Method for designing the body shape of a supersonic aircraft, program for designing the body shape of a supersonic aircraft, recording medium having recorded thereon the program for designing the body shape of a supersonic aircraft, and device for designing the body shape of a supersonic aircraft

Abstract

To provide a method for designing an airframe shape of a supersonic flying body that can take off-track boom into consideration on the basis of a theoretical basis.SOLUTION: A method for designing an airframe shape of a supersonic flying body comprises: a sinogram creation step S20 of creating a sinogram defined as a result of performing a Radon transform on singular points representing an airframe of a supersonic flying body; and a shape reconstruction step S30 of reconstructing the airframe shape from the sinogram. The sinogram is defined using a function that satisfies the Helgason-Ludwig consistency condition for a non-circular region. The sinogram is obtained from physical quantities defined in a cylindrical coordinate system coaxial with the airframe.SELECTED DRAWING: Figure 3

Description

[Technical Field]

[0001] The present invention relates to a method for designing the body shape of a supersonic flight vehicle, a program for designing the body shape of a supersonic flight vehicle, a recording medium on which a program for designing the body shape of a supersonic flight vehicle is recorded, and an apparatus for designing the body shape of a supersonic flight vehicle. [Background technology]

[0002] The supersonic Concorde proved that there was a profitable market for supersonic transport on major business routes. However, sonic booms prevented supersonic flight over populated areas, severely limiting route options. Therefore, reducing sonic booms is one of the keys to the commercial success of supersonic aircraft.

[0003] Therefore, various methods for designing the airframe shape of a supersonic aircraft to reduce sonic booms have been proposed (for example, Patent Document 1 and Non-Patent Documents 1 to 5). A sonic boom is an acoustic phenomenon that occurs when flying at supersonic speeds and affects people, animals, or structures such as buildings on the ground.

[0004] Design methods for reducing sonic booms (hereafter referred to as low-boom design methods) can be broadly divided into two categories: direct optimization and inverse design. Direct optimization searches for an aircraft shape that minimizes sonic boom indicators. Inverse design searches for the near-field pressure waveform or equivalent cross-sectional area of ​​the target sonic boom, and then reconfigures the shape or adjusts the existing shape so that the final shape achieves the target. Inverse design is primarily used in recent research. The near-field pressure waveform is expressed using a function called the F-function. The equivalent cross-sectional area is the cross-sectional area of ​​a rotating body when it is assumed that the physical quantities at a distance from an aircraft flying at supersonic speed are created by the rotating body. Physical quantities at a distance from the aircraft due to lift and other factors can also be expressed using the equivalent cross-sectional area distribution of the rotating body.

[0005] Most research on inverse design has focused on the sonic boom directly below an aircraft. That is, the target F-function is defined in two dimensions: the distance directly below the aircraft axis and the distance along the aircraft axis. However, it was known that the sonic boom other than directly below the aircraft (hereinafter referred to as "off-track") can be larger than the sonic boom directly below. In the following, the sonic boom off-track is referred to as "off-track boom."

[0006] Therefore, low-boom design methods that take off-track booms into account have been proposed. For example, Non-Patent Document 1 expands the F-function to three dimensions to enable off-track boom evaluation. Non-Patent Document 2 describes a design method that incorporates off-track boom calculations. Non-Patent Document 3 describes a design method that defines a target equivalent cross-sectional area distribution other than directly below in order to minimize off-track and directly below sonic booms. Non-Patent Document 4 considers the off-track boom by imposing constraints on the second derivative of the off-track equivalent cross-sectional area. Non-Patent Document 5 considers the off-track boom by incorporating additional equivalent cross-sectional areas for both the directly below and off-track Mach planes and adjusting the fuselage cross section. [Prior art documents] [Patent documents]

[0007] [Patent Document 1] International Publication No. 2019 / 187828 [Non-patent literature]

[0008] [Non-Patent Document 1] Plotkin, KJ, “Sonic Boom Shaping in Three Dimensions,” 15th AIAA / CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), 2009. https: / / doi.org / 10.2514 / 6.2009-3387 [Non-patent document 2] Ordaz, I., and Li, W., “Integration of Off-Track Sonic Boom Analysis for Supersonic Aircraft Conceptual Design,” Journal of Aircraft, Vol. 51, No. 1, 2014, pp. 23-28. https: / / doi.org / 10.2514 / 1.C031511 [Non-patent document 3] Ordaz, I., Wintzer, M., and Rallabhandi, SK, “Full-Carpet Design of a Low-Boom Demonstrator Concept,” 33rd AIAA Applied Aerodynamics Conference, 2015. https: / / doi.org / 10.2514 / 6.2015-2261 [Non-patent document 4] Ueno, A., Kanamori, M., and Makino, Y., “Robust Low-Boom Design Based on Near-Field Pressure Signature in Whole Boom Carpet,” Journal of Aircraft, Vol. 54, No. 3, 2017, pp. 918-925. https: / / doi.org / 10.2514 / 1.C033972 [Non-Patent Document 5] Ueno, A., and Makino, Y., “RobustLow-Boom Design in Primary Boom Carpet,” AIAA Scitech 2021 Forum, 2021. https: / / doi.org / 10.2514 / 6.2021-1270 Summary of the Invention [Problem to be solved by the invention]

[0009] Although the advances in low-boom design methods mentioned above have resulted in significant improvements in sonic booms, there are still issues that need to be resolved. That is, the methods used to consider off-track booms so far lack a theoretical basis, and there is no guarantee that the target can be achieved.

[0010] For example, the method described in Non-Patent Document 3 achieved some improvement, but the waveform obtained did not match the target waveform. This is thought to be because the three-dimensional F-function could not be freely defined, making the target waveform unrealizable.

