CN108629143A - The direct solving method of electromagnetic finite member-boundary element based on low-rank decomposition - Google Patents
The direct solving method of electromagnetic finite member-boundary element based on low-rank decomposition Download PDFInfo
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Abstract
The direct solving method of the invention discloses a kind of electromagnetic finite member boundary element based on low-rank decomposition, method include the following steps：Build the domain of electromagnetic field, the Finite Element Matrix equation and boundary element matrix equation of electromagnetic field are established respectively, it is coupled together by the continuity of electromagnetic field, simultaneous obtains the finite element-boundary element hybrid system matrix equation being made of Finite Element Matrix and boundary element matrix；The H- matrix expressions for building Finite Element Matrix and boundary variable matrix in finite element-boundary element hybrid system matrix equation respectively using near field mapping and low-order decomposition method, to establish the H- matrix expressions of finite element-boundary element hybrid system；Using H- matrix formats LU decomposition algorithms and it is front and back to back-substitution algorithm to entire finite element-boundary element hybrid system matrix equation carry out direct solution, obtain the electric field and current distribution of domain, and therefrom obtain electromagnetic parameter to be asked；Computation complexity and storage demand can be reduced to close to linear by the method for the present invention.
Description
Technical field
The present invention relates to electromagnetism Fast simulation algorithm field more particularly to a kind of electromagnetic finite members-based on low-rank decomposition
The direct solving method of boundary element.
Background technology
Finite element-boundary element mixed method fully combines the advantage of FInite Element and boundary element method, at FInite Element
Interior of articles complexity composition and medium are managed, boundary element method is avoided and is difficult to solving complexity problem of materials；Using Boundary Element Analysis
The open domain problem of body surface avoids the problem of increasing using the unknown quantity that FInite Element truncated boundary condition is brought, in electricity
The open domains simulation analysis such as magnetic scattering and aerial radiation field is applied widely.Finite element method-boundary element method sytem matrix equation
Coefficient matrix be that part is dense and the matrix of the sparse square in part, it is always finite element-side that Efficient Solution how is carried out to it
The difficult point and research hotspot in the boundary fields Yuan Fa.The existing solution side common for finite element method-boundary element method sytem matrix equation
Method includes：Interior sight equation, appearance equation and mixed formulation method.The above method belongs to iterative solution method, not only easily by iteration
The puzzlement of convergence rate problem, and when handling multiple activation problem there are bulk redundancy calculate (X.Q.Sheng, J.M.Jin,
J.Song,C.C.Lu,W.C.Chew,On the formulation of hybrid finite‐element and
boundary‐integral methods for 3‐D scattering,IEEE Trans.Antennas Propagat.,
vol.46,pp.303‐311,1998).Preconditioning method can be alleviated but can not completely eliminate iteration convergence to a certain extent
The problem of speed.Direct solution refers to the method for solving directly decomposed based on matrix to be asked, and efficiency is independently of matrix condition, phase
Iterative solution method is more stablized, and more efficient than iterative solution method for the solution of multiple activation problem.However, direct solution has
Higher computation complexity and memory consumption, therefore extensive finite element method-boundary element method system square is solved using direct solution
Battle array equation is typically considered infeasible.
Invention content
The direct solution of the main purpose of the present invention is to provide a kind of electromagnetic finite member-boundary element based on low-rank decomposition
Computation complexity and memory requirement can be reduced to close to linear by method, direct solving method through the invention, both can be with
The iteration convergence sex chromosome mosaicism of conventional iterative solution is avoided, and energy Efficient Solution multiple activation problem, specific technical solution are as follows：
A kind of direct solving method of electromagnetic finite member-boundary element based on low-rank decomposition, the method includes walking as follows
Suddenly：
S1：The domain for building electromagnetic field, establishes Finite Element Matrix equation and the boundary variable matrix side of electromagnetic field respectively
Journey is coupled together by the continuity of electromagnetic field, and simultaneous obtains the finite element-side being made of Finite Element Matrix and boundary element matrix
Boundary's member hybrid system matrix equation；
S2：It is built respectively in finite element-boundary element hybrid system matrix equation using near field mapping and low-order decomposition method
The H- matrix expressions of Finite Element Matrix and boundary variable matrix, to establish the H- matrix tables of finite element-boundary element hybrid system
Up to formula；
S3：Using H- matrix formats LU decomposition algorithms and front and back entire finite element-boundary element is mixed to back-substitution algorithm
Sytem matrix equation carries out direct solution, obtains the electric field and current distribution of domain, and therefrom obtains electromagnetism ginseng to be asked
Number.
