CN109918744A - A kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function - Google Patents
A kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function Download PDFInfo
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Abstract
The invention discloses a kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function, belong to Fluid Mechanics Computation field, by introducing Semi Lagrangian scheme and Interpolation Property of Radial Basis Function method, so that Lattice Boltzmann Method (LBM) is got rid of the limitation of uniform grid, further expands application of the LBM in Complex Flows problem.The present invention introduces SL method first in Lattice Boltzmann Method, finds the starting point of migration, i.e. x-eαδ t, then starting point (x-e is obtained by introducing part RBF interpolation methodαδ t) distribution function；The part RBF interpolation method are as follows:WhereinIt is target position functional value, φ (| | x-xi| |) it is radial basis function, | | x-xi| | Euclid norm, p (x) they are multinomials, and n is supporting point number, and m is the order of multinomial, and λ and γ are interpolation coefficients.
Description
Technical field
The invention belongs to Fluid Mechanics Computation technical fields, more particularly to one kind to be based on semi-Lagrange and radial basis function
Mesh free Lattice Boltzmann Method.
Background technique
Fluid Mechanics Computation (CFD) is the Men Xueke that flow field problem is solved using numerical method.Traditional calculating stream
Mechanics method is usually to Fluid Control Equation, i.e. Na Wei-Stokes (N-S) equation is directly carried out discrete and solved.Lattice
A kind of Fluid Mechanics Computation method of the sub- Boltzmann's method (LBM) as novel meso-scale, it is simple by the movement of fluid
Collision and the travel motion of fluid particles are turned to, so that the solution of flow field problem is greatly simplified.Therefore, LBM has been increasingly becoming separately
A kind of widely used CFD approach, and the Complex Flows problem such as be successfully applied to turbulent flow and multiphase flow.Compared to direct
Traditional CFD approach of N-S equation is solved, LBM has many advantages, such as that solution procedure is simple, high convenient for parallel computation, method universality.
But traditional LBM can only be implemented in uniform grid, this severely limits application of the LBM in Complex Flows problem.
If zoning is separated into limited node, the difficult point that LBM is implemented at this time be how accurately to obtain from
Distribution function value at scatterplot.Because physical space and lattice space usually mismatch, i.e., grid is after the migration of some node
The position reached will not be discrete nodes position.At this point, the collision transition process of tradition LBM is difficult to implement.Half
Ge Lang (SL) method is a kind of effective method for solving convection problem.In the method, along characteristic curve since starting point
Integral can be obtained the functional value of target position.In LBM, fluid particles travel motion is linear, it is easy to find migration
Move corresponding characteristic curve and starting point.So can use SL method, obtained by migrating the distribution function value of starting point
The distribution function value of discrete nodes after migration.For migrating the distribution function value of starting point, radial basis function (RBF) can be passed through
Interpolation obtains.RBF interpolation method is a kind of scatterplot data interpolating method being simple and efficient, and has O (hd+1) precision, wherein h
For spatial mesh size, d is Spatial Dimension.But currently, RBF interpolation is mainly used in terms of calculating grid reconstruction, SL method is also main
For solving traditional N-S equation.
Summary of the invention
The present invention provides a kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function leads to
Introducing Semi Lagrangian scheme and Interpolation Property of Radial Basis Function method are crossed, Lattice Boltzmann Method (LBM) is made to get rid of uniform grid
Limitation, further expand application of the LBM in Complex Flows problem.
To achieve the goals above, the invention adopts the following technical scheme:
A kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function, for can not be adhesive
Flow field problem, the governing equation of Lattice Boltzmann Method are as follows:
Wherein fαIt is distribution function, corresponding migration velocity is eα,It is fαCorresponding equilibrium state, x are physical spaces
Coordinate, δ t is time step, and τ is single slack time coefficient.
Under the premise of not influencing calculated result, above-mentioned governing equation can resolve into two parts, it may be assumed that
1. collision process:
2. transition process:
If zoning is separated into limited node, because physical space and lattice space usually mismatch, i.e. lattice
The position that son is reached after the migration of some node will not be discrete nodes position.At this point, the collision of tradition LBM migrates
Process will be difficult to carry out.For this purpose, the present invention is firstly introduced into SL method, the starting point of migration, i.e. x-e are foundαδ t is (real as shown in figure 1
Shown in heart square icon).Point because distribution function is constant in grid transition process in LBM, after migration in discrete nodes
Cloth functional value is the distribution function value being equal at migration starting point.If retaining collision process to occur in discrete nodes, go out
Send out point (x-eαδ t) distribution function can be obtained by interpolation.For this purpose, present invention introduces local RBF interpolation methods, i.e.,
WhereinIt is target position functional value, φ (| | x-xi| |) it is radial basis function, | | x-xi| | Euclid's model
Number, p (x) is multinomial, and n is supporting point number, and m is the order of multinomial, and λ and γ are interpolation coefficients.Pass through structure
Reasonable radial basis function and multinomial are built, can use the distribution function that local RBF interpolation accurately obtains migration starting point
Value, and then the distribution function value after being migrated at discrete nodes.