[0011] The present invention has been made in consideration of the above-mentioned problems, and provides a method for designing the airframe shape of a supersonic aircraft that is capable of taking into account an off-track boom based on theoretical grounds, a program for designing the airframe shape of a supersonic aircraft, a recording medium on which the program for designing the airframe shape of a supersonic aircraft is recorded, and an apparatus for designing the airframe shape of a supersonic aircraft. [Means for solving the problem]

[0012] A first aspect of the present invention is a preparation step (S10) of expressing a target cylindrical surface pressure distribution as a three-dimensional F function; The lift sinogram and the thickness sinogram are calculated from the three-dimensional F function expressed in the preparation step (S10). a sinogram creation step (S20) for creating a sinogram; A two-dimensional distribution of lift is calculated from the sinogram of lift created in the sinogram creation step (S20), a two-dimensional distribution of thickness is calculated from the sinogram of thickness created in the sinogram creation step (S20), and the two-dimensional distribution of lift is converted into a two-dimensional distribution of camber, a shape reconstruction step (S30) of reconstructing the shape of the airframe; A method for designing the airframe shape of a supersonic flight vehicle, comprising: the three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is The singular points representing the body of a supersonic aircraft were transformed using the Radon transform. It is something, The shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber. The present invention relates to a method for designing the fuselage shape of a supersonic flying vehicle.

[0013] A second aspect of the present invention is The three-dimensional F function, which is a function that expresses the target pressure distribution on a cylindrical surface, is used as input to calculate the lift sinogram and thickness sinogram. a sinogram creation step (S20) for creating a sinogram; A two-dimensional distribution of lift is calculated from the sinogram of lift created in the sinogram creation step (S20), a two-dimensional distribution of thickness is calculated from the sinogram of thickness created in the sinogram creation step (S20), and the two-dimensional distribution of lift is converted into a two-dimensional distribution of camber, A shape reconfiguration step (S30) of reconfiguring the shape of the aircraft. A design program for the airframe shape of a supersonic flight vehicle, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram isThe singular points representing the body of a supersonic aircraft were transformed using the Radon transform. It is something, The shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber. It is a design program for the fuselage shape of a supersonic flight vehicle.

[0014] A third aspect of the present invention is The three-dimensional F function, which is a function that expresses the target pressure distribution on a cylindrical surface, is used as input to calculate the lift sinogram and thickness sinogram. a sinogram creation step (S20) for creating a sinogram; A two-dimensional distribution of lift is calculated from the sinogram of lift created in the sinogram creation step (S20), a two-dimensional distribution of thickness is calculated from the sinogram of thickness created in the sinogram creation step (S20), and the two-dimensional distribution of lift is converted into a two-dimensional distribution of camber, A shape reconfiguration step (S30) of reconfiguring the shape of the aircraft. A recording medium on which a design program for the body shape of a supersonic aircraft is recorded, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is The singular points representing the body of a supersonic aircraft were transformed using the Radon transform. It is something, The shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber. The recording medium contains a program for designing the fuselage shape of a supersonic aircraft.

[0015] A fourth aspect of the present invention is The three-dimensional F function, which is a function that expresses the target pressure distribution on a cylindrical surface, is used as input to calculate the lift sinogram and thickness sinogram. a sinogram creation unit (S20) that creates a sinogram; A two-dimensional distribution of lift is calculated from the sinogram of lift created by the sinogram creation unit (S20), a two-dimensional distribution of thickness is calculated from the sinogram of thickness created by the sinogram creation unit (S20), and the two-dimensional distribution of lift is converted into a two-dimensional distribution of camber, A shape reconfiguration unit (S30) that reconfigures the shape of the aircraft body. A design device for the airframe shape of a supersonic flying vehicle, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created by the sinogram creation unit (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is The singular points representing the body of a supersonic aircraft were transformed using the Radon transform. It is something, The shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber. It is a design device for the body shape of a supersonic flying vehicle. [Effects of the Invention]

[0016] According to the first aspect of the present invention, the airframe shape is reconstructed from a sinogram, making it possible to design the airframe shape of a supersonic flight vehicle taking into account an off-track boom. There is a theoretical basis for the feasibility of the sinograms created in the sinogram creation process (see "2. Theoretical Basis of the Sinogram Creation Process S20" described below). Therefore, it is possible to provide a method for designing the airframe shape of a supersonic flight vehicle that can take into account an off-track boom based on theoretical grounds.

[0017] According to the second aspect of the present invention, it is possible to obtain the same effects as those of the first aspect, and therefore it is possible to provide a design program for the airframe shape of a supersonic flight vehicle that is capable of taking into account an off-track boom based on theoretical grounds.

[0018] According to the third aspect of the present invention, it is possible to obtain the same effects as those of the first aspect, and therefore it is possible to provide a recording medium having recorded thereon a design program for the airframe shape of a supersonic flight vehicle that is capable of taking into account an off-track boom based on theoretical grounds.

[0019] According to the fourth aspect of the present invention, it is possible to obtain the same effects as those of the first aspect, and therefore it is possible to provide a design system for the airframe shape of a supersonic flight vehicle that is capable of taking into account an off-track boom based on theoretical grounds.

