JP2020047015A - Analysis device - Google Patents

Abstract

To provide an analysis device that can appropriately identify a position of a flow field around a structure greatly contributing to wind noise by analyzing a relationship between the flow field around the structure including a vehicle and the wind noise reaching the inside of the structure.SOLUTION: An analysis device executes non-steady CFD calculation processing for moving a structure model that models a structure. The analysis device calculates a first index (IDS) that indicates a contribution degree of a flow field in a space nodal point to sound intensity (Iin) incident on an observation point of an analysis surface of the structure in a normal direction, and calculates a second index (APwDS) that indicates a contribution degree of the flow field in the space nodal point to sound power reaching the analysis surface. Furthermore, the analysis device displays information on size distribution of the second index in a predetermined region in the flow field around the structure model.SELECTED DRAWING: Figure 6

Description

  The present invention relates to an analysis device for analyzing a relationship between a flow field (for example, a flow velocity and a vorticity) around a moving structure and a wind noise reaching the inside of the structure.

  2. Description of the Related Art Techniques have been developed for analyzing sounds generated during the movement of structures such as vehicles. For example, a conventional apparatus acquires “amplitude of pressure fluctuation” and “average flow velocity” at an arbitrary position on the surface of a vehicle model by a simulation of running the vehicle model, and radiates from the vehicle model surface based on these values. The sound power, which is the total energy of the sound to be played, is calculated (for example, see Patent Document 1).

JP 2017-062727 A

  By the way, during the movement of a structure (for example, a vehicle), noise generated by the flow of fluid around the structure reaches inside the structure. In this specification, such noise that reaches the inside of the structure is referred to as “wind noise”. Wind noise changes under the influence of the flow field around the structure. For this reason, in order to reduce the wind noise, which part (position) of the flow field around the structure has a strong influence on the wind noise, the flow field of that part is changed. (That is, the shape of the component of the structure must be changed). However, there is no unified method for “methods to identify the part of the flow field that greatly contributes to the wind noise”. In practice, engineers use the flow field based on their own know-how and knowledge. Part is specified. For this reason, there has been a problem that the results of the examination vary depending on the engineer, and trial and error often occurs.

  The present invention has been made to solve the above problems, and one of its objects is to provide a flow field around a structure including a vehicle and a wind noise reaching the inside of the structure. An object of the present invention is to provide a technique capable of appropriately specifying the position of a flow field around a structure that greatly contributes to wind noise by analyzing the relationship of the wind noise.

The analyzer of the present invention (hereinafter, sometimes referred to as “the present device”) is an analyzer (10) that analyzes a wind noise generated in a moving structure.
The analyzer is
A non-stationary CFD calculation process for moving a structure model (20) obtained by modeling the structure is executed, and a flow field in a predetermined region (21) of a flow field around the structure model (20) is calculated. The average flow velocity (U−) and the average vorticity (Ω−) in a predetermined time are calculated for each spatial node (z) which is a node in the predetermined area (21), and the angle to be analyzed for wind noise is calculated. In the angular frequency band of interest ([θ _l , θ _h ]), which is a frequency band, a value based on the amplitude (u to _θ ) of the turbulent flow velocity in the predetermined area (21) is calculated for each spatial node (z). First calculating means (11, step 605, step 610, step 615)
The structure based on the average flow velocity (U−) calculated by the first calculation unit, the average vorticity (Ω−), and a value based on the amplitude of the turbulent flow velocity (u to _θ ). The first index (IDS) indicating the contribution of the flow field at the spatial node (z) to the sound intensity (Iin _n ) incident along the normal direction at the observation point (x) on the analysis surface at is calculated. Second calculating means (11, step 620);
A sound power (Pin) reaching the analysis surface is calculated based on the first index calculated by the second calculation means, and a sound power (Pin) at the space node (z) with respect to the sound power reaching the analysis surface is calculated. Third calculating means (11, step 630) for calculating a second index (APwDS) indicating the contribution of the flow field;
Display means (13, 19, step 640) for displaying information on the distribution of the size of the second index in the predetermined area;
Is provided.

  According to the device of the present invention, the information on the distribution of the magnitude of the second index indicating the contribution of the flow field at the spatial node (z) to the acoustic power reaching the analysis surface is displayed by the display means. Therefore, the user can appropriately specify the position of the flow field around the structure that largely contributes to the wind noise reaching the inside of the structure.

  In the above description, in order to facilitate understanding of the invention, the reference numerals used in the embodiments are appended in parentheses to the configuration of the invention corresponding to the embodiment, but each constituent element of the invention is the same as the reference numeral. It is not limited to the embodiment defined by.

FIG. 1 is a block diagram illustrating a configuration of an analyzer according to an embodiment of the present invention (hereinafter, referred to as “present embodiment”). FIG. 2 is a schematic diagram of a vehicle model and a flow domain to be subjected to an unsteady CFD simulation. It is a figure explaining acoustic intensity used as an index at the time of analyzing wind noise (in-vehicle sound) in this embodiment. It is a figure used for explaining the derivation method of APDS calculation formula, and is a schematic diagram of a rigid body model and a flow domain. It is a schematic diagram of a rigid body model and a flow domain. 9 is a flowchart illustrating an “analysis routine” executed by a CPU of the present embodiment. It is a distribution diagram of the sound intensity on the analysis surface (left front side glass) of the vehicle model. It is an example of an isosurface distribution of APwDS image-displayed on the display screen based on the first reference value of APwDS. It is an example of an iso-surface distribution of APwDS image-displayed on the display screen based on the second reference value of APwDS. FIG. 4 is a diagram for explaining a plane (evaluation section) on which a distribution map of parameters constituting an IDS is created. FIG. 11 is a distribution diagram of an average flow velocity around a left side mirror in an evaluation section (see a broken line L in FIG. 10). FIG. 11 is a distribution diagram of a turbulent flow velocity around a left side mirror in an evaluation cross section (see a broken line L in FIG. 10). (A) is a “distribution diagram of acoustic intensity on an analysis surface (left front side glass)” when an analysis process is performed on a first vehicle model using the present embodiment device. FIG. 10 is a “distribution diagram of acoustic intensity on an analysis surface (left front side glass)” when an analysis process is performed on a second vehicle model using the embodiment device. “The difference between the actually measured vehicle sound between the first vehicle and the second vehicle”, “the acoustic power calculated by analyzing the first vehicle model using the present device, and FIG. 7 is a graph showing a difference from acoustic power calculated by analyzing a second vehicle model.

