CN109753690A - Nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation - Google Patents
Nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation Download PDFInfo
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Abstract
The invention discloses a kind of nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation, the technical issues of for solving the existing linear unsteady aerodynamic force order reducing method low precision based on Fluid Mechanics Computation.Technical solution is to realize that algorithm obtains the linear segment of non-linear reduced-order model based on Volterra series theory and system minimal characteristic, the non-linear partial that process obtains non-linear reduced-order model is minimized by output error, linear reduced-order model compared to background technique based on Volterra series, it can predict to obtain more accurate aeroelasticity response and without recognizing Second-Order Volterra kernel function, not only having reduced calculating cost but also improving computational accuracy.In addition to this, which also maintains succinct form, it may be convenient to be applied in engineering practice.
Description
Technical field
The present invention relates to a kind of linear unsteady aerodynamic force order reducing method based on Fluid Mechanics Computation, in particular to it is a kind of
Nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation.
Background technique
Document " An Efficient Approach for Solving Nonlinear Aeroelastic
Phenomenon using Reduced-Order Modeling, 45th AIAA/ASME/ASCE/AHS/ASC
Structures,Structural Dynamics and Materials Conference,AIAAPaper 2004-2037,
2004, pp.19-22. " discloses a kind of linear unsteady aerodynamic force order reducing method based on Fluid Mechanics Computation.This method benefit
It is recognized with approximate single order Volterra core of the Volterra series theory to aeroelastic system, and sets up unsteady gas
Power reduced-order model realizes that reduced-order model is converted into state space form by algorithm using system minimal characteristic, by pneumatic
Elastic system is analyzed, and preferable prediction has obtained the Flutter Boundaries of aeroelastic system.This method utilizes limited calculating
Hydrodynamics Flow Field Calculation data establish unsteady aerodynamic force reduced-order model, can significantly improve the solution of aeroelastic system
Efficiency.The unsteady aerodynamic force reduced-order model that document is established is linear reduced-order model, when the nonlinear degree of aeroelastic system
When stronger, the aeroelasticity response accuracy predicted is inadequate, be additionally based on Volterra series theory establish it is non-linear non-fixed
Normal aerodynamic reduced order model needs to recognize the Second-Order Volterra kernel function of aeroelastic system, and required calculation amount can exponentially increase
It grows and can not bear, thus can not be applied to engineering problem.
Summary of the invention
In order to overcome the shortcomings of the existing linear unsteady aerodynamic force order reducing method low precision based on Fluid Mechanics Computation, this
Invention provides a kind of nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation.This method is based on Volterra grades
Mathematics opinion and system minimal characteristic realize that algorithm obtains the linear segment of non-linear reduced-order model, are minimized and are flowed by output error
Journey obtains the non-linear partial of non-linear reduced-order model, the linear depression of order mould compared to background technique based on Volterra series
Type can be predicted to obtain more accurate aeroelasticity response and be not necessarily to recognize Second-Order Volterra kernel function, both reduce calculating
It spends and improves computational accuracy again.In addition to this, which also maintains succinct form, it may be convenient to be applied to engineering reality
In trampling.
The technical solution adopted by the present invention to solve the technical problems: a kind of non-linear non-fixed based on Fluid Mechanics Computation
Normal aerodynamic force order reducing method, expression-form are as follows:
Wherein, Aa,Ba,Ca,DaFor the linear component part of reduced-order model, EaFor non-linear component part.Nonlinear equation
φ () is the hyperbolic sine function about input ξ (n).Its main feature is that the following steps are included:
Step 1: by identification aeroelastic system approximation single order Volterra kernel function establish it is linear based on
The reduced-order model of Volterra series.Euler equation and N-S equation have small nonlinearity characteristic under microvariations, therefore non-fixed
Normal aerodynamic force Precise Representation in the form of Second-Order Volterra series:
Select step response energized gas bullet system, established by recognizing approximate single order kernel function it is linear based on
The reduced-order model of Volterra series.Approximate single order Volterra kernel functionIt is defined as follows:
In formula, s (n) is step response, and n is discrete time step, ξ0For the amplitude of step response.
