Patent Publication Number: US-2005116107-A1

Title: Area ruling for vertical stabilizers

Description:
BACKGROUND OF THE INVENTION  
      The area rule is an important concept related to the drag on an aircraft or other body in transonic and supersonic flight. The area rule was developed in the early 1950s when production fighter designs began pushing ever closer to the sound barrier. Designers had found that the drag on these aircraft increased substantially when the planes traveled near Mach 1, a phenomenon known as the transonic drag rise. The increase in drag is due to the formation of shock waves over portions of the vehicle, which typically begins around Mach 0.8. The drag increase reaches a maximum near Mach 1. Because the drag results from the shock waves, this type of drag is referred to as wave drag.  
      Since the physics of supersonic flight were still largely a mystery to manufacturers, designers had no idea how to address this problem except to provide their aircraft with more powerful engines. Even though jet engine technology was rapidly advancing in those days, the first generation of jet-powered fighters was hampered by relatively low-thrust engines which limited them to subsonic flight. The US Air Force hoped to overcome this deficiency with its first dedicated supersonic fighter, the F-102 Delta Dagger.  
      Since the transonic drag rise was still not fully understood, the F-102&#39;s designers chose an engine they believed would provide enough thrust to reach a maximum speed of about Mach 1.2. However, initial flight tests of the YF-102 prototype indicated that the aircraft could not reach Mach 1. The aircraft&#39;s designers did not understand this lack of performance until a NACA researcher named Dr. Richard Whitcomb developed the area rule.  
      Whitcomb experimented with several different axisymmetric bodies and wing-body combinations in a transonic wind-tunnel. He discovered that the drag created on these shapes was directly related to the change in cross-sectional area of the vehicle from the nose to the tail. The shape itself was not as critical in the creation of drag, but the rate of change in that shape had the most significant effect. Stated differently, the wave drag is related to the second-derivative (or curvature) of the volume distribution of the vehicle.  
      To illustrate the point, Whitcomb&#39;s experimental models included (1) a simple cylindrical fuselage, (2) a cylindrical fuselage with wings attached, (3) a bulged fuselage, and (4) a “wasp-waisted” fuselage with wings. The addition of wings to the basic cylinder produced twice as much drag as the cylinder alone. Additionally, the drag rose by the same amount if a simple bulge were added to the cylinder, the bulge having equivalent volume to the wings. However, if the cross-sectional area of the fuselage was reduced over the region where the wings were attached, the total drag was about the same as that of the cylinder alone.  
      The conclusion of this research was that the drag on an aircraft could be drastically reduced by shaping the vehicle to create a smooth cross-sectional area distribution from the nose to the tail. The area rule dictates that the volume of the body should be reduced in the presence of a wing, tail surface, or other projection so that there are no discontinuities in the cross-sectional area distribution of the vehicle shape.  
      Whitcomb&#39;s findings are related to a more theoretical concept called the Sears-Haack body. This shape yields the lowest possible wave drag for a given length and volume. The variation in cross-sectional area for a Sears-Haack body results in minimized wave drag when the curvature of the volume distribution is minimized. The closer the volume distribution of an aircraft or other high-speed vehicle comes to the ideal Sears-Haack body, the lower its wave drag will be.  
      The area rule has found greater application to subsonic aircraft, particularly commercial airliners since they cruise at the lower end of the transonic regime. A good example is the Boeing 747, known for its distinctive “hump.” This hump, which houses the cockpit and upper passenger deck, increases the cross-sectional area of the forward fuselage and has the effect of evening the volume distribution over the length of the aircraft. As a result, the 747 is able to cruise efficiently at a slightly higher speed than most other airliners since the increase in transonic wave drag is delayed. It is very difficult, however, to achieve realistic vehicle configurations that completely meet the ideal Sears-Haack body.  
     SUMMARY OF THE INVENTION  
      The goal is to find new ways to apply the area rule, which is a method to achieve vehicle configurations that meet the ideal Sears-Haack minimum wave drag body for a given mach number, to the design of a vertical tail of a supersonic configuration.  
