CN109190169B - Three-dimensional time domain electromagnetism hybrid time domain intermittent Galerkin numerical method - Google Patents

Three-dimensional time domain electromagnetism hybrid time domain intermittent Galerkin numerical method Download PDF

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CN109190169B
CN109190169B CN201810869354.8A CN201810869354A CN109190169B CN 109190169 B CN109190169 B CN 109190169B CN 201810869354 A CN201810869354 A CN 201810869354A CN 109190169 B CN109190169 B CN 109190169B
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徐立
李星
李斌
杨中海
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University of Electronic Science and Technology of China
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Abstract

The invention belongs to the technical field of three-dimensional time domain electromagnetics numerical solution, and relates to a three-dimensional time domain electromagnetics hybrid time domain intermittent Galerkin numerical method based on a high-order laminated basis function. The invention introduces the cross-over discontinuous Galerkin method of the frequency domain into the time domain, and forms a full discrete form in a simpler way to improve the calculation performance; and constructing a high-order interpolation laminated basis function to obtain a high-precision numerical simulation result, and then constructing preprocessing of a p-type multiple grid to accelerate the solution of a global linear matrix so as to improve the calculation performance. When the method is used for solving the non-structural local encryption grid, the calculation performance is improved on the premise of ensuring the high-order precision.

Description

Three-dimensional time domain electromagnetism hybrid time domain intermittent Galerkin numerical method
Technical Field
The invention belongs to the technical field of three-dimensional time domain electromagnetics numerical solution, and relates to a three-dimensional time domain electromagnetics hybrid time domain intermittent Galerkin numerical method based on a high-order laminated basis function.
Background
Under a complex time domain electromagnetic environment, the transient electromagnetic field of multi-scale equipment is required to be solved. These multi-scale devices include various unstructured grids, both fine grids with dimensions much smaller than the wavelength and uniform grids that resemble or even far larger than the wavelength. Such as military aircraft, ships and warships, etc., the whole structure is large, but contains a lot of fine structures such as apertures and antennas. The local encryption grid with the fine grid has a serious influence on the time step and the computing performance of the time domain numerical algorithm. For example, the explicit time iteration format requires a very small time step to ensure its conditional stability, but greatly increases the computation time; although the implicit time iteration format is unconditionally stable and can increase the time step, it needs to solve a global linear system, resulting in memory consumption. In particular, for the high-order problem, as the number of model meshes increases, the dimension of the global matrix may be larger, and the global matrix is likely to be highly ill-conditioned, which causes great difficulty in matrix solving. Therefore, how to choose an efficient time domain electromagnetic numerical method to solve the local encryption grid is crucial.
However, conventional numerical methods, such as the finite difference time domain method, finite element time domain method, and finite volume time domain method, are often limited due to the grid size and convergence accuracy. In recent years, a time domain discontinuous Galerkin method appears in the field of time domain computational electromagnetics numerical values, and the method is widely applied due to the fact that a non-structural grid is supported and a high-precision and natural high-performance parallel technology exists, so that the method becomes a more popular research field at present. Therefore, we can see that the time-domain discontinuous galois method has many advantages over the traditional numerical method.
However, all the advantages of the time-domain discontinuous Galerkin method are based on the condition: although the field is discontinuous at the cell interface, each cell needs to maintain its own basis function. This condition leads to a fatal drawback thereof: the unknowns on the cell interfaces are repetitive, resulting in significantly more global unknowns being required to achieve the same accuracy than in classical time-domain finite element methods. For a uniform grid with a large size, the explicit time iteration format does not have a large problem, but for a large-scale problem with a large number of grids, the unknowns of the time-domain discontinuous galois method become large. Particularly, when the local encryption grid is solved, a smaller time step is needed by adopting an explicit time format, and further, the calculation time and the memory consumption are seriously increased. Therefore, for such local encryption grids, the existing methods mostly incorporate the implicit time iteration format.
However, how to solve the global matrix quickly and efficiently as described above remains a significant difficulty for the implicit time iterative format. And with the complicated structure of analysis, the precision of the low-order basis function can not meet the requirement of a designer, and the construction of the high-order basis function invisibly increases the number of global unknowns and the difficulty of matrix solution, so that an efficient time domain electromagnetic numerical method is urgently needed to solve the problem of the non-structural local encryption grid, and the calculation performance is improved on the premise of ensuring the high-order precision.
Disclosure of Invention
Aiming at the problems or the defects, when the high-efficiency time domain electromagnetic numerical method is used for solving the unstructured local encryption grid, the calculation performance is improved on the premise of ensuring high-order precision. The invention provides a three-dimensional time domain electromagnetism hybridization time domain intermittent Galerkin numerical method. By the method, the three-dimensional time domain Maxwell equation set of the unstructured local encryption grid can be efficiently solved, and the method has less global unknown quantity, higher precision and remarkable calculation performance.
