Abstract:
A computer-aided method suitable for assisting in the design of an aircraft by providing the values of dimensional variables, dependant of a predefined set of parameters, for the complete aircraft or an aircraft component, comprising the following steps: a) Defining a parametric space grid; b) Obtaining a suitable Reduced Order Model (ROM) model, particularly a Proper Orthogonal Decomposition (POD) model, for calculating said variables for whatever point over the parametric space through an iterative process. Computer Fluid Dynamics (CFD) is used to calculate said variables for an appropriately selected set of points over the parametric space, which are used to approximate, via ROM and ad hoc interpolation, the variables in any other point over the parametric space. The method minimizes the required number of CFD calculations (to minimize the computational cost, which dramatically depends on this number) for a given level of error.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention refers to methods for assisting in the design of aircrafts by making cost-optimized calculations of the aerodynamic forces experimented by the complete aircraft or an aircraft component. 
       BACKGROUND OF THE INVENTION 
       [0002]    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 aerodynamic forces experimented by aircraft components 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. 
         [0003]    Thanks to the increase of the use of the Computer Fluid Simulation Capability, nowadays, the determination of the aerodynamic forces on an aircraft is commonly done by solving numerically the Reynolds Averaged Navier-Stokes equations (RANS equations from now onwards) that model the movement of the flow around the aircraft, using discrete finite elements or finite volume models. With the demand of accuracy posed in the aeronautical industry, each one of these computations requires important computational resources. 
         [0004]    The dimensioning aerodynamic forces are not known a priori, and since the global magnitude of the forces may depend on many different flight parameters, like angle of attack, angle of sideslip, Mach number, control surface deflection angle, it has been necessary to perform many lengthy and costly computations to properly calculate the maximum aerodynamic forces experimented by the different aircraft components or the complete aircraft. 
         [0005]    In order to reduce the overall number of these lengthy computations, approximate mathematical modelling techniques for obtaining a Reduced Order Model (ROM) have been developed in the past, like Single Value Decomposition (SVD) as a mean to perform intelligent interpolation, or the more accurate Proper Orthogonal Decomposition (POD from now onwards) that takes into account the physics of the problem by using a Galerkin projection of the Navier-Stokes equations. 
         [0006]    The idea of these techniques is to define the new analytical solution as a combination of the information obtained before. POD defines several modes that include the solution obtained by Computational Fluid Dynamics (CFD) and then uses those modes to reproduce solutions not obtained by CFD. The application of this techniques may require many CFD calculations involving a large computational cost. 
         [0007]    The present invention is intended to solve this drawback. 
       SUMMARY OF THE INVENTION 
       [0008]    It is an object of the present invention to provide methods for making analytical calculations of the aerodynamic forces experimented by a complete aircraft or an aircraft component which are dependant of a significant number of parameter, minimizing the computational costs. 
         [0009]    It is another object of the present invention to provide methods for making analytical calculations of the aerodynamic forces experimented by a complete aircraft or an aircraft component which are dependant of a significant number of parameters, minimizing the number of CFD computations. 
         [0010]    These and other objects are met by a computer-aided method suitable for assisting in the design of an aircraft by providing the values of one or more dimensional variables, such as the pressure distribution along a wing surface, for the complete aircraft or an aircraft component, being said one or more variables dependant of a predefined set of parameters, such as a set including the angle of attack and the Mach number, comprising the following steps:
       Defining a parametric space grid setting predetermined distances between its values.   Obtaining a suitable model for calculating said one or more dimensional variables for whatever point over the parametric space through an iterative process with respect to a reduced group of points, of increasing number of members in each iteration, comprising the following sub-steps:
           Calculating the values of said one or more dimensional variables for an initial group of points using a CFD model.   Obtaining an initial ROM model from said CFD computations and calculating the values of said one or more dimensional variables for said initial group of points using the initial ROM model.   Selecting the e-point of the group with the largest deviation ε between the results provided by the CFD and the ROM models and finishing the iterative process if ε is lesser than a predefined value ε 0 .   Selecting new points over the parametric space to be added to the group of points as those points placed inside the parametric space grid at a predefined distance from said e-point.   Calculating the values of said one or more dimensional variables for the new points using the CFD and the ROM model and going back to the third sub-step.   
               