      In accordance with some embodiments, a vertical stabilizer is configured to minimize the rate of change of cross-sectional area of a vehicle or device to which the vertical stabilizer is mounted. One or more “waisted” areas can be included at the tip and/or the root of the vertical stabilizer. In some embodiments, a strake is mounted on the vehicle or device, such as an aircraft, and the vertical stabilizer is mounted to the tip of the strake. The strake can also be area ruled with one or more “waisted” sections at the juncture of the vertical stabilizer. The strake may also be called dorsal.  
      According to other embodiments, an aircraft design system comprises logic instructions operable to apply area ruling theory to the tail section of the aircraft, including configuring a vertical fin with at least one “waisted” area to minimize the overall rate of change of cross-sectional area of the aircraft.  
      In some embodiments, a number of vertical stabilizer configurations can be optimized for minimum drag at different Mach numbers. The configurations can be weighted according to selected criteria, and the average of the weighted configurations can be taken as a final configuration. The weighting factors can be based on any suitable constraint(s) or criteria. For example, for an aircraft, the weighting factors can be based on percent of typical flight profile spent at each Mach number, minimum drag, the difference between thrust available and thrust required (referred to as pinch-point), and/or minimum sonic boom. Applying area ruling to the vertical stabilizer helps to further reduce the wave drag of the vehicle or device to operate efficiently over expected operating conditions, low boom climb and cruise, high speed cruise, and landing of an aircraft. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      Embodiments of the invention relating to both structure and method of operation, may best be understood by referring to the following description and accompanying drawings.  
       FIG. 1A  is a top view of an embodiment of a rear portion of an aircraft including an area ruled vertical fin extending from a strake.  
       FIG. 1B  shows the difference in cross-sectional area for components of an example of an aircraft body versus the ideal Sears-Haack conceptual body shape for minimizing wave drag.  
       FIG. 1C  is a top view of the tip of the area ruled vertical fin of  FIG. 1A  coupled to the inverted V-tail.  
       FIG. 1D  is a top view of a cross-section of the root of the vertical fin of  FIG. 1A  mounted at the tip of the strake of  FIG. 1A .  
       FIG. 1E  is a side view of the rear portion of the aircraft with the vertical fin and strake of  FIG. 1A .  
       FIG. 1F  is a graph showing thickness profiles for an embodiment of the vertical fin and dorsal fuselage portion of  FIG. 1A .  
       FIG. 2A  is a flow diagram of an embodiment of a method for configuring the area rule shape for the vertical fin and strake of  FIG. 1A .  
       FIG. 2B  is a diagram of the vertical fin reflected above the inverted V-tail of  FIG. 1A .  
       FIGS. 3A through 3C  are side, front, and top views, respectively, of an embodiment of an aircraft that can utilize the area ruled vertical fin and dorsal fuselage portion of  FIG. 1A .  
       FIG. 4  is a block diagram of an embodiment of a design system for configuring the area rule shape for the vertical fin and dorsal fuselage portion of  FIG. 1A . 
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS  
      Referring to  FIGS. 1A through 1D ,  FIG. 1A  is a top view of an embodiment of a rear portion of aircraft  100  including vertical fin  102  extending from strake  104 , rear fuselage portion  106 , inverted V-tail  108 , and wings  110 . While both the tip and root of vertical fin  102  are shown shaped according to area ruling concepts, in some embodiments, either the tip or root, but not both, of vertical fin  102  can be area ruled. Additionally, the area ruled shape of vertical fin  102  and strake  104  can include more than one “wasp-waist,” or reduced cross-sectional areas. Applying area ruling to aircraft  100 , including vertical fin  102  and strake  104 , helps reduce wave drag by reducing the rate of change in the aircraft&#39;s overall cross-sectional area, and minimizing the maximum cross-sectional area of the aircraft. Additionally, reducing the volume in the fuselage  106  in the vicinity of vertical fin  102 , strake  104 , inverted V-tail  108 , and wings  110  facilitates application of area ruling as well as reducing sonic boom disturbance.  