The method specifically comprises the following steps:
a, according to the physical structure of a target electronic device, combining a working environment and boundary conditions to carry out simulation modeling on the target electronic device;
Step B, a tetrahedron unit is adopted to subdivide a three-dimensional solving area, and the surface dispersion and the volume dispersion must be compatible;
step C, selecting a finite element scalar laminated basis function, and unfolding the electromagnetic field and the hybridization quantity by using the basis function;
three-dimensional time domain electromagnetism hybridization time domainThe method of breaking Galerkin numerical value needs to solve
Figure BDA0001751747900000021
I.e., first forming a complex with only the amount of hybridization Λ h The global linear system is used to obtain the hybridization amount, and the electromagnetic field E is obtained according to the local system h And H h ，
Figure BDA0001751747900000022
Is a discontinuous finite element function space,
Figure BDA0001751747900000023
a finite element trace space;
step D, establishing a semi-discrete form of a three-dimensional hybrid time domain intermittent Galerkin numerical method in space;
introducing a hybridization amount to replace a numerical trace of a time domain intermittent Galerkin method, and adding a third conservation equation to ensure the stability of a global system, wherein the three-dimensional hybridization time domain intermittent Galerkin numerical method adopts a semi-discrete form on a space:
Figure BDA0001751747900000024
ε is the relative permittivity of the medium in the calculation region Ω, μ is the relative permeability of the medium in the calculation region Ω, n is the calculation region boundary
Figure BDA00017517479000000211
Above the outer normal unit vector, τ > 0 is the local stability factor. g inc A boundary equation representing the Absorption Boundary Condition (ABC),
Figure BDA0001751747900000025
and
Figure BDA0001751747900000026
is a test function.
Figure BDA0001751747900000027
Is tangential toA magnetic field.
Step E, in terms of time, considering the hybridization amount as a constant, only considering the time dispersion of the electromagnetic field, and combining the step D to form a full dispersion equation form, thereby obtaining a global linear system;
defining time step delta T, and comparing the total simulation time [0, T]Time steps t discrete to equal intervals n ＝nΔt,
Figure BDA0001751747900000028
Wherein the maximum value of N is N t I.e. the total number of time iterations. Let t n Electromagnetic field of time of day
Figure BDA0001751747900000029
t n+1 Electromagnetic field of time of day
Figure BDA00017517479000000210
With the implicit Crank-Nicolson time format, the time partial derivatives in the formula (1) are:
Figure BDA0001751747900000031
without the time partial derivative term, the approximation is:
Figure BDA0001751747900000032
in time, considering the time dispersion (2) and (3) of the electromagnetic field only and considering the constant hybridization quantity, combining the space semi-discrete form generated by the step D, and further deducing the space semi-discrete form
Figure BDA0001751747900000033
Wherein the equation of formula (4) is all with respect to t to the left n+1 Time of day, and y 1 And y 2 Is about t n At a time of y only 3 Containing t n And t n+1 The time of day. The specific right-end term form is:
Figure BDA0001751747900000034
first consider a locally linear system due to a tetrahedron
Figure BDA0001751747900000035
Above, there are 4 triangular face units, we define τ i The hybridization amount of all face units onΛ e . From the first two equations of equation (4), a locally linear system on a tetrahedron is derived as
Figure BDA0001751747900000036
Figure BDA0001751747900000037
Is t n+1 The electromagnetic field at the moment of time,
Figure BDA0001751747900000038
is t n The electromagnetic field at the moment.
Figure BDA0001751747900000039
And
Figure BDA00017517479000000310
is a local matrix that can be obtained by the basis function action. Combining the last equation of the formula (4) to obtain a basic equation forming a global linear system
Figure BDA00017517479000000311
Wherein
Figure BDA0001751747900000041
Is a local matrix obtainable by the action of basis functions, b e Is the local right-hand term. According to the formula (7),after stacking each tetrahedral unit in turn, a global linear system is obtained, i.e.
Figure BDA0001751747900000042
Matrix array
Figure BDA0001751747900000043
In the form of a global linear matrix, the matrix,yis the right end item.
And F, solving the hybridization quantity of the global linear system.
And further, after the step E is finished, preprocessing the p-type multiple grids constructed by the step E, and then performing the step F. The method comprises the following specific steps:
starting from equation (8), the global matrix is reconstructed according to the properties of the stacking basis functions
Figure BDA0001751747900000044
As follows
Figure BDA0001751747900000045
Sub-matrix
Figure BDA0001751747900000046
The self-acting matrix, which represents the low-order part of the stacked basis functions, can be seen as a coarse-grid matrix.
