INTER-NOISE 2016
Proposal of the evaluation function for selection of optimal
measurement location in inverse-numerical acoustic analysis
Masahiro AKEI1; Nobutaka TSUJIUCHI2; Daisuke KUBOTA3; Akihito ITO4; Takayuki YAMAUCHI5
15
YANMAR CO.,LTD. Japan
234
Doshisha University, Japan
ABSTRACT
The noise reduction of industrial products is required for dealing with noise pollution. To reduce such noise,
we focused on a predicting method for an identified sound source using inverse-numerical acoustic analysis
(INA) to identify the sound source. INA identifies the surface vibration of the sound source by using acoustic
transfer functions and actual sound pressures, which are measured at field points located near the sound
source. To measure sound pressures using INA, the field points must be arranged. Increased field points lead
to longer test and analysis time. Therefore, guidelines for field point arrangement are needed to conduct INA
efficiently. In this study, we focused on the standard deviations of distance between the sound source
elements and field points, and proposed a new evaluation function for the optimal selection of the field points
based on these standard deviations. The effectiveness of the new evaluation function was verified using a
plate model. As a result, we confirmed that the selection of optimal field point arrangement was achieved
when using two guidelines; specifically, the condition number and the new evaluation function that we
proposed.
Keywords: Theoretical sound sources, Inverse-numerical acoustic analysis, Analytical models
1. INTRODUCTION
Alleviating noise pollution and providing comfortable living environments requires the noise
reduction of industrial products. In order to achieve noise reduction efficiently, sound source
vibration must be identified correctly and take measures to prevent the noise. Examination of
soundproof structures using numerical simulations, such as a boundary element method in the design
stage of the product, is often conducted. To examine the soundproof structure using numerical
simulations, a model that identifies the surface vibration of the sound source to be targeted for sound
insulation and sound absorption is needed.
Inverse-numerical acoustic analysis (INA) has been proposed as a method of identifying a sound
source (1,2,6). INA is a method that identifies the surface vibration of the sound source by using
acoustic transfer functions and actual sound pressures that are measured at field points located near
the sound source. With INA, the measured sound pressures are used as the input data. Increased field
points lead to longer test and analysis time. Therefore, guidelines for field point arrangement are
needed to carry out INA efficiently.
The location of field points has been investigated㸦3,4). Martinez et al. proposed using three
indicators of condition number N , completeness number (CN), and wavelength of the sound. N is
defined as the ratio of the largest singular value and the minimum singular value when the singular
value decomposition (SVD) of the transmission function matrix is executed. CN is an indi cator with
a vector of the sum of each column in the acoustic transfer function matrix, which represents the size
of the identified sound source element. According to this study, the optimal arrangement of the field
points is low N and high CN. Furthermore, as a third index, the distance between the field points
1
2
3
4
5
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should be less than half of the wavelength of the sound analysis frequency. However, comparing
these three indicators simultaneously is complex and hard to use it and creates practical problems.
In this paper, we focused on the standard deviations of the distance between the sound source
elements and field points, and proposed a new evaluation function for the optimal selection of the
field points based on these standard deviations. We then tested whether it is possible to select
optimum field points using the two proposed indicators of the evaluation function and N .
2. THEORY
2.1 Inverse-numerical acoustic
When there is a structure composed of a sound source, a system of equations relating the normal
surface velocities to the sound pressure in the field can be constructed:
^p` >H @^v`
(1)
where ^p` are the sound pressures in the field of the m row, ^v` are the normal velocities on the
surface of the n column structure, and >H @ is the acoustic transfer function matrix of the structure
surface of the vibration velocity and field point pressure of the (m n) matrix. INA is an inverse problem
solving Eq. (1), as ^p` are known and ^v` are unknown. Since >H @ is almost never regular, a
pseudo-inverse matrix is used to identify the vibration velocity. Equation (1), multip lying
pseudo-inverse matrix >H @ of >H @ from the left, is expressed as:
^v` >H @ ^p`
The calculation of the pseudo-inverse matrix is often used for SVD. Using SVD,
expressed as:
>H @ >U @>V @>V @T
>H @
(2)
can be
(3)
where >U @ and >V @ are the unitary matrix, which is a matrix of each(n × n) and (m × m), and
>V @ is the singular value of >H @ of the (m n) matrix. >V @ consists of singular values V i 㸦i = 1, ,
min (m,n)㸧and only the diagonal elements have a value.
