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Engineering Optimization
ISSN: 0305-215X (Print) 1029-0273 (Online) Journal homepage: http://www.tandfonline.com/loi/geno20
Synthesis of porous–acoustic absorbing systems
by an evolutionary optimization method
F. I. Silva & R. Pavanello
To cite this article: F. I. Silva & R. Pavanello (2010) Synthesis of porous–acoustic absorbing systems by an evolutionary optimization method, Engineering Optimization, 42:10, 887-905, DOI: 10.1080/03052150903477184
To link to this article: https://doi.org/10.1080/03052150903477184
Published online: 14 Jun 2010.
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Vol. 42, No. 10, October 2010, 887–905
Synthesis of porous–acoustic absorbing systems by an
evolutionary optimization method
F.I. Silvaa* and R. Pavanellob
aDepartment of Mechanical Engineering and Manufacturing, Federal University of Ceará - UFC,
Fortaleza, Ceará, Brazil;bDepartment of Computational Mechanics, Faculty of Mechanical Engineering, State University of Campinas - Unicamp, Campinas, São Paulo, Brazil
(Received 20 November 2008; final version received 09 November 2009 )
Topology optimization is frequently used to design structures and acoustic systems in a large range of engineering applications. In this work, a method is proposed for maximizing the absorbing performance of acoustic panels by using a coupled finite element model and evolutionary strategies. The goal is to find the best distribution of porous material for sound absorbing panels. The absorbing performance of the porous material samples in a Kundt tube is simulated using a coupled porous–acoustic finite element model. The
equivalent fluid model is used to represent the foam material. The porous material model is coupled to a
wave guide using a modal superposition technique. A sensitivity number indicating the optimum locations for porous material to be removed is derived and used in a numerical hard kill scheme. The sensitivity number is used to form an evolutionary porous material optimization algorithm which is verified through examples.
Keywords: porous material; evolutionary optimization; acoustic devices
1. Introduction
The goal of this article is to propose a systematic methodology to design acoustic sound absorbing panels. The acoustic panels must be as light as possible and the acoustic absorption in the low-frequency domain must be maximized for certain values. The design of acoustic panels has been an important research topic for many years. In particular, for aircraft structures interior sound radiation problems have been an active research area (Herdic 2005), where the majority of the techniques are based on numerical simulations.
In the last thirty years, several numeric techniques based on the finite element formulation for sound absorbing materials have been developed. The acoustical analysis of cavities with absorption boundary conditions and the equivalent fluid concept was introduced by Craggs (1978, 1979, 1985). Impedance techniques have been used to model absorption of normal and random incidence waves in acoustic domains (Bliss 1981, Beranek 1992). Coupled fluid-structure models *Corresponding author. Email: [email protected]
ISSN 0305-215X print/ISSN 1029-0273 online © 2010 Taylor & Francis
DOI: 10.1080/03052150903477184 http://www.informaworld.com
based on the Biot theory have been developed and improved by Kang (1995), Panneton and Atalla (1996) and Lamary et al. (2001). In Atalla et al. (1998), the classical Biot–Allard equations are rewritten in terms of the solid phase macroscopic displacement vector and interstitial fluid phase macroscopic pressure.
The performance of perforated material used in acoustic absorbing material design was tested by Sgard et al. (2001, 2005). Using double porosity theory, the problem can be formulated on the meso-scale (Olny and Boutin 2003). A modal decomposition scheme to represent the acoustic wave guide can be used (Sgard et al. 2001, 2005). Using these techniques, foam materials can be simulated with good accuracy and reasonable computational cost.
Pioneering works in finite element based topology optimization are to be found in Bendsoe and Kikuchi (1988), Bendsoe (1989) and Susuki and Kikuchi (1991). They used the homogenization method to find the best distribution of material and holes in several structures.