It realizes that the reduced-order model based on Volterra is converted into state space form by algorithm using system minimal characteristic, obtains
To the linear component part of non-linear reduced-order model.The linearly invariant discrete state that system minimal characteristic realizes that algorithm obtains is empty
Between form it is as follows:
Aa,Ba,Ca,DaSytem matrix, input matrix, output matrix and the feedforward matrix of pneumatic power system are corresponded respectively to,
xaFor the state variable of system, ξ is the input of system, FaIt is exported for the unsteady aerodynamic force of system, q is dynamic pressure.The pulse of system
Response output are as follows:
Step 2: being realized after establishing the obtained reduced-order model based on Volterra series by system minimal characteristic
Series expression-form is converted state space form by algorithm.NoteValue for the matrix of M × N, parameter M and N is 4.Structure
It is as follows to make Hankel matrix
Wherein, the value that the value of α is 800, β is 60.K=1 is enabled, to Hαβ(0) make singular value decomposition
Hαβ(0)=U Σ VT (7)
Define matrix Aa,Ba,Ca,DaExpression formula it is as follows
Wherein,WithIt is defined as follows
Matrix A is calculateda,Ba,Ca,Da, the linear composition of the nonlinear unsteady aerodynamics reduced-order model as foundation
Part.
Step 3: determining nonlinear terms EaParameter.Using Gauss-Newton method discernibility matrixes EaParameter value.By EaIn
Value sorts top to bottom to obtain a unknown vector g=vec (Ea), following quadrature discrete equation is solved until error delta g=gnew-gold
It is sufficiently small:
In formula, UfExpression formula it is as follows
By equation (10), output error e (k)=F of each time step is obtaineda,CFD(k)-Fa,g(k), Jacobi
The expression formula of matrix is
By solving following least square problem renewal vector g in each iteration step
JTJ Δ g=-JTea (13)
Wherein, eaFor error matrix, indicate empty in each time step Fluid Mechanics Computation calculated result and nonlinear state
Between error between reduced-order model prediction result.
ea=[e (1) e (2) ... e (N)] (14)
After obtaining vector g, sort top to bottom back to matrix EaIn, obtain nonlinear unsteady aerodynamics reduced-order model
Nonlinear terms.
Step 4: in the linear segment and non-linear partial method for solving that determine nonlinear unsteady aerodynamics reduced-order model
Afterwards, training data necessary to establishing reduced-order model is calculated using Fluid Mechanics Computation solver.Use the step of selection
Excitation motivates 4 rank mode before model, obtains the corresponding pneumatic force-responsive of unsteady broad sense.It is obtained using based on vibration frequency
It obtains training data method and training data method is obtained based on incoming flow dynamic pressure, training number needed for obtaining identification reduced-order model nonlinear terms
According to.
Obtaining training data method based on vibration frequency, steps are as follows.
Using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range (q1,
q2) and predict dynamic pressure boundary q1,q2Corresponding unsteady aerodynamic force response；
Dynamic pressure boundary q is obtained using Fast Fourier Transform (FFT)1,q2The vibration frequency of corresponding unsteady aerodynamic force response, is obtained
Obtain vibration frequency range (f1,f2)；
A finite impulse response (FIR) narrowband low-pass filter is designed based on the frequency range；
Generate a white noise signal and it be filtered by designed filter, adjustment signal amplitude extremely with
Aero-elastic response amplitude is close, obtains pumping signal；
It is input in Fluid Mechanics Computation solver using pumping signal, obtains corresponding output response；
Pumping signal and corresponding output response are as training data.
Obtaining training data method based on incoming flow dynamic pressure, steps are as follows.
Using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range (q1,
q2)；
Optional one in (q1,q2) incoming flow dynamic pressure value q in rangec, phase is calculated using the method that CFD/CSD coupling calculates
The generalized displacement response answered and the pneumatic force-responsive of broad sense；
Generalized displacement response and the pneumatic force-responsive of broad sense are as training data.
