CN111597744A - Rapid frequency sweep simulation method based on region decomposition - Google Patents

Rapid frequency sweep simulation method based on region decomposition Download PDF

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CN111597744A
CN111597744A CN202010402451.3A CN202010402451A CN111597744A CN 111597744 A CN111597744 A CN 111597744A CN 202010402451 A CN202010402451 A CN 202010402451A CN 111597744 A CN111597744 A CN 111597744A
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宛汀
唐佳东
朱剑
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a rapid sweep frequency simulation method based on region decomposition, which comprises the steps of firstly establishing a model of a microwave device, dividing the whole region to be solved into a plurality of non-overlapping sub-regions, and dispersing each sub-region model by utilizing a tetrahedral mesh; then, constructing a finite element coefficient matrix which does not contain frequency items and corresponds to the sub-domains according to the vector wave equation and the boundary condition, butting adjacent sub-domains to construct a Krylov subspace of the global finite element system, and reducing the coefficient matrix which does not contain the frequency items and the right vector by utilizing the constructed Krylov subspace; further, adding a frequency term to form a finite element matrix equation after reduction, and rapidly solving the small-scale matrix equation by adopting a direct solver; and finally, rapidly recovering the subfield electric field values on frequency points of a frequency sweeping interval by using a Krylov subspace, thereby completing frequency sweeping calculation.

Description

Rapid frequency sweep simulation method based on region decomposition
Technical Field
The invention relates to the field of rapid frequency sweep simulation in microwave engineering, in particular to a rapid frequency sweep simulation method based on area decomposition.
Background
In the field of microwave engineering, accurate electromagnetic simulation of microwave devices with large electrical size and complex geometric structure in a wide frequency band is an inevitable task. When a device model is swept by a traditional finite element method, each frequency point needs to be modeled again and solved repeatedly, when the size of the model is too complex or the requirement on subdivision precision is too high, unknown quantity is too much, the requirement on the internal memory of a computer is often not met by adopting a direct solution, and the convergence can be realized by adopting an iterative solution in a long time.
The predecessor combines the model reduction method into the finite element method, establishes the finite element model reduction method, converts the repeatedly solved original model matrix equation into the repeatedly solved reduced small matrix equation, and greatly improves the frequency sweeping efficiency. However, the essence of order reduction is that the matrix with the same scale as the original finite element matrix is solved for a plurality of times in a circulating way, so that when a model with large unknown quantity and complex structure is encountered, the direct solution cannot be competent for the task of order reduction, and only iterative solution can be adopted, so that the order reduction time is very slow.
Compared with the traditional finite element method, the area decomposition method is a method which is very suitable for being applied to large-scale complex structures, particularly periodic structures, to carry out large-scale numerical calculation. The algorithm decomposes the region to be solved into a plurality of sub-domains, carries out independent calculation in each sub-domain, carries out iterative transmission operation between the sub-domains through boundary continuity conditions, converts the solution of the original large-scale matrix into independent solutions of a plurality of sub-domains, and has the advantages of reduced calculation scale, high preprocessing iteration efficiency and the like, so the solution speed is very high.
In order to enable the finite element model reduction method to be suitable for more structures, a region decomposition method and the finite element model reduction method are combined, the efficiency of reducing large-scale complex structures, particularly periodic structure models, is greatly improved, the method is wider in adaptability compared with a simple finite element model reduction method, and has the characteristics of higher frequency sweeping speed and controllable time precision compared with a traditional finite element method.
Disclosure of Invention
The invention aims to provide a rapid frequency sweep simulation method based on area decomposition. The method utilizes the characteristic that the adjacent subdomains of the conformal partition surface have the common edge information and can be mutually transmitted between the adjacent subdomains to carry out order reduction operation on the finite element model, ensures high result precision, has higher speed compared with the traditional finite element method frequency sweeping, and improves the efficiency of model order reduction due to the adoption of the area decomposition technology, thereby leading the finite element model to have wider order reduction adaptability.
