CN107194088A - A kind of high order finite difference time domain electromagnetic-field simulation method based on ICCG - Google Patents
A kind of high order finite difference time domain electromagnetic-field simulation method based on ICCG Download PDFInfo
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Abstract
The present invention provides a kind of high order finite difference time domain electromagnetic-field simulation method based on ICCG, including：It is based on Fang forms and the expansion of Taylor radixes and discrete to Maxwell equation progress come approximation space derivative using the central difference schemes of quadravalence or higher order on direction in space；On time orientation, Maxwell equation is carried out based on pungent operator form discrete；Sparse matrix in Electromagnetic Simulation is solved using ICCG methods；For sparse matrix A, all neutral elements present in matrix A are rejected, and are revised as 3 one-dimension arrays AAA, ND1 and NC1 that retrieval is set up as set by one-dimensional contracted form stores each element in A, incomplete triangle decomposition are then carried out, to constitute preconditioning matrix.
Description
Technical field
The present invention relates to a kind of Numerical Calculation of Electromagnetic Fields technology, particularly a kind of high order finite difference time domain based on ICCG
Electromagnetic-field simulation method.
Background technology
Traditional FDTD algorithms have two shortcomings：First, it can not accurately simulation it is complex-curved, simulate it is discontinuous
Had any problem in terms of material.Second, with prolonged emulation, it has in terms of numerical stability, dispersivity and anisotropy
Significant accumulated error.Because the precision that these shortcomings can cause Electromagnetic Simulation to calculate is substantially reduced, final calculating knot is influenceed
Really, so needing a kind of improved FDTD algorithms to solve the above problems.
So, since the initial stage nineties, the conventional FDTD algorithm used for calculating Electrically large size object calculates interior
Deposit big, the shortcomings of numerical dispersion is poor, higher order FDTD method is suggested, and can effectively reduce numerical dispersion error, is being met
The grid cell thicker than conventional FDTD algorithm can be used in the case of same accuracy.It is engaged in the difference scheme of algorithm in itself
See, in conventional FDTD algorithm, the field component of Yee forms uses the distribution of non-concurrent point staggered-mesh, then passes through Taylor
Series expansion, obtains Second-Order Central Difference form approximate as space；
And in higher order FDTD method, there is various higher-order wave equations, in the hope of the high-precision of electromagnetic-field simulation calculating
Degree and low numerical dispersion error.Some use the recessive difference scheme of quadravalence, and Taylor series exhibitions are based on staggered-mesh
Open, using the central difference schemes of quadravalence come approximation space derivative；Some construct difference scheme using non-standard difference；Some
Yee staggered-meshes and Bi concurrent grids are combined, a kind of more preferable form of the dispersion characteristics than the above two is derived；Some
Each component of electromagnetic field is deployed with the tight scaling functions of Battle-Lemarie and wavelet function in space dimension, and in time dimension
Deployed with Harr small echos impulse function, i.e. the multi-resolution time-domain based on multiple dimensioned resolution ratio；Some are deduced base
In the variable differential form of discrete convolution (DSC-Discrete Singular Convolution) method, pass through DSC delta
Core, such as Shannon cores, Poisson cores and Lagrange cores, the field amount to space are sampled, and then reconstruct height
Jump cellular, can thus reach very high precision.
The content of the invention
It is an object of the invention to provide a kind of high order finite difference time domain electromagnetic-field simulation method based on ICCG, bag
Include：
On direction in space, based on Fang forms and the expansion of Taylor radixes and the centered difference for utilizing quadravalence or higher order
It is discrete to Maxwell equation progress that form carrys out approximation space derivative；
On time orientation, Maxwell equation is carried out based on pungent operator form discrete；
When solving the sparse matrix in Electromagnetic Simulation using ICCG methods；
For sparse matrix A, all neutral elements present in matrix A are rejected, and be revised as retrieval by one-dimensional contracted form
Set 3 one-dimension arrays AAA, ND1 and the NC1 set up of each element in A are stored, incomplete triangle decomposition are then carried out, with structure
Into preconditioning matrix, wherein
(1) AAA is used for depositing the nonzero element of compressed format coefficient matrix；
(2) ND1 is used for depositing the diagonal entry sequence number in former coefficient matrices A.
(3) NC1 is used for depositing row number of each element in former coefficient matrices A in compressed format coefficient matrices A AA.
The present invention compared with prior art, with advantages below：(1) high order finite difference time domain form is on direction in space
By higher difference, relatively low numerical dispersion error can be obtained, more preferable computational accuracy is obtained；(2) lead on time orientation
Cross and introduce pungent operator, it is ensured that the conservation of energy of whole maxwell equation group；(3) it can be solved because of sparse square using ICCG
The conditional number of battle array is larger to cause iterative convergence speed to cause operation time long defect slowly excessively；(4) internal memory needed for computer
Space is substantially reduced, and can reach the purpose for saving calculator memory, and improve the speed of calculating.
With reference to Figure of description, the invention will be further described.
Brief description of the drawings
Fig. 1 is flow chart of the method for the present invention.
