CN104951580A - Unconditional stability and conditional stability mixed time domain spectral element electromagnetic analyzing method - Google Patents

Abstract

The invention discloses an unconditional stability and conditional stability mixed time domain spectral element electromagnetic analyzing method. In a traditional time domain spectral element method, for complex models, especially under the condition of containing multiscale problems, the sizes of meshes obtained through hexahedral subdivision are not uniform. In the subsequent time iteration, in order to ensure that an algorithm is stable and not diverging, the time step is set according to the size of the minimum subdivision mesh, and therefore the whole solution process wastes a large amount of time. The invention provides the self-adapting unconditional stability and conditional stability mixed time domain spectral element method. Small meshes and large meshes are automatically found out through the self-adapting method, an ordinary central difference format is used for the large mesh area, and a Newmark-beta difference format is used for the small mesh area, so that the unconditionally stable iteration format is obtained, the whole time step can be very large, and the solution time is greatly shortened.

Description

The time domain spectral element emi analysis method of unconditional stability and condition stability mixing

Technical field

The invention belongs to the time domain approach, particularly a kind of express-analysis technology for multiple dimensioned electromagnetic problem of condition stability and unconditional stability mixing.

Background technology

1864, Maxwell (Maxwell), on the Research foundation of forefathers, proposed math equation group---the famous Maxwell equation can summarized the basic return road of macroscopical electromagnetic field, has established the basis of electromagnetic theory research thus.Numerous technological sciences fields such as the research of Theory of Electromagnetic Field is learned with gradually penetrating into, life science, medical science, material science and information science, facilitate the development of science and technology and the change of human lives greatly.

The early stage a very long time, the analytic solution obtaining some problems are devoted in the research of Theory of Electromagnetic Field, but are very limited by the problem of analytical method solving completely, can not solve what problem.So, in order to solve the electromagnetic problems in science and technology, developed again some approximation methods and numerical method.But, be limited to design conditions at that time, the effect of these methods cannot be given full play to, make some problem can not get substantial solution.Along with the develop rapidly of electronic computer technology, with high-performance calculation machine technology for means, in conjunction with the various numerical methods that Theory of Electromagnetic Field and computational mathematics provide, arisen at the historic moment a cross discipline---Computational electromagnetics.

When using Computational electromagnetics to analyze electromagnet phenomenon, first set up corresponding electromagnetism, mathematical model according to the feature of analytic target.Then, select suitable algorithm and realize on computers.Current Computational electromagnetics method is divided by solution territory can be divided into frequency domain method and time domain approach.Frequency domain method mainly contains: the method for moment (MOM) based on the integral equation of electromagnetic problems and the finite element method (FEM) etc. based on variational principle; Time domain approach mainly contains: the pseudo-spectral method (PSTD) of Finite-Difference Time-Domain Method (FDTD), time-domain finite element method (FETD), time-domain integration method (TDIE) and time domain 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, May2007) a kind of special time-domain finite element method can be thought, due to difference centered by the differential mode that time domain spectral element method adopts, matrix of coefficients is only containing mass matrix, again because basis function selected in the method is orthogonal basis function, so matrix of coefficients is block diagonal angle, what matrix inversion can become is easy to, with time domain finite element method ratio, this will greatly reduce computing time.Time domain spectral element method is that bent hexahedron is discrete to the discrete way that grid adopts, this can the electromagnetic structure of the various complexity of matching well, again can be very large because of the size that time domain spectral element net of justice lattice are discrete, 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, and these methods are applicable to the large scale problem with labyrinth and uneven ature of coal.These methods have benefited from solving of the Neutron Transmission equation that the eighties, the first half had Reed and Xi Er to propose to a great extent.The most important feature of these methods allows basis function (therefore, numerical solution) discontinuous on the interface of different units.Each unit is introduced a set of local basis function instead of in whole zoning, and dissimilar unit such as hexahedron, prism or tetrahedron can coexist in a model, the equation on both sides can be allowed to use different difference schemes simultaneously, calculate more flexible like this.

Mainly there is following two problems in the time domain spectral element method of the multiple dimensioned electromagnetic problem of existing analysis:

(1) adopt hexahedron with tetrahedron mixing subdivision, but hexahedron part adopts time domain spectral element method, tetrahedral portions adopts time domain finite element method, and the mass matrix that time domain finite element method generates is sparse disease, invert and spend the plenty of time to add the very little of time step setting, so solving speed is slower.

