# Patent application title: Poroelastic acoustical foam having enhanced sound-absorbing performance

##
Inventors:
Yoon Young Kim (Seoul, KR)
Yoon Young Kim (Seoul, KR)
Yeon-June Kang (Seongnam-Si, KR)
Joong-Seok Lee (Seoul, KR)

Assignees:
SEOUL NATIONAL UNIVERSITY INDUSTRY FOUNDATION

IPC8 Class: AE04B184FI

USPC Class:
181286

Class name: Sound-modifying means sound absorbing panels specific acoustical properties or dimensions

Publication date: 2008-09-18

Patent application number: 20080223653

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## 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.## 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; anda bowl-shaped bowl unit where formed at one end of the wedge unit where the cross section of the wedge unit is smallwherein 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 χ

_{e}of the air layer domain 31 is zero, the design variables χ

_{e}of the porous material domain 32 is set to one. Here, the subscript e indicates identification number of elements. The design variable χ

_{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 χ

_{e}can have a value between zero and one, various intermediate materials can be represented according to the value of the design variable χ

_{e}. In this case, properties of the intermediate materials can be interpolated in the form of a function of the design variable χ

_{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 χ

_{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 χ

_{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 ] = ∂ 2 ∂ t 2 ( ρ 11 u + ρ 12 U ) + b ∂ ∂ t ( u - U ) [ Q e + R ] = ∂ 2 ∂ t 2 ( ρ 12 u + ρ 22 U ) - b ∂ ∂ t ( u - U ) . ( 1 )

**[0042]**Parameters used in Equation (1) are defined by Table 1 (∇ indicates gradient).

**TABLE**-US-00001 TABLE 1 Symbol Description u Solid-phase displacement vector U Fluid-phase displacement vector e ∇ u ε ∇ U N Elastic shear modulus A Lame' constant Q, R Coupling coefficients ρ

_{11}Density of solid phase ρ

_{22}Density of fluid phase ρ

_{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

_{e}of an intermediate material may be represented by a function of a design variable χ

_{e}as in Equation (2). The material property M

_{e}denotes any one of N, A, Q, R, ρ

_{11}, ρ

_{22}, ρ

_{12}, and b.

**M**

_{e}=χ

_{e}

^{r}(M

_{foam}-M

_{air})+M

_{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

_{foam}is smaller than M

_{air}, M

_{e}follows pattern A of FIG. 6. Conversely, when M

_{foam}is greater than M

_{air}, M

_{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 (χ

_{e}=1) is gradually reduced (to an intermediate material). Ultimately, the element is in a state (χ

_{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

_{1}, L

_{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 α

_{n}and the reflection coefficient R have the following relationship defined by Equation (3).

**n**=1-|R|

^{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 Σχ

_{e}of design variables for all elements in a design domain is less than a predetermined value V

_{0}is used.

**L**= min χ e [ w 1 i = 1 m 1 α n ( f i , χ e ) + w 2 e = 1 N e χ e ( 1 - χ e ) ] subject to e = 1 N e χ e ≦ V 0 , ( 4 )

**where w**

_{1}and w

_{2}indicate weights, and the sound absorption coefficient α

_{n}is represented by a function of a considering frequency f

_{i}and design variables χ

_{e}. An explicit penalty function:

**e**= 1 N e χ e ( 1 - χ e ) ,

**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

_{0}is 0.6, initial value of the design variables χ

_{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 α

_{n}must be calculated. The sound absorption coefficient α

_{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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