>V @
ªV 1 0 «0 V
2
«
«
«
«
«0 ¬
V min(m,n)
0º
»»
»
»
»
0»¼
It should be noted that V i satisfies the following relation:
V 1 t V 2 t t V min(m,n) t 0
(4)
(5)
From Eq. (3), the pseudo-inverse matrix can be reconstructed by:
>H @ >V @>V @1 >U @T
>V @
1
(6)
can be expressed as:
>V @1
0
ª1 V 1
« 0 1V
2
«
« «
«
«
«
¬« 0
º
»
»
»
»
»
1 V min(m, n ) »
»
0
¼»
0
(7)
^v` is:
(8)
^v` >V @>V @1>U @T ^p`
If there are similar components in >H @ , it is often because the linear independence of the column
Substituting Eq. (6) into equation (2),
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vector of >H @ is low and >H @ creates an ill-conditioned matrix (5). If there is an ill-conditioned
matrix, >V @ has a small value. When an error mixes in the sound pressure measurement, the
solution of the inverse problem under the influence of this small singular value becomes unstable. In
this case, the condition number is often used as an indication of the solution’s instability. Condi tion
number N is defined as the ratio of the largest singular value and the minimum singular value:
N V max V min
(9)
which represents the maximum magnification of the relative error of the measured value with respect
to the relative error of the solution. In the worst case scenario, the error of the measured value will
affect the relative error of the solution N times.
2.2 Acoustic transfer matrix
In the infinite baffle, when the vibrating planar diaphragm radiates sound waves, the velocity
potential of the radiated sound waves, I , is expressed as (7):
Q j (Zt kr )
e
2Sr
I
(10)
where Z is the angular frequency, k is the wavenumber, r is the distance between the sound
source elements and field points, and Q is the volume velocity. When only the diaphragm with a
portion of the baffle surface is radiating sound, it appears that the d iaphragm is divided into small
area elements and the respective area element is radiating sound as a sound source point. Sound
pressure p and the excluded volume velocity V at the field point r away from a point source is
expressed as:
p
U wI wt
jZU
4Sr 2Qe jZt
Q j (Zt kr )
e
2Sr
(11)
Qe jZt
(12)
where U is the air density and v is the vibration velocity of the small area elements of the
diaphragm. From Eqs. (11) and (12), where c is the sound velocity, O is the wavelength, and
S is the area of the flat plate element, the acoustic transfer function H between a p oint source and
the field points is expressed as:
V
H
p
S
V
jSZU
j 2Sr
exp( )
O
rO
(13)
2.3 Proposed evaluation function for measurement location selection
In this section, we describe our proposed new evaluation function, the Standard deviation of
Distance per Wave length number (SDW). From Eq. (13), if c and U are constant, the components
constituting H are determined by O and the distance from the sound source to the field points.
When focusing attention on only a certain wavelength, the value of the acoustic transfer function
increases because the distance is small. In other words, the sound pressure of a certain field point is
influenced by the nearest sound source. On the other hand, from the sound source side, a point sound
source likely influences the sound pressure of the closest field point. Therefore, we focused only on the
distance from a point source as the smallest field point. The distance matrix from a point source to the
field point >R @ is defined as:
>R@
ª r11
«r
« 21
« «
«
«rm1
¬
r12
r22
r1n º
»
»
»
»
»
rmn »¼
(14)
where rmn is the distance between the m-th field point and the n-th point sound source. To extract
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>R@ :
the smallest value rn min in each column of
^Rmin ` ^r1min
r2 min
rn min `
(15)
In order to extend to other frequencies, we divide by the O of each sound wave component of
>R@ to dimensionless, which is expressed as:
^Rmin O` ^r1min O
constituting ^Rmin ` O
r2 min O
rn min O`
(16)
If the components
are uniform, then the variation of the distance of the
nearest field point from each point source is small. Therefore, to evaluate the variation, the standard
deviation is used. The standard deviation is defined as the evaluation function SDW for the field
point selection, which can be expressed as:
SDW
1
n
n
¦(
i 1
ri min
O
M )2
(17)
where M is the average value of the components constituting the column vector ^Rmin ` O .
