the target microwave component, introducing boundary conditions and excitations to establish a corresponding electromagnetic simulation model. Then, a tetrahedral mesh is used to divide the solution domain, and finite element electromagnetic simulation analysis is performed on the model to calculate the electric field at the centroid of the four faces of the tetrahedral mesh. The least squares principle is then used to refit the electric field on the tetrahedral mesh to reconstruct a first-order linear fitting expression. Next, the reconstructed solution at the nodes is calculated using the reconstructed fitting expression, and posterior error estimation is performed based on the electric field at the nodes. Finally, the mesh is refined according to the error on the tetrahedral mesh, and finite element electromagnetic simulation analysis is performed again until the finite element solution meets the accuracy requirements. This invention has the advantages of simple engineering implementation, low computational cost, and high universality.

Description

Technical Field

[0001] This invention belongs to the field of numerical solution of electromagnetic fields, specifically a grid adaptive method based on restorative posterior error estimation. Background Technology

[0002] With the development of computer technology, using simulation technology to pre-analyze the electromagnetic field distribution of microwave components is an economical and effective method. Therefore, using electromagnetic simulation software to simulate the electromagnetic characteristics of microwave components during the design phase is an essential step in the product lifecycle. For computer simulation, solution accuracy is a major concern, and improving solution accuracy is a crucial development direction. For the finite element method, generally, the smaller the size of the discrete mesh of the model, the more accurate the simulation results. However, relying solely on experience to refine the mesh in certain computational regions lacks scientific basis and can lead to many uncertain results. Therefore, adaptive mesh refinement has been proposed to address this issue. Using the obtained posterior error estimate indicator, the current mesh is refined or coarsened, thus providing the mesh for the next calculation. The convergence of the results is then determined through adaptive error analysis.

[0003] Posterior error estimation is a crucial step in adaptive learning. Since the analytical solution is unknown, error calculation cannot be performed directly using the results; alternative methods are needed. The reconstructive posterior error estimation method is a commonly used approach. This method first constructs a reconstructive solution to replace the analytical solution, then constructs a certain element norm as a measure of error, uses this norm to obtain a mesh refinement indicator, and finally performs mesh refinement.

[0004] The core of restorative posterior error estimation methods lies in the construction of the restored solution. Traditional methods for constructing restored solutions, such as the SPR (Scraping Replacing Proof) method based on superconvergence and its improved algorithms, suffer from problems such as high computational cost, complex processing, and the need for special handling at the boundaries. Therefore, it is necessary to construct a restored solution with lower computational cost, simpler construction, and universal applicability for posterior error estimation to achieve adaptive mesh refinement. Summary of the Invention

[0005] To address the aforementioned problems and solve the issues of high computational cost, complex processing, and low universality (requiring special handling at boundaries) in existing mesh adaptive densification methods due to the construction of recoverable solutions, this invention provides a mesh adaptive method based on recoverable posterior error estimation. It constructs recoverable solutions using the least squares principle, performs posterior error estimation, and achieves mesh adaptive densification.

[0006] A grid adaptive method based on restorative posterior error estimation includes the following steps:

[0007] A. Perform finite element modeling on the target microwave component, and introduce boundary conditions and excitations to establish the corresponding electromagnetic simulation model.

[0008] B. The electromagnetic simulation model established in step A is divided into a tetrahedral mesh to solve the domain, and the coordinate information of the tetrahedral mesh nodes is obtained.

[0009] C. Using vector basis functions, perform standard finite element electromagnetic simulation analysis to obtain the electric field on the tetrahedral mesh obtained in step B.

[0010] D. Calculate the centroid coordinates of the four faces of the tetrahedral mesh based on the tetrahedral mesh node coordinates obtained in step B, and obtain the electric field at the centroid position based on the finite element interpolation obtained in step C.

