CN110457791B - Hydromechanical finite element algorithm based on characteristic line of Longge-Kutta time dispersion - Google Patents

Hydromechanical finite element algorithm based on characteristic line of Longge-Kutta time dispersion Download PDF

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CN110457791B
CN110457791B CN201910679877.0A CN201910679877A CN110457791B CN 110457791 B CN110457791 B CN 110457791B CN 201910679877 A CN201910679877 A CN 201910679877A CN 110457791 B CN110457791 B CN 110457791B
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characteristic line
coordinate system
kutta
time dispersion
navier
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CN110457791A (en
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廖绍凯
王德海
高金良
潘卓民
徐超
李伟
黄小亮
李玉印
张煜
钱锦锋
徐俊
顾歆翊
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Abstract

The invention discloses a characteristic line-based hydromechanical finite element algorithm for Longge-Kutta time dispersion, which comprises the following steps of: s1, establishing a Navier-Stokes equation without convection terms under a moving coordinate system along a characteristic line; s2, introducing Runge-Kutta time dispersion under a dynamic coordinate system to carry out time dispersion on a Navier-Stokes equation without a convection term, and S3, converting the quantity under the dynamic coordinate system into the quantity under a static coordinate system through Taylor expansion along a characteristic line; s4, carrying out interpolation dispersion spatially by adopting a Galerkin method; finally, a hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained. The method can improve the calculation accuracy and efficiency of the Navier-Stokes equation and simulate the complicated fluid mechanics problem.

