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Design optimization of a cellular-type noise insulation panel to improve transmission loss at low frequency
Accepted Manuscript Design optimization of a cellular-type noise insulation panel to improve transmission loss as low frequency Hyun-Guk Kim, Seongyoel Goo, Jaesoon Jung, Semyung Wang PII:
S0022-460X(19)30062-8
DOI:
https://doi.org/10.1016/j.jsv.2019.01.046
Reference:
YJSVI 14620
To appear in:
Journal of Sound and Vibration
Received Date: 31 January 2018 Revised Date:
24 January 2019
Accepted Date: 25 January 2019
Please cite this article as: H.-G. Kim, S. Goo, J. Jung, S. Wang, Design optimization of a cellular-type noise insulation panel to improve transmission loss as low frequency, Journal of Sound and Vibration (2019), doi: https://doi.org/10.1016/j.jsv.2019.01.046. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
Submitted to Journal of Sound and Vibration
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Design optimization of a cellular-type noise insulation panel to improve transmission loss as
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low frequency Hyun-Guk Kim, Seongyoel Goo, Jaesoon Jung, Semyung Wang*
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School of Mechanical Engineering, Gwangju Institute of Science and Technology, 123
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Cheomdangwagi-ro, Buk-gu, Gwangju, 61005, Republic of Korea
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* Corresponding author
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Postal Address: School of Mechanical Engineering, Gwangju Institute of Science and Technology,
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123 Cheomdan-gwagiro, Buk-gu, Gwangju, 61005, Republic of Korea
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E-mail address: [email protected]; Tel.: +82 62 715 2390; Fax: +82 62 715 2384
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Abstract
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In this paper, a systematic design method is proposed to improve the sound transmission
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loss of a cellular-type noise insulation panel, which is a thin rectangular plate with
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supporting frame. The unit cell of the noise insulation panel generates an anti-resonance
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effect at a certain frequency and yields a higher sound transmission loss than a flat plate
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of equivalent mass. Previous studies confirm that the mass ratio between the thin plate
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and the supporting frame has an important influence on the frequency at which the
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maximum sound transmission loss occurs. However, no design method to maximize the
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sound transmission loss at a target frequency has yet been reported, although this issue
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is very important for noise insulation of mechanical systems. In this regard, sizing
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optimization is performed using the thickness of the noise insulation panel as a design
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parameter. Hence, the optimal thickness distribution to maximize the sound transmission
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loss at a given target frequency is determined. To calculate the sound transmission loss, a
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finite element model of the noise insulation panel is constructed, considering the vibro-
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acoustic effect. Several numerical examples are presented to verify the proposed design
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method. The optimization results confirm that the designed unit cell exhibits high sound
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transmission loss at given frequencies of interest.
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Keywords
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Transmission loss; Acoustic metamaterial; Design optimization
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1. Introduction Recently, the development of noise insulation structures to reduce the noise generated
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from mechanical systems has received considerable attention. The noise insulation
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performance of such structures is generally determined by the mass law, i.e., the mass of
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the structure should be increased to improve the noise insulation performance [1].
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However, the noise insulation panel designed by the mass law yields increase in the total
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mass of the system; thus, the manufacturing cost also increases. To overcome this
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problem, several studies of acoustic metamaterials have been conducted and the findings
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have been implemented in noise reduction applications [2, 3].
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An acoustic metamaterial is an artificial structure having an extraordinary wave
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propagation property. Liu et al. first applied an acoustic metamaterial in the design of a
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noise insulation structure [4]. The sonic crystal proposed in that study is a 3D periodic
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structure comprised of lead balls coated with a silicone rubber layer. At a specific
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frequency, the lead balls locally resonate and the transmission loss increases. Following
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the development of this sonic crystal, several studies were conducted to develop noise
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insulation structures with improved sound transmission loss by the local resonance of a
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resonator [5, 6]. In addition, several studies were conducted on the plate and membrane-
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based acoustic metamaterials [7, 8]. In 2008, Yang et al. proposed a membrane-type
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acoustic metamaterial composed of a membrane and a small attached mass [9]. Those
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researchers showed that the noise insulation performance increases at the anti-resonance
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frequency, because of the interaction between the membrane and mass. In 2010, Yang et
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al. also experimentally verified that noise insulation performance increases through the
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use of periodically arranged unit cells [10]. In 2018, Langfeldt et al. proposed an
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analytical model to explain the behavior of multi-celled membrane-type acoustic
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metamaterials panels with periodically arranged unit cell structure [11]. Each unit cell
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structure shows a local resonance at the anti-resonance frequency. Owing to this local
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resonance, the averaged vibration response decreases and the sound transmission loss
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increases. In addition, as shown in [8], the noise insulation performance of the
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membrane-type acoustic metamaterials increases at the anti-resonance frequency
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between the resonant modes.
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However, it is difficult to manufacture a local-resonator-based acoustic metamaterial.
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Further, it is also difficult to maintain the tension of a membrane-type acoustic
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metamaterial. To overcome these difficulties, Varanasi et al. have proposed a cellular-type 2
ACCEPTED MANUSCRIPT noise insulation panel composed of a periodic array of unit cells [12]. The unit cell of this
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panel is composed of a thick supporting frame with a thin plate in the center. Hence, the
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cellular-type noise insulation panel is structurally robust compared to the membrane-type
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and local-resonator-based acoustic metamaterials. In 2014, Kim et al. applied the cellular-
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type noise insulation panel in a study of floor noise [13]. In 2017, Varanasi et al.
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proposed a sound normalizing layer to increase the sound transmission loss of a cellular-
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type acoustic metamaterial in a diffuse field and verified its performance [14].
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The existing research on the cellular-type noise insulation panel has the merit that the
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thickness distribution is simple and easy to be fabricated because the design parameters
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are only the mass ratio between the edge and the center part of the cellular-type noise
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insulation panel [12]. However, it is difficult to maximize the transmission loss under a
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constant volume constraint at the given frequency. The thickness distribution of thin
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panel at the center and the supporting frame will affect its dynamic characteristics,
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thereby altering the anti-resonance frequency.
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In this study, the sizing optimization method for a given thickness distribution on the
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panel has been proposed to maximize the sound transmission loss of a cellular-type
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noise insulation panel at given target frequencies. To improve the computation efficiency,
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the unit cell of the cellular-type noise insulation panel is first modeled using the finite
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element method based on a plate element considering a vibro-acoustic model. The
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design sensitivity is computed using the adjoint variable method [15] and the optimizer
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of the sizing optimization is the MMA (method of moving asymptotes) algorithm [16].
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Finally, the designed unit cell structure is extended to a multi-celled structure consisting
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of a periodic array of unit cell structures, and the validity of the design method is verified
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through several numerical examples.
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This section provides a general description of the cellular-type noise insulation panel [12]. A conceptual diagram of which is shown in Fig. 1.
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2. General description of cellular-type noise insulation panel
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Fig. 1. Conceptual diagram of the noise insulation panel; (a) a multi-celled structure
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composed of multiple unit cell structures; (b) the boundary condition to describe the
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behavior of a unit cell in a periodic array.
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Fig. 1(a) shows that the noise insulation panel is composed of a periodic array of the unit
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cell structures, while Fig. 1(b) shows the boundary condition to describe the behavior of
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a unit cell in a multi-celled structure. The vectors x = [ x
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are the position and displacement vectors, respectively. The variables
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indicate the rotations on the normal to the un-deformed middle surface in the
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y
z]
T
and u = βx
βx
β y w and
βy
xz
and
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yz planes, respectively. The variable w is the translational displacement along the z-
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axis. According to the boundary conditions described in this section, there is no
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displacement in the x-direction at ∂Ω x , where the edge of the unit cell structure is
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parallel to the x-axis. Similarly, there is no displacement in the y-direction at ∂ Ω y , where
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the edge is parallel to the y-axis. As briefly mentioned in Section 1, the unit cell of a 4
ACCEPTED MANUSCRIPT cellular-type noise insulation panel consists of a thin plate at the center and a thick
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supporting frame at the edge. As the thick supporting frames of the unit cell are much
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stiffer than the thin plate at the center, the local response of each unit cell structure
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dominates over the global response of the entire panel in which the unit cell structures
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are periodically arranged. If the thickness distribution is in the form of a stiffened panel,
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the behavior of the panel in the low frequency band does not directly affect adjacent
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unit structures due to a relatively stiff supporting frame. This phenomenon can be
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defined as the in-plane boundary condition at the edge of the unit cell structure. The
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supporting frame of the unit cell structure is considerably thicker than the thin plate. The
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thickness of the thin plate and the supporting frame of the initial model in section 4 are
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2 mm and 15 mm, respectively, and the thickness of the supporting frame and the thin
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plate of the optimization result in section 5 are 0.5 mm and 15 mm, respectively. Thus,
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because the local response of the panel is larger than the global response in the low
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frequency range, the low frequency behavior of the unit cell in the multi-celled structure
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can be expressed by the boundary condition defined in this section.