The beneficial effects of the present invention are: this method is based on Volterra series theory and system minimal characteristic realizes that algorithm obtains
The linear segment for obtaining non-linear reduced-order model minimizes the non-linear that process obtains non-linear reduced-order model by output error
Point, the linear reduced-order model compared to background technique based on Volterra series can be predicted to obtain more accurate aeroelasticity
It responds and without recognizing Second-Order Volterra kernel function, not only having reduced calculating cost but also improving computational accuracy.In addition to this, should
Model also maintains succinct form, it may be convenient to be applied in engineering practice.The knot that the present invention is calculated with CFD/CSD coupling
Fruit is reference, and the response results of Volterra series reduced-order model prediction linear compared to background technique, present invention only requires one
Secondary additional Fluid Mechanics Computation calculates, so that it may significantly improve precision of prediction, test shows that maximum relative response error reduces
To 5% hereinafter, and the reduced-order model expression-form it is simple, be highly suitable for practical implementation.
It elaborates with reference to the accompanying drawings and detailed description to the present invention.
Detailed description of the invention
Fig. 1 is the flow chart of the nonlinear unsteady aerodynamics order reducing method the present invention is based on Fluid Mechanics Computation.
Fig. 2 is the approximate single order Volterra kernel function figure that the embodiment of the present invention recognizes.
Fig. 3 is the implementation flow chart that the embodiment of the present invention obtains training data method based on vibration frequency.
Fig. 4 is the training data curve graph that the embodiment of the present invention obtains that training data method is obtained based on vibration frequency.
Fig. 5 is the implementation flow chart that the embodiment of the present invention obtains training data method based on incoming flow dynamic pressure.
The time domain aeroelasticity that Fig. 6, which is dynamic pressure of the embodiment of the present invention, to be emulated using three kinds of numerical methods when being 7000 is rung
Answer comparative result figure.
The time domain aeroelasticity that Fig. 7, which is dynamic pressure of the embodiment of the present invention, to be emulated using three kinds of numerical methods when being 7650 is rung
Answer comparative result figure.
Specific embodiment
Referring to Fig.1-7.Analog simulation is carried out to international standard aeroelasticity example AGARD 445.6, the present invention is carried out
It is described in detail.445.6 wing of AGARD is in NASA Langley Research Center transonic wind tunnel for studying the elasticity of buffet characteristic
Model has become the modular, pneumatically powered elastic example of research nonlinear unsteady aerodynamics.The wing is 45 ° of swept-back wings, aerofoil profile
For NACA64A004, taper ratio 0.6576 shows than being 1.65, and the calculating state used is Mach number 0.499.Base of the present invention
In the nonlinear unsteady aerodynamics order reducing method of Fluid Mechanics Computation, expression-form is as follows:
Wherein Aa,Ba,Ca,DaFor the linear component part of reduced-order model, EaFor non-linear component part.Nonlinear equation φ
() is the hyperbolic sine function about input ξ (n).Specific step is as follows:
1, it is established by identification aeroelastic system approximation single order Volterra kernel function linear based on Volterra grades
Several reduced-order models.Euler equation and N-S equation have small nonlinearity characteristic, therefore unsteady aerodynamic force energy under microvariations
It is enough to carry out Precise Representation in the form of Second-Order Volterra series:
Select step response to carry out energized gas bullet system, established by recognizing approximate single order kernel function it is linear based on
The reduced-order model of Volterra series.Approximate single order Volterra kernel functionIt is defined as follows:
In formula, s (n) is step response, and n is discrete time step, ξ0For the amplitude of step response.
