CN106528984A - Radial point interpolation type meshless method shape function construction method - Google Patents
Radial point interpolation type meshless method shape function construction method Download PDFInfo
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Abstract
The invention discloses a radial point interpolation type meshless method shape function construction method. The method comprises the steps of firstly establishing a coordinate axis; secondly performing structure modeling; thirdly performing space conversion; fourthly searching for nearest neighbors; and finally obtaining a shape function. According to the method, an ellipsoidal support domain is adopted, so that the requirement of the number of nodes required for long and thin structure modeling is lowered and the number of to-be-calculated unknowns is greatly reduced; and a mapping space concept is introduced and the ellipsoidal support domain in a real space is converted into a spherical support domain in a new space, so that a nearest neighbor search algorithm used by a conventional RPIM algorithm can be directly used.
Description
Technical field
The present invention relates to a kind of radial point interpolation method type gridless routing shape function building method.
Background technology
In the numerical computations of electromagnetic field, traditional numerical algorithm needs to be built using the grid cell of given shape
Mould, such as finite difference calculus (FDTD) adopt hexahedral element, FInite Element (FEM) and moment method (MOM) then to adopt triangular facet
Unit.But, for electrically large sizes or fine structure, the process of modeling takes very much, in some instances it may even be possible to can exceed simulation time.
In addition, for Time variable structure, once structure changes, modeling process is accomplished by repeating, and causes simulation time drastically to increase
Plus.Different from traditional method, gridless routing is the numerical algorithm that a class does not adopt that grid cell is modeled, including mobile minimum
Square law (MLS), divergence unit method (DEM), Element-Free Galerkin method (EFG) and radial point interpolation method type gridless routing (RPIM)
Deng.Gridless routing is modeled using the node that flexibly can be placed, due to not being connected with each other between node, thus without the need for picture
Traditional algorithm equally processes the topological relation between grid.In addition, when structure changes, it is only necessary in the region of structure change
Reconfigure node.
Radial point interpolation method type gridless routing (RPIM) enters row interpolation in spatial domain using RBF (RBF), with higher
Numerical precision.RPIM needs to carry out pretreatment after modeling, calculates the shape function at each node.In pretreatment, be
Spherical support domain, and the adjacent node searched within supporting domain are constructed at each node.In general, supporting domain radius is got over
Greatly, then precision is higher, but the efficiency of algorithm is accordingly reduced, thus the selection of supporting domain radius is needed between precision and efficiency
Make balance.
Traditional RPIM methods spherical support domain used in pretreatment.For slim-lined construction, if calculate space adopted
The more uniform node of spatial distribution carries out discrete, although can adopt less supporting domain radius, but due to node total number it is many,
Cause the unknown quantity number of final calculating a lot.If conversely, which is compared for discrete node on the longer direction of structure
Its direction is sparse, in order to ensure precision, needs to increase the radius of supporting domain so that the nodes in supporting domain increase so that whole
The efficiency of individual algorithm declines.
The content of the invention
It is an object of the invention to provide a kind of radial point interpolation method type gridless routing shape function building method.
To achieve the object of the present invention, the present invention adopts technical scheme：A kind of radial point interpolation method type gridless routing shape function
Building method, which comprises the following steps：
The first step：X, y, z coordinate axess are set up, and along physical dimension most length direction, x-axis, and structure are denoted as in x, y and z axes
Size on direction is denoted as L respectivelyx、LyAnd Lz；
Second step：Structure is modeled using series of discrete node, the node average distance ratio along three axial directions is
Lx:Ly:LzSo that in x-axis direction, node density is minimum；Due to LxMaximum, so as to x directions average distance is maximum, so as to node is close
Degree is minimum；
3rd step：Linear transformation is carried out to the discrete nodes coordinate (x, y, z) in real space Ω, Linear Mapping is converted to
Space Ω*Interior coordinate (x*,y*,z*), transformation for mula is：
By linear transformation, the elliposoidal supporting domain in Ω is mapped to Ω*Under spherical support domain, wherein supporting domain half
The selection in footpath may be referred to document 1：G.R.Liu and Y.T.Gu,An Introduction to Meshfree Methods
and Their Programming.Dortrecht,The Netherlands:Springer,2005.)