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Fig. 2. Sound transmission loss of the unit cell structure and the displace response of (a)
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rigid body mode, (b) anti-resonance response and (c) Monopole-like mode
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Fig. 2 shows the sound transmission loss and displacement responses of the unit cell
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structure. Figs. 2(a) and (c) indicate the resonant modes of the panel. In these in-phase
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modes, the displacement response and the transmitted acoustic pressure are increased 5
ACCEPTED MANUSCRIPT while the sound transmission loss is decreased. The interaction between two resonant
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modes generates an anti-resonance response (Fig. 2(b)) in which the supporting frame
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and the thin plate vibrate out of phase. At this anti-resonance frequency, the
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displacement response and the transmitted acoustic pressure decrease while the sound
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transmission loss increase [12]. In Section 3, the finite element model using Reissner-
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Mindlin plate theory is developed.
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3. Finite element analysis using a vibro-acoustic model
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In this section, the finite element model used to calculate the sound transmission loss
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of a cellular-type noise insulation panel is presented. In order to describe the sound
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reflection and transmission caused by the behavior of the cellular-type noise insulation
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panel, the finite element model is derived from the differential equation based on
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Reissner Mindlin plate theory as follows [17]:
ρh
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∂2 w ∂ ∂w ∂ ∂w ∂ − Gκ h − Gκ h + ( Gκ hβ y ) 2 ∂t ∂x ∂x ∂y ∂y ∂x = P − ( x, y, z, t )
12 ∂t
2
−
∂β y ∂ ∂w − β x = 0, (1 −ν ) − Gκ h ∂y ∂x ∂x
(1)
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−
,
∂β x ∂ ∂ D ∂β x ∂ D ∂β y − (1 −ν ) ν D − (1 −ν ) ∂y ∂x ∂x 2 ∂y ∂x 2 ∂x ∂β ∂w ∂ − (1 −ν ) y − Gκ h − β y = 0, ∂y ∂y ∂y
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12 ∂t
2
z =0
∂β y D ∂β x ∂ ∂ ∂β x ∂ − ν D D − (1 −ν ) 2 ∂y ∂x ∂x ∂x ∂y ∂y −
2 ρ h3 ∂ β y
− P + ( x, y, z, t )
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ρ h3 ∂ 2 β x
z =0
ρ , ν , G , D , κ , and h are the mass density, Poisson s ratio, shear modulus,
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where
147
bending rigidity, shear correction factor( κ = 5 / 6 ), and thickness of the plate, respectively,
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and
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The first equation governs the dynamic behavior for normal displacement and the
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second and third equations govern the dynamic behaviors of rotation in x- and y-
β x , β y , and w are the displacement vector of the plate as mentioned in Fig. 1.
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directions, respectively. Here, the acoustic pressure normally incident on the plate is
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regarded as the distributed pressure load, as expressed on the right-hand side of the first
153
equation. The distributed pressure load is represented by the difference of acoustic
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pressure between the right P + ( x, y , z , t )
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Thus, the pressure difference acts as an excitation force on the plate.
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z =0
sides of the plate.
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and left P − ( x, y , z , t )
z =0
The sound transmission through the plate excited by normally incident acoustic pressure is described in the Fig. 3 [18].
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Fig. 3. Conceptual diagram of a flat plate model excited by the normally incident
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acoustic wave; (a) side-view and (b) iso view of the plate model under the fixed
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boundary condition (B.C.)
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By the plane wave assumption, the acoustic waves incident on and reflected from the
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plate are expressed as
AC C
Pin ( z, t ) = P%in e (
j ωt − k0 z )
,
ω j ωt + k z Pref ( z, t ) = P%ref e ( 0 ) where k0 = , c0
(2)
Pin ( z, t ) and Pref ( z, t ) express the acoustic pressures of the incident and
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where
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reflected waves, respectively, with regard to time and space, and
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the complex pressure magnitudes of the respective acoustic waves, and
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are the angular frequency, wave number, and speed of sound, respectively. 7
P%in
and
P%ref
denote
ω , c0 , and k0
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Using the velocity continuity on the plate, the relation between the normal velocity of
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& and the acoustic pressures is expressed as the plate w
Pin − Pref , Z0
w& =
(3)
where Z 0 is the characteristic impedance of air ( Z 0 = ρ0 c0 ), which is a function of the
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density of air ρ 0 , and the speed of sound in air c0 . Assuming the harmonic motion, the
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& is expressed as normal velocity of the plate, w
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%& jωt , w& = we
(4)
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% jωt . = jω we
Using Eqs. (3) and (4), the pressure difference between incident and reflected acoustic
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% − P% ) at z = 0 can be expressed as a function of the normal velocity of a waves ( P in ref
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plate and the characteristic impedance of air. This relation is described as follows:
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&% jωt = P%in e ( Z0 we Z w%& = P% − P% .
j ωt −k0 z )
0
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ref
j ωt +k0 z )
,
(5)
As stated in Eq. (1), the pressure difference excites the plate. The acoustic pressures radiated from the positive Fig. 3(a) are expressed as
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in
− P%ref e (
Prad+ , and negative Prad− , direction in
+ Prad ( z , t ) = P%rad+ e j(ωt − k z ) ,
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%e = jω Z 0 w
j(ω t − k 0 z )
,
(6)
− Prad ( z , t ) = P%rad− e j(ωt + k0 z ) ,
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j ω t + k0 z )
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= − jω Z 0 w% e (
.
Using Eqs. (5) and (6), the total acoustic pressures acting on the left and right hand sides of the plate are expressed as
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+
( z , t ) = Prad ( z , t ) +
+ j(ω t + k 0 z ) = P%rad e
%e = jω Z 0 w P
−
j( ω t + k 0 z )
,
( z , t ) = Pin ( z , t ) + P ( z , t ) ,
(7)
ref
j( ω t − k z ) j( ω t + k z ) = P%in e + P%ref e , 0
j( ω t − k z ) % ) e j(ωt + k0 z ) , = P%in e + ( P%in − jω Z 0 w 0
= 2 P%in ( cos k 0 z ) e
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.
From Eq. (7), the pressure difference acting on the plate described on the right hand side of Eq. (1) can be expressed as
P − ( z, t ) 184
j( ω t + k 0 z )
z =0
− P + ( z, t )
z =0
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%e − jω Z 0 w
(
)
= 2 P%in − 2 jω Z 0 w% e jωt .
(8)
Then, substituting Eq. (8) into Eq. (1) and applying the principle of virtual work [19], the
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jω t
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discretized finite element equation of the plate model is obtained as
Su = F,
(9)
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S = −ω 2 M + jωC + K T T K = ∫∫Ω Bb CbBb dΩ + ∫∫Ω B s Cs B s dΩ, T C = ∫∫Ω ( 2 Z 0 ) H w H wdΩ, M = ∫∫ ρ H T HdΩ, Ω F = ∫∫ 2 P%in H wT dΩ, Ω
(
)
where M , C , K and F are the mass, damping, stiffness matrices, and the external force
187
vector, respectively, and S is the dynamic stiffness matrix. The stiffness matrix is
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calculated using the bending and shear strain-displacement matrices, which are
189
represented by
190
calculated using the element shape function H ; H w is the shape function for the z-
191
direction. As shown in the expression of the damping matrix C , the impedance of air
192
acts as mechanical damping which is stated as air loading in [18].
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Bb
and
Bs
, respectively. The mass and damping matrices are
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The normal displacement of a plate excited by incident acoustic pressure can be
194
calculated from Eq. (9). Assuming that the acoustic wavelength is considerably smaller
195
than the plate, the transmitted acoustic pressure at the right-hand side of the plate is
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expressed in terms of the characteristic impedance of air as follows [18] 9
ACCEPTED MANUSCRIPT Ptr = jω Z 0 w. 197 198
Using Eq. (10), the transmission coefficient loading (i.e.
τ
(10) of the plate excited by the unit acoustic
P%in = 1Pa ) is expressed as P%
= ∫∫ jω Z0 w% dΩ . Ω
200
Finally, the sound transmission loss (STL) that represents the transmission coefficient on the log scale is expressed as
STL = −20log10 τ .
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(11)
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τ = ∫∫ %tr dΩ, Ω P in
(12)
In this study, the sound transmission loss is calculated using the finite element analysis
202
procedure explained in this section. In this procedure, only the normal displacement of a
203
two dimensional plate is considered, whereas the three-dimensional acoustic domain is
204
not considered. Therefore, the proposed finite element model is computationally efficient.
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In order to verify the finite element model described above, the sound transmission loss
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calculated from the proposed numerical model was compared with the experimental data
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obtained from the literature [12]. Fig. 4(b) shows that the sound transmission loss
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computed from the finite element model proposed in this paper is very similar to the
209
experimentally measured data in [12]. Thus, the accuracy and validity of the proposed
210
finite element model is verified.
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Fig. 4. Comparison of the sound transmission loss of the plate determined using the
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proposed method with the measured data reported in [12]: (a) model information used
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for calculation and (b) the sound transmission loss calculated using proposed the finite
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element model (solid line) and the measured data from [12] (circles).
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4. Sizing optimization
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4.1. Statement of the optimization problem
is defined in Fig. 5.
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In this section, the sizing optimization problem for the developed finite element model
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Fig. 5. Conceptual diagram of sizing optimization for cellular-type noise insulation panel:
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(a) design and non-design domains; (b) design variables corresponding to design domain
225
thickness. 11
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227
Fig. 5(a), all of the plate elements in this panel can move along the z-axis. Thus, the
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sound transmission loss of this panel is determined by the motion of all of the plate
229
elements. Fig. 5(a) indicates that the design variable is the thickness of the plate element
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corresponding to the design domain Ωd , while the thickness of the non-design domain
231
Ωnd is unchanged.
