CN107220212B - Boundary element calculation method for two-dimensional non-compact boundary acoustic scattering - Google Patents

Boundary element calculation method for two-dimensional non-compact boundary acoustic scattering Download PDF

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CN107220212B
CN107220212B CN201710265667.8A CN201710265667A CN107220212B CN 107220212 B CN107220212 B CN 107220212B CN 201710265667 A CN201710265667 A CN 201710265667A CN 107220212 B CN107220212 B CN 107220212B
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刘秋洪
蔡晋生
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Northwestern Polytechnical University
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Abstract

The invention discloses a boundary element calculation method for two-dimensional non-compact boundary acoustic scattering, and belongs to the technical field of acoustics. The method comprises the following steps: discretizing the non-compact boundary into a series of grid cells and determining the geometric parameters of each grid cell; for far field observation point x, and any two grid cells zmAnd znDetermining a free space Green's function G (x, z)n,ω)、G(zm,znω) and
Figure DDA0001275950540000011
according to G (x, z)n,ω)、G(zm,znω) and
Figure DDA0001275950540000012
determining a trend of sound sources towards znScattered sound field p (x, z) of non-compact boundariesnω); for observation point x, sound source point y and point z on boundarynDetermining the free space Green functions G (x, y, omega) and G (z)nY, ω) and
Figure DDA0001275950540000013
according to p (x, z)n,ω)、G(x,y,ω)、G(znY, ω) and
Figure DDA0001275950540000014
the sound field p (x, y, ω) at the observation point x is determined, which the sound waves generated by the sound source point y propagate to. The boundary element method of the invention enables the sound source to tend to the scattering boundary when solving the sound scattering of the non-compact boundary, when the distance between the sound source and the boundary is far smaller than the characteristic size of the grid unit, the boundary grid unit can still be processed as a normal unit, and the calculation precision is greatly improved compared with the traditional boundary element method under the same boundary grid discrete condition.

Description

Boundary element calculation method for two-dimensional non-compact boundary acoustic scattering
Technical Field
The invention relates to the field of acoustics, in particular to a boundary element calculation method for two-dimensional non-compact boundary acoustic scattering.
Background
In the frequency domain, the acoustic wave equation is generally expressed as
Figure BDA0001275950520000011
In the formula (I), the compound is shown in the specification,
Figure BDA0001275950520000012
for Laplace operator, ω, k, p, and γ are circle frequency, acoustic wave number, sound pressure, and sound source intensity, respectively, and k is ω/c0，c0Is the speed of sound.
Solid boundaries often exist in the space where the sound waves propagate, and if the geometric characteristic size of the boundaries is larger than or close to the wavelength of the sound waves, the boundaries are not compact acoustically, and the noise excited by the sound source is radiated on the boundaries to generate a scattering effect. The calculation of sound propagation can adopt a boundary element method, when a sound source is positioned at a y point and an observation point is positioned at an x point, the solution of an acoustic hard boundary sound scattering and acoustic wave equation is
Figure BDA0001275950520000013
In the formula: g (x, y, ω) is the frequency domain free space Green's function and z is the point on the scattering boundary. The key points of the boundary element method comprise the following steps: firstly, dispersing a solid boundary into a series of grid units, and assuming that the scattering sound pressure on each grid unit is the same in size; then, placing the observation point at a z point on a scattering boundary, and solving acoustic scattering p (z, y, omega) at the boundary by using the acoustic wave equation; finally, the sound field at the far-field x point is solved by using p (z, y, omega) and an acoustic wave equation. The boundary element method only needs discrete solid boundaries and has the advantage of high calculation efficiency. However, in the pneumatic noise problem, the distance between the sound source in the boundary layer and the solid boundary is far smaller than the characteristic size of the boundary grid unit, and the sound scattering of each point on the same boundary grid unit is different, that is, the grid unit cannot be processed as a normal unit, otherwise, a calculation result has a large error. In order to improve the calculation accuracy, the boundary grid needs to be encrypted, so that the characteristic size of the boundary grid is close to the distance between the sound source and the solid boundary, a large number of grid units are needed, the calculation efficiency is very low, and the application of the boundary element method in pneumatic acoustic analysis is limited.
