CN107526869B - Numerical method for reducing order of input/output window model of adaptive three-dimensional microwave tube based on function approximation - Google Patents

Numerical method for reducing order of input/output window model of adaptive three-dimensional microwave tube based on function approximation Download PDF

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CN107526869B
CN107526869B CN201710584032.4A CN201710584032A CN107526869B CN 107526869 B CN107526869 B CN 107526869B CN 201710584032 A CN201710584032 A CN 201710584032A CN 107526869 B CN107526869 B CN 107526869B
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徐立
李星
杨中海
李斌
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University of Electronic Science and Technology of China
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Abstract

The invention belongs to the field of computational electromagnetism numerical value solving, and provides a numerical value method for reducing an order of a three-dimensional microwave tube input/output window model based on function approximation self-adaptive error analysis.

Description

Numerical method for reducing order of input/output window model of adaptive three-dimensional microwave tube based on function approximation
Technical Field
The invention belongs to the field of computational electromagnetic numerical solution, and relates to a numerical method for reducing an input/output window model of a three-dimensional microwave tube based on function approximation adaptive error analysis.
Background
In a microwave system, an input/output window is the key for energy coupling between a microwave tube and the microwave system, and the performance of the input/output window directly affects the bandwidth, reliability, transduction efficiency, service life and the like of a device. With the rapid development of modern computer technology, microwave tubes are continuously advancing in the directions of higher frequency, higher power, wider frequency band and the like in a plurality of fields such as military application and the like, so that the requirements on the simulation design of the input and output structure of the microwave tube are stricter.
In order to accurately obtain the electromagnetic response characteristics of the microwave tube and calculate the S-parameters describing the transmission performance of the microwave tube, the experimental method is usually limited due to the problems of complex model, high cost, difficult operation, time consumption and the like, and the theoretical analysis cannot obtain an analytic solution due to the complexity and changeability of the actual environment, so that a numerical calculation method and simulation software are required to obtain a reliable result in most electromagnetic field problems. Considering that the S parameter is solved by adopting the discrete sweep frequency only, the matrix equation can be solved repeatedly at a plurality of discrete frequency points, so that the final calculated amount is large and the time consumption is long. Therefore, a great deal of literature researches a model order-reduced fast frequency sweeping numerical method, and aims to find some inherent properties or structures which can reduce the scale of an original system and keep the original problems, so that the calculation of a plurality of frequency points can be rapidly processed. The main process of solving the S parameter of the microwave tube by the method is to perform three-dimensional eigen analysis on the microwave tube by a vector finite element method, convert a complex electromagnetic problem into a large-scale mathematical matrix function equation, perform a series of order reduction processing on the equation, and perform post-processing solution to obtain the final S parameter.
Most of the existing model order reduction methods used in the rapid frequency sweep adopt Taylor series expansion, and relate to derivation operation of matrix functions, for nonlinear large matrix function equations in complex electromagnetic environments, the difficulty of linearization or pretreatment in the order reduction process is increased, and pathological matrixes are possibly generated, so that a large amount of memory and time are consumed, and the frequency bandwidth is narrower; on the other hand, most of the existing fast frequency sweeping technologies require a user to participate in setting at any time, the S parameter obtained only through one expanded frequency point cannot obtain accurate response, but the calculation time is increased by adopting a plurality of frequency point information, and the self-adaptive effect is obviously reduced. Therefore, the model order reduction technology in the fast sweep finite element analysis needs to be constructed by a stable and reliable numerical method.
Disclosure of Invention
Aiming at the problems or the defects, the invention provides a three-dimensional microwave tube input and output window model order reduction numerical method based on function approximation adaptive error analysis, which aims to solve the problems or the defects that in the existing electromagnetic model order reduction numerical method, the time consumption and the complexity of matrix function derivation operation are caused by Taylor series expansion, and the self-adaptive performance of calculation is improved. The method comprises the steps of carrying out Chebyshev function approximation on a finite element matrix obtained by vector finite element eigen analysis, adaptively selecting interpolation points and dividing a convergence interval of a full frequency band through error analysis of characteristic values to obtain an expansion subspace of a reduced order model, and accordingly, the full frequency band adaptive optimization simulation of the microwave tube is achieved.