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The sizing optimization problem that maximizes the sound transmission loss (STL) at
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the target angular frequency ( ωtarget ) with a limited volume usage is expressed as follows:
min . − STL (ωtarget )
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S (ωtarget ) u = F where ωtarget = 2π f target hmin ( = 0.5mm ) < hi < hmin ( = 0.5mm ) where hi : design variable subject to h dΩ V = ∫∫Ωd i d ≤ volfrac: volume constraint V 0 ∫∫Ωd hmax dΩd
(13)
234
where S is the dynamic stiffness matrix, which is a function of the target angular
235
frequency
236
variable of i th element. The bounded condition of the design variable is varied from
237
minimum value ( hmin ) 0.5 mm to maximum height ( hmax ) 20mm. The volume constraint
238
V / V0
239
the maximum volume
240
accelerates the convergence to the optimal solution, is used as the optimizer [16]. The
241
objective function to be minimized is the negative sound transmission loss, which is a
242
function of the frequency. As the MMA is a gradient-based optimization algorithm,
243
design sensitivity information that quantitatively indicates the effect of each design
244
variable on the objective function is necessary to conduct the sizing optimization.
ftarget
is the target frequency. The variable
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and
corresponds to the ratio of the volume V
hi
is the design
at a given iteration with respect to
In this paper, the MMA algorithm, which stabilizes and
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V0 .
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ωtarget
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4.2 Design sensitivity analysis In this section, the design sensitivity analysis procedure for the sizing optimization is
251
explained. A design sensitivity analysis is a quantitative evaluation of the effect of a
252
design variable on the objective function. Thus, by the design sensitivity analysis, the
253
effect of the thickness of each plate element in the design domain on the sound
254
transmission loss (STL) can be evaluated [15, 20, 21]. Using Eqs. (11) and (12), the
255
objective function can be expressed as
ψ = −STL = 20 log τ
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where τ = ∫∫ jω Z 0 w% dΩ Ω
(14)
256
As shown in Eq. (11), the transmission coefficient τ
257
% . And, τ normal displacement w
258
design sensitivity (i.e. first derivative) for the objective function is obtained by the chain
259
rule. Thus, the design sensitivity d ψ / d hi with respect to the i th element can be
260
expressed as
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is expressed as a function of the
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is an implicit function of the design variable
dψ d = 20 log τ , dhi dhi
1 dτ = 20 , τ ln10 dhi
hi .
The
(15)
261
where τ
262
calculated by the finite element analysis. By the chain rule, the design sensitivity for the
263
magnitude of the transmitted coefficient can be derived as follow [20]
TE D
is the magnitude of the transmission coefficient and the term that can be
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dτ dτ 1 = Re (τ ) Re τ d hi dhi
dτ + Im (τ ) Im d hi
.
(16)
The adjoint variable method can be used to calculate the derivatives of τ with respect
265
to the design variable components. The adjoint equation for calculating the adjoint
266
response is expressed as
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STλ = −
∂τ , ∂w%
= − jω Z 0 ∫∫ H w T d Ω ,
(17)
Ω
267
where λ is the adjoint response [21]. Using the adjoint response obtained from Eq. (17)
268
and the state response, which is the displacement obtained from Eq. (9), the derivative of
269
the transmission coefficient is expressed as 13
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(18)
270
Substituting Eq. (18) into Eq. (16), the design sensitivity for the objective function is
271
expressed as
272 273
dτ + Im (τ ) Im . d hi
(19)
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dτ dψ 1 = 20 2 Re (τ ) Re d hi τ ln10 dhi
5. Numerical examples
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5.1. Sizing optimization of cellular-type noise insulation panel with respect to target
276
frequency under constant volume constraint
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In this section, numerical examples for constant volume and several single target
278
frequencies are presented, and were used to evaluate the performance of the
279
optimization process with respect to the target frequency. An initial model used in the
280
sizing optimization is shown as in Fig. 6.
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Fig. 6. An initial model used in sizing optimization. (a) model information and (b) sound
283
transmission loss.
284
Fig. 6(a) shows the model information of the initial model. The finite element model 14
ACCEPTED MANUSCRIPT implemented in an area of 83.5mm x 83.5mm, and the area of the thin plate
286
corresponding to the design domain was 63.5mm x 63.5mm. The thicknesses of the thin
287
plate and supporting frames are 2mm and 15mm, respectively. ABS (Acrylonitrile
288
butadiene styrene) plastic is the material of this panel. As shown in Fig. 6(b), the sound
289
transmission loss of the initial model is maximized at 1036.5 Hz and the dip frequency
290
appears at 1325.3 Hz. The model in Fig. 6 is also used as the initial model for the sizing
291
optimization procedure described in this section. From the above results, if the sizing
292
optimization is applied to the noise insulation panel, the maximum sound transmission
293
loss at 1036.5 Hz can be shifted to the given target frequency.
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Fig. 7. Sizing optimization results with constant volume fraction (0.4) and various single
296
target frequencies. Target frequencies (
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600 Hz, and (e) 700 Hz.
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ftarget )
are (a) 300Hz, (b) 400 Hz, (c) 500 Hz, (d)
298
The sizing optimization is performed at single target frequencies of 300, 400, 500, 600,
299
and 700 Hz for a constant volume fraction of 0.4. The thickness distributions
300
corresponding to the optimization result for each case are shown in Fig. 7 and the
301
improvements in sound transmission loss are listed in Table 1. The edge thickness of 15
302
mm is constant because the edge is a non-design area, whereas the thickness of the
303
central part within the inner plate area converge to the minimum thickness, 0.5 mm. As 15
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shown in Fig. 7, as the target frequency increases, the width of the thin plate (yellow
305
region) in the center decreases gradually. The sound transmission loss is improved by
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14.32 dB on average for each optimization result.
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Table 1. Sound transmission losses at various target frequencies for the initial and the
309
optimal design and its improvement, for specific volume fraction (0.4). Initial design
Optimal design
[Hz]
[dB]
[dB]
300
24.31
36.69
400
26.99
42.14
500
29.62
600
33.89
700
31.71
Avg.
29.30
Improvement
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12.38 15.15
44.70
15.08
48.18
14.29
46.42
14.71
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Target freq.
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43.63
14.32
Fig. 8 shows the sound transmission loss for each optimization result in Fig. 7. The
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sound transmission loss of the optimally designed noise insulation panel is maximized by
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anti-resonance. Further, if the thickness distribution of the cellular-type noise insulation
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panel is changed, the frequency at which the monopole-like resonant mode appears and
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the anti-resonance frequency at which the maximum sound transmission loss appears are
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also changed. Thus, it is possible to change the frequency at which the sound
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transmission loss increases only by changing the thickness distribution without using an
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additional mass. As shown in Fig. 8, the mode shape of the monopole-like mode and the
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displacement response at the anti-resonance frequency for each result show that the
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local response of thin plate region near the center is dominant. Further, as shown in Fig.
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7, as the target frequency increases, the width of the thin plate region decreases. This is
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frequency of the panel to increase; thus, the anti-resonance frequency also increases. In
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this section, by constructing a thickness distribution that adjusts the anti-resonance
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frequency, a noise insulation structure can be designed to maximize the sound
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transmission loss at the desired frequency.
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Fig. 8. Sound transmission loss of the optimally designed noise insulation panel (square
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line) and the initial model (solid line), the displacement response at the peak frequency
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and the mode shape at the vibrating mode for target frequencies of (a) 300 Hz, (b)
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400Hz, (c) 500 Hz, (d) 600 Hz, and (e) 700 Hz, respectively. 18
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5.2. Sizing optimization of cellular-type noise insulation panel with respect to target
333
frequency band and specific volume constraint In Section 5.1, the sizing optimization for the single target frequency was performed.
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However, the noise generated in the mechanical system is generally composed of several
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frequency components. Thus, in this section, the sizing optimization was performed for
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the target frequency band of 300 700 Hz with specific volume fraction 0.4. The objective
338
function and the design sensitivity for the sizing optimization with the target frequency
339
band are defined as
ψ mean
and d ψ / d hi
mean
1 n dψ (ω j ) = ∑ , n j =1 dhi
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dψ dhi
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ψ mean =
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(20)
are averaged values of the objective function and design
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where
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sensitivity for the target frequency band, respectively, ψ ω j
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objective function computed at the j th frequency
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sensitivity for i the element in the design domain and j th frequency within the target
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frequency band.
( ) is the ωj , and dψ (ω ) / dh j
i
value of the is the design
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Fig. 9. Sizing optimization result for band target frequency (300 700Hz) and volume
347
constraint (0.4); (a) iso-view, (b) thickness distribution for sizing optimization result and
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color bar show thickness distribution, and (c) sound transmission losses of initial design
349
(solid line), optimization result (square line), and the flat panel with equivalent mass (star
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line). 19
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the design domain converges to the minimum value of the design variable, while the
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supporting frame tends to converge to a maximum. For the given target frequency range,
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the average sound transmission loss of the optimized model is improved by 8.71 and
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6.60 dB compared to the initial model and flat panel with equivalent mass, respectively,
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and the maximum sound transmission loss is at 635.6 Hz. As mentioned in Section 5.1,
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this improvement is due to the anti-resonance. A graphical description is given in Fig. 10.
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Fig. 10. Sound transmission loss of optimization result for target frequency band (300-
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700 Hz) with volume constraint (0.4), and illustrations of the anti-resonance response,
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rigid body mode, and monopole-like mode.