Figure BDA0001751747900000047
A self-action matrix representing the high order part of the stack basis functions. While
Figure BDA0001751747900000048
And
Figure BDA0001751747900000049
then it is the coupling matrix where the lower order interacts with the higher order basis functions. The whole
Figure BDA00017517479000000410
Can be seen as a fine grid matrix. Due to the fact that
Figure BDA00017517479000000411
Different basis functions are included, and the basis functions of different orders have different properties. Therefore, before solving the matrix, the matrix is solved
Figure BDA00017517479000000412
The diagonal bins of (a) are normalized. Then applying Schur decomposition, we can get
Figure BDA00017517479000000413
Wherein
Figure BDA00017517479000000414
Is the Schur complement matrix. First, an approximation is made to the Schur complement matrix, i.e.
Figure BDA00017517479000000415
Then, for two sub-matrixes
Figure BDA00017517479000000416
And
Figure BDA00017517479000000417
incomplete Choleski decomposition is performed and a number of numerical examples are shown for
Figure BDA00017517479000000418
And
Figure BDA00017517479000000419
the threshold values of the required rejection factors are 10 in each case when approximated -5 And 10 -2 Further obtain
Figure BDA00017517479000000420
And
Figure BDA00017517479000000421
combining the formula (10) to obtain the final p-type multi-grid preprocessing matrix
Figure BDA00017517479000000422
Before the time iteration starts, the global matrix is divided into
Figure BDA00017517479000000423
The time iteration solution can be used for each time after one decomposition; from equation (11), we can see that only the matrix needs to be aligned before the whole time iteration starts
Figure BDA00017517479000000424
And
Figure BDA00017517479000000425
performing incomplete Choleski decomposition for one time; the preprocessing technology of the p-type multiple grids proposed in the step is applied to the solving part of the global linear system of the step E, and the p-type multiple grids are obtained through the global linear system (8)ΛObtaining the mapping relation between the local hybridization amount and the global hybridization amount on each tetrahedronΛ e So as to obtain the electromagnetic field waiting coefficient of each unit according to the formula of a local linear system (6)
Figure BDA0001751747900000051
The invention introduces the cross-over discontinuous Galerkin method of the frequency domain into the time domain, and forms a full discrete form in a simpler way to improve the calculation performance; and constructing a high-order interpolation laminated basis function to obtain a high-precision numerical simulation result, and then constructing preprocessing of a p-type multiple grid to accelerate the solution of a global linear matrix so as to improve the calculation performance.
In conclusion, when the unstructured local encryption grid is solved, the calculation performance is improved on the premise of ensuring high-order precision.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a discontinuous finite element function space
Figure BDA00017517479000000510
And finite element trace space
Figure BDA00017517479000000511
Schematic representation of (a).
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
Referring to fig. 1, a numerical method for reducing the order of a three-dimensional microwave tube input/output window model based on function approximation adaptive error analysis includes the following steps:
a, carrying out simulation modeling on a target electronic device structure in combination with material characteristics;
and (3) according to the physical structure of the target electronic device, combining the working environment and the boundary condition to carry out simulation modeling on the target electronic device.
B, solving a domain by adopting tetrahedral mesh dispersion;
the use of tetrahedral units to subdivide the three-dimensional solution area is a well-known process and therefore this step is not described in detail. It should be noted that the surface dispersion and the volume dispersion must be compatible. Calculation region in the present embodiment
Figure BDA0001751747900000052
Is divided into N h A set of tetrahedral meshes
Figure BDA0001751747900000053
Wherein each unit is represented by i (i=1,2,3,…,N h ) Is shown, i.e.
Figure BDA0001751747900000055
And set of faces
Figure BDA0001751747900000056
Is formed by N f A triangular surface unit D f Is composed of, i.e.
Figure BDA0001751747900000057
C, selecting scalar laminated basis functions, and expanding the electromagnetic field and the hybridization quantity by using the basis functions;
firstly, in the three-dimensional time domain electromagnetism hybridization time domain intermittent Galerkin numerical method, the used intermittent finite element function space
Figure BDA0001751747900000058
And finite element trace space
Figure BDA0001751747900000059
Is a well-known space and its specific form will not be described in detail. To better illustrate the difference between the two, a schematic diagram of the two spatially different orders is given in fig. 2, where
Figure BDA0001751747900000061
And
Figure BDA0001751747900000062
representing the degrees of freedom of these two spaces, respectively.
