Abstract:
A computer-aided method suitable for assisting in the design of an aircraft by providing a variable distribution V over an aircraft surface inside a predefined parameter space by means of a reconstruction of the results obtained through computations for an initial group of points in the parameter space using a CFD model comprising in each of said computations the following steps: a) Selecting relevant shock wave structures being defined by its thickness δ sw , its position S sw and the variable jump ΔV; b) Decomposing said variable distribution V into a smooth variable distribution V smooth  and a shock wave variable distribution V jump  corresponding to said relevant shock wave structures; c) Obtaining the variable distribution V at any point of the parameter space as a reconstruction of the smooth variable distribution V smooth  and the shock wave variable distribution V jump  obtained for said point.

Description:
FIELD OF THE INVENTION 
     The present invention refers to methods for assisting in the design of aerodynamic surfaces by making cost-optimized calculations of variable distributions such as the pressure distribution over them and more particularly to methods for assisting in the design of aircraft components. 
     BACKGROUND OF THE INVENTION 
     A common situation in practical industrial applications related to product development is the need to perform many surveys inside a space of state parameters. In the specific case of aeronautics, the calculation of the pressure distribution and/or other variable distributions over an aerodynamic surface is an important feature, in order to optimally design its structural components so that the weight of the structure is the minimum possible, but at the same time being able to withstand the expected aerodynamic forces. 
     Thanks to the increase of the use of the Computer Fluid Simulation Capability, nowadays, the calculation of the pressure distributions and/or other variable distributions over an aerodynamic surface is commonly done by solving numerically the Reynolds Averaged Navier-Stokes equations that model the movement of the flow around the surface, using discrete finite elements, finite differences or finite volume models. With the demand of accuracy posed in the aeronautical industry, each one of these computations requires important computational resources. 
     As the pressure distribution and/or other variable distributions over an aerodynamic surface depend on many different flight parameters, like angle of attack and Mach number, it is necessary to perform many lengthy and costly computations to obtain all the required information. 
     Some methods for calculating the pressure distributions and/or other variable distributions over the surface of an aerodynamic surface such as an aircraft wing, inside a defined parameter space, using Computational Fluid Dynamics (CFD) and interpolation are known in the prior art. In particular a known interpolation method is disclosed in the article “ A Multilinear Singular Value Decomposition ”, Lieven De Lathauwer, Bart de Moor and Joos Vandewalle. SIAM  J. Matrix Annal. Appl . Vol. 21, No. 4, pp 1253-1278. 
     CFD is used to calculate the pressure distributions and/or other variable distributions in a predefined group of points of the parameter space. The shock wave phenomena causes difficulties at the interpolation step. Namely, the number of computations needed to accurately reproduce the shock wave phenomena must be larger than the quantity I/d, where I stands for the geometric distance between extreme positions of the shock wave (as the parameters are varied) and d stands for the thickness of the shock wave. However, regarding that, in typical aircraft wings, the thickness of shock waves is usually small and the parameter I can be of the order of the 50% of the chord length, the number of computations needed to perform the interpolation of the shock wave increases rapidly. 
     The present invention is intended to solve this drawback. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide methods to be used in the design or analysis of an aircraft or an aircraft component for obtaining a given variable distribution over an aerodynamic surface inside a predefined parameter space taking into account the shock wave phenomena and minimizing the computational costs. 
     This and other objects are met by a computer-aided method that provides a variable distribution V over the surface of an aircraft or an aircraft component inside a predefined parameter space by means of a reconstruction of the results obtained through computations for an initial group of points in the parameter space using a CFD model in connection with a structured mesh defined on said surface, comprising in each of said computations the following steps:
         Selecting relevant shock wave structures along mesh lines in the chord direction taking place between a first point and a last point in the mesh line, being defined by its thickness δ sw , its position s sw  in the mid point between said first point and said last point and the variable jump ΔV between the last and the first point.   Decomposing said variable distribution V into a smooth variable distribution V smooth  and a shock wave variable distribution V jump  corresponding to said relevant shock wave structures.   Obtaining the variable distribution V at any point of the parameter space as a reconstruction of the smooth variable distribution V smooth  and the shock wave variable distribution V jump  obtained for said point from the values obtained in the previous step.       