It realizes that the reduced-order model based on Volterra is converted into state space form by algorithm using system minimal characteristic, obtains
To the linear component part of non-linear reduced-order model.The linearly invariant discrete state that system minimal characteristic realizes that algorithm obtains is empty
Between form it is as follows:
Aa,Ba,Ca,DaSytem matrix, input matrix, output matrix and the feedforward matrix of pneumatic power system are corresponded respectively to,
xaFor the state variable of system, ξ is the input of system, FaIt is exported for the unsteady aerodynamic force of system, q is dynamic pressure.The pulse of system
Response output are as follows:
2, it after establishing the obtained reduced-order model based on Volterra series, since it is series expression-form, needs
Realize that algorithm is translated into state space form by system minimal characteristic.NoteFor the matrix of M × N, parameter M's and N
Value is 4.It is as follows to construct Hankel matrix
The value that wherein value of α is 800, β is 60.K=1 is enabled, to Hαβ(0) make singular value decomposition
Hαβ(0)=U Σ VT (7)
Define matrix Aa,Ba,Ca,DaExpression formula it is as follows
WhereinWithIt is defined as follows
Matrix A is calculateda,Ba,Ca,Da, the line of nonlinear unsteady aerodynamics reduced-order model established as the present invention
Property component part.
3, nonlinear terms E is next determinedaParameter.Since the reduced-order model that the present invention establishes is nonlinear model, no
Can use classical linear Identification method, thus we using Gauss-Newton method come discernibility matrixes EaParameter value.By EaIn
Value sorts top to bottom to obtain a unknown vector g=vec (Ea), following quadrature discrete equation is solved until error delta g=gnew-gold
It is sufficiently small:
U in formulafExpression formula it is as follows
By equation (10), output error e (k)=F of each time step is obtaineda,CFD(k)-Fa,g(k), Jacobi
The expression formula of matrix is
By solving following least square problem come renewal vector g in each iteration step
JTJ Δ g=-JTea (13)
Wherein eaFor error matrix, indicate in each time step Fluid Mechanics Computation calculated result and non-linear state space
Error between reduced-order model prediction result.
ea=[e (1) e (2) ... e (N)] (14)
After obtaining vector g, sort top to bottom back to matrix EaIn, obtain nonlinear unsteady aerodynamics reduced-order model
Nonlinear terms.
4, it after the linear segment and non-linear partial method for solving for determining nonlinear unsteady aerodynamics reduced-order model, needs
Training data necessary to establishing reduced-order model is calculated using Fluid Mechanics Computation solver.First using the rank selected
Jump excitation motivates 4 rank mode before model, obtains the corresponding pneumatic force-responsive of unsteady broad sense.In order to obtain identification depression of order
The method for providing two kinds of acquisition training datas: training data needed for model nonlinear item obtains training number based on vibration frequency
Training data method is obtained according to method and based on incoming flow dynamic pressure.
The first is to obtain training data based on vibration frequency, and principle is, for aeroelastic system, aeroelasticity
The vibration frequency of response is a very important parameter, can determine training data according to it.The vibration frequency of gas bullet system
Rate changes with the variation of dynamic pressure, by by matrix Aa,Ba,Ca,DaThe linear depression of order based on Volterra series characterized
Model, available one we compare the dynamic pressure range of care, by motivating the corresponding frequency range of dynamic pressure range can get
Required training data, Fig. 3 are the implementation flow chart of the method.
1) using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range
(q1,q2) and predict dynamic pressure boundary q1,q2Corresponding unsteady aerodynamic force response；
2) dynamic pressure boundary q is obtained using Fast Fourier Transform (FFT)1,q2The vibration frequency of corresponding unsteady aerodynamic force response,
Obtain vibration frequency range (f1,f2)；
3) a finite impulse response (FIR) narrowband low-pass filter is designed based on the frequency range；
4) it generates a white noise signal and it is filtered by designed filter, adjustment signal amplitude is extremely
It is close with aero-elastic response amplitude, obtain pumping signal；
5) it is input in Fluid Mechanics Computation solver using pumping signal, obtains corresponding output response；
Pumping signal and corresponding output response can be used as training data, and Fig. 4 is the training that the present invention is obtained using the method
Data.