<Configuration of the present embodiment>
The present embodiment analyzes wind noise generated when various structures move. In the present embodiment, a vehicle is described as an example of the structures.

  As illustrated in FIG. 1, the analysis device 10 includes a calculation unit 11, an input unit 12, and an output unit 13. The arithmetic unit 11 includes a CPU 14, a RAM 15, a ROM 16, a hard disk (HDD) 17, and an I / O interface 18.

  The ROM 16 and / or the HDD 17 store data of a three-dimensional model of a vehicle to be analyzed for wind noise, data of a flow domain indicating an analysis range, and data necessary for an unsteady CFD (computational fluid dynamics) simulation. Are stored in advance.

  Further, the ROM 16 stores instructions (programs, routines) executed by the CPU 14. The CPU 14 implements various functions described below by executing the instructions.

  The operation unit 11 is connected to the input unit 12 and the output unit 13 via the I / O interface 18. The input unit 12 is a device that receives various requests from a user, and includes a keyboard and a mouse. Therefore, the user can use the input unit 12 to specify a reference value, a parameter, and the like of the index (APwDS described later). The output unit 13 has a display screen 19 that can be visually recognized by a user.

<Configuration of vehicle model>
FIG. 2 is a diagram schematically showing a three-dimensional model 20 of a vehicle (hereinafter, simply referred to as “vehicle model 20”) and a flow domain 21. Note that the vehicle model 20 corresponds to an example of a “structure model”. As shown in FIG. 2, the space coordinate system of the vehicle model 20 includes an e1 axis (vehicle longitudinal direction), an e2 axis (vehicle width direction), and an e3 axis (height direction). In addition, the origin is set at a position separated from the reference point (for example, the center of gravity) of the vehicle model 20 by a predetermined distance (including a zero value) in a predetermined direction.

  The flow domain 21 is an area indicating a range in which the unsteady CFD calculation is performed in the space around the vehicle model 20. The flow domain 21 includes “a flow field of a part of the flow field around the vehicle model 20 that may affect the sound intensity (described later) incident on the left front side glass 22 in a direction normal to the surface thereof. Area ". Specifically, in the flow domain 21, a portion overlapping the vehicle model 20 is cut out from a substantially rectangular parallelepiped space including the left front side glass 22, the left side mirror 23, the left front pillar, the left half of the windshield, and the like. The shape has The flow domain 21 can be freely set depending on which part of the vehicle model 20 is to be analyzed. Note that the flow domain 21 corresponds to an example of a “predetermined area”.

<Analysis of wind noise>
Hereinafter, a specific method for analyzing the wind noise generated when the vehicle travels will be described. Hereinafter, the wind noise (noise arriving inside the vehicle) is referred to as “in-vehicle sound P _out ” (see FIG. 3).

(Relationship between vehicle interior sound and sound power reaching the vehicle surface)
As shown in the following formula (1), interior noise P _out is determined by the sound power P _in and the vehicle surface transmission loss TL reaching the vehicle surface. The vehicle surface transmission loss TL depends on the material properties of the vehicle surface. The vehicle surface transmission loss TL can be obtained by measurement in advance.

In order to analyze the relationship between the interior noise P _out and the vehicle surrounding flow field, to build a relational expression representing the relationship between the sound power P _in the vehicle surrounding flow field.

Sound power _{P in} can be represented by the following formula (2) and (3). Therefore, the sound power P _in is the sound pressure at the vehicle surface (hereinafter, sometimes simply referred to as "surface sound pressure".) Can be calculated using the.

The symbols in the formulas (2) and (3) are as follows.

The surface sound pressure pa can be evaluated by “APDS (Acoustic Pressure Density Source)” described below. The APDS is an index that quantitatively indicates the relationship between the flow field around the vehicle and the fluctuation of the surface sound pressure (in other words, the “flow field (state) at an arbitrary position around the vehicle” is “arbitrary on the vehicle surface”. Index indicating how much the surface sound pressure varies at the position "). The APDS has as parameters physical parameters such as an average flow velocity (U- described later), a turbulent flow velocity (u- _θ described later) and an average vorticity (Ω- described later) at an arbitrary position around the vehicle.

  In addition, the “fluctuation in surface sound pressure at an arbitrary position on the vehicle surface” can be calculated (predicted) based on a value obtained by spatially integrating the APDS with a flow field in a predetermined region (flow domain 21) around the vehicle. it can. For this reason, the present embodiment apparatus calculates the average flow velocity, the turbulent flow velocity, and the average vorticity of the flow field for each position in the flow domain 21 by the unsteady CFD calculation processing, and based on these values, calculates the APDS in the flow domain. Calculated for each position in 21.

More specifically, the autocorrelation function of the turbulent flow velocity (strictly speaking, the amplitude of the turbulent flow velocity) u to _θ is calculated for each frequency (angular frequency) by the unsteady CFD calculation processing (θ: angular frequency. = 2πf). The “autocorrelation function of the turbulent flow velocity amplitudes u to _θ ” corresponds to an example of “a value based on the turbulent flow velocity amplitude”.

The autocorrelation function is calculated in a frequency band to be analyzed for wind noise (hereinafter, the frequency band is also referred to as a “frequency band of interest”). Focusing frequency band is selectable by the user, for example, the center frequency _{f m} is 500 Hz, 1 kHz, the frequency band of 2kHz or 4kHz may be selected. Typically, a frequency band of 2 kHz or more can be selected. For example, when 2 kHz (lower limit frequency f ₁ = 1420 Hz, upper limit frequency f _h = 2840 Hz) is selected as the frequency band of interest, the autocorrelation function of the turbulent flow velocity is within the range of 1420 Hz to 2840 Hz and within the flow domain 21. It is calculated every time.

ここで、上記式（４）中のθ_ｌ及びθ_ｈは、着目角周波数帯の下限角周波数及び上限角周波数をそれぞれ表し、Ｆ_θ ^＊は、関数Ｆ_θの複素共役を表す。即ち、本明細書では、自己相関関数は、「任意の関数Ｆ_θとその複素共役関数Ｆ_θ ^＊との積（後述）を着目角周波数帯において積分した値」として定義される。 The autocorrelation function of an arbitrary function _Fθ is defined by the following equation (4).