     The variable distributions may comprise one or a combination of the following: the pressure distribution, the velocity components distribution, the mach number (euler computation) distribution, the friction components distribution, the temperature distribution, the density distribution, the energy distribution, the entropy distribution, the enthalpy distribution. 
     The parameter space may comprise any combination of the following parameters (if applicable to the aircraft or aircraft component being designed): the angle of attack, the Mach number, the sideslip angle, the wing aileron deflection angle, spoilers deflection, high lift devices deflection, canard deflection, landing gear deflected status, landing gear doors angle, APU inlet open angle, the vertical tailplane rudder deflection angle, the horizontal tailplane elevator deflection angle and the horizontal tailplane setting angle. The range of validity of said parameters is that of the aircraft typical flight envelope. 
     The above-mentioned method is applicable to the design of a classical aircraft formed by a cylindrical fuselage, wings in the centre fuselage either in the middle or high or low region of the fuselage, conventional tail or T-tail or Cruciform-tail or V-tail or H-tail or U-tail or canard and engines in the wing or at the rear fuselage and also to the design of a non-classical aircraft configuration like a blended wing body (BWB) or a flying wing. 
     Other characteristics and advantages of the present invention will be clear from the following detailed description of embodiments illustrative of its object in relation to the attached figures. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows the structured mesh defined over a wing surface which is used in a method according to this invention for the CFD calculations. 
         FIGS. 2   a  and  2   b  illustrate the three parameters used in the method according to this invention for characterizing a shock wave of, respectively, linear and sinusoidal shape. 
         FIG. 3  is the wing surface of  FIG. 1  including pressure iso-lines corresponding to “λ”-shape shock wave structures. 
         FIGS. 4-9  illustrate different features of the pressure distribution along a mesh line with two shock waves. 
         FIGS. 10-16  illustrate different features of the pressure distribution along a mesh line with one shock wave. 
         FIGS. 17 and 18  illustrate the improvements obtained with the method according to this invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     This invention is intended to provide a variable distribution over an aerodynamic surface such as for instance an aircraft wing, inside a predefined parameter space, using Computational Fluid Dynamics (CFD), a special treatment of the shock wave structures, and interpolation. CFD is used to calculate the variable distributions in a predefined group of points of the parameter space. The special treatment of the shock wave according to this invention allows performing accurate calculations of the variable distribution over the aerodynamic surface while keeping small the number of CFD computations, which carries an important CPU time reduction. The interpolation allows obtaining the variable distribution over the aerodynamic surface at all points inside the parametric space not considered in the initial computations. 
     Although the method shall be preferably applied to a structured mesh, it can be also used with a different mesh (hybrid, unstructured or Cartesian for example) by means of a user defined method to interpolate CFD solutions from the defined mesh to a user defined structured mesh. 
     An embodiment of a method according to the present invention will now be described in reference to an example for obtaining the pressure distribution over the surface of a 3D-wing with two free parameters, angle of attack (α), in the range −3° to +3°, and Mach number (M) in the range 0.40 to 0.80. 
     The pressure distribution over the surface of the wing will be firstly calculated for an initial group of points in the parametric space given by the following values:
         α=−1.50°, −1.00°, −0.50°, 0.00°, +0.50°, +1.00°, +1.50°   M=0.80, 0.81, 0.83, 0.85       

     Then, the pressure distribution in any point of the parametric space can be obtained by interpolation. The test points in which the pressure distribution will be interpolated from the results obtained are:
         α=−1.25°, −0.75°, −0.25°, +0.25°, +0.75°, +1.25°   M=0.82, 0.84       

