CN110457791A - The hydrodynamics finite element algorithm of Runge-Kutta time discrete based on characteristic curve - Google Patents

The hydrodynamics finite element algorithm of Runge-Kutta time discrete based on characteristic curve Download PDF

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CN110457791A
CN110457791A CN201910679877.0A CN201910679877A CN110457791A CN 110457791 A CN110457791 A CN 110457791A CN 201910679877 A CN201910679877 A CN 201910679877A CN 110457791 A CN110457791 A CN 110457791A
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characteristic curve
runge
coordinate system
discrete
kutta
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CN110457791B (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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Zhejiang Hengxin Design Group Co ltd
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Zhejiang Hengxin Building Tsukito Design Inc
Jiaxing University
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Abstract

The hydrodynamics finite element algorithm of the invention discloses a kind of Runge-Kutta time discrete based on characteristic curve, comprising the following steps: the Navier Stokes equation without convective term under moving coordinate system S1, is established along characteristic curve；S2, Runge-Kutta time discrete is introduced under moving coordinate system time discrete carried out to the Navier Stokes equation of no convective term, S3, the amount being converted into the amount under moving coordinate system by the Taylor expansion along characteristic curve under quiet coordinate system；S4, spatially use Galerkin method progress interpolation discrete；Finally obtain the hydrodynamics finite element algorithm format of the Runge-Kutta time discrete based on characteristic curve.The present invention can be improved the computational accuracy and efficiency of Navier Stokes equation, simulate complicated fluid mechanics problem.

