CN115392068A - Grid self-adaption method based on recovery type posterior error estimation - Google Patents
Grid self-adaption method based on recovery type posterior error estimation Download PDFInfo
- Publication number
- CN115392068A CN115392068A CN202210876687.XA CN202210876687A CN115392068A CN 115392068 A CN115392068 A CN 115392068A CN 202210876687 A CN202210876687 A CN 202210876687A CN 115392068 A CN115392068 A CN 115392068A
- Authority
- CN
- China
- Prior art keywords
- tetrahedral mesh
- electric field
- finite element
- tetrahedral
- mesh
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2111/00—Details relating to CAD techniques
- G06F2111/10—Numerical modelling
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention belongs to the field of electromagnetic field numerical solution, and particularly relates to a grid self-adaption method based on recovery type posterior error estimation. Firstly, carrying out finite element modeling on a target microwave component, introducing boundary conditions and excitation to establish a corresponding electromagnetic simulation model; then, a tetrahedral mesh subdivision solution domain is adopted, finite element electromagnetic simulation analysis is carried out on the model, electric fields at the gravity centers of four surfaces of the tetrahedral mesh are calculated, and a first-order linear fitting expression of electric field reconstruction on the tetrahedral mesh is re-fitted by adopting a least square principle; then, calculating a recovery solution at the node by using a reconstruction fitting expression, and carrying out posterior error estimation by combining an electric field at the node; and finally, encrypting the meshes according to errors on the tetrahedral meshes, and then performing finite element electromagnetic simulation analysis after encryption until the finite element solution meets the precision requirement. The method has the advantages of simple engineering realization, small calculated amount and high universality.
Description
Technical Field
The invention belongs to the field of electromagnetic field numerical solution, and particularly relates to a grid self-adaption method based on recovery type posterior error estimation.
Background
With the development of computer technology, it is an economic and effective means to analyze the electromagnetic field distribution of microwave components in advance by using simulation technology, so that it is an essential step in the product cycle to adopt electromagnetic simulation software to perform simulation calculation on the electromagnetic characteristics of microwave components in the design stage. For computer simulation, the solution accuracy is a problem of great concern, and improving the solution accuracy is also an important development direction of computer simulation. For finite element methods, generally, the smaller the mesh size of the model dispersion, the more accurate the simulation computation results. However, if the grid encryption is performed on some calculation areas only by experience, which is a scientific basis, many uncertain results are brought. Therefore, the grid adaptive encryption is proposed to solve the problem, the current grid is subjected to refined encryption or coarsening adjustment through the obtained posterior error estimation indicator, then the grid calculated in the next step is given, and whether the result is converged is judged through the adaptive error.
A posteriori error estimation is an important step in the adaptation. In error estimation, since an analytical solution is unknown, error calculation cannot be performed directly using the result, and error estimation by another means is necessary. The recovery type posterior error estimation method is a commonly used method. The method comprises the steps of firstly constructing a recovery solution to replace an analytic solution, then constructing a certain unit norm to be used as the measurement of an error, obtaining a grid encryption indication by utilizing the unit norm, and then carrying out grid encryption.
The core of the recovery type posterior error estimation method is the construction of a recovery solution. The traditional recovery solution construction method, such as the patch repair method SPR based on the super-convergence characteristic and the improved algorithm thereof, has the problems of large calculated amount, complex processing, special processing on the boundary and the like. Therefore, it is necessary to construct a recovery solution with small calculation amount, simple structure and universality for posterior error estimation to realize grid adaptive encryption.
Disclosure of Invention
Aiming at the existing problems, in order to solve the problems of large calculation amount, complex processing, low universality (special processing is needed on the boundary) and the like caused by the recovery solution structure of the existing grid self-adaptive encryption, the invention provides a grid self-adaptive method based on recovery type posterior error estimation, which constructs the recovery solution by using the principle of least square, completes the posterior error estimation and realizes the grid self-adaptive encryption.
A mesh self-adaption method based on recovery type posterior error estimation comprises the following steps:
A. finite element modeling is carried out on the target microwave component, boundary conditions are introduced, and a corresponding electromagnetic simulation model is established through excitation.