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The optimization result in Fig. 9 shows that the sound transmission loss at 635.6 Hz
363
significantly increases due to the anti-resonance, which is generated by the interaction
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between the rigid body mode at 0 Hz and the monopole-like mode at 1084.7 Hz [8].
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6. Sound transmission loss of the sizing-optimized extended array structure and
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verification of the sizing optimization for unit cell structure and the boundary
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condition In Section 5, several numerical examples of the sizing optimization to maximize the
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sound transmission loss of the unit cell structure were presented. In this study, the finite
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element model was developed by applying the specific boundary condition to explain
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the behavior of the unit cell structure within the entire panel. In this section, an
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investigation on multi-celled structure consisting of a periodic array of unit cell structures
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is presented to verify the validity of the optimization results and the boundary condition
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used in this research. The simplified model was calculated from the optimization result
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presented in Section 5.2 and extended to 5 x 5, 10 x 10, 15 x 15, and 20 x 20 multi-
380
celled structure. Then, the sound transmission loss for each case was calculated using
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Comsol Multiphysics [22] to verify the validity of the sizing optimization.
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shown in Fig. 11.
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A simplified model was established from the optimization result using the process
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Fig. 11. Simplified model based on the optimization result for the target frequency band
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(300 700 Hz) and the volume constraint (0.4): (a) and (c) are the optimization results; (b)
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and (d) are the simplified model.
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In Fig. 11(a), the domain Ω 1 is the non-design domain corresponding to the thick frame
389
on the edge of the initial model. The domains Ω2 , Ω 3 , Ω4 and Ω5 are the design
390
domain corresponding to the thin plate in the center. As shown in Figs. 11(a) and (b), the
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thickness of domains Ω 1 , Ω4 , and Ω5 have thicknesses of 15.0, 7.0, and 0.5 mm,
392
respectively. The thickness of a domain Ω2 is set to the maximum, i.e. 20.0 mm. The
393
averaged thickness of a domain Ω 3 is 14 mm. Thus, the thicknesses of Ω2 and Ω 3 are
394
approximated values, respectively. Using these approximated thicknesses, a simplified
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model was established as shown in Fig. 11(b). The sound transmission loss of the
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optimization result and the simplified model are shown in Fig. 12.
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Fig. 12. Comparison of the sound transmission losses for the optimized and simplified
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model; (a) the configuration of finite element model used in Comsol Multiphysics and (b)
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sound transmission losses for the optimization result and the simplified model.
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Fig. 12(a) shows the configuration used for the finite element model, which is composed
402
of the structural mechanics (shell) module to describe the behavior of the structural part
403
and the pressure acoustics module to consider the air-loading effect in the Comsol
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implementation for the simplified model. The upper boundary of a hemispherical
405
acoustic domain is considered, with the perfectly matched layer boundary condition [23],
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which means absorbing boundary condition to represent the infinite acoustic domain.
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The bottom boundary is applied to the wall boundary condition. The thickness of the
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bottom of the acoustic space with the boundary condition described in section 2,
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meaning that the displacement in the x-derection is zero on Ω x while the displacement in
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the y-direction is zero on Ω y . The simplified model corresponding to an acoustic-
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structure boundary is excited by the distributed load. As shown in Fig. 12(b), the sound
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transmission loss of the optimization result is very similar to that of the simplified model.
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The simplified model has been extended to 5 x 5, 10 x 10, 15 x 15 and 20 x 20 multi-
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celled structures. The dimensions of each structure are listed in Table. 2.
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Table 2. The size of each multi-celled structure
Unit cell
83.5×83.5
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5×5
417.5×417.5
10 × 10
835.0×835.0
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1670.0×1670.0
As the size of the multi-celled structure increases, the number of vibration modes
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increases and the behavior becomes complex. Thus, the sound transmission loss of each
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structure was calculated in a 1/3 octave band. The configuration of the finite element
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model configuration and the sound transmission losses for each structure are shown in
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Fig. 13.
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Fig. 13. Sound transmission loss of the multi-celled structures (5 x 5, 10 x 10, 15 x 15,
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and 20 x 20); (a) Configuration of finite element analysis model to calculate the sound
426
transmission loss of each multi-celled structure; (b) sound transmission loss of each
427
structure.
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As shown in Fig. 13(a), except for the boundary condition supporting each structure and
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size of the acoustic domain, the finite element model configuration is identical to that of
430
Fig. 12(a). The radius of the hemispherical acoustic domain corresponds to twice the
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length of each structure. The thickness of the perfectly matched layer has 0.15 m
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thickness in all cases. The fixed boundary condition on ∂Ω f means that the edges of
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each structure are clamped on the bottom side of the acoustic domain. As shown in Fig.
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differs from the behavior of the unit cell structure under the boundary condition
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described in section 2, because it is strongly influenced by clamped boundaries. However,
437
as the size of the array increases, the sound transmission loss of the multi-celled
438
structures converges to that of the unit cell structure. Therefore, the use of the boundary
439
condition to explain the behavior of the unit cell structure in the periodic array is
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reasonable. In addition, the performance of the sizing optimization under this boundary
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condition is obviously reasonable.
frequency corresponding to the maximum sound transmission loss in Fig. 13(b).
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Fig. 14 shows the z-directional displacement field of each multi-celled structure at the
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Fig. 14. The z-directional displacement field at the frequency corresponding to maximum
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sound transmission loss of (a) a unit cell structure and (b) 5 x 5, (c) 10 x 10, (d) 15 x 15,
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and (e) 20 x 20 sized multi-celled structure. The unit cell structure is under the boundary
448
condition to describe the unit cell behavior in the periodic array, whereas the multi-celled
449
structures are under the fixed boundary condition. Peak frequencies of each case are (a)
450
630 Hz, (b) 500 Hz, (c) 630 Hz, (d) 630Hz, and (e) 630 Hz.
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As shown in Fig. 14, the local response of each unit cell structure dominates over the
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global response of the whole structure. This means that each unit cell in the structure
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locally vibrates at the peak frequency, regardless of the size of the structure. This is
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because the edge thickness is 15 mm, whereas the thickness of the central part in the
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design domain is 0.5 mm; therefore, the center of the unit cell structure is more flexible
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than the edge. Thus, it is acceptable to analyze a unit cell structure under the boundary
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condition to express the dynamic behavior of unit cell structure, so as to maximize the
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sound transmission loss in the low frequency range.
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7. Conclusions
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In this paper, a design optimization method for a cellular-type noise insulation panel
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was presented. Firstly, an efficient finite element model of a cellular-type noise insulation
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panel was established based on the Reissner Mindlin plate theory by considering a
464
vibro-acoustic interaction between the plate and the acoustic domain. In addition, sizing
465
optimization was performed for several constraints using the design method proposed in
466
this paper. Several single target frequencies and a target frequency band, all under a
467
constant volume constraint, were considered. The results indicate that the design method
468
can increase the sound transmission loss at the target frequency set by the designer
469
without changing the mass of the panel. From a practical perspective, finally, the design
470
method was verified by calculating the sound transmission loss of the cellular-type noise
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insulation panel which is made of a periodic array of unit cells. This design method for a
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noise insulation panel proposed in this study is expected to be applicable to design of
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automobile dash panels and exterior panels of home appliances.
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Acknowledgements
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This work was supported by a National Research Foundation of Korea (NRF) grant
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funded by the Korean government [NRF-2017R1A2A1A05001326].
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in the 50 1000 Hz regime, Applied Physics Letters, 96 (2010) 041906.
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[11] F. Langfeldt, W. Gleine, O. von Estorff, An efficient analytical model for baffled, multi-celled
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membrane-type acoustic metamaterial panels, Journal of Sound and Vibration, 417 (2018) 359-375.
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[12] S. Varanasi, J.S. Bolton, T.H. Siegmund, R.J. Cipra, The low frequency performance of
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metamaterial barriers based on cellular structures, Applied Acoustics, 74 (2013) 485-495.
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[13] H.S. Kim, J.S. Kim, S.H. Lee, Y.H. Seo, Low frequency sound transmission of stiffened panels, in:
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Engineering, 2014, pp. 3049-3056.
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[14] S. Varanasi, J.S. Bolton, T. Siegmund, Experiments on the low frequency barrier characteristics
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of cellular metamaterial panels in a diffuse sound field, The Journal of the Acoustical Society of
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Multidisciplinary Optimization, 32 (2006) 263-275.
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[18] F.J. Fahy, P. Gardonio, Sound and structural vibration: radiation, transmission and response,
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[19] K.J. Bathe, Finite element procedures, Klaus-Jurgen Bathe, 2006.
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vibrating structures, Journal of Mechanical Design, 114 (1992) 166-173.
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[21] M.B. Dühring, J.S. Jensen, O. Sigmund, Acoustic design by topology optimization, Journal of
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sound and vibration, 317 (2008) 557-575.
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[22] COMSOL Acoustic Module User s Guide, COMSOL Inc. Sweden, 2016.
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[23] S. Johnson, Notes on perfectly matched layers (PMLs), in, Massachusetts Institute of
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Technology, Massachusetts, 2010.
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ACCEPTED MANUSCRIPT A finite element model of a cellular-type noise insulation panel was established. A systematic design method to improve the sound transmission loss was proposed.
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The finite element analysis of a celled array for the optimal result was carried out.