      To reduce the wave drag due to change in cross-sectional area, the area must follow a smooth bell curve shape known as a Sears-Haack curve  112  shown in  FIG. 1B .  FIG. 1B  also shows the difference in cross-sectional area for components of an example of an aircraft body versus the Sears-Haack conceptual body shape. The closer the cross-sectional area distribution of an aircraft comes to the ideal Sears-Haack curve  112 , the lower the wave drag. A smooth overall vehicle cross-sectional area in the vicinity of wing  110 , vertical fin  102  and inverted V-tail  108  can be more significant in reducing wave drag than a smooth fuselage area.  
       FIG. 1C  is a top view of the tip of vertical fin  102  of  FIG. 1A  coupled to inverted V-tail  108 . The area ruling of the tip of vertical fin  102  reduces the cross-sectional area of vertical fin  102  in the vicinity of inverted V-tail  108 , as well as the rate of change of the cross-sectional area. In some embodiments, the area ruling is only applied to vertical fin  102  at the juncture of inverted V-tail  108 . In other embodiments, the area ruling can extend all or part of the distance between the root and the tip of vertical fin  102 .  
       FIG. 1D  is a top view of the root of vertical fin  102  at a juncture with the tip of strake  104  of  FIG. 1A .  FIG. 1E  is a side view of vertical fin  102  and strake  104  of  FIG. 1A  with respect to fuselage portion  106 . Strake  104  includes an upper portion above fuselage  106  that is exposed to the airstream, and a lower portion that is mounted within fuselage  106  that carries loads from vertical fin  102  and inverted V-tail  108 . Strake  104  improves aircraft fuselage stiffness, enables a desired reduction in fuselage volume, and compensates for any reduction in stiffness that results from the reduced area fuselage.  
      In some embodiments, both the root of vertical fin  102  and the upper portion of strake  104  are shaped to reduce the cross-sectional area of aircraft  100 , as well as the rate of change of the cross-sectional area. In other embodiments, either vertical fin  102  or the upper portion of strake  104  is configured to help aircraft  100  achieve a shape that corresponds as closely to Sears-Haack curve  112  ( FIG. 1B ) as possible.  
      Referring to  FIGS. 1A and 1F ,  FIG. 1F  is a graph of thickness profiles for embodiments of tip and root cross-sections of vertical fin  102 , strake  104 , and the root of V-tail  108  at the juncture with the tip of vertical fin  102 , along stations of fuselage  106 . The thickness profile of the root of vertical fin  102  corresponds approximately to the shape of the tip of strake  104 . Note that the thickness profiles represent the thickness of the entire component.  
       FIG. 2A  is a flow diagram of an embodiment of a method for configuring the area rule shape for an aircraft including vertical fin  102  and/or strake  104  of  FIG. 1A . Process  202  includes determining design variables (DVs) to use to optimize a specified figure of merit, such as minimizing wave drag and/or sonic boom disturbance. Each design variable can be a single value, or a range of values for one or more parameters.  
       FIGS. 1D and 1E  show an example of locations of design variables DV 1 -DV 7  that can be used to determine an optimum configuration for vertical fin  102  and strake  104 . In some embodiments, design variables DV 1 -DV 7  are utilized to perform area rule optimization on the shape of vertical fin  102  and/or strake  104 . The optimization can be performed on individual components, or combinations of vertical fin  102 , strake  104 , fuselage  106 , inverted V-tail  108 , and/or wing  110 . Additional design variables can be used in certain analysis methods, such as computational fluid dynamics (CFD), to simultaneously determine optimum configurations for a combination of vertical fin  102 , strake  104 , fuselage  106 , inverted V-tail  108 , and wing  110 . Further, any suitable number and location of design variables can be used to configure vertical fin  102 , strake  104 , and inverted V-tail  108 , depending on the geometry of a specific configuration. In some instances, constraints are specified to maintain maximum possible thickness of vertical fin  102  for structural support considerations while still achieving minimum drag and zero closure angle at the trailing edge of vertical fin  102 .  
      The design variables are typically optimized along Mach angle lines  120 , which vary according to the Mach number. Thus, an optimum configuration for the components such as vertical fin  102 , strake  104 , and inverted V-tail  108  can vary between Mach numbers.  