B. And D, solving a domain by adopting the tetrahedral mesh subdivision on the electromagnetic simulation model established in the step A to obtain coordinate information of the tetrahedral mesh nodes.
C. And D, performing standard finite element electromagnetic simulation analysis by adopting a vector basis function to obtain the electric field on the tetrahedral mesh obtained in the step B.
D. And C, calculating the gravity center coordinates of four surfaces of the tetrahedral mesh based on the node coordinates of the tetrahedral mesh obtained in the step B, and interpolating to obtain the electric field of the gravity center position based on the finite element solution obtained in the step C.
E. And D, based on the gravity center coordinates of the four surfaces of the tetrahedral mesh and the electric field at the gravity center obtained in the step D, obtaining an electric field reconstruction first-order linear fitting expression on the tetrahedral mesh by adopting a least square method.
F. And E, calculating to obtain an electric field recovery solution on the tetrahedral mesh nodes based on the tetrahedral mesh node coordinates and the electric field reconstruction first-order linear fitting expression obtained in the step E.
G. And F, carrying out posterior error estimation based on the electric field recovery solution obtained in the step F to obtain a tetrahedral mesh encryption indication, and encrypting the tetrahedral mesh to obtain the coordinate information of the encrypted tetrahedral mesh and the nodes thereof.
H. And D, repeating the step C to the step G until the calculation result of the finite element electromagnetic simulation analysis meets the precision requirement.
Firstly, carrying out finite element modeling on a target microwave component, introducing boundary conditions and excitation to establish a corresponding electromagnetic simulation model; then, a tetrahedral mesh subdivision solution domain is adopted, finite element electromagnetic simulation analysis is carried out on the model, electric fields at the gravity centers of four surfaces of the tetrahedral mesh are calculated, and a first-order linear fitting expression of electric field reconstruction on the tetrahedral mesh is re-fitted by utilizing the electric fields at the gravity centers and adopting the least square principle; then, calculating a recovery solution at the node by using a reconstruction fitting expression, and carrying out posterior error estimation by combining an electric field at the node; and finally, encrypting the meshes according to errors on the tetrahedral meshes, and then performing finite element electromagnetic simulation analysis after encryption until the finite element solution meets the precision requirement.
In summary, in the present invention, a structure of a recovery solution on a single tetrahedral mesh can be realized based on a finite element solution on the single tetrahedral mesh and coordinate information of the tetrahedral mesh by using a least square method, without additional information; and a single tetrahedron is used as a minimum implementation unit, and data interaction with other tetrahedral meshes is not required, so that the information is relatively centralized, the problem of special processing on the boundary is avoided, and the universality is improved. The method has the advantages of simple engineering realization, small calculated amount and high universality.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a model diagram of an embodiment;
FIG. 3 is a tetrahedral mesh of an embodiment;
FIG. 4 is a graph comparing the number of meshes before and after adaptation in the embodiment.
Detailed Description
The technical solution of the present invention is described in detail below with reference to the accompanying drawings and examples.
Referring to fig. 1, a mesh adaptation method based on a recovery type a posteriori error estimation includes the following steps:
A. finite element modeling is carried out on the target microwave component, and boundary conditions and excitation are introduced to establish a corresponding electromagnetic simulation model.
The embodiment of the invention takes the electromagnetic transmission analysis of magic T as an example, establishes a model structure as shown in figure 2, introduces 4 wave ports for excitation, and takes a model boundary as an ideal electrical boundary.
B. And D, solving a domain by adopting the tetrahedral mesh subdivision on the electromagnetic simulation model established in the step A to obtain coordinate information of the tetrahedral mesh nodes.
The use of a tetrahedral mesh subdivision computational domain model is a well known process in finite element methods and therefore this step is not described in detail, with the tetrahedral mesh nodes being represented as four nodes 1,2,3,4 in a cartesian coordinate system as shown in figure 3.
