CN104951580B - The time domain spectral element emi analysis method of unconditional stability and condition stability mixing - Google Patents
The time domain spectral element emi analysis method of unconditional stability and condition stability mixing Download PDFInfo
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Abstract
The invention discloses a kind of time domain spectral element emi analysis methods that unconditional stability and condition stability mix.In traditional time domain spectral element method, for complicated model, especially comprising Issues On Multi-scales, the sizing grid obtained with hexahedron subdivision is very uneven.In subsequent time iteration, in order to guarantee that algorithmic stability does not dissipate, it is necessary to time step be arranged according to the size of minimum subdivision grid, will lead to entirely solution in this way and take considerable time.The present invention proposes a kind of adaptive time domain spectral element method mixed based on unconditional stability with condition stability, small grid and big grid are found out automatically by adaptive method, common central difference schemes are used in big net region, Newmark- β difference scheme is used in the region of grid very little, therefore the Iteration of unconditional stability is just obtained, so overall time step-length can take very greatly, the solution time will greatly reduce in this way.
Description
Technical field
The invention belongs to the time domain approach that condition stability and unconditional stability mix, especially a kind of to be directed to multiple dimensioned electricity
The quick analytical technology of magnetic problem.
Background technique
1864, for Maxwell (Maxwell) on the Research foundation of forefathers, proposing can be basic to macroscopical electromagnetic field
Thus the math equation group that return road is summarized --- famous Maxwell equation has established the base of electromagnetic theory research
Plinth.Numerous technologies such as the research of Theory of Electromagnetic Field learns with penetrating into gradually, life science, medicine, material science and information science
The development of science and technology and the variation of human lives is greatly facilitated in scientific domain.
A very long time early stage, the research of Theory of Electromagnetic Field are dedicated to obtaining the analytic solutions of some problems, however completely
Be with the problem of analytical method solving it is extremely limited, not can solve what problem.Then, in order to solve the electricity in science and technology
Magnetic field problem, and developed some approximation methods and numerical method.But it is limited to design conditions at that time, it is unable to give full play
The effect of these methods enables certain problems cannot get substantive solution.With the rapid development of electronic computer technology, with height
Project Computer technology is means, in conjunction with Theory of Electromagnetic Field and calculates the various numerical methods that mathematics provides, has come into being one
Door cross discipline --- Computational electromagnetics.
When analyzing using Computational electromagnetics electromagnet phenomenon, corresponding electricity is established according to the characteristics of analysis object first
Magnetic, mathematical model.Then, it selects suitable algorithm and realizes on computers.Current Computational electromagnetics method can by solution domain point
It is divided into frequency domain method and time domain approach.Frequency domain method mainly has: the moment method based on the integral equation of electromagnetic problems
(MOM) and the FInite Element based on variation principle (FEM) etc.；Time domain approach mainly has: Finite-Difference Time-Domain Method (FDTD), time domain
FInite Element (FETD), time-domain integration method (TDIE) and time domain puppet spectral method (PSTD) etc..
Time domain spectral element method (Joon-Ho Lee and Qing Huo Liu, " A3-D Spectral-Element Time-
Domain Method for Electromagnetic Simulation,”IEEE Transactions on Microwave
Theory and Techniques., vol.55, no.5, pp.983-991, May 2007) it is regarded as a kind of special time domain
FInite Element, for the differential mode used due to time domain spectral element method for center difference, coefficient matrix contains only mass matrix, and due to
Selected basic function is orthogonal basis function in this method, so coefficient matrix is that block is diagonal, the very appearance that matrix inversion can become
Easily, with time domain finite element method ratio, this, which will greatly reduce, calculates the time.The discrete way that time domain spectral element method uses grid is song
Hexahedron is discrete, this can be fitted the electromagnetic structure of various complexity well, and because the discrete size of time domain spectral element net of justice lattice can
With very big, compared with time-domain finite difference, this will greatly reduce the unknown quantity of calculating.
The development of discontinuous Galerkin's Procedure (Discontinuous Galerkin, DG) there has been significant progress, this
A little methods are suitable for the large scale problem with labyrinth and uneven coal quality.These methods largely have benefited from 80 years
For the solution for the Neutron Transmission equation that the first half has Reed and Xi Er to propose.The most important feature of these methods is to allow base letter
Number (therefore, numerical solution) is discontinuous on the interface of different units.A set of local basic function is introduced on each unit and
It is not in entire zoning, and different types of unit such as hexahedron, prism or tetrahedron can be total in a model
It deposits, while the equation that can permit both sides calculates more flexible and convenient in this way using different difference schemes.
The time domain spectral element method of the existing multiple dimensioned electromagnetic problem of analysis mainly has the following two problems:
(1) using hexahedron with tetrahedron mixing subdivision, but hexahedron part uses time domain spectral element method, tetrahedral portions
Time domain finite element method is used, and the mass matrix that time domain finite element method generates is sparse disease, inverts and takes a significant amount of time again
In addition the very little of time step setting, so solving speed is slower.
