U.S. patent application number 11/979583 was filed with the patent office on 2008-09-18 for poroelastic acoustical foam having enhanced sound-absorbing performance.
This patent application is currently assigned to Seoul National University Industry Foundation. Invention is credited to Yeon-June Kang, Yoon-Young Kim, Joong-Seok Lee.
Application Number | 20080223653 11/979583 |
Document ID | / |
Family ID | 39761527 |
Filed Date | 2008-09-18 |
United States Patent
Application |
20080223653 |
Kind Code |
A1 |
Kim; Yoon-Young ; et
al. |
September 18, 2008 |
Poroelastic acoustical foam having enhanced sound-absorbing
performance
Abstract
An optimal shape of a poroelastic acoustical foam which can
maximize sound-absorbing effect is disclosed. The poroelastic
acoustical foam is made of a wedge-shaped wedge unit where the
cross section is reduced in one direction, and a bowl-shaped bowl
unit where formed at one end of the wedge unit where the cross
section of the wedge unit is small, and the other end, where the
cross section of the wedge unit is large, is separated from the
wall, forming an air layer.
Inventors: |
Kim; Yoon-Young; (Seoul,
KR) ; Kang; Yeon-June; (Seongnam-si, KR) ;
Lee; Joong-Seok; (Seoul, KR) |
Correspondence
Address: |
NATH & ASSOCIATES PLLC
112 South West Street
Alexandria
VA
22314
US
|
Assignee: |
Seoul National University Industry
Foundation
Seoul
KR
|
Family ID: |
39761527 |
Appl. No.: |
11/979583 |
Filed: |
November 6, 2007 |
Current U.S.
Class: |
181/286 |
Current CPC
Class: |
G10K 11/16 20130101 |
Class at
Publication: |
181/286 |
International
Class: |
E04B 1/84 20060101
E04B001/84 |
Foreign Application Data
Date |
Code |
Application Number |
Mar 16, 2007 |
KR |
10-2007-0026059 |
Oct 22, 2007 |
KR |
10-2007-0106309 |
Claims
1. Poroelastic acoustical foam made of a porous material
comprising: a wedge-shaped wedge unit where the cross section is
reduced in one direction; and a bowl-shaped bowl unit where formed
at one end of the wedge unit where the cross section of the wedge
unit is small wherein the other end, where the cross section of the
wedge unit is large, is separated from the wall, forming an air
layer.
2. The foam of claim 1, wherein the shape of the poroelastic
acoustical foam is obtained using a topology optimization technique
which is based on Biot's equation.
3. The foam of claim 2, wherein material properties used in Biot's
equation are interpolated in the form of a polynomial expression
using design variables having values between zero and one.
4. The foam of claim 3, wherein, when a design variable of zero
indicates air, a design variable of one indicates a porous
material, and a design variable having a value between zero and one
indicates an intermediate material.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application claims priority from Korean Patent
Application No. 10-2007-0026059 filed on Mar. 16, 2007 and No.
10-2007-0106309 filed on October 22 in the Korean Intellectual
Property Office, the disclosure of which is incorporated herein by
reference in its entirety.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The present invention relates to poroelastic acoustical
foam, and, more particularly, to an optimal shape of poroelastic
acoustical foam which can improve a sound-absorbing effect of
low-frequency bands and middle-frequency bands.
[0004] 2. Description of the Related Art
[0005] Poroelastic acoustical foams are designed to reduce noise
and vibrations and are widely used in mechanical fields such as the
automobile, airplane, and construction industries. Generally,
poroelastic acoustical foams are porous materials having two
phases: air and solid.
[0006] FIG. 1 is an enlarged view of a porous material. Referring
to FIG. 1, a solid phase forms the frame of the porous material,
and air fills in pores of the porous material. In this state, the
two phases are physically coupled, thereby dynamically affecting
each other. As a result, acoustic waves are dissipated as heat. The
dissipation of thermal energy reduces the energy of sound waves
transmitting through or reflected by the poroelastic acoustical
foam, which, in turn, results in noise reduction.
