CN108536954A - A kind of high-precision Lattice Boltzmann Method based on intersection point interruption gal the Liao Dynasty gold - Google Patents
A kind of high-precision Lattice Boltzmann Method based on intersection point interruption gal the Liao Dynasty gold Download PDFInfo
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- CN108536954A CN108536954A CN201810305382.7A CN201810305382A CN108536954A CN 108536954 A CN108536954 A CN 108536954A CN 201810305382 A CN201810305382 A CN 201810305382A CN 108536954 A CN108536954 A CN 108536954A
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Abstract
The invention discloses a kind of high-precision Lattice Boltzmann Methods based on intersection point interruption gal the Liao Dynasty gold, pure convection equation is solved by using the golden method of intersection point interruption gal the Liao Dynasty, to obtain the result of calculation on grid cell, precision can be controlled by adjusting the number of hits in unit.The present invention introduces the golden method of intersection point interruption gal the Liao Dynasty in Lattice Boltzmann Method and improves the precision of Flow Field Calculation result by effectively adjusting the intersection point number calculated in grid cell.On this basis, in conjunction with other computing techniques such as such as immersed Boundary Method, Phase Field, achieved the purpose that simulate Various Complex fluid mechanics problem using Lattice Boltzmann Method.
Description
Technical field：
The present invention relates to a kind of high-precision Lattice Boltzmann Methods based on intersection point interruption gal the Liao Dynasty gold, belong to calculating fluid
Mechanical technology field.
Background technology：
Fluid Mechanics Computation is to utilize the discrete solution Fluid Control Equation of numerical method, to obtain information of flow and with this
One subject of pre- measured fluid movement rule.In traditional Fluid Mechanics Computation, the Fluid Control Equation of solution be Na Wei-this
Lentor (N-S) equation.But in recent decades, Lattice Boltzmann Method (LBM) is increasingly becoming another widely used meter
Fluid operator mechanics method.Compared to N-S equations, LBM is not comprising complicated nonlinear terms, and also without higher derivative item, it is only
There is simple algebraic operation process.Which strongly simplifies its calculating process, and improve the universality of application.But, due to
The limitation of this method internal factor, its computational accuracy theoretically can only achieve second order.As hydrodynamics is involved in the problems, such as getting over
Come more complicated, the required precision of logarithm analog result is also higher and higher.For this purpose, improving the computational accuracy of LBM, to be to maintain its competing
Strive the committed step of power.
As a kind of highly-accurate nephelometric titrimetry method, golden (DG) method of interruption gal the Liao Dynasty is in Fluid Mechanics Computation field
It is widely applied.For DG on the basis of remaining FInite Element high accurate scheme, the numerical value for introducing finite volume method is logical
The concept of amount.Therefore, the advantages of it has both methods simultaneously.In DG methods, there are two types of approach for control computational accuracy.
One is precision is changed by using different basic functions, another kind is come by adjusting the interpolation point calculated in grid cell
Change precision.Compared to the former, the latter is easier to construct and realize, it is also referred to as golden (NDG) method of intersection point interruption gal the Liao Dynasty.
But currently, DG methods are still only used for solving traditional N-S equations.Therefore, it is necessary to which high-precision DG methods are drawn
Enter into LBM to improve its computational accuracy, to meet the needs of current fluid mechanics problem research.
Invention content：
The present invention is provided a kind of based on intersection point interruption gal the Liao Dynasty gold to solve the above-mentioned problems of the prior art
High-precision Lattice Boltzmann Method is interrupted the golden method of gal the Liao Dynasty by introducing intersection point, and the present invention improves Lattice Boltzmann Method
Computational accuracy, to be used to simulate the fluid mechanics problem that becomes increasingly complex.
The technical solution adopted in the present invention has：A kind of high-precision Lattice Boltzmann method side based on intersection point interruption gal the Liao Dynasty gold
Method, set the Lattice Boltzmann Method governing equation based on more relaxation time models as：
gi(x+eiδ t, t+ δ t)=gi(x,t)-M-1S[R(x,t)-Req(x, t)], i=0,1 ... b-1
Wherein giIt is the distribution function in the velocity space, R is giOne group of physical quantity on momentum space, M is corresponding
Transformed matrix, ReqIt is the corresponding equilibrium state of R, S is a non-negative diagonal matrix, and b is the number of grid directional velocity, and τ is
Single slack time coefficient, x are position coordinates vectors, and t is the time, and δ t are time steps；
Under the premise of not influencing result of calculation, above-mentioned governing equation can resolve into two parts, i.e.,：
1. collision process：
2. transition process：
Transition process therein is reduced into pure convection equation：
Wherein eiIt is grid velocity vector, using the golden method of intersection point interruption gal the Liao Dynasty to its discrete solution；
Define flux Gi(g)=eigi, convection equation is rewritten as：
After being divided using triangle or tetrahedral grid cellular convection field space, above-mentioned equation intersection point is interrupted gal the Liao Dynasty
Golden method carries out discrete solution, and the discrete equation finally obtained on some grid cell is：
WhereinIt is the basic solution vector in k-th of grid cell, NpIt is intersection point in unit
Number, NkIt is mass matrix,It is stiffness matrix,It is the right item matrix, E is of entire flow field space lattice unit
Number；
Function g i on k-th of grid cell is by basic solution vectorIt is acquired, is changed by N rank multinomial interpolation
Number of hits Np, polynomial exponent number N changes, and the relationship between them is Np=(N+1) (N+2)/2；
Finally, it is based on giFlow field physical quantity needed for obtaining.
