CN105975645A - Quick calculation method of aircraft flow field containing a shock-wave area on the basis of multiple steps - Google Patents

Abstract

The invention puts forward a quick calculation method of an aircraft flow field containing a shock-wave area on the basis of multiple steps. The method comprises the following steps: firstly, estimating the flow field through a POD-RBF (Proper Orthogonal Decomposition-Radial Basis Function) method; secondly, further improving the precision of the flow field through a POD-ROM (Reduced Order model) method, and correcting a small area which contains shock wave; and finally, obtaining an airfoil flow field. The multistep solving method formed by the above steps realizes the calculation of a transonic flow field with the combination of an established POD model. The method overcomes the defect that computational fluid dynamics numerical simulation needs to consume a great quantity of time, can realize the high-fidelity calculation of the flow field which contains the shock wave, and solves the problem of low calculation efficiency since the number of the flow fields to be solved is enormous in engineering application.

Description

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method

Technical field

The present invention relates to a kind of aircraft flow field fast solution method, particularly relate to a kind of flow field fast solution method containing shock wave region, belong to applied aerodynamics research field.

Background technology

In aerodynamic scope, the most important thing of quick and precisely solving of aircraft flow field, always research work, especially contain flow field calculation the paying special attention to greater need for research worker in shock wave region.And the velocity interval of increasing Flight Vehicle Design be in transonic speed with supersonic speed region, the most hypersonic region, this most inevitably produces shock wave, and flowing becomes extremely complex.

Although high-fidelity Flow Field Calculation based on high accuracy physical model, namely Fluid Mechanics Computation (Computational Fluid Dynamics, CFD) numerical simulation, accurately solving of aircraft complex flowfield can be realized, but need to expend substantial amounts of calculating resource, especially when the required flow field huge number solved (the optimization design etc. of aircraft aerodynamic characteristic, aircraft Preliminary design or profile under different operating modes), the time expended is needed to tend not to be accepted in practical engineering application.

In order to improve the efficiency of aircraft Flow Field Calculation, (Proper Orthogonal Decomposition is decomposed in conjunction with Proper Orthogonal, POD) agent model (response surface model, Kriging model and RBF model etc.) method and reduced-order model (Reduced Order Model, the ROM) method of governing equation are arisen at the historic moment and obtain certain development.Two kinds of methods of relative maturity, one is the POD-RBF method being combined with POD by RBF (Radial Basis Function, RBF) model；A kind of is the model order reduction (POD-ROM) being combined with POD by governing equation.

POD-RBF method is applied to the specific practice of aircraft flow field prediction: calculated the flow field obtained under aircraft difference operating mode (including the most multiple variable of height, the angle of attack, Mach number, angle of rudder reflection, geometric shape) by experiment or CFD as sampling snapshot set, snapshot set Proper Orthogonal is decomposed and obtains POD base, each sampling snapshot is projected on POD base (POD base can block), obtain a series of POD base system numbers of correspondence, more just can be set up with operating mode/POD base system number POD-RBF model as input/output by RBF agent model.For any work condition state in duty parameter space, can be calculated by the POD-RBF model set up and solve.Contrast POD-RBF method, POD-ROM method by the flow field variable drop in governing equation on the subspace opened by minority POD base, the most complicated nonlinear partial differential equation can be converted into one group of nonlinear ordinary differential equation solving base system number, the dimension of governing equation known variables is substantially reduced, gained numerical result is the optimal solution found in lower dimensional space, also more conforms to actual physics problem.Although taking more calculating resource than POD-RBF method, but compared with solving original governing equation, the calculating time is greatly reduced.

It should be noted that, although POD is used to approximate non-linear problem, but it must be appreciated that POD itself is a linear process, for smooth flow field regions, POD-RBF method can obtain the flow field calculation result of higher fidelity, the flow field calculation result of POD-ROM method is more accurate, but two kinds of methods flow field calculation accuracy in shock wave region is all substantially reduced when flow field regions exists shock problems, and the situation being difficult to restrain even occurs in POD-ROM method.