4th step：In space of linear mapping Ω*It is interior, as traditional RPIM algorithms, using kNN algorithms in spherical support domain
Neighbor node is searched for one by one.To mapping space Ω*Interior i-th node, remembers that the collection of its neighbor node numbering is combined into Xi, hence for
I-th node in real space Ω, the set of the neighbor node numbering in elliposoidal supporting domain is still Xi；Wherein kNN is calculated
Method may be referred to document 2：J.H.Friedman,J.L.Bentley,and R.A.Finkel,“An algorithm for
finding best matches in logarithmic expected time,”ACM Trans.Math.Softw.,
vol.3,no.3,pp.209–226,1977.Document 3：N Kumar, Z Li, S Nayar, " What is a Good Nearest
Neighbors Algorithmfor Finding Similar Patches in Images？”European Conference
on Computer Vision,vol.5303,pp.364-378,2008.KNN algorithms are the general designations of k nearest neighbor algorithms.
5th step：In real space Ω, to i-th node (xi,yi,zi), by XiIt is calculated as follows vector：
RT=[r (d1),r(d2),…,rn(dn)]
PT=[1, xi,yi,zi]
Wherein r is RBF, djIt is node i and XiIn the distance between j-th node, n is set XiIn close on section
The number of point numbering, T represent transposition.
Following matrix is constructed in Ω equally：
Wherein Rkl=r (dkl), dklIt is XiIn the distance between k-th node and l-th node.P0In xj、yjAnd zjFor
XiIn j-th node coordinate figure；In Ω, the shape function of i-th node is Φi=RTSa+PTSb, wherein
6th step：All nodes said process being applied in Ω, obtain the corresponding shape function of each point to terminating.
On the basis of above-mentioned technical proposal, following attached technical scheme is further included：
Which is used for slim-lined construction modeling.
The present invention replaces spherical support domain using elliposoidal supporting domain, thus is processing elongated knot when shape function is calculated
During structure, can be while direction of elongate reduces node density, it is not necessary to increase supporting domain radius, so as to improve computational efficiency.
In addition, by introducing space of linear mapping so that the present invention can be adopted and tradition RPIM method identical nearest neighbor searches
Algorithm.
The method have the advantages that：
1st, using elliposoidal supporting domain, reduce slim-lined construction modeling needed for node number require so that it is to be calculated not
Know that several numbers greatly reduce；
2nd, mapping space concept is introduced, the elliposoidal supporting domain in real space is for conversion into spherical under new space
Supporting domain, such that it is able to the neighbor search algorithm for directly being used using tradition RPIM algorithms.
Description of the drawings
Below in conjunction with the accompanying drawings and embodiment the invention will be further described：
Fig. 1 is the schematic diagram of coordinate axess in the present invention.
Fig. 2 is the schematic diagram of Linear Transformation of the present invention.
Specific embodiment
Embodiment：As shown in Figure 1-2, the present invention provides a kind of radial point interpolation method type gridless routing shape function building method
Specific embodiment, which comprises the steps：.
1 sets up coordinate axess
Wherein one coordinate axess are along physical dimension most length direction (being denoted as x-axis).Structure size in the x, y and z directions
L is denoted as respectivelyx、LyAnd Lz(such as Fig. 1)；
2 structural modelings
Slim-lined construction is modeled using series of discrete node, on three directions, nodal distance ratio is Lx:Ly:Lz, make
Obtain minimum in x directions node density；
3 spatial alternations
In order to neighbor searching is rapidly carried out in ellipsoid supporting domain, the concept of mapping space is introduced.To real space Ω
Interior discrete nodes coordinate (x, y, z) carries out linear transformation, is converted to space of linear mapping Ω*Interior coordinate (x*,y*,z*), become
Changing formula is：
By linear transformation, the elliposoidal supporting domain in Ω is mapped to Ω*Under spherical support domain (such as Fig. 2).Wherein
The selection of spherical support domain radius, may be referred to document 1；
4 search neighbour
In space of linear mapping Ω*It is interior, as traditional RPIM algorithms, searched in spherical support domain one by one using kNN algorithms
Rope neighbor node.To mapping space Ω*Interior i-th node, remembers that the collection of its neighbor node numbering is combined into Xi, hence for true sky
Between i-th node in Ω, the set of the neighbor node numbering in ellipsoid supporting domain is still Xi；KNN algorithms may be referred to text
Offer 2；
5 calculate shape function
In real space Ω, to i-th node (xi,yi,zi), by XiIt is calculated as follows vector：
RT=[r (d1),r(d2),…,rn(dn)]
PT=[1, xi,yi,zi]
Wherein r is RBF, djIt is node i and XiIn the distance between j-th node, n is set XiIn close on section
The number of point numbering, T represent transposition.Following matrix is constructed in Ω equally：
Wherein Rkl=r (dkl), dklIt is XiIn the distance between k-th node and l-th node.P0In xj、yjAnd zjFor
XiIn j-th node coordinate figure；In Ω, the shape function of i-th node is Φi=RTSa+PTSb, wherein
6th step：All nodes said process being applied in Ω, obtain the corresponding shape function of each point to terminating.