S0022-460X(19)30062-8
DOI:
https://doi.org/10.1016/j.jsv.2019.01.046
Reference:
YJSVI 14620
To appear in:
Journal of Sound and Vibration
Received Date: 31 January 2018 Revised Date:
24 January 2019
Accepted Date: 25 January 2019
Please cite this article as: H.-G. Kim, S. Goo, J. Jung, S. Wang, Design optimization of a cellular-type noise insulation panel to improve transmission loss as low frequency, Journal of Sound and Vibration (2019), doi: https://doi.org/10.1016/j.jsv.2019.01.046. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT 1
Submitted to Journal of Sound and Vibration
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Design optimization of a cellular-type noise insulation panel to improve transmission loss as
3
low frequency Hyun-Guk Kim, Seongyoel Goo, Jaesoon Jung, Semyung Wang*
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School of Mechanical Engineering, Gwangju Institute of Science and Technology, 123
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Cheomdangwagi-ro, Buk-gu, Gwangju, 61005, Republic of Korea
7
* Corresponding author
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Postal Address: School of Mechanical Engineering, Gwangju Institute of Science and Technology,
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123 Cheomdan-gwagiro, Buk-gu, Gwangju, 61005, Republic of Korea
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E-mail address: [email protected]; Tel.: +82 62 715 2390; Fax: +82 62 715 2384
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Abstract
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In this paper, a systematic design method is proposed to improve the sound transmission
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loss of a cellular-type noise insulation panel, which is a thin rectangular plate with
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supporting frame. The unit cell of the noise insulation panel generates an anti-resonance
16
effect at a certain frequency and yields a higher sound transmission loss than a flat plate
17
of equivalent mass. Previous studies confirm that the mass ratio between the thin plate
18
and the supporting frame has an important influence on the frequency at which the
19
maximum sound transmission loss occurs. However, no design method to maximize the
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sound transmission loss at a target frequency has yet been reported, although this issue
21
is very important for noise insulation of mechanical systems. In this regard, sizing
22
optimization is performed using the thickness of the noise insulation panel as a design
23
parameter. Hence, the optimal thickness distribution to maximize the sound transmission
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loss at a given target frequency is determined. To calculate the sound transmission loss, a
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finite element model of the noise insulation panel is constructed, considering the vibro-
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acoustic effect. Several numerical examples are presented to verify the proposed design
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method. The optimization results confirm that the designed unit cell exhibits high sound
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transmission loss at given frequencies of interest.
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Keywords
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Transmission loss; Acoustic metamaterial; Design optimization
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1. Introduction Recently, the development of noise insulation structures to reduce the noise generated
33
from mechanical systems has received considerable attention. The noise insulation
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performance of such structures is generally determined by the mass law, i.e., the mass of
35
the structure should be increased to improve the noise insulation performance [1].
36
However, the noise insulation panel designed by the mass law yields increase in the total
37
mass of the system; thus, the manufacturing cost also increases. To overcome this
38
problem, several studies of acoustic metamaterials have been conducted and the findings
39
have been implemented in noise reduction applications [2, 3].
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An acoustic metamaterial is an artificial structure having an extraordinary wave
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propagation property. Liu et al. first applied an acoustic metamaterial in the design of a
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noise insulation structure [4]. The sonic crystal proposed in that study is a 3D periodic
43
structure comprised of lead balls coated with a silicone rubber layer. At a specific
44
frequency, the lead balls locally resonate and the transmission loss increases. Following
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the development of this sonic crystal, several studies were conducted to develop noise
46
insulation structures with improved sound transmission loss by the local resonance of a
47
resonator [5, 6]. In addition, several studies were conducted on the plate and membrane-
48
based acoustic metamaterials [7, 8]. In 2008, Yang et al. proposed a membrane-type
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acoustic metamaterial composed of a membrane and a small attached mass [9]. Those
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researchers showed that the noise insulation performance increases at the anti-resonance
51
frequency, because of the interaction between the membrane and mass. In 2010, Yang et
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al. also experimentally verified that noise insulation performance increases through the
53
use of periodically arranged unit cells [10]. In 2018, Langfeldt et al. proposed an
54
analytical model to explain the behavior of multi-celled membrane-type acoustic
55
metamaterials panels with periodically arranged unit cell structure [11]. Each unit cell
56
structure shows a local resonance at the anti-resonance frequency. Owing to this local
57
resonance, the averaged vibration response decreases and the sound transmission loss
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increases. In addition, as shown in [8], the noise insulation performance of the
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membrane-type acoustic metamaterials increases at the anti-resonance frequency
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between the resonant modes.
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However, it is difficult to manufacture a local-resonator-based acoustic metamaterial.
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Further, it is also difficult to maintain the tension of a membrane-type acoustic
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metamaterial. To overcome these difficulties, Varanasi et al. have proposed a cellular-type 2
ACCEPTED MANUSCRIPT noise insulation panel composed of a periodic array of unit cells [12]. The unit cell of this
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panel is composed of a thick supporting frame with a thin plate in the center. Hence, the
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cellular-type noise insulation panel is structurally robust compared to the membrane-type
67
and local-resonator-based acoustic metamaterials. In 2014, Kim et al. applied the cellular-
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type noise insulation panel in a study of floor noise [13]. In 2017, Varanasi et al.
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proposed a sound normalizing layer to increase the sound transmission loss of a cellular-
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type acoustic metamaterial in a diffuse field and verified its performance [14].
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The existing research on the cellular-type noise insulation panel has the merit that the
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thickness distribution is simple and easy to be fabricated because the design parameters
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are only the mass ratio between the edge and the center part of the cellular-type noise
74
insulation panel [12]. However, it is difficult to maximize the transmission loss under a
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constant volume constraint at the given frequency. The thickness distribution of thin
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panel at the center and the supporting frame will affect its dynamic characteristics,
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thereby altering the anti-resonance frequency.
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In this study, the sizing optimization method for a given thickness distribution on the
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panel has been proposed to maximize the sound transmission loss of a cellular-type
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noise insulation panel at given target frequencies. To improve the computation efficiency,
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the unit cell of the cellular-type noise insulation panel is first modeled using the finite
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element method based on a plate element considering a vibro-acoustic model. The
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design sensitivity is computed using the adjoint variable method [15] and the optimizer
84
of the sizing optimization is the MMA (method of moving asymptotes) algorithm [16].
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Finally, the designed unit cell structure is extended to a multi-celled structure consisting
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of a periodic array of unit cell structures, and the validity of the design method is verified
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through several numerical examples.
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This section provides a general description of the cellular-type noise insulation panel [12]. A conceptual diagram of which is shown in Fig. 1.
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2. General description of cellular-type noise insulation panel
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Fig. 1. Conceptual diagram of the noise insulation panel; (a) a multi-celled structure
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composed of multiple unit cell structures; (b) the boundary condition to describe the
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behavior of a unit cell in a periodic array.
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Fig. 1(a) shows that the noise insulation panel is composed of a periodic array of the unit
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cell structures, while Fig. 1(b) shows the boundary condition to describe the behavior of
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a unit cell in a multi-celled structure. The vectors x = [ x
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are the position and displacement vectors, respectively. The variables
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indicate the rotations on the normal to the un-deformed middle surface in the
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y
z]
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and u = βx
βx
β y w and
βy
xz
and
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yz planes, respectively. The variable w is the translational displacement along the z-
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axis. According to the boundary conditions described in this section, there is no
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displacement in the x-direction at ∂Ω x , where the edge of the unit cell structure is
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parallel to the x-axis. Similarly, there is no displacement in the y-direction at ∂ Ω y , where
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the edge is parallel to the y-axis. As briefly mentioned in Section 1, the unit cell of a 4
ACCEPTED MANUSCRIPT cellular-type noise insulation panel consists of a thin plate at the center and a thick
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supporting frame at the edge. As the thick supporting frames of the unit cell are much
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stiffer than the thin plate at the center, the local response of each unit cell structure
115
dominates over the global response of the entire panel in which the unit cell structures
116
are periodically arranged. If the thickness distribution is in the form of a stiffened panel,
117
the behavior of the panel in the low frequency band does not directly affect adjacent
118
unit structures due to a relatively stiff supporting frame. This phenomenon can be
119
defined as the in-plane boundary condition at the edge of the unit cell structure. The
120
supporting frame of the unit cell structure is considerably thicker than the thin plate. The
121
thickness of the thin plate and the supporting frame of the initial model in section 4 are
122
2 mm and 15 mm, respectively, and the thickness of the supporting frame and the thin
123
plate of the optimization result in section 5 are 0.5 mm and 15 mm, respectively. Thus,
124
because the local response of the panel is larger than the global response in the low
125
frequency range, the low frequency behavior of the unit cell in the multi-celled structure
126
can be expressed by the boundary condition defined in this section.
127
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112
128
Fig. 2. Sound transmission loss of the unit cell structure and the displace response of (a)
129
rigid body mode, (b) anti-resonance response and (c) Monopole-like mode
130
Fig. 2 shows the sound transmission loss and displacement responses of the unit cell
131
structure. Figs. 2(a) and (c) indicate the resonant modes of the panel. In these in-phase
132
modes, the displacement response and the transmitted acoustic pressure are increased 5
ACCEPTED MANUSCRIPT while the sound transmission loss is decreased. The interaction between two resonant
134
modes generates an anti-resonance response (Fig. 2(b)) in which the supporting frame
135
and the thin plate vibrate out of phase. At this anti-resonance frequency, the
136
displacement response and the transmitted acoustic pressure decrease while the sound
137
transmission loss increase [12]. In Section 3, the finite element model using Reissner-
138
Mindlin plate theory is developed.