[0020] In addition, the symbols in parentheses in the claims and the means for solving the problems indicate the correspondence with the specific means described in the embodiments described below, and do not limit the technical scope of the present invention. [Brief explanation of the drawings]

[0021] [Figure 1] FIG. 1 is a diagram illustrating a method for designing the airframe shape of a supersonic flying vehicle in one embodiment. [Figure 2] FIG. 1 is a diagram illustrating definitions of coordinates and projection views in an embodiment. [Figure 3] FIG. 1 illustrates the process of aircraft shape reconstruction in one embodiment. [Figure 4] FIG. 2 is a diagram showing a plane form of an airframe used as a design condition of one embodiment. [Figure 5] 10 is a graph comparing a projection of a lift force target with a projection of a reconstructed two-dimensional distribution in one embodiment. [Figure 6] 1 is a graph illustrating an optimized cubic F-function in one embodiment. [Figure 7]1 is a graph showing a sonic boom ground waveform calculated in one embodiment. [Figure 8] 1 is a graph illustrating the calculated sonic boom noise level in one embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0022] The method for designing the fuselage shape of a supersonic aircraft includes a sinogram creation step of creating a sinogram defined as a result of Radon transforming singular points representing the fuselage of the supersonic aircraft, and a shape reconstruction step of reconstructing the shape of the aircraft from the sinogram.

[0023] In the method for designing the airframe shape of a supersonic flight vehicle, the sinogram may be defined using a function that satisfies the Helgason-Ludwig consistency condition for a non-circular region. Since a sinogram suitable for supersonic aerodynamics can be created in the sinogram creation step, the shape reconstruction step can facilitate the reconstruction of the airframe shape.

[0024] In the method for designing the airframe shape of a supersonic flight vehicle, the sinogram may be obtained from physical quantities defined in a cylindrical coordinate system coaxial with the airframe, which can facilitate evaluation of sonic booms.

[0025] In the method for designing the airframe shape of a supersonic flight vehicle, the shape reconstruction step may use angle-deficient tomography to reconstruct the airframe shape, including areas where no sinogram exists. The airframe shape can be reliably reconstructed for sinograms defined only in a limited range.

[0026] In the method for designing the airframe shape of a supersonic flight vehicle, the shape reconstruction step may use a convex projection method as the angle-deficient tomography method, which repeats projections sequentially for multiple conditions including sinograms. This allows for more reliable reconstruction of the airframe shape from sinograms defined only within a limited range.

[0027] The design program for the body shape of a supersonic aircraft and the recording medium on which it is recorded include a sinogram creation step that creates a sinogram defined as the result of Radon transforming singular points representing the body of the supersonic aircraft, and a shape reconstruction step that reconstructs the shape of the aircraft from the sinogram.

[0028] In the design program for the airframe shape of a supersonic flight vehicle and the recording medium on which the program is recorded, the sinogram may be defined using a function that satisfies the Helgason-Ludwig consistency condition for a non-circular region. Since a sinogram suitable for supersonic aerodynamics can be created in the sinogram creation step, reconstruction of the airframe shape can be facilitated in the shape reconstruction step.

[0029] In the design program for the airframe shape of a supersonic flight vehicle and the recording medium on which the program is recorded, the sinogram may be calculated from physical quantities defined in a cylindrical coordinate system coaxial with the airframe, which can facilitate the creation of the sinogram in the sinogram creation step.

[0030] In the design program for the airframe shape of a supersonic flight vehicle and the recording medium on which the program is recorded, the shape reconstruction step may use angle-deficient tomography to reconstruct the airframe shape, including areas where no sinogram exists. The airframe shape can be reliably reconstructed for sinograms defined only in a limited range.

[0031] In the design program for the airframe shape of a supersonic flight vehicle and the recording medium on which the program is recorded, the shape reconstruction step may use a convex projection method as the angle-deficient tomography method, which repeats projections in order for multiple conditions including sinograms. This makes it possible to more reliably reconstruct the airframe shape from sinograms defined only in a limited range.

[0032] The design device for the body shape of a supersonic aircraft includes a sinogram creation unit that creates a sinogram defined as a result of Radon transforming singular points representing the body of the supersonic aircraft, and a shape reconstruction unit that reconstructs the shape of the aircraft from the sinogram.

[0033] In the design system for the airframe shape of a supersonic flight vehicle, the sinogram may be defined using a function that satisfies the Helgason-Ludwig consistency condition for a non-circular region. Since the sinogram creation unit can create a sinogram suitable for supersonic aerodynamics, the shape reconstruction step can facilitate the reconstruction of the airframe shape.

[0034] In the device for designing the body shape of a supersonic flight vehicle, the sinogram may be calculated from physical quantities defined in a cylindrical coordinate system coaxial with the body, which can facilitate creation of the sinogram in the sinogram creation unit.

[0035] In the design system for the airframe shape of a supersonic flight vehicle, the shape reconstruction unit may use angle-deficient tomography to reconstruct the airframe shape, including areas where no sinogram exists. The airframe shape can be reliably reconstructed for sinograms defined only in a limited range.

[0036] In the design system for the airframe shape of a supersonic flight vehicle, the shape reconstruction unit may use a convex projection method as the angle-deficient tomography method, which repeats projection in order for multiple conditions including a sinogram, thereby enabling more reliable reconstruction of the airframe shape from a sinogram defined only within a limited range.

[0037] 1. Overview of the design method for the fuselage shape of a supersonic flight vehicle An embodiment of the present invention will be described below with reference to Figures 1 to 8. As shown in Figure 1, the method for designing the body shape of a supersonic flight vehicle in this embodiment is roughly divided into four steps: a preparation step S10, a sinogram creation step S20, a shape reconstruction step S30, and an evaluation step S40.

[0038] Of these four steps, the sinogram creation step S20 and the shape reconstruction step S30 are programmed as a design program for the body shape of a supersonic aircraft. The design program for the body shape of a supersonic aircraft, in which the sinogram creation step S20 and the shape reconstruction step S30 are programmed, is executed by a design device for the body shape of a supersonic aircraft (specifically, a computer). That is, the design program for the body shape of a supersonic aircraft is recorded on a recording medium readable by the design device for the body shape of a supersonic aircraft, and the design device for the body shape of a supersonic aircraft reads the design program for the body shape of a supersonic aircraft from the recording medium and executes it.

