CN103902782A - POD (proper orthogonal decomposition) and surrogate model based order reduction method for hypersonic aerodynamic thermal models - Google Patents

Abstract

The invention relates to a POD (proper orthogonal decomposition) and surrogate model based order reduction method for hypersonic aerodynamic thermal models and belongs to the technical field of aerospace. A hypersonic aircraft aerodynamic thermal environment is predicted and obtained by the aid of a POD and surrogate model method, nonlinear characteristics of real gas effect, turbulence viscosity and the like of high-precision numerical calculation are reserved, the hypersonic aircraft aerodynamic thermal environment can be provided for design of the hypersonic aerocraft by the aid of model order reduction method which is high in precision and efficiency, related thermal boundary conditions are provided for aerodynamic thermal elastic design, the thermal environment is provided for thermal protection design of the hypersonic aerocraft, design efficiency can be greatly improved, design circle is shortened, and design cost is reduced.

Description

Based on the hypersonic Aerodynamic Heating model order reducing method of POD and agent model

Technical field

The present invention relates to a kind of hypersonic Aerodynamic Heating model order reducing method that decomposes (POD) and agent model based on Proper Orthogonal, belong to field of aerospace technology.

Background technology

Aeronautical and space technology is the outstanding feature of mankind's modern civilization, is the concentrated reflection of a national science and technology level and overall national strength.Hypersonic technology is the important component part of aerospace field, is current and following aeronautical and space technology important development direction.But front, countries in the world all get down to research hypersonic technology, have formulated hypersonic technology development plan, and in succession will develop hypersonic aircraft and realize as national objective.

Hypersonic aircraft generally refers to that flight Mach number is greater than 5, can and realize the aircraft of hypersonic flight across atmospheric envelope at atmospheric envelope.Can be divided into rocket-powered hypersonic aircraft and air suction type hypersonic aircraft according to the difference of propulsion system.Hypersonic aircraft have speed fast, be swift in response, the feature such as maneuverability, penetration ability are strong, can adapt to the demand of following high-tech war and military-civil fast transportation, have important strategic importance and high using value.Hypersonic aircraft configuration adopts slender bodies, lifting body or Waverider more.Its fuselage and control rudder face because heavy quantitative limitation often has larger structure flexibility.In addition, these hypersonic aircrafts often have larger flight Mach number envelope curve, conventionally can meet the flight in Mach number 0 to 15.In order to meet the requirement of airbreathing propulsion system, this class aircraft also needs in atmospheric envelope, to carry out the hypersonic flight of a period of time.Under the acting in conjunction of Aerodynamic Heating and aerodynamic loading, between incoming flow, aerodynamic force, structure, control and propulsion system, produce complicated interaction, very complicated hypersonic aircraft is pneumatic-and Re-structure-propelling coupled problem produces thereupon.In the research in the past of these coupled problems, do not cause enough concerns, simultaneously owing to cannot utilizing wind-tunnel scale model to carry out the conventional wind tunnel test aspect aeroelasticity and aerothermoelasticity in hypersonic speed flow, thereby hypersonic aeroelasticity and aerothermoelasticity simulation analysis seem incomparably important.

Aerothermoelasticity analysis is one of gordian technique of hypersonic aircraft design.Since various countries start to develop hypersonic aircraft, hypersonic aerothermoelasticity analysis is subject to researchist's very big attention always.The aerothermal accurate fast prediction of hypersonic aircraft is the important prerequisite that aerothermoelasticity is analyzed.The aerothermal Accurate Prediction of hypersonic aircraft is one of current important research topic.Mainly utilize at present two kinds of methods of Aerodynamic Heating engineering calculation and high precision numerical evaluation (CFD) to solve Aerodynamic Heating forecasting problem.Aerodynamic Heating engineering calculation based on simple geometry hypothesis has met the requirement of engineering primary design under certain condition, but real gas effect, air-flow viscosity etc. have inevitably been ignored in engineering calculation, and usable range is limited.High precision numerical evaluation (CFD) can take into full account air-flow viscosity, real gas effect etc., can solve preferably the nonlinear equation extensively existing in coupling analysis, but due to huge analysis degree of freedom and because double counting uncertain and that design optimization needs causes calculated amount huge, current value is calculated and is still not suitable for engineering application.