Here, theta _l and theta _h in the formula (4) in represents the lower limit angular frequency and the upper limit angular frequency of the target angular frequency band, respectively, F _theta ^* denotes a complex conjugate of the function F _theta. That is, in the present specification, the autocorrelation function is defined as "a value obtained by integrating a product (described later) of an arbitrary function _Fθ and its complex conjugate function _Fθ ^* in the angular frequency band of interest".

The surface of the left front side glass 22 of the vehicle model 20 is defined as “analysis surface”. Furthermore, the i-th node x counted from any reference node on the analysis surface is defined as x _i (i: 1 to m), and the j-th space node counted from any reference space node in the flow domain 21. z is defined as z _j (j: 1 to n). The APDS for the k-th node x _k (k: _{1 to} m) is calculated at all spatial nodes z _j in the flow domain 21.

Here, the value obtained by spatially integrating the APDS in the frequency band of interest for the node x _k with all the spatial nodes z _j in the flow domain 21 is “the surface sound pressure fluctuation pa _θ (x _k ) of the frequency band of interest at the node x _k . Of the autocorrelation function of (see later). Therefore, the APDS value of the frequency band of interest at each spatial node z _j is “the autocorrelation function of the surface sound pressure fluctuation pa _θ (x _k ) of the frequency band of interest at the node x _k ” per unit volume of the flow domain 21. Can be interpreted as representing an approximate value of In the following, when it is not necessary to distinguish a certain node x _k from another node x _i , the notation of “k” and “i” is omitted. The same applies to the space node z _i.

(Derivation of APDS calculation formula)
Next, a calculation formula of the APDS and a method of deriving the calculation formula will be described. Hereinafter, the amplitude of the pressure fluctuation, the amplitude of the sound pressure fluctuation, and the amplitude of the fluid flow pressure fluctuation on the structure surface are referred to as “surface pressure fluctuation”, “surface sound pressure fluctuation”, and “surface fluid flow pressure fluctuation”, respectively. . Further, the node x is referred to as “observation point x”, and the spatial node z is referred to as “evaluation point z”.

  APDS is a relational expression between the surface pressure fluctuation (that is, the sum of the surface sound pressure fluctuation and the surface fluid flow pressure fluctuation) in the rigid body model, which is calculated based on the well-known Powell equation, and the flow field around it. Is converted from the time domain to the frequency domain. However, the Powell formula is expressed in a format suitable for deriving the APDS calculation formula. Therefore, hereinafter, the rigid body model and the flow domain will be described first with reference to FIG. 4, and subsequently, the derivation of the Powell formula in a form suitable for deriving the APDS calculation formula will be described. Next, the derivation of the relational expression between the surface pressure fluctuation and the flow field in the time domain will be described, and then the derivation of the APDS calculation expression will be described.

1. Rigid Body Model and Flow Domain FIG. 4 is a schematic diagram of the rigid body model 41 and the flow domain 42. The rigid body model 41 (hereinafter, also simply referred to as “rigid body 41”) has an observation point x on its surface S. The flow domain 42 is a flow field of a portion of the flow field around the rigid body 41 that may affect the surface pressure fluctuation at the observation point x. The flow domain 42 has a plurality of evaluation points z therein. APDS (x, z) is an index representing the contribution of the flow field at the evaluation point z to the surface sound pressure fluctuation at the observation point x. The other characters in FIG. 4 will be described later.

ここで、ρは流れ場の密度であり、ｕは流れ場の流速（ベクトル量）であり、Ｄ／Ｄｔは物質微分であり、ｐは剛体４１の表面における圧力（圧力変動、表面圧力変動）であり、ηは流れ場の粘性係数である。 2. Derivation of Powell equation The following equation (5) shows the Navier-Stokes equation when the body force is zero.

Here, ρ is the density of the flow field, u is the flow velocity (vector quantity) of the flow field, D / Dt is the material derivative, and p is the pressure on the surface of the rigid body 41 (pressure fluctuation, surface pressure fluctuation). Where η is the viscosity coefficient of the flow field.

を用いて式変形すると、下記式（６）が得られる。なお、上記の関係式中のＡは任意のベクトルを表す。

ここで、νは流れ場の動粘性係数であり、ν＝η／ρである。 The material derivative of the above equation (5) is rewritten using partial differentiation, and then the following relational expression;

The following equation (6) is obtained by transforming the equation using Note that A in the above relational expression represents an arbitrary vector.

Here, ν is the kinematic viscosity coefficient of the flow field, and ν = η / ρ.

を用いて式変形すると、下記式（７）が得られる。なお、ωは流れ場の渦度（ベクトル量）であり、Ｂはエンタルピーである。

Subsequently, the above equation (6) is converted into the following relational equation;

The following equation (7) is obtained by transforming the equation using Here, ω is the vorticity (vector quantity) of the flow field, and B is the enthalpy.

が成立するため、上記式（７）は下記式（８）によって近似できる。

The flow field of the present embodiment is an incompressible fluid or a microcompressible fluid. In this case, the following relational expression:

Holds, the above equation (7) can be approximated by the following equation (8).

なお、上記式（９）中の「×」は乗算記号を表す。本明細書では基本的に乗算記号は省略して記載するが、明示したほうが理解し易い場合は例外的に記載する。 When divergence is obtained by multiplying both sides of the above equation (8) by the density ρ of the flow field, the following equation (9) is obtained.

Note that “x” in the above equation (9) represents a multiplication symbol. In the present specification, the multiplication symbol is basically omitted and described, but if it is easier to understand the description, the description is exceptionally described.

「流れ場の密度ρ」と「流れ場の流速ｕの時間についての偏微分」との積の発散（即ち、下記式（１１）の左辺）を上記式（１０）を用いて式変形すると、下記式（１１）の右辺のようになる。

ここで、ｃは流れ場の音速である。表面圧力変動ｐ、流れ場の密度ρ、及び、流れ場の音速ｃの間には以下の関係が成立する。

Here, the following equation (10) holds between the flow velocity u of the flow field and the density ρ of the flow field.

When the divergence of the product of the “flow field density ρ” and the “partial derivative of the flow field velocity u with respect to time” (ie, the left side of the following equation (11)) is transformed using the above equation (10), It becomes like the right side of the following equation (11).

Here, c is the sound speed of the flow field. The following relationship is established between the surface pressure fluctuation p, the density ρ of the flow field, and the sound velocity c of the flow field.