Further, in step s 2, further include step：
Respectively to the Finite Element Matrix, boundary variable matrix structure electric field, electric current group tree, and to Finite Element Matrix, boundary
Variable matrix and their coupling matrix are interacted by group tree generates block tree；
The lossless expression formula of H- matrixes of Finite Element Matrix is generated using near field mapping method, wherein far field block of decomposition is zero,
Then the low-rank decomposition block that far field is constructed using matrix compression algorithm, generates the H- matrix expressions of the boundary variable matrix,
And then using near field mapping and matrix compression algorithm, the H- matrix expressions of the coupling matrix of finite element-boundary element are constructed, finally
Obtain the H- matrix expressions of entire finite element-boundary element hybrid system matrix.
Further, further include step, using singular value decomposition method to the low-rank decomposition block of matrix compression algorithm construction
Further compression obtains the H- matrix expressions with more compact format.
Further, multilayer can be used and adaptively intersect approximate data, Multi-layer matrix decomposition algorithm, multilayer UV methods and more
Layer QR decomposition algorithm equal matrix compression algorithms carry out the far field low-rank decomposition block in tectonic boundary variable matrix.
The method for solving of electromagnetic finite member-boundary element based on low-rank decomposition of the present invention builds target object first
Electromagnetic field analysis region, establishes Finite Element Matrix and boundary variable matrix within the scope of electromagnetic field, and by Finite Element Matrix and
Boundary element Matrix coupling forms a finite element-boundary element hybrid system matrix；Then near field reflection method and low-rank decomposition are used
Method builds Finite Element Matrix, boundary variable matrix and the corresponding H- matrix expressions of finite element-boundary element coupling matrix respectively, and most
The H- matrix expressions of finite element-boundary element hybrid system matrix are obtained eventually；Finally by LU decomposition algorithms and front and back to back substitution
Algorithm solves the matrix equation of entire finite element-boundary element hybrid system, to obtain the electric field and electric current of electromagnetic field
Distribution, and therefrom obtain the related electromagnetic parameter of demand；Compared with prior art, the present invention is based on the direct solutions of low-rank decomposition
Method has the beneficial effect that：The computation complexity and memory consumption for significantly reducing direct solution improve electromagnetic finite member-
The solution efficiency of boundary element method；Direct solution proposed by the present invention is insensitive to matrix condition, efficiently solves iterative solution method receipts
Hold back slow the problem of not restraining even；The present invention is when solving multiple activation problem, it is only necessary to which it is more to carry out primary decomposition to coefficient matrix
Secondary back substitution avoids redundant computation amount.
Description of the drawings
Fig. 1 is that the flow diagram of direct solving method of the present invention is illustrated；
Fig. 2 is finite element-boundary meta analysis object delineation of the present invention；
Fig. 3 is half space upper dielectric cylinder target component schematic diagram of the embodiment of the present invention；
Fig. 4 is half space upper dielectric cylinder target Bistatic RCS curve synoptic diagram of the embodiment of the present invention.
Specific implementation mode
In order to enable those skilled in the art to better understand the solution of the present invention, below in conjunction in the embodiment of the present invention
Attached drawing, technical scheme in the embodiment of the invention is clearly and completely described.