The general time domain interrupted Galerkin method only needs to solve the electromagnetic field
Figure BDA0001751747900000063
However, the three-dimensional time-domain electromagnetism hybrid time-domain discontinuous Galerkin numerical method needs to be solved
Figure BDA0001751747900000064
I.e., first forming a complex with only the amount of hybridization Λ h The global linear system obtains the amount of hybridization, and then obtains an electromagnetic field (E) from the local system h ,H h )。
Selecting scalar stacking basis functions to correlate the electromagnetic field with the amount of hybridization (E) h ,H h ,Λ h ) By basis functionsAnd (4) unfolding. For each tetrahedron
Figure BDA0001751747900000065
Defining a local electromagnetic field as
Figure BDA00017517479000000617
And is
Figure BDA0001751747900000066
To achieve high-order precision and pre-processing techniques for p-type multigrid, scalar stacked basis functions are used
Figure BDA0001751747900000067
To spread out (14) the field components in each direction in the equation, i.e.
Figure BDA0001751747900000068
Wherein
Figure BDA0001751747900000069
And
Figure BDA00017517479000000610
is the desired coefficient of the electromagnetic field,
Figure BDA00017517479000000611
Of dimension d i (p +1) (p +2) (p + 3)/6. In this embodiment, a second order stacking base function is taken as an example for explanation. Therefore d i From the nature of the second order stacking basis function, we know that 10
Figure BDA00017517479000000612
Is of node type, and
Figure BDA00017517479000000613
is of the edge type. For a second order stacking base function specific form on a tetrahedronThe following were used:
Figure BDA00017517479000000614
defining the tetrahedron as W e ＝[E e ,H e ] T Corresponding to a coefficient to be solved ofW e And the number of unknowns is 6d i 。
Amount of hybridization Λ h Since the hybridization volume on each side is univalent, the hybridization volume on the set of sides can be expressed as:
Figure BDA00017517479000000615
where f is ∈ [1, N ∈ [ ] f ]Is a face unit
Figure BDA00017517479000000616
The corresponding local hybridization amount is Λ f The concrete form is as follows:
Figure BDA0001751747900000071
wherein u is f And w f Is a plane coordinate, and the unit vector n of the outer normal vector on the plane is u f ×w f . Here using scalar stacked basis functions
Figure BDA0001751747900000072
To spread out (18) the medium component of the formula, i.e.
Figure BDA0001751747900000073
Wherein
Figure BDA0001751747900000074
And
Figure BDA0001751747900000075
is the local hybridization quantity Λ f The coefficient to be calculated of (a) is,
Figure BDA0001751747900000076
has dimension d f (p +1) (p + 2)/2. In this embodiment, a second order stacking base function is taken as an example for explanation. Therefore d f From the nature of the quadratic stacking base function, we can see that the specific form of quadratic stacking base function for one plane is as follows:
Figure BDA0001751747900000077
defining local hybridization quantity Lambda f Coefficient to be solvedΛ f And the number of unknowns is 2d f 。
In summary, we have detailed the stacking basis functions for electromagnetic fields and hybridization. Finally, the advantage of the hybrid time domain intermittent Galerkin method compared with the time domain intermittent Galerkin method is demonstrated from the perspective of unknown quantity. As the unknown quantity number of the time domain interrupted Galerkin method is related to the electromagnetic field, the unknown quantity number on one tetrahedron is 6d i Thus over the entire solution area E h And H h The number of unknowns of (2) is 6d i N h . For the time-domain intermittent Galerkin method of hybridization, the unknown quantity number is only related to the hybridization quantity, and the unknown quantity number on one surface unit is 2d f Thus a over the entire panel set h The number of unknowns of (2 d) f N f . Note d i /d f In practical problems, the number of tetrahedrons after dispersion is often different from the number of planes (p +3)/3, but in general, we can estimate the relationship between the two as follows: n is a radical of h ＝N f /2。
The ratio of the number of unknowns in the time domain intermittent Galerkin method to the time domain intermittent Galerkin method in the cross is (p + 3)/2. The fact shows that along with the increase of the order, the hybrid time domain discontinuous Galerkin method needs less unknown quantity compared with the time domain discontinuous Galerkin method, and particularly for a complex model containing a plurality of grids, the advantage of the hybrid time domain discontinuous Galerkin method is more prominent.
D, establishing a semi-discrete form of a three-dimensional hybridization time domain intermittent Galerkin numerical method in space;
different from the spatial dispersion of the traditional time-domain intermittent Galerkin method, the hybrid time-domain intermittent Galerkin method needs to introduce hybrid quantity to replace the numerical trace of the time-domain intermittent Galerkin method, and needs to add a third conservation equation to ensure the stability of the global system. The derivation of the specific formula is similar to the frequency domain hybridization discontinuous Galerkin method, which is a well-known process and will not be further described here. The following only gives the semi-discrete form of the three-dimensional hybridization time domain discontinuous Galerkin numerical method:
Figure BDA0001751747900000081
where ε is the relative permittivity of the medium in the calculation region Ω, μ is the relative permeability of the medium in the calculation region Ω, and n is the calculation region boundary
Figure BDA00017517479000000812
Above the outer normal unit vector, τ > 0 is the local stability factor.