In approach described above, before carrying out part RBF interpolation to the starting point, minute of collision rear support point is utilized
Cloth functional value calculates interpolation coefficient λ and γ, and the supporting point refers to the m discrete sections nearest apart from the migration starting point
Point；The increased condition of interpolation coefficient λ and γ of calculating are as follows:
Wherein p (x) is obtained by Pascal triangle.
It is preferred that using Polyharmonic Splines (PHS) radial basis function φ (r)=r2k+1In conjunction with second order polynomial
(pT(x)={ 1, x, y, x2,xy,y2) obtain starting point (x-eαδ t) distribution function, wherein r=| | x-xj||2,
The utility model has the advantages that the present invention provides a kind of mesh free grid bohrs based on semi-Lagrange and radial basis function hereby
Graceful method introduces Semi Lagrangian scheme and Interpolation Property of Radial Basis Function method in Lattice Boltzmann Method, makes grid bohr
Hereby graceful method (LBM) gets rid of the limitation of uniform grid, realizes and solves stream using Lattice Boltzmann Method in the case of mesh free
The purpose of dynamic problem, further expands application of the LBM in Complex Flows problem.
Detailed description of the invention
Fig. 1 is the schematic diagram of inventive algorithm under two-dimensional case.
Fig. 2 is the motion pattern that two-dimentional lid-driven cavity flow under resulting different Reynolds number is simulated using the method for the present invention.
Fig. 3 be simulated using the method for the present invention flow field vorticity cloud atlas that TWO-DIMENSIONAL CIRCULAR CYLINDER under resulting different Reynolds number is streamed and
Motion pattern.
Fig. 4 is the flow field pressure coefficient that resulting two dimension NACA0012 profile flow is simulated using method of the present invention
Cloud atlas and motion pattern.
Specific embodiment
The present invention is described in detail with specific embodiment below in conjunction with the accompanying drawings:
How quasi- a kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function, key are
Distribution function value after really being migrated at discrete nodes.
For incompressible viscous flow problem, the governing equation of Lattice Boltzmann Method are as follows:
Wherein fαIt is distribution function, corresponding migration velocity is eα,It is fαCorresponding equilibrium state, x are physical spaces
Coordinate, δ t is time step, and τ is single slack time coefficient.
If zoning is separated into limited node, because physical space and lattice space usually mismatch, i.e. lattice
The position that son is reached after the migration of some node will not be discrete nodes position.At this point, the collision of tradition LBM migrates
Process will be difficult to carry out.For this purpose, the present invention is firstly introduced into SL method, the starting point of migration, i.e. x-e are foundαδ t, it is real as shown in figure 1
Shown in the rectangular mark of the heart.Distribution because distribution function is constant in grid transition process in LBM, after migration in discrete nodes
Functional value is the distribution function value being equal at migration starting point.In order to solve migration starting point (x-eαδ t) at distribution function
Value, present invention introduces local RBF interpolation methods, i.e.,
WhereinIt is target position functional value, φ (| | x-xi| |) it is radial basis function, | | x-xi| | Euclid's model
Number, p (x) is multinomial, and n is supporting point number, and m is the order of multinomial, and λ and γ are interpolation coefficients.
Before carrying out part RBF interpolation to starting point, needs to calculate interpolation coefficient λ and γ, needs to increase condition thus,
I.e.
Wherein p (x) can be obtained by Pascal triangle.For two-dimensional space, second order polynomial is pT(x)=1,
x,y,x2,xy,y2}.At this point it is possible to calculate interpolation coefficient λ and γ, supporting point using the distribution function value of collision rear support point
Refer to m nearest discrete nodes of distance translation starting point (as shown in figure 1 shown in the solid circles icon in chain-dotted line region).?
It, can be to migration starting point (x-e after obtaining interpolation coefficient λ and γαδ t) part RBF interpolation is carried out to obtain the distribution at this
Functional value, and then the distribution function value after being migrated at discrete nodes.