Description

Hydromechanical finite element algorithm based on characteristic line of Longge-Kutta time dispersion
Technical Field
The invention relates to the technical field of hydromechanics calculation, in particular to a hydromechanics finite element algorithm based on the longge-kutta time dispersion of a characteristic line.
Background
In computational fluid dynamics, a numerical method is generally employed to solve a flow control equation (a navier-stokes equation) to obtain flow field information and predict the law of fluid motion. In the case of conventional solved navier-stokes equations, the following difficulties are often encountered:
(1) the nonlinear convective terms make the numerical solution of the equations difficult.
(2) When a standard Galerkin finite element format is adopted, the convection item is a non-self-accompanying operator, an optimal solution cannot be obtained, pseudo oscillation of a numerical solution is caused in severe cases, and even a result is diverged.
(3) The Navier-Stokes equation and the continuity equation are not closely related, and the pressure term and the divergence term have different orders, so that when an improper shape function is adopted, the pseudo oscillation of the numerical value solution is also caused.
The traditional windward finite element method (SUPG method) achieves good effect by introducing an asymmetric weight function (i.e. an artificial sticky term), increasing the weights of the incoming flow and the outgoing flow direction, so as to eliminate the numerical oscillation caused by the flow term. The method has the defects that a stable term interpolation function and a windward coefficient need to be found by experience, popularization is not facilitated, the windward coefficient and a unit matrix need to be updated in each iteration step, and the calculation efficiency is reduced. A characteristic line Galerkin finite element method (CBS method) based on time splitting carries out coordinate transformation on a one-dimensional Navier-Stokes equation, eliminates convection terms of the equation, is popularized to a two-dimensional problem, then introduces middle auxiliary speed splitting solution on time dispersion, and finally obtains a speed field and a pressure field. The CBS method effectively avoids the difficulty of selecting a reasonable weight function and a windward factor by the SUPG method, has fixed format and definite physical significance, is widely concerned by various scholars, and has important application in the aspects of shallow water circulation, square cavity drive flow, blunt body streaming and the like.
However, the CBS method is time-discrete to a first-order precision single-step method (similar to the euler method), and the speed is required to be split, which reduces the precision. The learners consider a two-step method (similar to a midpoint method) and a pressure stabilizing measure, so that the improved method has second-order precision, but the total stiffness matrix needs to be updated twice in the iteration step, and the calculation efficiency is reduced. The applicant has previously adopted the time dispersion idea of the fourth-order longge-kutta method (longge-kutta method) to be applied to the navier-stokes equation, so that the accuracy is further improved, but the equation solution number is increased and the total stiffness matrix needs to be updated twice, so that the format is complex and the calculation efficiency is reduced.
Therefore, the calculation accuracy of numerical solution of the navier-stokes equation can be improved without reducing the calculation time cost, and further research is needed.
Disclosure of Invention
Aiming at the defects and defects of the prior art, the hydrodynamics finite element algorithm based on the longge-kutta time dispersion of the characteristic line is provided, the calculation accuracy and efficiency of the Navier-Stokes equation can be improved, and the complex hydrodynamics problem can be simulated.
In order to achieve the above object, the present invention provides the following technical solutions.
The hydrodynamics finite element algorithm of the Longge-Kutta time dispersion based on the characteristic line comprises the following steps:
s1, establishing a Navier-Stokes equation without convection terms under a moving coordinate system along a characteristic line;
s2, introducing the time dispersion of the Runge-Kutta to carry out the time dispersion on the Navier-Stokes equation without the convection term under the moving coordinate system,
s3, converting the quantity under the dynamic coordinate system into the quantity under the static coordinate system through Taylor expansion along the characteristic line;
s4, carrying out interpolation dispersion spatially by adopting a Galerkin method; finally, a hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained.
The invention has the beneficial effects that: according to the invention, coordinate transformation along the characteristic line direction of the flow line is introduced, so that a Navier-Stokes equation without convection terms is obtained, and the problems of nonlinearity and numerical oscillation caused by the convection terms are solved. High-order Longge-Kutta time dispersion is introduced under an action standard system to carry out time dispersion on a Navier-Stokes equation without a convection term, so that the calculation precision is improved. And (4) introducing Taylor expansion along the streamline to solve the problem of grid updating caused by the action mark. And the Taylor expansion along the uniform streamline is introduced, so that the rigidity matrix is only required to be updated once in each time iteration step, and the calculation efficiency is improved.
As an improvement of the invention, the Navier-Stokes equation based on convection-free terms under the characteristic line is as follows:
Figure BDA0002144393280000031
wherein, the characteristic line is defined as:
Figure BDA0002144393280000032
the Runge-Kutta time discrete format under the moving coordinate system:
Figure BDA0002144393280000033
wherein the content of the first and second substances,
Figure BDA0002144393280000034
Figure BDA0002144393280000035
the taylor expansion format along the feature line is:
Figure BDA0002144393280000036
Figure BDA0002144393280000037
Figure BDA0002144393280000038
Figure BDA0002144393280000041
interpolation dispersion is carried out spatially by adopting Galerkin:
Figure BDA0002144393280000042
wherein
Figure BDA0002144393280000043
And psi is a function of speed and pressure, respectively>
Finally, the hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained as follows:
Figure BDA0002144393280000044
Figure BDA0002144393280000045
Figure BDA0002144393280000046
Figure BDA0002144393280000047
wherein the content of the first and second substances,
Figure BDA0002144393280000048
Figure BDA0002144393280000049
Figure BDA00021443932800000410
Figure BDA00021443932800000411
θ 2 is 0 or 1/2 or 1,
the calculation flow is as follows:
solving equation using initial velocity and pressure
Figure BDA00021443932800000412
The pressure p at the next moment is obtained n+1 ；
Solving equation using initial velocity and pressure
Figure BDA0002144393280000051
To obtain
Figure BDA0002144393280000052
Solving equation using initial velocity and pressure
Figure BDA0002144393280000053
To obtain
Figure BDA0002144393280000054
Solution formula
Figure BDA0002144393280000055
The speed of the next moment is obtained>
Figure BDA0002144393280000056
Drawings
FIG. 1 is a graph of the coordinate transformations used in deriving the non-convective term Navier-Stokes equations.
Fig. 2 is a time-discrete flow chart of the longge-kutta method for the convection-free term-navy-stokes equation.
FIG. 3 is a convergence comparison graph of the algorithm under the coarse grid and the traditional CBS algorithm.
Fig. 4 is a graph comparing the centerline horizontal velocity obtained by the present algorithm and the conventional CBS algorithm under a coarse grid, where the result of Erturk is a solution under a fine grid.