      To shape vertical fin  102 , strake  104 , and inverted V-tail  108  for minimum drag, parameters such as thickness, height, and width can be used to minimize cross-sectional area, as well as the change (first derivative) of cross-sectional area, and rate of change (second derivative) of cross-sectional area. A combination of thicknesses and n-order derivatives can also be used as design variables in orthogonal functions such as Fourier series and Legendre polynomial functions, as well as non-orthogonal functions such as Taylor series. Use of first and second order derivatives as design variables typically achieves smoother shapes than cross-sectional area and thickness alone.  
      As an example, the following Table 1 shows examples of first derivative variables and equations that can be specified for the design variables:  
                           TABLE 1                           Axial   First Derivative           Design Variable   Location   of thickness   thickness                  DV1   X 1     t&#39; 1     0       DV2   X 2     t&#39; 2     t 1  = c 1 *t&#39; 1 *(X 2  − X 1 )       . . .   . . .   . . .   . . .       DV(n − 1)   X n−1     t&#39; 1     t n−1  = t n−2  + c n−1 * t&#39; n−1 *(X n−1 -X n−2 )       DVN   X n     t&#39; 1     t n  = t n−1  + c n * t&#39; n * (X n-X   n−1 )                  
 
 The coefficients c n  through c n  can be adjusted to match the maximum thickness constraints as well as achieve trailing edge thickness, t n , of zero. 
 
      Referring to  FIGS. 2A-2B , process  204  includes adjusting various parameters and constraints to determine optimum configurations of vertical fin  102 , strake  104 , and inverted V-tail  108  for minimum drag at different Mach numbers. A single configuration typically will not be optimal for all flight conditions. In some situations, a designer is able to segregate portions of an aircraft&#39;s flight regime into distinct phases during which the aircraft can be reconfigured to achieve an optimum result for a given flight segment. For example, while it is desirable to reduce sonic boom disturbances over land, louder sonic booms can be tolerated over an ocean or large unpopulated land mass. Sonic boom constraints are easier to meet at slower speed, such as 1.5 Mach, versus 1.8 Mach for supersonic cruise.  
      Accordingly, a designer can determine a first configuration that meets sonic boom constraints during climb and cruise flight over land, another configuration that optimizes range during cruise flight over unpopulated areas with relaxed sonic boom constraints, as well as other configurations that optimize specified parameters during a particular flight segment at different Mach numbers. Process  204  can also be used to help determine optimum configurations for takeoff, climb, and landing modes, based on the constraints to be met for each flight mode. The design variables are typically selected along Mach angle lines  302 , which vary according to the Mach number. Thus, an optimum configuration for aircraft  300  can vary between Mach numbers.  
      In some embodiments, process  204  utilizes linear theory techniques to analyze shapes that minimize the first and second rates of change of the cross-sectional area, thickness, to optimize the configuration of aircraft  205  for low drag. Another consideration in determining the shape of vertical fin  102  is that structural constraints generally require a minimum thickness between the sides of vertical fin  102 , typically over a specified percentage of the distance from the leading edge to the trailing edge of the vertical fin  102 . The term “chord”, denoted by “c”, refers to a straight line from the leading edge to the trailing edge of an airfoil. The structural requirements can be specified as a minimum closure angle θ at the trailing edge of vertical fin  102  that will result in the desired thickness d at the specified distance along chord c, such as 80 percent of the chord (0.8c).  
      An example of a suitable linear theory software program than can be used in process  204  is AWAVE, which was developed by NASA Langley. AWAVE determines fuselage cross sections which yield minimum wave drag by enforcing the supersonic area rule. Three-dimensional aircraft component definitions, such as wing, fuselage, nacelle and empennage are input into the program. Then, the far-field wave drag of the configuration is analyzed based on the equivalent bodies of those components, producing an optimum fuselage cross section distribution along its axis for design Mach numbers larger than 1.0. Once candidate configurations are identified, more complex non-linear computational fluid dynamic methods can be employed to further refine the configurations.  