Fig. 2 is that target object is put into field region by the present invention combined with Figure 1 and Figure 2, is divided into interior zone SiThe outside and
Region Se, Fig. 1 is that the present invention is based on the key step flows of the direct solving method of the electromagnetic finite of low-rank decomposition member-boundary element
Figure, in embodiments of the present invention, described method includes following steps：
S1：The domain for building electromagnetic field, establishes Finite Element Matrix equation and the boundary variable matrix side of electromagnetic field respectively
Journey is coupled together by the continuity of electromagnetic field, and simultaneous obtains the finite element-side being made of Finite Element Matrix and boundary element matrix
Boundary's member hybrid system matrix equation；
In embodiment, first in target object interior zone SiUsing finite element model for solving boundary value problem, functional is established
Variation is asked, uses tetrahedron seamed edge Basis Function and surface RWG base function expansions electric field and surface current respectively, can get one group of line
Property equationWherein, K is Finite Element Matrix, and B is finite element-boundary element coupling matrix, and K and B is dilute
Dredge matrix, E andUnknown electric field coefficient and current coefficient to be asked are indicated respectively；Then, in target object boundary SeWith using side
Boundary's member method structure improved Electric Field Integral Equation and magnetic field integral equation, are used in combination RWG basic functions to be tested, and can get another group of linearly side
JourneyWherein, Q is boundary variable matrix, and P is finite element-boundary element coupling matrix, and b is the right side that excitation generates
Edge-vector；Finally by formulaWith formulaSimultaneous generates final finite element-boundary element
Sytem matrix expression formulaIn formula, subscript I indicates the inside seamed edge collection in addition to boundary edge,
Subscript S indicates that boundary seamed edge collection, B ' and P ' indicate the non-zero submatrices in B and P matrixes respectively.
In embodiments of the present invention, it is sparse matrix that target object interior zone, which builds the Finite Element Matrix to be formed, external
It is dense matrix that region, which builds and to form boundary variable matrix, and the finite element-boundary element hybrid system matrix that coupling obtains is only in object
There are nonzero elements on body boundary.
S2：It is built respectively in finite element-boundary element hybrid system matrix equation using near field mapping and low-order decomposition method
The H- matrix expressions of Finite Element Matrix and boundary variable matrix, to establish the H- matrix tables of finite element-boundary element hybrid system
Up to formula；
In the present invention, in order to finally obtain the electromagnetic parameter of demand by direct solution, the present invention first has to have
The finite element-boundary element hybrid system matrix conversion that limit variable matrix, boundary variable matrix and the two couple is at corresponding H-
Matrix expression is as follows：
First, the H- matrix expressions of construction Finite Element Matrix K：Recurrence two is carried out to finite element seamed edge basic function collection first
Point, structure electric field group tree TE, and interacted by group tree and generate electric field-electric field block tree TE×E, then by the non-zero in K matrix
Element is inserted in block tree；In embodiment, the ranks number of K matrix is total number of edges nl(internal number of edges ninnWith boundary number of edges nbnd
The sum of), therefore TE×EFor Fang Shu.It introduces admissible condition and distinguishes nearly far field, field portion is used with full formation formula expression, far-field portion
Low-rank decomposition matrix form is expressed.Due to the local characteristics of differential operator, the far-field portion in Finite Element Matrix is zero, that is, is had
The first method low-rank decomposition block of limit is 0, therefore the non-zero element in Finite Element Matrix K need to only be mapped to filling TE×EIn near field block be
Can, concrete operations flow is：To generating the near field action edge l of nonzero element in K matrixi、lk, them are found out in H- matrix tree-shaped
Corresponding group n in structurei、nk, which, which is inserted corresponding H- matrixes, expires battle array block.
Then, the H- matrix expressions of tectonic boundary variable matrix K.Here boundary variable matrix is constructed with moment method.With construction
The H- matrix expressions of Finite Element Matrix are similar, build electric current group to dividing RWG basic function collections using recursive bisection first
Set TJ, and acted on by group tree and generate electric current-current block tree TJ×J, T hereJ×JIt is similarly Fang Shu.