Further, polynomial order N values are 3.
The present invention has the advantages that：The present invention introduces the distant gold side of intersection point interruption gal in Lattice Boltzmann Method
Method improves the precision of Flow Field Calculation result by effectively adjusting the intersection point number calculated in grid cell.On this basis,
In conjunction with other computing techniques such as such as immersed Boundary Method, Phase Field, reaches and simulated Various Complex using Lattice Boltzmann Method
The purpose of fluid mechanics problem.
Description of the drawings：
Fig. 1 is the intersection point schematic diagram in triangular mesh unit under two-dimensional case.
Fig. 2 is the streamline of two dimension cavity driven flow under the different Reynolds number using method of the present invention simulation gained
Figure.
Fig. 3 is the three-dimensional dragonfly model fortune that gained is simulated using method of the present invention and after combining immersed Boundary Method
Dynamic vorticity figure.
Fig. 4 is to simulate drop in two dimension microchannels of gained using method of the present invention and after combining Phase Field
Forming process.
Specific implementation mode：
The present invention will be further described below with reference to the drawings.
High-precision Lattice Boltzmann Method of the present invention, key are by using the golden method of intersection point interruption gal the Liao Dynasty
Solve pure convection equation, to obtain the result of calculation on grid cell, precision can by adjusting the number of hits in unit into
Row control.
For incompressible viscous flow problem, the Lattice Boltzmann Method governing equation based on more relaxation time models
For：
gi(x+eiδ t, t+ δ t)=gi(x,t)-M-1S[R(x,t)-Req(x, t)], i=0,1 ... b-1
Wherein giIt is the distribution function in the velocity space, R is giOne group of physical quantity (such as density, speed on momentum space
Degree, stress tensor etc.), M is corresponding transformed matrix, ReqIt is the corresponding equilibrium state of R, S is a non-negative diagonal matrix, b
It is the number of grid directional velocity, τ is single slack time coefficient, and x is position coordinates vector, and t is the time, and δ t are time steps.
Under the premise of not influencing result of calculation, above-mentioned governing equation can resolve into two parts, i.e.,：
1. collision process：
2. transition process：
Transition process therein can be reduced into pure convection equation：
Wherein eiIt is grid velocity vector.Above-mentioned convection equation may be used different numerical methods and carry out discrete solution.
In the present invention, using the golden method of intersection point interruption gal the Liao Dynasty to its discrete solution.
If defining flux Gi(g)=eigi, above-mentioned convection equation can be rewritten as：
After being divided using triangle (two-dimensional case) or tetrahedron (three-dimensional situation) grid cell stream field space, on
Discrete solution can be carried out with the golden method of intersection point interruption gal the Liao Dynasty by stating equation.Finally obtain the discrete equation on some grid cell
For：
WhereinIt is the basic solution vector in k-th of grid cell, NpIt is intersection point in unit
Number, NkIt is mass matrix,It is stiffness matrix,It is the right item matrix, E is of entire flow field space lattice unit
Number.
Above-mentioned ODE may be used different time discrete formats and solve.Finally, point on k-th of grid cell
Cloth function giIt can be by basic solution vectorIt is acquired by N rank multinomial interpolation.Change number of hits Np, polynomial exponent number N will send out
Changing, the relationship between them are Np=(N+1) (N+2)/2.Correspondingly, the computational solution precision in flow field can also be made accordingly
Variation.
In the present invention, above-mentioned ODE is solved using 4 rank Runge-Kutta discrete schemes, to obtain k-th of net
Basic solution vector on lattice unitThen, function g is acquired using N rank multinomial interpolationi.Finally, it is based on giObtain institute
The flow field physical quantity (such as density, speed, stress tensor) needed.
The present invention has found that effect is optimal when polynomial order N takes 3 by test.Continue to increase N, result of calculation is substantially not
It changes.