[0039] In the design program, the sinogram creation process S20 is programmed as a sinogram creation step, and the shape reconstruction process S30 is programmed as a shape reconstruction step. In the design device, the sinogram creation process S20 is configured as a sinogram creation unit, and the shape reconstruction process S30 is configured as a shape reconstruction unit.

[0040] In the preparation step S10, calculations necessary for the sinogram creation step S20 and the shape reconstruction step S30 are performed. In the sinogram creation step S20, which follows the preparation step S10, a sinogram satisfying predetermined conditions is created. The sinogram is defined as the result of Radon transform of singular points representing the body of a supersonic aircraft. In the shape reconstruction step S30, the body shape is reconstructed based on the sinogram created in the sinogram creation step S20. In the evaluation step S40, the noise level of the sonic boom on the ground is calculated and evaluated based on the reconstruction result in the shape reconstruction step S30. Then, based on the evaluation result in the evaluation step S40, the preparation step S10, the sinogram creation step S20, the shape reconstruction step S30, and the evaluation step S40 are repeated to optimize the body shape.

[0041] As described above, conventional design methods that take off-track booms into consideration have no theoretical basis, but in this embodiment, by creating a sinogram that satisfies predetermined conditions in the sinogram creation step S20, it is possible to perform a low boom design that has a theoretical basis. The theoretical basis will be explained below.

[0042] 2. Rationale for Sinogram Creation Process S20 The theoretical basis of the design method that takes into account the off-track boom in this embodiment will be explained. First, a cylindrical surface pressure distribution that can be realized in a linear supersonic flow is determined. Hereinafter, a function that expresses the cylindrical surface pressure distribution will be referred to as a three-dimensional F-function. Once a three-dimensional F-function that can be realized in a linear supersonic flow is formulated, it can be used to reconstruct the airframe shape.

[0043] Once the 3D F-function is defined, several important pieces of information can be derived, such as wave drag, volume, axial center of volume, lift, axial center of lift, and sonic boom waveforms both directly below and off-track, which can be used in optimization to create low boom, low drag, and trimmable designs without considering the shape.

[0044] The three-dimensionality of the F-function can be expressed by decomposing it into multipoles, but it is known that higher-order poles have strict constraints and cannot be defined arbitrarily. When linear theory is used and a nearly planar geometry is assumed, the far-field pressure distribution can be calculated by integrating singular points at the Mach cut (i.e., the intersection of the plane on which the aircraft lies and the Mach surface). This integral (also called a projection) is known as the Radon transform. It has been widely studied in fields such as medical imaging (e.g., CT scans), astronomy, and electron microscopy, but not in the field of supersonic aircraft. The result of the Radon transform (i.e., the sinogram) must satisfy the Helgason-Ludwig consistency condition. Functions that satisfy this condition and can be used to define the three-dimensional F-function are described below.

[0045] In a linear supersonic flow, assuming left-right symmetry, the strength of the source and the strength of the lift element can be expressed as the equivalent source strength (in other words, the singular point in linear supersonic theory) as shown in the following formula F1.

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[0046] FIG. 2 shows the definitions of coordinates and projections used in this embodiment. In FIG. 2, o is the origin in the Cartesian coordinate system. R is the radius [m] of the cylindrical surface. s is the coordinate axis in the length direction of the projection. t is the coordinate axis in the projection direction. φ is the projection angle [rad]. When projecting on a Mach cut, which is the intersection of the Mach plane with a circumferential angle θ and the z=0 plane, there is a relationship of tanφ=-βsinθ. ω represents a vector with cosφ in the x direction and sinφ in the y direction. ξ is the x coordinate [m] of the intersection of the Mach cut and the x axis. η is the coordinate [m] in the y direction. L is the aircraft length [m].

[0047] In the far field, according to a well-known theorem, the magnitude and gradient of the disturbance velocity potential at a certain azimuthal angle are constant for a finite movement of the source on the Mach plane. Assuming that the source and lift element are located near the z=0 plane, the total source on a specific Mach plane (specifically, a Mach cut) can be obtained as follows:

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[0048] The disturbance velocity potential is given by the following equation F3:

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[0049] The above formulas F1 to F3 show the relationship between the pressure distribution on the cylindrical surface and the distribution of the plane source and lift. The three-dimensional F function and the equivalent cross-sectional area are calculated using the following formulas F4 and F5.

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[0050] Since the 3D F-function can be calculated by H(ξ,θ), the problem of defining a realizable 3D F-function is reduced to the problem of defining a realizable H(ξ,θ).

[0051] The variables of H(ξ,θ) are changed by ξcosφ=s and η=ssinφ+tcosφ, and H(ξ,θ) is c By redefining it as (s,ω), formula F2 can be expressed as formula F6 below.

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[0052] In the following, h(x,y) refers to the source term or lift term. For the source term, substituting formula F6 into formula F5 amounts to applying an integral to formula F6. By changing the order of the integrals, it can be shown that the integral of the source, i.e., the thickness, can be used to derive the F-function, rather than the source itself. Therefore, the thickness distribution can be used instead of the source, which allows the introduction of some constraints during the optimization process, such as minimum thickness, total volume, etc.

[0053] The sinogram g(s,ω) cannot be arbitrarily defined and must satisfy the Helgason-Ludwig consistency condition, which is expressed in the projection-moment theorem, which implies that F7 is a homogeneous polynomial of degree m in the components of ω = (cosφ, sinφ), and in the symmetry condition of F8:

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[0054] In the past, orthogonal functions satisfying the Helgason-Ludwig consistency condition have been proposed in fields such as medical imaging, astronomy, and electron microscopy (i.e., fields outside of supersonic aerodynamics) and used as a method of interpolating sinograms when φ is not defined over a range. These orthogonal functions can be used to create sinograms for calculating thickness and lift distributions. However, while the orthogonal functions proposed so far define sinograms over circular regions, in supersonic aerodynamics, the sinograms are not necessarily defined over circular regions. 2 A non-circular region such as that shown in Figure 1, specifically a diamond shape surrounded by the rearward and forward Mach cones from the tip and tail, respectively, is suitable.