The technical solution for realizing the invention is as follows: a rapid frequency sweep simulation method based on regional decomposition comprises the following steps:
the first step is as follows: the model to be simulated is divided into a plurality of non-overlapping sub-regions, so that a large-scale complex problem is divided into a series of small-scale simple problems, and the solving efficiency is improved conveniently. When dividing the subdomains, the size of each zone needs to be ensured to be the same as much as possible to obtain the optimal solution efficiency, and the size of the subdomains does not exceed half of the minimum sweep wavelength, so as to obtain a subdomain finite element matrix with good performance. Discretizing the model with tetrahedral units;
the second step is that: establishing an electric field vector wave equation aiming at each sub-domain, introducing an electromagnetic field transmission condition on the interface of each sub-domain, constructing a finite element coefficient matrix equation of each sub-domain under a frequency expansion point by using a Whitney basis function, and solving an electric field unknown vector related to an edge in each sub-domain;
the third step: vector information about edges is transmitted between adjacent subdomains, and each column of a Krylov subspace of the global finite element system is constructed under the frequency expansion point;
the fourth step: reducing the finite element coefficient matrix of the global finite element system without the frequency item by using the constructed Krylov subspace;
the fifth step: adding the reduced finite element matrix equation into each frequency point in the frequency interval corresponding to the frequency expansion point, solving by adopting a direct solver, and sequentially recovering the global electric field value of the edge on each frequency point in the frequency band range corresponding to the frequency expansion point by utilizing a Krylov subspace.
Furthermore, in the second step, the number of frequency expansion points is determined according to the requirements of precision and efficiency, if k expansion points are set, the sweep frequency range is equally divided into k intervals, and each expansion point is selected as the middle point of the interval, so that the precision is consistent in the equal-width expansion range.
Further, in the third step, in the order reduction process of the finite element model, each column forming the Krylov subspace involves an inversion of a matrix of the same scale as the original finite element matrix, and the flow of the specific matrix inversion is as follows:
firstly, solving unknown vectors of each sub-domain by adopting a region decomposition method; then, transmitting exchange information between adjacent sub-domains, combining unknown vectors of each sub-domain into a global vector and storing the global vector into a corresponding column of a Krylov subspace; if the next column needs to be solved, the current column of the Krylov subspace is separated according to the edge number of the sub-domains and sent to each sub-domain, new equation right vectors of each sub-domain are constructed through vector operation of each sub-domain, under the condition that the coefficient matrix is not changed, new unknown vectors of each sub-domain are solved through a region decomposition method, the next column of global vectors are combined through transmission of each sub-domain, and the Krylov subspace of the global finite element system can be constructed sequentially.
Compared with the prior art, the invention has the following remarkable effects: (1) the efficiency of the rapid frequency sweep simulation of the microwave device is greatly improved, and the efficiency is higher compared with that of the traditional method; (2) the method is easy to realize on a parallel computing platform, can greatly improve the frequency sweep simulation efficiency by utilizing a distributed parallel computer, and is particularly suitable for large-scale complex microwave device frequency sweep simulation; (3) the sweep frequency simulation time and precision have good controllability, and the sweep frequency time can be shortened on the premise of ensuring the requirement on precision by setting the dimensionality of the reduced-order model.
Drawings
FIG. 1 is an overall process flow diagram of the method of the present invention;
fig. 2 is a schematic diagram of a specific structure of a simulation object waveguide filter in embodiment 1 of the present invention;
FIG. 3 is a comparison of S-parameter curves calculated in 10-15GHz band by the method of the present invention and the conventional method in example 1 of the present invention.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
FIG. 1 shows the main flow of the method of the present invention, and the steps of the method of the present invention are further described in detail with reference to FIG. 1:
s1: dividing a model to be simulated into a plurality of non-overlapping sub-regions, and ensuring the same size of each sub-region as much as possible during division so as to ensure load balance among different processes during parallel computing; the whole model is subdivided by using the tetrahedral mesh units, so that the subdivision density of each area is reasonable and uniform;
s2: sequencing global edges in sequence according to the sequence of area numbers and sending the edge information of each area to each sub-area, so that each sub-area has all the edge information of the corresponding sub-area, and adjacent sub-areas are connected through Dirichlet boundary conditions and impedance boundary conditions;
s3: each subdomain uses an electric field vector wave equation as a control equation, and the Whitney basis function is used for establishing S corresponding to the edge of the areaiMatrix and T without frequency termiMatrix, using Dirichlet boundary conditions to construct right vector B without frequency terms in each region for solving model excitationi；
S3.1: establishing S of corresponding region edge through Whitney basis functioniMatrix and T without frequency termiThe key to the matrix is as follows:
Figure BDA0002490014450000031
Figure BDA0002490014450000032
wherein nprog represents the number of divided regions; i represents a region number;
Figure BDA0002490014450000033
the total number of tetrahedrons divided for the ith sub-domain;
Figure BDA0002490014450000034
Figure BDA0002490014450000035
Figure BDA0002490014450000036
and
Figure BDA0002490014450000037
are respectively a cell matrix
Figure BDA0002490014450000038
And
Figure BDA0002490014450000039
m and n are the serial numbers of hexahedron edges;
s3.2: construction of right vector B without frequency terms in each region by using Dirichlet boundary conditions on solving model excitationiThe key points are as follows:
Figure BDA00024900144500000310
Figure BDA00024900144500000311
wherein the content of the first and second substances,
Figure BDA00024900144500000312
Eincrepresents an excitation source;
s4: by Si,Ti,BiSelecting a proper frequency expansion point, constructing a correlation coefficient matrix and a right vector of each subdomain under the frequency expansion point, and solving unknown vectors of edges in each subdomain by adopting a region decomposition iterative solver.