C. And D, performing standard finite element electromagnetic simulation analysis by adopting a vector basis function to obtain the electric field on the tetrahedral mesh obtained in the step B.
The control equation of the problem is a vector wave equation, and the heuristic function is a first-order vector basis function
Finite element electromagnetic simulation analysis is a well known process and will not be described in detail herein. Based on finite element solution and vector basis function of the edge, the vector electric field of any point on the tetrahedral mesh obtained by interpolation is as follows:
wherein the content of the first and second substances,
is the basis function of the jth edge of the e-th tetrahedral mesh,
is the solution of the j edge of the e-th tetrahedral mesh, and x, y and z are three coordinate direction components of a Cartesian coordinate system.
D. C, calculating gravity center coordinates of four surfaces of the tetrahedral mesh based on the node coordinates of the tetrahedral mesh obtained in the step B, and interpolating to obtain an electric field at the gravity center position based on the finite element solution obtained in the step C;
as shown in FIG. 3, four faces of the tetrahedron are respectively designated as F 1 ，F 2 ，F 3 ，F 4 In which F 1 Is a node2. 3,4 are linked, wherein F 2 Is a plane formed by connecting nodes 1, 3 and 4, wherein F 3 Is a plane formed by connecting nodes 1,2 and 4, wherein F 4 Is a plane formed by connecting the nodes 1,2 and 3. The center of gravity of four faces of the tetrahedral mesh is represented as 5,6,7,8 four points, where point 5 is face F 1 Point 6 is the face F 2 Point 7 is the face F 3 Point 8 is the plane F 4 The center of gravity of (1).
Tetrahedral mesh face F 1 Has a center of gravity of
x 5 ＝(x 2 +x 3 +x 4 )/3,y 5 ＝(y 2 +y 3 +y 4 )/3,z 5 ＝(z 2 +z 3 +z 4 )/3 (2)
Tetrahedral mesh face F 2 Has a center of gravity coordinate of
x 6 ＝(x 1 +x 3 +x 4 )/3,y 6 ＝(y 1 +y 3 +y 4 )/3,z 6 ＝(z 1 +z 3 +z 4 )/3 (3)
Tetrahedral mesh plane F 3 Has a center of gravity of
x 7 ＝(x 1 +x 2 +x 4 )/3,y 7 ＝(y 1 +y 2 +y 4 )/3,z 7 ＝(z 1 +z 2 +z 4 )/3 (4)
Tetrahedral mesh plane F 4 Has a center of gravity coordinate of
x 8 ＝(x 1 +x 2 +x 3 )/3,y 8 ＝(y 1 +y 2 +y 3 )/3,z 8 ＝(z 1 +z 2 +z 3 )/3 (5)
Substituting the barycentric coordinates into the formula (1) to obtain four surfaces F of the tetrahedron 1 ，F 2 ，F 3 ，F 4 The electric fields at the center of gravity are respectively
E. And D, based on the gravity center coordinates of the four surfaces of the tetrahedral mesh and the electric field at the gravity center, which are obtained in the step D, obtaining a first-order linear fitting expression of the electric field reconstruction on the tetrahedral mesh by adopting a least square method.
The electric field on the tetrahedral mesh is a vector, and the component values in each direction of x, y and z are complex numbers, so that first-order linear fitting needs to be performed on the components in the three directions of x, y and z and 6 total components of the real part and the imaginary part of the components respectively to obtain an electric field reconstruction expression. The method of constructing the reconstruction expression of these 6 components is the same, and here, the real part f of the x-direction component is taken as an example to describe in detail.
Constructing a first order linear fit expression of the following format
f＝a+bx+cy+dz (6)
Wherein a, b, c and d are coefficients to be solved, and four surfaces F of the tetrahedral mesh 1 ，F 2 ，F 3 ，F 4 Substituting the real part value of the x-direction component of the electric field at the gravity center and the gravity center coordinate into the formula (6) to obtain the residual square sum of gravity center points of the expression at four surfaces of the tetrahedral mesh as:
wherein f is i Representing four faces F of a tetrahedral mesh 1 ，F 2 ，F 3 ，F 4 The value of the real part of the x-direction component of the electric field at the center of gravity, s is the sum of the squares of the residuals of the fitted data.