In structural optimization and design, some topology optimization techniques have been intensively used to solve compliance minimization problems (Xie and Steven 1997, Bendsoe and Sigmund 2003, Hsu and Hsu 2005). The authors used a methodology based on a single interpolation material with penalization (SIMP) interpolation scheme. Among others works, evolutionary algorithms (EAs) have demonstrated efficiency and accuracy to find optimal solutions for a large range of optimization problems (Xie and Steven 1997, Farmani et al. 2005).
These methods are based on the concept of removing the elements that represent the parts of the structure that are not being efficiently used. This technique was developed by Xie and Steven (1993, 1996, 1997) and it is known as evolutionary optimization (EO). EO is the technique chosen for this work, with many new concepts and features adapted for the sound absorbing panels optimization problem.
Two design procedures applied to maximize the sound absorption capability of the insula-tion acoustic system have been recently proposed by Becot et al. (2006) and Tsay (2006). Tsay (2006) presented a systematic study of the influence of pyramid shaped foam inserts on the acoustic absorption properties. Different geometrical configurations of a pyramidal polyurethane foam system inside the impedance tube have been simulated. The numerical results were anal-ysed to find the optimal configuration. Becot et al. (2006) presented a study of the influence of the location of the absorbing materials on the sound radiated by an acoustic cavity backed plate.
In order to explore the concept of meso-perforations for optimal absorption properties, Sgard et al. (2005) presented a systematic review of theoretical, numerical and experimental works using the theory of poroelasticity and multi-scale models. Applications to design optimized noise control solutions were presented and practical design rules were established.
In this work, a systematic method is proposed for finding the optimal topology of a porous– acoustic system. The goal is to maximize the absorption properties for a specific frequency value. The method combines an equivalent fluid finite element model coupled to a classical acous-tic finite element model with an evolutionary topological algorithm (ETA). The concept of a meso-perforation is explored and the plane wave hypothesis has been adopted. This methodology has been implemented and tested for plane configurations to show the validity of the proposed techniques.
The outline of the rest of the article is as follows: in Section 2, the porous–acoustic problem is stated using the equivalent fluid method. In this context, the finite element method is applied to describe the system and a wave guide modal superposition technique is used to model the pure acoustic domain. In Section 3, an evolutionary topology optimization procedure is proposed for the coupled system. In Section 4, the numerical results are presented and the performance of the method is illustrated. The conclusions are outlined in Section 5.
2. Problem description
A plane absorbing panel is considered with a periodic geometrical perforation as shown in Figure 1. Because of the geometric periodicity, it is possible to find the global acoustic response considering only the behaviour of a local repetitive cell response. For this condition, just the local geometrical configuration can be considered (Sgard et al. 2001). Figure 2 shows the problem in meso-scale. The local geometry is a square Kundt tube with absorbing material placed on the bottom of the tube. It is assumed that only plane waves propagate in the wave guide, which is acceptable for a frequency range below the cut-off limit of the Kundt tube.
In Figure 2, one porous material sample is placed into a semi-infinite wave guide with rigid walls and is acoustically excited by plane waves. In this work, two configurations are tested: homogeneous absorbent lining and perforated absorbent lining.
The porous material acoustic performance in the meso-scale is evaluated by the absorption coefficient
α= diss inc
, (1)
where α is the acoustic absorption coefficient, dissrepresents the dissipated potential and incis the incidence potential (Sgard et al. 2005). The absorption coefficient α is a real and frequency-dependent function.
Figure 1. Absorbing panel in macro-scale.
The topology optimization of the lining system is performed using an evolutionary simula-tion design approach that combines a porous–acoustic finite element method with a hard-kill optimization technique.
The objective of this optimization is to improve the absorption value of a particular frequency. The finite element formulation implemented is based on the partial differential equations formu-lated and described by Depollier (1988), Allard (1993) and using a symmetric equivalent fluid formulation derived by the mixed formulation (u, p) of Atalla et al. (1998).
