CN111639447A - Any high-order mixed grid time domain discontinuous Galerkin method based on multistage local time stepping technology - Google Patents

Any high-order mixed grid time domain discontinuous Galerkin method based on multistage local time stepping technology Download PDF

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CN111639447A
CN111639447A CN202010365579.7A CN202010365579A CN111639447A CN 111639447 A CN111639447 A CN 111639447A CN 202010365579 A CN202010365579 A CN 202010365579A CN 111639447 A CN111639447 A CN 111639447A
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陈如山
丁大志
樊振宏
包华广
何姿
王蒙蒙
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Abstract

The invention discloses an arbitrary high-order mixed grid time domain discontinuous Galerkin method of a multistage local time stepping technology, which selects Maxwell equation set as a basic numerical model, combines an arbitrary high-order derivative (ADER) time stepping scheme, divides a calculation domain by adopting a reasonable tetrahedron/hexahedron mixed grid, each divided cell respectively and automatically determines a proper time iteration step length according to stability conditions, can realize the size of a plurality of arbitrary time iteration step lengths in any proportion, each cell electromagnetic field quantity is iteratively updated according to the time iteration step length thereof until all cell field quantities iterate to a specified time point, and carries out post-processing on the obtained time-varying electromagnetic field quantity to obtain corresponding S parameters, radar scattering cross section area and electromagnetic field spatial distribution. The invention relieves the problem of low calculation efficiency caused by the fact that the time step of the time domain electromagnetic analysis method is limited by the minimum discrete grid size, not only improves the calculation precision, but also reduces the calculation time, and is particularly suitable for the rapid analysis of the space multi-scale electromagnetic problem.