[0007] Since the 1930s, many studies have been conducted on the
development of poroelastic acoustical foams and the interpretation
of their properties. In particular, in the 1950s, Maurice A. Biot
conducted research on the propagation of elastic waves in porous
materials, and thus laid the foundation for the analysis of the
porous materials. Biot's study of porous materials not only
directly and indirectly affected various fields including civil
engineering, oil drilling engineering, soil engineering and marine
engineering, but also was later applied in the analysis of the
poroelastic acoustical foams.
[0008] Various studies on the sound-absorbing performance of
poroelastic acoustical foams have also been conducted using Biot's
theory, but most of the studies have relied on experiments. The
studies have found that poroelastic acoustical foam shows better
sound-absorbing performance when it has a wedge or trigonal pyramid
shape. Wedge-shaped poroelastic acoustical foams are still widely
used, mainly in anechoic chambers that require effective sound
absorption.
[0009] FIGS. 2A and 2B are graphs illustrating sound absorption
coefficients of a poroelastic acoustical foam having a simple
rectangular shape and a poroelastic acoustical foam having a wedge
shape, with respect to frequency. Referring to FIG. 2A, the lengths
and widths of two shapes are the same, but the amounts of
poroelastic acoustical foam used are different. In FIG. 2A, The
amount of a porous material of the wedge shape is 65% of that of
the rectangular shape. Referring to FIG. 2B illustrating sound
absorption coefficients with respect to frequency, the rectangular
poroelastic acoustical foam shows relatively better performance
than the wedge-shaped poroelastic acoustical foam in some
low-frequency bands. However, the wedge-shaped poroelastic
acoustical foam shows far better performance than the square
poroelastic acoustical foam in most frequency bands. That is,
considering that high performance is displayed despite less amount
of a porous material of wedge-shaped poroelastic acoustical foam,
the performance of poroelastic acoustical foams can be greatly
affected by the shape of the poroelastic acoustical foams when the
material properties of the porous material are the same.
[0010] Nevertheless, most conventional studies have analyzed and
experimented on poroelastic acoustical foams having conventional
shapes, and no study has been conducted to obtain the optimal shape
of poroelastic acoustical foams without initial shapes. That is,
most of the conventional studies have attempted to identify
properties of porous materials and interpret the performance of the
poroelastic acoustical foams having given shapes in order to
enhance the performance of the poroelastic acoustical foams.
Further, the conventional studies have focused on enhancing the
performance of poroelastic acoustical foams by repeating analyses
and experiments based on their initial shapes, such as a wedge
shape, and obtaining the optimal scales of the initial shapes.
SUMMARY OF THE INVENTION
[0011] Aspects of the present invention provide an apparatus and
method for designing the optimal shape of poroelastic acoustical
foam to obtain optimal performance under given conditions in a
state where no initial shape is given.
[0012] However, aspects of the present invention are not restricted
to the one set forth herein. The above and other aspects of the
present invention will become more apparent to one of ordinary
skill in the art to which the present invention pertains by
referencing the detailed description of the present invention given
below.
[0013] According to an aspect of the present invention, there is
provided poroelastic acoustical foam which is made of a porous
material, the poroelastic acoustical foam consisting of a
wedge-shaped wedge unit where the cross section is reduced in one
direction, and a bowl unit where formed at one end of the wedge
unit where the cross section of the wedge unit is small. Further,
the other end, where the cross section of the wedge unit is large,
is separated from the wall, forming an air layer.
[0014] a wedge-shaped wedge unit where the cross section is reduced
in one direction; and [0015] a bowl-shaped bowl unit where formed
at one end of the wedge unit where the cross section of the wedge
unit is small [0016] wherein the other end, where the cross section
of the wedge unit is large, is separated from the wall, forming an
air layer.