The design variable is represented by μ, defined as the relative density between the porous and acoustic phases. This variable must be evaluated in the domain. The objective function can be written as max μ (α)ω= maxμ diss inc ω = max μ
sdiss+ vdiss+ tdiss inc ω (2) such that V (μ)≤ fvolV0 (3) μ= {μi∈ Zn/μi= [0, 1]}, (4)
where V (μ) is the porous volume, V0 represents the initial design volume and fvolis the final volumetric fraction of porous material. The term dissis the dissipative potential in the porous media and inc represents the incident energy on the porous–acoustic system. In Biot–Allard theory (Allard 1993), three dissipative mechanisms can be noted: the structural, viscous and thermal damping, denoted by the superscripts s, v and t, respectively.
The constraint described in Equation (3) is the relation between the initial and final porous volume. The second constraint given by Equation (4) represents the lateral limits for the topological design variables.
For the evolutionary optimization technique, the design variable is discrete and binary, given by Equation (4), with values 0 in the acoustic domain and 1 in the porous domain.
It is possible to calculate the absorption coefficient α of the perforated material coupled to a wave guide using a numerical approximation. It is based on a finite element discretization of the porous–acoustic systems and a modal approach for the acoustic wave guide. In this case, the design variables are defined over the elements. For each element, constituent material properties and the corresponding differential equation are applied. A weak solution using a semi-discrete finite element method is used and coupled with a semi-analytic modal discretization of an acoustic wave guide.
2.1. Governing field equations
Two different domains can be found in a porous–acoustic system: the pure acoustic domain in the wave guide region and in the holes of meso-perforated material and the porous domain in the lining region as shown in Figure 3. For all domains, classical linear acoustic behaviour is assumed (Allard 1993). In this approach, wave propagation theory for the coupled porous medium is valid for low-frequency and fully saturated conditions. At low frequency, viscous forces dominate. The flow in the pores is governed by a dynamic model of permeability introduced in Auriault et al. (1985) and Johnson et al. (1987). Experimental tests were performed by Allard et al. (1994) to validate this model.
For the pure acoustic domain a, shown in Figure 3, the behaviour of the fluid is governed by the Helmholtz equation (Equation 5) and given as follows (Kinsler et al. 1995):
1 ω2ρ 0 ∇2p+ 1 ρ0c20 p= 0, (5)
Porous Domain Acoustic Domain p a
Figure 3. Domains in the porous–acoustic cell system.
where p is the acoustic pressure, ω is the angular frequency, ρ0is the fluid density and c0is the speed of sound in the fluid.
In the porous domain p, shown in Figure 3, all dependent quantities represent small fluctuations around a static reference value. For these conditions, the porous material properties, like porosity, tortuosity and resistivity, are continuous and isotropic. The porous material behaviour on the meso-scale is related to the behaviour of the discontinuous porous skeleton on the micro-scale using a homogenization technique. Some general hypotheses have been introduced by Olny and Boutin (2003). They considered a characteristic size smaller than 10−5m for the micro-porous medium to be pervious to acoustical waves. For homogenization, the characteristic size of the material micro-pores must be much smaller than the wavelength in the material. In this article, the equivalent fluid formulation is used for modelling the absorbing material (Panneton and Atalla 1996, 1997a,b). In the equivalent fluid model the solid phase vibration is neglected, hence this model is best applied to rigid porous materials, for example rock-wool and dense foams.