Fig. 2 is the numerical dispersion error of the high order finite difference time domain algorithm based on incomplete cholesky conjugate gradient method
With the change of space lattice resolution ratio and the comparison schematic diagram with multi-resolution time-domain.
Embodiment
With reference to Fig. 1, one kind is based on the high order finite difference time domain electromagnetism of incomplete cholesky conjugate gradient method (ICCG)
Field emulation mode, it is characterised in that including：
On direction in space, based on Fang forms and the expansion of Taylor radixes and the centered difference for utilizing quadravalence or higher order
It is discrete to Maxwell equation progress that form carrys out approximation space derivative；
On time orientation, Maxwell equation is carried out based on pungent operator form discrete；
When solving the sparse matrix in Electromagnetic Simulation using ICCG methods；
For sparse matrix A, all neutral elements present in matrix A are rejected, and be revised as retrieval by one-dimensional contracted form
Set 3 one-dimension arrays AAA, ND1 and the NC1 set up of each element in A are stored, incomplete triangle decomposition are then carried out, with structure
Into preconditioning matrix, wherein
(1) AAA is used for depositing the nonzero element of compressed format coefficient matrix；
(2) ND1 is used for depositing the diagonal entry sequence number in former coefficient matrices A.
(3) NC1 is used for depositing row number of each element in former coefficient matrices A in compressed format coefficient matrices A AA.
Carrying out discrete method to Maxwell equation on direction in space is
Wherein δ=x, y, z, h=i, j, k, q are exponent number, WrIt is the coefficient of space difference.
Discrete method is carried out to Maxwell equation on direction in space and specifically may refer to Jiayuan Fang, " A
locally conformed finite-difference time-domain algorithm of modeling
arbitrary shape planar metal strips,”IEEE Transactions on Microwave Theory
and Techniques,vol.41,No.5,pp:830-838,1989.
Carrying out discrete method to Maxwell equation on time orientation is
Wherein cl,dlFor pungent propagator coefficient, m, p are respectively the series and exponent number of Symplectic Algorithm, m >=p
Discrete method is carried out to Maxwell equation on time orientation and specifically may refer to Hans Van de
Vyver,“A Fourth-Order Symplectic Exponentially Fitted Integrator,”Computer
Physics Communications,Vol.74,No.4,pp.255-262,February 2006.
So, the high order finite difference time domain form can obtain relatively low number by higher difference on direction in space
It is worth error dispersion, obtains more preferable computational accuracy.And by introducing pungent operator on time orientation, it is ensured that whole Maxwell
The conservation of energy of equation group.This method, no matter calculating early stage or later stage, calculates essence in the analog simulation to electromagnetic wave
Degree is all very high.
But it is due to higher difference, although the precision of simulation result is improved, still, in calculating process, right
The demand of calculator memory is larger, and can reduce the speed of calculating.In order to overcome these shortcomings, incomplete cholesky is introduced
Conjugate gradient method is into electromagnetic-field simulation.
It is frequently not simple square or circle due to the scrambling of simulation object during Electromagnetic Simulation, but it is real
Present in medium, such as aircraft, antenna, photonic crystal etc..So, during calculating, generally without parsing
Solution, can only go to try to achieve numerical solution using electromagnetic calculations such as High-order FDTDs, but last is all to be attributed to solution greatly
Type sparse vectors (sparse matrix) problem.
In engineer applied, commonly use classical iterative method to solve sparse matrix, convergence rate often occur slowly, or even shake
Swing, not convergent situation.Accordingly, conjugate gradient method (CG-Conjugate GradientMethod) amount of storage is small, and program is multiple
Miscellaneous degree is low, most suitable to solve sparse matrix.But in actual applications, when the conditional number of matrix is larger, conjugate gradient
Method often make it that operation time is long slowly excessively because of iterative convergence speed, so as to limit its use.Therefore how coefficient is improved
Key of the condition of battle array into this problem of solution.Improved incomplete cholesky conjugate gradient method (ICCG-Incomplete
Cholesky Conjugate Gradient Method) defects of simple CG methods is compensate for well, solving Large Scale Sparse
There is very big advantage in terms of the numerical problem of matrix, no matter very ten-strike is obtained in memory requirements and calculating speed.
The convergence rate of incomplete cholesky conjugate gradient method is fast compared with Gaussian reduction, and amount of storage can subtract significantly
Few, especially to the sparse matrix produced during higher difference, this method can more embody its superiority.Specific method may refer to
Michele Benzi,“Preconditioning Techniques for Large Linear Systems:A Survey,”
Journal of Computational Physics,Vol.182,No.2,pp.418-477,November 2002.