By introducing Semi Lagrangian scheme and Interpolation Property of Radial Basis Function method, mesh free Lattice Boltzmann method of the invention
The specific implementation of method can be divided into following three step:
(1) collision process is completed in discrete nodes:
(2) distribution function of migration starting point is obtained by Interpolation Property of Radial Basis Function:
Wherein xL=x-eαδ t is migration starting point coordinate.
(3) distribution function after being migrated at discrete nodes:
The present invention is by test discovery, using Polyharmonic Splines (PHS) radial basis function, i.e. φ (r)=
r2k+1, wherein r=| | x-xj||2,In conjunction with second order polynomial (pT(x)={ 1, x, y, x2,xy,y2) when,
Calculated result and computational stability are optimal.
Embodiment 1
As shown in Fig. 2, the two-dimentional top cover using the process described above energy accurate and effective simulation different Reynolds number drives
Square chamber stream.
Embodiment 2
As shown in figure 3, can accurately and effectively simulating two-dimensional peripheral flow using the process described above.
Embodiment 4
As shown in figure 4, can accurately and effectively simulating two-dimensional NACA0012 profile flow using the process described above.
The above is only the preferred embodiment of the present invention, it will help and those skilled in the art further understands the present invention,
But the invention is not limited in any way.It should be noted that those skilled in the art, not departing from this hair
Under the premise of bright design, the several modifications and improvements made belong to protection scope of the present invention.
Claims (5)
1. a kind of mesh free Lattice Boltzmann Method based on semi-Lagrange and radial basis function, which is characterized in that first
SL method is introduced in Lattice Boltzmann Method, finds the starting point of migration, i.e. x-eαδ t, then inserted by introducing part RBF
Value method obtains starting point (x-eαδ t) distribution function；The part RBF interpolation method are as follows:
WhereinIt is target position functional value, φ (| | x-xi| |) it is radial basis function, | | x-xi| | Euclid norm, p
It (x) is multinomial, n is supporting point number, and m is the order of multinomial, and λ and γ are interpolation coefficients.
2. the mesh free Lattice Boltzmann Method according to claim 1 based on semi-Lagrange and radial basis function,
It is characterized in that, being counted before carrying out part RBF interpolation to the starting point using the distribution function value of collision rear support point
Interpolation coefficient λ and γ are calculated, the supporting point refers to the m discrete nodes nearest apart from the migration starting point.
3. the mesh free Lattice Boltzmann Method according to claim 2 based on semi-Lagrange and radial basis function,
It is characterized in that, calculating the increased condition of interpolation coefficient λ and γ before carrying out part RBF interpolation to the starting point are as follows:
Wherein p (x) is obtained by Pascal triangle.
4. the mesh free Lattice Boltzmann method side according to claim 1 or 3 based on semi-Lagrange and radial basis function
Method, which is characterized in that use Polyharmonic Splines (PHS) radial basis function φ (r)=r2k+1It is multinomial in conjunction with second order
Formula (pT(x)={ 1, x, y, x2,xy,y2) obtain starting point (x-eαδ t) distribution function, wherein r=| | x-xj||2,
5. the mesh free Lattice Boltzmann Method according to claim 1 based on semi-Lagrange and radial basis function,
It is characterized in that, the mesh free Lattice Boltzmann Method implementation steps are as follows:
(1) collision process is completed in discrete nodes:
(2) distribution function of migration starting point is obtained by Interpolation Property of Radial Basis Function:
Wherein xL=x-eαδ t is migration starting point coordinate.
(3) distribution function after being migrated at discrete nodes:
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112560358A (en) * | 2020-12-10 | 2021-03-26 | 中国空气动力研究与发展中心计算空气动力研究所 | DSMC calculation result noise reduction processing method based on radial basis function |
| CN113505518A (en) * | 2021-06-30 | 2021-10-15 | 同济大学 | Simulation method for preparation process of catalyst slurry of proton exchange membrane fuel cell |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112560358A (en) * | 2020-12-10 | 2021-03-26 | 中国空气动力研究与发展中心计算空气动力研究所 | DSMC calculation result noise reduction processing method based on radial basis function |
| CN113505518A (en) * | 2021-06-30 | 2021-10-15 | 同济大学 | Simulation method for preparation process of catalyst slurry of proton exchange membrane fuel cell |
| CN113505518B (en) * | 2021-06-30 | 2022-10-25 | 同济大学 | Simulation method for preparation process of catalyst slurry of proton exchange membrane fuel cell |
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