In summary, in the prior art, there is a problem that the conventional boundary element method is inefficient in calculating the sound scattering when the sound source is very close to the solid boundary.
Disclosure of Invention
The embodiment of the invention provides a boundary element calculation method for two-dimensional non-compact boundary acoustic scattering, which is used for solving the problem of low acoustic scattering calculation efficiency when a sound source is very close to a solid boundary in the traditional boundary element method.
The embodiment of the invention provides a boundary element calculation method for two-dimensional non-compact boundary acoustic scattering, which comprises the following steps:
establishing a boundary meta-model, and dispersing a non-compact boundary S into M boundary grid units;
determining center point z of each boundary grid cellmUnit outer normal direction n (z)m) And area s (z)m) And the intensity γ and circular frequency ω of the sound source;
determining an observation point x outside the non-compact boundary S, and determining a central point z of each grid cell according to the boundarymUnit outer normal direction n (z)m) And area s (z)m) And intensity gamma and circular frequency omega of the sound source, determining that the sound source tends to a boundary mesh cell z by formula (1)nScattered sound field p (x, z) of non-compact boundary Sn,ω)；
Trending toward boundary grid cell z according to sound sourcenScattered sound field p (x, z) of non-compact boundary Snω), the sound field p (x, y, ω) where the sound wave generated by the sound source point y propagates to the observation point x is determined by the formula (2);
the formula (1) is as follows:
Figure BDA0001275950520000021
wherein M, n is 1,2, …, M is the number of the boundary grid cell, M is the total number of the grid cells; p (x, z)nω) tends to boundary grid cell z for sound sourcenThe sound field of time observation point x; p is a radical of0(x,znω) tends to boundary grid cell z for sound sourcenDirect acoustic radiation of time to observation point x, and p0(x,zn,ω)＝γG(x,znω), γ is the sound source intensity, ω is the circular frequency; z is a radical ofmIs marginally different from znA grid cell center point of (a); g is a frequency domain free space Green function; n (z)m) Is zmUnit outer normal vector of boundary at point, s (z)m) Is zmThe unit area of the boundary grid where the point is located; s' is removal of znBoundary surfaces of the mesh cell portions.
The formula (2) is as follows:
Figure BDA0001275950520000031
wherein y is a sound source point; n is 1,2, …, M is the boundary grid cell number, M is the total number of grid cells; z is a radical ofnThe grid cell center point is numbered n; n (z)n) Is znUnit outer normal vector of boundary at point, s (z)n) Is znThe unit area of the boundary grid where the point is located; p is a radical of0(x, y, ω) is the direct acoustic radiation of the source point y to the observation point x, and p0(x, y, ω) ═ γ G (x, y, ω), G is the frequency domain free space green's function.
Preferably, the solving process of the formula (1) comprises:
a. discretizing equation (1) into a series of algebraic linear equations
Hp(x,z,ω)＝p0(x,z,ω) (3)
Wherein the sub-entries of the H matrix are
Figure BDA0001275950520000032
Wherein m, n is 1,2, …,180 is the number of the boundary grid cell; z is a radical ofmAnd znIs numbered asmAnd n boundary grid cell center points; s (z)m) Is the area of the grid cell numbered m, n (z)m) Is zmUnit outer normal vector of boundary at point, matrices p (x, z, ω) and p0(x, z, ω) are each
Figure BDA0001275950520000033
b. Determination of G (x, z)n,ω)、G(zm,znω) and
Figure BDA0001275950520000034
with respect to the problem of two-dimensional sound propagation,
Figure BDA0001275950520000035
Figure BDA0001275950520000041
Figure BDA0001275950520000042
wherein the content of the first and second substances,
Figure BDA0001275950520000043
and
Figure BDA00012759505200000413
the first-class Hankel functions are 0-order and 1-order respectively, j is an imaginary number unit, and k is omega/c0Is the acoustic wave number, ω is the circular frequency, c0Is the speed of sound;
c. determining a matrix H; when m is not equal to n, the number of the first and second groups,
Figure BDA0001275950520000044
wherein the content of the first and second substances,
Figure BDA0001275950520000045
Figure BDA0001275950520000046
wherein phi is znPoint and zmAn included angle between two end points of the boundary grid unit;
d. solving an algebraic linear equation set (3) by adopting a numerical method to obtain p (x, z)n,ω)。
Preferably, the solving process of the formula (2) comprises:
a. determining G (x, y, omega), G (z)nY, ω) and
Figure BDA0001275950520000047
for the two-dimensional sound propagation problem,
Figure BDA0001275950520000048
Figure BDA0001275950520000049
Figure BDA00012759505200000410
b. determination using equation (15)
Figure BDA00012759505200000411
Figure BDA00012759505200000412
Wherein the content of the first and second substances,
Figure BDA0001275950520000051
Figure BDA0001275950520000052
wherein phi is the y point and znAn included angle between two end points of the boundary grid unit;
c. p (x, y, ω) is calculated using equation (2).