Fig. 5 is a flow chart obtained by simulating a square cavity flow based on the present algorithm.
FIG. 6 is a comparison graph of the resistance coefficient of the cylindrical flow around in different areas simulated by the algorithm and the CBS method.
FIG. 7 is a periodic flow chart obtained by simulating cylindrical streaming based on the present algorithm.
Detailed Description
The invention is further explained with reference to the drawings.
Referring to fig. 1 to 7, the characteristic line-based lange-kutta time-discrete hydromechanical finite element algorithm includes the following steps:
s1, establishing a Navier-Stokes equation without convection terms under a moving coordinate system along a characteristic line;
s2, introducing the time dispersion of the Runge-Kutta to carry out the time dispersion on the Navier-Stokes equation without the convection term under the moving coordinate system,
s3, converting the quantity under the dynamic coordinate system into the quantity under the static coordinate system through Taylor expansion along the characteristic line;
s4, carrying out interpolation dispersion spatially by adopting a Galerkin method; finally, a hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained.
Specifically, in S1, the dimensionless navier-stokes equation expression, without considering the influence of physical strength, is:
Figure BDA0002144393280000061
wherein Ω × (0, T) is space domain and time domain, t is time, u is i Is the velocity in the i direction, p is the pressure, and Re is the Reynolds number;
wherein, the characteristic line is defined as:
Figure BDA0002144393280000062
with t n+1 The position of the particle at time establishes a coordinate system, and as the particle moves along the streamline, at t n The corresponding position of the time particle P (x, y) is P (x ', y'), which is a moving point, and for this reason, we describe the position relationship of two times by using the following formula: s' = s-U Δ t
The corresponding directional derivatives along the streamline feature line are:
Figure BDA0002144393280000063
wherein the content of the first and second substances,
Figure BDA0002144393280000064
is the velocity of the streamlines. />
Using the coordinate transformation described above, t can be obtained n+1 Time derivative of time instant:
Figure BDA0002144393280000065
substituting the formula and the directional derivative into the dimensionless formula can eliminate the convection term thereof to obtain the Navier-Stokes equation without the convection term, which is as follows:
Figure BDA0002144393280000071
wherein, x' i And s' is the moving rectangular coordinate and arc coordinate corresponding to the moving point P.
In S2, carrying out time dispersion on the Navier-Stokes equation without the convection term by adopting a Runge-Kutta method, wherein the corresponding dispersion format is as follows:
Figure BDA0002144393280000072
wherein the content of the first and second substances,
Figure BDA0002144393280000073
Figure BDA0002144393280000074
s3, taylor expansion format along the characteristic line, i.e. performing Taylor expansion on each value under the action system along the characteristic line, and converting the Taylor expansion into t n+1 Quantity under the moment-of-rest system:
Figure BDA0002144393280000075
Figure BDA0002144393280000076
Figure BDA0002144393280000077
Figure BDA0002144393280000078
wherein, the streamline characteristic line is defined as:
Figure BDA0002144393280000081
in S4, galerkin is adopted spatially to carry out interpolation dispersion:
Figure BDA0002144393280000082
wherein
Figure BDA0002144393280000083
And psi are the velocity and pressure shape functions, respectively.
Finally, the hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained as follows:
Figure BDA0002144393280000084
Figure BDA0002144393280000085
Figure BDA0002144393280000086
Figure BDA0002144393280000087
wherein the content of the first and second substances,
Figure BDA0002144393280000088
Figure BDA0002144393280000089
Figure BDA00021443932800000810
Figure BDA00021443932800000811
θ 2 is 0 or 1/2 or 1.
The calculation flow of the invention is as follows:
solving equation using initial velocity and pressure
Figure BDA00021443932800000812
To obtainPressure p at the next moment n+1 ；
Solving equation using initial velocity and pressure
Figure BDA0002144393280000091
To obtain
Figure BDA0002144393280000092
Solving equation using initial velocity and pressure
Figure BDA0002144393280000093
To obtain
Figure BDA0002144393280000094
Solution formula
Figure BDA0002144393280000095
The speed of the next moment is obtained>
Figure BDA0002144393280000096
/>
The non-convection term Navier-Stokes equation of the invention describes the flow particle from t n Time to t n+1 The equation of flow momentum at the moment, when dividing the grid, selects t n+1 And establishing a static coordinate system in the flow field area at the moment, wherein the positions of the same particles are not in the positions corresponding to the static coordinate system at other moments because the particles flow, and taking the coordinate system corresponding to the moving particles as a moving coordinate system. According to the characteristic that the particles move along the streamline, the invention establishes the transformation relation of the coordinate systems of the particles at different moments, as shown by the formula s' = s-U delta t. Based on the coordinate transformation formula and the directional derivative, the invention establishes a Navier-Stokes equation without a convection term, as shown in the formula
Figure BDA0002144393280000097
As shown.
The invention is introduced into the Longge-Kutta methodWhen the line time is discrete, the method is different from the traditional Runge-Kutta method. The difference is that the slope and velocity expressions in discrete format are not only time dependent, but also position dependent, which is the difference caused by the moving coordinate system. Therefore, the time discrete format of the invention is as the formula
Figure BDA0002144393280000098
As shown.
The invention utilizes Taylor expansion along the characteristic line to expand the slope and the speed of different moments under a moving coordinate system into t n+1 The Taylor expansion format of the invention along the characteristic line is as shown in the formula
Figure BDA0002144393280000101
Figure BDA0002144393280000102
Figure BDA0002144393280000103
Figure BDA0002144393280000104
As shown.
Because the flow lines at different moments have different speeds, taylor expansion formats along the characteristic lines at different moments are different, and further the stiffness matrix formats are different, so that the stiffness matrix needs to be updated for multiple times at one time step, and the calculation efficiency is reduced. To this end, the invention adopts the formula
Figure BDA0002144393280000105
The streamline is described at a uniform speed, so that repeated updating of the stiffness matrix is avoided, and the calculation cost is saved.
According to the invention, coordinate transformation along the characteristic line direction of the flow line is introduced, so that a Navier-Stokes equation without convection terms is obtained, and the problems of nonlinearity and numerical oscillation caused by the convection terms are solved. High-order Longge-Kutta time dispersion is introduced under an action standard system to carry out time dispersion on a Navier-Stokes equation without a convection term, so that the calculation precision is improved. And (4) introducing Taylor expansion along the streamline to solve the problem of grid updating caused by the action mark. And the Taylor expansion along the uniform streamline is introduced, so that the rigidity matrix is only required to be updated once in each time iteration step, and the calculation efficiency is improved.
The above description is only a preferred embodiment of the present invention, and all equivalent changes or modifications of the structure, characteristics and principles described in the present invention are included in the scope of the present invention.