139 140
3. Finite element analysis using a vibro-acoustic model
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133
In this section, the finite element model used to calculate the sound transmission loss
142
of a cellular-type noise insulation panel is presented. In order to describe the sound
143
reflection and transmission caused by the behavior of the cellular-type noise insulation
144
panel, the finite element model is derived from the differential equation based on
145
Reissner Mindlin plate theory as follows [17]:
ρh
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141
∂2 w ∂ ∂w ∂ ∂w ∂ − Gκ h − Gκ h + ( Gκ hβ y ) 2 ∂t ∂x ∂x ∂y ∂y ∂x = P − ( x, y, z, t )
12 ∂t
2
−
∂β y ∂ ∂w − β x = 0, (1 −ν ) − Gκ h ∂y ∂x ∂x
(1)
EP
−
,
∂β x ∂ ∂ D ∂β x ∂ D ∂β y − (1 −ν ) ν D − (1 −ν ) ∂y ∂x ∂x 2 ∂y ∂x 2 ∂x ∂β ∂w ∂ − (1 −ν ) y − Gκ h − β y = 0, ∂y ∂y ∂y
AC C
12 ∂t
2
z =0
∂β y D ∂β x ∂ ∂ ∂β x ∂ − ν D D − (1 −ν ) 2 ∂y ∂x ∂x ∂x ∂y ∂y −
2 ρ h3 ∂ β y
− P + ( x, y, z, t )
TE D
ρ h3 ∂ 2 β x
z =0
ρ , ν , G , D , κ , and h are the mass density, Poisson s ratio, shear modulus,
146
where
147
bending rigidity, shear correction factor( κ = 5 / 6 ), and thickness of the plate, respectively,
148
and
149
The first equation governs the dynamic behavior for normal displacement and the
150
second and third equations govern the dynamic behaviors of rotation in x- and y-
β x , β y , and w are the displacement vector of the plate as mentioned in Fig. 1.
6
ACCEPTED MANUSCRIPT 151
directions, respectively. Here, the acoustic pressure normally incident on the plate is
152
regarded as the distributed pressure load, as expressed on the right-hand side of the first
153
equation. The distributed pressure load is represented by the difference of acoustic
154
pressure between the right P + ( x, y , z , t )
155
Thus, the pressure difference acts as an excitation force on the plate.
157
z =0
sides of the plate.
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156
and left P − ( x, y , z , t )
z =0
The sound transmission through the plate excited by normally incident acoustic pressure is described in the Fig. 3 [18].
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159 160
Fig. 3. Conceptual diagram of a flat plate model excited by the normally incident
161
acoustic wave; (a) side-view and (b) iso view of the plate model under the fixed
162
boundary condition (B.C.)
164
By the plane wave assumption, the acoustic waves incident on and reflected from the
EP
163
plate are expressed as
AC C
Pin ( z, t ) = P%in e (
j ωt − k0 z )
,
ω j ωt + k z Pref ( z, t ) = P%ref e ( 0 ) where k0 = , c0
(2)
Pin ( z, t ) and Pref ( z, t ) express the acoustic pressures of the incident and
165
where
166
reflected waves, respectively, with regard to time and space, and
167
the complex pressure magnitudes of the respective acoustic waves, and
168
are the angular frequency, wave number, and speed of sound, respectively. 7
P%in
and
P%ref
denote
ω , c0 , and k0
ACCEPTED MANUSCRIPT 169
Using the velocity continuity on the plate, the relation between the normal velocity of
170
& and the acoustic pressures is expressed as the plate w
Pin − Pref , Z0
w& =
(3)
where Z 0 is the characteristic impedance of air ( Z 0 = ρ0 c0 ), which is a function of the
172
density of air ρ 0 , and the speed of sound in air c0 . Assuming the harmonic motion, the
173
& is expressed as normal velocity of the plate, w
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171
%& jωt , w& = we
(4)
SC
% jωt . = jω we
Using Eqs. (3) and (4), the pressure difference between incident and reflected acoustic
175
% − P% ) at z = 0 can be expressed as a function of the normal velocity of a waves ( P in ref
176
plate and the characteristic impedance of air. This relation is described as follows:
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174
&% jωt = P%in e ( Z0 we Z w%& = P% − P% .
j ωt −k0 z )
0
178 179
ref
j ωt +k0 z )
,
(5)
As stated in Eq. (1), the pressure difference excites the plate. The acoustic pressures radiated from the positive Fig. 3(a) are expressed as
TE D
177
in
− P%ref e (
Prad+ , and negative Prad− , direction in
+ Prad ( z , t ) = P%rad+ e j(ωt − k z ) ,
EP
0
%e = jω Z 0 w
j(ω t − k 0 z )
,
(6)
− Prad ( z , t ) = P%rad− e j(ωt + k0 z ) ,
181
j ω t + k0 z )
AC C
180
= − jω Z 0 w% e (
.
Using Eqs. (5) and (6), the total acoustic pressures acting on the left and right hand sides of the plate are expressed as
8
ACCEPTED MANUSCRIPT P
+
( z , t ) = Prad ( z , t ) +
+ j(ω t + k 0 z ) = P%rad e
%e = jω Z 0 w P
−
j( ω t + k 0 z )
,
( z , t ) = Pin ( z , t ) + P ( z , t ) ,
(7)
ref
j( ω t − k z ) j( ω t + k z ) = P%in e + P%ref e , 0
j( ω t − k z ) % ) e j(ωt + k0 z ) , = P%in e + ( P%in − jω Z 0 w 0
= 2 P%in ( cos k 0 z ) e
185
.
From Eq. (7), the pressure difference acting on the plate described on the right hand side of Eq. (1) can be expressed as
P − ( z, t ) 184
j( ω t + k 0 z )
z =0
− P + ( z, t )
z =0
SC
183
%e − jω Z 0 w
(
)
= 2 P%in − 2 jω Z 0 w% e jωt .
(8)
Then, substituting Eq. (8) into Eq. (1) and applying the principle of virtual work [19], the
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182
jω t
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0
discretized finite element equation of the plate model is obtained as
Su = F,
(9)
TE D
S = −ω 2 M + jωC + K T T K = ∫∫Ω Bb CbBb dΩ + ∫∫Ω B s Cs B s dΩ, T C = ∫∫Ω ( 2 Z 0 ) H w H wdΩ, M = ∫∫ ρ H T HdΩ, Ω F = ∫∫ 2 P%in H wT dΩ, Ω
(
)
where M , C , K and F are the mass, damping, stiffness matrices, and the external force
187
vector, respectively, and S is the dynamic stiffness matrix. The stiffness matrix is
188
calculated using the bending and shear strain-displacement matrices, which are
189
represented by
190
calculated using the element shape function H ; H w is the shape function for the z-
191
direction. As shown in the expression of the damping matrix C , the impedance of air
192
acts as mechanical damping which is stated as air loading in [18].
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Bb
and
Bs
, respectively. The mass and damping matrices are
193
The normal displacement of a plate excited by incident acoustic pressure can be
194
calculated from Eq. (9). Assuming that the acoustic wavelength is considerably smaller
195
than the plate, the transmitted acoustic pressure at the right-hand side of the plate is
196
expressed in terms of the characteristic impedance of air as follows [18] 9
ACCEPTED MANUSCRIPT Ptr = jω Z 0 w. 197 198
Using Eq. (10), the transmission coefficient loading (i.e.
τ
(10) of the plate excited by the unit acoustic
P%in = 1Pa ) is expressed as P%
= ∫∫ jω Z0 w% dΩ . Ω
200
Finally, the sound transmission loss (STL) that represents the transmission coefficient on the log scale is expressed as
STL = −20log10 τ .
SC
199
(11)
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τ = ∫∫ %tr dΩ, Ω P in
(12)
In this study, the sound transmission loss is calculated using the finite element analysis
202
procedure explained in this section. In this procedure, only the normal displacement of a
203
two dimensional plate is considered, whereas the three-dimensional acoustic domain is
204
not considered. Therefore, the proposed finite element model is computationally efficient.
205
In order to verify the finite element model described above, the sound transmission loss
206
calculated from the proposed numerical model was compared with the experimental data
207
obtained from the literature [12]. Fig. 4(b) shows that the sound transmission loss
208
computed from the finite element model proposed in this paper is very similar to the
209
experimentally measured data in [12]. Thus, the accuracy and validity of the proposed
210
finite element model is verified.
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Fig. 4. Comparison of the sound transmission loss of the plate determined using the
214
proposed method with the measured data reported in [12]: (a) model information used
215
for calculation and (b) the sound transmission loss calculated using proposed the finite
216
element model (solid line) and the measured data from [12] (circles).
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217 218
4. Sizing optimization
219
4.1. Statement of the optimization problem
is defined in Fig. 5.
AC C
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In this section, the sizing optimization problem for the developed finite element model
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220
222 223
Fig. 5. Conceptual diagram of sizing optimization for cellular-type noise insulation panel:
224
(a) design and non-design domains; (b) design variables corresponding to design domain
225
thickness. 11
ACCEPTED MANUSCRIPT Fig. 5 shows a unit cell structure of the cellular-type noise insulation panel. As shown in
227
Fig. 5(a), all of the plate elements in this panel can move along the z-axis. Thus, the
228
sound transmission loss of this panel is determined by the motion of all of the plate
229
elements. Fig. 5(a) indicates that the design variable is the thickness of the plate element
230
corresponding to the design domain Ωd , while the thickness of the non-design domain
231
Ωnd is unchanged.