In the embodiment of the invention, a two-dimensional non-compact boundary acoustic scattering boundary element calculation method is provided, and compared with the prior art, the boundary element calculation method has the following beneficial effects: in the traditional boundary element method, an observation point is placed on a boundary when sound scattering of a non-compact boundary is solved, and when the distance between a sound source and the boundary is far smaller than the size of a unit grid, the unit of the boundary grid cannot be processed as a common unit, so that a calculation result is inaccurate. The boundary element method of the invention puts the sound source on the boundary when solving the sound scattering of the non-compact boundary, when the distance between the sound source and the boundary is far smaller than the size of the unit grid, the boundary grid unit can still be processed as a normal unit, and the calculation precision is greatly improved compared with the traditional boundary element method under the same boundary grid discrete condition.
Drawings
FIG. 1 is znPoint and zmA schematic diagram of an included angle of an end point of a grid unit;
FIG. 2 shows the sound source points y and znA schematic diagram of an included angle of an end point of a grid unit;
FIG. 3 is a schematic diagram of two-dimensional non-compact cylindrical acoustic scattering;
fig. 4 is a far-field sound field directivity comparison diagram of a numerical solution of a boundary element method of cylindrical sound scattering, a numerical solution of a traditional boundary element method and a theoretical analytic solution.
Description of reference numerals:
1、zma grid cell; 2. z is a radical ofmAn outer normal vector of a grid cell; 3. z is a radical ofnPoint; 4. z is a radical ofnA grid cell; 5. z is a radical ofnAn outer normal vector of a grid cell; 6. a sound source point y; 7. a two-dimensional cylinder; 8. a viewpoint x; 9. a sound source point y; 10. point z on the cylindrical surface; 11. the cylindrical sound scattering space directivity theory is analyzed; 12. the cylindrical sound scattering space directivity is solved by the boundary element method; 13. the traditional boundary element method numerical solution of the directivity of the cylindrical sound scattering space.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment of the invention provides a boundary element calculation method for two-dimensional non-compact boundary acoustic scattering by taking two-dimensional non-compact cylindrical acoustic scattering as an example, and a figure 1 shows znPoint and zmA schematic diagram of an included angle of an end point of a grid unit; FIG. 2 shows the sound source points y and znA schematic diagram of an included angle of an end point of a grid unit; FIG. 3 is a schematic diagram of two-dimensional non-compact cylindrical acoustic scattering; fig. 4 is a far-field sound field directivity comparison diagram of a numerical solution of a boundary element method of cylindrical sound scattering, a numerical solution of a traditional boundary element method and a theoretical analytic solution.