The governing equation of pressure for the equivalent fluid model is derived using the Biot equations giving h2 ω2˜ρ 22 ∇2p+h 2 ˜Rp= 0, (6)
where h is the porosity of the material, defined as the ratio of the volume of the interstitial voids to the total volume of the porous medium. In Equation (6), the state variable p represents the interstitial fluid macroscopic pressure. The effective density for the porous system, ˜ρ22 is a complex and frequency-dependent variable given by Johnson et al. (1987):
˜ρ22= α∞ρ0
1+ σ h iωα∞ρ0
˜G(ω), (7)
where ‘i’ is the imaginary term, α∞is the tortuosity and σ is the flow resistivity coefficient of the porous material. The function ˜G(ω)is given by
˜G(ω) =1+4iα 2 ∞ηρ0ω σ2 2h2 1/2 , (8)
where η is the kinematic fluid viscosity and is the viscous characteristic length (Johnson et al. 1987). The term ˜Rin Equation (6) is the modified bulk modulus of the air in the porous medium,
i.e. ˜R= h ˜Kf and the original bulk modulus ˜Kf is defined as ˜ Kf = γ P0 γ− (γ − 1) 1+ σ h iPrωρ0α∞ ˜G (Prω) −1, (9)
where γ = Cp/Cv = 1.4 is the ratio of specific heats of the fluid, P0is the atmospheric pressure, Pr is the Prandtl number, σis the thermal resistivity coefficient and is the thermal characteristic length (Johnson et al. 1987). The function ˜G(Prω) is given by
˜G(Prω)=1+4iα2∞ηρ0Prω
σ2 2h2
1/2
. (10)
These material coefficients are derived from the Biot–Allard theory (Biot 1956, Allard 1993). The equivalent fluid formulation has been implemented in the frequency domain. For harmonic motion, the coupled system of Equations (5) and (6) can be written in a weak integral form. In the next section, the weak form for the acoustic and porous domains is developed.
2.2. The weak integral form
Using the Galerkin method, taking δp as the admissible virtual variation of the fluid phase pressure field (p) for the acoustic domain, the weak integral form of Equation (5) is given by
a 1 ω2ρ 0∇p a· δ∇p d − a 1 ρ0c02 paδpd− a 1 ω2ρ 0 (∇pa· na)δpd = 0, (11)
where a and adenote the acoustic domain and its boundary, respectively. The vector na is the unitary normal pointing outward from the boundary a and (∇pa· na)= ∂pa/∂na is the directional derivative of the acoustic pressure.
For the porous domain, the weak integral form of Equation (6) is given by
p h2 ω2˜ρ 22 ∇p · δ∇p d − p h2 ˜Rpδpd− p h2 ω2˜ρ 22 (∇p · n)δp d = 0, (12)
where pand pdenote the porous domain and its boundary, respectively. The vector n is the unitary normal pointing outward from the boundary pand (∇p · n) = ∂p/∂n is the directional derivative of the porous phase pressure.
This formulation has several advantages over others methods, such as low computational cost and direct compatibility among porous and acoustic formulations. Therefore, the coupling among porous and acoustic media eliminates the need for the calculation of the interface matrix. The interface conditions of the porous–acoustic system take into account the continuity of fluid normal displacements and pressures, Equations (13) and (14), respectively.
The relations among the acoustic normal displacement Unaand the normal porous displacement
Unin the porous–acoustic interface pacan be described as
Una= 1 ω2ρ 0 ∂pa ∂na = h2 ω2˜ρ 22 ∂p ∂n = hUn, (13)
where pa is the pressure in the acoustic domain and p is the pressure in the porous domain. The unitary normal vectors to the porous–acoustic interface pa are represented by na and n.
The equilibrium of pressures in the normal surface pais given by
p= pa. (14)
For this coupled model, all variables are complex. As a consequence, the problems of high computational costs and high-frequency limits are increased.
For the wave guide domain, an analytical model presented by Castel (2005) was used. The coupling of the porous–acoustic numeric model with the semi-analytic wave guide model is obtained by the imposition of pressure continuity at the porous–acoustic interface. Using the concept of the admittance operator A from point P1 to point P2, as defined by Equation (15) (Sgard et al. 2005), it is possible to represent the wave guide domain using a modal superposition given by A(P1, P2)= m,n kmn ρ0ωNmn ψmn(P1)ψmn(P2), (15)
where ω is the frequency, ψmnrepresents the orthogonal modal shape of the transversal section of the wave guide, Nmnis the quadratic norm of the modal shape and kmn is the wavenumber of the propagation wave in the Kundt tube. The acoustic pressure p is written as the sum of the incidence average pressure 2p0and the radiated pressure from the porous–acoustic interface.