     Computation for the Initial Group of Points in the Parametric Space 
     Step 1: Computation via CFD of the pressure distribution over the wing surface  11  for the initial group of points in the parametric space. 
     In each computation it is obtained the value of the pressure in each point of a structured mesh defined over the wing surface such as the mesh shown in  FIG. 1 . 
     The mesh is parameterized by two integer variables u and v. The mesh lines that go from the leading edge  13  to the trailing edge  15  are lines where v=constant and the mesh lines that extend over the span wise direction are such that u=constant. 
     Shock Wave Treatment: 
     Identification of the shock wave is made for all computations of said initial group of points. The pressure and the suction sides are treated separately. The flow just over the surface is a subsonic flow. Therefore it is impossible to find out shock waves attached to the wing surface. However, as a consequence of the presence of shock waves inside the fluid domain, just above the surface, pressure distribution is quite steep in some regions of the wing. These regions are called hereinafter shock waves. The shock wave in the surface of the 3D wing  11  is characterized in each mesh line along the chord as shown in  FIG. 2  by three parameters that depend on the span wise dimension coordinate, namely, the pressure jump in the shock wave (ΔP), the shock wave position in terms of arc length measured from the leading edge (s SW ) and the shock wave thickness (δ SW ). These three quantities are of the form:
 
 ΔP=ΔP ( v ; parameters)   (1)
 
 s   SW   =s   SW ( v ; parameters)   (2)
 
δ SW =δ SW ( v ; parameters)   (3)
 
where parameters stands for those parameters that identify each computation inside the parameter space. As it was already said, the parameters considered in the above-mentioned example are the angle of attack (α) and the Mach number (M).
 
     Thus the original pressure distribution is decomposed into two parts, a smooth distribution and a distribution associated with the shock waves:
 
 P=P   smooth   +P   jump   (4)
 
     This decomposition can be made considering a shape of the shock wave defined by the user. It can be a linear shape as shown in  FIG. 2   a  or a sinusoidal shape as shown in  FIG. 2   b.    
     Furthermore, two (or more) shock waves can co-exist in the same mesh line v=constant because, in 3D wings, “λ”-shape shock wave structures are common as can be observed in the pressure iso-lines plotted in  FIG. 3  that correspond to the computation of the example being considered. 
     Thus, the method must be able to deal with two (or more) shock waves in the same mesh line. In fact, the method considers that there are “always” two (or more) shock waves. For two shock wave case the shock wave number one is that one that is the closest to the leading edge and the shock wave number two is the other one. When there is only one shock wave, the shock wave is split into two shock waves with the same position and the same thickness but with a ΔP that is a half of the original ΔP. This is done to avoid discontinuities and to facilitate interpolation, which will be a part of the process. 
     To show explicitly the shock wave identification and disassembly, in the example being considered the method will be applied step by step to two different lines v=20 and v=40 for the computation corresponding to M=0.85 and α=+1.5°. 
     (Steps 2-9 are performed for each computation) 
     Step 2: Obtention of the derivative 
               ⅆ   P       ⅆ   s           
following the mesh lines v=constant, where P stands for pressure and s for arc length.
 
     The CFD pressure distribution in the first line (v*=20), illustrated in  FIG. 4  shows that there are two different shock waves. The CFD pressure distribution in a second line (v*=40), illustrated in  FIG. 10  shows that there is only one shock wave. 
     The distribution of 
               ⅆ   P       ⅆ   s           
over the first line (v*=20) is illustrated in  FIG. 5 .
 
     The distribution of 
               ⅆ   P       ⅆ   s           
over the second line (v*=40) is illustrated in  FIG. 11 .
 
(Steps 3-9 are performed for each line v=constant. We will show the results obtained in the example being considered for both lines v*=20 and v*=40)
 
     Step 3: Seek for the maximum value of 
     
       
         
           
             
               
                 ⅆ 
                 P 
               
               
                 ⅆ 
                 s 
               
             
             . 
           
         
       
         
         
           
             3a) If the maximum value of 
           
         
       
    
               ⅆ   P       ⅆ   s           
were smaller than a predetermined value P 1  (P 1 =0.22 in the example being considered), it would be assumed that no shock wave would exist in this line. Then pressure jumps would be ΔP 1 (v*;α*,M*)=0 and ΔP 2 (v*;α*,M*)=0, while the position and thickness of the shock wave remain undefined. Repeat Step 3 for the next line.
         3b) If the maximum value of       

               ⅆ   P       ⅆ   s           
is larger than P 1 , which is the case for both lines v*=20 and v*=40, proceed to Step 4.
 