Description

Any high-order mixed grid time domain discontinuous Galerkin method based on multistage local time stepping technology
Technical Field
The invention belongs to the technical field of electromagnetic simulation, in particular to a numerical calculation technology of a discontinuous Galerkin time domain finite element algorithm, which is an efficient algorithm for simulating a space multi-scale electromagnetic problem.
Background
With the increasing complexity of engineering design and electromagnetic simulation environments, when multiple spatio-temporal scales have research significance on the problem of interest, the analysis of electromagnetic transient problems of the.
Disclosure of Invention
The invention aims to provide an arbitrary high-order mixed grid time domain discontinuous Galerkin method based on a multistage local time stepping technology.
The technical solution for realizing the purpose of the invention is as follows: an arbitrary high-order mixed grid time domain discontinuous Galerkin method of a multilevel local time stepping technology comprises the following steps:
the method comprises the steps that firstly, a spatial multi-scale problem containing a local fine or high-electric structure is selected as an electromagnetic simulation model, and tetrahedral and hexahedral units with proper sizes are adopted to carry out spatial dispersion on corresponding areas to obtain structural information of the model;
secondly, taking a Maxwell equation set as a basic control equation, establishing a matrix equation by adopting a discontinuous Galerkin technology, and performing time dispersion by adopting an ADER iterative formula in time to obtain a display solution matrix iterative scheme;
thirdly, according to stability conditions approximately followed by the ADER explicit time scheme, sequentially dividing each subdivision unit into a plurality of different calculation domains according to any integer proportion, and automatically determining the iteration time step length met by each domain;
fourthly, sequentially carrying out time iteration on the calculation domains according to the time step from small to large;
and fifthly, finishing iterative solution of the electric field and the magnetic field according to the specified time steps, extracting electromagnetic field information on the observation surface, and acquiring corresponding parameters, spatial distribution and the like of the system.
Compared with the prior art, the invention has the following remarkable advantages:
the rapid analysis of the transient electromagnetic characteristics of the spatial multi-scale structure is realized from two aspects of spatial dispersion and time difference.
(1) By utilizing the method of tetrahedron and hexahedron mixed subdivision, the advantages of two kinds of grid modeling are fully utilized, and the non-conformal processing technology is combined, so that the calculation time is reduced, and the memory consumption is reduced.
(2) An arbitrary high order derivative (ADER) time difference format is used for the iteration of each sub-region using a flexible multi-stage local time stepping scheme. Each grid cell is updated according to the respective optimal time step, and can reach any precision in time and space.
Drawings
Fig. 1 is a schematic diagram of region division.
Fig. 2 is a schematic diagram of an Electromagnetic Bandgap (EBG) model. (a) A schematic array arrangement diagram, and (b) a hybrid subdivision grid diagram.
Fig. 3 is a reflection coefficient comparison graph of S11.
Detailed Description
In order to efficiently process the spatial multi-scale problem of fine structures containing patches or via holes and the like, the invention provides an arbitrary high-order mixed grid time domain discontinuous Galerkin scheme of a multi-stage local time stepping technology, which allows each grid unit to adopt different time steps to overcome the local stability limitation and relieve the problem of large unknown quantity brought by the fine structures; any high-order derivative (ADER) time stepping scheme is integrated into a flexible Local Time Stepping (LTS) technology, the electromagnetic simulation numerical precision is improved, and meanwhile, high overall calculation efficiency is obtained.
The present invention is described in further detail below with reference to the attached drawing figures.
The invention relates to a discontinuous Galerkin time domain finite element algorithm of a multilevel local time stepping technology, which comprises the following steps:
firstly, establishing an electromagnetic simulation model of a spatial multi-scale problem, dispersing a fine structure or a metal via edge by using a tetrahedral unit, and subdividing other flat areas or free spaces by using a hexahedron to obtain structural information of the model;
and step two, following the standard analysis step of the discontinuous Galerkin time domain finite element method, expanding the unknown electric field and the unknown magnetic field by using the laminated vector basis function:
Figure BDA0002476634330000021
and performing Galerkin test on both sides of the square program. For processing in surface integration, it is necessary to introduce numerical flux to emphasize the continuity of the field. Taking the introduction of the windward flux as an example, the expression is
Figure BDA0002476634330000022
Obtaining an explicit semi-discrete form matrix equation:
Figure BDA0002476634330000023
the above equation is written as:
Figure BDA0002476634330000031
wherein u ═ pe,ph]TIs the unknown vector that needs to be solved for,
Figure BDA0002476634330000032
is the inverse matrix of the diagonal characteristics of the block,
Figure BDA0002476634330000033
is the right matrix of the system.
Each matrix expression is:
Figure BDA0002476634330000034