Claims (1)

1. A hydrodynamics finite element algorithm of Longge-Kutta time dispersion based on characteristic lines is characterized in that: the method comprises the following steps:
s1, establishing a Navier-Stokes equation without convection terms under a moving coordinate system along a characteristic line;
s2, introducing the time dispersion of the Runge-Kutta to carry out the time dispersion on the Navier-Stokes equation without the convection term under the moving coordinate system,
s3, converting the quantity under the dynamic coordinate system into the quantity under the static coordinate system through Taylor expansion along the characteristic line;
s4, carrying out interpolation dispersion spatially by adopting a Galerkin method; finally, obtaining a hydromechanics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line;
the Navier-Stokes equation based on the convection-free term under the characteristic line is as follows:
Figure FDA0003968431860000011
wherein, the characteristic line is defined as:
Figure FDA0003968431860000012
the LONGG-Kutta time discrete format under the moving coordinate system:
Figure FDA0003968431860000013
wherein the content of the first and second substances,
Figure FDA0003968431860000014
Figure FDA0003968431860000015
the taylor expansion format along the feature line is:
Figure FDA0003968431860000021
Figure FDA0003968431860000022
Figure FDA0003968431860000023
Figure FDA0003968431860000024
interpolation dispersion is carried out by adopting Galerkin in space:
Figure FDA0003968431860000025
wherein
Figure FDA0003968431860000026
And psi are eachThe function of the shape of the velocity and the pressure,
finally, the hydrodynamics finite element algorithm format of the Longge-Kutta time dispersion based on the characteristic line is obtained as follows:
Figure FDA0003968431860000027
Figure FDA0003968431860000028
Figure FDA0003968431860000029
Figure FDA00039684318600000210
wherein the content of the first and second substances,
Figure FDA00039684318600000211
Figure FDA0003968431860000031
Figure FDA0003968431860000032
Figure FDA0003968431860000033
θ 2 is 0 or 1/2 or 1,
the calculation flow is as follows:
solving equation using initial velocity and pressure
Figure FDA0003968431860000034
The pressure p at the next moment is obtained n+1 ；
Solving equation using initial velocity and pressure
Figure FDA0003968431860000035
Get->
Figure FDA0003968431860000036
Solving equation using initial velocity and pressure
Figure FDA0003968431860000037
Get->
Figure FDA0003968431860000038
Solution formula
Figure FDA0003968431860000039
The speed of the next moment is obtained>
Figure FDA00039684318600000310
/>
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