[0011] E. Based on the centroid coordinates of the four faces of the tetrahedral mesh obtained in step D and the electric field at the centroid, the least squares method is used to obtain the first-order linear fitting expression for the electric field on the tetrahedral mesh.

[0012] F. Based on the tetrahedral mesh node coordinates and the first-order linear fitting expression for electric field reconstruction obtained in step E, the electric field recovery solution on the tetrahedral mesh nodes is calculated.

[0013] G. Based on the electric field recovery solution obtained in step F, perform posterior error estimation to obtain a tetrahedral mesh refinement indicator, and refine the tetrahedral mesh to obtain the coordinate information of the refined tetrahedral mesh and its nodes.

[0014] H. Repeat steps C through G until the calculation results of the finite element electromagnetic simulation analysis meet the accuracy requirements.

[0015] This invention first performs finite element modeling on the target microwave component, introducing boundary conditions and excitations to establish a corresponding electromagnetic simulation model. Then, a tetrahedral mesh is used to divide the solution domain, and finite element electromagnetic simulation analysis is performed on the model to calculate the electric field at the centroid of the four faces of the tetrahedral mesh. Using the electric field at the centroid, the least squares principle is used to refit the electric field on the tetrahedral mesh to reconstruct a first-order linear fitting expression. Next, the reconstructed fitting expression is used to calculate the recovered solution at the nodes, and combined with the electric field at the nodes, a posterior error is estimated. Finally, the mesh is refined based on the error on the tetrahedral mesh, and finite element electromagnetic simulation analysis is performed again after refinement until the finite element solution meets the accuracy requirements.

[0016] In summary, this invention employs the least squares method, enabling the construction of a recovered solution on a single tetrahedral mesh based solely on the finite element solution and the mesh's coordinate information, without requiring additional information. Furthermore, by using a single tetrahedron as the smallest implementation unit, data interaction with other tetrahedral meshes is unnecessary, resulting in relatively concentrated information. This avoids the need for special handling at boundaries and improves universality. The method of this invention is simple to implement in engineering, requires minimal computation, and has high universality. Attached Figure Description

[0017] Figure 1 This is a flowchart of the present invention;

[0018] Figure 2 This is a model diagram of an embodiment;

[0019] Figure 3 It is a tetrahedral mesh in the embodiment;

[0020] Figure 4 This is a comparison chart of the number of grids before and after the adaptive implementation of the example. Detailed Implementation

[0021] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0022] Reference Figure 1 A grid adaptive method based on restorative posterior error estimation includes the following steps:

[0023] A. Perform finite element modeling on the target microwave component, and introduce boundary conditions and excitations to establish the corresponding electromagnetic simulation model.

[0024] This invention takes the electromagnetic transmission analysis of a magic T as an example to establish, as follows: Figure 2 The model structure shown introduces four waveport excitations, and the model boundary is an ideal electric boundary.

[0025] B. The electromagnetic simulation model established in step A is divided into a tetrahedral mesh to solve the domain, and the coordinate information of the tetrahedral mesh nodes is obtained.

[0026] Using tetrahedral meshes to partition the computational domain is a well-known process in the finite element method, therefore this step will not be described in detail here. In Cartesian coordinates, the tetrahedral mesh nodes are as follows: Figure 3 The diagram shows four nodes: 1, 2, 3, and 4.

[0027] C. Using vector basis functions, perform standard finite element electromagnetic simulation analysis to obtain the electric field on the tetrahedral mesh obtained in step B.

[0028] The governing equations for this problem are vector wave equations, and the trial functions are first-order vector basis functions. Finite element electromagnetic simulation analysis is a well-known process and will not be elaborated upon here. Based on the edge-based finite element solution and vector basis functions, the vector electric field at any point on the tetrahedral mesh is obtained by interpolation as follows:

[0029]

[0030] in, It is the basis function of the j-th edge of the e-th tetrahedral mesh. It is the solution for the j-th edge of the e-th tetrahedral mesh, where x, y, and z are the three coordinate direction components of the Cartesian coordinate system.