      Process  204  can also utilize the Method of Reflections of linear fluid flow theory to analyze the drag characteristics of inverted V-tail  108  and vertical fin  102 . In applying the Method of Reflections to aircraft  205 , a virtual image of vertical fin  102  is “reflected” on the other side of inverted V-tail  108  as shown in  FIG. 2B  by reflected fin  102 ′. The Method of Reflections is used to model flow trapped in a channel formed between vertical fin  102  and inverted V-tail  108  by simulating a symmetry boundary condition in which a degree of freedom is removed from the flowfield. In some embodiments, the aircraft configuration is optimized for Mach numbers that are close to transonic conditions in order to avoid choking the flow in the channel formed between vertical fin  102  and inverted V-tail  108 .  
      With respect to area ruling, an optimum shape at one near-transonic Mach number generally will have more “waist” than shapes optimized for a higher Mach number. Process  206  includes weighting each configuration with a factor, summing the weighted values, and taking the average of the weighted values to configure aircraft  205  to operate efficiently over expected flight conditions, such as takeoff, climb, low boom climb and cruise, high speed cruise, and landing. The weighting factors can be based on any suitable constraint(s) or criteria, such as percent of typical flight profile, the difference between thrust available and thrust required (referred to as pinch-point), minimum drag, or minimum sonic boom, for example. In one embodiment, configurations are determined at Mach 1.2, 1.3, and 1.8. The configurations are weighted ½ on Mach 1.3, and ¼ on Mach 1.2 and 1.8. Mach numbers that are closer to transonic conditions can be weighted more heavily to avoid choking the flow in the channel formed between vertical fin  102  and inverted V-tail  108 . Configurations at other suitable Mach numbers and combination of weighting factors can be utilized.  
      The performance of the Mach-weighted configuration resulting from process  206  is analyzed in process  208 . In the early stages of a new aircraft design, several configurations may be analyzed to determine the best tradeoff between often-competing goals. Thus, in some embodiments, process  208  can perform linear analysis to evaluate multiple candidates. In the later design stages, however, process  208  can utilize non-linear techniques, which are computationally much more intensive, but provide more accurate results than linear techniques.  
      Referring to  FIGS. 3A through 3C , an embodiment of aircraft  300  shown is configured with area ruled vertical fin  102  and strake  104  of  FIG. 1A  for reduced drag and improved performance characteristics. Wings  110  includes gull dihedral portions  308  with inboard dihedral primarily at the trailing edges while maintaining straight inboard wing sections forward of the trailing edge dihedral. Gull dihedral portions  308  are configured to extend lifting length and reduce sonic boom effects. Gull dihedral portions  308  can be configured by adjusting the twist and camber of wings  110  to produce low-boom and low induced drag.  
      In some examples, aircraft  300  has twin non-afterburning turbofan engines (not shown) set below and behind wings  110 . The non-afterburning turbofan engines operate behind fixed-geometry axisymmetric external compression inlets. Nacelles  312  enclose the engines and are coupled to the lower surface of wings  110 .  
      Inverted V-tail  108  generates additional aft lift to improve aerodynamics and reduce sonic boom disturbances during cruise flight conditions, as well as providing structural support for wings  110  and engine nacelles  312 . Supersonic aircraft  300  creates a shaped pressure wave caused by overpressure at nose  304  and underpressure at tail  306 . Pressure rises rapidly at nose  304 , declines to an underpressure condition at tail  306 , and then returns to ambient pressure.  
      The shape of supersonic aircraft  300 , including aspects of wing  110 , V-tail  108 , and a reflexed airfoil portion (not shown) integrated with engine nacelle  312 , are adapted according to sonic boom signature and supersonic cruise drag considerations. In some embodiments, a portion of nose  304  can be blunted to create a pressure spike ahead of the aircraft forward shock, which raises local temperature and sound velocity, thereby extending the forward shock and slowing the pressure rise. Wings  110  can have a sharply swept delta configuration that reduces peak overpressure by spreading wing lift along the length of aircraft  300 . Wings  110  can also include reduced leading and trailing edge sweeps. Inverted V-tail control surfaces, termed ruddervators  324 , adjust aircraft longitudinal lift distribution throughout the flight envelope to maintain a low boom, low drag trim condition.  