BRIEF DESCRIPTION OF THE DRAWINGS
[0017] The above and other aspects and features of the present
invention will become apparent by describing in detail exemplary
embodiments thereof with reference to the attached drawings, in
which:
[0018] FIG. 1 is an enlarged view of a porous material;
[0019] FIG. 2 is a graph illustrating sound absorption coefficients
of a poroelastic acoustical foam having a simple rectangular shape
and a poroelastic acoustical foam having a wedge shape, with
respect to frequency;
[0020] FIG. 3 illustrates an example of an entire system set for
optimization of a poroelastic acoustical foam shape;
[0021] FIG. 4 illustrates various examples of interfaces between an
air layer domain and a porous material domain;
[0022] FIG. 5 illustrates an air layer domain and a porous material
domain according to values of design variables using a method
suggested in the present invention;
[0023] FIG. 6 illustrates a material property interpolation process
using Equation (2);
[0024] FIG. 7 is a diagram for explaining the concept of an
intermediate material between the air and the porous material;
[0025] FIG. 8 is a block diagram of an apparatus for topology
optimization of poroelastic acoustical foam according to an
exemplary embodiment of the present invention;
[0026] FIG. 9 sequentially illustrates a plurality of upgrade
processes according to an exemplary embodiment of the present
invention;
[0027] FIG. 10 illustrates optimized shapes as the amount of a
porous material used is changed; and
[0028] FIGS. 11A through 11D are graphs comparing sound absorption
coefficients of topology-optimized shapes of the four cases of FIG.
10 to those of conventional wedge shapes.
DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENTS
[0029] The present invention will now be described more fully with
reference to the accompanying drawings, in which exemplary
embodiments of the invention are shown. The invention may, however,
be embodied in many different forms and should not be construed as
being limited to the embodiments set forth herein; rather, these
embodiments are provided so that this disclosure will be thorough
and complete, and will fully convey the concept of the invention to
those skilled in the art. Like reference numerals in the drawings
denote like elements, and thus their description will be
omitted.
[0030] The present invention will hereinafter be described in
detail with reference to the accompanying drawings.
[0031] The present invention formulates the design of a poroelastic
acoustical foam as an issue of topology optimization design and
suggests a new methodology for designing the shape of the
poroelastic acoustical foam without a given initial shape.
[0032] In order to design the shape of a poroelastic acoustical
foam, a new analysis method of an acoustic system including a
porous material is required. The conventional analysis method
interprets a domain formed of a porous material (hereinafter,
referred to as a porous material domain) and an air layer domain,
to which sound waves propagate, using different governing
equations. Therefore, the conventional analysis method requires a
complicated process of adjusting interface conditions between the
two domains. The interface conditions include displacement
continuity and pressure continuity. In addition, as analysis
proceeds, if the location of the interface between the two domains
is changed, the process of adjusting the interface conditions
should be repeated. In this regard, it is difficult to design an
optimal shape of a poroelastic acoustical foam using the
conventional analysis method.
[0033] FIG. 3 illustrates an example of an entire system set for
optimization of a poroelastic acoustical foam shape. Referring to
FIG. 3, the entire system includes an air layer domain 31 in which
acoustic air exists and a porous material domain 32 in which a
porous material exists. The porous material domain 32 is a design
domain for optimization of shape. The air layer domain 31 and the
porous material domain 32 are surrounded by a strong support 34,
and an end of the air layer domain 31 is open. After a plane sound
wave is incident on the open end of the air layer domain 31, the
reflection coefficient can be calculated by measuring the reflected
wave. Generally, the reflection coefficient can be calculated
simply by dividing the amplitude of the reflected wave by that of
the incident wave. For example, if the amplitude of a reflected
wave is zero, the reflection coefficient is zero. If the amplitude
of the reflected wave is equal to that of an incident wave, the
reflection coefficient is one. In this case, the reflection
coefficient of zero indicates that poroelastic acoustical foam has
absorbed all incident waves, and the reflection coefficient of one
indicates that the poroelastic acoustical foam has not absorbed the
incident wave at all. The reflection coefficient varies according
to the frequency of the incident sound wave. Hence, the core of the
present invention lies in how to reduce the reflection
coefficient.
[0034] The conventional analysis method uses a Helmholtz equation
as a governing equation for the air layer domain 31 and uses Biot's
equation as a governing equation for the porous material domain 32.