In order to achieve the purpose, the numerical method for reducing the order of the input and output window model of the adaptive three-dimensional microwave tube based on function approximation is characterized by comprising the following steps of:
a, according to the physical structure and the working frequency range of a target electronic device, carrying out simulation modeling on the target electronic device;
b, solving a domain by adopting tetrahedral mesh dispersion; the universe Ω is divided into M subdomains using tetrahedral mesh discretization, denoted as: omegae(e-1, 2,3, …, M), the subfields being referred to generically hereinafter as a unit;
c, selecting a finite element vector basis function, and establishing a matrix equation of the eigenvalue problem: a (f) x (f) b (f), where a (f) is a large N × N sparse matrix function, b (f) is an N-dimensional column vector, x (f) is an N-dimensional column vector to be solved, and N represents a degree of freedom;
step D, selecting a Chebyshev polynomial zero point as an interpolation point, and normalizing the frequency band to be obtained to an interval of [ -1,1 ]:
for the original frequency f and the normalized frequency
Figure BDA0001353025300000021
The correspondence transformation of (a) is:
Figure BDA0001353025300000022
the corresponding A (f), b (f) and x (f) are transformed into
Figure BDA0001353025300000023
And
Figure BDA0001353025300000024
in the normalized frequency interval [ -1,1 [)]3 Chebyshev nodes are selected in the interval:
Figure BDA0001353025300000025
to carry outChebyshev interpolation with n being 2, n being the order of the Chebyshev interpolation polynomial, i.e. using
Figure BDA0001353025300000026
To pair
Figure BDA0001353025300000027
And
Figure BDA0001353025300000028
performing Lagrange quadratic interpolation to obtain AjAnd bjThen, the original matrix function equation in step C is transformed as follows:
Figure BDA0001353025300000029
step E, generating orthogonal basis by using moment matching technology
Figure BDA00013530253000000210
Spaces, i.e. mapping matrices
Figure BDA00013530253000000211
Reduced order model space of (2):
aiming at a specific frequency band, generating an initial reduced-order model vector space in the frequency band by adopting a polynomial moment matching method
Figure BDA0001353025300000031
Wherein p isi+1 is the vector space
Figure BDA0001353025300000032
Number of medium vectors, in respect of solving
Figure BDA0001353025300000033
The system of linear equations of (1) is:
Figure BDA0001353025300000034
to pair
Figure BDA0001353025300000035
Performing Schmidt unit orthogonalization to obtain wiFurther obtain the orthogonal base space
Figure BDA0001353025300000036
Step F, solving the unknown vector
Figure BDA0001353025300000037
Solving unknown vector by adopting model reduced GAWE known solution form
Figure BDA0001353025300000038
Setting pi+1 linearly independent vectors wiAnd coefficient gammaiWherein i is 0,1, …, pi(ii) a Defining dimension as piVector of +1
Figure BDA0001353025300000039
Combining what has been obtained in step E
Figure BDA00013530253000000310
By wiCoefficient of sum gammaiApproximating the unknown vector to obtain:
Figure BDA00013530253000000311
further, in the step E
Figure BDA00013530253000000312
And the number p of its space vectorsiThe +1 specific calculation process is:
defining the balance:
Figure BDA00013530253000000313
expanding the margin by Taylor series, i.e.