S4.1: if order
s＝jk0(S4.1.1)
The coefficient matrix for subfield i can be expressed as:
Ai＝Si+s2Ti(S4.1.2)
the right vector for subfield i can be assigned as:
bi＝s·Bi(S4.1.3)
s4.2: according to the tearing and butting algorithm in the region decomposition method, a Dirichlet boundary condition is utilized to connect linear systems in each sub-domain, and the following forms are adopted:
AD·xD＝bD(S4.2.1)
wherein A isD,xD,bDRespectively in the form of blocks:
Figure BDA0002490014450000041
Figure BDA0002490014450000042
Figure BDA0002490014450000043
in which the edges inside each sub-region and the edges adjoining the two sub-regions at the boundary are indicated by the superscript r and the edges at the corners adjoining the plurality of sub-regions are indicated by the superscript c, and thus in which formula (S4.2.2) the edges and the edges adjoining the two sub-regions are indicated by the superscript c
Figure BDA0002490014450000044
The matrix is shown as a matrix relating edges inside and at the boundaries of the sub-field i, i.e. A in equation (S4.1.2)i，
Figure BDA0002490014450000045
Is a matrix associated with the edges at the corners,
Figure BDA0002490014450000046
indicating the coupling between them, and determining the edge information of each subfield
Figure BDA0002490014450000047
Matrix array
Figure BDA0002490014450000048
Referred to as a boolean matrix, which is,
Figure BDA0002490014450000049
showing sonMapping of the edge on the interface of field i within the entire subfield to the number on the interface, e.g. the number of an edge on the interface is a and the number in subfield i is b, then
Figure BDA00024900144500000410
Otherwise
Figure BDA00024900144500000411
Figure BDA00024900144500000412
The same process is carried out;
Figure BDA00024900144500000413
representing the projection of the corner-related matrix element within the sub-field i over the entire set of corners. In formula (S4.2.4)
Figure BDA00024900144500000414
The right vector representing the subfield i, i.e. b in equation (S4.1.3)i. In formula (S4.2.3)
Figure BDA00024900144500000415
I electric field values of the edges inside the subdomain and the edges on the interface, and λ is an unknown vector defined on the subdomain interface, jointly solving the participation matrix equation. Thus, in AiKnowing that the vector on the right and the Boolean matrix are known, only xDIs the only unknown vector.
The field value on the interface, i.e. lambda, can be determined by using a region decomposition iterative solver and then each subdomain can be determined
Figure BDA00024900144500000416
And then each sub-field
Figure BDA00024900144500000417
Eliminating the remaining sorted edge field values except the i-1 th sub-field interface edge field value, and finally forming a real global vector x to be solved by transmission combination among the sub-fieldsNThe concrete form is as follows:
Figure BDA00024900144500000418
xN＝Gather(xN+2) (S4.2.6)
all edges are sorted according to the area numbers in the previous step, so that on the premise that the interface is conformal, two adjacent subdomains have the same edge sequence on the interface, and the value can be directly transmitted as long as the corresponding relation of the two areas on the upper side of the dividing plane is found. In the formula
Figure BDA00024900144500000419
Indicating the remaining sorted edge field values except the edge field value of the interface with the (i-1) th sub-field,
Figure BDA0002490014450000051
is the global candidate vector we really require. Also, the following steps are performed:
Figure BDA0002490014450000052
bN+2＝Split(xN) (S4.2.8)
wherein b isN+2Is a new right vector b, a new matrix equation can be formed by substituting equation (S4.2.1).