That is, since the conventional analysis method uses two different
governing equations for the entire system, interface conditions
must be adjusted at an interface 33 between the air layer domain 31
and the porous material domain 32. In addition, as the conventional
analysis method proceeds, the interface conditions should be
readjusted whenever the porous material domain 32, which is a
design domain, is changed to various forms 41 through 43 as
illustrated in FIG. 4. That is, the conventional analysis method
can be used to change some scales of a poroelastic acoustical foam
having a given initial shape, it is almost impossible to apply the
conventional analysis method to optimization of arbitrary shape
design which requires many repeat calculations.
[0035] Therefore, in order to optimize the shape of a poroelastic
acoustical foam, it is required to analyze the entire system using
a single governing equation. Accordingly, the present invention
suggests a method to analyze a porous material and an air layer
using the same governing equation, i.e., the Biot's equation which
is widely used to analyze porous materials, with adopting the
concept of a material property interpolation used in topology
optimization design.
[0036] To this end, it is required to assign an independent design
variable to each of a plurality of elements or meshes that form a
design domain and to set material properties of a porous material
as functions of design variables. By this process, each element of
the design domain can represent both states of the porous material
and the air layer according to values of the design variables.
Then, since a porous material domain and an air layer domain can be
analyzed using the same Biot's equation instead of different
governing equations, the complicated process of adjusting interface
conditions at an interface between the porous material domain and
the air layer domain is not required. Therefore, the optimal shape
of a poroelastic acoustical foam can be obtained even without its
initial shape.
[0037] Referring to FIG. 5, in the present invention, since the
design variables .chi..sub.e of the air layer domain 31 is zero,
the design variables .chi..sub.e of the porous material domain 32
is set to one. Here, the subscript e indicates identification
number of elements. The design variable .chi..sub.e not only
indicates whether a corresponding element is an air or a porous
material, but also can express an intermediate material that does
not actually exist. Since the design variable .chi..sub.e can have
a value between zero and one, various intermediate materials can be
represented according to the value of the design variable
.chi..sub.e. In this case, properties of the intermediate materials
can be interpolated in the form of a function of the design
variable .chi..sub.e.
[0038] In summary, the present invention performs numeral
interpretation, such as a finite element method and a finite
difference method, using a single governing equation (the Biot's
equation) for the entire system. For that, material properties of
each element that forms a design domain are interpolated using the
design variable .chi..sub.e. In addition, topology optimization
according to the present invention is a process of finding an
optimal design variable (a converged value: zero or one), which
satisfies limit conditions and an objective function, by repeating
numerical interpretation while varying the design variable
.chi..sub.e. The Biot's equation, material property interpolation,
and topology optimization will be described in detail in the
following.
[0039] Biot's Equation
[0040] The solid phase in a porous material is deformed by external
pressure, which results in strain and stress. The external pressure
also acts on the fluid phase. That is, the external pressure causes
the volume change of the fluid phase and the change of internal
pressure. More important aspect is that a change in the solid phase
results in a change in the fluid phase. Also, a change in the fluid
phase results in a change in the solid phase. That is, the two
phases are coupled to each other.
[0041] In order to describe this phenomenon, Biot suggested
Equation (1) below for the propagation of an elastic wave in a
porous material having its pores saturated by viscous fluid. The
propagation of a sound wave in a porous material saturated by air
can also be explained using the Biot's equation. The first equation
in Equation (1) is an equation of motion which is based on the
equilibrium of force acting on the solid phase, and the second
equation is an equation of motion which is based on the equilibrium
of force acting on the fluid phase. The last common term to the two
equations has been added in consideration of heat dissipation due
to pores in the solid.
N 2 u + [ ( A + N ) e + Q ] = .differential. 2 .differential. t 2 (
.rho. 11 u + .rho. 12 U ) + b .differential. .differential. t ( u -
U ) [ Q e + R ] = .differential. 2 .differential. t 2 ( .rho. 12 u
+ .rho. 22 U ) - b .differential. .differential. t ( u - U ) . ( 1
) ##EQU00001##
[0042] Parameters used in Equation (1) are defined by Table 1
(.gradient. indicates gradient).