Description

The hydrodynamics finite element algorithm of Runge-Kutta time discrete based on characteristic curve
Technical field
The present invention relates to hydromechanical computing technique fields, in particular to the Runge-Kutta time based on characteristic curve from Scattered hydrodynamics finite element algorithm.
Background technique
In Fluid Mechanics Computation, numerical method is generallyd use to solve Fluid Control Equation (Navier Stokes equation (Navier Stokes equation)), to obtain the rule of information of flow and pre- measured fluid movement.The Na Wei-of traditional solution this When lentor equation, following difficulty is typically encountered:
1. nonlinear convection item brings very big difficulty to the numerical solution of equation.
2. use standard Galerkin finite element scheme, convective term is non-self-adjoint operator, cannot get optimal solution, seriously When will cause the pseudo- oscillation or even result diverging of numerical solution.
3. Navier Stokes equation and continuity equation connect not close, and pressure term and divergence item not same order each other, When using inappropriate shape function, the pseudo- oscillation of numerical solution is likewise resulted in.
Traditional Upwind Finite Element Method (SUPG method) increases and by introducing asymmetric weight function (i.e. artificial viscous term) Stream and reduction go the weight in stream direction to achieve good effect to eliminate numerical oscillation caused by convective term.This method is insufficient It is that stable item interpolating function and upwind coefficient need to be found by rule of thumb, is unfavorable for promoting, and each iteration step is both needed to update windward Coefficient and cell matrix, reduce computational efficiency.The golden FInite Element (CBS method) of characteristic curve gal the Liao Dynasty based on time splitting is to one-dimensional Navier Stokes equation be coordinately transformed, its convective term is eliminated, and extend to two-dimensional problems, then in time discrete On, it introduces intermediate auxiliary speed division and solves, finally obtain velocity field and pressure field.CBS method effectively prevents the selection of SUPG method Reasonable weight function and the windward difficulty of the factor, format is fixed and explicit physical meaning, by scholars' extensive concern, in shallow water Circulation, cavity driven flow and flow around bluff bodies etc. have major application.
However CBS method in time discrete for single order precision single -step method (be similar to Euler method), and speed need to be carried out Division, reduces precision.There is scholar to consider two-step method (being similar to mid-point method), pressure stabilizing measures, makes improved method It with second order accuracy, but needs to update global stiffness matrix twice in iteration step, reduces computational efficiency.Once four are used before applicant The time discrete thought of rank Runge-Kutta method (Runge-Kutta method) is applied in Navier Stokes equation, is further increased Precision, but increase equation solution number and global stiffness matrix need to be updated twice, so that format is complicated and reduces computational efficiency.
Therefore, while not reducing calculating time cost, numerical solution Navier Stokes equation equation can be improved Computational accuracy, it is also necessary to further investigate.
Summary of the invention
In view of the deficiencies of the prior art and defect, a kind of fluid of Runge-Kutta time discrete based on characteristic curve is provided Mechanics finite element algorithm can be improved the computational accuracy and efficiency of Navier Stokes equation, simulate complicated fluid force knowledge Topic.
To achieve the above object, the present invention provides following technical scheme.
The hydrodynamics finite element algorithm of Runge-Kutta time discrete based on characteristic curve, comprising the following steps:
S1, the Navier Stokes equation without convective term under moving coordinate system is established along characteristic curve；
S2, Navier Stokes equation progress of the Runge-Kutta time discrete to no convective term is introduced under moving coordinate system Time discrete,
S3, the amount being converted into the amount under moving coordinate system by the Taylor expansion along characteristic curve under quiet coordinate system；
S4, spatially use Galerkin method progress interpolation discrete；Finally obtain the Runge-Kutta time based on characteristic curve from Scattered hydrodynamics finite element algorithm format.
The invention has the benefit that having obtained present invention introduces the coordinate transform along streamline characteristic curve direction without convection current Navier Stokes equation, solve that convective term bring is non-linear and numerical value oscillation problem.It is introduced under movement mark system The Runge-Kutta time discrete of high-order carries out time discrete to the Navier Stokes equation of no convective term, improves calculating essence Degree.The Taylor expansion along streamline is introduced, solves the problems, such as movement mark bring grid updating.Introduce Taylor's exhibition along uniform streamline It opens, so that it is primary that only stiffness matrix need to be updated in each time iteration step, improves computational efficiency.
As an improvement of the present invention, based on the Navier Stokes equation without convective term under characteristic curve are as follows:
Wherein, characteristic curve is defined as:
Runge-Kutta time discrete format under moving coordinate system:
Wherein,
Along the Taylor expansion format of characteristic curve are as follows:
It is discrete that interpolation is spatially carried out using gal the Liao Dynasty gold:
WhereinIt is respectively speed and pressure shape function with ψ,
The hydrodynamics finite element algorithm format for finally obtaining the Runge-Kutta time discrete based on characteristic curve is as follows:
Wherein, θ2It is 0 or 1/2 or 1,
Calculation process is as follows:
Utilize initial velocity and Pressure solution formulaObtain the pressure p of subsequent timen+1；
Utilize initial velocity and Pressure solution formula
It obtains
Utilize initial velocity and Pressure solution formula
It obtains
Solution formula Obtain the speed of subsequent time
Detailed description of the invention
Fig. 1 is being derived without coordinate transform figure used in convective term Navier Stokes equation.
Fig. 2 is the time discrete flow chart that Runge-Kutta method is carried out to no convective term Navier Stokes equation.
Fig. 3 is that this algorithm and traditional CBS Algorithm Convergence compare figure under coarse grid.
Fig. 4 is that the middle line horizontal velocity that this algorithm and traditional CBS algorithm obtain under coarse grid compares figure, wherein Erturk Result be fine grid blocks under solution.