     Step 4: Definition of the shock wave structure as the set of all the mesh points around the maximum value of 
                 ⅆ   P       ⅆ   s       ⁢           ⁢   where   ⁢           ⁢       ⅆ   P       ⅆ   s             
is larger than a predetermined value P 2  (P 2 =0.15 in the example being considered) plus two additional mesh points at each side of the interval.
 
     In  FIG. 5  it has been highlighted for line v*=20 the region affected by the shock wave between the two dashed lines  21 ,  23 , namely, all the mesh points where 
               ⅆ   P       ⅆ   s           
is larger than P 2  plus two additional mesh points added at each side of the interval.
 
     In  FIG. 11  it has been highlighted for line v*=40 the region affected by the shock wave between the two dashed lines  31 ,  33 , namely, all the mesh points where 
               ⅆ   P       ⅆ   s           
is larger than P 2  plus two additional mesh points.
 
     Step 5: Definition of the shock wave position as the mid point of the shock wave structure obtained in step 4. 
     Step 6: The pressure jump in the shock wave is defined as the pressure difference between the last and the first point of the shock wave structure.
         6a) If the pressure jump is smaller than a third predetermined value P 3  (P 3 =0.080 in the example being considered) it is assumed that there is no shock wave in this line; thus, ΔP 1  (v*;α*,M*)=0 and ΔP 2 (v*;α*,M*)=0 is set, while the position and thickness of the shock wave remain undefined. Go to Step 3 for the next line.   6b) If the pressure jump is greater than P 3 , which is the case illustrated in  FIG. 6 , then it is considered that there is a shock wave with the characterizing parameters calculated as defined above.   As can be observed in  FIG. 6 , for line v*=20,
 
ΔP=0.22&gt;P3=0.08
   and thus a shock wave exists whose characterizing parameters have been just calculated.   As can be observed in  FIG. 12  for line v*=40,
 
ΔP=0.40&gt;P3=0.08
   and thus a shock wave exists whose characterizing parameters are ΔP=0.40, s SW =1.32 and δ SW =0.72.       

     Step 7: Excluding the zone affected by the shock wave just detected, repeat from Step 3 to Step 6 to detect a possible second shock wave. 
     For line v*=20  FIG. 7  shows the zone affected by a second shock wave as the region in between of the dashed lines  25 ,  27 . As illustrated in  FIG. 8 , the pressure jump in this case ΔP=0,165 is greater than P 3 , thus, it is concluded that there are two shock waves in this line v*=20. 
     For line v*=40, as shown in  FIGS. 13 and 14  the maximum of the 
               ⅆ   P       ⅆ   s           
derivative (after excluding that zone affected by the firstly detected shock wave) is compared with P 1 . In this case, the maximum value of
 
               ⅆ   P       ⅆ   s           
is larger than P 1 , and thus, Step 4 and 5 are addressed. Once the zone affected by the shock wave (the zone between the dashed lines  35 ,  37 ) has been defined, the pressure jump is computed. As illustrated in  FIG. 15  the pressure jump through the shock wave is ΔP=0.015&lt;P 3 =0.08. According to Step 6a), there is no shock wave in this position.
 
     Step 8: Depending on the existence of one or two shock waves:
         8a) If there is only one shock wave with characterizing parameters ΔP, s SW  and δ SW , the shock wave is split into two shock waves with the same position and thickness but with a pressure jump equal to a half of the total value:
 
 ΔP   1 ( v*;α*,M* )=Δ P   2 ( v*;α*,M* )= ΔP/ 2   (5)
 
s 1     SW   =s 2     SW   =s SW    (6)
 