Therefore, pin current hypersonic aircraft Aerodynamic Heating engineering calculation and high precision numerical evaluation are not all well positioned to meet the problem of engineering demand, and how obtaining quickly and efficiently hypersonic aircraft Aerodynamic Heating environment is one of gordian technique of current hypersonic aerothermoelasticity analysis and hypersonic aircraft design.

The core concept that Proper Orthogonal is decomposed (POD) method is to utilize the result of calculation of full rank model to set up the orthogonal basis that can fully describe full-order system dynamics of one group of the best, and then by blocking POD base vector number, model is carried out to depression of order reduction, thereby realize the object of system order reduction.After POD basic mode state after being blocked, still need to obtain original system sample point and the one-to-one relationship of projection coefficient under POD base after blocking.Utilize above-mentioned one-to-one relationship can obtain test design point at the projection coefficient that blocks POD base, and then obtain blocking the predicted value under POD base.Agent model method has good approximation quality and efficiency, can obtain quickly and efficiently the corresponding approximation relation one by one of systematic sample point and projection coefficient under POD base.

Summary of the invention

The object of the invention is the problem in order to solve all can not the fine adaptation current engineering application of current hypersonic Aerodynamic Heating engineering calculation and high precision numerical evaluation, in conjunction with the basic thought of POD method and agent model, utilize POD method to obtain the POD basic mode state of reduced order system, the approximation relation of projection coefficient and sample point under agent model method processing reduced order system basic mode state, a kind of model order reducing method (POD-Surrogate) that decomposes (POD) and agent model technology (Surrogate) in conjunction with Proper Orthogonal has been proposed, and be successfully applied to the prediction of hypersonic Aerodynamic Heating.

Based on the aircraft Aerodynamic Heating model order reducing method of Proper Orthogonal decomposition and agent model, specifically comprise the steps:

Step 1, selects the physical model of hypersonic Aerodynamic Heating prediction and determines design variable and design space.Physical model is elected wing as, and design variable is elected flight Mach number, flying height and flying drilling angle as, and design space is the bound scope of flight Mach number, flying height and flying drilling angle.

Step 2, uses Latin hypercube experimental design method to obtain the sample point I in design space ⁽ⁱ⁾, i=1...n, n is total nodes; Sample point has spatially uniform and orthogonal space in design space.

Utilize numerical evaluation (CFD) to obtain the different flight Mach numbers in each sample point place, flying height and flying drilling angle lower aerofoil to calculate the temperature-responsive value U of i node ⁽ⁱ⁾, i=1...n, utilizes sample point response tectonic system eigenmatrix $S=\left\{U^{\left(i\right)}\right\}{|}_{i=1}^n,$

In above formula, in system features matrix S, n is sampled point number, and p is that the CFD under sampled point operating mode calculates aerofoil nodes, and T is aerofoil node temperature.

Step 3, the system features matrix S that step 2 is obtained is carried out svd (SVD), obtains the POD base vector ψ of system matrix S ₀, S ^tsV=V Λ, ψ ₀=SV; In formula, Λ is the diagonal matrix that the eigenwert ξ that obtains of SVD forms; V is by respectively classifying S as ^tthe matrix of the proper vector composition of S.

Step 4, POD method is by blocking POD base vector ψ ₀realize the depression of order to whole model, the blocking according to being energy Ratios principle of POD base vector, the energy after blocking with block before energy Ratios be approximately 1.The contribute energy of each POD base vector is weighed by its eigenwert, and the base vector contribute energy of the larger correspondence of eigenwert is larger.The eigenwert that step 3 is obtained sorts by size, then choose continuously m dimension (m<<n) eigenwert by order from big to small, and block from m+1 dimension, block the m dimension POD base ψ of rear reservation, the principal character that has comprised whole samples, the energy after blocking with block before energy Ratios be greater than 90%.Carry out the whole sample space characteristic of matching with the POD base ψ choosing.