[0031] D. Calculate the centroid coordinates of the four faces of the tetrahedral mesh based on the tetrahedral mesh node coordinates obtained in step B, and obtain the electric field at the centroid position based on the finite element interpolation obtained in step C.

[0032] like Figure 3 As shown, the four faces of the tetrahedron are named F1, F2, F3, and F4, where F1 is the face formed by connecting nodes 2, 3, and 4; F2 is the face formed by connecting nodes 1, 3, and 4; F3 is the face formed by connecting nodes 1, 2, and 4; and F4 is the face formed by connecting nodes 1, 2, and 3. The centroids of the four faces of the tetrahedron mesh are represented by points 5, 6, 7, and 8, where point 5 is the centroid of face F1, point 6 is the centroid of face F2, point 7 is the centroid of face F3, and point 8 is the centroid of face F4.

[0033] The centroid coordinates of tetrahedral mesh surface F1 are:

[0034] x5=(x2+x3+x4) / 3, y5=(y2+y3+y4) / 3, z5=(z2+z3+z4) / 3 (2)

[0035] The centroid coordinates of tetrahedral mesh surface F2 are:

[0036] x6=(x1+x3+x4) / 3, y6=(y1+y3+y4) / 3, z6=(z1+z3+z4) / 3 (3)

[0037] The centroid coordinates of tetrahedral mesh surface F3 are:

[0038] x7=(x1+x2+x4) / 3, y7=(y1+y2+y4) / 3, z7=(z1+z2+z4) / 3 (4)

[0039] The centroid coordinates of tetrahedral mesh surface F4 are:

[0040] x8=(x1+x2+x3) / 3, y8=(y1+y2+y3) / 3, z8=(z1+z2+z3) / 3 (5)

[0041] Substituting the coordinates of the centroid into formula (1), we obtain the electric fields at the centroids of the four faces F1, F2, F3, and F4 of the tetrahedron, respectively.

[0042] E. Based on the centroid coordinates of the four faces of the tetrahedral mesh obtained in step D and the electric field at the centroid, the least squares method is used to obtain the first-order linear fitting expression for the electric field on the tetrahedral mesh.

[0043] The electric field on the tetrahedral mesh is a vector, and the components in the x, y, and z directions are complex numbers. Therefore, it is necessary to perform first-order linear fitting on the components in the x, y, and z directions, as well as their real and imaginary parts, for a total of six components, to obtain the electric field reconstruction expression. The reconstruction expressions for these six components are constructed in the same way. Here, we will take the real part f of the x-direction component as an example for detailed explanation.

[0044] Construct a first-order linear fitting expression in the following format:

[0045] f = a + bx + cy + dz (6)

[0046] Where a, b, c, and d are coefficients to be determined, substituting the real part of the electric field x-direction component at the centroid of the four faces F1, F2, F3, and F4 of the tetrahedral mesh and the centroid coordinates into equation (6), we obtain the expression for the sum of squares of the residuals at the centroid points of the four faces of the tetrahedral mesh:

[0047]

[0048] Where f i Let represent the real part of the electric field x-direction component at the centroid of the four faces F1, F2, F3, and F4 of the tetrahedral mesh, and s be the sum of squared residuals of the fitted data.

[0049] To minimize the sum of squared residuals s, we have:

[0050]

[0051] Solving the system of equations (8) will yield the coefficients of the fitted expression (6).

[0052] F. Based on the tetrahedral mesh node coordinates and the first-order linear fitting expression for electric field reconstruction obtained in step E, the electric field recovery solution on the tetrahedral mesh nodes is calculated.

[0053] Based on the fitted expression (6) and the coordinates of the tetrahedral mesh nodes, the fitted electric fields (i.e., the recovered solutions) at nodes 1, 2, 3, and 4 are calculated. The superscript "*" indicates that the electric field is reconstructed, which is generally referred to as the recovered solution.