 
         [0018]    In particular, said one or more dimensional variables includes one or more of the following: aerodynamic forces, skin values and values distribution around the complete aircraft or aircraft component; said set parameters includes one or more of the following: angle of attack and Mach number; and said aircraft component is one of the following: a wing, an horizontal tail plane, a vertical tail plane. 
         [0019]    In a preferred embodiment, said complete aircraft or an aircraft component is divided into blocks and said CFD and ROM models are applied block by block. Hereby an accurate method for providing the values of one or more dimensional variables of an aircraft or an aircraft component is achieved. 
         [0020]    In another preferred embodiment said ROM model is a POD model. CFD is used to calculate the pressure distributions for an appropriately selected set of points over the parametric space, which are used to approximate, via POD and ad hoc interpolation, the dimensional variables in any other point over the parametric space. In addition, the method minimizes the required number of CFD calculations (to minimize the computational cost, which dramatically depends on this number) for a given level of error. This is made using POD and interpolation on the already calculated points. New points are selected iteratively, either one by one or in groups. Hereby a method for providing the values of one or more dimensional variables of an aircraft or an aircraft component dependant of a predefined set of parameters, optimizing the computing costs, is achieved. 
         [0021]    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 
         [0022]      FIG. 1  shows views of the suction side, the pressure side, the leading edge and the tip of an aircraft wing divided in blocks. 
           [0023]      FIG. 2  shows a graphic representation of a local sub-grid in the parametric space grid for selecting new points to be added to the group of points used for obtaining the POD model according to this invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0024]    An embodiment of a method according to the present invention will now be described for obtaining a POD model that allows calculating the steady pressure distribution over the surface of the wing of an aircraft, being said pressure distribution dependant of two free parameters: angle of attack (a) and Mach number (M). 
       Initiation Steps: 
       [0025]    Step 1: Division of the wing into several blocks according to the geometry of the object. CFD tools usually divide the 3D computational domain into blocks, as illustrated in  FIG. 1  showing the wing divided into 16 main blocks. This is a convenient but non-essential part of the method, which can be applied with just one block. 
         [0026]    Step 2: A definition of a parametric space grid is carried out by setting an initial value of the minimal distance in each parameter in the parametric space, d l , l=1, . . . , parameter #, which comes from a first guess of the smallest distance between points in the parametric space in the subsequent steps and could need some calibration. Such distance will be reduced by the method during the iteration, if needed. Then an equispaced grid is defined in parametric space based on these distances. Such grid will evolve during the process and can become non-equispaced. 
         [0027]    For instance, if angle of attack (α), in the range −3° to +3°, and Mach number (M), in the range 0.40 to 0.80, are the parameters being considered, the parametric space grid can be defined setting the distances d α =0.5 and d M =0.05. 
         [0028]    Step 3: Initiation of the process for an initial group of points over the parametric space selected by the user, such as the following 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
               
               
                 Initial 
                   
                   
               
               
                 Group 
                 Mach 
                 Alpha 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 P1 
                 0.400 
                 −3.00 
               
               
                 P2 
                 0.600 
                 −3.00 
               
               
                 P3 
                 0.800 
                 −3.00 
               
               
                 P4 
                 0.400 
                 0.00 
               
               
                 P5 
                 0.600 
                 0.00 
               
               
                 P6 
                 0.800 
                 0.00 
               
               
                 P7 
                 0.400 
                 3.00 
               
               
                 P8 
                 0.600 
                 3.00 
               
               
                 P9 
                 0.800 
                 3.00 
               
               
                   
               
             
          
         
       
     
       Introduction of the New Group of Points 
       [0029]    Step 4: Application, block by block, of POD to the initial group of points. A block-dependent set of modes is obtained for each block: 
         [0000]    
       
         
           
             
               
                 P 
                  
                 
                   ( 
                   
                     
                       
                         
                           x 
                           i 
                         
                         _ 
                       
                       ; 
                       
                         α 
                         j 
                       
                     
                     , 
                     
                       M 
                       k 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     P 
                     ijk 
                   
                    
                   
                     → 
                     POD 
                   
                    
                   
                     P 
                     ijk 
                   
                 
                 = 
                 
                   
                     ∑ 
                     p 
                   
                    
                   
                     
                       
                         A 
                         p 
                       
                        
                       
                         ( 
                         
                           
                             α 
                             j 
                           
                           , 
                           
                             M 
                             k 
                           
                         
                         ) 
                       
                     
                      
                     
                       φ 
                       ip 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where P is the pressure distribution, x i  are the spatial coordinates, α is the angle of attack, M is the Mach number, A p  are the mode amplitudes, and the columns of the matrix φ ip  are the POD modes. Each mode has an associated singular value, which results from application of POD. 
         [0030]    Step 5: Classification of modes:
       A first classification (in each block) of the modes into two parts is as follows: (a) those modes yielding a RMSE smaller than some threshold value ε 1  (depending on ε 0 , after some calibration) are neglected; (b) the n 1  retained modes are called main modes.   Main modes, in turn, are classified into two groups, namely n primary modes and n 1 −n secondary modes with, with n obtained after some calibration, say       
 
         [0000]    
       
         
           
             
               n 
               = 
               
                 
                   4 
                   5 
                 
                  
                 
                   n 
                   1 
                 
               
             
             , 
             . 
           
         
       
     
         [0033]    The root mean squared error (RMSE), is defined as 
         [0000]    
       
         
           
             RMSE 
             = 
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         N 
                         p 
                       
                     
                      
                     
                       error 
                       i 
                       2 
                     
                   
                   
                     N 
                     p 
                   
                 
                 , 
               
             
           
         
       
     
         [0034]    where N p  is the total number of points of the mesh that defines the wing, and error i  is the difference between the pressure of the approximation and the pressure of the CFD solution at i-th the point of the mesh. 
         [0035]    Step 6: POD reconstruction of the pressure distribution for each of the already computed group of points using the (n) main primary modes in each block. Then each point is further approximated using the neighboring points via least squares. 
         [0036]    Step 7: Comparison between the CFD calculated and the POD+interpolation-approximated pressure profiles, and estimation of the RMSE in each block, for each already computed points. 
         [0000]    The RMSE for the above-mentioned initial group of nine points is the following: 
         [0000]    
       
         
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 RMSE 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 P1 
                 0.0371 
               
               
                   
                 P2 
                 0.0298 
               
               
                   
                 P3 
                 0.0887 
               
               
                   
                 P4 
                 0.0273 
               
               
                   
                 P5 
                 0.0190 
               
               
                   
                 P6 
                 0.0756 
               
               
                   
                 P7 
                 0.0605 
               
               
                   
                 P8 
                 0.0930 
               
               
                   
                 P9 
                 0.1758 
               
               
                   
                   
               
             
          
         
       
     
         [0037]    Step 8: Selection of the point with largest RMSE. 
         [0000]    As shown in the above table in the first iteration this point is P9. 
         [0038]    Step 9: Definition, as shown in  FIG. 2 , of a local sub-grid of the total parametric space grid in the vicinity of the point 21 of maximum error. Such local sub-grid consists of three levels, at distances d l  (first level), 2·d l  (second level) and 4·d l  (third level). 
         [0039]    Step 10: Selection of the level in which the new point will be introduced. If there are some points in between of two levels (see below), they are considered to belong to the inner level.
       If no points are present in the whole sub-grid, then the new point is introduced in the third level.   If only the third level exhibits points, then the new point is introduced in the second level.   If there are no points in the first level and there is only one point in the second level, the new point is introduced in the second level.   If there are no points in the first level and there are at least two points in the second level, the new point is introduced in the first level.   If at least one point is present in the first level, then the new point is introduced in the first level with one exception that leads to the introduction of a sub-level in the local grid. This occurs when (a) at least five points are present in the first level, and (b) at least four of these show the largest RMSE among all points in the three levels. In that case, the distances in the local sub-grid are divided by two and step 9 is repeated again with the resulting new subgrid. Note that this step means that each point will generally have a different set of minimal distances d l .
 
In the example being considered, the new point P10 will be introduced in the third level because none of the points of the initial group is present in the whole sub-grid in the vicinity of P9.
 
Step 11: Once the target level has been chosen, the most space-filling point in this level is selected as follows. The minimum distance, D, from each possible candidate to the remaining, already selected points is computed. That candidate that shows the largest value of D is selected. D is the distance in the parametric space. In this example, the distance between two points of the parametric space (labeled 1 and 2) is defined as follows:
       
 
         [0000]        D   12 =√{square root over (α 12   2   +M   12   2 )} 
         [0000]    where 
         [0000]    
       
         
           
             
               α 
               12 
             
             = 
             
               
                 
                   
                     
                       α 
                       2 
                     
                     - 
                     
                       α 
                       1 
                     
                   
                   
                     Δ 
                      
                     
                         
                     
                      
                     α 
                   
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   M 
                   12 
                 
               
               = 
               
                 
                   
                     M 
                     2 
                   
                   - 
                   
                     M 
                     1 
                   
                 
                 
                   Δ 
                    
                   
                       
                   
                    
                   M 
                 
               
             
           
         
       
     
         [0000]    are the distances in the parameters α and M, and Δα and ΔM are the corresponding total ranges in these parameters.
 
In the example being considered the distance between third level points and the closest point belonging to the group is shown in the following table.
 
         [0000]                                                                                                Third level       Closest point               points       of the group            Mach   Alpha   Mach   Alpha   Distance                    0.650   3.00   0.600   3.00   0.1250       0.650   2.50   0.600   3.00   0.1502       0.650   2.00   0.600   3.00   0.2083       0.650   1.50   0.600   0.0   0.2795       0.700   1.50   0.600   0.0   0.3536       0.750   1.50   0.800   0.0   0.2795       0.800   1.50   0.800   0.0   0.2500                    
Therefore the new point to be introduced is P10: Mach=0.700, Alpha=1.50.
 
         [0045]    Step 12: If more than one point is introduced in each iteration, then the process is repeated from step 8 with the already selected points excluded. 
       Update of the Set of Modes: 
       [0046]    Once the new point (or group of points) has been computed, the set of modes for each block is updated. 
         [0047]    Step 13: Application of POD to the group of points, ignoring those modes that show a RMSE smaller than ε 1 . 
         [0048]    Step 14: Computation of some pseudo-points, defined block by block, which consists of two groups:
       The n 1  main modes of each block, multiplied by their respective singular values.   The POD modes obtained upon application of POD to the new points resulting from last iteration, multiplied by their respective singular values.       
 
         [0051]    Steps 13 and 14 may be collapsed into just only one step. In this case pseudo-points are defined adding together the main modes of the already computed points, multiplied by their respective singular values, and the new points. Division into steps 13 and 14, as above, is made to filter out numerical errors from the process, which is a well known benefit of the POD method. 
         [0052]    Step 15: Application of POD to the set of all pseudo-points, block by block. 
         [0053]    Step 16: Repetition of the process from step 5. 
         [0000]    To illustrate this iterative process a brief description of the second iteration in the example being considered follows:
 
The RMSE for the group of then points in the second iteration is the following:
 
         [0000]                                                      RMSE                                        P1   0.0313           P2   0.0242           P3   0.0723           P4   0.0275           P5   0.0167           P6   0.0569           P7   0.0853           P8   0.0458           P9   0.1421           P10   0.0260                        
El maximum error point is still P9 and the new point P11 will be introduced in the second level because there is not any point in the group in levels 1 and 2 and there is a point in level 3 (P10 introduced in the first iteration).
 
The distance between second level points and the closest point belonging to the group is shown in the following table:
 
         [0000]                                                                          Second level       Closest point               points       of the group            Mach   Alpha   Mach   Alpha   Distance               0.700   3.00   0.800   3.00   0.2500       0.700   2.50   0.700   1.50   0.1667       0.700   2.00   0.700   1.50   0.0833       0.750   2.00   0.700   1.50   0.1502       0.750   2.00   0.800   3.00   0.1662                    
Therefore the new point to be introduced is P11: Mach=0.700, Alpha=2.50.
 
       Stop Criteria: 
       [0054]    Step 17: The process is completed when the RMSE, computed in step 7 using POD and both linear and a quadratic least squares interpolation, are both smaller than ε 0 . 
       Results 
       [0055]    In the execution of the method in the example being considered the initial group of points over the parametric space was, as said before, the following: 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 Mach 
                 Alpha 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 P1 
                 0.400 
                 −3.00 
               
               
                 P2 
                 0.600 
                 −3.00 
               
               
                 P3 
                 0.800 
                 −3.00 
               
               
                 P4 
                 0.400 
                 0.00 
               
               
                 P5 
                 0.600 
                 0.00 
               
               
                 P6 
                 0.800 
                 0.00 
               
               
                 P7 
                 0.400 
                 3.00 
               
               
                 P8 
                 0.600 
                 3.00 
               
               
                 P9 
                 0.800 
                 3.00 
               
               
                   
               
             
          
         
       
     
         [0056]    Along the iteration process, the following points were added to the group: 
         [0000]    
       
         
               
               
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 P10 
                 0.700 
                 1.50 
               
               
                   
                 P11 
                 0.700 
                 2.50 
               
               
                   
                 P12 
                 0.800 
                 2.00 
               
               
                   
                 P13 
                 0.500 
                 1.50 
               
               
                   
                 P14 
                 0.750 
                 2.50 
               
               
                   
                 P15 
                 0.400 
                 2.00 
               
               
                   
                 P16 
                 0.700 
                 −1.00 
               
               
                   
                 P17 
                 0.750 
                 1.50 
               
               
                   
                 P18 
                 0.750 
                 3.00 
               
               
                   
                 P19 
                 0.800 
                 −1.50 
               
               
                   
                 P20 
                 0.500 
                 2.50 
               
               
                   
                 P21 
                 0.800 
                 2.50 
               
               
                   
                 P22 
                 0.800 
                 1.50 
               
               
                   
                 P23 
                 0.700 
                 0.50 
               
               
                   
                 P24 
                 0.750 
                 1.00 
               
               
                   
                 P25 
                 0.700 
                 3.00 
               
               
                   
                 P26 
                 0.750 
                 2.00 
               
               
                   
                 P27 
                 0.450 
                 2.50 
               
               
                   
                 P28 
                 0.800 
                 1.00 
               
               
                   
                 P29 
                 0.450 
                 3.00 
               
               
                   
                 P30 
                 0.750 
                 −0.50 
               
               
                   
                   
               
             
          
         
       
     
         [0057]    An evaluation of the model obtained according to the method of this invention can be done comparing the results obtained in 16 test points using said model in several iterations with the results obtained using the CFD model that are shown in the following tables: 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Invention Model Results 
               
             
          
           
               
                 Test 
                   
                   
                   
                 10 
                 15 
                 20 
                 25 
                 30 
               
               
                 Point 
                 Mach 
                 Alpha 
                 CFD 
                 Points 
                 Points 
                 Points 
                 Points 
                 Points 
               
               
                   
               
             
          
           
               
                 Lift Coefficient 
                   
               
             
          
           
               
                 Tp1 
                 0.800 
                 2.25 
                 0.1965 
                 0.1922 
                 0.1966 
                 0.1965 
                 0.1971 
                 0.1966 
               
               
                 Tp2 
                 0.800 
                 1.25 
                 0.1045 
                 0.1061 
                 0.1082 
                 0.1075 
                 0.1054 
                 0.1058 
               
               
                 Tp3 
                 0.800 
                 −1.25 
                 −0.1077 
                 −0.1089 
                 −0.1085 
                 −0.1073 
                 −0.1082 
                 −0.1088 
               
               
                 Tp4 
                 0.800 
                 −2.25 
                 −0.1920 
                 −0.1871 
                 −0.1925 
                 −0.1927 
                 −0.1928 
                 −0.1936 
               
               
                 Tp5 
                 0.775 
                 2.25 
                 0.1895 
                 0.1899 
                 0.1899 
                 0.1903 
                 0.1910 
                 0.1900 
               
               
                 Tp6 
                 0.775 
                 1.25 
                 0.1012 
                 0.1036 
                 0.1051 
                 0.1031 
                 0.1023 
                 0.1018 
               
               
                 Tp7 
                 0.775 
                 −1.25 
                 −0.1048 
                 −0.1018 
                 −0.1121 
                 −0.1057 
                 −0.1066 
                 −0.1068 
               