Thus the present invention has advantages below：
1st, using elliposoidal supporting domain, to slim-lined construction on the direction of longer dimension, node density requires to reduce, that is, drop
Node number needed for low slim-lined construction modeling is required so that unknown number number to be calculated greatly reduces；
2nd, mapping space concept is introduced, the elliposoidal supporting domain in real space is for conversion into spherical under new space
Supporting domain, carries out neighbor search in the mapping space after Linear Transformation, such that it is able to directly be used using tradition RPIM algorithms
Neighbor search algorithm.
Certainly above-described embodiment technology design only to illustrate the invention and feature, its object is to allow and are familiar with technique
People will appreciate that present disclosure and implement according to this, can not be limited the scope of the invention with this.It is all according to this
Equivalent transformation or modification that the spirit of bright main technical schemes is done, should all be included within the scope of the present invention.
Claims (2)
1. a kind of radial point interpolation method type gridless routing shape function building method, it is characterised in which comprises the following steps：
The first step：X, y, z coordinate axess are set up, and along physical dimension most length direction, x-axis, and structure are denoted as in x, y and z axes direction
On size be denoted as L respectivelyx、LyAnd Lz；
Second step：Structure is modeled using series of discrete node, the node average distance ratio along three axial directions is Lx:Ly:
LzSo that in x-axis direction, node density is minimum；
3rd step：Linear transformation is carried out to the discrete nodes coordinate (x, y, z) in real space Ω, space of linear mapping is converted to
Ω*Interior coordinate (x*,y*,z*), transformation for mula is：
By the linear transformation, the elliposoidal supporting domain in Ω is mapped to Ω*Under spherical support domain；
4th step：In space of linear mapping Ω*It is interior, neighbor node is one by one searched in spherical support domain using kNN algorithms, to linear
Mapping space Ω*Interior i-th node, remembers that the collection of its neighbor node numbering is combined into Xi, hence for i-th in real space Ω
Node, the set of the neighbor node numbering in elliposoidal supporting domain is still Xi；
5th step：In real space Ω, to i-th node (xi,yi,zi), by XiIt is calculated as follows vector：
RT=[r (d1),r(d2),…,rn(dn)]
PT=[1, xi,yi,zi]
Wherein r is RBF, djIt is node i and XiIn the distance between j-th node, n is set XiMiddle neighbor node is compiled
Number number, T represents transposition；
And following matrix is constructed in real space Ω：
Wherein Rkl=r (dkl), dklIt is XiIn the distance between k-th node and l-th node, P0In xj、yjAnd zjFor XiIn
The coordinate figure of j-th node；And the shape function of i-th node is Φ in real space Ωi=RTSa+PTSb, wherein
6th step：All nodes said process being applied in Ω, obtain the corresponding shape function of each point to terminating.
2. a kind of radial point interpolation method type gridless routing shape function building method as claimed in claim 1, it is characterised in that its use
Model in slim-lined construction.
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| CN110824558A (en) * | 2019-11-20 | 2020-02-21 | 中国石油大学(华东) | Seismic wave numerical simulation method |
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| CN110824558A (en) * | 2019-11-20 | 2020-02-21 | 中国石油大学(华东) | Seismic wave numerical simulation method |
| CN110824558B (en) * | 2019-11-20 | 2021-07-16 | 中国石油大学(华东) | Seismic wave numerical simulation method |
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Application publication date: 20170322 |