Step 5, design sample point I ⁽ⁱ⁾, in i=1...n, each sample point all can be described under the POD base ψ after blocking, and the temperature-responsive value of i node meets approximation relation

utilize least square method to calculate respectively n sample point coefficient under POD base ψ after blocking each sample point has m coefficient after blocking under POD base.According to sample point and block the corresponding relation between the coefficient of POD base, utilize agent model technology to set up and obtain design sample point

with the coefficient blocking under POD base $\left\{a_j^{\left(i\right)}\right\}{|}_{i=1}^n,j=1,...,m$ Approximate matching relation.

Step 6, any given test sample book point I (p) in design space, utilizes the approximate matching relation of the agent model that step 5 constructed to obtain predicting accordingly POD base system number

blocking the predicated response value that obtains this sample point under POD base

Beneficial effect

Utilize the inventive method prediction to obtain hypersonic Aerodynamic Heating environment and there is higher precision and efficiency, effectively made up the deficiency of hypersonic Aerodynamic Heating engineering calculation and high precision numerical evaluation.The present invention utilizes Proper Orthogonal to decompose and the prediction of agent model method obtains hypersonic aircraft Aerodynamic Heating environment, retain the real gas effect in high precision numerical evaluation, the nonlinear characteristics such as air-flow viscosity, utilize model reduction thought to there is higher precision and efficiency simultaneously, can provide hypersonic aircraft thermal environment in hypersonic aircraft design, for aerothermoelasticity design provides associated hot boundary condition, for hypersonic aircraft heat protection design provides thermal environment, improve greatly design efficiency, shorten the design cycle, save design cost, have broad application prospects.

POD-Surrogate method of the present invention can utilize POD method to realize the depression of order of full-order system, also can utilize agent model under difference test operating mode, to obtain the predicted value of full-order system simultaneously.

Accompanying drawing explanation

Fig. 1 is the schematic flow sheet of POD-Surrogate order reducing method of the present invention;

Fig. 2 is the geometric model schematic diagram of F104 fighter plane wing in embodiment;

Fig. 3 is wing Fluid Computation grid chart in embodiment;

Fig. 4 is flying height 30km in embodiment, 0 ° of angle of attack, the wing aerofoil temperature CFL3D solver of Mach number 5.0 and Fastran solver contrast distribution, wherein (a), for CFL3D calculates aerofoil Temperature Distribution, is (b) Fastran calculating aerofoil Temperature Distribution;

Fig. 5 be 100 design sample point temperature system matrixes of embodiment carry out Proper Orthogonal decomposition POD eigenwert with POD basic mode state situation of change;

Fig. 6 is that in embodiment, POD-Kriging method and the POD-RBF method under 10 test operating modes predicted the aerofoil temperature-averaging relative error obtaining, wherein (a) is POD-Kriging aerofoil temperature-averaging relative error, is (b) POD-RBF aerofoil temperature-averaging relative error;

Fig. 7 is that in embodiment, typical case tests CFD calculating aerofoil temperature under operating mode V=2614.5m/s, H=25783.9m, AOA=4.0;

Fig. 8 is that in embodiment, typical case tests POD-Kriging method reduced-order model aerofoil temperature relative error and POD-RBF method reduced-order model aerofoil temperature relative error under operating mode V=2614.5m/s, H=25783.9m, AOA=4.0, wherein, (a) POD-Kriging aerofoil temperature-averaging relative error, (b) POD-RBF aerofoil temperature-averaging relative error;

Fig. 9 is the L of POD-Kriging and POD-RBF method under 10 test operating modes in embodiment _∞average error is with the variation diagram of POD basic mode state quantity;

Figure 10 be in embodiment under 10 test operating modes the NRSME average error of POD-Kriging and POD-RBF method with the variation diagram of POD basic mode state quantity.