Summary of the invention

It is an object of the invention to provide a kind of aircraft flow field fast solution method, the method overcome the drawback that Computational fluid mechanics numerical simulation needs to take considerable time, the high-fidelity being capable of comprising shock wave flow field calculates, and solves flow field to be asked huge number in engineer applied and the low problem of computational efficiency.

The technical scheme is that

It is described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: comprise the following steps:

Step 1: determine operating condition design space:

The flight parameter of the aircraft of analytical calculation as required, it is thus achieved that n, aerocraft flying parameter design space sample point x_i, i=1,2 ..., n；

Step 2: in obtaining step 1, the flow field sampling of each sample point solves w_i, i=1,2 ... n, and solved structure sampling snapshot matrix by the flow field sampling of all n sample points

Step 3: the sampling snapshot matrix A obtaining step 2 carries out POD decomposition:

Step 3.1: calculate the spatial correlation matrix R, R=A on n × n rank^HA；

Step 3.2: eigenvalue matrix Λ of solution room correlation matrix R and eigenvectors matrix V；

Step 3.3: calculate POD base vector matrix Φ=AV Λ^-1/2；Wherein base vector matrixφ_iFor POD base vector；

Step 3.4: calculate the base system matrix number X of sampling snapshot matrix A_r, X_r=Φ^HA；Wherein base system matrix numberx_riW is solved for the sampling of corresponding flow field_iPOD base system number；

Step 4: the RBF agent model setting up flow field sampling solution POD base system number is:

Wherein x_rAerocraft flying parameter design point x for needing analytical calculation is corresponding, RBF agent model the POD base system number exported；Radial base interpolation coefficient matrix

Radial distance r_kj=r (| | x_k-x_j||,θ_j), θ_jFor width parameter, r () is RBF；

And according to formula w_rbf=Φ x_rCalculate and under the design point x obtained by RBF agent model, predict Flow Field Solution w_rbf；

Step 5: extract the eigenvalue λ of spatial correlation matrix R_i, i=1,2 ... in n, value is maximum to n-th_wBig eigenvalue, and this n_wIndividual eigenvalue sum is more than precision threshold；Obtain this n_wThe POD base vector φ that individual eigenvalue is corresponding_j', j=1,2 ... n_w；

Step 6: the n obtained according to step 5_wThe POD base vector φ that individual eigenvalue is corresponding_j', the flow field w to be asked obtained under design point x projects to POD base vector φ_j' upper expression formulaWhereinPOD base system number x to be asked for corresponding design point x_r' jth element value；

Step 7: will flow field be asked w expression formulaSubstituting into flow field control equation, stream field governing equation carries out discrete, obtains POD base system number x to be asked_r' for the residual error governing equation R (Φ ' of variable_wx_r′；X)=0, wherein ${\mathrm{\Φ}}_w^{\′}=\left\{{\mathrm{\φ}}_j^{\′}\right\}{|}_{j=1}^{n_w};$

Step 8: use gauss-newton method to residual error governing equation R (Φ '_wx_r′；X)=0 solves:

Step 8.1: residual error governing equation is projected in test space L opened by one group of base Ψ；Obtain containing n_wStatic determinacy equation group Ψ of individual unknown quantity^TR(Φ′_wx_r′；X)=0, wherein Ψ=J Φ '_w, J is Jacobian matrix；

Step 8.2: to formula

Ψ^TJ^kΦ′_wp^k=-Ψ^TR^k

It is iterated solving, obtains POD base system number x_r′；Wherein k=1 ..., K is Newton iteration step, and K is determined by convergence, p^kWithRespectively POD base system number increment during kth step iteration and Jacobian matrix, α^kIt is at p^kStep-length in the direction of search,For initial value, by formulaObtain；

Step 8.3: according to POD base system number x_r', obtain the flow field w under design point x_rom=Φ '_wx_r′；

Step 9: the CFD of local flow field revises:

In step 8, residual error R of last iteration step is as foundation, selects residual error to carry out CFD correction more than the mesh point setting threshold value: with w_romAs first field, calling CFD fluid diagnosis and solve residual error more than the mesh point setting threshold value, other mesh point flow field values holdings are constant and conduct solves boundary condition；Stop solving when residual error R drops to set below threshold value；

Step 10: the flow field combination under the design point x obtain flow field and the step 8 of step 9 local correction, obtains the final Flow Field Solution of design point x.