[0055] In this embodiment, an orthogonal function g that spans the range from the tip to the tail and satisfies the Helgason-Ludwig consistency condition is used. c (r c,ω) is defined as the following formula F9.

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[0056] First, we will explain the projection-moment theorem. Substituting formula F9 into formula F7, we get the following formula F10.

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[0057] Formula F10 r c m can be expanded in Gegenbauer polynomials as shown in the following equation F11.

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[0058] Therefore, the formula F10 can be expressed as the following formula F12.

number

[0059] The Gegenbauer polynomials have an orthogonal relationship as shown in the following formula F13.

number

number

[0060] cos m-n Since φ is always positive, we can drop the absolute value sign. k + |n| is by definition even, and since the odd and even functions in F11 are expanded in terms of odd and even polynomials, respectively, k + m is also even. Therefore, mn is even.

[0061] Since exp(jnφ) can be expressed as a homogeneous polynomial of degree n in cosφ and sinφ, formula F14 is a homogeneous polynomial of degree m in the components of ω = (cosφ, sinφ). Therefore, formula F9 satisfies the projection-moment theorem of the Helgason-Ludwig consistency condition.

[0062] Next, we explain the symmetry condition. w(r c ) is an even function, and by substituting formula F9 into the left side of formula F8, we obtain the following formula F15.

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[0063] Since k+|n| is even by definition, formula F15 is equal to formula F9, which confirms that formula F8, the symmetry condition of the Helgason-Ludwig consistency condition, is satisfied.

[0064] In addition to the above conditions, the orthogonal function formula F9 is c=-1,1 because the intersection of the Mach plane with the planform at these locations has no length. By definition, α>1 / 2, so w(r c )=(1-r c 2 ) α―1 / 2 is r c =-1,1 is zero.

[0065] In this embodiment, Formula F9 is used to generate a new shape, but it can also be used to modify an existing shape. That is, when Formula F9 is used to modify an existing shape, the Radon transform of the existing shape is performed and expressed in the form of Formula F9. Then, by changing some coefficients, a new target 3D F-function can be defined.

[0066] 3. Shape reconstruction process S30 Once a feasible sinogram is defined, the corresponding two-dimensional distribution can be reconstructed. The following describes the process of reconstructing the airframe shape based on the sinogram created in the sinogram creation step S20.

[0067] The Fourier transform of g(s,ω) is expressed by the following formula F16.

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[0068] The sinogram g(s,ω) is H c When given as (s, ω) / cosφ, formula F16 becomes the following formulas F17 to F18.

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[0069] H F (u,v) is the two-dimensional Fourier transform of h(x,y). Equation F17 shows that the one-dimensional Fourier transform of the projection is the same as one direction of the two-dimensional Fourier transform of the two-dimensional distribution. This is called the Fourier slice theorem or the projection-slice theorem. Equations F16-F17 and s to r c By changing the variables to, the following formula F20 is obtained.

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[0070] g in formula F20 c (r c ,ω) is given by formula F9, so H F It is possible to calculate (u,v) and look at its inverse Fourier transform to be h(x,y) according to equation F18.

[0071] However, H F (u,v) is defined only in the limited range -(π / 2-μ)≦φ≦π / 2-μ by the relation tanφ=-βsinθ. In other words, H F (u,v) is undefined in the range π / 2-μ<φ≦π / 2 and -π / 2≦φ<-(π / 2-μ), where μ is the Mach angle [rad].

[0072] In other words, the Mach cut does not exist at all angles; when θ = 0°, it is parallel to the y-axis, and when θ = 90, -90°, it is at an angle of μ with respect to the x-axis; outside this range, the Mach cut does not exist geometrically.

[0073] This H F Due to the partial lack of (u,v), H FThe two-dimensional distribution h(x,y) cannot be calculated by a two-dimensional inverse Fourier transform of (u,v). This type of problem is known as angular deficit tomography, and has been studied outside of supersonic aerodynamics. There are three main categories of methods for solving this problem: iterative methods, estimation methods, and algebraic methods.

[0074] Iterative methods utilize iteration between object space and projection space. At each iteration, known regions of the sinogram are replaced with known sinograms to approach the solution. Estimation methods use the Helgason-Ludwig consistency condition to estimate missing sinograms using known sinograms. Algebraic methods solve large systems of linear equations obtained by discretizing the Radon transform.

[0075] In this embodiment, we use the convex projection onto convex surfaces (POCS) and apply an iterative method for reconstruction because it is possible to incorporate constraints such as minimum thickness and planar shape constraints. The convex projection onto convex surfaces is a method of iteratively projecting multiple conditions, including the sinogram, in order.

[0076] The reconstruction process is shown in Figure 3. The reconstruction process consists of steps 1 to 7. In step 1, the target sinogram is converted to the frequency domain using a one-dimensional fast Fourier transform (also known as FFT) and the Fourier slice theorem. In step 2, an arbitrary two-dimensional distribution is prepared as the initial state. In this embodiment, a uniform distribution within a planar shape is used. In step 3, a two-dimensional FFT is performed. In step 4, in the frequency domain, values ​​at φ in the range -(π / 2-μ)≦φ≦π / 2-μ, given the target sinogram, are replaced with the target sinogram. In step 5, a two-dimensional inverse FFT is performed. In step 6, a constraint is applied to the spatial domain. In this embodiment, the outside of the planar shape is set to zero. In step 7, the process returns to step 3 until a certain convergence criterion is met. This method allows for a two-dimensional distribution that matches the target sinogram.