が成立する。なお、レイノルズ数Ｒｅは、Ｒｅ＝ｕＬ／νとして定義され、Ｌは流れ場の代表長さを表す。上記式（１１）の右辺の括弧内の項を以下に示すように式変形して上記関係式を適用すると、式（１１）の右辺の括弧内の項は下記式（１２）によって近似される。

By the way, the flow field of the present embodiment is a high Reynolds number flow (that is, a flow having a relatively high flow velocity, for example, 100 km / h to 140 km / h). For high Reynolds number flows, the following relation:

Holds. The Reynolds number Re is defined as Re = uL / ν, and L represents the representative length of the flow field. When the terms in the parentheses on the right side of the above equation (11) are modified as shown below and the above relational expression is applied, the terms in the parentheses on the right side of the equation (11) are approximated by the following equation (12). .

When the above equation (9) is transformed (approximately) using the above equations (11) and (12), the following equation (13) is obtained.

が成立するため、上記式（１３）をこれらの関係式を用いて式変形すると、下記式（１４）に示すパウエル式が得られる。なお、上記の関係式のｃ_０は静止状態のときの流れ場の音速であり、ρ_０は静止状態のときの流れ場の密度である。

Here, the flow field of the present embodiment is a low Mach number flow (that is, a flow field in which the ratio of the flow velocity of the fluid to the speed of sound is relatively low). For low Mach number flows:

Is satisfied, the above equation (13) is transformed using these relational equations to obtain the Powell equation shown in the following equation (14). In the above relational expression, c ₀ is the sound velocity of the flow field in the stationary state, and ρ ₀ is the density of the flow field in the stationary state.

ここで、観測点ｘ及び評価点ｚは上述したとおりであり（図４参照）、τは過去の時刻である。なお、δ（ｘ）はディラックデルタ関数である。加えて、上記式（１５）中のグリーン関数Ｇは、自由空間においては下記式（１６）のように表される。

3. Derivation of Relational Expression between Surface Pressure Fluctuation and Flow Field in Time Domain The Green function G satisfies the following expression (15).

Here, the observation point x and the evaluation point z are as described above (see FIG. 4), and τ is a past time. Note that δ (x) is a Dirac delta function. In addition, the Green function G in the above equation (15) is expressed in free space as the following equation (16).

Ｈ（ｚ）はヘビサイド関数（Heaviside Unit Function）であり、下記式（１８）によって定義される。

ここで、図４に示すように、Ｖはフロードメイン４２を表すかたまりであり、Ｓ＋は剛体４１の表面Ｓを取り囲む仮想表面である。上記式（１８）中の「z in V」とは、評価点ｚがフロードメイン４２の内部に位置している場合を示し、「z inside S+」とは、評価点ｚが仮想表面Ｓ＋の内部に位置している場合を示す。 Considering the object boundary condition, the Powell equation shown in the above equation (14) is expressed as the following equation (17).

H (z) is a Heaviside Unit Function, and is defined by the following equation (18).

Here, as shown in FIG. 4, V is a lump representing the flow domain 42, and S + is a virtual surface surrounding the surface S of the rigid body 41. “Z in V” in the above equation (18) indicates that the evaluation point z is located inside the flow domain 42, and “z inside S +” means that the evaluation point z is inside the virtual surface S +. The case where it is located in is shown.

ここで、式中の（・）は任意の値との内積を表す。加えて、図４に示すように、剛体４１は、その表面Ｓのうち観測点ｘとは異なる位置に評価点ｙを有する。式中のｎは、当該評価点ｙの法線ベクトルｎ（ｙ）を表し、ｄＳは、評価点ｙ近傍の表面面積ｄＳ（ｙ）を表す。 This Heaviside function H (z) satisfies the integral equation of the following equation (19).

Here, (·) in the expression represents an inner product with an arbitrary value. In addition, as shown in FIG. 4, the rigid body 41 has an evaluation point y at a position different from the observation point x on the surface S thereof. In the formula, n represents a normal vector n (y) of the evaluation point y, and dS represents a surface area dS (y) near the evaluation point y.

One item, two items and the right side of the left side of the above equation (17) can be modified as in the following equations (17-1), (17-2) and (17-3), respectively.

By substituting the above equations (17-1) to (17-3) into the above equation (17), the following equation (20) is obtained.

The two items on the right side of the above equation (20) (the minus sign is omitted) can be transformed into the following equation (20-1).

When the above equation (20-1) is substituted into the above equation (20), the following equation (21) is obtained.

When the above equation (16) is used, the integral expression of the above equation (21) is as shown in the following equation (22).

When one item on the right side of the above equation (22) is expanded using the above equation (21), the following equation (22-1) is obtained. In the following, for convenience of notation, when describing the equation deformation of each term on the right side of the above equation (22), time integration is omitted.

When the two items and the three items on the right side of the above equation (22) are expanded using the above equation (21), the following equation (22-2) is obtained.

The four items on the right side of the above equation (22) can be transformed into the following equation (22-3).

By substituting the above equations (22-1) to (22-3) into the above equation (22), the following equation (23) is obtained.

When the virtual surface S + converges on the surface S of the rigid body 41, the above equation (23) can be rewritten as the following equation (24).

Here, in the high Reynolds number flow, the three items on the right side of the above equation (24) are so small as to be negligible with respect to the four items. In addition, when the rigid body 41 is stationary, there is no flow velocity, so the two items on the right side are equal to zero. Therefore, the above equation (24) can be simplified as the following equation (25).

Since there is no flow velocity on the surface of the stationary object, the enthalpy B is expressed by the following equation.

By defining the right side p / ρ of the above equation as p / ρ = p _* and substituting the enthalpy B into the above equation (25), the following equation (26) is obtained. This relational expression is “the relational expression between the surface pressure fluctuation and the flow field in the time domain”.

4. Derivation of APDS calculation formula Since the relational expression of the above formula (26) includes a value (function) p _* (= p / ρ) based on the surface pressure fluctuation p on both sides thereof, the surface pressure fluctuation p at the observation point x is expressed by the flow velocity It cannot be calculated directly from u and vorticity ω. For this reason, in the following, the term including p _* is deleted from the right side by equation deformation, and a relational expression between “surface pressure fluctuation p at observation point x” and “flow velocity u and vorticity ω of the flow field” is derived.