Figure BDA0001751747900000082
A boundary equation representing the Absorption Boundary Condition (ABC),
Figure BDA0001751747900000083
and
Figure BDA0001751747900000084
is a function of the tests that are to be performed,
Figure BDA0001751747900000085
is a tangential magnetic field.
(21) The formula is only a semi-discrete form in space, and the time partial derivative term in the formula is also required to be processed, namely the time discrete process of the following step E, so that a fully discrete form is obtained, and a global linear system is formed.
Step E, in terms of time, considering the hybridization quantity as a constant, only considering the time dispersion of the electromagnetic field, and combining the step D to form a full dispersion equation form so as to obtain a global linear system;
Defining time step length delta T, and comparing the total simulation time [0, T]Time steps dispersed as equal intervals
Figure BDA0001751747900000086
Wherein the maximum value of N is N t I.e. the total number of time iterations. Let t n Electromagnetic field of time of day
Figure BDA0001751747900000087
t n+1 Electromagnetic field of time of day
Figure BDA0001751747900000088
The second-order implicit Crank-Nicolson time format is adopted, and for the time partial derivative term in the formula (21):
Figure BDA0001751747900000089
without the time partial derivative term, the approximation is:
Figure BDA00017517479000000810
considering that the hybridization amount only exists on the surface unit and keeps a single value, the invention considers the hybridization amount as a constant to be solved, and further provides a simpler mode for constructing a global linear system. Namely, in time, considering only the time dispersion (22) and (23) of the electromagnetic field according to the constant hybridization quantity, combining the space semi-discrete form generated by the step D, and further deducing the following steps:
Figure BDA00017517479000000811
wherein (A) and (B)24) Equation of formula all with respect to t to the left n+1 Time of day, and y 1 And y 2 Is about t n At a time of y only 3 Containing t n And t n+1 The time of day. The specific right-end term form is:
Figure BDA0001751747900000091
from the above discrete process, we can see that the fully discrete form of the hybrid time domain discontinuous galois method includes two systems: a local linear system and a global linear system. In fact, the first two equations (24) can form a local linear system, while the third equation is the conservation condition of the hybrid time-domain intermittent Galerkin method, and a global linear system only containing the hybrid quantity can be obtained according to the conservation condition. Once the hybridization amount is resolved, the electromagnetic field of each unit can be obtained by a local linear system. We proceed further from the perspective of these two systems.
First consider a locally linear system. Due to a tetrahedron
Figure BDA0001751747900000092
Above, there are 4 triangular face units, we define τ i The hybridization amount of all face units onΛ e . From the first two equations of equation (24), we can derive a locally linear system on a tetrahedron as
Figure BDA0001751747900000093
Here, the
Figure BDA0001751747900000094
Is t n+1 The electromagnetic field at the moment of time,
Figure BDA0001751747900000095
is t n The electromagnetic field at the moment.
Figure BDA0001751747900000096
And
Figure BDA0001751747900000097
is a local matrix that can be obtained by the basis function action. In combination with the last equation of equation (24), we can obtain the basic equation forming a global linear system
Figure BDA0001751747900000098
Wherein
Figure BDA0001751747900000099
Is a local matrix obtainable by the action of basis functions, b e Is the local right-hand term. After stacking each tetrahedral unit in turn according to equation (27), we can obtain a global linear system, i.e.
Figure BDA00017517479000000910
Matrix array
Figure BDA00017517479000000911
It is a global linear matrix that is,yis the right end item. Once we have obtained through the global linear system (28)ΛAccording to the mapping relation between the local hybridization amount and the global hybridization amount on each tetrahedron, the local hybridization amount and the global hybridization amount on each tetrahedron can be obtainedΛ e So as to obtain the electromagnetic field coefficient of each unit according to the formula of a local linear system (26)
Figure BDA00017517479000000912
So far, we have detailed the whole process of the three-dimensional hybridization time domain discontinuous Galerkin numerical method.
Step F, constructing a preprocessing technology of p-type multiple grids to accelerate the solving of the global linear system of the step E
Although step E has shown a fully discrete form of the three-dimensional hybrid time-domain discontinuous galois numerical method, it is a key aspect of the invention how to efficiently accelerate the solution of the global linear system (28). As mentioned above, for such local encryption grids, although the implicit time iteration format is unconditionally stable and can increase the time step, it needs to solve a global linear system, resulting in memory consumption. In particular, for the high-order problem, as the number of model meshes increases, the dimension of the global matrix may be larger, and the global matrix is likely to be highly ill-conditioned, which causes great difficulty in matrix solving. For the solution of a matrix equation, a good preprocessing matrix is the key to accelerate convergence and stability of the numerical iteration method. However, some existing preprocessing techniques often cause very slow convergence or even non-convergence of the iterative method due to the large dimension of the matrix or the negative qualitative and highly ill-conditioned nature of the matrix, so that it is extremely difficult to efficiently solve the large sparse matrix. For example, the direct method performs LU decomposition on the negative ill-conditioned matrix and then performs back substitution solution, which results in limited sparsity.