Different from Finite Element Matrix K, boundary element integral operator is global, therefore boundary variable matrix Q is full battle array, i.e.,：Closely
Field and far-field portion are dense matrix.Near field matrix uses moment method to be filled with full formation formula.Due to generating far field matrix
Field group and the interaction of source group are weaker, and far field matrix can be compressed using matrix compression algorithm, to generate low-rank decomposition
Form.Here, different matrix compression algorithms can be used, such as：Multilayer adaptively intersects approximation (Adaptive Cross
Approximation, ACA) algorithm, matrix decomposition algorithm, multilayer UV methods and multilayer QR decomposition algorithms etc..It is with ACA algorithms
Example, the form that original matrix is compressed into low-rank decomposition matrix are Hmn=Amr×Brn；Wherein, m, n are respectively the ranks number of matrix, r
Order is blocked in ＜ ＜ m, n expressions, for controlling the compression accuracy of ACA algorithms, to expire battle array HmnIt is smaller to be decomposed into two dimensions
Matrix AmrAnd BrnStorage and operation are carried out, calculation amount and memory consumption are substantially reduced.It is pointed out that tradition ACA algorithm bases
In octree structure, and H- matrix algorithms are generally basede on binary tree structure, it is therefore desirable to establish two kinds of the corresponding of tree and close
System.Due to three layers of one layer of corresponding binary tree of Octree, three layers of H- matrixes are equivalent to one layer of ACA algorithm trees.It establishes
After tree structure, in TJ×JIt is middle to fill corresponding near field matrix and far field matrix, you can to obtain the H-matrix expression of boundary variable matrix Q
Formula.Since the low-rank decomposition block of classical matrix compression algorithm construction still has redundancy, singular value decomposition method is used here
The low-rank decomposition block of ACA algorithm constructions is further compressed, the H- matrix expressions with more compact format are obtained.
Finally, the H- matrix expressions of the H- matrix expressions and coupling matrix P of coupling matrix B are built respectively；Wherein, B
The corresponding electric field of matrix-current block tree TE×JBy electric field group tree TEWith electric current group tree TJInteraction generates, therefore TE×JNot side
Tree, it is total number of edges n that line number, which corresponds to K matrix,l, columns is then boundary number of edges nbnd.Such as formulaAnd formulaIt is described, the preceding internal number of edges n of B matrixesinnRow is neutral element, and B ' is nbnd×nbndIt is dilute
Dredge matrix.Similar with Finite Element Matrix K forms, the far-field portion of coupling matrix B is 0, i.e., low in its H- matrix expression
Order block of decomposition is 0.Nonzero element in B matrixes is acted on by the boundary edge in finite element and boundary element part to be generated, according to side
Boundary side is corresponding group in H- matrix trees, you can the nonzero element in coupling matrix B is packed into corresponding full battle array block,
To generate the H- matrix expressions of B matrixes；The corresponding electric current of P matrixes-electric field block tree TJ×ENor Fang Shu, line number is boundary
Number of edges nbnd, columns is total number of edges nl.Such as formulaAnd formulaIt is described, in P
Preceding internal number of edges n in matrixinnRow are all neutral element, and P ' matrixes are that ranks number is boundary number of edges nbndSquare formation.With Q matrix phases
Seemingly, P ' matrixes are dense matrix, can be divided near field and far field, and its far-field portion can also use ACA algorithm construction low-ranks
Decompose expression formula.Row group of the field portion in H- matrixes in P matrixes is identical as Q matrixes, but the corresponding near field group of row group
For the group belonging to boundary edge in FInite Element.The near field element that moment method generates is inserted TJ×EIn full battle array block in.For far field
Partial filling, such as [al1…almab1…abn] shown in formula, wherein al1…almFor boundary edge corresponding element, a in groupb1…abnThen
To remove the element except boundary edge in group.The nonzero element of P matrix far-field portions is finally inserted into low-rank decomposition block, generates P
The H- matrix expressions of matrix.After completing Finite Element Matrix, boundary variable matrix and the corresponding H- matrix expressions of coupling matrix, i.e.,
It can get the H- matrix expressions of entire finite element-boundary element hybrid system matrix.
Preferably, in embodiments of the present invention, using singular value decomposition method to the low-rank point of matrix compression algorithm construction
Deblocking further compression, obtains the H- matrix expressions with more compact format；Further, multilayer can be used adaptively to intersect
Approximate data, Multi-layer matrix decomposition algorithm, multilayer UV methods and multilayer QR decomposition algorithm equal matrix compression algorithms, to this present invention
It is not limited and is fixed, can be selected according to actual conditions.
S3：Using H- matrix formats LU decomposition algorithms and front and back entire finite element-boundary element is mixed to back-substitution algorithm
Sytem matrix equation carries out direct solution, obtains the electric field and current distribution of domain, and therefrom obtains electromagnetism ginseng to be asked
Number.