Embodiment

In analyzing with aerothermoelasticity, (wing structure geometric model is shown in Fig. 2 to the wing of conventional typical F104 fighter plane, wing Fluid Computation grid model is shown in that Fig. 3 is example, utilize CFD-FASTRAN solver Wings aerofoil Temperature Distribution, utilize the aerothermal reduced-order model of POD-Surrogate method construct aerofoil of the present invention, further analyze efficiency and the precision of the method.

Step 1: the selection of physical model.Hypersonic aircraft adopts slender bodies, lifting body or Waverider more, the dynamic and static stability of paying close attention to wing or rudder face when aerothermoelasticity is analyzed more.The present invention is take the wing of typical F104 fighter plane conventional in aerothermoelasticity analysis as example, and interrelated geometrical parameters as shown in Figure 2.

Step 2: design variable and design space determine.As shown in Figure 1, POD-Surrogate method is being determined after physical model, and then definite design variable and design space, the reduced-order model building in design space.For typical hypersonic aircraft wing, design variable is defined as flight Mach number, flying height and flying drilling angle, specifically in table 1:

Table 1 design variable and design space

Step 3: test design point determine.Initially choose 100 sample points, adopt Latin hypercube experimental design method to produce, guarantee spatially uniform and the orthogonal space of design sample point in design space.

Step 4: design determining of response.For hypersonic aircraft, choose suitable solver and could simulate and obtain hypersonic Aerodynamic Heating distribution situation.Hypersonic Aerodynamic Heating of the present invention calculates chooses the distribution of CFD-FASTRAN solver calculating Aerodynamic Heating.

Most of CFD solvers are the method for solving based on pressure, this is mainly because nature that can not baric flow equation determines, and variable density is very important at a high speed can baric flow, CFD-FASTRAN is coupled together the compressible Euler equations based on density or N-S equation with multi-body movement, limited response rate chemistry and non-equilibrium heat transfer, can solve a series of and complicated Aero-Space problem.CFD-FASTRN adopts multiple dynamic mesh gridding technique, compressible Euler equations based on density or N-S equation, can take into full account the Aerodynamic Heating effect being caused by hypersonic speed flow, decomposition and the ionization etc. of real gas, huge advantage aspect processing aircraft thermoelasticity has larger adaptability in research hypersonic speed flow.

Step 5: the structure of temperature system matrix.Utilize the temperature-responsive value of a CFD-FASTRAN numerical evaluation acquisition sample point, tectonic system matrix S.Aerofoil Temperature Distribution and contrasting as shown in Figure 4 with CFL3D solver.

In above formula, in system features matrix S, n is sampled point number, and p is that the CFD under sampled point operating mode calculates aerofoil nodes; T is aerofoil node temperature.

Step 6: the proper vector ψ that carries out svd (SVD) and obtain system matrix S for system matrix S ₀, S ^tsV=V Λ, ψ ₀=SV, in formula, Λ is the diagonal matrix that eigenwert forms.As shown in Figure 5, due to 20 most features that base vector has comprised all samples above, choose sub-fraction base vector above according to eigenwert ξ and block rear 20 POD base ψ, be similar to all sample space characteristics of matching.

Step 7: under the POD base ψ after blocking, by approximation relation

utilize least square method to calculate respectively the coefficient that blocks rear POD base under each sample point

due to sample point and block the corresponding relation between the coefficient of POD base, utilize the agent model such as Kirging or radial basis function to set up design sample point

with the coefficient blocking under POD base

approximate matching relation;

Step 8: utilize Latin hypercube experimental design method to obtain test sample book point I in design space ^(p), utilize the agent model of having constructed can obtain predicting accordingly that POD base system counts a ^(p), blocking the predicated response value that obtains this sample point under POD base

Order reducing method Research on Accuracy

In order quantitatively to weigh the quality predicting the outcome, adopt root-mean-square error NRMSE and maximal value error L _∞, its formula is:

$\mathrm{NRMSE}[\%]=\frac{\sqrt{\frac{1}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{({\mathrm{ReducedModel}}_i-{\mathrm{Full}}_i)}^2}}{\max \left(\mathrm{Full}\right)-\min \left(\mathrm{Full}\right)}$

$L_{\&infin;}[\%]=\frac{\max \left(\right|\mathrm{ReducedModel}-\mathrm{Full}\left|\right)}{\max \left(\mathrm{Full}\right)-\min \left(\mathrm{Full}\right)}$

In formula, i represents to predict the temperature value of i node under operating mode.ReducedModel is illustrated in the predicted value of the reduced-order model that POD-Surrogate method obtains, and Full represents the temperature value that time CFD calculates.