  As shown in FIG. 5, a region near the observation point x on the surface S of the rigid body 41 is defined as a surface region Sx, and a normal vector of the surface region Sx is set to n (x). The surface area Sx is a sufficiently large plane. In addition, two points x + and x− having the same distance from the observation point x are defined on the normal vector n (x). The point x + is located outside the rigid body 41, and the point x− is located inside the rigid body 41. Other characters will be described later. By substituting the points x + and x− into the above equation (25) as observation points, the following equations (25-1) and (25-2) are obtained.

When the two sides of the equations (25-1) and (25-2) are added to each other to take the limit when approaching x with respect to x− and x +, the following equation (27) is obtained. In the following equation modification, H (x +) = 1 and H (x −) = 0 are used (see equation (18)).

One and two items on the right side of the above equation (27) can be transformed into the following equations (27-1) and (27-2), respectively. Note that S＼Sx in the equation represents a portion of the surface S of the rigid body 41 that does not include the surface region Sx (see FIG. 5).

By substituting the above equations (27-1) and (27-2) into the above equation (27), the following equation (28) is obtained.

ここで、式中のＡ_θ（ｘ）は角周波数θにおける関数Ａ（ｘ，ｔ）の大きさ（振幅）を表す。Ａ_θ（ｘ）は複素数であり、その複素共役Ａ_θ ^＊（ｘ）は下記（２９）の関係式を満たす。

The Fourier transform is defined by the following equation.

Here, A _θ (x) in the expression represents the magnitude (amplitude) of the function A (x, t) at the angular frequency θ. _Aθ (x) is a complex number, and its complex conjugate _Aθ ^* (x) satisfies the following relational expression (29).

By applying the Fourier transform to the above equation (27) and using the Fourier transform result of the convolution integral, the following equation (30) is obtained.

ここで、ｋ_０は角周波数θ成分の波数（いわゆる音響波数）であり、ｋ_０＝θ／ｃ_０である。ｒは観測点ｘと評価点ｚとの距離である。 Here, G _θ (x, z) obtained by Fourier-transforming the Green's function G is represented by the following equation (31), and satisfies the relational equation of the following equation (32) (the so-called Helmholtz equation).

Here, k ₀ is the wave number (so-called acoustic wave number) of the angular frequency θ component, and k ₀ = θ / c ₀ . r is the distance between the observation point x and the evaluation point z.

The gradient of G _θ (x, z) can be expressed by the following equation (33) using the above equation (31). The vector r in the equation is a vector from the observation point x to the evaluation point z.

Here, the pressure generated on the surface of the rigid body 41 is constituted by the fluid flow pressure fluctuation and the sound pressure. Since the fluid flow pressure fluctuations are not propagated to move so much distant place in the fluid flow rate, satisfying the acoustic compact condition (k 0 _r << 1). Therefore, the portion of the gradient of G _θ (x, z) in the above equation (33) that contributes to the surface fluid flow pressure fluctuation depends on the angular frequency θ as expressed by the following equation (33-1). It can be approximated only by the function of the distance r.

On the other hand, since the sound pressure moves at the sound speed and propagates to a distant place, the acoustic compact condition is not satisfied (that is, (k ₀ r >> 1) at the distant point). Therefore, the portion of the gradient of G _θ (x, z) in the above equation (33) that contributes to the surface sound pressure fluctuation is expressed as in the following equation (33-2).

The autocorrelation function of the “function p _{* θ} (x) based on the surface pressure fluctuation p _θ (x) at the observation point x” (see the above equation (4)) on the left side of the above equation (30) is represented by the following equation (34). ).

The above equation (34) is expressed as the following equation (35) using the acoustic compact condition of the gradient of G _θ (x, z) (see equation (33)).

Here, the correlation distance l _{x, cr} , the correlation area S _{x, cr} (see FIG. 5), and the correlation volume V _{x, cr} of the physical quantity of the arbitrary observation point x in the target angular frequency band [θ _l , θ _h ] are respectively defined. It is defined by the following equations (36-1), (36-2) and (36-3). Incidentally, the theta _m in the formula is the center angular frequency of the target angular frequency band (θ _m = 2πf _m).

That is, the correlation area S _{x, cr} is proportional to the square of the average flow velocity, and the correlation volume V _{x, cr} is proportional to the cube of the average flow velocity.

Since the angular frequency band of interest in the present embodiment is relatively large, the correlation distance l _{x, cr} is relatively small. Therefore, the correlation area S _{x, cr} and the correlation volume V _{x, cr} are also relatively small. Therefore, as shown in FIG. 5, the correlation area S _{x, cr} is included in the surface region Sx (S _{x, cr} ｃSx). Here, the integrand of one item on the right side of the above equation (35) includes the evaluation point y on the surface S of the rigid body 41, and a region (S＼Sx) of the surface S that does not include the surface region Sx ). Therefore, the physical quantity of the observation point x is uncorrelated with the physical quantity of the evaluation point y, and as a result, one item on the right side of the above equation (35) becomes equal to zero.

即ち、自己相関関数（式（４）参照）は、任意の関数Ｆ_θのノルムを着目角周波数帯において積分した値である。 In addition, in the following, the product of arbitrary function F _theta and its complex conjugate function F _theta ^* is referred to as "norm of the function F _theta" is defined as the following equation (37).

That is, the autocorrelation function (see equation (4)) is the integral value at the focus angle frequency band norm of any function F _theta.

Thereby, the above equation (35) can be expressed as the following equation (38).

This autocorrelation function can be modified as shown in the following equation (39).

Here, z1 and z2 in the expression are arbitrary evaluation points in the flow domain 42, and z1 ≠ z2. The correlation between the physical quantity of the evaluation point z1 and the physical quantity of the evaluation point z2 in the angular frequency band of interest can be ignored when the evaluation point z1 is not included in the correlation volume V _{z2, cr} of the evaluation point z2. Therefore, the above equation (39) can be further modified to the following equation (40).

The part contributing to the surface sound pressure fluctuation in the “part integrated in the angular frequency band of interest” on the right side of the above equation (40) is approximated by the following equation (41) using the above equation (33-2). it can.

The flow velocity u in the above equation (41) can be decomposed into an average flow velocity U− as an average component and a turbulent flow velocity u as a turbulence component, as shown in the following equations (42-1) and (42-2). The vorticity ω can be decomposed into an average vorticity Ω− as an average component and a turbulence vorticity ω− as a turbulence component. In addition, "-" of "U-" and "Ω-" is a notation representing an average, and replaces a bar attached to letters U and Ω in the formula described later. "~" Of "u ~" is a notation representing a turbulence component when the flow velocity u is averaged, and replaces a tilde added to a letter u in an expression described later. “To” of “ω−” is a notation representing a turbulence component when the vorticity ω is averaged, and replaces a tilde added to a character ω in an expression described later.