It in the present invention, can be according to formulaIt can by finite element-boundary element sytem matrix
It is expressed as 2 × 2 matrix in block form form, passes through formulaIts LU factorization can be such as formulaShown in matrix in block form formula pass
Return execution, can therefrom obtain, H-LU decomposition computation complexities are O (k2Nlog2N), memory consumption is O (kNlogN), and wherein k is institute
Have an average order of low-rank decomposition matrix, in the present invention, average order k by formulaCome into
Row defines, in formula, kiOrder is blocked for i-th of matrix, j is the total number of low-rank decomposition matrix, mi,niRespectively low-rank decomposition square
The line number and columns of battle array.Direct solving method provided by the invention can be controlled by adjusting and controlling the size of average order k
The time and memory consumption that the precision and H-LU that H-LU processed is decomposed are decomposed.After L the and U matrixes for obtaining H- matrix formats, use
H- matrix formats it is front and back to back substitution, you can complete to the direct solution of entire finite element-boundary element sytem matrix equation, H- squares
Grid array formula it is front and back to the computation complexity of back substitution be O (kNlogN).And solution of the present invention for multiple activation problem, such as：It is single
Radar cross section of standing calculating etc., the process that H-LU is decomposed need to only execute once, then for each different driving source, only need
Non trivial solution can be quickly obtained to back substitution by executing the front and back of H- matrix formats.
After above-mentioned steps S1, S2 and S3, you can obtain electric field and current coefficient according to solution, obtain region to be asked
Electric field and current distribution, and then electromagnetic parameter of concern can be calculated.
It is half space upper dielectric cylinder target component diagram meaning in conjunction with Fig. 3 and Fig. 4, Fig. 3, Fig. 4 is to be situated between above half space
Matter cylinder target Bistatic RCS curve graph, electromagnetic finite member-boundary element of the invention based on low-rank decomposition it is straight
Connect the analysis that method for solving is also applied for the Bistatic RCS of half space upper dielectric cylinder target, wherein the cylinder
Bottom surface radius r=0.3m, high h=0.4m, permittivity εr=3.0, conductivityσ=0.01S/m consumes half space soil positioned at having
Above earth at h=0.5m, the permittivity ε of soilrg=(5.0, -0.2), conductivityσg=0.0S/m, the incidence angle of plane wave
Spend θinc=0 ° and φinc=0 °, incident frequencies f=700MHZ.For finite element and boundary element part, be respectively adopted tetrahedron and
Triangular element progress is discrete, ultimately generates 6799 inner edges and 1668 boundary edges.Then according to provided by the invention direct
Method for solving builds the finite element-boundary element direct solution based on low-rank decomposition；Approximate calculation is adaptively then intersected using multilayer
The far field of method negative margin member part, the total number of plies of H- matrixes constructed are 6 layers, in H- LU Decomposition algorithms, block precision
It is set as 10-4.The method of the present invention is can be seen that from the curve tendency result of Fig. 4 and popular business software FEKO emulation is tied
Fruit is compared, and the two with quite well, that is, can illustrate that the method for the present invention has good solution efficiency and higher asks
Solve precision.
In conjunction with following Tables 1 and 2, wherein table 1 provides using direct solution of the present invention and uses conventional iterative
The solution time of solution compares, and the calculating of one Monostatic RCS with 360 angular samples points of analysis belongs to one
Kind multiple activation problem；As can be seen that direct solution of the present invention has higher computational efficiency.Table 2 then gives finite element-
Average order situation of change of the boundary element sytem matrix before and after executing H-LU and decomposing, it can be seen that because of the present invention direct
There is solution stable calculated performance and superior solution efficiency, H-LU to decompose the rising for not causing average order, have instead
A degree of decline.