Further preferred version, described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: step 1 use Latin hypercube experimental design method obtain aerocraft flying parameter design space sample point x_i, i=1,2 ..., n.

Further preferred version, described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: step 2 use the flow field sampling of each sample point in CFD fluid diagnosis obtaining step 1 solve w_i, i=1,2 ... n.

Further preferred version, described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: width parameter in step 4d_maxFor maximum Euclidean distance between sample point in aerocraft flying parameter design space in step 1.

Further preferred version, described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: RBF r () employing gauss of distribution function in step 4:D is Euclidean distance.

Further preferred version, described a kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterized in that: in step 9, in step 8, residual error R of last iteration step is as foundation, Domain Decomposition Method, selection residual error is used to be more than the region of the mesh point composition setting threshold value as the region needing correction；The grid needing the region revised individually is taken out, with w from former grid_romAs first field and boundary condition, call CFD fluid diagnosis and the grid needing the region revised individually is solved, stop solving when residual error R drops to set below threshold value.

Beneficial effect

Beneficial effects of the present invention embodies as follows:

1, present method solves the most accurately calculating in extensive operating mode aircraft flow field in engineer applied.

2, in this method, POD-RBF method provides just field can accelerate the convergence rate of reduced-order model POD-ROM for POD-ROM method, and two kinds of methods combine (Two-step) and can realize the most accurately calculating of low speed flow field (without shock wave region).

3, the application of Region Decomposition technology, can combine the CFD approach of the Two-step method and shock wave region that are applied to smooth domain, finally realizes the most accurately calculating of transonic speed or supersonic flow field；By the mesh point that whole flow field residual values is big is marked, reapplies CFD and solve the effect that can also reach similar.

The additional aspect of the present invention and advantage will part be given in the following description, and part will become apparent from the description below, or is recognized by the practice of the present invention.

Accompanying drawing explanation

Above-mentioned and/or the additional aspect of the present invention and advantage will be apparent from easy to understand, wherein from combining the accompanying drawings below description to embodiment:

Fig. 1 aircraft multistep quick calculation method schematic flow sheet

The c-type grid chart of Fig. 2 two dimension example NACA0012 aerofoil profile

Fig. 3 sampled point operating mode distribution schematic diagram

The grid of the CFD modification region that Fig. 4 Region Decomposition obtains and the position view in original mesh

The pressure distribution comparison diagram of Fig. 5 distinct methods

Fig. 6 mach line contrast schematic diagram (wide dotted line be CFD result, solid line be multistep fast solution method, dotted line be Two-step method)

In figure, symbol description is as follows:

X/c is aerofoil profile dimensionless x coordinate；Y/c is the dimensionless y-coordinate of aerofoil profile；Variable Ma represents Mach number；Variables A OA represents the angle of attack；C_PFor airfoil surface pressure coefficient；POD-RBF is radial base interpolation method based on POD；POD-ROM is model order reduction based on POD；Two-step is the method that POD-RBF and POD-ROM combines；Multistep is aircraft flow field based on multistep quick calculation method.

Detailed description of the invention

Embodiments of the invention are described below in detail, and described embodiment is exemplary, it is intended to is used for explaining the present invention, and is not considered as limiting the invention.

Based on multistep in the present embodiment containing shock wave region aircraft flow field quick calculation method flow chart as it is shown in figure 1, with around NACA0012 aerofoil profile without viscous Transonic Flows as example, in conjunction with accompanying drawing, the present invention is described in further detail.

Step 1: determine operating condition design space:

The flight parameter of the aircraft of analytical calculation as required, it is thus achieved that n, aerocraft flying parameter design space sample point x_i, i=1,2 ..., n；Described flight parameter can select from flying height, Mach number, the angle of attack etc..