The sum of squared residuals s is minimized, i.e.:
and solving the equation system (8) to obtain the coefficient of the fitting expression (6).
F. And E, calculating to obtain an electric field recovery solution on the tetrahedral mesh nodes based on the tetrahedral mesh node coordinates and the electric field reconstruction first-order linear fitting expression obtained in the step E.
Calculating the fitting electric field (i.e. recovery solution) at the nodes 1,2,3,4 according to the fitting expression (6) and the coordinates at the nodes of the tetrahedral mesh
Where the superscript "indicates that the electric field is reconstructed, it is generally referred to as a recovery solution.
G. And F, carrying out posterior error estimation based on the electric field recovery solution obtained in the step F to obtain a tetrahedral mesh encryption indication, and encrypting the tetrahedral mesh to obtain the coordinate information of the encrypted tetrahedral mesh and the nodes thereof.
Calculating the electric field at the nodes 1,2,3 and 4 according to the formula (1)
Binding recovery solution
Obtaining the e-th tetrahedral grid posterior error norm [ xi ] e I is
Where | | | denotes a modulus value, V e Is the volume of the e-th tetrahedral mesh,
for the recovery of the solution at node 1,2,3,4
The components in the x, y, z directions,
is an electric field at a node 1,2,3,4
The components in the x, y, z directions.
Calculating the posterior error of all tetrahedral meshes, adopting a mesh encryption strategy to carry out mesh encryption judgment to obtain the tetrahedral mesh needing to be encrypted, and calculating the encryption indication of the corresponding mesh. The encryption of a mesh based on the encryption indication to obtain coordinate information of the encrypted tetrahedral mesh and its nodes is a well-known process, and therefore this step is not described in detail.
H. And D, repeating the step C to the step G until the calculation result of the finite element electromagnetic simulation analysis meets the precision requirement.
Fig. 4 shows a comparison of the mesh before and after the adaptation in this embodiment, and the result shows that the invention can implement mesh adaptive encryption.
According to the embodiment, the finite element solution on the single tetrahedral mesh and the coordinate information of the tetrahedral mesh are firstly obtained, and the structure of the recovery solution on the single tetrahedral mesh can be realized by combining the least square method without additional information; in addition, a single tetrahedron is used as a minimum implementation unit in the whole method, so that data interaction with other tetrahedron meshes is not needed, information is relatively centralized, the problem that special processing is needed on the boundary is avoided, and universality is improved. The method has the advantages of simple engineering realization, small calculated amount and high universality.
Claims (2)
1. A mesh adaptive method based on recovery type posterior error estimation is characterized by comprising the following steps:
A. finite element modeling is carried out on a target microwave component, boundary conditions and excitation are introduced, and a corresponding electromagnetic simulation model is established;
B. b, solving a domain of the electromagnetic simulation model established in the step A by adopting tetrahedral mesh subdivision to obtain coordinate information of nodes of the tetrahedral mesh;
C. b, performing standard finite element electromagnetic simulation analysis by adopting a vector basis function to obtain an electric field on the tetrahedral mesh obtained in the step B;
D. c, calculating gravity center coordinates of four surfaces of the tetrahedral mesh based on the node coordinates of the tetrahedral mesh obtained in the step B, and interpolating to obtain an electric field at the gravity center position based on the finite element solution obtained in the step C;
E. d, based on the gravity center coordinates of the four surfaces of the tetrahedral mesh and the electric field at the gravity center, which are obtained in the step D, obtaining an electric field reconstruction first-order linear fitting expression on the tetrahedral mesh by adopting a least square method;
F. e, calculating to obtain an electric field recovery solution on the tetrahedral mesh nodes based on the tetrahedral mesh node coordinates and the electric field reconstruction first-order linear fitting expression obtained in the step E;
G. f, performing posterior error estimation based on the electric field recovery solution obtained in the step F to obtain a tetrahedral mesh encryption indication, and encrypting the tetrahedral mesh to obtain the coordinate information of the encrypted tetrahedral mesh and the nodes thereof;
H. and D, repeating the step C to the step G until the calculation result of the finite element electromagnetic simulation analysis meets the precision requirement.