(2) if zones of different all adopts hexahedral mesh subdivision, when using time domain spectral element method, for ensureing algorithm stability, time step must set according to minimum size of mesh opening, the time iteration step number of such entirety will be a lot, cause solving the time slow.

Summary of the invention

The object of the present invention is to provide a kind of time domain spectral element emi analysis method of unconditional stability and condition stability mixing, thus the complicated multiple dimensioned electromagnetic problem of express-analysis.

The technical solution realizing the object of the invention is: a kind of time domain spectral element emi analysis method of unconditional stability and condition stability mixing, and step is as follows:

The first step, carries out Geometric Modeling to the electromagnetic problem that will analyze, and is adopted by block mold bent hexahedron to carry out subdivision, obtains the numbering of the summit numbering of each individual cell, coordinate and body after subdivision;

Second step, in adopt the method comparing the length of side to find out grid that subdivision obtains, size is less than the hexahedron of setting value, and be labeled as small size region, residue grid is designated as large scale region;

3rd step, electric field value is defined on each point in the grid after block mold subdivision, and electric field is launched in XYZ tri-directions as Basis Function with the GLL polynomial expression in time domain spectral element method, substitute into time domain wave equation, and adopt the gold test of gal the Liao Dynasty, namely test basis function identical with expansion basis function, obtain matrix equation.

4th step, time term time difference in equation is launched, the Newmark-β difference scheme with unconditional stability is adopted in the undersized region be marked, other regions adopt the central difference schemes of condition stability, when carrying out time iteration, overall time step sets by central difference region, according to each step direct solution of total time step number, finally try to achieve time domain electric field value.

The bent hexahedral element length of side that in step one, subdivision adopts is 1/10 λ, λ is electromagnetic wavelength.

In step 2 setting value choose relevant with the size obtaining grid after subdivision, should ensure that the number of the grid being less than this setting value accounts for the ratio of integral grid number little as far as possible, ensure that the size of minimum grid in large scale region is large as far as possible with the ratio of the size of whole region minimum grid again.

The GLL basis function form used in step 3 is as follows:

${\mathrm{\Φ}}_{\mathrm{rst}}^{\mathrm{\ξ}}=\widehat{\mathrm{\ξ}}{\mathrm{\φ}}_r^{\left(N_{\mathrm{\ξ}}\right)}\left(\mathrm{\ξ}\right){\mathrm{\φ}}_s^{\left(N_{\mathrm{\η}}\right)}\left(\mathrm{\η}\right){\mathrm{\φ}}_t^{\left(N_{\mathrm{\ζ}}\right)}\left(\mathrm{\ζ}\right)$

${\mathrm{\Φ}}_{\mathrm{rst}}^{\mathrm{\η}}=\widehat{\mathrm{\η}}{\mathrm{\φ}}_r^{\left(N_{\mathrm{\ξ}}\right)}\left(\mathrm{\ξ}\right){\mathrm{\φ}}_s^{\left(N_{\mathrm{\η}}\right)}\left(\mathrm{\η}\right){\mathrm{\φ}}_t^{\left(N_{\mathrm{\ζ}}\right)}\left(\mathrm{\ζ}\right)$

${\mathrm{\Φ}}_{\mathrm{rst}}^{\mathrm{\ζ}}=\widehat{\mathrm{\ζ}}{\mathrm{\φ}}_r^{\left(N_{\mathrm{\ξ}}\right)}\left(\mathrm{\ξ}\right){\mathrm{\φ}}_s^{\left(N_{\mathrm{\η}}\right)}\left(\mathrm{\η}\right){\mathrm{\φ}}_t^{\left(N_{\mathrm{\ζ}}\right)}\left(\mathrm{\ζ}\right)$

Wherein, ${\mathrm{\Φ}}_j^{\left(N\right)}\left(\mathrm{\ξ}\right)=\frac{1}{N(N+1)L_N\left({\mathrm{\ξ}}_j\right)}\frac{(1-{\mathrm{\ξ}}^2)L_N^{\′}\left(\mathrm{\ξ}\right)}{\mathrm{\ξ}-{\mathrm{\ξ}}_j},$ J=0,1, LN, L _n(ξ) be N rank Legendre polynomial expressions, by the node { ξ in ξ ∈ [-1,1] _j, j=0,1, LN} are as GLL point, and they are equations $(1-{\mathrm{\ξ}}_j^2)L_N^{\′}\left({\mathrm{\ξ}}_j\right)=0$ (N+1) individual root;

By electric field base function expansion, be updated to time domain wave equation

Employing Galerkin method is tested, and namely trial function is identical with basis function, obtains overall coefficient matrix, solving equation