[0077] The sinogram defined by equation F9 is given in polar coordinates. Therefore, in a standard inverse FFT, H is given in Cartesian coordinates. F In this embodiment, for each angle of each grid point, the inverse FFT is applied to the formula F9, and only the corresponding point in the Cartesian coordinate system is defined as H F (u,v) are kept to define the shape. The shape can be reconstructed by calculating h(x,y) separately for thickness and lift. The thickness distribution can be used as is, but the lift distribution needs to be converted to a camber distribution. To avoid introducing large variations in the spanwise camber, we apply a viscous vortex-like filter.

[0078] 4. Verification of the sinogram creation process and shape reconstruction process The target sinogram of the sample lift force is defined by formula F9, and a two-dimensional distribution can be reconstructed from the target sinogram. It is confirmed that the reconstructed sinogram of the two-dimensional distribution matches the target sinogram. In this example, the flight condition is Mach number 1.6. The plane shape in this example is shown in Figure 4. To define the lift sinogram, the coefficient b kn In this example, the weight α of the Gegenbauer polynomial is set to 1.5. In this example, a 128x128 grid is used to define the two-dimensional distribution. In this example, the number of iterations is 1,000.

[0079] Figure 5 shows the target lift projection and the actual projection of the reconstructed 2D distribution. The target lift projection and the actual projection match well. For small errors, if an exact match of the projection at a specific angle is required, the difference in the projection at that angle can be calculated by dividing it by the length of the intersection line between the planform and the Mach surface, and adding this value to the 2D distribution. However, the error at other angles will increase. In Figure 5, the dashed line shows an example of an exact match of the projection at φ=0.0 deg.

[0080] 5. Each step in the design method for the fuselage shape of a supersonic flight vehicle As described above, the method for designing the airframe shape of a supersonic flight vehicle in this embodiment is broadly divided into four steps: a preparation step S10, a sinogram creation step S20, a shape reconstruction step S30, and an evaluation step S40. Each of the preparation step S10, sinogram creation step S20, shape reconstruction step S30, and evaluation step S40 will be described below.

[0081] (1) Preparation process S10 In the preparation step S10, the aging variables and waveguide cross-sectional area for sonic boom propagation are calculated, and an equivalent cross-sectional area conversion step, an interpolation step, a lift distribution conversion step, and a definition step are performed in this order.

[0082] In the equivalent cross-sectional area conversion process, the input total F function directly below the fuselage is linearly interpolated and converted into an equivalent cross-sectional area. In the interpolation process, the input equivalent cross-sectional area directly below the fuselage due to the volume is interpolated using a non-uniform rational B-spline (so-called NURBS). In the lift distribution conversion process, the equivalent cross-sectional area due to the volume is subtracted from the total equivalent cross-sectional area to define the equivalent cross-sectional area due to the lift, and then it is converted into a lift distribution. In the definition process, the coefficient b that defines the sinogram is used. kn Using (n ≥ 2), we define the thickness and lift sinograms using formula F9, subtract them from the thickness and lift projections directly below the aircraft, and use the coefficient b kn Define the sinogram resulting from (n<2). The sinogram used is the sinogram with coefficient b kn (n≧2) and coefficient b kn is the sum of sinograms (n<2).

[0083] (2) Sinogram creation process S20 In the sinogram creation process S20, the coefficient b for lift kn Create a sinogram of the lift force from the coefficient b for thickness kn Create a thickness sinogram from the

[0084] (3) Shape reconstruction process S30 In the shape reconstruction step S30, the above-described process for reconstructing the airframe shape is applied to the thickness and lift.

[0085] (4) Evaluation process S40 The evaluation process S40 involves the vortex drag calculation process, three-dimensional F-function calculation process, volume and volume center calculation process, lift and lift center calculation process, wave resistance calculation process, and ground sonic boom calculation process. Of these processes, the three-dimensional F-function calculation process must be performed first in the evaluation process S40, but the order of the other processes does not matter. This is because the three-dimensional F-function is the input for wave resistance and sonic boom calculations, so it must be calculated first.

[0086] In the vortex resistance calculation process, the spanwise lift distribution is calculated and vortex resistance is calculated. In the three-dimensional F function calculation process, a sinogram is calculated from the two-dimensional distribution and a three-dimensional F function is calculated. In the volume and volume center calculation process, the volume and volume center are calculated from the zeroth-order pole of the equivalent cross-sectional area, i.e., the cross-sectional area distribution. In the lift and lift center calculation process, the lift and lift center are calculated from the first-order pole of the equivalent cross-sectional area, i.e., the lift distribution. In the wave resistance calculation process, wave resistance is calculated.

[0087] In the ground sonic boom calculation process, a three-dimensional F-function is propagated to calculate the ground sonic boom waveform and calculate the noise level. In other words, to evaluate the noise level of a sonic boom, it is necessary to propagate the pressure waveform near the aircraft to the ground using a method that can obtain information on the internal structure of the shock wave. In this embodiment, although a detailed explanation is omitted, a method is used in which the diffusion term of the Burgers equation is simplified and solved using the Cole-Hopf transformation.

[0088] Then, optimization is performed by repeating the preparation step S10, sinogram creation step S20, shape reconstruction step S30, and evaluation step S40. There are several points to note during this process. The center of volume is used as a rough estimate of the center of gravity. The projection directly below the aircraft is corrected so that it matches the target value exactly. For evaluation, values ​​calculated from two-dimensional distribution are used. However, it is also possible to calculate from the sinogram, except for vortex drag.