Figure BDA00013530253000000314
Then, w calculated in step E is addediSubstituting the formula into the formula, and enabling:
Figure BDA00013530253000000315
solving to obtain the corresponding coefficient gammai；
Determined by conjugate gradient method
Figure BDA00013530253000000316
Maximum eigenvalue modulo lambda ofmax|，
Will be provided with
Figure BDA0001353025300000041
Schmidt unitization to obtain wiSubstituting into a linear equation system to be left-multiplied by A0And right end item is recorded as tj,j＝0,1,...,piI.e. by
Figure BDA0001353025300000042
Defining the overall relative error:
Figure BDA0001353025300000043
defining local relative error:
Figure BDA0001353025300000044
adaptive solving reduced order model space
Figure BDA0001353025300000045
And the number p of its space vectorsiThe +1 process is:
initialization of pi＝0，
First, the method is obtained by using the matrix matching and the Schmidt orthogonalization
Figure BDA0001353025300000046
Find out
Figure BDA0001353025300000047
Maximum eigenvalue modulo lambda ofmaxL, |; calculating error1, and performing the first step of judgment:
if the absolute value of error1 is less than or equal to value1 and value1 are preset overall accuracy judgment threshold values, determining that
Figure BDA0001353025300000048
And the number p of its space vectorsi+ 1; otherwise, let pi＝pi+1, second step determination:
if p isi＞pmax，pmaxTo preset
Figure BDA0001353025300000049
Maximum number of space vectors, then determining
Figure BDA00013530253000000410
And the number p of its space vectorsi+ 1; otherwise, new ones are generated from the moment matches in step E
Figure BDA00013530253000000411
After Schmidt orthogonalization, the new one is obtained
Figure BDA00013530253000000412
Carrying out local error analysis by an equation (10), and judging in the third step:
if it satisfies
Figure BDA00013530253000000413
If the value2 is a preset local precision judgment threshold, returning to the first step of judgment and carrying out overall error judgment; otherwise, continue to increase pi＝pi+1, update
Figure BDA00013530253000000414
Until it meets
Figure BDA00013530253000000415
And returning to the first step of judgment to judge the whole error.
Furthermore, the numerical method for reducing the order of the input and output window model of the adaptive three-dimensional microwave tube based on function approximation further comprises frequency band division, and the specific process is as follows:
the initial band through step D is [ -1,1 [)]The total number of the scanning frequency points is parts, and the frequency point spacing distance is
Figure BDA0001353025300000051
According to
Figure BDA0001353025300000052
Maximum eigenvalue modulo lambda ofmaxAnd I, performing band division:
if lambdamaxIf | ≧ 1.0, divergence will occur, the band [ -1,1 [ ]]Solving in half, dividing the frequency band into: [ -1,0]And [0, 1]]The number of the sweep frequency points on each frequency band is part/2;
otherwise, let the convergence length convRAD be | λmaxA frequency band of
Figure BDA0001353025300000053
Wherein
Figure BDA0001353025300000054
New number of sweep points
Figure BDA0001353025300000055
The remaining interval is the non-convergence band, and the band range needs to be updated next time for solving.
The invention has the beneficial effects that: the numerical method for reducing the order of the three-dimensional microwave tube input/output window model is discussed from the Chebyshev function approximation angle, so that the dragon lattice phenomenon of the traditional high-order interpolation and the time consumption and complexity of matrix function derivation operation caused by Taylor series expansion in the existing electromagnetic model order reduction numerical method are avoided; and meanwhile, adaptively selecting interpolation points, dividing the convergence interval of the full frequency band to obtain a reduced-order model space, and performing post-processing to obtain a final S parameter so as to realize the full frequency band adaptive optimization simulation of the microwave tube. The invention can effectively avoid the complexity that a single frequency point in the traditional method can not obtain accurate response and calculate space and time, and simultaneously, the self-adaption performance of the invention widens the bandwidth widening capability and the solution stability in a wide frequency band range.
Drawings
FIG. 1 is a flow chart of a three-dimensional microwave tube input/output window model order reduction numerical method based on function approximation adaptive error analysis.