Since the convergence rate of the multigrid method is almost independent of the dimension of the matrix, a p-type multigrid preprocessing technology is adopted to accelerate the solution of the global linear system of the E step. Compared with the traditional h-type multiple grid, the p-type multiple grid is limited by the grid size, the two sets of thick and thin grids are constructed by the stacking property of the basis function, and the flexibility is stronger. Based on the equation (28), the global matrix is reconstructed according to the property of the second-order stacking base function
Figure BDA0001751747900000101
As follows
Figure BDA0001751747900000102
Sub-matrix here
Figure BDA00017517479000001020
A self-action matrix representing the basis functions of the nodes of the first order, and mayAs a coarse grid matrix.
Figure BDA0001751747900000104
A self-acting matrix representing the high order portion of the second order stacking-type basis function. While
Figure BDA0001751747900000105
And
Figure BDA0001751747900000106
then it is the coupling matrix where the lower order interacts with the higher order basis functions. The whole
Figure BDA0001751747900000107
Can be seen as a fine grid matrix. Due to the fact that
Figure BDA0001751747900000108
Different basis functions are included, and the basis functions of different orders have different properties. Therefore, before solving the matrix, the matrix is solved
Figure BDA0001751747900000109
The diagonal bins of (a) are normalized. Then, using Schur decomposition, we can obtain:
Figure BDA00017517479000001010
wherein
Figure BDA00017517479000001011
Is the Schur complement matrix. It is well known that the closer the preprocessing matrix is to the
Figure BDA00017517479000001012
The inverse matrix of (2) is more efficient in preprocessing, but the computational cost is increased. Therefore, to reduce the computational overhead, we first approximate the Schur complement matrix, i.e.
Figure BDA00017517479000001013
Then to twoSub-matrix
Figure BDA00017517479000001014
And
Figure BDA00017517479000001015
incomplete Choleski decomposition is performed and a number of numerical examples are shown for
Figure BDA00017517479000001016
And
Figure BDA00017517479000001017
the threshold values of the required rejection factors are 10 in each case when approximated -5 And 10 -2 Further obtain
Figure BDA00017517479000001018
And
Figure BDA00017517479000001019
and combining the formula (30) to obtain a final p-type multi-grid preprocessing matrix:
Figure BDA0001751747900000111
the three-dimensional hybridization time domain intermittent Galerkin numerical method provided by the invention has the advantage that the global matrix is formed in each time iteration
Figure BDA0001751747900000112
Is invariant and is a sparse symmetric matrix. Therefore, the global matrix is only required to be set before the time iteration starts
Figure BDA0001751747900000113
And the decomposition is carried out once, so that the time iteration solution of each time can be used. From equation (31), we can see that only the matrix needs to be aligned before the whole time iteration starts
Figure BDA0001751747900000114
And
Figure BDA0001751747900000115
an incomplete Choleski decomposition was performed once. By applying the preprocessing technology of the p-type multiple grids provided by the step to the solving part of the global linear system in the step E, the calculation time and the memory consumption can be further reduced, so that the solving speed of the global linear system is remarkably increased, and the calculation performance of the whole three-dimensional hybrid time domain intermittent Galerkin numerical method is improved.