in the above formula, mu represents the dielectric constant and permeability of the discrete unit, Ni、NjTesting the basis functions and the expanded basis functions of the field magnitudes for a finite element tetrahedral or hexahedral vector stack, Nj +Representing the relationship between the wave impedance and admittance of the test basis function of the adjacent body
Figure BDA0002476634330000035
The ADER iterative format is adopted to complete the dispersion in time, and the main idea of the scheme is that the time partial derivative term of the field value on each discrete unit is subjected to Taylor series expansion:
Figure BDA0002476634330000036
through repeated derivation and back substitution, the required time derivative is converted into a problem that a space derivative value is obtained through solving a differential value:
Figure BDA0002476634330000037
finally, an iterative formula of an n-order ADER time stepping scheme can be obtained as
Figure BDA0002476634330000041
And thirdly, dividing the calculation domain by adopting reasonable criteria to finish the automatic selection of the time step of each region. For the tetrahedral subdivision grid, calculating the inscribed sphere radius of each unit, dividing the calculation domain into n parts according to the integer proportion of the minimum inscribed sphere radius in sequence, and determining the iteration time step length delta t of each calculation domain, wherein the calculation formula approximately satisfied is as follows:
Figure BDA0002476634330000042
where N is the order of the spatial basis function expansion, c0Is the speed of light in free space，rIs the dielectric constant of the medium, d is the diameter of the inscribed sphere, and has the following definition:
Figure BDA0002476634330000043
and fourthly, sequentially performing time iteration on the calculation domains by using an ADER format according to the time step from small to large, wherein the calculation domains can be arbitrarily divided into n calculation domains. The iteration for dividing the two calculation domains is implemented as follows:
(1) first, let Δ t of two regions be small time steps Δ t respectivelysAnd a large time step Δ tlAnd the relation satisfied between the two time steps is Δ ts:ΔtlN is any positive integer, as shown in fig. 1.
(2) Time step of Δ tsUpdating of small cell area field values
Figure BDA0002476634330000044
If the adjacent cells of the region satisfy the time step Δ tlThe second term requires the relevant interpolation information of the large unit. Since there is no intermediate time t + (n-1) Δ t in the adjacent large cellsTherefore, the unknown field value u (t + (n-1) Δ t cannot be directly obtaineds). Then the expression in equation (10) will not be able to continue to solve, and the term needs to be represented in the original form of a taylor series expansion:
Figure BDA0002476634330000045
solving by using initial values of adjacent large units:
Figure BDA0002476634330000046
(4) time step of Δ tlThe same principle of updating format of large unit area field value can be obtained
Figure BDA0002476634330000051
At this time, the field value of the small unit required by the interface of the two regions is already calculated in (10), and only corresponding transmission is needed.
And fifthly, finishing iterative solution of the electric field and the magnetic field according to the specified time steps, extracting electromagnetic field information on the observation surface, and acquiring corresponding parameters, spatial distribution and the like of the system.
In order to verify the correctness and effectiveness of the present invention, the electromagnetic propagation characteristics of the Electromagnetic Bandgap (EBG) structure are analyzed below, and the electric field information on the observation surface is extracted to obtain the reflection coefficient thereof (S11).
Calculation example: considering a periodic array of infinitely long double-layer conductor round bars, as shown in fig. 2(a), a modulated gaussian signal with a center frequency of 17.5GHz and a bandwidth of 25GHz is incident vertically. When the mixed grid is used for discretely calculating the space, the PML layer and a part of air area are subdivided by adopting 1mm hexahedral grids, the periphery of the metal conductor bar needs to be finely subdivided, the subdivision size is 0.35mm, the total number of subdivision units is 1187, and a subdivision schematic diagram of the periodic unit is shown in fig. 2 (b). The observation points (0mm, -4mm,12mm) are arranged on the reflecting surface. Minimum time step Δ t of the entire calculation regions9.175740e-15s, a local solution can be achieved up to 5 time steps. FIG. 3 shows the comparison of the reflection coefficient at each time step with the simulation results for CST; and, table 1 shows the data result with the time scale base number set to 2, comparing the iteration time used by the divided regions of different levels.
TABLE 1 iterative time comparison for dividing time steps of different levels
Figure BDA0002476634330000052
The method selects Maxwell equation set as a basic numerical model, combines any high-order derivative (ADER) time stepping scheme, divides a calculation domain by adopting a reasonable tetrahedron/hexahedron mixed grid, each divided cell respectively and automatically determines a proper time iteration step length according to stability conditions, the size of any multiple time iteration step lengths in any proportion can be realized, each cell electromagnetic field quantity is iteratively updated according to the time iteration step length thereof until all cell field quantities iterate to a specified time point, and the obtained time-varying electromagnetic field quantity is post-processed to obtain corresponding S parameters, radar scattering cross section and electromagnetic field spatial distribution.