Using the SDW value We can evaluate a variation of the distance from the nearest field point to the
identified point sound sources by using the SDW value. In other words, for each point sound source
to be identified, it is possible to evaluate whether the field points are evenly distributed.
3. NUMERICAL ANALYSIS
3.1 Numerical analysis model
The newly proposed indicator is verified using a numerical analysis model to determine whether
they are valid for the selection of the field point arrangement. A schematic of the model used in the
numerical analysis is shown in Figure 1. This study is considering vibrating flat diaphragm plate in the
infinity baffle. The size of the flat plate model to which the sound source was set was 200 mm × 360
mm. The flat plate model, with a total of 2993 5 mm × 5 mm elements, was divided using each area
element as a point sound source. The field point, with a plane size of 210 mm × 420 mm so as to cover
the plate model, was divided at equal intervals and placed above the L-mm flat plate. Four conditions
determined the distance between the field points. The distances between a field point and the
measurement number are shown in Table 1. The layout of the field points in each of the conditions is
shown in Figure 2.
First, the vibration velocity was introduced in a direction perpendicular (Fig. 3) to each of the flat
plate model’s elements to calculate the sound pressure at each of the field points using >H @ . The
components constituting >H @ were calculated using Eq. (13). In this study, sound velocity c =
340000 mm/s and the density of the air was set to U = 1.2 × 10 -9 kg/mm3 . The area S of each
element of the flat plate was 25 mm2 . Next, the vibration velocity of each of the slab model’s elements
was calculated by INA using the calculated sound pressure. MathWorks' MATLAB version R2015a
was used for the numerical analysis.
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Figure 1 – Numerical analysis model
Table 1 – Number of field points and field point spacing
Number of field points
Field point spacing: D㸦mm㸧
91
66
45
28
(7×13)
(6×11)
(5×9)
(4×7)
35
42
52.5
70
Field point
420 mm
42 mm
35 mm
210 mm
210 mm
420 mm
91 point
66 point
35 mm
42 mm
420 mm
45 point
52.5 mm
70 mm
52.5 mm
210 mm
210 mm
420 mm
28 point
Figure 2 – Field point arrangement
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Figure 3 – Surface vibration distribution
3.2 Analysis condition
There were four conditions (25㸪50㸪100㸪150 mm) for the vertical distance between the
observation point and the flat plate model L, four conditions (35㸪42㸪52.5㸪70 mm) for the distance
between the observation points, and four conditions (650㸪1634㸪3000㸪4000 Hz) for the frequency.
Thus, there were a total of 64 approaches. After identification of the surface vibration in each
condition, we calculated the identification error by the value entered with the obtained results. The
accuracy of the INA in each condition was evaluated using the identification error. Identification
error Eident is defined as:
Eident
§1
¨
¨n
©
n
¦
i 1
vinput_ i vident _ i ·¸
u 100
¸
vinput_ i
¹
(18)
where n is the dividing element number of the flat plate model, vinput_ i is the vibration velocity
amplitude entered in the i-th element, and vident _ i is the vibration velocity amplitude identified in
the i-th element.