In the present embodiment, for given NACA0012 aerofoil profile, flight parameter elects Mach number Ma and angle of attack AOA as, uses Latin hypercube experimental design method uniform sampling to obtain aerocraft flying parameter design space sample point x_i=[AOA_i Ma_i]^T, angle of attack AOA is 2 °, 3 °, 4 ° and 5 ° respectively, and Mach number Ma is 0.74,0.76,0.78 and 0.80 respectively, then sampled point x_i, i=1,2 ..., the number n=16 of n, Fig. 3 is shown in sampling operating mode distribution.

Step 2: in obtaining step 1, the flow field sampling of each sample point solves w_i, i=1,2 ... n, and solved structure sampling snapshot matrix by the flow field sampling of all n sample pointsThe most various middle n is the number of sampled point, then matrix A has n to arrange, and the dimension N of every string is the flow field mesh point number and the product of each mesh point flow field variable that under sampled point operating mode, CFD calculates.

The flow field sampling using each sample point in CFD fluid diagnosis obtaining step 1 in the present embodiment solves w_i, i=1,2 ... n.Solving of flow field uses a c-type grid, has 193 × 57 mesh points, sees Fig. 2.Without viscous Transonic Flows and CFD fluid diagnosis, finite volume method based on the lattice heart is used for two dimension, then has 4 conservation variablees [ρ, ρ u, ρ v, ρ E].The dimension N that the sampling of each flow field solves is the flow field mesh point number and the product of each mesh point flow field variable that under sampled point operating mode, CFD calculates.It is deconstructed into sampling snapshot matrix by these discrete samplingsThen A is the matrix of N × n.

Step 3: the sampling snapshot matrix A obtaining step 2 carries out POD decomposition, i.e. sets up POD model in engineer applied, POD model comprises data below: a) all sampled points in operating condition design space；B) POD base vector and characteristic of correspondence value thereof；C) sampling solution in each flow field projects to base system number (POD base system number) corresponding on POD base.Set up solution procedure following (H is the conjugate transpose of matrix):

Step 3.1: calculate the spatial correlation matrix R, R=A on n × n rank^HA；

Step 3.2: eigenvalue matrix Λ of solution room correlation matrix R and eigenvectors matrix V, A^HAV=V Λ；

Step 3.3: calculate POD base vector matrix Φ=AV Λ^-1/2；Wherein base vector matrixφ_iFor POD base vector；POD base vector φ_iEigenvalue λ with R_iOne_to_one corresponding；

Step 3.4: calculate the base system matrix number X of sampling snapshot matrix A_r, X_r=Φ^HA；Wherein base system matrix numberx_riW is solved for the sampling of corresponding flow field_iPOD base system number.

Step 4: application POD-RBF method, it was predicted that just field w_rbf:

Set up flow field sampling according to POD model and solve the RBF agent model of POD base system number, i.e. with sampling operating mode/POD base system number as input/output, then for a certain aerocraft flying parameter design point x needing analytical calculation in operating condition design space, the available corresponding prediction operating mode POD base system number of agent model of utilization structure:

Wherein x_rAerocraft flying parameter design point x for needing analytical calculation is corresponding, RBF agent model the POD base system number exported；Radial base interpolation coefficient matrix

Radial distance r_kj=r (| | x_k-x_j||,θ_j), θ_jFor width parameter, width parameter takesd_maxFor maximum Euclidean distance between sample point in aerocraft flying parameter design space in step 1, r () is RBF, uses gauss of distribution function:D is Euclidean distance；Then according to formula w_rbf=Φ x_rCalculate and under the design point x obtained by RBF agent model, predict Flow Field Solution w_rbf。

Below step 5 to step 9 applies POD-ROM method, revises to obtain Flow Field Solution w further_rom:

Step 5: extract the eigenvalue λ of spatial correlation matrix R_i, i=1,2 ... in n, value is maximum to n-th_wBig eigenvalue, and this n_wIndividual eigenvalue sum is more than precision threshold；Obtain this n_wThe POD base vector φ that individual eigenvalue is corresponding_j', j=1,2 ... n_w。

It is 99.8% that the present embodiment takes POD base relevant information capacity, then when using n_wDuring=12 POD bases, eigenvalue sum just accounts for more than 99.8%.

