Efficient sound insulation prediction models forhollow core masonry walls

Jan Van den Wyngaert, Edwin ReyndersKU Leuven Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium.

Mattias SchevenelsKU Leuven Department of Architecture, Kasteelpark Arenberg 1, B-3001 Leuven, Belgium.

Summary

Bricks are often perforated to decrease weight, resulting in a decreased material cost and an easier

handling on site. However, walls made of perforated brick have a lower airborne sound insulation than

solid walls with similar weight and thickness. This can be partially explained by thickness resonances

and the orthotropy introduced by the perforations. Numerical sound insulation prediction models that

include the detailed perforation geometry of all individual bricks are computationally demanding since

a rened mesh is needed in order to properly capture this geometry. However, the length scale of the

perforations is generally much smaller than the wavelength of deformation. Both length scales can

therefore be considered separately: rst the behavior of individual bricks is considered (micro-scale),

upon which this behavior is employed for constructing a homogeneous solid model of the entire wall

(macro-scale). The most simple macro-scale models only take into account the bending behavior

of an isotropic thin plate. More advanced macro-scale models account for thickness resonances and

orthotropic behavior. An example of such a model that has been proposed in literature is a TMM

model with spatial windowing. This model does not include the modal behavior of the wall, leading

to a poor prediction of the sound insulation at low frequencies. In this paper, an alternative macro-

scale sound transmission model that takes thickness, orthotropy and modal behavior into account,

is developed within the hybrid deterministic-statistical energy analysis framework. The sound elds

in both rooms are modelled as diuse in order to reduce the problem size. The vibration eld in

the wall is modelled deterministically, using an analytical formulation of the eigenmodes of a thick,

simply supported orthotropic plate to achieve high computational eciency. The model is validated

with experimental data obtained at the KU Leuven Laboratory of Acoustics. The computed results

show a good correspondence with the data, while retaining a low computation time.

PACS no. 43.55.Rg

1. Introduction

Load bearing wall systems composed of bricks withhollows are often used in the construction industrybecause of their light weight and handling ease. How-ever, its light weight and perforation geometry lead toa strong reduction of the sound insulation. Due to theperforations, the sound insulation of this type of wallis generally lower than for a solid wall with the samesurface mass, as will be shown in Sect. 4.3. The pre-diction of the airborne sound insulation of hollow corewalls is challenging since it is characterized by manyparameters and physical phenomena. Experimentalparametric studies by Fringuellino and Smith [1] andGuillen et al. [2] have shown that the airborne sound

(c) European Acoustics Association

insulation of a hollow core wall depends mainly on theleaf thickness, its surface mass, material parametersand perforation pattern. All these parameters denean array of physical phenomena. The coincidence fre-quency for example, i.e. where the free bending wave-length of an innite leaf matches the projected freewavelength in air, depends on the material parame-ters of the leaf, its perforation pattern and thickness.As does the rst thickness resonance, i.e. where thetwo leaf interfaces resonate with a phase shift of 90.All these dierent phenomena and input parametersmake it hard to accurately predict the airborne soundinsulation using a simplied model.

Existing prediction models can be dividedinto semi-analytical and numerical models. Semi-analytical models are used to gain insight into themain physical phenomena, e.g. coincidence. A lowcomputational cost is achieved by assuming diuse

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conditions in the transmission rooms and by usinga simplied model for the vibrational behavior ofthe leaf. This often leads to an inaccurate predic-tion of the sound insulation due to this simpliedrepresentation of the wall and since some physicalphenomena are omitted, e.g. thickness resonances.Element-based numerical models, e.g Finite elements,manage to capture the vibratory behavior of both thewall and the transmission rooms in full detail makingit possible to accurately predict the single numberrating. However, these models are not practicaldue to the large number of boundary and/or niteelements needed at high frequencies. A concise reviewof these existing models will be provided in the nextsection.The goal of this work is to close the gap between

these two types of models. A prediction model for theairborne sound insulation of hollow core walls is devel-oped with both a high prediction accuracy and a lowcomputational cost. The model can therefore be usedas a practical and reliable design tool for the soundinsulation of hollow core brick walls.The wall is modelled deterministically to accurately

capture its vibratory behavior while the sound eldsin the sending and receiving rooms are modeled as dif-fuse. The wall leaf is modeled in a semi-analytic wayto reduce computational cost. A model of a homo-geneous thick orthotropic plate with equivalent ma-terial parameters is used. The hybrid Deterministic-Statistical Energy Analysis (DET-SEA) framework isadopted in this work to rigorously couple the diusesound elds to the deterministic wall model by em-ploying the diuse eld reciprocity relation [3]. Thisframework has been chosen due to its low computa-tional cost and high prediction accuracy [4, 5]. Theprediction model presented in this work is validatedagainst experimental results obtained in the KULeu-ven Laboratory of Acoustics.A review of existing prediction models is given in

section 2. The deterministic model of the wall is ex-plained in full detail and is subsequently coupled tothe sound elds in the adjacent transmission roomsin section 3. In section 4, the experimental setup andtest results for the measurements of the sound insu-lation of a hollow core brick wall is presented. Theresults of the model are compared with experimentalresults and are discussed. The paper concludes withsome nal remarks and conclusions in section 5.