TABLE-US-00001 TABLE 1 Symbol Description u Solid-phase
displacement vector U Fluid-phase displacement vector e .gradient.
u .epsilon. .gradient. U N Elastic shear modulus A Lame' constant
Q, R Coupling coefficients .rho..sub.11 Density of solid phase
.rho..sub.22 Density of fluid phase .rho..sub.12 Mass effect of
fluid flowing through pores of solid b Viscous coupling coefficient
or Darcy coefficients
[0043] In order to solve a governing equation, such as Equation
(1), by numerical analysis, the Galerkin method may be applied to
the governing equation, and the governing equation is changed by
finite element method. The Galerkin method is a technology for
generating a finite element model based on a specified governing
equation, which is well known to those of ordinary skill in the
art.
[0044] Material Property Interpolation
[0045] In order to apply topology optimization, an intermediate
material whose design variable has a value between zero and one
needs to be taken into consideration. The intermediate material
does not exist in real, and is eventually removed. However, in the
numerical analysis process of the iterative topology optimization,
the intermediate material is regarded as an existent material.
[0046] Therefore, a key issue here is how to represent properties
(N through b in Table 1) of the intermediate material. In the
present invention, the properties of the intermediate material are
represented by continuous functions of design variables. Like this,
expressing the material state in an element by a continuous
function may be defined as material property interpolation.
[0047] According to an exemplary embodiment of the present
invention, a property M.sub.e of an intermediate material may be
represented by a function of a design variable .chi..sub.e as in
Equation (2). The material property M.sub.e denotes any one of N,
A, Q, R, .rho..sub.11, .rho..sub.22, .rho..sub.12, and b.
M.sub.e=.chi..sub.e.sup.r(M.sub.foam-M.sub.air)+M.sub.air, (2)
where the subscript foam indicates a porous material, and the
subscript air indicates air. In addition, r indicates a degree
related to the curvature of a function and may have a different
value for each material property. When M.sub.foam is smaller than
M.sub.air, M.sub.e follows pattern A of FIG. 6. Conversely, when
M.sub.foam is greater than M.sub.air, M.sub.e follows pattern B of
FIG. 6.
[0048] Topology Optimization
[0049] Optimization can be defined variously in many different
fields. From the engineering perspective, optimization is defined
as a process and a method of finding a solution that can produce
the optimal performance under given circumstances. From the
structural perspective, in particular, optimization is classified
shape optimization, size optimization, and topology
optimization.
[0050] Shape optimization refers to designing an optimal structure,
which serves a purpose, using performance differences according to
the shape of all or part of a structure. Size optimization refers
to finding out which part of a structure should be changed by how
much to achieve better performance. That is, both of shape
optimization and size optimization require basic layouts at the
beginning of design process. Shape optimization requires a basic
shape and is conducted by modifying the basic shape. Similarly,
size optimization requires a basic size. Since both of the methods
can be used in limited design domains, the room for optimization is
reduced.
[0051] Topology optimization makes it possible to design a
structure, which optimally serves a purpose, from a state such as a
black box, without any basic layout or initial assumption. In
topology optimization, the entire shape and detailed dimensions are
designed at one time. That is, using topology optimization, an
optimal structure that is physically and mathematically reasonable
can be designed regardless of whether a designer is experienced,
skilled, or prejudiced.
[0052] Topology optimization was first suggested by Bendsoe and
Kikuchi. An initial study of topology optimization was applied
mainly in the optimal design of a structure under a static load.
However, application of topology optimization in various fields
have recently been reported.
[0053] A design technique using topology optimization, that is,
topology optimization design, requires an understanding of
"existence and non-existence of materials," which is the most
important concept for this technique. In the case of a structure,
"existence and non-existence of materials" denotes existence or
non-existence of materials that form the structure. In the case of
optimal design of poroelastic acoustical foams, "existence and
non-existence of a material" denotes the existence or non-existence
of porous materials. Topology optimization based on the above basic
concept may be defined as "a process of obtaining the distribution
of materials which optimally serves a purpose under given
constraint conditions in a design domain."
[0054] The existence/non-existence of materials denotes the
distribution of the materials. In order to variously represent the
distribution of materials within a design domain, the design domain
needs to be divided into smaller units. In the finite element
method, the units are defined as elements or meshes.