Step 6: the n obtained according to step 5_wThe POD base vector φ that individual eigenvalue is corresponding_j', the flow field w to be asked obtained under design point x projects to POD base vector φ_j' upper expression formulaWhereinPOD base system number x to be asked for corresponding design point x_r' jth element value.

Step 7: will flow field be asked w expression formulaSubstituting into flow field control equation, stream field governing equation carries out discrete, obtains POD base system number x to be asked_r' for the residual error governing equation R (Φ ' of variable_wx_r′；X)=0, wherein ${\mathrm{\Φ}}_w^{\′}=\left\{{\mathrm{\φ}}_j^{\′}\right\}{|}_{j=1}^{n_w}.$

The present embodiment flow field control equation uses the ordinary differential equation of the discrete conservation form of compressible Navier-Stokes equations halfWill flow field be asked w expression formulaSubstitute into flow field control equation, obtainTime term is carried out discrete, it is considered to the approximate shceme form residual equation (in the case of permanent, not considering time-derivative item) in a certain moment, obtain POD base system number x to be asked_r' for the residual error governing equation R (Φ ' of variable_wx_r′；X)=0, the number of number unknown quantity the to be far longer than base system number of equation group, it is an over-determined systems, has N number of nonlinear equation, n_wIndividual unknown quantity (POD base system number), it is possible to use gauss-newton method solves.

Step 8: use gauss-newton method to residual error governing equation R (Φ '_wx_r′；X)=0 solves:

Step 8.1: residual error governing equation is projected in test space L opened by one group of base Ψ；Obtain containing n_wStatic determinacy equation group Ψ of individual unknown quantity^TR(Φ′_wx_r′；X)=0, wherein Ψ=J Φ '_w, J is Jacobian matrix；

Step 8.2: to formula

Ψ^TJ^kΦ′_wp^k=-Ψ^TR^k

$x_r^{\′k+1}=x_r^{\′k}+{\mathrm{\α}}^k{p}^k$

It is iterated solving, obtains POD base system number x_r′；Wherein k=1 ..., K is Newton iteration step, and K is determined by convergence, p^kWithRespectively POD base system number increment during kth step iteration and Jacobian matrix, α^kIt is at p^kStep-length in the direction of search,For initial value, by formulaObtain；

Step 8.3: according to POD base system number x_r', obtain the flow field w under design point x_rom=Φ '_wx_r′。

So far, obtain non-linear reduced-order model equation (POD-ROM) based on POD and solve mode, the prediction w obtained with POD-RBF_rbfAs first fieldPOD-ROM iterative steps can be made to reduce and relatively rapid convergence, and precision improves further.

Step 9: the CFD of local flow field revises:

In step 8, residual error R of last iteration step is as foundation, selects residual error to carry out CFD correction more than the mesh point setting threshold value: with w_romAs first field, calling CFD fluid diagnosis and solve residual error more than the mesh point setting threshold value, other mesh point flow field values holdings are constant and conduct solves boundary condition；Stop solving when residual error R drops to set below threshold value.

Region Decomposition (Domain Decomposition, DD) method can also be used, select the region that residual error is revised as needs more than the region of the mesh point composition of setting threshold value；The grid needing the region revised individually is taken out, with w from former grid_romAs first field and boundary condition, call CFD fluid diagnosis and the grid needing the region revised individually is solved, stop solving when residual error R drops to set below threshold value.As shown in Figure 4, the left side be modification region grid schematic diagram, the right is modification region position view in original mesh.For this region, first field is by POD-ROM method solving result w_romThering is provided, docking border, region is with corresponding w_romSolution as boundary condition, then application CFD solver solve so that residual error drops to suitable with the residual error of the whole flow field CFD solving result contrasted.