2. A review of existing models

A concise review of existing models for the predictionof the airborne sound insulation of brick walls is pro-vided here. Watters [6] used the model for single leafwalls of Cremer [7] to predict the sound insulation ofhollow brick walls. This model represents the wall asa thin isotropic plate and can be used when the twobending stinesses of the leaf are roughly equal. The

model is unable to predict the thickness resonancesand will give an overestimation of the sound insula-tion at higher frequencies.

Hopkins [8] models the sound insulation of hollowcore masonry using a simplied model of a thin or-thotropic wall. This model allows for two critical fre-quencies to occur when the bending stinesses in thetwo orthogonal directions are not the same. This 'crit-ical frequency region' is often observed in measure-ments of the sound insulation. As it was the case forthe classical models of London and Cremer, this modelwill give an overestimation at higher frequencies sincethe thickness resonances are not included.

Jacqus et al. [9] model the wall as a thick or-thotropic plate using a transfer matrix method(TMM) with spatial windowing. The equivalent or-thotropic material parameters are determined by re-lating forces and displacements in a virtual loadingtest performed on a single brick in nite elements.This approach is valid when the perforations are smallcompared to half the wavelength of the vibrations.The model shows good results for the sound insula-tion in a frequency range of 500 - 5000 Hz. At lowfrequencies, the prediction of the sound insulation ispoor since the modal behavior is not taken into ac-count.

Jean and Villot [10] compute the sound insulationbased on a 3D nite element model, taking into ac-count the wall geometry in full detail. Due to the largeproblem size, only blocks with simple perforation pat-ters and a frequency range up to 1000 Hz are possibleto model with a reasonable computational cost of acouple of hours. del Coz Diaz et al. [11] and Ratniekset al. [12] model sound insulation using a 2D nite el-ement model of a horizontal cross-section of the wallwith the assumption of plane strain to reduce the com-putational cost. To further reduce the computationalcost, del Coz Diaz et al. [11] only model the sound in-sulation at the 1/3 octave band center frequencies andat ±5% of these center frequencies to avoid spuriouseects in the computed sound insulation.

Dijckmans et al. [13] model the sound insulation us-ing a wave based-transfer matrix model to include themodal behavior of the plate. The leaf is modelled as athick isotropic plate with equivalent material param-eters and an equivalent wall thickness. The equivalentmaterial parameters and wall thickness are based onthe surface mass of the wall, and the measured co-incidence and thickness resonance frequencies. Reyn-ders et al. [5] use a hybrid nite element-SEA (FE-SEA) model based on a nite element model of athick isotropic plate to compute the sound insula-tion. The equivalent material parameters were deter-mined based on approach of Dijckmans et al. [13].The transmission rooms are modelled as diuse SEAsubsystems to reduce the computational cost and arecoupled to the wall model using the diuse eld reci-procity relationship [3]. Decraene et al. [14] employ a

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dedicated hybrid modal transfer matrix-statistical en-ergy analysis method (mTMM-SEA) to even furtherreduce the computational cost of the hybrid FE-SEAmodel. The mTMMmodel of the wall takes the modalbehavior of the into account, resulting in a better pre-diction at low frequencies with a low computationalcost.

3. An ecient and accurate model

From the previous discussion, it is clear that exist-ing models for the sound insulation prediction of hol-low core brick walls are either insuciently accurateor have a high computational cost. Therefore, in thissection, a forward model with both a high predictionaccuracy and a low computational cost is developed.The wall is modelled deterministically to capture

its vibration behavior while the sound elds in thesending and receiving rooms are modelled as diuse.To reduce the computational cost, the leaf is modelledas a homogeneous thick plate. Assuming the the ma-terial is periodic, its eective properties may be fullydetermined by an analysis of the smallest repetitiveunit, the base cell [15]. The eective properties arefound by homogenization of the micro-structure. In arst step, the equivalent orthotropic material param-eters of the leaf are determined with a single brickas the base cell. These equivalent material parame-ters are in term used for a modal analysis using asemi-analytical model for a thick orthotropic plate.The hybrid DET-SEA modelling framework [4, 5] isused to rigorously couple the diuse sound elds in thetransmission rooms to the deterministic wall model byemploying the diuse eld reciprocity relationship [3].In what follows, the deterministic model of the wallsystem is explained rst. Subsequently, the interactionbetween the wall and the sound elds in the adjoiningrooms is described.