FIG. 2 is a diagram of the adaptive solution reduced order model space of the present invention
Figure BDA0001353025300000056
And the number p of its space vectorsi+1 scheme.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The embodiment provides a numerical method for reducing the order of a three-dimensional microwave tube input/output window model based on function approximation adaptive error analysis, as shown in fig. 1, comprising the following steps:
A. carrying out simulation modeling on the structure of the target electronic device by combining the material characteristics;
setting a corresponding working frequency range according to the physical structure of the target electronic device by combining a working environment and boundary conditions, and carrying out simulation modeling on the working frequency range;
B. solving a domain by adopting tetrahedral mesh dispersion;
in any finite element analysis, region discretization is the first step, and the global domain Ω is divided into M subdomains using tetrahedral mesh discretization, represented as: omegae(e-1, 2,3, …, M), the subfields being referred to generically hereinafter as a unit;
in the three-dimensional simulation model, considering that a tetrahedron is the simplest unit which is most suitable for dispersing any volume area, the tetrahedron units are adopted in the three-dimensional area for area division, and the triangular units are adopted in the two-dimensional area; it should be noted that the surface and volume dispersions must be compatible, that is, the cell edges resulting from the volume dispersion must also be the edges of the surface discrete cells;
C. selecting finite element vector basis functions, and establishing a matrix equation A (f) x (f) b (f) of an eigenvalue problem;
based on the finite element method of the edge vector in the electromagnetic field, after the intrinsic analysis of the finite element, the obtained matrix function equation is expressed as:
A(f)x(f)＝b(f) (1)
wherein A, x and b are both matrix functions with respect to frequency f, A (f) is a N × N large sparse matrix function, b (f) is an N-dimensional column vector, and x (f) is an N-dimensional column vector to be solved, wherein N represents a degree of freedom, obtained from model dimensions and finite element analysis; the specific formula derivation and solution is a well-known process in the vector finite element method, and is not explained here;
D. selecting a Chebyshev polynomial zero point as an interpolation point, and normalizing the frequency band to be obtained to an interval of [ -1,1 ];
because the traditional model order reduction method needs to select an expansion point to carry out Taylor series expansion on the required frequency band, which is difficult to differentiate between a large sparse matrix function A (f) and a column vector b (f), the Taylor series expansion is replaced by adopting a function approximation mode, and the Taylor series expansion is also beneficial to the prior art;
in order to avoid the dragon lattice phenomenon of high-order interpolation, Chebyshev polynomial zero-point interpolation is adopted to ensure the convergence of the whole interval; because the Chebyshev polynomial is in the interval [ -1,1 [)]As defined above, for the desired frequency band [ fmin,fmax]To be normalized to the interval [ -1,1] by a variable substitution transform]Namely:
for the original frequency f and the normalized frequency
Figure BDA0001353025300000061
The correspondence transformation of (a) is:
Figure BDA0001353025300000062
then f e f is realizedmin,fmax]And
Figure BDA0001353025300000071
the mutual transformation of (1); corresponding A (f), b (f) and x (f) also become
Figure BDA0001353025300000072
Figure BDA0001353025300000073
And
Figure BDA0001353025300000074
in the normalized frequency interval [ -1,1 [)]3 Chebyshev nodes are selected in the interval:
Figure BDA0001353025300000075
to perform Chebyshev interpolation with n being 2, n being the order of the Chebyshev interpolation polynomial, i.e. to use
Figure BDA0001353025300000076
To pair
Figure BDA0001353025300000077
And
Figure BDA0001353025300000078
performing Lagrange quadratic interpolation to obtain AjAnd bjThen, the original matrix function equation in step C is transformed as follows:
Figure BDA0001353025300000079
the specific formula derivation and solution is a known process in Chebyshev polynomial zero-point interpolation, and is not explained here; it can be seen that A is obtained in this way using Chebyshev interpolationiAnd biDifferent from the Taylor series expansion adopted by the traditional model reduction, the method reduces the calculation time and complexity;
E. moment matching technique for generating orthogonal basis
Figure BDA00013530253000000710
Spaces, i.e. matrices
Figure BDA00013530253000000711
A reduced order model space of (a);
aiming at a specific frequency band, generating an initial reduced-order model vector space in the frequency band by adopting a polynomial moment matching method
Figure BDA00013530253000000712
Wherein p isi+1 is the vector space
Figure BDA00013530253000000713
The number of medium vectors, applying order matching and recursive thinking, can be obtained about solving
Figure BDA00013530253000000714
The system of linear equations of (1) is as follows:
Figure BDA00013530253000000715
to pair
Figure BDA00013530253000000716
Performing Schmidt unit orthogonalization to obtain wiFurther obtain the orthogonal base space
Figure BDA00013530253000000717
From equation (4), it can be seen that the reduced order space generated by the method only needs to be A0The inversion is carried out once, and compared with the discrete frequency sweep, much time is saved; the specific formula derivation and solution is a well-known process of moment matching in model reduction, and is not described here.