(2) if different zones are all made of hexahedral mesh subdivision, when using time domain spectral element method, to guarantee that algorithm is steady
Qualitative, time step must be set according to the smallest size of mesh opening, and whole time iteration step number will be many in this way, cause to ask
It is slow to solve the time.
Summary of the invention
The purpose of the present invention is to provide the time domain spectral element emi analysis that a kind of unconditional stability and condition stability mix
Method, thus the quickly complicated multiple dimensioned electromagnetic problem of analysis.
The technical solution for realizing the aim of the invention is as follows: a kind of Time Domain Spectrum of unconditional stability and condition stability mixing
First emi analysis method, steps are as follows:
The first step carries out Geometric Modeling to the electromagnetic problem to be analyzed, and overall model is cutd open using bent hexahedron
Point, the vertex number of each individual cell, the number of coordinate and body are obtained after subdivision；
Second step finds out the hexahedron that size in the grid that subdivision obtains is less than setting value using the method for comparing side length,
It is marked as small size region, remaining grid is designated as large scale region；
Third step, on each point in the grid after electric field value to be defined on to overall model subdivision, and with time domain spectral element method
In GLL multinomial electric field is unfolded in tri- directions XYZ as Basis Function, substitute into time domain wave equation, and adopt
With the gold test of gal the Liao Dynasty, i.e. test basic function is identical as expansion basic function, obtains matrix equation.
Time term in equation is unfolded 4th step with time difference, in the region for the small size being marked using tool
There is the Newmark- β difference scheme of unconditional stability, other regions use the central difference schemes of condition stability, when carrying out
Between iteration when overall time step-length by centered difference region set, according to total each step direct solution of time step number, finally ask
Obtain time domain electric field value.
The bent hexahedral element side length that subdivision uses in step 1 is electromagnetic wavelength for 1/10 λ, λ.
It obtains that the size of grid is related in step 2 after the selection of setting value and subdivision, should guarantee less than this setting value
The ratio that the number of grid accounts for integral grid number is as small as possible, guarantees the size of minimum grid in large scale region with whole again
The ratio between the size of a region minimum grid is as big as possible.
GLL basic function form used in step 3 is as follows:
Wherein,J=0,1 ... N, LN(ξ) is N rank Legendre
Multinomial, by the node { ξ in ξ ∈ [- 1,1]j, j=0,1 ... N } and it is used as GLL point, they are equations(N+1) a root；
By electric field base function expansion, it is updated to time domain wave equation
It is tested using Galerkin method, i.e., test function is identical as basic function, obtains overall coefficient matrix, solves equation
In step 4, the equation after the region of small size is using Newmark- β difference scheme are as follows:
([T]+Δt2β[S])en+1=(2 [T]-Δ t2(1-2β)[S])en-([T]+Δt2β[S])en-1
Equation becomes after large scale region is using central difference schemes:
[T]en+1=(2 [T]-Δ t2[S])en-[T]en-1
Equation is solved, in each step time iteration, first solves the electric field at centered difference region, then solve Newmark-
The electric field of β differential area finally obtains total time domain electric field value.
Compared with prior art, the present invention its remarkable advantage: (1) by the bent hexahedron subdivision of model, can be very good to intend
Close the shape of complex object.(2) whole time step is no longer limited by minimum subdivision size of mesh opening, can be set compared with
Greatly, solving speed is greatly speeded up.
Detailed description of the invention
Fig. 1 is the structural schematic diagram of medium annulus resonant cavity.
Fig. 2 is the method for the present invention with traditional counted spectral contrast figure of central difference method.
Specific embodiment
The present invention is based on a kind of time domain spectral element emi analysis methods that unconditional stability and condition stability mix, and step is such as
Under:
Unified Model is carried out subdivision using bent hexahedral mesh by the first step, model facetization.
Second step is adaptively found out the hexahedron that size in grid is less than setting value, is marked.
Electric field is unfolded third step with basic function, substitutes into vector wave equation, and using the gold test of gal the Liao Dynasty.
Time term is unfolded with time difference for 4th step, solution matrix equation, is used in the region of small size with no item
The stable Newmark- β difference scheme of part, other regions use centered difference, first the solution in the progress region Newmark- β, then into
The subregional solution of the row equation of the ecentre.
The present invention will be further described with reference to the accompanying drawing.
The first step models the labyrinth of analysis, obtains required geological information.Then by model using song six
Face volume mesh carries out subdivision, and subdivision size hexahedral element side length is 1/10 λ (λ is electromagnetic wavelength).It is obtained after subdivision each
Vertex number and coordinate and the number of body of individual cell etc..
Second step sets out a threshold value, adaptively finds out the hexahedron that size in grid is less than setting value, is marked
Out, it is denoted as small size grid, it is remaining to be denoted as large scale grid.
Electric field is unfolded at node with GLL basic function third step,
In 1-D canonical reference unit ξ ∈ [- 1,1], we define N rank GLL (Gauss-Lobatto-Legendre, height
Si-Luo Batuo-Legendre) basic function are as follows:
Wherein, j=0,1 ... N, LN(ξ) is N rank Legnedre polynomial, L 'N(ξ) is its derivative.It will be in ξ ∈ [- 1,1]
Mesh point { ξj, j=0,1 ... N } and it is used as GLL point, they are equation (1- ξj 2)L′N(ξj(N+1) a root of)=0,
Basic function meets φj(ξii)=δijCharacteristic.