ここで、ｒ１、ｒ２及びｒ３は、観測点ｘを原点としたときのベクトルｒのｅ１成分、ｅ２成分及びｅ３成分である。 In addition, in the space coordinate system of the rigid body 41, the vector r is defined by the following equation (43).

Here, r1, r2, and r3 are the e1, e2, and e3 components of the vector r when the observation point x is the origin.

When these equations (42-1), (42-2) and (43) are substituted into the right side of the above equation (41), the right side can be transformed into the following equation (44). Since the product of the turbulence components is so small that it can be ignored, its description is omitted in the following equation (44).

Here, a correlation function between two arbitrary functions E _θ and F F _θ is defined as in the following equation (45) in the angular frequency band of interest.

ここで、ｋは流速ｕ_θと渦度ω_θとの関係を規定した定数である。 Using the above equations (4) and (45), the following equations (46-1) to (46-6) are obtained from the vortex uniform isotropic hypothesis.

Here, k is a constant that defines the relationship between the flow velocity u _θ and the vorticity ω _θ .

が成立するため、乱れ渦度ω〜の自己相関関数は下記式(46-7)のように近似できる。

When calculating the degree of contribution to the surface fluid flow pressure fluctuation, the following relational expression between the spatial derivative and the time derivative:

Holds, the autocorrelation function of the turbulence vorticity ω can be approximated by the following equation (46-7).

が成立するため、乱れ渦度ω〜の自己相関関数は下記式(46-8)のように近似できる。

On the other hand, when calculating the contribution to the surface sound pressure fluctuation, the following relational expression is obtained between the spatial derivative and the time derivative of the sound pressure moving at the sound speed;

Holds, the autocorrelation function of the turbulence vorticity ω〜 can be approximated by the following equation (46-8).

By rearranging the right side of the above equation (44) using the above equations (46-1) to (46-8), the left side of the above equation (41) is finally expressed as the following equation (47). .

The right side of the above equation (47) corresponds to a part contributing to the surface sound pressure fluctuation among the “part integrated in the angular frequency band of interest” on the right side of the above equation (40). Therefore, the following equation (48) is obtained by substituting the right side of equation (47) into the corresponding part of equation (40).

When the integrand on the right side of the above equation (48) is multiplied by “square ρ ² of the density of the flow field” and the evaluation point z2 is rewritten into a general notation z, the APDS calculation defined by the following equation (49) is obtained. An expression is obtained. In the present specification, the equation (49) derived in this manner is defined as “the flow field at the evaluation point z in the flow domain 42” with respect to “the surface sound pressure fluctuation pa _θ (x) at the observation point x of the rigid body 41”. Is defined as an index indicating the degree of contribution. Incidentally, to multiply the [rho ² is by using a p ^{* =} p / ρ in the process of deriving the APDS formula.

Note that the following approximate expression (50) holds between the surface sound pressure fluctuation pa _θ (x) and APDS (x, z) at the observation point x (expressions (38) and (48)). And equation (49)).

That is, the integral value of the “norm of the surface sound pressure fluctuation pa _θ (x) at the observation point x” in the angular frequency band of interest (in other words, the autocorrelation function of the surface sound pressure fluctuation pa _θ (x) at the observation point x) ) Can be approximated as “a value obtained by spatially integrating the APDS for the observation point x in the flow domain”. For this reason, the APDS functions with high accuracy as an index indicating the contribution of the flow field at the evaluation point z to the surface sound pressure fluctuation pa _θ (x) at the observation point x. The above is the description of the derivation of the APDS calculation formula.

(Calculation of IDS)
From Equations (49) and (50), it can be considered that the APDS represents “sound pressure generated by the spatial flow (sound source) of the flow field at the evaluation point z and reaching the observation point x via spatial propagation”. it can. Therefore, “a unit vector in the sound pressure propagation direction that is generated by the spatial flow (sound source) of the flow field at the evaluation point z and reaches the observation point x via the spatial propagation” can be expressed by the following equation (51). .

なお、

は、観測点ｘにおける剛体４１の表面の法線ベクトルである。 From Expression (2), Expressions (50), and Expression (51), the acoustic intensity Iin _n (that is, the angular frequency band of interest [θ]) incident at the observation point x along the normal direction of the surface of the rigid body 41 is obtained. ₁ , θ _h ], the spectrum average of the sound intensity at an arbitrary observation point x can be expressed as the following equation (52).

In addition,

Is a normal vector of the surface of the rigid body 41 at the observation point x.

Further, by substituting equation APDS shown in equation (49) into the equation (52), sound intensity Iin _n can be expressed by the following equation (53).

The integrand in Expression (53) represents “the contribution of the flow field at the evaluation point z to the acoustic intensity Iin _n incident along the normal direction of the surface of the rigid body 41 at the observation point x”. This integrand is defined as an IDS (Intensity Density Source) as in the following equation (54). Hereinafter, the IDS may be referred to as a “first index”.

(Calculation of sound power)
Sound power P _in, as in the following equation (55), a sound intensity Iin _n can be calculated by the surface integration at the surface of the rigid body 41.

The area component of equation (55) is defined as APwDS (Acoustic Pressure Power Density Source) as in equation (56) below.

上記式（５７）より、ＡＰｗＤＳは、「剛体４１の表面に到達する音響パワーＰｉｎに対する評価点ｚにおける流れ場の寄与度」と考えることができる。以降、ＡＰｗＤＳは「第２指標」と称呼される場合がある。以上で説明した指標（ＩＤＳ及びＡＰｗＤＳ等）を用いることにより、音響パワーＰ_ｉｎと車両周囲の流れ場との関係から、車内音Ｐ_ｏｕｔと車両周囲の流れ場との関係を分析／解析することが可能になる。 Therefore, equation (55) can be rewritten as equation (57) below.

From the above equation (57), APwDS can be considered as “the contribution of the flow field at the evaluation point z to the acoustic power Pin reaching the surface of the rigid body 41”. Hereinafter, APwDS may be referred to as a “second index”. By using the index described above (IDS and APwDS etc.), the relationship between the sound power P _in the vehicle surrounding flow field, analyzing / analyzing the relationship between the interior noise P _out and the vehicle surrounding flow field Becomes possible.