The boundary integral form of the acoustic interface, shown in Equation (11), can be calculated using the modal discretization of the wave guide, shown as follows:
a 1 ω2ρ 0 ∂p ∂nδpd = 1 iω pa pa A(P1, P2) p(P2) δp(P1)d 2d 1 − 1 iω pa pa 2A(P1, P2) p0(P2) δp(P1)d 2d 1. (16)
In Castel (2005), the admittance operator A is projected on the element mesh of the porous– acoustic interface. The effects of the local admittance matrix can be assembled in the global numeric model, resulting in an equivalent fluid coupling represented by coupling wave guide matrix A.
2.3. The finite element analysis
In order to solve the porous–acoustic problem expressed by the weak formulation in Equations (11) and (12) it is possible to use a numerical approximation of the acoustic pressure. Using a classical finite element interpolation, the discrete form of the dynamic equation can be derived as
⎡ ⎣1/ω 2H a− Qa 0 0 0 1/ω2Hn− Qn+ A 0 0 0 1/ω2H− Q ⎤ ⎦ ⎧ ⎨ ⎩ pa pn p ⎫ ⎬ ⎭= ⎧ ⎨ ⎩ 0 2Ap0 0 ⎫ ⎬ ⎭, (17)
where Hais the volumetric or mass matrix of the acoustic domain, H is the volumetric or mass matrix of the porous domain, Qaand Q are the compressibility or stiffness matrices of the acous-tic and porous domains, respectively. The discrete pressures are as follows: pa represents the acoustic domain pressure, pn the pressure in the porous–acoustic interface and p the porous domain pressure. The matrix A is the wave guide interface matrix given by the discrete form of Equation (15).
The matrices Ha, Qa, H and Q can be calculated as integral terms described as follows: δp Hap= 1 ω2ρ 0 a ∇p ∇δp d, (18) δp Qap= 1 ρ0c02 a p δp d, (19) δp H p= h 2 ω2˜ρ 22 p ∇p ∇δp d, (20) δp Q p=h 2 ˜R p p δp d, (21)
where the matrices Hnand Qnrefer to the porous finite elements on the porous–acoustic interface. In compact form, Equation (17) can be rewritten as
ZAU= F, (22)
where the dynamic matrix ZA can be expressed as ZA= Z + A, the dynamic matrix of the
porous phase is Z and the contribution of the coupling wave guide is expressed as A, defined in Equation (15) and determined by a discrete formulation described in Castel (2005).
The equation system (22) is solved to determine the dynamic response U, using a direct method for any frequency. The dynamic matrix ZAis complex and frequency dependent. The
computa-tional cost of this solution can be prohibitive for large 3D problems. In this case, partitioning techniques must be used.
3. Evolutionary topology optimization in porous–acoustic systems
In this article, evolutionary porous material optimization (EPO) has been developed. The EPO method is based on the simple concept of systematically removing the (unwanted) porous material in acoustic media. In order to determine the optimum locations for removing material, a sensitivity analysis is based on compliance modification effects. The effect of changing a single element is evaluated and the gain in the objective function is determined and compared to the initial solution. The sensitivity is calculated by the finite difference method. Porous elements that increase the cost function must be changed to acoustic elements. The cost function is the acoustic absorption of the porous sample placed into the Kundt tube. In Section 3.2, an expression for the sensitivity number is determined by sensitivity analysis of the acoustic absorption function.
3.1. The evolution porous–acoustic optimization process
In the following subsection, the evolution porous–acoustic optimization process (EPO) is pre-sented. The EPO algorithm was devised using the procedures for the evolutionary model presented in Xie and Steven (1993, 1996, 1997).
In Sgard et al. (2001, 2005), the optimization process of one periodic cell can be used to find an optimum configuration of absorbing panels built of multiple acoustic–porous periodic cells. By use of homogenization techniques, it is possible to expand the meso-scale results to the macro-scale results.