Claims (2)

1. A three-dimensional time domain electromagnetism hybridization time domain intermittent Galerkin numerical method specifically comprises the following steps:
A, according to the physical structure of a target electronic device, combining a working environment and boundary conditions to carry out simulation modeling on the target electronic device;
step B, a tetrahedron unit is adopted to subdivide a three-dimensional solving area, and the surface dispersion and the volume dispersion must be compatible;
step C, selecting a finite element scalar laminated basis function, and unfolding the electromagnetic field and the hybridization quantity by using the basis function;
three-dimensional time domain electromagnetism hybrid time domain intermittent Galerkin numerical method needs to be solved
Figure FDA0003635998430000011
I.e., first forming a complex with only the amount of hybridization Λ h The global linear system is used to obtain the hybridization amount, and the electromagnetic field E is obtained according to the local system h And H h ，
Figure FDA0003635998430000012
Is a discontinuous finite element function space,
Figure FDA0003635998430000013
a finite element trace space;
step D, establishing a semi-discrete form of a three-dimensional hybrid time domain intermittent Galerkin numerical method in space;
introducing a hybridization amount to replace a numerical trace of a time domain intermittent Galerkin method, and adding a third conservation equation to ensure the stability of a global system, wherein the three-dimensional hybridization time domain intermittent Galerkin numerical method adopts a semi-discrete form on a space:
Figure FDA0003635998430000014
ε is the relative permittivity of the medium in the calculation region Ω, μ is the relative permeability of the medium in the calculation region Ω, n is the calculation region boundary
Figure FDA0003635998430000015
Outer normal unit vector of (1), τ>0 is a local stability factor which is,
Figure FDA0003635998430000016
A boundary equation representing the absorption boundary condition,
Figure FDA0003635998430000017
and
Figure FDA0003635998430000018
is a function of the tests that are to be performed,
Figure FDA0003635998430000019
is a tangential magnetic field and is characterized in that,
Figure FDA00036359984300000110
is a calculation region
Figure FDA00036359984300000111
An approximation of;
step E, in terms of time, considering the hybridization amount as a constant, only considering the time dispersion of the electromagnetic field, and combining the step D to form a full dispersion equation form, thereby obtaining a global linear system;
defining time step length delta T, and converting the total simulation time [0, T]Time steps t discrete to equal intervals n ＝n△t,
Figure FDA00036359984300000112
Wherein the maximum value of N is N t I.e. the total number of time iterations; let t n Electromagnetic field of time of day
Figure FDA00036359984300000113
t n+1 Electromagnetic field of time of day
Figure FDA00036359984300000114
With the implicit Crank-Nicolson time format, the time partial derivatives in the formula (1) are:
Figure FDA00036359984300000115
without the time partial derivative term, the approximation is:
Figure FDA00036359984300000116
in time, considering only the time dispersion (2) and (3) of the electromagnetic field, considering the constant hybridization quantity, combining the space semi-discrete form generated by the D step, and further deducing the following steps:
Figure FDA0003635998430000021
wherein the equation of formula (4) is all with respect to t to the left n+1 Time of day, and y 1 And y 2 Is about t n At a time of y only 3 Containing t n And t n+1 Time of day; the specific right-end term form is:
Figure FDA0003635998430000022
first consider a locally linear system due to a tetrahedron
Figure FDA0003635998430000023
Contains 4 triangular surface units, define tau i The hybridization amount of all face units onΛ e From the first two equations of equation (4), the local linear system on a tetrahedron is derived as:
Figure FDA0003635998430000024
Figure FDA0003635998430000025
Is t n+1 The electromagnetic field at the moment of time,
Figure FDA0003635998430000026
is t n An electromagnetic field at a time;
Figure FDA0003635998430000027
and
Figure FDA0003635998430000028
the method is characterized in that a local matrix obtained through the function of a basis function is combined with the last equation of the formula (4) to obtain a basic equation forming a global linear system:
Figure FDA0003635998430000029
wherein
Figure FDA00036359984300000210
Is a local matrix obtainable by the action of basis functions, b e Is a local right-hand term; according to the formula (7), after stacking each tetrahedral unit in turn, a global linear system is obtained, i.e.
Figure FDA00036359984300000211
Matrix array
Figure FDA00036359984300000212
In the form of a global linear matrix, the matrix,yis the right end term;
and F, solving the hybridization quantity of the global linear system.
2. The three-dimensional time-domain electromagnetism hybrid time-domain intermittent Galerkin numerical method of claim 1, characterized in that: and D, after the step E is finished, preprocessing the constructed p-type multiple grids, and then performing the step F, wherein the steps are as follows:
starting from equation (8), the global matrix is reconstructed according to the properties of the stacking basis functions
Figure FDA0003635998430000031
As follows
Figure FDA0003635998430000032
Sub-matrix
Figure FDA0003635998430000033
A self-action matrix representing the low-order part of the stacking basis functions and can be regarded as a coarse grid matrix;
Figure FDA0003635998430000034
a self-action matrix representing a high order portion of the stack basis functions; while
Figure FDA0003635998430000035
And
Figure FDA0003635998430000036
then is a coupling matrix where the low order and high order basis functions interact; the whole
Figure FDA0003635998430000037
Can be seen as a fine grid matrix, since
Figure FDA0003635998430000038
The different base functions are contained, and the base functions with different orders have different properties, so that the matrix is solved before
Figure FDA0003635998430000039
Normalizing the angle element and then applying Schur decomposition to obtain:
Figure FDA00036359984300000310
wherein
Figure FDA00036359984300000311
Is a Schur complement matrix; first, an approximation is made to the Schur complement matrix, i.e.