Claims (4)

1. An arbitrary high-order mixed grid time domain discontinuous Galerkin method of a multilevel local time stepping technology is characterized in that:
the method comprises the steps that firstly, a spatial multi-scale problem containing a local fine or high-electric structure is selected as an electromagnetic simulation model, and tetrahedral and hexahedral units are adopted to carry out spatial dispersion on corresponding regions to obtain structural information of the model;
secondly, establishing a matrix equation by using a Maxwell equation set as a basic control equation and adopting a discontinuous Galerkin technology, and performing time dispersion by adopting an arbitrary high-order derivative ADER iterative formula in time to obtain a display solution matrix iterative scheme;
thirdly, according to stability conditions approximately followed by the ADER explicit time scheme, sequentially dividing each subdivision unit into a plurality of different calculation domains according to any integer proportion, and automatically determining the iteration time step length met by each domain;
fourthly, sequentially carrying out time iteration on the calculation domains according to the time step from small to large;
and fifthly, finishing iterative solution of the electric field and the magnetic field according to the specified time steps, extracting electromagnetic field information on the observation surface, and acquiring corresponding parameters and spatial distribution of the system.
2. The arbitrary high-order mixed-lattice time-domain discontinuous Galerkin method of multistage local time stepping technique according to claim 1, characterized by: in the second step, following the standard analysis step of the discontinuous Galerkin time domain finite element method, an explicit semi-discrete form matrix equation is obtained:
Figure FDA0002476634320000011
the above equation is written as:
Figure FDA0002476634320000012
wherein u ═ e, h]TIs the unknown vector that needs to be solved for,
Figure FDA0002476634320000013
is the inverse matrix of the diagonal characteristics of the block,
Figure FDA0002476634320000014
is the right matrix of the system.
The specific expression of each matrix element:
Figure FDA0002476634320000021
Figure FDA0002476634320000022
Figure FDA0002476634320000023
Figure FDA0002476634320000024
Figure FDA0002476634320000025
Figure FDA0002476634320000026
in the above formula, mu represents the dielectric constant and permeability of the discrete unit, Ni、NjAs a finite element tetrahedral or hexahedral vector stackSpread basis functions of layer test basis functions and field magnitudes, Nj +Representing the relationship between the wave impedance and admittance of the test basis function of the adjacent body
Figure FDA0002476634320000027
And (3) completing dispersion by adopting an ADER iterative format in time, and performing Taylor series expansion on a time partial derivative term of a field value on each discrete unit:
Figure FDA0002476634320000028
through repeated derivation and back substitution, the required time derivative is converted into a problem that a space derivative value is obtained through solving a differential value:
Figure FDA0002476634320000029
finally, the iterative formula for obtaining the ADER time stepping scheme of n orders is
Figure FDA00024766343200000210
3. The arbitrary high-order mixed-lattice time-domain discontinuous Galerkin method of multistage local time stepping technique according to claim 1, characterized by: in the third step, computing domains are divided by adopting criteria, and the automatic selection of the time step length of each region is completed; for the tetrahedral subdivision grid, calculating the inscribed sphere radius of each unit, dividing the calculation domain into n parts according to the integer proportion of the minimum inscribed sphere radius in sequence, and determining the iteration time step length delta t of each calculation domain, wherein the calculation formula approximately satisfied is as follows:
Figure FDA0002476634320000031
where N is the order of the spatial basis function expansion, c0Is free spaceThe speed of light in between is reduced,ris the dielectric constant of the medium, d is the diameter of the inscribed sphere, and has the following definition:
Figure FDA0002476634320000032
4. the arbitrary high-order mixed-lattice time-domain discontinuous Galerkin method of multistage local time stepping technique according to claim 1, characterized by: in the fourth step, the calculation domains are respectively subjected to time iteration by using an ADER format from small to large according to the time step length, and can be arbitrarily divided into n calculation domains; the iteration for dividing the two calculation domains is implemented as follows:
(1) let Δ t of two regions be small time step Δ t respectivelysAnd a large time step Δ tlAnd the relation satisfied between the two time steps is Δ ts:ΔtlN is any positive integer;
(2) time step of Δ tsUpdating of small cell area field values
Figure FDA0002476634320000033
If the adjacent cells of the region satisfy the time step Δ tlThe second term needs to use the relevant interpolation information of the big unit; the term is represented in the original form of a taylor series expansion:
Figure FDA0002476634320000034
solving by using initial values of adjacent large units:
Figure FDA0002476634320000035
(3) time step of Δ tlThe same principle of updating format of large unit area field value can be obtained
Figure FDA0002476634320000036
At this time, the field value of the small unit required by the interface of the two regions is already calculated in (8), and corresponding transmission is carried out.
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CN112307644A (en) * 2020-11-20 2021-02-02 金陵科技学院 RCS (Radar Cross section) calculation method for electrically large-size target
CN113158492A (en) * 2021-05-14 2021-07-23 中国空气动力研究与发展中心计算空气动力研究所 Full-implicit double-time-step calculation method of time-varying electromagnetic field
CN113361138A (en) * 2021-07-08 2021-09-07 电子科技大学 Numerical solving method for nano surface plasmon non-local effect simulation
CN113591346A (en) * 2021-07-09 2021-11-02 上海大学 Self-adaptive region decomposition finite element method for electromagnetic field solution
CN116127699A (en) * 2022-11-23 2023-05-16 南京理工大学 Local time step arbitrary high-order precision full-wave solving method based on time domain spectral element method
CN116306036A (en) * 2023-05-18 2023-06-23 南京理工大学 Load balancing parallel method and medium of space-time mixed discrete DGTD
CN116522624A (en) * 2023-04-21 2023-08-01 南京理工大学 Efficient microwave device breakdown threshold prediction method based on discontinuous Galerkin
CN117452081A (en) * 2023-12-26 2024-01-26 国网天津市电力公司营销服务中心 Electromagnetic interference calculation method and device, storage medium and electronic terminal