As shown in fig. 3, there is a two-dimensional cylinder with a diameter D of 100mm in space, a monopole point sound source of unit intensity is located at the y point on the x axis, the distance D from the surface of the cylinder is 0.05mm, the circular frequency of the sound source is ω, the distance r from the center of the cylinder is the observation point xx＝12800mm。
The specific process of the embodiment includes the following steps:
step 1, dividing grids and establishing a boundary meta-model. In this embodiment, the two-dimensional cylinder has a simple structure, adopts a self-programming program to complete mesh division, and disperses 180 discrete nodes
Figure BDA0001275950520000061
(m-1, 2, …,180 stands for the discrete point numbered m) is uniformly arranged on the cylindrical surface, so that the two-dimensional cylindrical surface is discrete into 180 wire grid units, the length dimension of which is about 1.745mm, which is far larger than the distance between the sound source and the cylindrical surface by 0.05 mm.
Step 2, extracting the central point coordinate z of each boundary grid unitmUnit outer normal direction n (z)m) And area s (z)m) And m is 1,2, …,180 represents a grid cell numbered m. For this embodiment, the origin of coordinates is located at the center of the cylinder, and the extraction process includes:
a. and extracting the coordinates of the discrete nodes. Extracting each discrete node vector from the established boundary element model
Figure BDA0001275950520000062
Where m-1, 2, …,180 represents a discrete point numbered m,
Figure BDA0001275950520000063
and
Figure BDA0001275950520000064
coordinate values of the mth discrete point in the x and y directions, respectively.
b. And determining the coordinates of the central point of each wire grid cell. The m-th line grid unit is composed of discrete points
Figure BDA0001275950520000071
And
Figure BDA0001275950520000072
determination of the center point vector
Figure BDA0001275950520000073
Has the coordinates of
Figure BDA0001275950520000074
c. And determining the unit external normal direction vector of each line grid cell. The m-th wire grid unit external normal direction vector
Figure BDA0001275950520000075
Has the coordinates of
Figure BDA0001275950520000076
d. The area of each wire grid cell is determined. For the m-th wire grid cell, the area is equal to the length of the wire grid cell
Figure BDA0001275950520000077
Step 3, for observation point x and point z on boundary SnThe following integral equation is solved by adopting a boundary element method for calculating the trend z of the sound sourcenScattered sound field p (x, z) of point non-tight boundary Sn,ω)
Figure BDA0001275950520000078
Wherein m, n is 1,2, …,180 is the number of the boundary grid cell; p (x, z)nω) tends to boundary grid cell z for sound sourcenThe sound field of time observation point x; p is a radical of0(x,znω) tends to boundary grid cell z for sound sourcenDirect acoustic radiation of time to observation point x, and p0(x,zn,ω)＝G(x,znω), ω is the circular frequency; z is a radical ofmIs marginally different from znA grid cell center point of (a); g is a frequency domain free space Green function; n (z)m) Is zmUnit outer normal vector of boundary at point, s (z)m) Is zmThe unit area of the boundary grid where the point is located; s' is removal of znBoundary surfaces of the mesh cell portions.
The specific calculation process is as follows:
a. discretizing equation (1) into a series of algebraic linear equations
Hp(x,z,ω)＝p0(x,z,ω) (3)
Wherein the sub-entries of the H matrix are
Figure BDA0001275950520000081
Wherein m, n is 1,2, …,180 is the number of the boundary grid cell; z is a radical ofmAnd znThe cell center points are the boundary grid cell center points with the serial numbers of m and n; s (z)m) Is the area of the grid cell numbered m, n (z)m) Is zmUnit outer normal vector of boundary at point, matrices p (x, z, ω) and p0(x, z, ω) are each
Figure BDA0001275950520000082
b. Determination of G (x, z)n,ω)、G(zm,znω) and
Figure BDA0001275950520000083
with respect to the problem of sound propagation in two dimensions,
Figure BDA0001275950520000084
Figure BDA0001275950520000085
Figure BDA0001275950520000086
wherein the content of the first and second substances,
Figure BDA0001275950520000087
and H1 (1)The first-class Hankel functions are 0-order and 1-order respectively, j is an imaginary number unit, and k is omega/c0Is the acoustic wave number, ω is the circular frequency, c0Is the speed of sound;
c. determining a matrix H; when m is not equal to n, the number of the first and second groups,
Figure BDA0001275950520000088
wherein the content of the first and second substances,
Figure BDA0001275950520000089
Figure BDA00012759505200000810
wherein phi is znPoint and zmThe angle between the two end points of the boundary grid cell is shown in fig. 1.
d. Solving an algebraic linear equation set (3) by adopting a numerical method to obtain p (x, z)n,ω)。
And 4, calculating a sound field p (x, y, omega) of the sound wave generated by the sound source at the point y and transmitted to the point x by using the following formula (2) for the observation point x and the sound source point y.