F. Solving for unknown vectors
Figure BDA00013530253000000718
Solving unknown vector by adopting model reduced GAWE known solution form
Figure BDA00013530253000000719
Suppose there is pi+1 linearly independent vectors wiAnd coefficient gammaiWherein i is 0,1, …, pi(ii) a Defining dimension as piVector of +1
Figure BDA0001353025300000081
Combining what has been obtained in step E
Figure BDA0001353025300000082
By wiCoefficient of sum gammaiThe unknown vector is approximated as follows:
Figure BDA0001353025300000083
the specific formula derivation and solution is a well-known process in model reduction, and is not explained here; it can be seen that the matrix of dimension N x N is directly aligned compared to that of step C
Figure BDA00013530253000000814
Inversion, the reduced order system being solved for
Figure BDA0001353025300000084
It is simple and convenient.
On the basis of this, the method is suitable for the production,
Figure BDA0001353025300000085
performing inverse normalization to obtain x (f), and calculating to obtain S parameters:
S11＝b(f)H·x(f)
according to the formula, other parameters including reflection coefficient and transmission coefficient can be obtained through further post-processing, so that the performance factors of the microwave tube, such as output power, dispersion characteristic, coupling impedance characteristic, attenuation characteristic and energy conversion efficiency of the input and output window of the microwave tube, are further analyzed.
Further, solving for p in the above step EiSpecific value of + 1: and defining relative error by combining characteristic values, and realizing self-adaptive search of the positive value of each sub-frequency band calculated in the E-step moment matchingCross space of base
Figure BDA0001353025300000086
Number p of (2)i+ 1; as shown in FIG. 2, from the aspect of eigenvalue, error convergence is defined, and the reduced order model space of each sub-band can be calculated in an adaptive manner
Figure BDA0001353025300000087
Each of wiAnd the number pi+1；
Defining the balance:
Figure BDA0001353025300000088
expanding the margin by Taylor series, i.e.
Figure BDA0001353025300000089
Then, w calculated in step E is addediSubstituting into the formula (6), and enabling:
Figure BDA00013530253000000810
i.e. finding the corresponding coefficient gammaiSolving for a well-known process, which is not further elaborated herein;
simultaneously using conjugate gradient method to obtain
Figure BDA00013530253000000811
Maximum eigenvalue modulo lambda ofmaxNote that step E moment matching computation
Figure BDA00013530253000000812
When is only to A0Once inverse, will
Figure BDA00013530253000000813
Schmidt unitization to obtain wiSubstituting into formula (4), left-multiplying by A0And right end item is recorded as tj,j＝0,1,...,piI.e. by
Figure BDA0001353025300000091
Defining the overall relative error:
Figure BDA0001353025300000092
defining local relative error:
Figure BDA0001353025300000093
adaptive solving reduced order model space
Figure BDA0001353025300000094
And the number of space vectors pi+1 main idea:
when p isiWhen the value is 0, the value is obtained by orthogonalizing the value by using moment matching and Schmidt
Figure BDA0001353025300000095
Find out
Figure BDA0001353025300000096
Maximum eigenvalue modulo lambda ofmaxL, |; carrying out the first step of judgment by taking the formula (9) to obtain error 1:
by user-defined calculations
Figure BDA0001353025300000097
Overall accuracy decision threshold value1 values are compared:
if the overall accuracy judgment threshold values of | error1| ≦ value1 and value1 which are preset (defined by a user in advance) are met, determining that
Figure BDA0001353025300000098
And the number p of its space vectorsi+ 1; otherwise, let pi＝pi+1, second step determination:
if p isi＞pmax，pmaxTo preset
Figure BDA0001353025300000099
Maximum number of space vectors, then determining
Figure BDA00013530253000000910
And the number p of its space vectorsi+ 1; otherwise, new ones are generated from the moment matches in step E
Figure BDA00013530253000000911
After Schmidt orthogonalization, the new one is obtained
Figure BDA00013530253000000912
Carrying out local error analysis by an equation (10), and judging in the third step:
if it satisfies
Figure BDA00013530253000000913
If the value2 is a preset (user defines in advance) local precision judgment threshold, returning to the first step of judgment to judge the overall error; otherwise, continue to increase pi＝pi+1, update
Figure BDA00013530253000000914
Until it meets
Figure BDA0001353025300000101
And returning to the first step of judgment to judge the whole error.