      In some embodiments, the leading edges of wings  110  have a substantially straight geometry to accommodate Krueger flaps  334  that extend substantially the full length of wings  110 . Wings  110  integrate with nacelles  312  and diverters  336 , and follow the fuselage contour with a substantially normal intersection to reduce interference drag. An inboard wing flap hinge line is fully contained within the wing contour with upper and lower wing surfaces configured as planar as possible to facilitate aerodynamic performance.  
      Gull dihedral portions  308  raise the engines to increase available tip-back angle and reduce thrust-induced pitching moments. Gull dihedral portions  312  lower fuselage  106  to reduce the cabin door height above the ground and reduce entry stair length. Fuselage  106  sets a low aircraft center of gravity, reducing tip-over angle and promoting ground stability. Gull dihedral portions  308  form partial shrouds around nacelles  312 , enhancing favorable interaction between nacelles  312  and wings  110 . In addition, gull dihedral portions  308  enhance the aircraft low-boom signature by vertically staggering the longitudinal lift distribution of wings  110 .  
      In some embodiments, supersonic aircraft  300  can include canards  306  on both sides of fuselage  106  that operate primarily as longitudinal power control devices, particularly during takeoff and high-speed flight. Canards  306  also fine tune the aircraft longitudinal trim condition and augment rudder operation by supplying yaw control power when canards  306  are deflected differentially.  
      Supersonic aircraft  300  further includes segmented ailerons  338  that supply roll control power and automatic wing camber control to improve lift and drag conditions through the flight envelope. High-speed spoilers  340  supplement aileron roll power at transonic and supersonic speeds where Mach and aeroelastic effects reduce aileron effectiveness. Supersonic aircraft  300  has a high lift system including inboard trailing edge flaps  342  and full-wingspan leading edge Krueger flaps  334 .  
      Canards  306  and symmetric defections of the ruddervators  324  control pitch power. Rudder  350  controls yaw. Inboard, mid-board and outboard ailerons  348 , and high speed spoilers  340  control roll at both subsonic and supersonic speeds. In an illustrative embodiment, trailing edge (TE) flaps  342  are deployed 30° down to generate additional lift during landing. TE flap deployment reduces angle-of-attack by approximately 2° during landing. During a subsequent segment of the climb, the TE flaps  332  can be extended up to 40° to improve the lift-to-drag ratio for better performance.  
      Leading edge (LE) Krueger flaps  334  can be extended up to 130° for low speed operations including takeoff, approach and landing. The LE Krueger flaps  334  can improve lift-to-drag ratio by 1.5, resulting in better climb performance that facilitates second-segment climb in case of engine malfunction.  
      Some of the control surfaces on supersonic aircraft  300  can perform multiple functions. For example, ruddervators  332  enable continued operation and landing following single actuator failure or jammed control surface. Differential deflection of canards  306  can generate a yawing moment to counter a jammed rudder  350 . Ailerons  338  and ruddervators  324  include multiple surfaces, increasing fault tolerant capability and supplying redundant control elements for improved reliability.  
      Referring to  FIGS. 2 and 4 ,  FIG. 4  is a block diagram of an embodiment of an aircraft design system  450  for configuring the area rule shape for vertical fin  102  and/or strake  104  of  FIG. 1A . Aircraft design system  450  includes logic instructions that implement at least some of functions of processes  202  through  208  in  FIG. 2A .  
      Logic instructions are executed in computer processor  452  for various functions such as input all aspect, multi-point goals and constraints instructions  454 ; constraint comparison instructions  456 ; convergence check instructions  458 ; optimization method  460 ; set design variables instructions; and analysis model  464 . Note that in other embodiments, additional sets of instructions can be performed by computer processor  452  in addition to, or instead of, instructions shown in  FIG. 4 .  