Figure FDA00036359984300000312
Then, for two sub-matrixes
Figure FDA00036359984300000313
And
Figure FDA00036359984300000314
incomplete Choleski decomposition is performed and a number of numerical examples are shown for
Figure FDA00036359984300000315
And
Figure FDA00036359984300000316
the threshold values of the required rejection factors are 10 in each case when approximated -5 And 10 -2 Further obtain
Figure FDA00036359984300000317
And
Figure FDA00036359984300000318
and (5) combining the formula (10) to obtain a final p-type multi-grid preprocessing matrix:
Figure FDA00036359984300000319
before the time iteration starts, the global matrix is divided into
Figure FDA00036359984300000320
The time iteration solution can be used for each time after one decomposition; from equation (11), we can see that only the matrix needs to be aligned before the whole time iteration starts
Figure FDA00036359984300000321
And
Figure FDA00036359984300000322
performing incomplete Choleski decomposition for one time; the preprocessing technology of the p-type multiple grids proposed in the step is applied to the solving part of the global linear system of the step E, and the preprocessing technology is obtained through the global linear system (8)ΛObtaining the mapping relation between the local hybridization amount and the global hybridization amount on each tetrahedronΛ e So as to obtain the electromagnetic field waiting coefficient of each unit according to the formula of a local linear system (6)
Figure FDA00036359984300000323
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CN109684740B (en) * 2018-12-27 2022-10-14 电子科技大学 Electromagnetic multi-scale calculation method based on mixed grid and time step
CN111079278B (en) * 2019-12-10 2022-06-03 电子科技大学 Processing method for three-dimensional time domain hybridization discontinuous Galerkin method with additional electromagnetic source item
CN111241728B (en) * 2020-01-03 2023-05-05 电子科技大学 Intermittent Galerkin finite element numerical solution method of Euler equation
CN113221392B (en) * 2021-01-26 2023-12-19 中国地震局工程力学研究所 Construction method of non-uniform viscous acoustic wave propagation model in infinite domain
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
CN115374673A (en) * 2022-08-26 2022-11-22 宁波德图科技有限公司 Time domain heat conduction simulation method and storage medium

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107515955A (en) * 2016-06-16 2017-12-26 南京理工大学 Based on the EB time domain finite element methods that continuously discontinuous gal the Liao Dynasty gold mixes
CN107526856A (en) * 2016-06-22 2017-12-29 南京理工大学 Can be parallel aobvious implicitly mix the golden time-domain finite element method of discontinuous gal the Liao Dynasty
CN107944141A (en) * 2017-11-24 2018-04-20 电子科技大学 Time-domain calculation electromagnetism numerical method based on hybridization time-discontinuous Galerkin method
CN108052738A (en) * 2017-12-13 2018-05-18 电子科技大学 The golden analysis method of high-order part unconditional stability time-discontinuous gal the Liao Dynasty of dispersive medium

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
EP2378444B1 (en) * 2010-04-13 2015-07-29 CST-Computer Simulation Technology AG Method, device and computer program product for determining an electromagnetic near field of a field excitation source for an electrical system
GB2510338A (en) * 2013-01-30 2014-08-06 Shravan Hanasoge Wave propagation using the Lattice Boltzmann Method and imaging using the propagated wave

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107515955A (en) * 2016-06-16 2017-12-26 南京理工大学 Based on the EB time domain finite element methods that continuously discontinuous gal the Liao Dynasty gold mixes
CN107526856A (en) * 2016-06-22 2017-12-29 南京理工大学 Can be parallel aobvious implicitly mix the golden time-domain finite element method of discontinuous gal the Liao Dynasty
CN107944141A (en) * 2017-11-24 2018-04-20 电子科技大学 Time-domain calculation electromagnetism numerical method based on hybridization time-discontinuous Galerkin method
CN108052738A (en) * 2017-12-13 2018-05-18 电子科技大学 The golden analysis method of high-order part unconditional stability time-discontinuous gal the Liao Dynasty of dispersive medium

Non-Patent Citations (7)

* Cited by examiner, † Cited by third party
Title
Li Kun 等."A Reduced-Order Discontinuous Galerkin Method Based on POD for Electromagnetic Simulation".《IEEE Transactions on Antennas and Propagation ( Volume: 66, Issue: 1, Jan. 2018) 》.2017, *
Qiang Ren 等."An Improved Subdomain Level Nonconformal Discontinuous Galerkin Time Domain (DGTD) Method for Materials With Full-Tensor Constitutive Parameters".《https://ieeexplore.ieee.org/document/7862232》.2017, *
买文鼎等.改进的二维三维混合时域不连续伽辽金方法.《电波科学学报》.2018,(第01期), *
刘利民."不连续伽辽金时域面积分方程方法的研究".《中国优秀硕士学位论文全文数据库 信息科技辑》.2017, *
叶珍宝等.基于高阶叠层矢量基函数的E-H时域有限元方法分析谐振腔和波导结构.《计算物理》.2016,(第03期), *
吕宏强等.间断伽辽金方法在可压缩流数值模拟中的应用研究综述.《空气动力学学报》.2017,(第04期), *
吴欣等.刚塑性无网格伽辽金方法近似方案研究.《塑性工程学报》.2008,(第03期), *

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