S5: based on the characteristic that edges on the interface of the conformal grid are public edges, edge information of adjacent subfields can be combined into global edge information by utilizing transmission, receiving and sending of the corresponding subfields, and each column of cyclic iteration realizes the construction of a global finite element system Krylov subspace under the frequency expansion point;
s5.1: linearizing the global finite element system as a frequency dependent only linear system, the following form can be obtained:
Figure BDA0002490014450000053
wherein
Figure BDA0002490014450000054
Figure BDA0002490014450000055
s0Representing the frequency spread points, K and s0b represents a new coefficient matrix with the dimension of N and a right vector formed after the frequency expansion point is added. A. the2NA matrix of dimension 2N is shown in relation to the K coefficient matrix. Sigma is a frequency-dependent infinitesimal quantity, the unknown vector y is related to the vector x to be solved, and only a temporary column does not need to be solved;
s5.2: the specific implementation flow for constructing the Krylov subspace (dimension [ N × q ]) of the global finite element system is as follows:
input quantity: frequency expansion point s0S with frequency terms not included in each sub-fieldi,Ti,BiS, T, b globally without frequency term;
(1) using frequency expansion points s0Forming a coefficient matrix K for each sub-fieldiAnd the right vector biSolving x by using a region decomposition iterative solverN+2
(2) Let p be0＝Gather(Bi)；q0＝xN＝Gather(xN+2)；
(3)β＝||q0||
(4)
Figure BDA0002490014450000056
(5)for(j＝1；j＜q；j++)
(6)
Figure BDA0002490014450000057
(7)for(i＝1；i＜j+1；i++)
(8)
Figure BDA0002490014450000061
(9)
Figure BDA0002490014450000062
(10)endfor
(11)
Figure BDA0002490014450000063
(12)if(hj+1,j≡0)
(13)breakdown
(14)
Figure BDA0002490014450000064
(15)endfor
(16)VN×q＝{q1,q2,q3,…,qq}
Wherein, will open up
Figure BDA0002490014450000065
The back formula substitutes the front formula with:
qj+1＝-K-1(Tqj-1+2s0Tqj) (S5.2.2)
to make equation (S5.2.2) suitable for solving the area decomposition solver, the specific steps are as follows:
(1)bD＝Split(Tqj-1+2s0Tqj)
(2)
Figure BDA0002490014450000066
(3)qj+1＝Gather(xD)
s6: and reducing the order of the corresponding coefficient matrix which does not contain the frequency item on the whole and the right vector by using the constructed Krylov subspace. And circularly sweeping, adding a frequency item to form a finite element matrix equation after reduction, solving by adopting a direct solver, and sequentially recovering the global electric field value on the frequency point by utilizing a Krylov subspace.
Example 1
The method of the invention is adopted to carry out simulation analysis on a waveguide filter, the interior of the waveguide is not filled with media, and except two ports, the other outer surfaces are all ideal metal surfaces. The model sizes are respectively: 19.05mm for a, 9.525mm for b, 2.86mm for c, 2.01mm for d, 18.9mm for h and 17.23mm for L, as shown in fig. 2.
Dividing the whole waveguide filter model into three subdomains, distributing a process on a simulation computer for each subdomain, applying incident waves of a TE10 mode to the left end, and adding a Perfect Matching Layer (PML) with the maximum wavelength thickness of 1/4 to the right end so as to achieve the purpose of complete absorption and no reflection; setting the subdivision accuracy to 1/15 minimum wavelength; the sweep frequency range is 10 GHz-15 GHz, 200 frequency points are set, and the spacing is 25 MHz;
respectively calculating S parameters, namely scattering parameters of the model at 10 GHz-15 GHz by using the method and a traditional finite element model reduction method, wherein the traditional method specifically adopts an iterative solution to solve;
as shown in fig. 3, the S parameter of the waveguide filter calculated by the method of the present invention and the conventional method is given, where S21 is the forward transmission coefficient, i.e. gain, and S11 is the input reflection coefficient, i.e. input return loss, which are not easy to see, and the two are well matched, thus proving the correctness of the method of the present invention;
as can be seen from table 1 below, the solving efficiency of the conventional method is different from that of the method of the present invention, and under the condition of solving the same model and the number of unknowns, the method of the present invention only needs 1658 seconds, while the conventional method needs 30398 seconds, and the comparison shows that the solving efficiency of the present invention is 18.5 times higher than that of the conventional method, thereby proving the high efficiency of the method of the present invention compared with the conventional method.