4. ANALYSIS RESULTS AND DISCUSSION
4.1 Relation between each parameter and the condition number
In this section, the relation of condition number N and the three parameters (the distance between
the vertical direction of the sound source and the field points, frequency, and the field point spacing)is
discribed. In Figure 4, when the field point spacing is 35 mm, which shows a relation between N
and the distance between the vertical direction of the sound source and field points. As the distance of
the vertical direction increases, N increases. Furthermore, as the frequency lessens, N increases. In
Figure 5, when the distance of the vertical direction is 50 mm, which shows a relation between N and
the field point spacing. The field point spacing is small; in other words, as the field points increase, N
increases. Therefore, N increases because there is more information that is similar to the components
constituting >H @ and it becomes lower linearly independent of the vertical vector. As mentioned
above in Section 2-1, N represents the maximum magnification of the relative error of the measured
value for the solution’s relative error. Therefore, in order to suppress INA’s identification error, we
must reduce N . In order to reduce N , the vertical direction distance must be small and the field
point spacing must increase.
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Condition Number κ
1010
650 Hz
1634 Hz
3000 Hz
4000 Hz
10 8
10 6
10 4
10 2
0
25
50
75
100
125
Distance mm
150
175
Figure 4 – Relation between condition number and distance
Condition Number κ
10 5
650 Hz
1634 Hz
3000 Hz
4000 Hz
10 4
10 3
10 2
10
1
25
30
35
40 45 50 55 60
Field point spacing mm
65
70
75
Figure 5 – Relation between condition number and field point spacing
4.2 Relation between evaluation function SDW and the identification error
As described in the previous section, one way to reduce N is to increase the field point spacing.
However, increasing the field point spacing results in an insufficient amount of information for
identifying the precise surface vibration distribution. Our proposed SDW can determine whether the
field points are evenly distributed. In this section, the verification results for SDW is described. As
shown in Figure 6, when the distance between the vertical direction of the sound source and the fiel d
point is 50 mm, which shows a relation between SDW and the identification error. The numbers in the
figure illustrate the field point spacing. Figure 7 shows that surface vibration distribution is identified
at a frequency of 4000 Hz and that the actual surface vibration distribution is a true solution. The
numbers marked on the top of each surface vibration distribution diagram indicate the field point
spacing. If the field point spacing increases, a larger element for the distance to the field points in each
divided sound source element appears and SDW increases. Furthermore, even if the field point spacing
is the same, the SDW differs depending on the frequency. This is due to the wavelength ratio, as shown
in Eq. (16). Figures 6 and 7 show that, the more SDW increases, the more the identification error
increases.
In Figure 8, when the distance of the vertical direction is 50 mm and 150 mm, which shows a
relation between SDW and the identification error at frequencies of 3000 Hz and 4000 Hz. When the
frequency is 4000 Hz and the field point spacing is 70 mm, 52.5 mm, or 42 mm, even if the distance to
the field points is the same, SDW and the identification error are smaller for a distance of 150 mm than
50 mm. Figure 9 shows the surface vibration distribution when the field point spacing is 70 mm. This
figure demonstrates that, for either frequency, the identified surface vibration distribution is better
when at a distance of 150 mm than 50 mm because it is similar to the true surface vibration distributi on.
When the frequency is 3000 Hz and the field point spacing is 42 mm or 35 mm, and when the frequency
is 4000 Hz and the field point spacing is 35 mm, SDW is smaller at a distance of 150 mm than 50 mm,
but the identification error is larger. This is caused by N . In this study, the sound pressures of the
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field points were calculated by the Acoustic analysis and INA was performed using these calculations.
Rounding error occurs when you enter the sound pressure. Under these conditions, the identified error
is large, the value of N is 10 6 from 10 7 , as shown in Figure 4, and the rounding error in the
calculation process increases; thus, it can be said that the identification has been affected by N . The
effect of N will be described in detail in Section 4.4.