Fig. 5 is the motion pattern obtained based on this algorithm simulation side chamber stream.
Fig. 6 is the peripheral flow resistance coefficient comparison diagram under this algorithm and CBS method simulation different zones.
Fig. 7 is a cycle motion pattern obtained based on this algorithm simulation peripheral flow.
Specific embodiment
It is further illustrated in conjunction with attached drawing to of the invention.
Referring to Fig. 1 to the hydrodynamics finite element algorithm of the Runge-Kutta time discrete shown in Fig. 7 based on characteristic curve, The following steps are included:
S1, the Navier Stokes equation without convective term under moving coordinate system is established along characteristic curve；
S2, Navier Stokes equation progress of the Runge-Kutta time discrete to no convective term is introduced under moving coordinate system Time discrete,
S3, the amount being converted into the amount under moving coordinate system by the Taylor expansion along characteristic curve under quiet coordinate system；
S4, spatially use Galerkin method progress interpolation discrete；Finally obtain the Runge-Kutta time based on characteristic curve from Scattered hydrodynamics finite element algorithm format.
Specifically, not considering the influence of physical strength, the Navier Stokes equation expression formula of nondimensionalization in S1 are as follows:
Ω × (0, T) is spatial domain and time-domain in formula, and t is time, uiIt is the speed in the direction i, p is pressure, and Re is thunder Promise number；
Wherein, characteristic curve is defined as:
With tn+1Coordinate system is established in position where moment particle, since particle is moved along streamline, so in tnMoment grain The corresponding position of sub- P (x, y) is P (x ', y '), is a dynamic point, for this purpose, we are closed using the position that following formula describes two moment System: s '=s-U Δ t
Along the corresponding directional derivative of streamline characteristic curve are as follows:
Wherein,It is flow line speed.
Using above-mentioned coordinate transform, t can be obtainedn+1The time-derivative at moment:
It based on above formula and directional derivative, is updated in nondimensionalization, its convective term can be eliminated, obtain receiving for no convective term Dimension-stokes equation, such as following formula:
Wherein, x 'iIt is that dynamic point P moves rectangular co-ordinate and arc coordinate accordingly with s '.
In S2, time discrete is carried out using Runge-Kutta method to the Navier Stokes equation of no convective term, accordingly Discrete scheme is as follows:
Wherein,
In S3, along the Taylor expansion format of characteristic curve, i.e., each magnitude under movement mark system is subjected to Taylor's exhibition along characteristic curve It opens, is converted to tn+1Amount under moment quiet coordinate system:
Wherein, streamline characteristic curve is defined as:
In S4, it is discrete that interpolation is spatially carried out using gal the Liao Dynasty gold:
WhereinIt is respectively speed and pressure shape function with ψ.
The hydrodynamics finite element algorithm format for finally obtaining the Runge-Kutta time discrete based on characteristic curve is as follows:
Wherein, θ2It is 0 or 1/2 or 1.
Calculation process of the invention is as follows:
Utilize initial velocity and Pressure solution formulaObtain the pressure p of subsequent timen+1；
Utilize initial velocity and Pressure solution formula
It obtains
Utilize initial velocity and Pressure solution formula
It obtains
Solution formula Obtain the speed of subsequent time
No convective term Navier Stokes equation of the present invention describes fluidized particle from tnMoment is to tn+1Moment The equation of momentum is flowed, in grid division, selects tn+1The flow field regions at moment establish quiet coordinate system, since particle is flowing, because , in the position of remaining moment identical particle not in the corresponding position of quiet coordinate system, we are corresponding by the particle moved at this time for this Coordinate system is moving coordinate system.According to the feature that particle is moved along streamline, the present invention establishes the change of particle different moments coordinate system Relationship is changed, as shown in formula s '=s-U Δ t.Based on the coordinate transform formula and directional derivative, the present invention establishes no convective term Navier Stokes equation, such as formulaIt is shown.
The present invention is when introducing Runge-Kutta method progress time discrete, different from traditional Runge-Kutta method.It is distinguished It is, the slope and velocity expression in discrete scheme are not only with time correlation, but also related to position, this is by moving coordinate system Caused difference.For this purpose, time discrete format such as formula of the inventionInstitute Show.
The present invention carries out Taylor expansion using along characteristic curve, and the slope of moving coordinate system lower different moments and speed are launched into tn+1Expression formula under moment corresponding quiet coordinate system, Taylor expansion format such as formula of the present invention along characteristic curve
It is shown.
Since the speed of different moments streamline is different, different moments the Taylor expansion format along characteristic curve not Together, and then it will cause stiffness matrix format difference, so that a time step need to repeatedly update stiffness matrix, reduction computational efficiency. For this purpose, the present invention uses formulaEven speed streamline is described, to avoid stiffness matrix It repeatedly updates, saves and calculate cost.
Present invention introduces the coordinate transform along streamline characteristic curve direction, Na Wei-Stokes side without convective term has been obtained Journey, solves that convective term bring is non-linear and numerical value oscillation problem.The Runge-Kutta time of high-order is introduced under movement mark system The discrete Navier Stokes equation to no convective term carries out time discrete, improves computational accuracy.Introduce the Taylor along streamline Expansion solves the problems, such as movement mark bring grid updating.The Taylor expansion along uniform streamline is introduced, so that each time iteration walks In only need to update stiffness matrix primary, improve computational efficiency.
The above description is only a preferred embodiment of the present invention, thus it is all according to the configuration described in the scope of the patent application of the present invention, The equivalent change or modification that feature and principle are done, is included in the scope of the patent application of the present invention.