A harmonic frequency dependence is assumed for the fields in the porous and acoustic domain. For a specific frequency value or a specific frequency range, it is possible to improve the absorption
Figure 4. Evolution of the porous–acoustic system for a specific frequency optimization problem.
of conventional sound absorbing panels by the introduction of acoustic inclusions in the porous domain.
For each iteration, the elements that give the ‘best’ improvement in the cost function are the ones that are removed. In Figure 4, a typical evolution of a topological optimization process of one porous–acoustic cell is presented. This shows the absolute gain in the objective function until a reduction in the volume limit is reached.
For each element, its sensitivity is calculated with respect to the objective function. The elements with the bigger element sensitivity should be removed. The sensitivity number is an indicator of the absorption value change due to the change status of the element. In Figure 5, a simplified process is presented for obtaining an optimized topology for one unitary cell in a porous–acoustic absorbing system.
The optimization process initiates with a design domain full of porous material – see Figure 5(a). During the evolutionary process, some porous elements are replaced by acoustic elements based on the sensitivity values. This method is based on the simple concept that by slowly removed inefficient material from the porous domain, the residual shape evolves in the direction of making a better acoustic panel – see Figure 5(b). The layout obtained at the end of the evolutionary procedure is defined by some segmental boundary curves – see Figure 5(c). In order to identify patterns and generate boundary curves from the output of the topology optimization procedure, an interfacing module using image processing techniques is required – see Figure 5(d).
In Figure 6, a flowchart is presented of the process to aid understanding of the evolutionary porous material optimization method proposed in this article.
In the next section, the mathematical development of the sensitivity number to the acoustic absorption objective function is presented.
3.2. Sensitivity number to acoustic absorption objective function
In order to identify the best locations to eliminate porous material, a parameter called the sensitivity number is needed. The finite element dynamic analysis can be represented by a forced vibration problem, described in Equation (22).
Suppose that a porous element is removed from the domain and replaced by an acoustic element. The change of the dynamic response U due to such an element removal may be derived from
Figure 5. Changing the porous elements by sensitivity number value. Design Domain Porous Volume Evolutionary Optimization Parameters Objective Function Sensitivity Number t Iteration - i=i+1 Element Transformation STOP
FEM - Z U=F YES
NO
Converged?
Dyn. ResponseA
Figure 6. Flowchart.
the first variation of Equation (22) and described by
U= −(ZA+ Z)−1Z U. (23)
The cost function is a derived result and is obtained using the Z matrix, the global dynamic response U and the potential of the incidence, inc= S|p0|2/2ρ0c0, for the plane wave guide excitation, i.e.
α= 1
2incω[U
TZU], (24)
where represents the imaginary part of the function, p0 is the amplitude of the excitation pressure, S is the section of the tube, ρ0is the air density, c0is the speed of sound in air and α is the absorption coefficient defined in Equation (1).
In order to obtain the value of the discrete sensitivity, α, it is assumed that the variation of the dynamic response is as described in Equation (23). Therefore, the discrete sensitivity is given by
α= 1
2incω[U T
Figure 7. Pseudo-code.
Using Equation (25), it is possible to determine the element sensitivity to the acoustic absorption objective function. This parameter indicates which element should be removed from the porous domain so that the absorption will increase toward a desired maximum value.
The evolutionary porous material optimization (EPO) algorithm has been presented in this section. In order to clarify the optimization process, a pseudo-code of the computational program developed is presented in Figure 7.
In the following, the numerical analysis results are presented. Firstly, a validation is presented of the numeric code by comparison with analytical solutions. A mesh dependency analysis is per-formed using four different meshes with the same domain by the proposed hard-kill optimization technique. For each analysis, the objective function evolution is performed. By analysis of the results, it is possible to observe monotonic convergence.
4. Numeric results
The verification of this method is an application of the present evolutionary procedure in a rectan-gular cell of design dimensions 0.06× 0.085 m. The domain is divided into successive meshes of four-node plane elements. In Figure 8, the porous–acoustic system coupled to the acoustic wave guide excitation is presented. In this article, the numeric tests were performed in two dimensions and the porous properties of the material used in these simulations are given in Table 1.