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226
232
The sizing optimization problem that maximizes the sound transmission loss (STL) at
233
the target angular frequency ( ωtarget ) with a limited volume usage is expressed as follows:
min . − STL (ωtarget )
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S (ωtarget ) u = F where ωtarget = 2π f target hmin ( = 0.5mm ) < hi < hmin ( = 0.5mm ) where hi : design variable subject to h dΩ V = ∫∫Ωd i d ≤ volfrac: volume constraint V 0 ∫∫Ωd hmax dΩd
(13)
234
where S is the dynamic stiffness matrix, which is a function of the target angular
235
frequency
236
variable of i th element. The bounded condition of the design variable is varied from
237
minimum value ( hmin ) 0.5 mm to maximum height ( hmax ) 20mm. The volume constraint
238
V / V0
239
the maximum volume
240
accelerates the convergence to the optimal solution, is used as the optimizer [16]. The
241
objective function to be minimized is the negative sound transmission loss, which is a
242
function of the frequency. As the MMA is a gradient-based optimization algorithm,
243
design sensitivity information that quantitatively indicates the effect of each design
244
variable on the objective function is necessary to conduct the sizing optimization.
ftarget
is the target frequency. The variable
TE D
and
corresponds to the ratio of the volume V
hi
is the design
at a given iteration with respect to
In this paper, the MMA algorithm, which stabilizes and
EP
V0 .
AC C
245
ωtarget
246 247 248 12
ACCEPTED MANUSCRIPT 249
4.2 Design sensitivity analysis In this section, the design sensitivity analysis procedure for the sizing optimization is
251
explained. A design sensitivity analysis is a quantitative evaluation of the effect of a
252
design variable on the objective function. Thus, by the design sensitivity analysis, the
253
effect of the thickness of each plate element in the design domain on the sound
254
transmission loss (STL) can be evaluated [15, 20, 21]. Using Eqs. (11) and (12), the
255
objective function can be expressed as
ψ = −STL = 20 log τ
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250
where τ = ∫∫ jω Z 0 w% dΩ Ω
(14)
256
As shown in Eq. (11), the transmission coefficient τ
257
% . And, τ normal displacement w
258
design sensitivity (i.e. first derivative) for the objective function is obtained by the chain
259
rule. Thus, the design sensitivity d ψ / d hi with respect to the i th element can be
260
expressed as
SC
is expressed as a function of the
M AN U
is an implicit function of the design variable
dψ d = 20 log τ , dhi dhi
1 dτ = 20 , τ ln10 dhi
hi .
The
(15)
261
where τ
262
calculated by the finite element analysis. By the chain rule, the design sensitivity for the
263
magnitude of the transmitted coefficient can be derived as follow [20]
TE D
is the magnitude of the transmission coefficient and the term that can be
EP
dτ dτ 1 = Re (τ ) Re τ d hi dhi
dτ + Im (τ ) Im d hi
.
(16)
The adjoint variable method can be used to calculate the derivatives of τ with respect
265
to the design variable components. The adjoint equation for calculating the adjoint
266
response is expressed as
AC C
264
STλ = −
∂τ , ∂w%
= − jω Z 0 ∫∫ H w T d Ω ,
(17)
Ω
267
where λ is the adjoint response [21]. Using the adjoint response obtained from Eq. (17)
268
and the state response, which is the displacement obtained from Eq. (9), the derivative of
269
the transmission coefficient is expressed as 13
ACCEPTED MANUSCRIPT dτ ∂S = λT u. d hi ∂hi
(18)
270
Substituting Eq. (18) into Eq. (16), the design sensitivity for the objective function is
271
expressed as
272 273
dτ + Im (τ ) Im . d hi
(19)
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dτ dψ 1 = 20 2 Re (τ ) Re d hi τ ln10 dhi
5. Numerical examples
275
5.1. Sizing optimization of cellular-type noise insulation panel with respect to target
276
frequency under constant volume constraint
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274
In this section, numerical examples for constant volume and several single target
278
frequencies are presented, and were used to evaluate the performance of the
279
optimization process with respect to the target frequency. An initial model used in the
280
sizing optimization is shown as in Fig. 6.
AC C
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277
281 282
Fig. 6. An initial model used in sizing optimization. (a) model information and (b) sound
283
transmission loss.
284
Fig. 6(a) shows the model information of the initial model. The finite element model 14
ACCEPTED MANUSCRIPT implemented in an area of 83.5mm x 83.5mm, and the area of the thin plate
286
corresponding to the design domain was 63.5mm x 63.5mm. The thicknesses of the thin
287
plate and supporting frames are 2mm and 15mm, respectively. ABS (Acrylonitrile
288
butadiene styrene) plastic is the material of this panel. As shown in Fig. 6(b), the sound
289
transmission loss of the initial model is maximized at 1036.5 Hz and the dip frequency
290
appears at 1325.3 Hz. The model in Fig. 6 is also used as the initial model for the sizing
291
optimization procedure described in this section. From the above results, if the sizing
292
optimization is applied to the noise insulation panel, the maximum sound transmission
293
loss at 1036.5 Hz can be shifted to the given target frequency.
294
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285
Fig. 7. Sizing optimization results with constant volume fraction (0.4) and various single
296
target frequencies. Target frequencies (
297
600 Hz, and (e) 700 Hz.
AC C
295
ftarget )
are (a) 300Hz, (b) 400 Hz, (c) 500 Hz, (d)
298
The sizing optimization is performed at single target frequencies of 300, 400, 500, 600,
299
and 700 Hz for a constant volume fraction of 0.4. The thickness distributions
300
corresponding to the optimization result for each case are shown in Fig. 7 and the
301
improvements in sound transmission loss are listed in Table 1. The edge thickness of 15
302
mm is constant because the edge is a non-design area, whereas the thickness of the
303
central part within the inner plate area converge to the minimum thickness, 0.5 mm. As 15
ACCEPTED MANUSCRIPT 304
shown in Fig. 7, as the target frequency increases, the width of the thin plate (yellow
305
region) in the center decreases gradually. The sound transmission loss is improved by
306
14.32 dB on average for each optimization result.
307
Table 1. Sound transmission losses at various target frequencies for the initial and the
309
optimal design and its improvement, for specific volume fraction (0.4). Initial design
Optimal design
[Hz]
[dB]
[dB]
300
24.31
36.69
400
26.99
42.14
500
29.62
600
33.89
700
31.71
Avg.
29.30
Improvement
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[dB]
12.38 15.15
44.70
15.08
48.18
14.29
46.42
14.71
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310
Target freq.
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308
43.63
14.32
Fig. 8 shows the sound transmission loss for each optimization result in Fig. 7. The
312
sound transmission loss of the optimally designed noise insulation panel is maximized by
313
anti-resonance. Further, if the thickness distribution of the cellular-type noise insulation
314
panel is changed, the frequency at which the monopole-like resonant mode appears and
315
the anti-resonance frequency at which the maximum sound transmission loss appears are
316
also changed. Thus, it is possible to change the frequency at which the sound
317
transmission loss increases only by changing the thickness distribution without using an
318
additional mass. As shown in Fig. 8, the mode shape of the monopole-like mode and the
319
displacement response at the anti-resonance frequency for each result show that the
320
local response of thin plate region near the center is dominant. Further, as shown in Fig.
321
7, as the target frequency increases, the width of the thin plate region decreases. This is
AC C
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311
16
ACCEPTED MANUSCRIPT because the reduction in the area of the thin plate (yellow) cause the local resonance
323
frequency of the panel to increase; thus, the anti-resonance frequency also increases. In
324
this section, by constructing a thickness distribution that adjusts the anti-resonance
325
frequency, a noise insulation structure can be designed to maximize the sound
326
transmission loss at the desired frequency.
AC C
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17
327
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328
Fig. 8. Sound transmission loss of the optimally designed noise insulation panel (square
329
line) and the initial model (solid line), the displacement response at the peak frequency
330
and the mode shape at the vibrating mode for target frequencies of (a) 300 Hz, (b)
331
400Hz, (c) 500 Hz, (d) 600 Hz, and (e) 700 Hz, respectively. 18
ACCEPTED MANUSCRIPT 332
5.2. Sizing optimization of cellular-type noise insulation panel with respect to target
333
frequency band and specific volume constraint In Section 5.1, the sizing optimization for the single target frequency was performed.
335
However, the noise generated in the mechanical system is generally composed of several
336
frequency components. Thus, in this section, the sizing optimization was performed for
337
the target frequency band of 300 700 Hz with specific volume fraction 0.4. The objective
338
function and the design sensitivity for the sizing optimization with the target frequency
339
band are defined as
ψ mean
and d ψ / d hi
mean
1 n dψ (ω j ) = ∑ , n j =1 dhi
SC
dψ dhi
1 n ∑ψ (ω j ), n j =1
M AN U
ψ mean =
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334
(20)
are averaged values of the objective function and design
340
where
341
sensitivity for the target frequency band, respectively, ψ ω j
342
objective function computed at the j th frequency
343
sensitivity for i the element in the design domain and j th frequency within the target
344
frequency band.