Fig. 6 (a) has shown that POD-Kriging method obtains the aerofoil temperature-averaging relative error cloud atlas of 10 test sample book points, and result shows that aerofoil error everywhere is all less than 5%, and leading edge temperature error is less, and the place that error is larger appears at aerofoil middle part.Fig. 6 (b) has shown that POD-RBF method obtains the aerofoil temperature-averaging relative error cloud atlas of 10 test sample book points, and result shows that leading edge temperature error is less, reaches 5% left and right, and the place that error is larger appears at aerofoil postmedian.Average relative error maximum reaches 13%, and minimum average B configuration relative error is 5% left and right, leading edge place.

Fig. 7, Fig. 8 (a) and Fig. 8 (b) shown typical case and tested under operating mode V=2614.5m/s, H=25783.9m, AOA=4.0, and CFD calculates aerofoil temperature, POD-Kriging method reduced-order model aerofoil temperature relative error and POD-RBF method reduced-order model aerofoil temperature relative error.Contrast finds that POD-Kriging method and POD-RBF method reduced-order model obtain aerofoil temperature prediction value and all can meet preferably CFD calculating aerofoil temperature trend, and leading edge place predicated error is all less.POD-kriging method maximum relative error is only 1.6%, and the most of relative error of aerofoil temperature is all in 1%, and POD-RBF method maximum relative error reaches 18%, and the most of relative error of aerofoil temperature is in 10% left and right.

Fig. 9 shows the L of 10 test sample book points _∞average error is along with the situation of change of the mode number of POD reservation, and hence one can see that in the time that POD reservation mode number is greater than 20, and the POD basic mode state quantity that continues increase reservation can not improve the approximation quality of reduced-order model effectively.The aerofoil L that POD-Kriging method obtains in this example is found in contrast _∞average relative error can reach below 6% after reservation mode number reaches 20; The L that POD-RBF method obtains _∞average relative error reaches 20 rear minimum value at reservation mode number and maintains 14% left and right.With L _∞average error contrast, the prediction effect of POD-Kriging method is better than POD-RBF method.

Figure 10 has shown that POD-Surrogate method obtains the Averaged Square Error of Multivariate NRSME of 10 test sample book points with the situation of change of POD basic mode state quantity, result shows, after POD basic mode state quantity reaches 20, continue to increase POD basic mode state quantity and can not continue to reduce Averaged Square Error of Multivariate NRSME.Contrast knownly, the Averaged Square Error of Multivariate NRSME of POD-Kriging method can reach 4% left and right, and the Averaged Square Error of Multivariate NRSME of POD-RBF method reaches 12% left and right, and obviously, in the time processing Aerodynamic Heating prediction, POD-Kriging method is better than POD-RBF method.

Order reducing method efficient studies

Table 2 order reducing method and numerical evaluation contrast

Caculationoni5-2320,3.0GHz,4.00GBRAM.

As shown in table 2, the 19.1h of being about consuming time while utilizing high precision CFD-Fastran numerical evaluation while calculating single sample point (snapshot), and consuming time while adopting POD-Kriging method construct reduced-order model and prediction test point aerofoil temperature be only 0.68s, the only 0.04s consuming time of reduced-order model prediction test point aerofoil of POD-RBF method construct, is about respectively 1/101550 and 1/1726275 of CFD calculating.Point out that POD-RBF method has higher efficiency, in the time of same design sample point and prediction test point, POD-RBF is consuming time is 1/17 of POD-Kriging simultaneously.This is because Kriging model needs global optimization procedure, more consuming time than radial basis function (RBF) like this.