Step 10: the flow field combination under the design point x obtain flow field and the step 8 of step 9 local correction, obtains the final Flow Field Solution of design point x.

The present embodiment first passes through POD-RBF method and estimates flow field, improves flow field precision further secondly by POD-ROM method, then revises the zonule containing shock wave, finally give the flow field of aerofoil profile.The multistep method for solving being made up of above step, in conjunction with the POD model set up, it is achieved that the calculating of transonic flow field.

The surface pressure distribution of distinct methods contrasts as shown in Figure 5, as can be seen from the figure aircraft flow field quick calculation method (Multistep) based on multistep fits like a glove with CFD result, and other two kinds of methods (POD-RBF and Two-step method) are only capable of capturing shock exterior domain.Fig. 6 gives the mach line of distinct methods and compares, wide dotted line be CFD result, solid line be aircraft quick calculation method based on multistep, dotted line be POD-ROM and POD-RBF associated methods (Two-step method), this figure further illustrates aircraft flow field based on the multistep quick calculation method effectiveness to solving containing shock wave flow field.

Table 1 gives the calculated performance of distinct methods, n in table₁And n₂Represent that POD-RBF, POD-ROM process uses the number (n of POD base respectively₁=n, n₂=n_w), n₃Represent CFD modification region gridding dimension.As seen from table: a) when introducing POD-RBF method and providing initial value for POD-ROM method, Newton iteration step number drops to 2 from 7, computational efficiency improves 3.5 times, and POD-RBF method itself calculates time-consuming (only 0.19s) and is time-consumingly negligible compared to POD-ROM；B) compared to CFD method for solving, aircraft flow field based on multistep quick calculation method efficiency improves about 9 times.

The calculated performance contrast of table 1 distinct methods

Although above it has been shown and described that embodiments of the invention, it is understandable that, above-described embodiment is exemplary, being not considered as limiting the invention, above-described embodiment can be changed in the case of without departing from the principle of the present invention and objective, revises, replace and modification by those of ordinary skill in the art within the scope of the invention.

Claims (6)

1. one kind based on multistep containing aircraft flow field, shock wave region quick calculation method, it is characterised in that: include following step Rapid:

Step 1: determine operating condition design space:

The flight parameter of the aircraft of analytical calculation as required, it is thus achieved that n, aerocraft flying parameter design space sample Point x_i, i=1,2 ..., n；

Step 2: in obtaining step 1, the flow field sampling of each sample point solves w_i, i=1,2 ... n, and by all n samples The flow field sampling of point solves and builds sampling snapshot matrix

Step 3: the sampling snapshot matrix A obtaining step 2 carries out POD decomposition:

Step 3.1: calculate the spatial correlation matrix R, R=A on n × n rank^HA；

Step 3.2: eigenvalue matrix Λ of solution room correlation matrix R and eigenvectors matrix V；

Step 3.3: calculate POD base vector matrix Φ=AV Λ^-1/2；Wherein base vector matrixφ_iFor POD base vector；

Step 3.4: calculate the base system matrix number X of sampling snapshot matrix A_r, X_r=Φ^HA；Wherein base system matrix numberx_riW is solved for the sampling of corresponding flow field_iPOD base system number；

Step 4: the RBF agent model setting up flow field sampling solution POD base system number is:

$x_r^T=\left[\begin{array}{cccc}r\left(\right||x-x_1||,{\mathrm{\θ}}_1)& r\left(\right||x-x_2||,{\mathrm{\θ}}_2)& ...& r\left(\right||x-x_n||,{\mathrm{\θ}}_n)\end{array}\right]\[B\]$

Wherein x_rAerocraft flying parameter design point x for needing analytical calculation is corresponding, RBF agent model export POD base system number；Radial base interpolation coefficient matrix

$A_r=\left(\begin{array}{cccc}r(0,{\mathrm{\θ}}_1)& r_{12}& ...& r_{1n}\\ r_{21}& r_{22}& ...& r_{2n}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ r_{n1}& r_{n2}& ...& r(0,{\mathrm{\θ}}_n)\end{array}\right)$