140
650 Hz
1630 Hz
3000 Hz
4000 Hz
Identification error %
120
100
70
70
80
60
52.5
40
42
52.5
20
0
35 42 52.5
70
35 42
0
35
42
70
52.5
35
0.01
0.02
0.03
0.04
0.05
SDW number, standard deviation of rmmin / λ
0.06
Figure 6 – Relation between identification error and SDW number, distance 50 mm
Actual
35 mm
0.15 m/s
0.10
70 mm
52.5 mm
42 mm
0.05
0.0
Figure 7 – Surface vibration distribution at 4000 Hz, distance 50 mm
350
1000
35
Identification error %
300
150mm - 3000 Hz
150mm - 4000 Hz
50mm - 3000 Hz
50mm - 4000 Hz
35
250
200
150
70
100
70
70
50
52.5
42
42
0
0
52.5
35
70
42
42
52.5
52.5
35
0.01
0.02
0.03
0.04
0.05
SDW number, standard deviation of rmmin/λ
0.06
Figure 8 – Comparison between 50 mm and 150 mm distances
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L=50 mm,3000 Hz
L=50 mm,4000 Hz
Actual
0.15 m/s
0.10
L=150 mm,4000 Hz
L=150 mm,3000 Hz
0.05
0.0
Figure 9 – Surface vibration distribution, field point spacing 70 mm
4.3 When an error is mixed into the sound pressure of the field points
When actually performing INA, the measurement error is mixed into the sound pressure.
Therefore, assuming measurement error, identification was performed on the input sound pressure
mixing error by mixing a normal random number generated within 5% of the true value. In Figure 10,
when the vertical direction distance is 50 mm, which shows a relation between SDW and the
identification error. Furthermore, in Figure 11 shows the surface vibration distribution at 4000 Hz.
Compared to when there is no mixing error (Figure 6), the identification error increases when SDW
is smaller than 0.02. On the other hand, when the SDW area is larger than 0.02, the identification
error is almost unchanged whether there is an error or not. This can be explained by the differences
in N , as shown in Figure 4. Based on the above, in the worst case scenario, the measurement value
error that affects the relative error of the solution is N doubled. This time, the conditions’
identified error increases, and N is 10 4 from 10 2 , which is larger than the unchanged
identification error (SDW 0.02 or higher). Therefore, the identification solution is affected because
the mixed measurement error increased during the inverse analysis.
1000
35
Identification error %
900
800
650 Hz
1630 Hz
3000 Hz
4000 Hz
35
700
600
500
400
42
300
42
35
200
52.5
70
0
0
52.5 35
70
70
42
100
52.5
42 70
52.5
0.01
0.02
0.03
0.04
0.05
SDW number, standard deviation of rmmin/λ
0.06
Figure 10 – Relation between identification error and SDW number, distance 50 mm
Actual
35 mm
0.15 m/s
0.10
42 mm
52.5 mm
70 mm
0.05
0.0
Figure 11 – Surface vibration distribution at 4000 Hz, distance 50 mm
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4.4 The arrangement of the field points by κ and SDW
Figure 12 shows the relation between SDW, N , and the identification error. In this figure, which
is plotted the results for all 64 conditions. The input sound pressure has a mixed normal random
number error within 5% of the true value. The more SDW or N increases, the more the
identification error increases. In addition, the more SDW and N decrease, the more the
identification error decreases. Therefore, by selecting the field points when both evaluation
functions N and SDW decrease, analysis with the suppressed identification error is possible.
SDW number, standard deviation of rm min/ λ
0.1
Identification error
0䡚25 %
50䡚75 %
75䡚100 %
Ӎ100 %
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
102
104
108
106
Condition number κ
1010
Figure 12 – Relation of identification error, condition number N , and SDW number
5. CONCLUSIONS
In this study, we evaluated the arrangement of the optimal field points in INA using the condition
number N and our new evaluation function SDW. The resulting findings are as follows:
1㸬 Condition number N represents the maximum magnification of the relative error of the
measured value on the relative error of the solution. Therefore, when there is measurement error,
the more N increases, the more the identification error increases. They were confirmed by
numerical analysis model that, in order to reduce N , it is necessary to increase the field point
spacing and to decrease the distance in the vertical direction between the sound source and the
field points.
2㸬The more the newly proposed SDW in this study increases, the more the identification error
increases. The SDW can evaluate whether there is even distribution when the surface vibration is
identified.
3㸬If an error include in the sound pressure of the field points, the effect of N becomes significant
and the mixed error will increase during inverse analysis.
4㸬When N and SDW are both small, the identification error is small. Using the two evaluation
functions of N and SDW, it is possible to analyze with a suppressed identification error.
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