In this problem, the Kundt tube length is considered to be infinite. Therefore, the acous-tic impedance is infinite. In paracous-ticular, the impedance of the porous–acousacous-tic system can be
Table 1. Properties of the porous material. Material h α∞ σ[Ns m−4] [μm] [μm] Rock–wool 0.94 2.1 135,000 49 166 Foam A 1 1 1000 – – Foam B 1 1 5000 – – Foam C 1 1 10,000 – – Foam D 1 1 20,000 – – Foam E 1 1 50,000 – – 0 500 1000 1500 2000 2 3 4 5 6 7 frequency Re(Z n /Zc ) 0 500 1000 1500 2000 −50 −40 −30 −20 −10 0 frequency Im(Z n /Zc )
Figure 9. Acoustic impedance of the rock-wool sample.
evaluated by analytical models assuming unidimensional propagation in porous media. In Figure 9, the acoustic impedance spectrum of the 60 mm rock-wool sample is presented. In this way, an overdamped system response can be noted.
4.1. Validation of the numeric code
In Craggs (1979), an impedance tube analysis is proposed by a one-dimensional model. The analytical model is based on a Rayleigh acoustic equivalent model (Craggs 1978). For one-dimensional propagation in porous media inside to the Kundt tube, the absorption acoustic is determined by the acoustic wave reflection coefficient R, given as
α= 1 − |R|2, (26)
where the coefficient R is defined as
|R| =Zn− Zc
Zn+ Zc
, (27)
where Zcis the impedance of air, given by
Zc= ρ0c0. (28)
The surface acoustic impedance of the porous material Zninterfaced with the acoustic domain is given as
where Raand Xaare the resistance and the reactance of the surface impedance in the porous– acoustic interface, respectively. The values of these parameters are given as
Ra= {Z0coth(iγ d)}, (30)
Xa= {Z0coth(iγ d)}, (31)
where d is the thickness of the porous material in the impedance tube. The values of Z0and γ are defined as
Z0= ρ0c0 h1/2 α∞− i σ ρ0ω 1/2 , (32) γ = ω c0 h1/2 α∞− i σ ρ0ω 1/2 . (33)
Figure 10. Numerical (squares) and analytical (solid line) solutions for the acoustic absorption function. (a) Coarse mesh – 5 elements in the thickness direction, (b) Fine mesh – 50 elements in the thickness direction.
Substituting Equations (29) and (27) into (26), an expression for the acoustic absorption coefficient α can be arrived at:
α= 4ZcRa
(Ra+ Zc)2+ X2a
. (34)
In Craggs (1979), the thickness of the porous material sample is 25 mm. The properties of the foams A, B, C, D and E used for the frequency analysis are described in Table 1. The fictitious porous materials have different flow resistivity coefficients (σ ). For each material analysis, two meshes are tested. The absorption function is determined by numerical simulations and the results compared with analytical solutions.
In Figure 10, the curves formed by the square symbols are numerical solutions for the foams described in Table 1. The numerical solutions were compared with the analytical solutions (solid lines). For the coarse meshes, the results show high error values in the high-frequency domain. By mesh refining, accurate results are reached and a satisfactory description of the dynamical porous material behaviour is given by the finite element code developed.
4.2. Mesh dependency
In this section, a mesh dependency analysis is performed for the optimization process. Four finite element meshes are used to evaluate for a final volume reduction to improve the acoustic absorption response of the absorbing system at 300 Hz. For all analyses, the limit volume reduction value is 10% of the initial volume. The results are presented in Figures 11 and 12.
For the coarse meshes presented in Figures 11(a) and 11(c), few elements were removed and the topology aspect is not so good, i.e. the geometric details of the optimized porous–acoustic system cannot be found. By mesh refining, as presented in Figures 12(a) and 12(c), convergence to an optimized topology can be recognized. For the fine meshes, the evolution curves of the objective function are continuous and a monotonic convergence to the topology presented by the 30× 30 finite element mesh can be noted.