<Specific operation of the present embodiment>
A specific operation of the present embodiment will be described. The CPU 14 (hereinafter simply referred to as “CPU”) of the calculation unit 11 executes the “analysis routine” shown in FIG. 6 every time a predetermined time elapses.

When starting this routine, the CPU reads from the ROM 16 and / or the HDD 17 the data of the vehicle model 20 to be analyzed for wind noise, the data of the flow domain 21 indicating the analysis range, and the unsteady CFD (computational fluid dynamics). ) simulation necessary data, and necessary for the analysis of wind noise formula (APDS, IDS, reads data of Iin _n and formulas such as APwDS), stores these data in the RAM 15. In this example, the surface of the left front side glass 22 of the vehicle model 20 is defined as an “analysis surface”.

  At a predetermined timing, the CPU starts the routine of FIG. 6 from step 600, sequentially performs the processing of steps 605 to 650, and then proceeds to step 695 to end this routine.

  Step 605: The CPU determines the time history data u (z, t) of the flow velocity of the flow field and the time history data ω (z, t) of the vorticity for each spatial node z within the analysis range (flow domain 21). Each is calculated over time. The position of the space node z in the flow domain 21 is determined so that the space node z does not exist on the surface of the vehicle model 20 (that is, the space node z is separated from the surface of the vehicle model 20). It is set in advance.

  Step 610: The CPU averages each of the time history data u (z, t) of the flow velocity and the time history data ω (z, t) of the vorticity to calculate the average for each spatial node z in the flow domain 21. The flow velocity U- (z) and the average vorticity Ω- (z) are calculated.

Step 615: CPU is time data u (z, t) of the flow rate calculated in step 605 is treated fast Fourier transform (FFT), the autocorrelation of the disturbance flow rate u~ θ _(z) at the target angular frequency band Calculate the function. This autocorrelation function is calculated for each spatial node z. For example, when 2 kHz is selected as the frequency band of interest, the autocorrelation function of the turbulence velocities u to _θ (z) is calculated for each spatial node z in the flow domain 21 within a range from 1420 Hz to 2840 Hz. The processing from step 605 to step 615 corresponds to “unsteady CFD calculation processing”.

Step 620: The CPU calculates the IDS for the observation point x at each evaluation point z in the flow domain 21 based on the APDS calculation equation (Equation (49)) and the IDS calculation equation (Equation (54)). Is performed for all observation points x on the analysis surface. As described above, when the node x on the analysis surface is defined as x _i (i: 1 to m) and the spatial node z in the flow domain 21 is defined as z _j (j: 1 to n), the CPU: the IDS for the observation point _{x k} n pieces (the number of evaluation points z) is calculated. The CPU, for all the observation points _{x k} Since do this, calculates the mn number of IDS in total.

Step 625: CPU, based on IDS calculated in step 620, according to the following equation (58), calculates the sound intensity Iin _n.

As described above, “the value obtained by spatially integrating the IDS at an arbitrary observation point x in the flow domain 21” represents “the sound intensity Iin _{n at the} observation point x”. Therefore, the CPU extracts n IDSs (x _k , z _j ) for the observation point x _k and adds the extracted IDSs (x _k , z _j ) (that is, IDS (x _k , z _j )). ) by spatial integration) in the flow domain 21, and calculates the acoustic intensity Iin _n at the observation point _{x k.} The CPU calculates m (the number of observation points x) sound intensities Iin _n .

Furthermore, CPU sends a display command to the image display data of the calculated sound intensity Iin _n on the display screen 19 to the output unit 13. When receiving the display command, the output unit 13 displays the data on the display screen 19 as an image. Figure 7 is an example of a distribution diagram of sound intensity Iin _n in the image displayed analyzed surface on the display screen 19 based on the display command (left front side window 22). The frequency band of interest is set to 2 kHz. In the example of FIG. 7, the larger observation point sound intensity Iin _n, dark color data are correlated. According to this display, the user can recognize the "out of the analysis surface, sound intensity Iin _n is greater position". That is, the user can for any position of the analysis surface is sound intensity Iin _n contributes a large degree in interior noise to recognize whether the incident.

Step 630: CPU, according to equation (55), to calculate the sound power _{P in.} That, CPU, by surface integral of the sound intensity Iin _n calculated in step 625 by analyzing the surface, to calculate the sound power _{P in.} Further, the CPU calculates APwDS from the relationship between Expressions (56) and (57). As described above, APwDS is "contribution of the flow field at the evaluation point z _j for the acoustic power Pin reaching the vehicle surface." CPU calculates at each evaluation point _{z j} in the flow domain 21 APwDS.

  Step 635: The CPU displays a screen for inputting the reference value of APwDS on the display screen 19. The user inputs a reference value of APwDS (hereinafter, referred to as “first reference value”).

Step 640: The CPU displays a screen showing the isosurface distribution of the APwDS on the display screen 19 based on the first reference value. FIG. 8 is an example of an isosurface distribution of APwDS image-displayed on the display screen 19 based on the first reference value. Points of the evaluation points z _j where the value of APwDS is larger than the first reference value are displayed in dark gray. That is, the position of the flow field having a high contribution to the sound power Pin is displayed in dark gray. Therefore, the user can recognize which position in the flow domain 21 the spatial flow greatly contributes to the sound power Pin.

  Note that the CPU may cause the display screen 19 to display a screen for inputting a plurality of APwDS reference values. In this configuration, it is assumed that the user has input a first reference value and a second reference value that is greater than the first reference value. FIG. 9 is an example of an isosurface distribution of APwDS image-displayed on the display screen 19 based on the second reference value. In the image of FIG. 9, the size of the dark gray area is smaller than that of the image of FIG. Therefore, the user can specify the position of the spatial flow that contributes more to the sound power Pin.

Step 645: The CPU displays, on the display screen 19, a screen for designating parameters constituting the IDS and a screen for designating an evaluation section of the designated parameters. In this example, the parameters that can be specified are the average flow velocity U−, the turbulent flow rates u to _θ , and the average vorticity Ω−. It should be noted that other parameters constituting the IDS (for example, a vector r connecting the observation point and the evaluation point) may be specified. The user isosurface distribution APwDS displayed by sound intensity Iin _n distribution and step 640, which is displayed in step 625 as a reference, specify the parameters to be focused.

  Further, the user specifies an evaluation section of the specified parameter. In this example, it is assumed that the user has designated the cross section indicated by the broken line L in FIG.