[0055] Topology optimization is not performed simply based on the
existence/non-existence of elements. Rather, topology optimization
obtains a physically reasonable, mathematically stable, and
cost-effectively optimal solution. One of the most widely used
algorithms for topology optimization is a sensitivity analysis
algorithm. Sensitivity represents a change in the performance of
the entire structure (a change in objective function value) when
the properties of one of a plurality of elements are slightly
changed. In mathematics, sensitivity is a differential value. In
order to differentiate a function, the function must be continuous
over the domain where the function is defined.
[0056] To this end, an intermediate element (whose design variable
is greater than zero and less than one) is taken into consideration
when the above-mentioned two states, i.e., existence and
non-existence, of materials are represented.
[0057] Referring to FIG. 7, a material that fills an element at the
beginning (.chi..sub.e=1) is gradually reduced (to an intermediate
material). Ultimately, the element is in a state (.chi..sub.e=0)
without any material. That is, each element that forms a design
domain can represent an intermediate material that continuously
changes between a porous material and air.
[0058] As described above, the properties of an intermediate
material are represented by continuous functions of a corresponding
design variable. If optimization begins in a state where all
elements in a design domain have the same design variables,
sensitivity analysis can be conducted using an objective function
and constraint conditions at that time. Then, upgraded design
variables can be obtained. Although the design variables are
upgraded, the constraint conditions in the entire design domain are
maintained unchanged.
[0059] After this upgrade process is iterated a number of times, a
convergence state in which the value of the objective function no
longer changes is reached. Here, design variables in the
convergence state can be understood as the optimal solution.
[0060] Based on the above technical description, the configuration
of an apparatus for topology optimization of poroelastic acoustical
foam according to an exemplary embodiment of the present invention
will now be described. FIG. 8 is a block diagram of an apparatus
100 for topology optimization of poroelastic acoustical foam
according to an exemplary embodiment of the present invention.
Referring to FIG. 8, the apparatus 100 includes a governing
equation determination unit 110, a numerical analysis unit 120, a
material property interpolation unit 130, a design domain setting
unit 140, an objective function setting unit 150, and a topology
optimization unit 160.
[0061] The design domain setting unit 140 sets an entire system as
illustrated in FIG. 3 and then sets a design domain which is
assumed to be filled with a porous material. In this case, a right
side of the design domain is fixed, and the vertical displacements
of upper and lower sides of the design domain are limited by rigid
supports. Therefore, the design domain setting unit 140 can set a
design domain by selecting L.sub.1, L.sub.2 and H.
[0062] The governing equation determination unit 110 determines a
governing equation that can well represent energy properties of a
porous material, and generates a finite element model based on the
determined governing equation. In the finite element model, the set
design domain is divided into a plurality of elements or meshes.
The Biot's equation may be used as the determined governing
equation, and the Galerkin method may be used to generate the
finite element model.
[0063] The objective function setting unit 150 sets an objective
function which is a basis to perform topology optimization. A goal
of topology optimization of poroelastic acoustical foam is to
design the shape of a poroelastic acoustical foam which has a
maximized sound absorption coefficient or minimized reflection
coefficient in a frequency range of interest. Generally, the sound
absorption coefficient .alpha..sub.n and the reflection coefficient
R have the following relationship defined by Equation (3).
.alpha..sub.n=1-|R|.sup.2 (3)
[0064] An objective function L according to an exemplary embodiment
of the present invention may be given by Equation (4). The
objective function has to include limit conditions, and mass limit
condition is used as the constraint condition. That is, a condition
in which a sum .SIGMA..chi..sub.e of design variables for all
elements in a design domain is less than a predetermined value
V.sub.0 is used.
L = min .chi. e [ w 1 i = 1 m 1 .alpha. n ( f i , .chi. e ) + w 2 e
= 1 N e .chi. e ( 1 - .chi. e ) ] subject to e = 1 N e .chi. e
.ltoreq. V 0 , ( 4 ) ##EQU00002##
where w.sub.1 and w.sub.2 indicate weights, and the sound
absorption coefficient .alpha..sub.n is represented by a function
of a considering frequency f.sub.i and design variables
.chi..sub.e. An explicit penalty function:
e = 1 N e .chi. e ( 1 - .chi. e ) , ##EQU00003##
is added to the objective function in order to increase convergence
in topology optimization, and thus guarantee the stability of the
result of topology optimization.