$\left[T\right]\frac{d^{2}e}{{\mathrm{dt}}^2}+\left[S\right]e=0$

In step 4, after undersized region adopts Newmark-β difference scheme, equation is:

([T]+Δt ²β[S])e ⁿ⁺¹＝(2[T]-Δt ²(1-2β)[S])e ⁿ-([T]+Δt ²β[S])e ^n-1

After large scale region adopts central difference schemes, equation becomes:

[T]e ⁿ⁺¹＝(2[T]-Δt ²[S])e ⁿ-[T]e ^n-1

Solving equation, in each step time iteration, first solves the electric field at central difference region place, then solves the subregional electric field of Newmark-β difference, finally obtains total time domain electric field value.

The present invention compared with prior art, its remarkable advantage: (1), can well the profile of matching complex object by the bent hexahedron subdivision of model.(2) overall time step is no longer by the restriction of minimum subdivision size of mesh opening, and what can arrange is comparatively large, greatly accelerates solving speed.

Accompanying drawing explanation

Fig. 1 is the structural representation of medium annulus resonator cavity.

Fig. 2 is the spectral contrast figure that the inventive method follows traditional central difference method to calculate.

Embodiment

The present invention is based on a kind of time domain spectral element emi analysis method of unconditional stability and condition stability mixing, step is as follows:

The first step, model facetization, adopts bent hexahedral mesh to carry out subdivision by Unified Model.

Second step, self-adaptation finds out the hexahedron that size in grid is less than setting value, is marked.

3rd step, launches electric field basis function, substitutes into vector wave equation, and adopts the gold test of gal the Liao Dynasty.

4th step, solution matrix equation, launches time term time difference, the Newmark-β difference scheme with unconditional stability is adopted in undersized region, other regions adopt central difference, first carry out solving of Newmark-β region, then carry out that the equation of the ecentre is subregional to be solved.

Below in conjunction with accompanying drawing, the present invention will be further described.

The first step, carries out modeling to the labyrinth analyzed, obtains required geological information.Then adopted by model bent hexahedral mesh to carry out subdivision, the subdivision size hexahedral element length of side is 1/10 λ (λ is electromagnetic wavelength).The summit numbering of each individual cell and the numbering etc. of coordinate and body is obtained after subdivision.

Second step, set out a threshold value, self-adaptation finds out the hexahedron that size in grid is less than setting value, is marked, and is designated as small size grid, is remainingly designated as large scale grid.

3rd step, launches electric field at node place GLL basis function,

In 1-D canonical reference unit ξ ∈ [-1,1], we define N rank GLL (Gauss-Lobatto-Legendre, Gauss-Luo Batuo-Legendre) basis function and are:

${\mathrm{\φ}}_j^{\left(N\right)}\left(\mathrm{\ξ}\right)=\frac{-1}{N(N+1)L_N\left({\mathrm{\ξ}}_j\right)}\frac{(1-{\mathrm{\ξ}}^2){L_N}^{\′}\left(\mathrm{\ξ}\right)}{(\mathrm{\ξ}-{\mathrm{\ξ}}_j)}$

Wherein, j=0,1, LN, L _n(ξ) be N rank Legendre polynomials, L _n' (ξ) be its derivative.By the net point { ξ in ξ ∈ [-1,1] _j, j=0,1, LN} are as GLL point, and they are equations (N+1) individual root, basis function meets φ _j(ξ _i)=δ _ijcharacteristic.

Solve vector wave equation

$\▿\×\▿\×\stackrel{\→}{E}=-\mathrm{\μ\ϵ}\frac{{\∂}^2\stackrel{\→}{E}}{{\∂t}^2}$

Adopt the gold test of gal the Liao Dynasty, namely test basis function identical with expansion basis function, obtain compact schemes

$\left[S\right]e+\left[T\right]\frac{d^{2}e}{{\mathrm{dt}}^2}=0$

Wherein, ${\left[S\right]}_{\mathrm{ij}}^e=\underset{V^e}{\∫\∫\∫}\▿\×{\stackrel{\mathrm{\Γ}}{N}}_u^e\·\▿\×{\stackrel{\mathrm{\Γ}}{N}}_j^e\mathrm{dxdydz}$

${\left[T\right]}_{\mathrm{ij}}^e=\mathrm{\μ\ϵ}\underset{V^e}{\∫\∫\∫}{\stackrel{\mathrm{\Γ}}{N}}_i^e\·{\stackrel{\mathrm{\Γ}}{N}}_j^e\mathrm{dxdydz}$