( ) is the ωj , and dψ (ω ) / dh j
i
value of the is the design
AC C
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mean
345 346
Fig. 9. Sizing optimization result for band target frequency (300 700Hz) and volume
347
constraint (0.4); (a) iso-view, (b) thickness distribution for sizing optimization result and
348
color bar show thickness distribution, and (c) sound transmission losses of initial design
349
(solid line), optimization result (square line), and the flat panel with equivalent mass (star
350
line). 19
ACCEPTED MANUSCRIPT Fig. 9 shows the result of the optimization problem shown in Eq. (20). The central part of
352
the design domain converges to the minimum value of the design variable, while the
353
supporting frame tends to converge to a maximum. For the given target frequency range,
354
the average sound transmission loss of the optimized model is improved by 8.71 and
355
6.60 dB compared to the initial model and flat panel with equivalent mass, respectively,
356
and the maximum sound transmission loss is at 635.6 Hz. As mentioned in Section 5.1,
357
this improvement is due to the anti-resonance. A graphical description is given in Fig. 10.
358
TE D
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351
Fig. 10. Sound transmission loss of optimization result for target frequency band (300-
360
700 Hz) with volume constraint (0.4), and illustrations of the anti-resonance response,
361
rigid body mode, and monopole-like mode.
362
The optimization result in Fig. 9 shows that the sound transmission loss at 635.6 Hz
363
significantly increases due to the anti-resonance, which is generated by the interaction
364
between the rigid body mode at 0 Hz and the monopole-like mode at 1084.7 Hz [8].
AC C
365
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359
366 367 368 20
ACCEPTED MANUSCRIPT 369
6. Sound transmission loss of the sizing-optimized extended array structure and
370
verification of the sizing optimization for unit cell structure and the boundary
371
condition In Section 5, several numerical examples of the sizing optimization to maximize the
373
sound transmission loss of the unit cell structure were presented. In this study, the finite
374
element model was developed by applying the specific boundary condition to explain
375
the behavior of the unit cell structure within the entire panel. In this section, an
376
investigation on multi-celled structure consisting of a periodic array of unit cell structures
377
is presented to verify the validity of the optimization results and the boundary condition
378
used in this research. The simplified model was calculated from the optimization result
379
presented in Section 5.2 and extended to 5 x 5, 10 x 10, 15 x 15, and 20 x 20 multi-
380
celled structure. Then, the sound transmission loss for each case was calculated using
381
Comsol Multiphysics [22] to verify the validity of the sizing optimization.
SC
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shown in Fig. 11.
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A simplified model was established from the optimization result using the process
AC C
382
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372
21
384
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ACCEPTED MANUSCRIPT
Fig. 11. Simplified model based on the optimization result for the target frequency band
386
(300 700 Hz) and the volume constraint (0.4): (a) and (c) are the optimization results; (b)
387
and (d) are the simplified model.
388
In Fig. 11(a), the domain Ω 1 is the non-design domain corresponding to the thick frame
389
on the edge of the initial model. The domains Ω2 , Ω 3 , Ω4 and Ω5 are the design
390
domain corresponding to the thin plate in the center. As shown in Figs. 11(a) and (b), the
391
thickness of domains Ω 1 , Ω4 , and Ω5 have thicknesses of 15.0, 7.0, and 0.5 mm,
392
respectively. The thickness of a domain Ω2 is set to the maximum, i.e. 20.0 mm. The
393
averaged thickness of a domain Ω 3 is 14 mm. Thus, the thicknesses of Ω2 and Ω 3 are
394
approximated values, respectively. Using these approximated thicknesses, a simplified
395
model was established as shown in Fig. 11(b). The sound transmission loss of the
396
optimization result and the simplified model are shown in Fig. 12.
AC C
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385
22
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397
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ACCEPTED MANUSCRIPT
Fig. 12. Comparison of the sound transmission losses for the optimized and simplified
399
model; (a) the configuration of finite element model used in Comsol Multiphysics and (b)
400
sound transmission losses for the optimization result and the simplified model.
401
Fig. 12(a) shows the configuration used for the finite element model, which is composed
402
of the structural mechanics (shell) module to describe the behavior of the structural part
403
and the pressure acoustics module to consider the air-loading effect in the Comsol
404
implementation for the simplified model. The upper boundary of a hemispherical
405
acoustic domain is considered, with the perfectly matched layer boundary condition [23],
406
which means absorbing boundary condition to represent the infinite acoustic domain.
407
The bottom boundary is applied to the wall boundary condition. The thickness of the
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398
23
ACCEPTED MANUSCRIPT perfectly matched layer is 0.15 m. The simplified model of Fig. 12 is applied at the
409
bottom of the acoustic space with the boundary condition described in section 2,
410
meaning that the displacement in the x-derection is zero on Ω x while the displacement in
411
the y-direction is zero on Ω y . The simplified model corresponding to an acoustic-
412
structure boundary is excited by the distributed load. As shown in Fig. 12(b), the sound
413
transmission loss of the optimization result is very similar to that of the simplified model.
414
The simplified model has been extended to 5 x 5, 10 x 10, 15 x 15 and 20 x 20 multi-
415
celled structures. The dimensions of each structure are listed in Table. 2.
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Table 2. The size of each multi-celled structure
Unit cell
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5×5
417.5×417.5
10 × 10
835.0×835.0
20 × 20 417
1252.5×1252.5
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15 × 15
1670.0×1670.0
As the size of the multi-celled structure increases, the number of vibration modes
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increases and the behavior becomes complex. Thus, the sound transmission loss of each
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structure was calculated in a 1/3 octave band. The configuration of the finite element
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model configuration and the sound transmission losses for each structure are shown in
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Fig. 13.
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Fig. 13. Sound transmission loss of the multi-celled structures (5 x 5, 10 x 10, 15 x 15,
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and 20 x 20); (a) Configuration of finite element analysis model to calculate the sound
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transmission loss of each multi-celled structure; (b) sound transmission loss of each
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structure.
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As shown in Fig. 13(a), except for the boundary condition supporting each structure and
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size of the acoustic domain, the finite element model configuration is identical to that of
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Fig. 12(a). The radius of the hemispherical acoustic domain corresponds to twice the
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length of each structure. The thickness of the perfectly matched layer has 0.15 m
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thickness in all cases. The fixed boundary condition on ∂Ω f means that the edges of
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each structure are clamped on the bottom side of the acoustic domain. As shown in Fig.
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differs from the behavior of the unit cell structure under the boundary condition
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described in section 2, because it is strongly influenced by clamped boundaries. However,
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as the size of the array increases, the sound transmission loss of the multi-celled
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structures converges to that of the unit cell structure. Therefore, the use of the boundary
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condition to explain the behavior of the unit cell structure in the periodic array is
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reasonable. In addition, the performance of the sizing optimization under this boundary
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condition is obviously reasonable.
frequency corresponding to the maximum sound transmission loss in Fig. 13(b).
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Fig. 14 shows the z-directional displacement field of each multi-celled structure at the
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Fig. 14. The z-directional displacement field at the frequency corresponding to maximum
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sound transmission loss of (a) a unit cell structure and (b) 5 x 5, (c) 10 x 10, (d) 15 x 15,
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and (e) 20 x 20 sized multi-celled structure. The unit cell structure is under the boundary
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condition to describe the unit cell behavior in the periodic array, whereas the multi-celled
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structures are under the fixed boundary condition. Peak frequencies of each case are (a)
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630 Hz, (b) 500 Hz, (c) 630 Hz, (d) 630Hz, and (e) 630 Hz.
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As shown in Fig. 14, the local response of each unit cell structure dominates over the
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global response of the whole structure. This means that each unit cell in the structure
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locally vibrates at the peak frequency, regardless of the size of the structure. This is
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because the edge thickness is 15 mm, whereas the thickness of the central part in the
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design domain is 0.5 mm; therefore, the center of the unit cell structure is more flexible
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than the edge. Thus, it is acceptable to analyze a unit cell structure under the boundary
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condition to express the dynamic behavior of unit cell structure, so as to maximize the
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sound transmission loss in the low frequency range.
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7. Conclusions
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In this paper, a design optimization method for a cellular-type noise insulation panel
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was presented. Firstly, an efficient finite element model of a cellular-type noise insulation
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panel was established based on the Reissner Mindlin plate theory by considering a
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vibro-acoustic interaction between the plate and the acoustic domain. In addition, sizing
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optimization was performed for several constraints using the design method proposed in
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this paper. Several single target frequencies and a target frequency band, all under a
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constant volume constraint, were considered. The results indicate that the design method
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can increase the sound transmission loss at the target frequency set by the designer
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without changing the mass of the panel. From a practical perspective, finally, the design
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method was verified by calculating the sound transmission loss of the cellular-type noise
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insulation panel which is made of a periodic array of unit cells. This design method for a
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noise insulation panel proposed in this study is expected to be applicable to design of
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automobile dash panels and exterior panels of home appliances.
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Acknowledgements
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This work was supported by a National Research Foundation of Korea (NRF) grant
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funded by the Korean government [NRF-2017R1A2A1A05001326].
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a new method for structural optimization,
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ACCEPTED MANUSCRIPT A finite element model of a cellular-type noise insulation panel was established. A systematic design method to improve the sound transmission loss was proposed.
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The finite element analysis of a celled array for the optimal result was carried out.