The following is detailed by examples, and the specific steps are as follows:
①pi0, l is 0, calculated in step E
Figure BDA0001353025300000102
Schmidt unitization thereof to obtain w0At this time
Figure BDA0001353025300000103
Then calculate out
Figure BDA0001353025300000104
Maximum eigenvalue modulo lambda ofmaxL, |; next, an overall error determination is made by first determining w0Substituting the balance
Figure BDA0001353025300000105
Let the constant term of the residual Taylor series be 0
Figure BDA0001353025300000106
Equation two-end left multiplication
Figure BDA0001353025300000107
Calculate out
Figure BDA0001353025300000108
At this time t0＝b0Will be gamma0And t0Substituting into the overall error (9) formula to calculate piEntering into step ② when the error1 is 0;
② if | error1| ≦ value1, that is, the overall relative error determination condition is satisfied, then the loop is stopped to get the final result
Figure BDA0001353025300000109
Otherwise, update pi＝pi+1, at this time, pi≤pmaxFrom step E, a new one is generated
Figure BDA00013530253000001010
Go to step ③;
③pi1, l is 1, in this case
Figure BDA00013530253000001011
t0＝b0，t1＝b1-A1w0Then, the local relative error is substituted into the local relative error (10) equation to obtain
Figure BDA00013530253000001012
If it is not
Figure BDA00013530253000001013
That is, the local relative error determination condition is satisfied, the process proceeds to step ④If the total relative error is not reached, the ⑤ step is entered;
④ similar to the overall error judgment of step ①, p obtained in step ③ is usedi1, l 1 and
Figure BDA00013530253000001014
calculate out
Figure BDA00013530253000001015
Maximum eigenvalue modulo lambda ofmaxL, |; the overall error determination is made by first determining
Figure BDA00013530253000001016
Substituting the balance
Figure BDA00013530253000001017
Let the constant term and the first term of the residual Taylor series be 0, that is
Figure BDA00013530253000001018
Left-multiplying both ends of the equation respectively
Figure BDA00013530253000001019
And
Figure BDA00013530253000001020
after shifting one's neck there is
Figure BDA00013530253000001021
Obtaining new gamma by solving the above equation system0、v1(ii) a At this time t0＝b0，t1＝b1-A1w0Will be gamma0、γ1、t0And t1Substituting the integral error (9) formula to calculate piAn error1 when the value is 1, and entering the integral error judgment;
⑤ when the local relative error judgment condition is not satisfied, updating pi＝pi+1, new ones are generated by step E
Figure BDA00013530253000001022
Until it appears
Figure BDA0001353025300000111
Carrying out integral error judgment;
thus far, the adaptive search of each sub-band to compute orthogonal basis space in E-step moment matching has been discussed
Figure BDA0001353025300000112
Number p of (2)i+1 and find out the corresponding
Figure BDA0001353025300000113
In addition, it should be noted that, in the microwave electromagnetic calculation, especially for devices with fine and complex structures, when the working frequency range is large, if the whole frequency band is calculated only once, an ideal S parameter numerical solution is difficult to obtain, and the precision is often limited, and the effective solution bandwidth is narrow; to address this problem, the present invention performs effective band division:
defining convergence radius, adaptively dividing full frequency band
The calculation of a frequency band f has been described abovemin,fmax]Normalized to [ -1,1 [ ]]Time solving unknown vector
Figure BDA0001353025300000114
The method of (1); since we convert the original band to the interval [ -1,1] at the time of the calculation]Therefore, we need to define the convergence radius to divide the full frequency band into the convergence frequency band and the non-convergence frequency band;
at the initial calculation, the frequency band is [ -1,1 [ ]]The total number of the scanning frequency points is parts, and the frequency point spacing distance is