Consider computational accuracy and efficiency, the hypersonic Aerodynamic Heating reduced-order model of POD-Kriging method construct more has superiority compared with POD-RBF method.Calculate for hypersonic aircraft Aerodynamic Heating, the present invention proposes a kind of Proper Orthogonal and decomposes the model order reducing method (POD-Surrogate) combining with agent model.Example shows, this model order reducing method and reduced-order model framework can be successfully applied to hypersonic Aerodynamic Heating and calculate, and have higher precision and efficiency.

In the above, background of invention, technical scheme and beneficial effect are had been described in detail; institute is understood that; determining just for practical application of the present invention is described of the choosing of above physical model, design variable and design space; the usable range being not intended to limit the present invention; within the spirit and principles in the present invention all; any modification of making, be equal to replacement, improvement etc., within protection scope of the present invention all should be included in.

Claims (3)

1. the hypersonic Aerodynamic Heating model order reducing method based on POD and agent model, is characterized in that: specifically comprise the steps:

Step 1, selects the physical model of hypersonic Aerodynamic Heating prediction and determines design variable and design space; Physical model is elected wing as, and design variable is elected flight Mach number, flying height and flying drilling angle as, and design space is the bound scope of flight Mach number, flying height and flying drilling angle;

Step 2, uses Latin hypercube experimental design method to obtain the sample point I in design space ⁽ⁱ⁾, i=1...n, n is total nodes;

Utilize numerical evaluation to obtain the different flight Mach numbers in each sample point place, flying height and flying drilling angle lower aerofoil to calculate the temperature-responsive value U of i node ⁽ⁱ⁾, i=1...n, utilizes sample point response tectonic system eigenmatrix $S=\left\{U^{\left(i\right)}\right\}{|}_{i=1}^n:$

In above formula, in system features matrix S, n is sampled point number, and p is the numerical evaluation aerofoil nodes under sampled point operating mode, and T is aerofoil node temperature;

Step 3, the system features matrix S that step 2 is obtained is carried out svd, obtains the POD base vector ψ of system matrix S ₀, S ^tsV=V Λ, ψ ₀=SV; In formula, Λ is the diagonal matrix that the eigenwert ξ that obtains of svd forms; V is by respectively classifying S as ^tthe matrix of the proper vector composition of S;

Step 4, the blocking according to being energy Ratios principle of POD base vector; The eigenwert that step 3 is obtained sorts by size, then choose continuously m dimensional feature value by order from big to small, m<<n, and block from m+1 dimension, block the m dimension POD base ψ of rear reservation, the principal character that has comprised whole samples, the energy after blocking with block before energy Ratios be greater than 90%; Carry out the whole sample space characteristic of matching with the POD base ψ choosing;

Step 5, design sample point I ⁽ⁱ⁾, in i=1...n, each sample point all can be described under the POD base ψ after blocking, and the temperature-responsive value of i node meets approximation relation

utilize least square method to calculate respectively n sample point coefficient under POD base ψ after blocking

each sample point has m coefficient after blocking under POD base; According to sample point and block the corresponding relation between the coefficient of POD base, utilize agent model technology to set up and obtain design sample point

with the coefficient blocking under POD base $\left\{a_j^{\left(i\right)}\right\}{|}_{i=1}^n,j=1,...,m$ Approximate matching relation;

Step 6, any given test sample book point I in design space ^(p), utilize the approximate matching relation of the agent model that step 5 constructed to obtain predicting accordingly POD base system number

blocking the predicated response value that obtains this sample point under POD base

2. the hypersonic Aerodynamic Heating model order reducing method based on POD and agent model according to claim 1, is characterized in that: in step 2, choose sample point and in design space, have spatially uniform and orthogonal space.

3. the hypersonic Aerodynamic Heating model order reducing method based on POD and agent model according to claim 1, is characterized in that: the contribute energy of each POD base vector is weighed by its eigenwert, and the base vector contribute energy of the larger correspondence of eigenwert is larger.

Images