[0065] The topology optimization unit 160 upgrades design variables
using a sensitivity-based topology optimization algorithm, and
obtains optimal topology of a poroelastic acoustical foam by
repeating this upgrade process. Specifically, the topology
optimization unit 160 initially sets the design variables to the
same value within a range satisfying the constraint condition. For
example, if the predetermined value V.sub.0 is 0.6, initial value
of the design variables .chi..sub.e for all elements in a design
domain is set within 0.6.
[0066] Next, the topology optimization unit 160 calculates the
sensitivity for each element. The sensitivity calculation includes
an operation in which the topology optimization unit 160 changes
the design variables a little, an operation in which the material
property interpolation unit 130 performs material property
interpolation using the changed design variable as shown in
Equation (2), and an operation in which the numerical analysis unit
120 performs numerical analysis by applying the interpolated
material properties to an analysis model (such as a finite element
model and a finite difference model) according to the determined
governing equation and calculates an output value of the objective
function (hereinafter, referred to as an objective function value).
In order to obtain the objective function value, the sound
absorption coefficient .alpha..sub.n must be calculated. The sound
absorption coefficient .alpha..sub.n can be easily calculated
through the above numerical analysis based on the reflection
coefficient R, that is, a ratio of the amplitude of a reflected
wave to that of the incident sound wave (see Equation (3)).
[0067] If the sensitivities for all elements included in the design
domain are calculated in the above operations, the topology
optimization unit 160 adjusts the design variable of each element
according to the sensitivities. In this case, an average of the
design variables for all elements is limited within 0.6.
[0068] As described above, the design variables (a design variable
set) in the design domain are upgraded by adjusting the design
variables according to sensitivity (a first upgrade process).
[0069] Then, the numerical analysis unit 120 calculates the sound
absorption coefficient based on the upgraded design variables, and
thus recalculates the objective function value. Accordingly, the
topology optimization unit 160 recalculates the sensitivities, and
thus upgrades the design variables again (a second upgrade
process).
[0070] The above upgrade processes are repeated until the change of
the objective function is converged within a predetermined
range.
[0071] Each component described above with reference to FIG. 8 may
be implemented as a software component, such as a task, a class, a
subroutine, a process, an object, an execution thread or a program
performed in a predetermined region of a memory, or a hardware
component, such as a Field Programmable Gate Array (FPGA) or an
Application Specific Integrated Circuit (ASIC). In addition, the
components may be composed of a combination of the software and
hardware components. The components may be reside on a
computer-readable storage medium or may be distributed over a
plurality of computers.
[0072] FIG. 9 sequentially illustrates iterative upgrade processes
described above according to an exemplary embodiment of the present
invention. Referring to FIG. 9, an initial design domain 91 is
filled with the same design variables. Then, as the upgrade
processes are repeated, finally converged design variables are
obtained. Therefore, the design domain 96 has a finally converged
shape, and the design variables included in the design domain 96
are optimal design variables. It is to be noted that while gray
elements indicating intermediate materials are widely distributed
in an initial upgrade process, they hardly exist in the finally
converged design domain 96. That is, in the result, each element is
determined to be either a porous material (black element) or air
(white element).
[0073] FIG. 10 illustrates shapes of poroelastic acoustical foam
capable of optimal performance using the above mentioned topology
optimization design technique. In FIG. 10, each case is the result
of performing optimization by limiting the amount of a porous
material that can fill the design domain as in FIG. 3. Case 1 shows
the result when the amount of a porous material is limited to 50%
of the design domain; case 2, the case of 55%; case 3, the case of
60%; and case 4, the case of 65%.
[0074] Generally, in high-frequency bands, the performance of
wedge-shaped poroelastic acoustical foams is very high, and the
sound absorption coefficient is close to 1. Hence, in FIG. 10, the
optimization was conducted by selecting a frequency range of
100-1500 Hz that includes low-frequency and middle-frequency bands
in which the wedge shape has poor absorption performance. This
setting is intended to design a new optimized shape of poroelastic
acoustical foam which differs from the wedge shape having a low
absorption performance in low-frequency bands.