Claims (2)

1. the hydrodynamics finite element algorithm of the Runge-Kutta time discrete based on characteristic curve, it is characterised in that: including following step It is rapid:
S1, the Navier Stokes equation without convective term under moving coordinate system is established along characteristic curve；
S2, Runge-Kutta time discrete is introduced under moving coordinate system to the Navier Stokes equation progress time of no convective term It is discrete,
S3, the amount being converted into the amount under moving coordinate system by the Taylor expansion along characteristic curve under quiet coordinate system；
S4, spatially use Galerkin method progress interpolation discrete；Finally obtain the Runge-Kutta time discrete based on characteristic curve Hydrodynamics finite element algorithm format.
2. the hydrodynamics finite element algorithm of the Runge-Kutta time discrete according to claim 1 based on characteristic curve, It is characterized in that: based on the Navier Stokes equation without convective term under characteristic curve are as follows:
Wherein, characteristic curve is defined as:
Runge-Kutta time discrete format under moving coordinate system:
Wherein,
Along the Taylor expansion format of characteristic curve are as follows:
It is discrete that interpolation is spatially carried out using gal the Liao Dynasty gold:
WhereinIt is respectively speed and pressure shape function with ψ,
The hydrodynamics finite element algorithm format for finally obtaining the Runge-Kutta time discrete based on characteristic curve is as follows:
Wherein,
θ2It is 0 or 1/2 or 1,
Calculation process is as follows:
Utilize initial velocity and Pressure solution formulaObtain the pressure p of subsequent timen+1；
Utilize initial velocity and Pressure solution formulaIt obtains
Utilize initial velocity and Pressure solution formulaIt obtains
Solution formulaIt obtains The speed of subsequent time
CN201910679877.0A 2019-07-26 2019-07-26 Hydromechanical finite element algorithm based on characteristic line of Longge-Kutta time dispersion Active CN110457791B (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113192566A (en) * 2021-04-30 2021-07-30 西安交通大学 Molten pool transient phase change simulation method under severe reactor accident

Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2005111872A2 (en) * 2004-05-11 2005-11-24 Siemens Aktiengesellschaft Method for solving the navier-stokes equation
US20070239414A1 (en) * 2006-04-05 2007-10-11 Oh-Young Song Method of simulating detailed movements of fluids using derivative particles
WO2012031398A1 (en) * 2010-09-09 2012-03-15 Tianjin Aerocode Engineering Application Software Development Inc. Numerical method for simulating subsonic flows based on euler equations in lagrangian formulation

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2005111872A2 (en) * 2004-05-11 2005-11-24 Siemens Aktiengesellschaft Method for solving the navier-stokes equation
US20070239414A1 (en) * 2006-04-05 2007-10-11 Oh-Young Song Method of simulating detailed movements of fluids using derivative particles
WO2007114548A1 (en) * 2006-04-05 2007-10-11 Seoul National University Industry Foundation Method of simulating detailed movements of fluids using derivative particles
WO2012031398A1 (en) * 2010-09-09 2012-03-15 Tianjin Aerocode Engineering Application Software Development Inc. Numerical method for simulating subsonic flows based on euler equations in lagrangian formulation

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
廖绍凯: "粘性流基于特征线的四阶Runge-Kutta有限元法", 《计算力学学报》 *

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113192566A (en) * 2021-04-30 2021-07-30 西安交通大学 Molten pool transient phase change simulation method under severe reactor accident
CN113192566B (en) * 2021-04-30 2022-12-09 西安交通大学 Molten pool transient phase change simulation method under severe reactor accident

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