Performance between 1 direct solution of the present invention of table and conventional iterative solution compares
| Method | The LU resolving times (second) | Solve the time (second) | Total time (second) |
| Conventional iterative solution | Nothing | 25.9 | 9310 |
| Direct solution of the present invention | 395.5 | 0.85 | 702 |
In 2 direct solution of the present invention of table order variation is blocked before and after matrix decomposition
| The matrix decomposition stage | Low-rank decomposition block number | Average order | Maximum order |
| Before decomposition | 192 | 21 | 28 |
| After decomposition | 272 | 12 | 30 |
It to sum up describes, the method for solving of electromagnetic finite member-boundary element of the invention based on low-rank decomposition builds mesh first
The electromagnetic field analysis region for marking object, establishes the Finite Element Matrix and boundary variable matrix within the scope of electromagnetic field, and will be limited
Variable matrix and boundary element Matrix coupling form a finite element-boundary element hybrid system matrix；Then use near field reflection method and
Low-rank decomposition method builds Finite Element Matrix, boundary variable matrix and the corresponding H- matrix tables of finite element-boundary element coupling matrix respectively
Up to formula, and finally obtain the H- matrix expressions of finite element-boundary element hybrid system matrix；Finally by LU decomposition algorithms with before
Backward back-substitution algorithm solves the matrix equation of entire finite element-boundary element hybrid system, to obtain the electricity of electromagnetic field
Field and current distribution, and therefrom obtain the related electromagnetic parameter of demand；Compared with prior art, the present invention is based on low-rank decompositions
Direct solving method has the beneficial effect that：The computation complexity and memory consumption for significantly reducing direct solution, improve electricity
The solution efficiency of magnetic finite element method-boundary element method；Direct solution proposed by the present invention is insensitive to matrix condition, efficiently solves repeatedly
The problem of not restrained even slowly for solution convergence；The present invention is when solving multiple activation problem, it is only necessary to carry out one to coefficient matrix
It is secondary to decompose multiple back substitution, avoid redundant computation amount.
The foregoing is merely a prefered embodiment of the invention, the scope of the claims of the present invention is not intended to limit, although with reference to aforementioned reality
Applying example, invention is explained in detail, still can be to aforementioned each tool for those skilled in the art comes
Technical solution recorded in body embodiment is modified, or carries out equivalence replacement to which part technical characteristic.Every profit
The equivalent structure made of description of the invention and accompanying drawing content is directly or indirectly used in other related technical areas,
Similarly within scope of patent protection of the present invention.
Claims (4)
1. the direct solving method of electromagnetic finite member-boundary element based on low-rank decomposition, which is characterized in that the method includes such as
Lower step：
S1：The domain for building electromagnetic field, establishes the Finite Element Matrix equation and boundary element matrix equation of electromagnetic field respectively, leads to
The continuity for crossing electromagnetic field is coupled together, and simultaneous obtains the finite element-boundary element being made of Finite Element Matrix and boundary element matrix
Hybrid system matrix equation；
S2：It is built respectively using near field mapping and low-order decomposition method limited in finite element-boundary element hybrid system matrix equation
The H- matrix expressions of variable matrix and boundary variable matrix, to establish the H- matrix expressions of finite element-boundary element hybrid system；
S3：Using H- matrix formats LU decomposition algorithms and it is front and back to back-substitution algorithm to entire finite element-boundary element hybrid system
Matrix equation carries out direct solution, obtains the electric field and current distribution of domain, and therefrom obtain electromagnetic parameter to be asked.
2. the direct solving method of electromagnetic finite member-boundary element according to claim 1 based on low-rank decomposition, feature
It is, further includes step in step s 2：
Respectively to the Finite Element Matrix, boundary variable matrix structure electric field, electric current group tree, and to Finite Element Matrix, boundary element square
Battle array and their coupling matrix are interacted by group tree generates block tree；
The lossless expression formula of H- matrixes of Finite Element Matrix is generated using near field mapping method, wherein far field block of decomposition is zero, then
The low-rank decomposition block that far field is constructed using matrix compression algorithm generates the H- matrix expressions of the boundary variable matrix, in turn
Using near field mapping and matrix compression algorithm, the H- matrix expressions of the coupling matrix of finite element-boundary element are constructed, it is final to obtain
The H- matrix expressions of entire finite element-boundary element hybrid system matrix.
3. the direct solving method of electromagnetic finite member-boundary element according to claim 2 based on low-rank decomposition, feature
It is, further includes step, the low-rank decomposition block of matrix compression algorithm construction is further compressed using singular value decomposition method, is obtained
Obtain the H- matrix expressions with more compact format.
4. the direct solving method of electromagnetic finite member-boundary element according to claim 2 based on low-rank decomposition, feature
It is, multilayer can be used and adaptively intersect approximate data, Multi-layer matrix decomposition algorithm, multilayer UV methods and multilayer QR decomposition algorithms
Equal matrix compression algorithm carrys out the far field low-rank decomposition block in tectonic boundary variable matrix.
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