As shown in Fig. 2, the two-dimentional cavity driven flow of method energy accurate and effective simulation different Reynolds number of the present invention.
As shown in figure 3, in conjunction with immersed Boundary Method, method of the present invention also can accurate and effective simulation three-dimensional dragonfly model fortune
It is dynamic.As shown in figure 4, in conjunction with Phase Field, method of the present invention can also accurately and effectively drop in simulating two-dimensional microchannel
Forming process.
The above is only a preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art
It for member, can also make several improvements without departing from the principle of the present invention, these improvement also should be regarded as the present invention's
Protection domain.
Claims (2)
1. a kind of high-precision Lattice Boltzmann Method based on intersection point interruption gal the Liao Dynasty gold, it is characterised in that：
Set the Lattice Boltzmann Method governing equation based on more relaxation time models as：
gi(x+eiδ t, t+ δ t)=gi(x,t)-M-1S[R(x,t)-Req(x, t)], i=0,1 ... b-1
Wherein giIt is the distribution function in the velocity space, R is giOne group of physical quantity on momentum space, M are to convert accordingly
Matrix, ReqIt is the corresponding equilibrium state of R, S is a non-negative diagonal matrix, and b is the number of grid directional velocity, and τ is Dan Song
Relaxation time coefficient, x are position coordinates vectors, and t is the time, and δ t are time steps；
Under the premise of not influencing result of calculation, above-mentioned governing equation can resolve into two parts, i.e.,：
1. collision process：
2. transition process：
Transition process therein is reduced into pure convection equation：
Wherein eiIt is grid velocity vector, using the golden method of intersection point interruption gal the Liao Dynasty to its discrete solution；
Define flux Gi(g)=eigi, convection equation is rewritten as：
After being divided using triangle or tetrahedral grid cellular convection field space, above-mentioned equation intersection point is interrupted the distant gold side of gal
Method carries out discrete solution, and the discrete equation finally obtained on some grid cell is：
WhereinIt is the basic solution vector in k-th of grid cell, NpIt is of intersection point in unit
Number, NkIt is mass matrix,It is stiffness matrix,It is the right item matrix, E is the number of entire flow field space lattice unit；
Function g i on k-th of grid cell is by basic solution vectorIt is acquired by N rank multinomial interpolation, changes number of hits
Np, polynomial exponent number N changes, and the relationship between them is Np=(N+1) (N+2)/2；
Finally, it is based on giFlow field physical quantity needed for obtaining.
2. the high-precision Lattice Boltzmann Method as described in claim 1 based on intersection point interruption gal the Liao Dynasty gold, it is characterised in that：
Polynomial order N values are 3.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109299569A (en) * | 2018-10-24 | 2019-02-01 | 广州市香港科大霍英东研究院 | A kind of Large eddy simulation method of the incompressible viscous flows body based on coherent structure |
| CN111241728A (en) * | 2020-01-03 | 2020-06-05 | 电子科技大学 | Intermittent Galerkin finite element numerical solution method of Euler equation |
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| CN105241911A (en) * | 2015-09-23 | 2016-01-13 | 中国石油大学(北京) | LBM-based simulated low field nuclear magnetic resonance fluid analysis method and device |
| CN106501357A (en) * | 2016-12-06 | 2017-03-15 | 华中科技大学 | A kind of method of measurement mixed gas molecule multi-mode relaxation process |
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2018
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Patent Citations (4)
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|---|---|---|---|---|
| CN101673393A (en) * | 2009-09-25 | 2010-03-17 | 上海大学 | Image de-noising method based on lattice Boltzmann model |
| US20140296842A1 (en) * | 2013-04-02 | 2014-10-02 | Siemens Corporation | Patient Specific Planning and Simulation of Ablative Procedures |
| CN105241911A (en) * | 2015-09-23 | 2016-01-13 | 中国石油大学(北京) | LBM-based simulated low field nuclear magnetic resonance fluid analysis method and device |
| CN106501357A (en) * | 2016-12-06 | 2017-03-15 | 华中科技大学 | A kind of method of measurement mixed gas molecule multi-mode relaxation process |
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| 吴杰等: ""一种模拟不可压流的节点间断有限元—格子波尔兹曼方法"，", 《南京航空航天大学学报》 * |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109299569A (en) * | 2018-10-24 | 2019-02-01 | 广州市香港科大霍英东研究院 | A kind of Large eddy simulation method of the incompressible viscous flows body based on coherent structure |
| CN111241728A (en) * | 2020-01-03 | 2020-06-05 | 电子科技大学 | Intermittent Galerkin finite element numerical solution method of Euler equation |
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Application publication date: 20180914 |