Step 650: The CPU causes the display screen 19 to display a distribution diagram of the parameters designated in step 645. FIG. 11 shows a distribution diagram of the average flow velocity U- (z) around the left side mirror 23 in the evaluation section (see the broken line L in FIG. 10). Further, FIG. 12 shows a distribution diagram of the turbulent flow rates u to _θ (z) around the left side mirror 23 in the evaluation cross section (see the broken line L in FIG. 10). By referring to these distribution maps, the user can determine the characteristics of the spatial flow contributing to the in-vehicle sound (that is, the average flow velocity U- (z), the turbulent flow rates u to _θ (z), and the average vorticity Ω). − (Z)), and the design (position and shape, etc.) of the left side mirror 23 can be examined.

<Reliability of index (IDS and ADwDS)>
Next, the reliability of the first index (IDS) and the second index (ADwDS) described above will be described. The inventor has prepared an actual vehicle having the same shape as the vehicle model 20 shown in FIG. Hereinafter, the actual vehicle is referred to as a “first vehicle”, and the vehicle model 20 corresponding to the first vehicle is referred to as a “first vehicle model”.

  The inventor measured the interior sound of the first vehicle. Further, the inventor performed the above-described analysis processing on the first vehicle model using the present embodiment device. Then, the inventor examined the result of the analysis processing, and moved the position of the left side mirror of the first vehicle by 200 mm toward the front side of the vehicle to mount it. The vehicle whose design has been changed in this manner is referred to as a “second vehicle”. Further, the inventor has prepared a vehicle model corresponding to the second vehicle. This vehicle model is referred to as a “second vehicle model”.

  The inventor measured the in-vehicle sound of the second vehicle. The inventor has confirmed that the in-vehicle sound is reduced as compared with the first vehicle. In this way, by changing the design (position) of the left side mirror, the interior sound could be reduced. The specific effect of improving the in-vehicle sound is shown in FIG. 14 described later.

Furthermore, the inventor performed the above-described analysis processing on the second vehicle model using the present embodiment device. Figure 13 (a) is a "distribution diagram of sound intensity Iin _n in the analysis surface (left front side window)" in the case of executing the analysis processing on the first vehicle model. Figure 13 (b) is a "distribution diagram of sound intensity Iin _n in the analysis surface (left front side window)" in the case of executing the analysis processing for the second vehicle model. As shown in FIGS. 13A and 13B, by changing the design (position) of the left side mirror, the magnitude of the sound intensity Iin _n is reduced throughout the analysis surface. In particular, in the second vehicle model, region of the sound intensity Iin _n degree of contribution is large interior noise (black area) is smaller as compared with the case of the first vehicle model.

  FIG. 14 is a diagram illustrating “difference in actually measured vehicle sound between the first vehicle and the second vehicle”, “acoustic power calculated by analyzing the first vehicle model using the present embodiment, and And the difference from the acoustic power calculated by analyzing the second vehicle model using the above. As shown in the graph, the calculation result of the improvement effect by the analysis processing using the present embodiment has a smaller error as compared with the actually measured improvement effect of the in-vehicle sound. Therefore, it can be understood that the indices (IDS and ADwDS) described above are highly reliable in analyzing the in-vehicle sound. Therefore, the wind noise can be analyzed with higher accuracy.

  As described above, according to the present embodiment, the position of the flow field that greatly contributes to the in-vehicle sound is specified using the above-described indices (IDS and ADwDS), so that the in-vehicle sound can be efficiently and reliably generated. Can be reduced.

  Specifically, in the present embodiment, the isosurface distribution of APwDS is displayed on the display screen 19. By appropriately setting the reference value of the APwDS, the user can specify the position of the spatial flow that contributes more to the acoustic power Pin.

Then, the user specifies parameters (average flow velocity U−, turbulent flow velocity u to _θ , and average vorticity Ω−) that relatively contribute to the APwDS value with reference to the isosurface distribution of APwDS. . The user can easily grasp the parameter that causes the APwDS to increase, and can more efficiently study and change the shape of the vehicle.

  As described above, the analyzer according to the embodiment has been described. However, the present invention is not limited to the above embodiment, and various modifications can be made without departing from the object of the present invention.

  For example, in the above embodiment, a vehicle model is used as a structure model to be analyzed. However, the present invention is not limited to this. For example, a structure model such as an aircraft and a ship may be used.

Information displayed on the display screen 19 at step 625 is not limited to the example of FIG. 7, may be information about the size distribution of the acoustic intensity Iin _n in the analysis surface. For example, the CPU may display, on the display screen 19, the position (observation point x) of the analysis surface _where the magnitude of the sound intensity Iinn is _equal to or larger than a predetermined value.

  The information displayed on the display screen 19 in step 640 is not limited to the examples of FIGS. 8 and 9 and may be any information as long as the information is related to the distribution of the size of the APwDS in the flow domain 21. For example, the CPU may display on the display screen 19 images that are color-coded for each size of APwDS.

10: analysis device, 11: arithmetic unit, 12: input unit, 13: output unit, 14: CPU, 15: RAM, 16: ROM, 17: hard disk (HDD), 18: I / O interface, 19: display screen , 20: vehicle model, 21: flow domain, 22: left front side glass, 23: left side mirror.

Claims (1)

An analyzer for analyzing wind noise generated in a moving structure,
An unsteady CFD calculation process of moving a structure model obtained by modeling the structure is performed, and an average flow velocity and an average vorticity of the flow field in a predetermined region of a flow field around the structure model at a predetermined time are executed. Is calculated for each spatial node that is a node in the predetermined region, and in the angular frequency band of interest that is the angular frequency band to be analyzed for wind noise, a value based on the amplitude of the turbulent flow velocity in the predetermined region is calculated. First calculating means for calculating for each of the spatial nodes;
Based on the average flow velocity calculated by the first calculation means, the average vorticity, and a value based on the amplitude of the turbulent flow velocity, along a normal direction at an observation point on the analysis surface of the structure. Calculating means for calculating a first index indicating a degree of contribution of the flow field at the spatial node to the sound intensity incident thereon;
Based on the first index calculated by the second calculating means, the sound power reaching the analysis surface is calculated, and the contribution of the flow field at the spatial node to the sound power reaching the analysis surface is calculated. Third calculating means for calculating a second index to be indicated,
Display means for displaying information about the distribution of the size of the second index in the predetermined area;
An analysis device comprising:

Images