2. The recovery-based a posteriori error estimation trellis adaptation method of claim 1 wherein: in the step E, the electric field on the tetrahedral mesh is a vector, and component values in each direction of the three coordinates x, y, and z of the cartesian coordinate system are complex, so that first-order linear fitting needs to be performed on the components in the three directions of x, y, and z and 6 total components of the real part and the imaginary part thereof, respectively, to obtain an electric field reconstruction expression.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210876687.XA CN115392068A (en) | 2022-07-25 | 2022-07-25 | Grid self-adaption method based on recovery type posterior error estimation |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210876687.XA CN115392068A (en) | 2022-07-25 | 2022-07-25 | Grid self-adaption method based on recovery type posterior error estimation |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN115392068A true CN115392068A (en) | 2022-11-25 |
Family
ID=84117236
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202210876687.XA Pending CN115392068A (en) | 2022-07-25 | 2022-07-25 | Grid self-adaption method based on recovery type posterior error estimation |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN115392068A (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN115935671A (en) * | 2022-12-20 | 2023-04-07 | 安徽大学 | Regional decomposition electromagnetic simulation method and system |
-
2022
- 2022-07-25 CN CN202210876687.XA patent/CN115392068A/en active Pending
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN115935671A (en) * | 2022-12-20 | 2023-04-07 | 安徽大学 | Regional decomposition electromagnetic simulation method and system |
| CN115935671B (en) * | 2022-12-20 | 2023-08-18 | 安徽大学 | Regional decomposition electromagnetic simulation method and system |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Yu et al. | Treatment of geometric singularities in implicit solvent models | |
| Zhang et al. | A novel training approach for parametric modeling of microwave passive components using Padé via Lanczos and EM sensitivities | |
| CN111859795B (en) | Polynomial aided neural network behavior modeling system and method for power amplifier | |
| CN113158527B (en) | Method for calculating frequency domain electromagnetic field based on implicit FVFD | |
| CN109033021B (en) | Design method of linear equation solver based on variable parameter convergence neural network | |
| Dault et al. | The generalized method of moments for electromagnetic boundary integral equations | |
| CN111541255B (en) | Low-frequency oscillation mode identification method and system based on dynamic system | |
| Koziel et al. | Space-mapping-based interpolation for engineering optimization | |
| CN116451584B (en) | Thermal stress prediction method and system based on neural network | |
| CN115392068A (en) | Grid self-adaption method based on recovery type posterior error estimation | |
| Hendrickx et al. | Sequential design and rational metamodelling | |
| CN117374985A (en) | Quantum algorithm of optimal power flow problem of power system and device comprising quantum algorithm | |
| CN111625946B (en) | Output signal mode component decomposition method for three-dimensional conformal full-electromagnetic particle simulation | |
| CN115130340A (en) | Pipeline modeling method based on fractional Brownian motion | |
| Leszczynska et al. | Fast full-wave multilevel zero-pole optimization of microwave filters | |
| Goussard et al. | An improvement of the Lee and Schetzen cross-correlation method | |
| Gál et al. | Modified bacterial memetic algorithm used for fuzzy rule base extraction | |
| CN112182739A (en) | Aircraft structure non-probability credible reliability topological optimization design method | |
| CN117528573B (en) | Wireless network optimization method, device, equipment and computer program product | |
| Peik et al. | A novel design approach for microwave planar filters | |
| CN117610386A (en) | High-frequency electromagnetic field simulation method, system and computer based on finite volume method | |
| TWI783713B (en) | Method and system for building dipole moment model | |
| Schulz | LPV system identification and order reduction with application to turbocharged SI engines | |
| CN116663379A (en) | Operation system for reconstructing three-dimensional component of PCB radiation magnetic field | |
| Antonova et al. | Recognition of Nonlinear Functions in τ LP-Search with averaging |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination |