Abstract:
Variational calculus principles are applied directly to the radiation integral to minimize the radar signature of a two- or three-dimensional geometry. In the preferred embodiment, the radiation integral is minimized through the solution to a differential equation generated by Euler&#39;s calculus of variations (CoV) equation. When used in conjunction with a minimizing sequence, the analysis affords a broad search of all possible coefficient values to ultimately arrive at global minima. Compared to existing techniques, the approach locates local extrema quickly and accurately using fewer impedance matrix calculations, and optimization using the invention is possible over a wide band of frequencies and angles. The method is applicable to a wide variety of situations, including the design of stealth platforms.

Description:
REFERENCE TO RELATED APPLICATION 
     This application claims priority from U.S. provisional application Ser. No. 60/152,687, filed Sep. 7, 1999, the entire contents of which are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to radar signature and, in particular, to a method of reducing radar cross-section and/or echo length. 
     BACKGROUND OF THE INVENTION 
     The use of radar is now widespread, for both commercial and military uses. In military applications, particularly during times of war, it may be essential that a vehicle such as an aircraft go undetected. 
     The radar signature or “cross-section” of an object is a measure of how much radar energy is reflected back or “returned” to a source or system searching for the object. The greater the signature, the easier it is to detect, track and potentially direct weapon systems against that object. 
     Radar cross section is also a function of the direction from which an object is “viewed.” With regard to aircraft, reduced aircraft radar cross-section is most important when viewed from the front, or in the “frontal sector.” Radar cross section is also typically increased in the presence of externally supported appendages such as weapons, which are typically mounted on pylons or against the body of the aircraft. 
     There are several techniques that may be employed to minimize the radar cross section. Broadly, one class of techniques is used to design aircraft having an inherently low radar signature, whereas other approaches seek to modify existing aircraft to achieve this same purpose. Of course, both broad principles may be applied to the same structure. 
     As discussed in U.S. Pat. No. 5,717,397, radar cross-section may be minimized using any combination of the following: 
     1. Shaping the exterior of the aircraft or external features, including leading/trailing edges, gaps, and seams, such that radar energy is reflected away from potential enemy radars; 
     2. Aligning leading and trailing edges, gaps and seams at a minimum number of similar angles (especially in the top or “plan” view of the aircraft), such that the radar returns from these various features are concentrated into fewer angles or sectors. 
     3. Concealing or hiding highly radar reflective aircraft components from the “view” of potential enemy radars; and 
     4. Utilizing materials and coatings in the construction of aircraft components that absorb or diffuse radar energy. 
     Whether designing a craft for a low radar signature in the first place, or modifying existing craft to achieve a reduced cross. section, the problem is complex and often mathematically intensive. Cross-section optimization using manual and empirical methods is labor intensive and, although computer methods may be employed to find local minima, global optimization is often elusive. Current automated techniques use Z-matrix calculations, often requiring numerous iterations to achieve dubious results. Although such techniques have improved in recent years, existing methods often still require mechanisms to avoid stagnating in local minima. 
     SUMMARY OF THE INVENTION 
     Broadly, this invention applies variational calculus principles directly to the radiation integral to minimize radar cross-section and/or echo length. The radiation integral, which is well known to those of skill in antenna design and other disciplines, may be used to determine the electromagnetic field scattering of a body given the surface current. In the preferred embodiment of this invention, the radiation integral is minimized through the solution to a differential equation generated by Euler&#39;s calculus of variations (CoV) equation. When used in conjunction with a minimizing sequence, the analysis affords a broad search of all possible coefficient values to ultimately arrive at. global minima. 
     Compared to existing techniques, the approach locates local extrema quickly and accurately using fewer impedance matrix calculations. The method is applicable to a wide variety of situations, including the design of stealth platforms. A thorough analysis of the applicable design equations is disclosed, which indicate that optimization over a wide band of frequencies and angles is possible. Although the examples presented are in two dimensions, the procedure is readily extensible to three dimensions. 
    
    
     BRIEF DESCRIPTION OF THE INVENTION 
     FIG. 1 is a drawing of a geometry used to illustrate the principles of the invention; 
     FIG. 2 is a drawing of a two-dimensional geometry defined by a single, continuous variable;, 
     FIG. 3 is a block diagram of an iterative optimization process according to the invention; 
     FIGS. 4 a  through  4   c  illustrate the way in which a two-dimensional shape is iteratively optimized according to the invention; 
     FIGS. 5 a  through  5   c  depict echo length corresponding to the optimizations of FIGS. 4 a  through  4   c;    
     FIGS. 6 a  through  6   c  illustrate the way in which a two-dimensional shape is iteratively optimized according to the invention using polar domain processing; and 
     FIGS. 7 a  through  7   c  depict echo length corresponding to the optimizations of FIGS. 6 a  through  6   c.   
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     This invention exploits Euler&#39;s equation to locate local extrema quickly and accurately. When used in conjunction with a minimizing sequence, the analysis affords a broad search of all possible coefficient values to ultimately arrive at the global minimum. 
     Introduction to the Design Equations 
     The analysis begins with the well-known radiation integral in two dimensions [3] given by:                    E   z   s          (     ρ   _     )       =         ω                   μ   0       4            ∫   s              J   z   s          (       ρ   _     ′     )              H   0     (   2   )            (     k               ρ   _     -       ρ   _     ′              )                          l             ,     
                    E   z   s          (     ρ   _     )       ≡                  TM   z                   radiated                 field                   (       m     -   1          V     )                       J   z   s          (       ρ   _     ′     )       ≡                z   -     directed                 surface                 current                   (       m     -   1          A     )                     ω   ≡                angular                 frequency                   (       s     -   1          rad     )                   k   ≡                wave                 number                   (     m     -   1       )                     μ   0     ≡                permittivity                 of                 free                 space                   (       m     -   1          H     )                       ρ   _     ′     ≡                vector                 from                 origin                 to                 geometry                 surface                   (   m   )                     ρ   _     ≡                vector                 from                 origin                 to                 observation                 point                   (   m   )                       (   1   )                                
     The geometry in question is depicted in FIG.  1 . 
     Introduction to the Calculus of Variations 
     If a two-dimensional integral equation can be constructed of the form                M        [   y   ]       =       ∫   a   b                     F   (     x   ,   y   ,         y   )     .                        x       ,                 (   2   )                                
     then it can be shown that the differential equation,                    F   y     -               x            F     y   .           =   0     ,           (   3   )                                
     when solved for y, will ensure that M[y] is a relative extrema. When the equation approaches zero, it ensures that a maximizing or minimizing solution is being obtained in an optimum sense. 
     As an example, consider a simplified form of problem that can arise when charged particles travel in an electromagnetic field or near a line charge:                  M        [   y   ]       =       ∫     x   1       x   2                  1   +       y   .     2         y                        x           ,           (   4   )                                
     where M[y] represents energy. Fundamental laws of physics dictate that charged particles will seek the path requiring the minimum amount of energy to traverse. The problem is to determine what that path may be, and this lends itself directly to a CoV solution. Here,                  F   y     =     -         1   +       y   .     2           y   2           ,           (   5   )                   F     y   .       =       y   .       y          1   +       y   .     2               ,     such                 that             (   6   )                             x            F     y   .         =           y   ¨        y          1   +       y   .     2           -       y   .          (         y   .            1   +       y   .     2           +     y                   y   .            y   ¨     /       1   +       y   .     2               )             (     y          1   +       y   .     2           )     2         ,   and           (   7   )                         F   y     -               x            F     y   .           =                  -           1   +       y   .     2                (       1   +       y   .     2         )     2             y   2          (       1   +       y   .     2         )       2         -                                      y   ¨        y          1   +       y   .     2           -       y   .          (         y   .            1   +       y   .     2           +     y                   y   .            y   ¨     /       1   +       y   .     2               )             (     y          1   +       y   .     2           )     2       .                 =              0                 (   8   )                                
     This reduces ultimately to 
     
       
           ÿy+{dot over (y)}   2 +1=0.  (9) 
       
     
     Solving for this differential equation produces as a solution, 
     
       
         ( x−C   1 ) 2   +y   2   =C   2   2 ,  (10) 
       
     
     an offset circular arc. The constants depend on the choices of x 1 , x 2 , y(x 1 ) and y(x 2 ). 
     Construction of the Design Equations 
     According to the invention, equation (1) is placed into a format amenable to the solution of the Euler equation. 
     Cartesian Format Design Equations 
     Assume that a two-dimensional geometry can be defined by a single continuous variable, y(x). If this is the case, the entire geometry may be defined according to curves of FIG.  2 . For this geometry, it is assumed that the two curves [y(x), {tilde over (y)}(x)] must meet at some common point (in this case, B, where both are zero). As a common point for optimization problems involving minimization, this is a judicious choice because we will assume that the primary minimization should occur about θ=0. Note that {tilde over (y)}(x) appears to depend on y(x) in the sense that it is of the opposite sign. Although this is not required under the invention, for convenience this assumption will be used in the following calculations. 
     We begin by recasting the radiation integral of equation (1) into a more amenable format for the application of the Euler equation:                    E   z   s          (   ρ   )       =           ω                   μ   0       4            ∫   A   B              J   z   s          [         x   2     +       y   2          (   x   )           ]              H   0     (   2   )            (     k        [     ρ   -     x                 cos                   θ   0       -       y        (   x   )          sin                   θ   0         ]       )              1   +         y   .     2          (   x   )                              x           +         ω                   μ   0       4            ∫   A   B                J   ~     z   s          [         x   2     +         y   ~     2          (   x   )           ]              H   0     (   2   )            (     k        [     ρ   -     x                 cos                   θ   0       -         y   ~          (   x   )          sin                   θ   0         ]       )              1   +       y       ∼   .     2            (   x   )                              x               ,           (   11   )                                
     where              y   .          (   x   )       =               x            y        (   x   )           ,                          
     θ 0  is the angle of observation, ρ→∞, and [A,B] is the range over x on the surface represented by the symmetric geometry above. From here on, the notation y(x) will be dropped in favor of y. 
     Using this new equation, now we can begin to consolidate the nomenclature. First, incorporate the large argument approximation for the Hankel function                    H   0     (   2   )            (     k        [     ρ   -   X     ]       )       ≅           2      j       k                   π        [     ρ   -   X     ]                  exp        [       -   j                     k        [     ρ   -   X     ]         ]           ,           (   12   )                                
     for ρ→∞, and X=x cos θ 0 +y sin θ 0 . 
     Equation (12) can be further reduced according to                      2      j       k                   π        [     ρ   -   X     ]                  exp        [       -   j                     k        [     ρ   -   X     ]         ]              →   lim                2      j       k                 π                 ρ              exp        [       -   j                   k                 ρ     ]              exp        [     j                 kX     ]       .               (   13   )                                
     Next, write                               E   z   s          (   ρ   )       =       M        [   y   ]       =       k   0            ∫   s                     F   (     x   ,   y   ,         y   )     .                        x       ,                       (   14   )                                
     and for convenience write, 
     
       
           F ( x,y,{dot over (y)} )= J   z   s ( x,y ) A ( x ) B ( x,y ) C ( x,{dot over (y)} ), where  (15) 
       
     
     
       
         
           
             
               k 
               0 
             
             ≡ 
             
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     μ 
                     0 
                   
                 
                 2 
               
                
               
                 
                   
                     2 
                      
                     j 
                   
                   
                     k 
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     ρ 
                   
                 
               
                
               
                 exp 
                  
                 
                   [ 
                   
                     
                       - 
                       j 
                     
                      
                     
                         
                     
                      
                     k 
                      
                     
                         
                     
                      
                     ρ 
                   
                   ] 
                 
               
             
           
         
                 
         
             
         
      
     
     A(x)≡exp[jkx cos θ 0 ] 
     B(x,y)≡exp[jky sin θ 0 ]         C        (     x   ,     y   .       )       ≡       1   +       y   .     2                                
     The optimization can now be more compactly described. Begin by finding 
     
       
           F   y =( J   z   s ) y   ABC+J   z   s   AB   y   C and   (16) 
       
     
     F {dot over (y)}   =J   z   x   ABC   {dot over (y)} , such that                    x            F     y   .         =         ABC     y   .                    x            J   z   s       +       J   z   s          BC     y   .                    x          A     +       J   z   s        A                   C     y   .                    x          B     +       J   z   s        AB                x              C     y   .       .                                
     Now calculate,                    (     J   z   s     )     y     =       y   r          ∂     ∂   r              J   z   s          (   r   )           ,   where           (   17   )                                r={square root over (x 2   +y   2 +L )},  and  (18) 
     
       
           B   y   =jk  sin θ 0  exp[ jky  sin θ 0   ]=jk  sin θ 0   B , such that  (19) 
       
     
     
       
         
           
             
               
                 
                   
                     F 
                     y 
                   
                   = 
                   
                     
                       AB 
                       C 
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               C 
                               2 
                             
                              
                             
                               y 
                               r 
                             
                              
                             
                               ∂ 
                               
                                 ∂ 
                                 r 
                               
                             
                              
                             
                               J 
                               z 
                               s 
                             
                           
                           + 
                           
                             j 
                              
                             
                                 
                             
                              
                             k 
                              
                             
                                 
                             
                              
                             
                               C 
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               
                                 J 
                                 z 
                                 s 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     In a similar fashion, it is straightforward to calculate,                            x            J   z   s       =         x   +     y                   y   .         r          ∂     ∂   r            J   z   s         ,           (   21   )                             x          A     =       jk                 cos                   θ   0          exp        [     j                 kx                 cos                   θ   0       ]         =     jk                 cos                   θ   0        A         ,   and           (   22   )                             x          B     =       jk                   y   .                   sin                   θ   0          exp        [     j                 ky                 sin                   θ   0       ]         =     jk                   y   .                   sin                   θ   0        B         ,   and           (   23   )                   C     y   .       =         y   .         1   +       y   .     2           =       y   .     C         ,     such                 that             (   24   )                               x            C     y   .         =       y   ¨       C   3         ,   and          
                      x            F     y   .         =       AB   C            (         y   .            x   +     y                   y   .         r          ∂     ∂   r            J   z   s       +     jk                   y   .                   cos                   θ   0          J   z   s       +     jk                     y   .     2        sin                   θ   0          J   z   s       +       J   z   s            y   ¨       C   2           )     .                 (   25   )                                
     At this point, the Euler equation can now be calculated as                  F   y     -               x            F     y   .           =       AB   C            (           y   -     x                   y   .         r          ∂     ∂   r            J   z   s       +       J   z   s          [         -     y   ¨         1   +       y   .     2         +     jk                 sin                   θ   0       -     jk                   y   .                   cos                   θ   0         ]         )     .               (   26   )                                
     Finally, the design equation for minimization reduces to                         D        ∂     ∂   r            J   z   s       +       (     jE   +   F     )          J   z   s              →   0     ,       for        
        D     =       y   -     x                   y   .         r       ,   and           (   27   )                                E=k[sin θ   0   −{dot over (y)} cos θ 0 ], and        F   =         -     y   ¨         1   +       y   .     2         .                             
     We may want to choose to allow the aft end of the geometry to have freedom of movement in some cases. To allow this, require [1] 
     
       
           F   {dot over (y)} | A   B   =J   z   s   ABC   {dot over (y)} | A   B =0.  (28) 
       
     
     Using previous calculations, this forces the condition 
     
       
           {dot over (y)} ( A )= {dot over (y)} ( B ).  (29) 
       
     
     The relevance of these assignments will become apparent shortly. 
     It should be apparent that the current (J z   s ) is not a priori information in the MoM calculation. This implementation thus requires some form of iteration. The advantage of this technique versus techniques seeking a similar end is that the optimization relationship is directly between the surface current and shape. As such, optimization may be attained without performing costly impedance matrix calculations for each iteration, so long as the shape solution does not change so radically as to significantly change the initialization current, thus invalidating the solution. The iteration thus requires some control scheme. 
     Construction of a Minimizing Sequence 
     Akhiezer (at p. 143) demonstrates a reasonable method devised by V. Ritz for the construction of a minimizing sequence. This sequence has enjoyed success in a variety of engineering applications [1]. Salient features of what are contained in the text are revisited here. 
     Start again with the functional formula of equation (2), subject to the conditions 
     
       
           y ( A )= a   1   , y ( B )= b   1 .  (30) 
       
     
     Assume that the functional argument, ƒ(x,y,{dot over (y)}), is continuous in all its arguments and assume further that the function can be bounded such that 
     
       
         ƒ( x,y,{dot over (y)} )≧α| {dot over (y)}|   p +β.  (31) 
       
     
     for α&gt;0, β, p&gt;1. It is shown in Akhiezer [1] that these conditions guarantee the existence of a minimizing sequence when combined with a judicious choice of basis functions. Further, and more importantly, this condition guarantees a limit on the bounds of the minimization coefficients. This is extremely significant since no other RCS minimization approach can guarantee that its results can contain the solution to a global minimum considering the infinite possible combinations of series coefficients. 
     The series and basis functions are constructed according to the following conditions: 
     a. φ 0 (A)=a 1 , φ 0 (B)=b 1    
     b. φ k (A)=φ k (B)=0 (k=1,2,3, . . . ) 
     c. φ 0 (x) lies in the region defined by equation (31) with the possible exception of its endpoints 
     d. basis function first derivatives are linearly independent 
     Based on these conditions, the basis functions chosen for this work were                      φ   0          (   x   )       =       a   1     +           b   1     -     a   1         B   -   A            (     x   -   a     )           ,     
        and          
              φ   k          (   x   )       =         (     x   -   A     )     k          (     x   -   B     )              
        for        
          k   &gt;   0.             (   32   )                                
     Not only does this choice of basis functions satisfy conditions a-d above, but the condition of equation (29) is satisfied as well (as a simple examination can show). Now if the series coefficients are chosen such that 
     
       
           y   n(   x )=φ 0 ( x )+ C   1 φ 1 ( x )+ C   2 φ 2 ( x )+ C   3 φ 3 ( x )+ . . . + C   n φ n ( x ),  (33) 
       
     
     then the original function equation (2) is adequately represented by 
     
       
           M[y   n ]=Φ( C   1   ,C   2   ,C   3   , . . . C   n )=Φ( {overscore (C)} ).  (34) 
       
     
     Since this is the case, we can assume that the only valuable solutions after an initial trial, where Φ({overscore (C)})=M, are those that subsequently have a solution, Φ({overscore (C)})≦M. 
     The analysis thus proceeds starting with the reorganization of equation (31) leading to                      ∫   A   B                         φ   .     0          (   x   )       +       ∑     i   =   1     n            C   i              φ   .     i          (   x   )                  p                        x         ≤       M   -     β        (     B   -   A     )         α       =     M   1       ,           (   35   )                                
     and thus                    {       ∫   A   B                     ∑     i   =   1     n            C   i              φ   .     i          (   x   )                p                        x         }       1   /   p       ≤       M   1     1   /   p       +       {       ∫   A   B                       φ   .     0          (   x   )            p                        x         }       1   /   p           =       M   2     .             (   36   )                                
     Now the left side of the above equation can be put into the form                      C   1   2     +     C   2   2     +     C   3   2     +   …   +     C   n   2                {       ∫   A   B                     ∑     i   =   1     n            K   i              φ   .     i          (   x   )                p                        x         }       1   /   p         ,   where           (   37   )                   K   i     =       C   i           C   1   2     +     C   2   2     +     C   3   2     +   …   +     C   n   2             ,     and                   it   &#39;        s                 easy                 to                 see                 that             (   38   )                   ∑     i   =   1     n          K   i   2       =   1.           (   39   )                                
     Because of this final convenient condition, the function                {       ∫   A   B                     ∑     i   =   1     n            K   i              φ   .     i          (   x   )                p                        x         }       1   /   p             (   40   )                                
     is continuous on the unit circle and, according to a Weierstrass theorem, assumes a minimum value of 6 on it. All of this leads to the final significant condition                    C   1   2     +     C   2   2     +     C   3   2     +   …   +     C   n   2         ≤         M   2     δ     .             (   41   )                                
     Ergo, the coefficients used to construct the geometry for RCS minimization have an upper bound on their combined value. 
     For this work, values of p=2 and β=0 were used for the inequality. The coefficient, α, was computed using a Total Least Squares (TLS) technique combined with the computation of the functional integrand and {dot over (y)}. This does not guarantee a bound which will only contain minimization solutions, but rather approximates that bound. 
     The advantage of CoV for the analysis of these problems should be clear by this point. Euler&#39;s equation offers the ability to locate local extrema quickly and accurately. When combined in this fashion to form a minimizing sequence, the analysis affords a broad search of all possible coefficient values to ultimately arrive at the global minimum. What remains is to study the effects of coefficient granularity in the application of these solutions. Studies so far have not shown that a single solution will arise out of every iterative approach. Certainly, genetic algorithms and other acceptable search schemes are applicable to this problem as well. A block diagram of the overall iterative scheme is depicted in FIG.  3 . 
     Example of Cartesian Domain Processing 
     The following example shows results from this technique during a typical run on a model order of 4. Note the boundary boxes in the figures. These are physical constraints placed on the geometry (a “can&#39;t be larger than” box on the outside and a “cannot be smaller than” box on the inside). The derived shape in this example is optimized for a single frequency and angle, where the shape is dictated during optimization by the model order and choice of basis functions. For this particular case, the routine obtains a fairly wide-well solution, but is limited in depth. The theoretical limit (−∞ at a single angle/single frequency) is not obtained, however. This limit would be more easily approached for higher model orders. What is particularly interesting about the final iteration of the shaping approach here is that it does not approach the inner boundary. Often, minimization approaches will tend to iterate closely to the limiting contour, but in this case additional space is provided for the boundary box. 
     Polar Format Design Equations 
     Assume that a two-dimensional geometry may be defined in the polar domain by r(θ) for θ=0 to θ max . Assume further that we desire the geometry to be symmetric such that {{tilde over (r)}(θ)ε[2 π,−θ max ]}={r(θ)ε[0,θ max ]}, where {tilde over (r)} represents the symmetric side. With this assumption in mind, we can rewrite (1) as                    E   z   s          (   ρ   )       =         ω                   μ   0       4            ∫   s              J   z   s          [     r        (   θ   )       ]              H   0     (   2   )            (     k        [     ρ   -       r        (   θ   )            cos        (     θ   -     θ   0       )           ]       )                  r   2          (   θ   )       +         r   .     2          (   θ   )                              θ             ,           (   42   )                                
     where              r   .          (   θ   )       =               θ            r        (   θ   )           ,                          
     θ 0  is the angle of observation, ρ→∞, and S is the surface represented by the symmetric geometry above. From here on, the notation r(θ) will be dropped for simply r. Now, Euler&#39;s equation becomes                  F   r     -               θ            F     r   .           =   0.           (   43   )                                
     First, incorporate the large argument approximation for the Hankel function                  H   0     (   2   )            (     k        [     ρ   -     r                   cos        (     θ   -     θ   0       )           ]       )       =           2      j       k                   π        [     ρ   -     r                   cos        (     θ   -     θ   0       )           ]                  exp        [       -   j                     k        [     ρ   -     r                   cos        (     θ   -     θ   0       )           ]         ]                 (   44   )                                
     for ρ→∞. 
     Next, write 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         E 
                         z 
                         s 
                       
                        
                       
                         ( 
                         ρ 
                         ) 
                       
                     
                     = 
                     
                       
                         M 
                          
                         
                           [ 
                           r 
                           ] 
                         
                       
                       = 
                       
                         
                           k 
                           0 
                         
                          
                         
                           
                             ∫ 
                             s 
                           
                            
                           
                               
                           
                            
                           
                             F 
                             ( 
                             
                               θ 
                               , 
                               r 
                               , 
                               
                                 
                                   
                                     r 
                                     ) 
                                   
                                   . 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   θ 
                                 
                               
                               , 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     and for convenience write, 
     
       
           F (θ, r,{dot over (r)} )= J   z   s   A (θ, r ) B (θ, r,{dot over (r)} ),where  (46) 
       
     
     
       
         
           
             
               k 
               0 
             
             ≡ 
             
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     μ 
                     0 
                   
                 
                 4 
               
                
               
                 
                   
                     2 
                      
                     j 
                   
                   
                     k 
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     ρ 
                   
                 
               
                
               
                 exp 
                  
                 
                   [ 
                   
                     
                       - 
                       j 
                     
                      
                     
                         
                     
                      
                     k 
                      
                     
                         
                     
                      
                     ρ 
                   
                   ] 
                 
               
             
           
         
                 
         
             
         
      
     
     A(θ,r)≡exp[jkr cos(θ−θ 0 )]         B        (     θ   ,   r   ,     r   .       )       ≡         r   2     +       r   .     2                                
     The optimization can now be more compactly described. Begin by finding 
     
       
           F   r =( J   z   s ) r   AB+J   z   s   A   r   B+J   z   s   AB   r , and  (47) 
       
     
     F {dot over (r)} =J z   s AB {dot over (r)} , such that                    θ            F     r   .         =         AB     r   .                    θ            J   z   s       +       J   z   s          B     r   .                    θ          A     +       J   z   s        A                θ              B     r   .       .                                
     Now one can calculate, 
     
       
           A   r   =jk  cos(θ−θ 0  ) A , and  (48) 
       
     
     
       
         
           
             
               
                 
                   
                     
                       B 
                       r 
                     
                     = 
                     
                       
                         r 
                         
                           
                             
                               r 
                               2 
                             
                             + 
                             
                               
                                 r 
                                 . 
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         r 
                         B 
                       
                     
                   
                   , 
                   
                     such 
                      
                     
                         
                     
                      
                     that 
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     r 
                   
                   = 
                   
                     
                       A 
                       B 
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               B 
                               2 
                             
                              
                             
                               ∂ 
                               
                                 ∂ 
                                 r 
                               
                             
                              
                             
                               J 
                               z 
                               s 
                             
                           
                           + 
                           
                             jk 
                              
                             
                                 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   θ 
                                   - 
                                   
                                     θ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               B 
                               2 
                             
                              
                             
                               J 
                               z 
                               s 
                             
                           
                           + 
                           
                             rJ 
                             z 
                             s 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     In a similar fashion, it is straightforward to calculate,                            θ            J   z   s       =       r   .          ∂     ∂   r            J   z   s         ,           (   51   )                             θ          A     =       jk        [         r   .                     cos        (     θ   -     θ   0       )         -     r                   sin        (     θ   -     θ   0       )           ]          A       ,   and           (   52   )                   B     r   .       =         r   .           r   2     +       r   .     2           =       r   .     B         ,     such                 that             (   53   )                               θ            B     r   .         =           r   ¨                     r   2       -     r                     r   .     2           B   3         ,   and          
                    θ            F     r   .         =       A   B            (           r   .     2          ∂     ∂   r            J   z   s       +       jk        [       r        r   .          cos        (     θ   -     θ   0       )         -         r   .     2          sin        (     θ   -     θ   0       )           ]            J   z   s       +             r   ¨                     r   2       -     r                     r   .     2           B   2            J   z   s         )     .               (   54   )                                
     At this point, the Euler equation can now be calculated as                  F   r     -               θ            F     r   .           =       A   B          (         r   2          ∂     ∂   r            J   z   s       +       jk        [         (       r   2     +       r   .     2     -     r                   r   .         )          cos        (     θ   -     θ   0       )         +         r   .     2          sin        (     θ   -     θ   0       )           ]            J   z   s       +           r   3     +     2      r                     r   .     2       -       r   ¨                     r   2             r   2     +       r   .     2              J   z   s         )               (   55   )                                
     Finally, the design equation for minimization reduces to                     r   2          ∂     ∂   r            J   z   s       +     jDJ   z   s     +     EJ   z   s            →   0     ,   for                          D=k [( r   2   +{dot over (r)}   2   −r{dot over (r)} )cos(θ−θ 0 )+ {dot over (r)}   2  sin(θ−θ 0 )], and  (56) 
     
       
         
           
             E 
             = 
             
               
                 
                   
                     r 
                     3 
                   
                   + 
                   
                     2 
                      
                     r 
                      
                     
                         
                     
                      
                     
                       
                         r 
                         . 
                       
                       2 
                     
                   
                   - 
                   
                     
                       r 
                       ¨ 
                     
                      
                     
                         
                     
                      
                     
                       r 
                       2 
                     
                   
                 
                 
                   
                     r 
                     2 
                   
                   + 
                   
                     
                       r 
                       . 
                     
                     2 
                   
                 
               
               . 
             
           
         
                 
         
             
         
      
     
     Example of Polar Domain Processing 
     The following example shows results from this technique during an ideal run. In general, the polar domain processing approach was far more sensitive in its ability to arrive at a successful result. The derived shape in this example is optimized for a single frequency and angle, which explains the awkward appearance. Essentially, the routine is attempting to develop competing scatterers on the fore and aft of the target thereby causing cancellation. For this successful run, the theoretical limit of −∞ is approached at 0° (off the fore end of the structure). 
     Extensibility of the Technique 
     In order to apply this technique to three dimensions, a modified version of the Euler equation may be used in two dimensions. In effect, the Euler equation of equation (3) is expanded according to                    f   u     -       ∂     ∂   x            f     u   x         -       ∂     ∂   y            f     u   y           =   0     ,           (   57   )                                
     for a integral equation defined according to                  M        [   u   ]       =       ∫   D          ∫       f        (     x   ,   y   ,   u   ,     u   x     ,     u   y       )               x             y             ,           (   58   )                                
     where x and y are the variates. The method is extensible to an arbitrary number of independent variables. The minimization that would arise from this equation would be directly analogous to its two dimensional counterpart. 
     It is also desirable to perform the optimization over a broad range of angles and frequencies in some cases. This is performed by creating another set of optimization equations at selected angles and frequencies of observation. The minimization according to equations (27) and (56) is then accomplished for each of those selected angles and frequencies (e.g., angles could be every 1° along the well or at 5 strategic locations throughout the well, frequencies could be similarly chosen). To accomplish this correctly, the user must remember that the surface current will change with observation angle and frequency as well, in effect making 
     
       
           J   z   x   ≡J   z   x ( r,k,θ   0 ).  (59) 
       
     
     Some costing function scheme would have to be applied to cause a successful minimization. The total number of equations that would have to be minimized in a three dimensional optimization would be identical to the number of equations requiring minimization in two dimensions {(#frequencies)×(#angles)}. 
     REFERENCES 
     1. Akheizer, Naum I (translation from the Russian by Aline H. Frink).  Calculus of Variations , Blaisdell Publishing Company, New York/London, 1962. 
     2. Balanis, Constantine A.  Advanced Engineering Electromagnetics , John Wiley &amp; Sons, New York, 1989. 
     3. Skinner, Dr Paul. “AFIT Notes from Course #EE630 Part II”, November 1991.