[0054] G. Based on the electric field recovery solution obtained in step F, perform posterior error estimation to obtain a tetrahedral mesh refinement indicator, and refine the tetrahedral mesh to obtain the coordinate information of the refined tetrahedral mesh and its nodes.

[0055] The electric fields at nodes 1, 2, 3, and 4 are calculated according to equation (1). Combined with recovery solution The posterior error norm ||ξ of the e-th tetrahedral mesh is obtained. e ||for

[0056]

[0057] Where || represents the modulo value, V e Let e ​​be the volume of the e-th tetrahedral mesh. Recover solutions at nodes 1, 2, 3, and 4. Components in the x, y, and z directions Electric fields at nodes 1, 2, 3, and 4 The components in the x, y, and z directions.

[0058] The posterior error of all tetrahedral meshes is calculated, and a mesh refinement strategy is used to determine the mesh refinement criteria, thereby obtaining the tetrahedral meshes that need to be refined, and the refinement indicator for the corresponding mesh is calculated. The meshes are then refined according to the refinement indicator to obtain the refined tetrahedral meshes and the coordinate information of their nodes. This is a well-known process, so this step will not be described in detail here.

[0059] H. Repeat steps C through G until the calculation results of the finite element electromagnetic simulation analysis meet the accuracy requirements.

[0060] Figure 4 The comparison between the mesh before and after adaptation in this embodiment is shown, and the results indicate that the present invention can achieve adaptive mesh encryption.

[0061] As can be seen from the above embodiments, this invention first obtains the finite element solution and coordinate information of a single tetrahedral mesh, and then combines it with the least squares method to construct the recovered solution on a single tetrahedral mesh without requiring additional information. Furthermore, the entire method uses a single tetrahedron as the smallest implementation unit, thus eliminating the need for data interaction with other tetrahedral meshes. This centralized information avoids the need for special handling at boundaries and improves universality. This invention has the advantages of simple engineering implementation, low computational cost, and high universality.

Claims

1. A grid adaptive method based on restorative posterior error estimation, characterized in that, Includes the following steps: A. Perform finite element modeling on the target microwave component, and introduce boundary conditions and excitations to establish the corresponding electromagnetic simulation model; B. The electromagnetic simulation model established in step A is divided into a tetrahedral mesh to obtain the coordinate information of the tetrahedral mesh nodes. C. Using vector basis functions, perform standard finite element electromagnetic simulation analysis to obtain the electric field on the tetrahedral mesh obtained in step B; D. Calculate the centroid coordinates of the four faces of the tetrahedral mesh based on the tetrahedral mesh node coordinates obtained in step B, and obtain the electric field at the centroid position based on the finite element interpolation obtained in step C. E. Based on the centroid coordinates of the four faces of the tetrahedral mesh obtained in step D and the electric field at the centroid, the least squares method is used to reconstruct the first-order linear fitting expression of the electric field on the tetrahedral mesh. F. Based on the tetrahedral mesh node coordinates and the first-order linear fitting expression for electric field reconstruction obtained in step E, the electric field recovery solution on the tetrahedral mesh nodes is calculated. G. Based on the electric field recovery solution obtained in step F, perform posterior error estimation to obtain a tetrahedral mesh refinement indicator, and refine the tetrahedral mesh to obtain the coordinate information of the refined tetrahedral mesh and its nodes; H. Repeat steps C through G until the calculation results of the finite element electromagnetic simulation analysis meet the accuracy requirements.

2. The grid adaptive method based on restorative posterior error estimation as described in claim 1, characterized in that: In step E, the electric field on the tetrahedral mesh is a vector, and the three coordinates of the Cartesian coordinate system are... x,y,z The component values ​​in each direction are complex numbers, for x,y,z The electric field reconstruction expression is obtained by performing first-order linear fitting on the components in the three directions, as well as the real and imaginary parts, totaling six components.

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