Figure BDA0001275950520000091
Wherein y is a sound source point; n is 1,2, …, M is the boundary grid cell number, M is the total number of grid cells; z is a radical ofnThe grid cell center point is numbered n; n (z)n) Is znUnit outer normal vector of boundary at point, s (z)n) Is znThe unit area of the boundary grid where the point is located; p is a radical of0(x, y, ω) is the direct acoustic radiation of the y-point source to the x-point, and p0(x, y, ω) is G (x, y, ω), which is the frequency domain free space green's function.
The specific calculation process is as follows:
a. determining G (x, y, omega), G (z)nY, ω) and
Figure BDA0001275950520000092
for the two-dimensional sound propagation problem,
Figure BDA0001275950520000093
Figure BDA0001275950520000094
Figure BDA0001275950520000095
b. determination using equation (15)
Figure BDA0001275950520000096
Figure BDA0001275950520000097
Wherein the content of the first and second substances,
Figure BDA0001275950520000098
Figure BDA0001275950520000099
wherein phi is the y point and znThe angle between the two end points of the boundary grid cell is shown in fig. 2.
c. P (x, y, ω) is calculated using equation (2).
When the acoustic wavenumber k is 20, the ratio of the acoustic wavelength to the diameter of the cylinder is about 3, and the cylinder is acoustically non-compact. Under the same boundary grid condition, the far-field sound field directivity pair of the numerical solution of the boundary element method of the patent, the numerical solution of the traditional boundary element method and the theoretical analytic solution is shown in fig. 3, wherein a curve 11 is the result of the theoretical analytic solution, a curve 12 is the numerical solution of the boundary element method of the patent, and a curve 13 is the numerical solution of the traditional boundary element method. The result of the traditional boundary element method is obviously different from the theoretical analytic solution, and the numerical solution of the boundary element method is consistent with the result of the theoretical analytic solution, so that on one hand, the correctness of the method is proved, and on the other hand, the superiority of the method is reflected.
The above disclosure is only a few specific embodiments of the present invention, and those skilled in the art can make various modifications and variations of the present invention without departing from the spirit and scope of the present invention, and it is intended that the present invention encompass these modifications and variations as well as others within the scope of the appended claims and their equivalents.

Claims (3)

1. A boundary element calculation method for two-dimensional non-compact boundary acoustic scattering is characterized by comprising the following steps:
establishing a boundary meta-model, and dispersing a non-compact boundary S into M boundary grid units;
determining center point z of each boundary grid cellmUnit outer normal direction n (z)m) And area s (z)m) And the intensity γ and circular frequency ω of the sound source;
determining an observation point x outside the non-compact boundary S, and determining a central point z of each grid cell according to the boundarymUnit outer normal direction n (z)m) And area s (z)m) And the intensity gamma and the circular frequency omega of the sound source, determining the scattered sound field p (x, z) of the non-compact boundary S when the sound source approaches the boundary grid cell by the formula (1)n,ω)；
According to the scattered sound field p (x, z) of the non-compact boundary S when the sound source tends to the boundary grid cellnω), by formula (2), determining the sound field p (x, y, ω) in which the sound wave generated by the sound source point y propagates to the observation point x, the sound source point y being placed on the boundary;
the formula (1) is as follows:
Figure FDA0002633739450000011