[0075] An improved wedge according to the present invention is
obtained through two-dimensional (2D) design. Therefore, it can be
understood that a three-dimensional (3D) wedge has a uniform shape
in a direction perpendicular to the cross section of FIG. 10.
Alternatively, the 3D wedge may be designed to have a shape as if
obtained by rotating the cross section of FIG. 10 about a
longitudinal axis.
[0076] A new type of poroelastic acoustical foam with improved
sound-absorbing performance in low-frequency and middle-frequency
bands according to the present invention is described in the
following with reference to FIG. 10.
[0077] First, there is a wedge-shaped wedge unit 210 where the
cross section is reduced in one direction. In case 1 of FIG. 10,
the wedge unit 210 is wedge-shaped where the cross section is
reduced in one direction (in the left direction in the drawing)
though the border is not straight as the result of optimization
process.
[0078] Further, a bowl-shaped bowl unit 220 is formed in one end
where the cross section of the wedge unit 210 is reduced.
[0079] Further, the other end of the wedge unit is separated from
the wall, thereby forming an air layer.
[0080] Characteristics in configuration of case 1 are similar in
cases 2, 3 and 4 which have gradually increased amounts of a porous
material. Only the characteristics in configuration of the
poroelastic acoustical foams have significantly disappeared in case
4 because the amount of a porous material becomes a significant
factor in the sound-absorbing performance as the amount of a porous
material increases.
[0081] FIGS. 11A through 11D are graphs comparing sound absorption
coefficients in the four cases of FIG. 10 to those of conventional
shapes. In each drawing, the optimized shapes and conventional
wedge shapes use the same amount of a porous material. That is, in
FIGS. 11A, 11B, 11C and 11D, the poroelastic acoustical foam
occupies 50%, 55%, 60% and 65%, in the design domain respectively.
FIGS. 11A through 11D show that the sound-absorbing performance is
generally improved in topology-optimized shapes, compared with
conventional shapes. Especially, it is shown that the
sound-absorbing performance is significantly improved in
low-frequency and middle-frequency bands.
[0082] Referring to the graph of FIG. 11A showing sound absorption
coefficients with respect to frequency, it is shown that the first
peak is located near 350 Hz in the graph of the optimized shape.
The air layer described with reference to FIG. 10 generates
vibrating pattern with repeat peaks in the graph. Further, the air
layer lowers the location of the peak to the low-frequency
location, thereby improving the sound-absorbing performance.
[0083] Further, the bowl unit formed in one side of the wedge unit
improves the sound-absorbing performance in the middle-frequency
area. For the graph of a vibrating pattern by the air layer, the
bowl unit reduces the width shaken by the vibration in the
middle-frequency area, and increases the value of the sound
absorption coefficient in the middle-frequency area, which can be
understood by comparing the results after removing the bowl unit
from the optimized shape.
[0084] The present invention suggests a design technique for
optimizing the shape of a poroelastic acoustical foam using
topology optimization design so that the poroelastic acoustical
foam have maximized sound-absorbing capability in a wide range of
audible frequency bands. The present invention is based on a
technique for representing an air layer as a porous material having
particular material properties by using a material property
interpolation technique of topology optimization design.
Consequently, the present invention makes it possible to design the
shape of a poroelastic acoustical foam, which can attain a desired
performance, without any basic layout or initial shape. Optimized
shapes according to the technique for designing the shape of a
poroelastic acoustical foam using topology optimization design
suggested by the present invention can significantly enhance the
sound-absorbing performance of the poroelastic acoustical foam as
compared to conventional shapes. Especially, the sound-absorbing
performance can be significantly enhanced in low-frequency and
middle-frequency bands.
[0085] While the present invention has been particularly shown and
described with reference to exemplary embodiments thereof, it will
be understood by those of ordinary skill in the art that various
changes in form and detail may be made therein without departing
from the spirit and scope of the present invention as defined by
the following claims. The exemplary embodiments should be
considered in descriptive sense only and not for purposes of
limitation.
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