Table 1:
method of producing a composite material Unknown quantity Number of expansion points Number of rows of V matrix Order-reducing time (seconds) Total solution time (seconds)
Conventional methods 42967 2 20 30384 30398
The method of the invention 42967 2 20 1641 1658

Claims (5)

1. A rapid frequency sweep simulation method based on area decomposition is characterized by comprising the following steps:
the first step is as follows: dividing a model to be simulated into a plurality of non-overlapping sub-areas, ensuring that the size of each area does not exceed half of the minimum sweep wavelength, and dispersing the model by utilizing a tetrahedron unit;
the second step is that: establishing an electric field vector wave equation aiming at each sub-domain, introducing an electromagnetic field transmission condition on the interface of each sub-domain, constructing a finite element coefficient matrix equation of each sub-domain under a frequency expansion point by using a Whitney basis function, and solving an electric field unknown vector related to an edge in each sub-domain;
the third step: vector information about edges is transmitted between adjacent subdomains, and each column of a Krylov subspace of the global finite element system is constructed under the frequency expansion point;
the fourth step: reducing the finite element coefficient matrix of the global finite element system without the frequency item by using the constructed Krylov subspace;
the fifth step: adding the reduced finite element matrix equation into each frequency point in the frequency interval corresponding to the frequency expansion point, solving by adopting a direct solver, and sequentially recovering the global electric field value of the edge on each frequency point in the frequency band range corresponding to the frequency expansion point by utilizing a Krylov subspace.
2. A fast frequency sweep simulation method based on region decomposition as claimed in claim 1, characterized in that in the second step, the number of frequency expansion points is determined according to the requirements of accuracy and efficiency, if k expansion points are set, the frequency sweep range is equally divided into k intervals, and each expansion point is selected as the midpoint of the interval to ensure the accuracy is consistent in the equal width expansion range.
3. A method for fast frequency sweep simulation based on region decomposition as claimed in claim 1, wherein in the third step, during the finite element model downscaling, each column forming Krylov subspace involves inverting a matrix of the same size as the original finite element matrix.
4. A fast frequency sweep simulation method based on area decomposition as claimed in claim 3, characterized in that the flow of specific matrix inversion is as follows:
firstly, solving unknown vectors of each sub-domain by adopting a region decomposition method; then, transmitting exchange information between adjacent sub-domains, combining unknown vectors of each sub-domain into a global vector and storing the global vector into a corresponding column of a Krylov subspace; if the next column needs to be solved, the current column of the Krylov subspace is separated according to the edge number of the sub-domains and sent to each sub-domain, new equation right vectors of each sub-domain are constructed through vector operation of each sub-domain, under the condition that the coefficient matrix is not changed, new unknown vectors of each sub-domain are solved through a region decomposition method, the next column of global vectors are combined through transmission of each sub-domain, and the Krylov subspace of the global finite element system can be constructed sequentially.
5. A fast frequency sweep simulation method based on region decomposition as claimed in any one of claims 1 to 4, characterized in that in the second step the sub-regions are connected by boundary conditions in a specific way of Dirichley boundary conditions or Robin boundary conditions.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113887102A (en) * 2021-09-30 2022-01-04 北京智芯仿真科技有限公司 Full-wave electromagnetic simulation method and system for integrated circuit under lossless frequency dispersion medium
CN115935671A (en) * 2022-12-20 2023-04-07 安徽大学 Regional decomposition electromagnetic simulation method and system

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113887102A (en) * 2021-09-30 2022-01-04 北京智芯仿真科技有限公司 Full-wave electromagnetic simulation method and system for integrated circuit under lossless frequency dispersion medium
CN113887102B (en) * 2021-09-30 2022-03-11 北京智芯仿真科技有限公司 Full-wave electromagnetic simulation method and system for integrated circuit under lossless frequency dispersion medium
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

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