Figure BDA0001353025300000115
By reducing the model space
Figure BDA0001353025300000116
And a space vectorNumber piOverall error analysis at +1, from which the maximum eigenvalue modulo lambda is recordedmaxSetting a new convergence interval:
if lambdamaxIf | ≧ 1.0, divergence will occur, the band [ -1,1 [ ]]Solving in half, dividing the frequency band into: [ -1,0]And [0, 1]]The number of the sweep frequency points on each frequency band is part/2;
otherwise, let the convergence length convRAD be | λmaxA frequency band of
Figure BDA0001353025300000117
Wherein
Figure BDA0001353025300000118
New number of sweep points
Figure BDA0001353025300000119
The remaining interval is the non-convergence band, and the band range needs to be updated next time for solving.
While the invention has been described with reference to specific embodiments, any feature disclosed in this specification may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps.

Claims (2)

1. A numerical method for reducing the order of an input/output window model of a self-adaptive three-dimensional microwave tube based on function approximation is characterized by comprising the following steps:
a, according to the physical structure and the working frequency range of a target electronic device, carrying out simulation modeling on the target electronic device;
b, solving a domain by adopting tetrahedral mesh dispersion; the universe Ω is divided into M subdomains using tetrahedral mesh discretization, denoted as: omegae(e-1, 2,3, …, M), the subfields being referred to generically hereinafter as a unit;
c, selecting a finite element vector basis function, and establishing a matrix equation of the eigenvalue problem: a (f) x (f) b (f), where a (f) is a large N × N sparse matrix function, b (f) is an N-dimensional column vector, x (f) is an N-dimensional column vector to be solved, and N represents a degree of freedom;
step D, selecting a Chebyshev polynomial zero point as an interpolation point, and normalizing the frequency band to be obtained to an interval of [ -1,1 ]:
for the original frequency f and the normalized frequency
Figure FDA0002207139000000011
The correspondence transformation of (a) is:
Figure FDA0002207139000000012
the corresponding A (f), b (f) and x (f) are transformed into
Figure FDA0002207139000000013
And
Figure FDA0002207139000000014
in the normalized frequency interval [ -1,1 [)]3 Chebyshev nodes are selected in the interval:
Figure FDA0002207139000000015
the Chebyshev interpolation is carried out with j equal to 0,1,2 and n equal to 2, n is the order of the Chebyshev interpolation polynomial, that is, the method adopts
Figure FDA0002207139000000016
To pair
Figure FDA0002207139000000017
And
Figure FDA0002207139000000018
performing Lagrange quadratic interpolation to obtain AjAnd bjThen, the original matrix function equation in step C is transformed as follows:
Figure FDA0002207139000000019
step E, generating orthogonal basis by using moment matching technology
Figure FDA00022071390000000110
Spaces, i.e. mapping matrices
Figure FDA00022071390000000111
Reduced order model space of (2):
aiming at a specific frequency band, generating an initial reduced-order model vector space in the frequency band by adopting a polynomial moment matching method
Figure FDA00022071390000000112
Wherein p isi+1 is the vector space
Figure FDA00022071390000000113
Number of medium vectors, in respect of solving
Figure FDA00022071390000000114
The system of linear equations of (1) is:
Figure FDA0002207139000000021
to pair
Figure FDA0002207139000000022
Performing Schmidt unit orthogonalization to obtain wiFurther obtain the orthogonal base space
Figure FDA0002207139000000023
The above-mentioned
Figure FDA0002207139000000024
And the number p of its space vectorsiThe +1 specific calculation process is:
defining the balance:
Figure FDA0002207139000000025
expanding the margin by Taylor series, i.e.