Radial distance r_kj=r (| | x_k-x_j||,θ_j), θ_jFor width parameter, r () is RBF；

And according to formula w_rbf=Φ x_rCalculate and under the design point x obtained by RBF agent model, predict Flow Field Solution w_rbf；

Step 5: extract the eigenvalue λ of spatial correlation matrix R_i, i=1,2 ... in n, value is maximum to n-th_wBig feature Value, and this n_wIndividual eigenvalue sum is more than precision threshold；Obtain this n_wThe POD base vector that individual eigenvalue is corresponding φ′_j, j=1,2 ... n_w；

Step 6: the n obtained according to step 5_wThe POD base vector φ ' that individual eigenvalue is corresponding_j, obtain under design point x Flow field w to be asked projects to POD base vector φ '_jUpper expression formulaWhereinFor treating of corresponding design point x Seek POD base system number x '_rJth element value；

Step 7: will flow field be asked w expression formulaSubstituting into flow field control equation, stream field governing equation enters Row is discrete, obtains POD base system number x ' to be asked_rResidual error governing equation R (Φ ' for variable_wx′_r；X)=0, wherein ${\mathrm{\Φ}}_w^{\′}=\left\{{\mathrm{\φ}}_j^{\′}\right\}{|}_{j=1}^{n_w};$

Step 8: use gauss-newton method to residual error governing equation R (Φ '_wx′_r；X)=0 solves:

Step 8.1: residual error governing equation is projected in test space L opened by one group of base Ψ；Obtain containing n_w Static determinacy equation group Ψ of individual unknown quantity^TR(Φ′_wx′_r；X)=0, wherein Ψ=J Φ '_w, J is Jacobian matrix；

Step 8.2: to formula

Ψ^TJ^kΦ′_wp^k=-Ψ^TR^k

$x_r^{\′k+1}=x_r^{\′k}+{\mathrm{\α}}^k{p}^k$

It is iterated solving, obtains POD base system number x '_r；Wherein k=1 ..., K is Newton iteration step, and K is by restraining mark Quasi-decision, p^kWithRespectively POD base system number increment during kth step iteration and Jacobian matrix, α^kBe p^kStep-length in the direction of search,For initial value, by formulaObtain；

Step 8.3: according to POD base system number x '_r, obtain the flow field w under design point x_rom=Φ '_wx′_r；

Step 9: the CFD of local flow field revises:

In step 8, residual error R of last iteration step is as foundation, selects residual error to click on more than the grid setting threshold value Row CFD revises: with w_romAs first field, call CFD fluid diagnosis to residual error more than the mesh point setting threshold value Solving, other mesh point flow field values holdings are constant and conduct solves boundary condition；When residual error R drops to set threshold value Stop time following solving；

Step 10: the flow field combination under the design point x obtain flow field and the step 8 of step 9 local correction, obtains The Flow Field Solution that design point x is final.

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is special Levy and be: step 1 uses Latin hypercube experimental design method obtain aerocraft flying parameter design space sample This some x_i, i=1,2 ..., n.

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is special Levy and be: step 2 uses the flow field sampling of each sample point in CFD fluid diagnosis obtaining step 1 solve w_i, i=1,2 ... n.

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is special Levy and be: width parameter in step 4d_maxSet for aerocraft flying parameter in step 1 Count the maximum Euclidean distance between sample point in space.

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is special Levy and be: RBF r () employing gauss of distribution function in step 4:D be Euclidean away from From.

A kind of based on multistep containing aircraft flow field, shock wave region quick calculation method, it is special Levying and be: in step 9, in step 8, residual error R of last iteration step is as foundation, uses Region Decomposition Method, selects the region that residual error is revised as needs more than the region of the mesh point composition of setting threshold value；To need The grid in the region revised individually takes out, with w from former grid_romAs first field and boundary condition, call CFD The grid needing the region revised individually is solved by fluid diagnosis, stops when residual error R drops to and sets below threshold value Only solve.