wherein M, n is 1,2, …, M is the number of the boundary grid cell, M is the total number of the grid cells; p (x, z)nω) tends to boundary grid cell z for sound sourcenThe sound field of time observation point x; p is a radical of0(x,znω) tends to boundary grid cell z for sound sourcenDirect acoustic radiation of time to observation point x, and p0(x,zn,ω)＝γG(x,znω), γ is the sound source intensity, ω is the circular frequency; z is a radical ofmIs marginally different from znA grid cell center point of (a); g is a frequency domain free space Green function; n (z)m) Is zmUnit outer normal vector of boundary at point, s (z)m) Is zmThe unit area of the boundary grid where the point is located; s' is removal of znBoundary surfaces of the mesh cell portions; g (x, z)nω) are point x and point znInter frequency domain free space green's function, G (z)m,znω) is the point zmAnd point znInter-frequency domain free space green's function;
the formula (2) is as follows:
Figure FDA0002633739450000012
wherein y is a sound source point; n is 1,2, …, M is the boundary grid cell number, M is the total number of grid cells; z is a radical ofnThe grid cell center point is numbered n; n (z)n) Is znUnit outer normal vector of boundary at point, s (z)n) Is znThe unit area of the boundary grid where the point is located; p is a radical of0(x, y, ω) is the direct acoustic radiation of the source point y to the observation point x, and p0(x, y, ω) ═ γ G (x, y, ω), G is the frequency domain free space green's function; g (x, y, ω) is the frequency domain free space Green's function between point x and point y, G (z)nY, ω) are point y and point znInter-frequency domain free space green's function.
2. The method for boundary element computation of two-dimensional non-compact boundary acoustic scattering according to claim 1, wherein the solving of the formula (1) comprises:
a. discretizing equation (1) into a series of algebraic linear equations
Hp(x,z,ω)＝p0(x,z,ω) (3)
Wherein the sub-entries of the H matrix are
Figure FDA0002633739450000021
Wherein m, n is 1,2, …,180 is the number of the boundary grid cell; z is a radical ofmAnd znThe cell center points are the boundary grid cell center points with the serial numbers of m and n; s (z)m) Is the area of the grid cell numbered m, n (z)m) Is zmUnit outer normal vector of boundary at point, matrices p (x, z, ω) and p0(x, z, ω) are each
Figure FDA0002633739450000022
b. Determination of G (x, z)n,ω)、G(zm,znω) and
Figure FDA0002633739450000023
with respect to the problem of sound propagation in two dimensions,
Figure FDA0002633739450000024
Figure FDA0002633739450000025
Figure FDA0002633739450000026
wherein the content of the first and second substances,
Figure FDA0002633739450000031
and
Figure FDA0002633739450000032
the first-class Hankel functions are 0-order and 1-order respectively, j is an imaginary number unit, and k is omega/c0Is the acoustic wave number, ω is the circular frequency, c0Is the speed of sound;
c. determining a matrix H; when m is not equal to n, the number of the first and second groups,
Figure FDA0002633739450000033
wherein the content of the first and second substances,
Figure FDA0002633739450000034
Figure FDA0002633739450000035
wherein phi is a point znAnd point zmThe included angle between two end points of the boundary grid unit;
z1,z2,…zMrespectively the central point of the grid cell with the number of 1, the central point of the grid cell with the number of 2 and the central point of the grid cell with the number of M of … …; z is z1,z2,…zMA set of (a);
d. solving an algebraic linear equation set (3) by adopting a numerical method to obtain p (x, z)n,ω)。
3. The method for calculating boundary elements of two-dimensional non-compact boundary acoustic scattering according to claim 2, wherein the solving process of the formula (2) comprises:
a. determining G (x, y, omega), G (z)nY, ω) and
Figure FDA0002633739450000036
for the problem of sound propagation in two dimensions,
Figure FDA0002633739450000037
Figure FDA0002633739450000038
Figure FDA0002633739450000039
b. determination using equation (15)
Figure FDA00026337394500000310
Figure FDA00026337394500000311
Wherein the content of the first and second substances,
Figure FDA0002633739450000041
Figure FDA0002633739450000042
wherein phi is the y point and znThe included angle between two end points of the boundary grid unit where the point is located;
c. p (x, y, ω) is calculated using equation (2).
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