4.3. Topological optimization of porous–acoustic cells
In this section, two optimization processes at two different frequencies were performed. The domain was divided into 40× 40 four-node plane elements.
The first example aims to increase the absorption response of the porous material at 300 Hz. In the second example, the optimization increases the absorption value at 500 Hz. The porous material eliminated corresponds to 10% of the total material, obtained in 80 iterations. In this process, two elements are removed per iteration.
Figures 13(a) and 13(b) present the topology obtained by the evolutionary process and the objective function evolution for the 300 Hz problem.
The results of the pressure for the optimized configuration are shown in Figures 14(a) and 14(b), for the real and absolute pressures, respectively. Figure 14(c) presents the absorption distribution in the porous domain. The spatial distribution of absorption is calculated by the element results.
Figures 15(a) and 15(b) present the optimized topology and the objective function evolution, respectively, for the 500 Hz problem.
In Figures 16(a) and 16(b), the results of the pressure for the optimized configuration are presented, the real and absolute pressures, respectively. In Figure 16(c), the absorption distribution in the porous domain is shown for the optimization problem at 500 Hz.
For both optimization cases, the topologies converged to inclined surfaces. The perforations are symmetric due the number and position of removed elements per iteration in the optimization process.
The nature of the acoustic excitation imposed by the wave guide model is an imaginary pressure value on the porous–acoustic interface. In fact, the regions near to the wave guide excitation present
Figure 13. Optimization process at 300 Hz.
Figure 15. Optimization process at 500 Hz.
Figure 16. Diagrams of pressure and acoustic absorption at 500 Hz.
smaller values of absolute pressure, otherwise these regions present high values of real pressure. The perforations change the pressure distribution in the porous domain. Therefore, the acoustic waves in the porous domain do not propagate as planes waves.
By analysis of acoustic absorption diagrams – Figures 14(c) and 16(c) – the elements with bigger absorption potential are concentrated closer to the perforations. By evolution process analysis, it is possible to verify that the elements with poor absorption potential are not the elements with bigger sensitivity. In general, the elements with high absorption potential must be removed to obtain the maximum variation of the objective function. Otherwise, for mass reduction with constant global absorption, the elements with low absorption potential must be removed from the system.
In Figures 17(a) and 17(b), the new absorption curves are compared with the non-perforated configuration absorption. The gains at 300 and 500 Hz confirm the good performance of the proposed methodologies.
5. Conclusions
The evolutionary procedure can effectively solve a frequency optimization problem applied to absorbing material design.
In the evolutionary optimization process, the number of elements eliminated in each iteration is very important and must represent a small percentage of the total number of elements. In this work, the removed elements per iteration are related to the structure symmetry. The cost function evolution is smooth for a fine mesh. For high frequencies, the optimized porous cell showed satisfactory absorption potential.
Further work will include multi-physics modelling in order to study a coupled formulation problem. This can be accomplished by combining the Helmholtz acoustic model with an elastic solid in vacuo (mixed formulation, [u,p]) in order to analyse the changes in the optimized results. Another application of the EPO should be to maximize the absorbing potential over a frequency range. Solutions obtained with this method should be compared with analytical and experimental results.
Acknowledgements
The authors would like to thank Fapesp (Fundação de Amparo à pesquisa do Estado de São Paulo) for financial support for this work (Process 03/08469-6).
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![Table 1. Properties of the porous material. Material h α ∞ σ [Ns m −4 ] [μm] [μm] Rock–wool 0.94 2.1 135,000 49 166 Foam A 1 1 1000 – – Foam B 1 1 5000 – – Foam C 1 1 10,000 – – Foam D 1 1 20,000 – – Foam E 1 1 50,000 – – 0 500 1000 1500 2000234567 f](https://thumb-us.123doks.com/thumbv2/123dok_es/6758105.265805/13.740.165.574.124.554/table-properties-porous-material-material-foam-foam-foam.webp)