Central difference schemes is adopted in large-sized region

$\left[S\right]e^n+\left[T\right]\frac{e^{n+1}-2e^n+e^{n-1}}{\mathrm{\Δ}{t}^2}=0$

[T]e ⁿ⁺¹＝(2[T]-Δt ²[S])e ⁿ-[T]e ^n-1，

Newmark-β difference scheme is adopted in undersized region

$\left[S\right](\mathrm{\β}{e}^{n+2}+(1-2\mathrm{\β})e^n+\mathrm{\β}{e}^{n-1})+\left[T\right]\frac{e^{n+1}-2e^n+e^{n-1}}{\mathrm{\Δ}{t}^2}=0$

([T]+Δt ²β[S])e ⁿ⁺¹＝(2[T]-Δt ²(1-2β)[S])e ⁿ-([T]+Δt ²β[S])e ^n-1

First carry out solving of Newmark-β region, then carry out that the equation of the ecentre is subregional to be solved, finally can obtain the electric field value of each point.

In order to verify the validity of the inventive method, analyze the typical examples of a dielectric resonant chamber below.

The dielectric ring of a specific inductive capacity 9.8 is placed with in resonator cavity as shown in Figure 1, a1=207.25mm, a2=440.75mm, b=242mm, c=43mm, r1=9.0mm, r2=10.0mm, h=14.0mm, the overall frequency spectrum adopting central difference and adopt mixed method of the present invention to calculate as shown in Figure 2, it is fine that coincide by both results visible.Calculate and consuming timely press shown in table 1, adopt 32 respectively, 16,8,4 process working procedures, can find out that the inventive method is saved than traditional method computing time greatly.

Process number	Mixed method	Central difference
			32	212s	359s
16	287s	589s
			8	518s	1103s
4	768s	2299s

Table 1

Claims (4)

1. a time domain spectral element emi analysis method for unconditional stability and condition stability mixing, its step is as follows:

The first step, carries out Geometric Modeling to the electromagnetic problem that will analyze, and is adopted by block mold bent hexahedron to carry out subdivision, obtains the numbering of the summit numbering of each individual cell, coordinate and body after subdivision;

Second step, in adopt the method comparing the length of side to find out grid that subdivision obtains, size is less than the hexahedron of setting value, and be labeled as small size region, residue grid is designated as large scale region;

3rd step, electric field value is defined on each point in the grid after block mold subdivision, and electric field is launched in XYZ tri-directions as Basis Function with the GLL polynomial expression in time domain spectral element method, substitute into time domain wave equation, and adopt the gold test of gal the Liao Dynasty, namely test basis function identical with expansion basis function, obtain matrix equation.

4th step, time term time difference in equation is launched, the Newmark-β difference scheme with unconditional stability is adopted in the undersized region be marked, other regions adopt the central difference schemes of condition stability, when carrying out time iteration, overall time step sets by central difference region, according to each step direct solution of total time step number, finally try to achieve time domain electric field value.

2. self-adaptation according to claim 1 unconditional/condition stability mixed time domain spectral element method emi analysis method, it is characterized in that: the bent hexahedral element length of side that in step one, subdivision adopts is 1/10 λ, λ is electromagnetic wavelength.

3. the time domain spectral element emi analysis method of unconditional stability according to claim 1 and condition stability mixing, is characterized in that: the GLL basis function form used in step 3 is as follows:

Wherein, j=0,1, LN, L _n(ξ) be N rank Legendre polynomial expressions, by the node { ξ in ξ ∈ [-1,1] _j, j=0,1, LN} are as GLL point, and they are equations (N+1) individual root;

By electric field base function expansion, be updated to time domain wave equation

Employing Galerkin method is tested, and namely trial function is identical with basis function, obtains overall coefficient matrix, solving equation

。

4. the time domain spectral element emi analysis method of unconditional stability according to claim 1 and condition stability mixing, is characterized in that: in step 4, and after undersized region adopts Newmark-β difference scheme, equation is:

([T]+Δt ²β[S])e ⁿ⁺¹＝(2[T]-Δt ²(1-2β)[S])e ⁿ-([T]+Δt ²β[S])e ^n-1

After large scale region adopts central difference schemes, equation becomes:

[T]e ⁿ⁺¹＝(2[T]-Δt ²[S])e ⁿ-[T]e ^n-1

Solving equation, in each step time iteration, first solves the electric field at central difference region place, then solves the subregional electric field of Newmark-β difference, finally obtains total time domain electric field value.