When the incomplete cholesky conjugate gradient method of application solves the Large sparse matrix produced during higher difference,
Need to be compressed storage to matrix by rows order.Due to the introducing of sparse storage pattern so that programming is rather complicated, so
Under actual conditions, realized with computer or with suitable difficulty.Because A is sparse, A non-zero entry is only stored
Element, can so greatly save the memory space of computer.For Description Matrix AN×N, reject all null elements existing in matrix A
Element, and be revised as retrieval as 3 one-dimension arrays set up in the A that one-dimensional contracted form is stored set by each element, AAA, ND1 and
NC1, then carries out incomplete triangle decomposition, to constitute preconditioning matrix.Because A is symmetrical, pair of this method only to A
Diagonal element and lower triangle element carry out pressing row sequential storage.
(1) AAA (NNC1), for depositing the nonzero element of compressed format coefficient matrix, NNC1 is amended matrix A
Tighten the sum of storage element.
(2) ND1 (NND1), for depositing the diagonal entry sequence number in former coefficient matrices A, NND1 is unknown number sum,
Namely the number of diagonal entry, including nonzero element.
(3) NC1 (NNC1), for depositing row of each element in former coefficient matrices A in compressed format coefficient matrices A AA
Number.
This is a very cleverly storage method, only requires the amount of storage of twice nonzero element number.
Assuming that matrix A is：
Storage rule more than, certain above-mentioned matrix A can be described by 3 one-dimension arrays AAA, AND and ANC, be seen
Table 1：
Table 1 is directed to the compression storage form of certain matrix A
So, the memory headroom needed for computer is substantially reduced, and can reach the purpose for saving calculator memory, and
Improve the speed calculated.
For the present invention, the situation that numerical dispersion changes with propagation angle is analyzed first：10 ranks are based on incomplete Qiao Liesi
The numerical dispersion error value and ripple of the high order finite difference time domain algorithm of base CG methods and corresponding 3 rank multi-resolution time-domain
Propagation angle φ variation relation figure is as shown in the figure.Higher-Order Time-Domain as can be seen from Figure 2 based on incomplete cholesky CG methods has
Difference algorithm is limited relative to multi-resolution time-domain under different propagation angles, numerical dispersion error value is smaller, in number
There is very big advantage in terms of value stabilization and numerical value dispersivity.
Claims (3)
1. a kind of high order finite difference time domain electromagnetic-field simulation method based on ICCG, it is characterised in that including：
On direction in space, based on Fang forms and the expansion of Taylor radixes and the central difference schemes for utilizing quadravalence or higher order
Carry out approximation space derivative discrete to Maxwell equation progress；
On time orientation, Maxwell equation is carried out based on pungent operator form discrete；
Sparse matrix in Electromagnetic Simulation is solved using ICCG methods；
For sparse matrix A, all neutral elements present in matrix A are rejected, and be revised as retrieval by one-dimensional contracted form storage A
Set 3 one-dimension arrays AAA, ND1 and the NC1 set up of middle each element, then carry out incomplete triangle decomposition, to constitute pre- place
Matrix is managed, wherein
(1) AAA is used for depositing the nonzero element of compressed format coefficient matrix；
(2) ND1 is used for depositing the diagonal entry sequence number in former coefficient matrices A.
(3) NC1 is used for depositing row number of each element in former coefficient matrices A in compressed format coefficient matrices A AA.
2. according to the method described in claim 1, it is characterised in that Maxwell equation is carried out on direction in space discrete
Method is
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Wherein δ=x, y, z, h=i, j, k, q are exponent number, WrIt is the coefficient of space difference.
3. according to the method described in claim 1, it is characterised in that Maxwell equation is carried out on time orientation discrete
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Cited By (3)
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| CN115065049A (en) * | 2022-06-24 | 2022-09-16 | 南方电网科学研究院有限责任公司 | Execution time calculation method, device and equipment for row reduced order model in power grid simulation |
| CN116861711A (en) * | 2023-09-05 | 2023-10-10 | 北京航空航天大学 | Double-isotropy medium simulation method based on time-domain intermittent Galerkin method |
| CN117421955A (en) * | 2023-10-24 | 2024-01-19 | 上海慕灿信息科技有限公司 | Second-order accurate interpolation method applying non-uniform grid in FDTD |
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Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN115065049A (en) * | 2022-06-24 | 2022-09-16 | 南方电网科学研究院有限责任公司 | Execution time calculation method, device and equipment for row reduced order model in power grid simulation |
| CN115065049B (en) * | 2022-06-24 | 2023-02-28 | 南方电网科学研究院有限责任公司 | Execution time calculation method, device and equipment for row reduced order model in power grid simulation |
| CN116861711A (en) * | 2023-09-05 | 2023-10-10 | 北京航空航天大学 | Double-isotropy medium simulation method based on time-domain intermittent Galerkin method |
| CN116861711B (en) * | 2023-09-05 | 2023-11-17 | 北京航空航天大学 | Double-isotropy medium simulation method based on time-domain intermittent Galerkin method |
| CN117421955A (en) * | 2023-10-24 | 2024-01-19 | 上海慕灿信息科技有限公司 | Second-order accurate interpolation method applying non-uniform grid in FDTD |
| CN117421955B (en) * | 2023-10-24 | 2024-04-26 | 上海慕灿信息科技有限公司 | Second-order accurate interpolation method applying non-uniform grid in FDTD |
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