[0089] In addition, when the sinogram creation process S20 and the shape reconstruction process S30 are programmed as a design program for the fuselage shape of a supersonic aircraft, the input parameters when executing the design program include the Mach number, a grid for defining the two-dimensional distribution of singular points representing the fuselage shape of the supersonic aircraft, a plane shape, and a coefficient b for lift for defining the sinogram using formula F9. kn and the coefficient b for thickness kn , and the coefficient α of each Gegenbauer polynomial are input. Then, as a result of executing the design program, the reconstructed lift distribution and thickness distribution are output.

[0090] 6. Design Example A design example using the above-mentioned method for designing the body shape of a supersonic flight vehicle will be described. The supersonic flight vehicle in this design example is a supersonic aircraft. The flight conditions are a Mach number of 1.6, an altitude of 50,000 ft, and a lift of 1.317 x 10 6 N. The fuselage diameter is 3.0 meters. The ground reflection coefficient of the sonic boom is 1.9, and the frequency used to calculate the noise level is 10 kHz. The plane geometry shown in Figure 4 is used. The coefficient b kn defines the sinograms of thickness and lift distribution up to n=11. The weight α of the Gegenbauer polynomials is set to 1.5, and a 128x128 grid is used to define the two-dimensional distribution. The number of reconstruction iterations is 1,000. A well-known genetic algorithm is used for optimization.

[0091] The objective functions are minimizing the maximum sonic boom noise level at azimuth angles θ of 0, 10, 20, 30, and 40 degrees, minimizing the sum of wave drag and vortex drag, and minimizing the distance between the volume center and the lift center. The total F function directly below the fuselage is defined by linear interpolation of 12 points, and the equivalent cross-sectional area due to the volume directly below the fuselage is defined by NURBS with 12 control points. The lift and thickness b kn Including these, the total number of design variables is 85. 600 individuals are evolved for 800 generations.

[0092] The lift and thickness distributions of the solution obtained through optimization allow the calculation of the F function for each azimuth angle θ, and the F function is then used to calculate the sonic boom ground waveform. At first glance, the calculated ground waveform appears similar to a conventional N wave; however, due to its longer pressure rise time, it is quieter than the conventional N wave (see the dashed-dotted line in Figure 7, described later). An N wave is a waveform shaped like the letter N, where the positive pressure portion of the waveform merges forward and the negative pressure portion merges aft as the sonic boom propagates over long distances, resulting in forward and aft shock waves and an expansion wave between them. While the pressure rise time of a conventional 1 psf (47.88 Pa) stepped sonic boom waveform is on the order of 1 ms, the pressure rise time of the ground waveform of this solution is on the order of 10 ms. This optimization achieved minimal noise levels not only directly below but also off-track (see the solid line in Figure 8, described later).

[0093] The shape obtained in this way is based on linear theory. To obtain the desired pressure distribution in a real environment, corrections using nonlinear analysis are required. Therefore, a shape correction process is carried out to match the F-function obtained from computational fluid dynamics (CFD), a nonlinear analysis, with the target value obtained in the above process. This process involves the shape definition process, CAD model creation, mesh creation process, target F-function definition process, inverse design process, and post-processing process, in this order. Additional information on each process is provided below.

[0094] In the shape definition step, the camber distribution is calculated from the lift distribution based on a known method. In the target F-function definition step, the target F-function of the near field is defined. That is, the target three-dimensional F-function is propagated to the nearby cylindrical surface.

[0095] In the inverse design process, a CFD analysis is performed to create an inverse design that matches the target 3D F-function. The target is optimized using a gradient method to minimize the square root of the difference between the actual 3D F-function (i.e., the 3D F-function obtained from the CFD analysis) and the target 3D F-function within a specified azimuthal angle range. A multipole analysis is performed to compare the 3D F-function with the target. When multipole analysis and a near-field target F-function definition are used, the 3D F-functions from linear theory and CFD have similar nonlinear and azimuthal flow effects, making them suitable for comparison. Shape deformation is performed using free-form deformation with B-splines (so-called FFD) as an interpolation method.

[0096] In the post-processing step, the pressure distribution on the cylinder surface is extracted from the CFD results, the multipole analysis described above is applied, and the sonic boom propagation is calculated using the method described above.

[0097] An example of this shape modification process is shown below. The results obtained in the above design example are used for the shape definition and target F-function definition. The calculation conditions are the same as those in the above design example.

[0098] The optimization was completed after 92 iterations. Figure 6 shows the three-dimensional F-function for each azimuth angle θ. After the shape was modified, the three-dimensional F-function closely matched the target. Figure 7 shows the sonic boom ground waveform. After the shape was modified, the waveform closely matched the target.

[0099] The noise levels of the sonic boom are shown in Figure 8. As can be inferred from the ground waveform, the noise levels at circumferential angles of 0, 10, and 20 degrees match the target well after shape modification, and although they are slightly higher at 30 and 40 degrees, the noise levels have decreased at all angles compared to before shape modification, demonstrating the validity of the process of this embodiment.

[0100] In this embodiment, using a sample target, it was confirmed that the projection of the reconstructed shape matched the target at multiple circumferential angles. In the design example, a design with a low boom at multiple circumferential angles was obtained in the first stage, and in the second stage, the shape of the design obtained in the first stage was modified using nonlinear CFD.

[0101] This embodiment solves the long-standing problem of feasible 3D F-functions and shape reconstruction. That is, this embodiment provides a method for designing the airframe shape of a supersonic flight vehicle that can take into account an off-track boom based on theoretical grounds.

[0102] The present invention is not limited to the above-described embodiments, and can be applied to various embodiments within the scope of the present invention.