δ 1     SW   =δ 2     SW   =δ SW    (7)
   For line v*=40 there is only one shock wave. The shock wave is split into two shock waves with the same positions and thicknesses but with pressure jumps equal to a half of the total pressure jump. The characterizing parameters of both shock waves are:   Shock wave # 1 : ΔP 1 =0.20, s SW     1   =1.32 and δ SW     1   =0.72.   Shock wave # 2 : ΔP 2 =0.20, s SW     2   =1.32 and δ SW     2   =0.72.   8b) If there are two shock waves, the shock wave that is closest to the leading edge is named as shock wave # 1  and the other shock wave as shock wave # 2 .   In the example being considered for line v*=20 the shock wave # 1 , which is closest to the leading edge, is the shock wave detected in second place, whereas the shock wave detected in first place is the shock wave # 2 . Thus, the characterizing parameters of the two shock waves are:   Shock wave # 1 : ΔP 1 =0.165, s SW     1   =0.84 and δ SW     1   =0.56.   Shock wave # 2 : ΔP 2 =0.220, s SW     2   =2.95 and δ SW     2   =0.80.       

     Step 9: Disassembling of the two shock waves line by line in the smooth pressure distribution and the shock wave pressure distribution. 
     For line v*=20, the smoothed pressure profile is showed in  FIG. 9 . 
     For line v*=40 the smoothed pressure profile is showed in  FIG. 16 . 
     Interpolation: 
     Step 10: Point by point, interpolation (by a user defined method) of the smooth pressure distribution at any point of the parameter space. 
     Step 11: Interpolation of the shock wave pressure distribution for each shock wave in the parameter space. 
     Step 12: When there is no shock wave, the position and thickness are both undefined. This problem is overcome using two new parameters, ΔPxs sw  and ΔPxδ sw . These two parameters are set to zero when there is no shock wave. Interpolation of these two new parameters in the parameter space. 
     Step 13: Once the smooth pressure distribution and the shock wave pressure distribution are determined at a point of the parameter space, assembly of both distribution is made to reconstruct the whole pressure distribution. 
     Results 
     The results in terms of mean error (ME) of the reconstruction in the example being considered with respect to a CFD computation are presented in the next table for a set of test points in the parameter space, where the ME is defined as follows: 
                     M   ⁢           ⁢     E   ⁡     (   %   )         =           ∑     i   =   1       I   max       ⁢       ∑     j   =   1       J   max       ⁢          error   ij                  I   max     ×     J   max         ×     1       P   max     -     P   min         ×   100             (   8   )               
where i and j are the indexes that identify each point inside the mesh and I max  and J max  are the total number of points in each direction. Note that the error is referred to the total variation of the pressure in each test point.
 
     
       
         
               
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 Test points 
                   
               
             
          
           
               
                 α (°) 
                 M 
                 ME (%) 
               
               
                   
               
               
                 −1.25 
                 0.82 
                 0.21 
               
               
                 −0.75 
                 0.82 
                 0.19 
               
               
                 −0.25 
                 0.82 
                 0.33 
               
               
                 +0.25 
                 0.82 
                 0.43 
               
               
                 +0.75 
                 0.82 
                 0.40 
               
               
                 +1.25 
                 0.82 
                 0.62 
               
               
                 −1.25 
                 0.84 
                 0.23 
               
               
                 −0.75 
                 0.84 
                 0.23 
               
               
                 −0.25 
                 0.84 
                 0.48 
               
               
                 +0.25 
                 0.84 
                 0.34 
               
               
                 +0.75 
                 0.84 
                 0.37 
               
               
                 +1.25 
                 0.84 
                 0.53 
               
               
                   
               
             
          
         
       
     
     The improvements obtained with the method according to this invention with respect to the prior art are illustrated in  FIGS. 17 and 18  showing two comparative examples of the pressure distribution  53  obtained according to a known High Order Single Value Decomposition (HOSVD)+interpolation method and the pressure distribution  55  obtained including the shock wave treatment according to this invention with respect to the CFD distribution  51 . As it can be seen the shock wave treatment improves the quality of final results. 
     Modifications may be introduced into the preferred embodiment just set forth, which are comprised within the scope defined by the following claims.