               
                 Tp8 
                 0.775 
                 −2.25 
                 −0.1867 
                 −0.1853 
                 −0.1884 
                 −0.1908 
                 −0.1912 
                 −0.1916 
               
               
                 Tp9 
                 0.725 
                 2.25 
                 0.1773 
                 0.1849 
                 0.1778 
                 0.1788 
                 0.1777 
                 0.1774 
               
               
                 Tp10 
                 0.725 
                 1.25 
                 0.0966 
                 0.0971 
                 0.0980 
                 0.0965 
                 0.0970 
                 0.0970 
               
               
                 Tp11 
                 0.725 
                 −1.25 
                 −0.1002 
                 −0.0962 
                 −0.1078 
                 −0.1022 
                 −0.1029 
                 −0.1022 
               
               
                 Tp12 
                 0.725 
                 −2.25 
                 −0.1785 
                 −0.1812 
                 −0.1816 
                 −0.1829 
                 −0.1867 
                 −0.1864 
               
               
                 Tp13 
                 0.525 
                 2.25 
                 0.1577 
                 0.1565 
                 0.1267 
                 0.1563 
                 0.1561 
                 0.1585 
               
               
                 Tp14 
                 0.525 
                 1.25 
                 0.0868 
                 0.0722 
                 0.0845 
                 0.0847 
                 0.0873 
                 0.0854 
               
               
                 Tp15 
                 0.525 
                 −1.25 
                 −0.0897 
                 −0.0749 
                 −0.0960 
                 −0.0786 
                 −0.0964 
                 −0.1084 
               
               
                 Tp16 
                 0.525 
                 −2.25 
                 −0.1600 
                 −0.1580 
                 −0.1598 
                 −0.1196 
                 −0.1199 
                 −0.1197 
               
             
          
           
               
                 X Momentum Coefficient 
                   
               
             
          
           
               
                 Tp1 
                 0.800 
                 2.25 
                 +0.2062 
                 0.1979 
                 0.2054 
                 0.2054 
                 0.2068 
                 0.2061 
               
               
                 Tp2 
                 0.800 
                 1.25 
                 +0.1109 
                 0.1124 
                 0.1181 
                 0.1174 
                 0.1128 
                 0.1127 
               
               
                 Tp3 
                 0.800 
                 −1.25 
                 −0.1018 
                 −0.1023 
                 −0.1024 
                 −0.1010 
                 −0.1016 
                 −0.1022 
               
               
                 Tp4 
                 0.800 
                 −2.25 
                 −0.1866 
                 −0.1810 
                 −0.1867 
                 −0.1866 
                 −0.1866 
                 −0.1870 
               
               
                 Tp5 
                 0.775 
                 2.25 
                 +0.1991 
                 0.1957 
                 0.1984 
                 0.1992 
                 0.2010 
                 0.1995 
               
               
                 Tp6 
                 0.775 
                 1.25 
                 +0.1078 
                 0.1102 
                 0.1140 
                 0.1117 
                 0.1090 
                 0.1085 
               
               
                 Tp7 
                 0.775 
                 −1.25 
                 −0.0987 
                 −0.0953 
                 −0.1067 
                 −0.0993 
                 −0.0999 
                 −0.1000 
               
               
                 Tp8 
                 0.775 
                 −2.25 
                 −0.1812 
                 −0.1790 
                 −0.1824 
                 −0.1846 
                 −0.1848 
                 −0.1850 
               
               
                 Tp9 
                 0.725 
                 2.25 
                 +0.1849 
                 0.1910 
                 0.1858 
                 0.1875 
                 0.1853 
                 0.1849 
               
               
                 Tp10 
                 0.725 
                 1.25 
                 +0.1036 
                 0.1041 
                 0.1060 
                 0.1029 
                 0.1036 
                 0.1037 
               
               
                 Tp11 
                 0.725 
                 −1.25 
                 −0.0939 
                 −0.0894 
                 −0.1018 
                 −0.0955 
                 −0.0959 
                 −0.0954 
               
               
                 Tp12 
                 0.725 
                 −2.25 
                 −0.1728 
                 −0.1746 
                 −0.1749 
                 −0.1760 
                 −0.1798 
                 −0.1796 
               