Second is to obtain training data based on incoming flow dynamic pressure, and principle is, in Practical Project problem, usually more closes
Incoming flow dynamic pressure value near heart flutter point.On the other hand, it is contemplated that directly solve aeroelasticity using the method for CFD/CSD coupling
All possible affecting parameters have been contained when response.Therefore a kind of side that training data is obtained based on incoming flow dynamic pressure is provided
Method, Fig. 5 are the implementation flow chart of the method.
1) using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range
(q1,q2)；
2) optional one in (q1,q2) incoming flow dynamic pressure value q in rangec, calculated using the method that CFD/CSD coupling calculates
Corresponding generalized displacement response and the pneumatic force-responsive of broad sense；
The pneumatic force-responsive of generalized displacement and broad sense obtained using the method can be used as training data.The present invention is directed to the example
Identified dynamic pressure range (q1,q2)=(7000,7650), optional incoming flow dynamic pressure value qc=7300.
Next the pneumatic bomb predicted using the nonlinear unsteady aerodynamics reduced-order model that the present invention establishes is provided
Property response results and between traditional full rank model result of linear reduced-order model and CFD/CSD based on Volterra series
Comparison.Fig. 6 is incoming flow dynamic pressure calculated result comparison diagram when being 7000, it is found that when dynamic pressure is 7000, aeroelasticity is rung
Convergence state should be presented, the difference between result that three kinds of numerical methods obtain at this time is unobvious, but passes through observation partial approach
Figure is still it can be found that the result predicted of the nonlinear unsteady aerodynamics reduced-order model that the present invention obtains is closer to CFD/
The result that the full rank model of CSD calculates.Fig. 7 is incoming flow dynamic pressure calculated result comparison diagram when being 7650, model is in flutter shape at this time
State.The knot that the result and the full rank model of CFD/CSD that linear reduced-order model based on Volterra series is calculated are calculated
There are significant differences between fruit, on the contrary, the result and the full rank mould of CFD/CSD of the prediction of nonlinear unsteady aerodynamics reduced-order model
The calculated result difference of type is small, can further demonstrate that the nonlinear unsteady that the present invention establishes by partial enlarged view observation
Aerodynamic reduced order model has very high computational accuracy.
Table 1 is the maximum relative error between time domain aeroelasticity peak value of response, when dynamic pressure is 7000, ClinearThan
CnonlinearValue it is slightly larger, when incoming flow dynamic pressure increases to 7650, ClinearValue be about 20% but CnonlinearValue still less than
0.8%.
Table 1
As it can be seen from table 1 the unsteady aerodynamic force reduced-order model that the method for the present invention is established can significantly improve prediction essence
Degree.In addition to this, using based on vibration frequency obtain training data method with based on dynamic pressure acquisition training data method show it is non-
Chang Hao shows that the method for provide two kinds of acquisition training datas is feasible and effective.
Claims (1)
1. a kind of nonlinear unsteady aerodynamics order reducing method based on Fluid Mechanics Computation, it is characterised in that including following step
It is rapid:
Its expression-form is as follows:
Wherein, Aa,Ba,Ca,DaFor the linear component part of reduced-order model, EaFor non-linear component part；Nonlinear equation φ
() is the hyperbolic sine function about input ξ (n)；Its main feature is that the following steps are included:
Step 1: being established by identification aeroelastic system approximation single order Volterra kernel function linear based on Volterra grades
Several reduced-order models；Euler equation and N-S equation have a small nonlinearity characteristic under microvariations, thus unsteady aerodynamic force with
The form Precise Representation of Second-Order Volterra series:
Step response energized gas bullet system is selected, it is linear based on Volterra grades to establish by recognizing approximate single order kernel function
Several reduced-order models；Approximate single order Volterra kernel functionIt is defined as follows:
In formula, s (n) is step response, and n is discrete time step, ξ0For the amplitude of step response；