[0103] For example, in the sinogram creation step S20 of the above embodiment, a sinogram that satisfies the Helgason-Ludwig consistency condition is used, but the sinogram does not necessarily have to satisfy the Helgason-Ludwig consistency condition. For example, a sinogram that can achieve excellent results by approximating the Helgason-Ludwig consistency condition, even if it is not possible to achieve it, may be used. For example, the sinogram in the above embodiment does not define φ=90°, but if a sinogram is defined so that the spanwise distribution of lift (φ=90°) becomes elliptical (i.e., induced drag is minimized), the spanwise distribution will approach the target and induced drag can be reduced.

[0104] For example, in the shape reconstruction step S30 of the above embodiment, angle defect tomography (specifically, convex projection) is used, but the present invention is not limited to this. For example, the shape may be changed so that the sinogram approaches the target using some optimization method.

[0105] A supersonic flight vehicle having an airframe shape designed by at least one of the above-described design methods, design programs, and design devices can effectively reduce off-track boom.

[0106] In a manufacturing method for manufacturing a supersonic aircraft, if a supersonic aircraft having an aircraft shape designed by at least one of the above-mentioned design methods, design programs, and design devices is manufactured, it is possible to manufacture a supersonic aircraft airframe with reduced off-track boom. [Explanation of symbols]

[0107] S10 Preparation process S20 Sinogram creation process (sinogram creation step, sinogram creation part) S30 Shape reconstruction process (shape reconstruction step, shape reconstruction unit) S40 Evaluation process

Claims

1. A preparation step (S10) of expressing a target cylindrical surface pressure distribution as a three-dimensional F function; a sinogram creation step (S20) of creating a lift sinogram and a thickness sinogram from the three-dimensional F-function expressed in the preparation step (S10); a shape reconstruction step (S30) of calculating a two-dimensional distribution of lift from the lift sinogram created in the sinogram creation step (S20), calculating a two-dimensional distribution of thickness from the thickness sinogram created in the sinogram creation step (S20), and converting the two-dimensional distribution of lift into a two-dimensional distribution of camber, thereby reconstructing the shape of the airframe, the three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition; Here, the sinogram is a Radon transform of singular points representing the body of a supersonic aircraft, A method for designing the airframe shape of a supersonic flight vehicle, wherein the airframe shape is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber.

2. A method for designing the body shape of a supersonic aircraft as described in claim 1, wherein the three-dimensional F function is expressed using a function that satisfies the Helgason-Ludwig consistency condition for non-circular regions, based on the lift sinogram and the thickness sinogram created in the sinogram creation process (S20).

3. A method for designing the body shape of a supersonic flying vehicle as described in claim 1 or 2, wherein the lift sinogram and the thickness sinogram can be obtained from physical quantities expressed in a cylindrical coordinate system coaxial with the aircraft.

4. 2. The method for designing the airframe shape of a supersonic flight vehicle according to claim 1, wherein in the shape reconstruction step (S30), the shape of the airframe is reconstructed including areas where the lift sinogram and the thickness sinogram do not exist by using angle-deficient tomography.

5. 5. The method for designing the airframe shape of a supersonic flight vehicle according to claim 4, wherein in the shape reconstruction step (S30), a convex projection method is used as the angle-deficient tomography method, in which projection is repeated in order for the lift sinogram against a plurality of conditions including the lift sinogram, and projection is repeated in order for the thickness sinogram against a plurality of conditions including the thickness sinogram.

6. A sinogram creation step (S20) of creating a lift sinogram and a thickness sinogram using a three-dimensional F function, which is a function that expresses a target cylindrical surface pressure distribution, as an input; a shape reconstruction step (S30) of calculating a two-dimensional distribution of lift from the lift sinogram created in the sinogram creation step (S20), calculating a two-dimensional distribution of thickness from the thickness sinogram created in the sinogram creation step (S20), and converting the two-dimensional distribution of lift into a two-dimensional distribution of camber, thereby reconstructing the shape of the airframe, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is a Radon transform of singular points representing the body of a supersonic aircraft, A design program for the airframe shape of a supersonic flight vehicle, in which the shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber.

7. A sinogram creation step (S20) of creating a lift sinogram and a thickness sinogram using a three-dimensional F function, which is a function that expresses a target cylindrical surface pressure distribution, as an input; a shape reconstruction step (S30) of reconstructing the shape of the airframe by calculating a two-dimensional distribution of lift from the lift sinogram created in the sinogram creation step (S20), calculating a two-dimensional distribution of thickness from the thickness sinogram created in the sinogram creation step (S20), and converting the two-dimensional distribution of lift into a two-dimensional distribution of camber, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created in the sinogram creation step (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is a Radon transform of singular points representing the body of a supersonic aircraft, A recording medium having recorded thereon a design program for the airframe shape of a supersonic flight vehicle, the design program designing the airframe shape based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber.

8. A sinogram creation unit (S20) that creates a lift sinogram and a thickness sinogram using a three-dimensional F function, which is a function that expresses a target cylindrical surface pressure distribution, as an input; a shape reconstruction unit (S30) that calculates a two-dimensional distribution of lift from the lift sinogram created by the sinogram creation unit (S20), calculates a two-dimensional distribution of thickness from the thickness sinogram created by the sinogram creation unit (S20), and converts the two-dimensional distribution of lift into a two-dimensional distribution of camber, thereby reconstructing the shape of the airframe, The three-dimensional F-function is expressed so that the lift sinogram and the thickness sinogram created by the sinogram creation unit (S20) satisfy the Helgason-Ludwig consistency condition, Here, the sinogram is a Radon transform of singular points representing the body of a supersonic aircraft, A design device for the airframe shape of a supersonic flight vehicle, wherein the shape of the airframe is designed based on the two-dimensional distribution of thickness and the two-dimensional distribution of camber.

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