               
                 Tp13 
                 0.525 
                 2.25 
                 +0.1654 
                 0.1644 
                 0.1279 
                 0.1637 
                 0.1637 
                 0.1658 
               
               
                 Tp14 
                 0.525 
                 1.25 
                 +0.0943 
                 0.0809 
                 0.0926 
                 0.0928 
                 0.0953 
                 0.0933 
               
               
                 Tp15 
                 0.525 
                 −1.25 
                 −0.0827 
                 −0.0668 
                 −0.0879 
                 −0.0704 
                 −0.0879 
                 −0.1001 
               
               
                 Tp16 
                 0.525 
                 −2.25 
                 −0.1534 
                 −0.1499 
                 −0.1514 
                 −0.1100 
                 −0.1100 
                 −0.1096 
               
             
          
           
               
                 Y Momentum Coefficient 
                   
               
             
          
           
               
                 Tp1 
                 0.800 
                 2.25 
                 −0.1068 
                 −0.1044 
                 −0.1076 
                 −0.1074 
                 −0.1081 
                 −0.1076 
               
               
                 Tp2 
                 0.800 
                 1.25 
                 −0.0345 
                 −0.0377 
                 −0.0392 
                 −0.0387 
                 −0.0361 
                 −0.0363 
               
               
                 Tp3 
                 0.800 
                 −1.25 
                 +0.1270 
                 0.1278 
                 0.1279 
                 0.1266 
                 0.1273 
                 0.1278 
               
               
                 Tp4 
                 0.800 
                 −2.25 
                 +0.1914 
                 0.1877 
                 0.1921 
                 0.1921 
                 0.1923 
                 0.1928 
               
               
                 Tp5 
                 0.775 
                 2.25 
                 −0.1036 
                 −0.1036 
                 −0.1038 
                 −0.1044 
                 −0.1054 
                 −0.1043 
               
               
                 Tp6 
                 0.775 
                 1.25 
                 −0.0340 
                 −0.0374 
                 −0.0384 
                 −0.0367 
                 −0.0351 
                 −0.0347 
               
               
                 Tp7 
                 0.775 
                 −1.25 
                 +0.1232 
                 0.1215 
                 0.1295 
                 0.1241 
                 0.1247 
                 0.1248 
               
               
                 Tp8 
                 0.775 
                 −2.25 
                 +0.1858 
                 0.1853 
                 0.1878 
                 0.1892 
                 0.1896 
                 0.1898 
               
               
                 Tp9 
                 0.725 
                 2.25 
                 −0.0960 
                 −0.1017 
                 −0.0970 
                 −0.0982 
                 −0.0967 
                 −0.0965 
               
               
                 Tp10 
                 0.725 
                 1.25 
                 −0.0335 
                 −0.0344 
                 −0.0356 
                 −0.0337 
                 −0.0338 
                 −0.0338 
               
               
                 Tp11 
                 0.725 
                 −1.25 
                 +0.1171 
                 0.1151 
                 0.1241 
                 0.1188 
                 0.1193 
                 0.1188 
               
               
                 Tp12 
                 0.725 
                 −2.25 
                 +0.1770 
                 0.1800 
                 0.1805 
                 0.1807 
                 0.1833 
                 0.1831 
               
               
                 Tp13 
                 0.525 
                 2.25 
                 −0.0868 
                 −0.0877 
                 −0.0618 
                 −0.0849 
                 −0.0847 
                 −0.0867 
               
               
                 Tp14 
                 0.525 
                 1.25 
                 −0.0321 
                 −0.0233 
                 −0.0302 
                 −0.0302 
                 −0.0321 
                 −0.0307 
               
               
                 Tp15 
                 0.525 
                 −1.25 
                 +0.1029 
                 0.0911 
                 0.1067 
                 0.0924 
                 0.1078 
                 0.1172 
               
               
                 Tp16 
                 0.525 
                 −2.25 
                 +0.1564 
                 0.1542 
                 0.1548 
                 0.1219 
                 0.1221 
                 0.1218 
               
               
                   
               
             
          
         
       
     
         [0058]    Modifications may be introduced into the preferred embodiment just set forth, which are comprised within the scope defined by the following claims.