It realizes that the reduced-order model based on Volterra is converted into state space form by algorithm using system minimal characteristic, obtains non-
The linear component part of linear reduced-order model；System minimal characteristic realizes the linearly invariant separate manufacturing firms shape that algorithm obtains
Formula is as follows:
Aa,Ba,Ca,DaCorrespond respectively to sytem matrix, input matrix, output matrix and the feedforward matrix of pneumatic power system, xaFor
The state variable of system, ξ are the input of system, FaIt is exported for the unsteady aerodynamic force of system, q is dynamic pressure；The pulse of system is rung
It should export are as follows:
Step 2: realizing algorithm by system minimal characteristic after establishing the obtained reduced-order model based on Volterra series
State space form is converted by series expression-form；NoteValue for the matrix of M × N, parameter M and N is 4；Construction
Hankel matrix is as follows
Wherein, the value that the value of α is 800, β is 60；K=1 is enabled, to Hαβ(0) make singular value decomposition
Hαβ(0)=U Σ VT (7)
Define matrix Aa,Ba,Ca,DaExpression formula it is as follows
Wherein,WithIt is defined as follows
Matrix A is calculateda,Ba,Ca,Da, the linear component part of the nonlinear unsteady aerodynamics reduced-order model as foundation；
Step 3: determining nonlinear terms EaParameter；Using Gauss-Newton method discernibility matrixes EaParameter value；By EaIn value press
Column sequence obtains a unknown vector g=vec (Ea), following quadrature discrete equation is solved until error delta g=gnew-goldEnough
It is small:
In formula, UfExpression formula it is as follows
By equation (10), output error e (k)=F of each time step is obtaineda,CFD(k)-Fa,g(k), Jacobin matrix
Expression formula be
By solving following least square problem renewal vector g in each iteration step
JTJ Δ g=-JTea (13)
Wherein, eaFor error matrix, indicate in each time step Fluid Mechanics Computation calculated result and non-linear state space depression of order
Error between model prediction result；
ea=[e (1) e (2) ... e (N)] (14)
After obtaining vector g, sort top to bottom back to matrix EaIn, obtain the non-thread of nonlinear unsteady aerodynamics reduced-order model
Property item；
Step 4: after the linear segment and non-linear partial method for solving for determining nonlinear unsteady aerodynamics reduced-order model,
Training data necessary to establishing reduced-order model is calculated using Fluid Mechanics Computation solver；Use the step excitation of selection
4 rank mode before model are motivated, the corresponding pneumatic force-responsive of unsteady broad sense is obtained；It is instructed using based on vibration frequency
Practice data method and training data method is obtained based on incoming flow dynamic pressure, training data needed for obtaining identification reduced-order model nonlinear terms；
Obtaining training data method based on vibration frequency, steps are as follows；
Using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range (q1,q2)
And predict dynamic pressure boundary q1,q2Corresponding unsteady aerodynamic force response；
Dynamic pressure boundary q is obtained using Fast Fourier Transform (FFT)1,q2The vibration frequency of corresponding unsteady aerodynamic force response, is shaken
Dynamic frequency range (f1,f2)；
A finite impulse response (FIR) narrowband low-pass filter is designed based on the frequency range；
Generate a white noise signal and it be filtered by designed filter, adjustment signal amplitude extremely with gas bullet
Response amplitude is close, obtains pumping signal；
It is input in Fluid Mechanics Computation solver using pumping signal, obtains corresponding output response；
Pumping signal and corresponding output response are as training data；
Obtaining training data method based on incoming flow dynamic pressure, steps are as follows；
Using by Aa,Ba,Ca,DaThe linear reduced-order model of characterization determines an engineering actually interested dynamic pressure range (q1,q2)；
Optional one in (q1,q2) incoming flow dynamic pressure value q in rangec, calculated using the method that CFD/CSD coupling calculates corresponding
Generalized displacement response and the pneumatic force-responsive of broad sense；
Generalized displacement response and the pneumatic force-responsive of broad sense are as training data.
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| CN114611416A (en) * | 2022-05-12 | 2022-06-10 | 中国空气动力研究与发展中心计算空气动力研究所 | LS-SVM modeling method for nonlinear unsteady aerodynamic characteristics of missile |
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