Figure FDA0002207139000000026
Then, w calculated in step E is addediSubstituting the formula into the formula, and enabling:
Figure FDA0002207139000000027
solving to obtain the corresponding coefficient gammai；
Determined by conjugate gradient method
Figure FDA0002207139000000028
Maximum eigenvalue modulo lambda ofmax|，
Will be provided with
Figure FDA0002207139000000029
Schmidt unitization to obtain wiSubstituting into a linear equation system to be left-multiplied by A0And right end item is recorded as tj,j＝0,1,...,piI.e. by
Figure FDA00022071390000000210
Defining the overall relative error:
Figure FDA0002207139000000031
defining local relative error:
Figure FDA0002207139000000032
adaptive solving reduced order model space
Figure FDA0002207139000000033
And the number p of its space vectorsiThe +1 process is:
initialization of pi＝0，
Firstly, obtaining W by using moment matching and Schmidt orthogonalizationpiTo find out
Figure FDA0002207139000000034
Maximum eigenvalue modulo lambda ofmaxL, |; calculating error1, and performing the first step of judgment:
if the absolute value of error1 is less than or equal to value1 and value1 are preset overall accuracy judgment threshold values, determining that
Figure FDA0002207139000000035
And the number p of its space vectorsi+ 1; otherwise, let pi＝pi+1, second step determination:
if p isi＞pmax，pmaxTo preset
Figure FDA0002207139000000036
Maximum number of space vectors, then determining
Figure FDA0002207139000000037
And the number p of its space vectorsi+ 1; otherwise, new ones are generated from the moment matches in step E
Figure FDA0002207139000000038
After Schmidt orthogonalization, the new one is obtained
Figure FDA0002207139000000039
Carrying out local error analysis by an equation (10), and judging in the third step:
if it satisfies
Figure FDA00022071390000000310
If value2 is the preset local precision judgment threshold, go back to the first step judgment and perform the integrationJudging the volume error; otherwise, continue to increase pi＝pi+1, update
Figure FDA00022071390000000311
Until it meets
Figure FDA00022071390000000312
Returning to the first step of judgment to judge the whole error;
step F, solving the unknown vector
Figure FDA00022071390000000313
Solving unknown vector by adopting model reduced GAWE known solution form
Figure FDA00022071390000000314
Setting pi+1 linearly independent vectors wiAnd coefficient gammaiWherein i is 0,1, …, pi(ii) a Defining dimension as piVector of +1
Figure FDA00022071390000000315
Combining what has been obtained in step E
Figure FDA00022071390000000316
By wiCoefficient of sum gammaiApproximating the unknown vector to obtain:
Figure FDA0002207139000000041
2. the numerical method for reducing the order of the input-output window model of the adaptive three-dimensional microwave tube based on the function approximation as claimed in claim 1, wherein the numerical method for reducing the order of the input-output window model of the adaptive three-dimensional microwave tube based on the function approximation further comprises frequency band division, and the specific process is as follows:
the initial band through step D is [ -1,1 [)]Total number of swept frequency pointsAre partitions, the frequency point spacing distance is
Figure FDA0002207139000000042
According to
Figure FDA0002207139000000043
Maximum eigenvalue modulo lambda ofmaxAnd I, performing band division:
if lambdamaxIf | ≧ 1.0, divergence will occur, the band [ -1,1 [ ]]Solving in half, dividing the frequency band into: [ -1,0]And [0, 1]]The number of the sweep frequency points on each frequency band is part/2;
otherwise, let the convergence length convRAD be | λmaxA frequency band of
Figure FDA0002207139000000044
Wherein
Figure FDA0002207139000000045
New number of sweep points
Figure FDA0002207139000000046
The remaining interval is the non-convergence band, and the band range needs to be updated next time for solving.
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