Solve vector wave equation
Using the gold test of gal the Liao Dynasty, i.e. test basic function is identical as expansion basic function, obtains compact schemes
Wherein,
Central difference schemes are used in large-sized region
[T]en+1=(2 [T]-Δ t2[S])en-[T]en-1,
Newmark- β difference scheme is used in the region of small size
([T]+Δt2β[S])en+1=(2 [T]-Δ t2(1-2β)[S])en-([T]+Δt2β|[S])en-1
The solution in the region Newmark- β is first carried out, then carries out the subregional solution of the equation of the ecentre, the electricity of each point finally can be obtained
Field value.
In order to verify the validity of the method for the present invention, the typical examples of a dielectric resonant chamber are analyzed below.
The dielectric ring of a dielectric constant 9.8, a1=207.25mm, a2=are placed in resonant cavity as shown in Figure 1
440.75mm, b=242mm, c=43mm, r1=9.0mm, r2=10.0mm, h=14.0mm, it is whole to use centered difference and adopt
The frequency spectrum being calculated with mixed method of the present invention is as shown in Figure 2, it is seen that the two result is coincide fine.It calculates time-consuming by table 1
It is shown, 32,16,8,4 process operation programs are respectively adopted, it can be seen that the method for the present invention is bigger than traditional method calculating time
It is big to save.
| Process number | Mixed method | Centered difference |
| 32 | 212s | 359s |
| 16 | 287s | 589s |
| 8 | 518s | 1103s |
| 4 | 768s | 2299s |
Table 1
Claims (4)
1. a kind of time domain spectral element emi analysis method of unconditional stability and condition stability mixing, its step are as follows:
The first step carries out Geometric Modeling to the electromagnetic problem to be analyzed, and overall model is carried out subdivision using bent hexahedron, is cutd open
/ after obtain the vertex number of each individual cell, the number of coordinate and body；
Second step finds out the hexahedron that size in the grid that subdivision obtains is less than setting value using the method for comparing side length, by it
Labeled as small size region, remaining grid is designated as large scale region；
Third step, on each point in the grid after electric field value to be defined on to overall model subdivision, and in time domain spectral element method
Electric field is unfolded in tri- directions XYZ as Basis Function for GLL multinomial, substitutes into time domain wave equation, and use gal
Distant gold test, i.e. test basic function is identical as expansion basic function, obtains matrix equation；
Time term in equation is unfolded 4th step with time difference, and using in the region for the small size being marked has nothing
The Newmark- β difference scheme of conditional stability, other regions use the central difference schemes of condition stability, change in the progress time
For when overall time step-length by centered difference region set, according to total each step direct solution of time step number, when finally acquiring
Domain electric field value.
2. the time domain spectral element emi analysis method of unconditional stability according to claim 1 and condition stability mixing,
Be characterized in that: the bent hexahedral element side length that subdivision uses in step 1 is electromagnetic wavelength for 1/10 λ, λ.
3. the time domain spectral element emi analysis method of unconditional stability according to claim 1 and condition stability mixing,
Be characterized in that: GLL basic function form used in step 3 is as follows:
Wherein,J=0,1 ... N, LN(ξ) is that N rank Legendre is multinomial
Formula, by the node { ξ in ξ ∈ [- 1,1]j, j=0,1 ... N } and it is used as GLL point, they are equations(N+1) a root；
By electric field base function expansion, it is updated to time domain wave equation
It is tested using Galerkin method, i.e., test function is identical as basic function, obtains overall coefficient matrix, solves equation
Wherein,
4. the time domain spectral element emi analysis method of unconditional stability according to claim 1 and condition stability mixing,
It is characterized in that: in step 4, the equation after the region of small size is using Newmark- β difference scheme are as follows:
([T]+Δt2β[S])en+1=(2 [T]-Δ t2(1-2β)[S])en-([T]+Δt2β[S])en-1
Wherein,
Equation becomes after large scale region is using central difference schemes:
[T]en+1=(2 [T]-Δ t2[S])en-[T]en-1
Equation is solved, in each step time iteration, first solves the electric field at centered difference region, then to solve Newmark- β poor
Subregional electric field finally obtains total time domain electric field value.
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| CN107239586B (en) * | 2016-03-29 | 2020-12-04 | 南京理工大学 | Area decomposition parallel method effective to unconditional stable time domain finite difference method |
| CN107305536B (en) * | 2016-04-22 | 2020-11-13 | 南京理工大学 | Mixed order time domain discontinuous Galois field gold method |
| CN107526856B (en) * | 2016-06-22 | 2022-01-14 | 南京理工大学 | Parallel explicit-implicit mixed discontinuous Galerkin time domain finite element method |
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| CN107729608B (en) * | 2017-09-13 | 2021-05-14 | 南京理工大学 | Short-gap gas discharge numerical simulation method based on time domain spectral element method |
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