      Input all aspect, multi-point goals and constraints instructions  454  receives input from users regarding the different flight modes at which the aircraft configuration will be optimized. For example, modes such as takeoff, climb, low sonic boom mode at Mach 1.5, and high speed cruise mode at Mach 1.8 can be entered for analysis. The results of system  450  can include different settings for control surfaces and other reconfigurable aspects of the aircraft, as well as optimize the shape of fixed components such as wing camber; fuselage shape; wing, strake, and vertical fin shape; and engine nacelle position and shape, among others. In addition to various flight modes, the user may also enter different aspect angles at which to analyze and optimize the aircraft&#39;s configuration. For example, while a particular aircraft configuration may meet low sonic boom goals directly under the aircraft during flight, the sonic boom disturbance generated to the sides of the aircraft may be objectionable. Therefore, the user can enter different flight modes as well as the distance/angle from centerline at which the sonic boom disturbance will be analyzed to determine the aircraft&#39;s configuration during various flight modes.  
      Constraint comparison instructions  456  receives input from all aspect, multi-point goals and constraints instructions  254  and analysis model  464 . The geometry and performance of one or more configurations is compared to the constraints, such as lift to drag ratio, sonic boom, and geometry constraints. Convergence check instructions  458  receive the constraint comparisons and determine whether a solution has converged to meet all of the desired constraints. If all the constraints have not been met, optimization method  460  is invoked.  
      Optimization method  460  proceeds through iterative changes to the geometry and perturbations that, for example, optimize a defined cost function (e.g. drag) while satisfying geometric and aerodynamic constraints (e.g. fuel volume, cruise lift coefficient). Configurations can be optimized for single or multiple conditions. Target pressure distributions can be specified for inverse design, or as a constraint during optimization of other performance drivers. Any suitable optimization method can be utilized, including one or a combination of methods.  
      Set design variables instructions  462  allow the user to specify locations on the aircraft&#39;s geometry, parameters at each location, and perturbation values for each parameter. The perturbations values can be single values, or a range of values, for each parameter.  
      Analysis model  464  performs linear and/or computational fluid analysis (CFD) on candidate configurations and provides performance results to constraint comparison instructions  456 . Any suitable analysis program or set of programs can be utilized. The complete iterative design process for performance optimization, geometry shaping and CFD analysis may be automated.  
      The logic instructions can be implemented as software programs and distributed over an information network or suitable computer-readable media as a software application program that can be installed on a personal computer, a centralized server, or other suitable computer system. The logic instructions can also be implemented in hardware, firmware, and/or a combination of hardware, firmware and software. One or more user input devices  466  can be provided, such as a keyboard, mouse, light pen, or a component such as a disk drive that can read data input files from a disk, to enable a designer to enter suitable constraints and design parameters. One or more output devices  468  such as a display device, printer, plotter, or other suitable output device can be coupled to receive information from processor  452 . A user interface can also be included that provides instructions for using system  450 , possible design parameters that can be varied, as well as assistance in interpreting the results. The results can be formatted and output for use in other design systems, such as computer-aided design and drawing systems via network interface  470 , to easily share the results of the design effort. Processor  452  can be configured to access a database  472  either directly or via network interface  470  for mass data storage and retrieval.  
      In addition to providing systems and methods for configuring aircraft with area ruling to minimize drag, system  450  and processes  202  through  208  can be adapted to apply area ruling to the shape of any type of apparatus that includes surfaces subject to some sort of flow. For example, aquatic vehicles used on the surface and/or underneath water can be analyzed and configured to minimize cross-sectional area as well as first and second order rates of change in cross sectional area to reduce wave drag. Additionally, embodiments of the systems and methods disclosed herein can be configured to apply to T-tails as well as inverted V-tails on aircraft and other vehicles.  
      While the present disclosure describes various embodiments, these embodiments are to be understood as illustrative and do not limit the claim scope. Many variations, modifications, additions and improvements of the described embodiments are possible. For example, those having ordinary skill in the art will readily implement the processes necessary to provide the structures and methods disclosed herein. Variations and modifications of the embodiments disclosed herein may also be made while remaining within the scope of the following claims. The functionality and combinations of functionality of the individual modules can be any appropriate functionality. In the claims, unless otherwise indicated the article “a” is to refer to “one or more than one”.