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CN110580365A (en) * 2018-06-08 2019-12-17 南京理工大学 dynamic p-adaptive DG-FETD method based on laminated vector basis function

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US20150293246A1 (en) * 2014-04-09 2015-10-15 Thomas A. Dickens Frequency-domain augmented time-domain full wavefield inversion
CN110580365A (en) * 2018-06-08 2019-12-17 南京理工大学 dynamic p-adaptive DG-FETD method based on laminated vector basis function

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CN112307644A (en) * 2020-11-20 2021-02-02 金陵科技学院 RCS (Radar Cross section) calculation method for electrically large-size target
CN113158492A (en) * 2021-05-14 2021-07-23 中国空气动力研究与发展中心计算空气动力研究所 Full-implicit double-time-step calculation method of time-varying electromagnetic field
CN113158492B (en) * 2021-05-14 2021-08-20 中国空气动力研究与发展中心计算空气动力研究所 Full-implicit double-time-step calculation method of time-varying electromagnetic field
CN113361138A (en) * 2021-07-08 2021-09-07 电子科技大学 Numerical solving method for nano surface plasmon non-local effect simulation
CN113361138B (en) * 2021-07-08 2022-11-29 电子科技大学 Numerical solving method for nano surface plasmon non-local effect simulation
CN113591346A (en) * 2021-07-09 2021-11-02 上海大学 Self-adaptive region decomposition finite element method for electromagnetic field solution
CN116127699A (en) * 2022-11-23 2023-05-16 南京理工大学 Local time step arbitrary high-order precision full-wave solving method based on time domain spectral element method
CN116127699B (en) * 2022-11-23 2023-07-25 南京理工大学 Local time step arbitrary high-order precision full-wave solving method based on time domain spectral element method
CN116522624A (en) * 2023-04-21 2023-08-01 南京理工大学 Efficient microwave device breakdown threshold prediction method based on discontinuous Galerkin
CN116522624B (en) * 2023-04-21 2023-10-10 南京理工大学 Efficient microwave device breakdown threshold prediction method based on discontinuous Galerkin
CN116306036A (en) * 2023-05-18 2023-06-23 南京理工大学 Load balancing parallel method and medium of space-time mixed discrete DGTD
CN116306036B (en) * 2023-05-18 2023-08-11 南京理工大学 Load balancing parallel method and medium of space-time mixed discrete DGTD
CN117452081A (en) * 2023-12-26 2024-01-26 国网天津市电力公司营销服务中心 Electromagnetic interference calculation method and device, storage medium and electronic terminal
CN117452081B (en) * 2023-12-26 2024-04-30 国网天津市电力公司营销服务中心 Electromagnetic interference calculation method and device, storage medium and electronic terminal

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