3.1. Deterministic model of the wall

The considered wall system consists of a single thickleaf. In the rst instance, the leaf is decoupled fromthe transmission rooms. The displacement eld of theleaf, u(x, ω), is approximated using a nite set of basisfunctions Φ(x), that satisfy the boundary conditions,and corresponding generalized coordinates q(ω):

u(x, ω) ≈n∑

k=1

φk(x)qk(ω) = Φ(x)q(ω) (1)

The choice of the basis functions Φ(x) of the decou-pled wall leaf will be discussed in detail below. Thiswill result in a system of equations in terms of thegeneralized coordinates of the wall:

Ddq = f (2)

with f the external uid loading onto the wall leaf bythe acoustic pressure in the adjoining rooms, and Dd

the dynamic stiness matrix of the leaf.

y1

y2

x1

x2

Γd

Γtt

fΩ

Y

f¥

ν p

Γd

Γtt

f

(a) (b)

(c) (d)

Figure 1. (a) A solid domain Ω consisting of (b) a re-peating microstructure composed of (c) a base cell Y withdimensions

]0,Y1

[x]0,Y2

[x]0,Y3

[and (d) the homoge-

neous solid domain with equivalent material parameters

3.1.1. Micro-scale model and homogenization

If a structure is built from a periodic material, it isoften inconvenient or even impossible to model everydetail of the micro-structure. The micro-structure istherefore replaced with average or smeared out prop-erties that model the material on a macro scale. Thisprocess is called homogenization. This approach canbe used to adequately model the wall leaf since thewavelength of the transmitted sound is larger thanthe perforation length in the acoustic frequency do-main (50-5000 Hz) [9, 16, 17]. The homogenizationtechnique for solids has been reviewed by Hassani andHinton [15] and is shortly presented in what follows.In the homogenization approach, a numerical equiv-

alence is written between the virtual work formulationof a body Ω, as shown in Fig. 1.(a), consisting of a pe-riodic repeating structure in terms of the micro-scalecoordinates y, as shown in Fig. 1.(c), and the virtualwork formulation of a homogeneous body in terms ofthe macro-scale coordinates x, as shown in Fig. 1.(d).Following this approach, the 36 elements of the ho-mogenized stiness tensor in Voigt notation, CH

ijkl, ofan equivalent homogeneous body based on a base cellyield [15]

CHijkl =

1

|Y|

∫Y

[Cijkl(y)− Cijpq(y)

∂uklp (y)

∂yq

]dY (3)

with |Y| the volume of the base cell, Cijkl(y) theelements of the stiness tensor of the base cell at eachlocation y and ukl(y) the displacement eld obtainedfrom [15]∫

Y

Cijpq(y)∂ukl

p (y)

∂yq

∂δui(y)

∂yjdY

=

∫Y

Cijkl(y)∂δui(y)

∂yjdY (4)

Note that uklp (y) equals component (p) of the displace-

ment eld ukl(y) and δui(y) equals component (i) ofa virtual displacement eld δu(y).

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y1

y2y3

u0(11) u

0(22)

u0(33)

u0(12)

u0(13)

u0(23)

(y)

(y)

(y)

(y) (y)

(y)

Figure 2. The six displacement elds, ukl,0(y) correspond-ing to strain eld εkl,0(y), for a cuboid 3D base cell. Theundeformed base cell is depicted in black and the deformedbase cell in red.

Eq. (4) represent the equilibrium conditions of aunit cell. This unit cell can be discretized using niteelements; and by imposing the equilibrium conditionson the discretized unit cell, the following classic sys-tem of equations is obtained:

Kukl = fkl (5)

with K the stiness matrix of the base cell with staticperiodic boundary conditions, i.e. equal displacementsat opposite faces. The load vector, fkl, can be shown tocorrespond to a loading case that results in a uniformunit strain eld εkl,0 with corresponding displacementeld ukl,0(y) [15]:

fkl =∑e

∫Ye

BTe Ce(y)εkl,0(y)dYe (6)

εkl,0pq (y) =1

2

(∂ukl,0

p (y)

∂yq+∂ukl,0

q (y)

∂yp

)= δkl,pq (7)

Note that εkl,0pq (y) equals component (pq) of the strain

eld εkl,0(y), Be denotes the strain-displacement ma-trix and Ce the stiness tensor of element e. The sixdisplacement elds ukl,0(y) are given for a 3D basecell in Fig. 2.Eq. (3) can be reformulated in terms of strains as

CHijkl =

1

|Y|

∫Y

Cpqrs(y)(εij,0pq (y)− εij,∗pq (y)

).(εkl,0rs (y)− εkl,∗rs (y)

)dY (8)

εkl,∗pq (y) =1

2

(∂ukl

p (y)

∂yq+∂ukl

q (y)

∂yp

)(9)

By discretizing Eq. (8), using a nite element ap-proach, and combining it with Eqs. (5) - (7), the ho-mogenized stiness tensor can be retrieved from

CHijkl =

1

|Y|∑e

(uij,0 − uij

)TKe

(ukl,0 − ukl

)(10)

where Ke represents the stiness matrix of element ein the base cell.

3.1.2. Macro-scale model

The wall leaf is modelled as a homogeneous thick platebased on the equivalent material parameters derivedin the previous section. Simply supported boundaryconditions are assumed for the nodes located at theneutral plane for bending; this assumption is fairly ac-curate in practice [5, 13]. To reduce the computationalcost, a semi-analytical formulation of the eigenmodesand eigenfrequencies of the decoupled in-vacuo plateis used. The leaf cannot be adequately be modelled asa thin orthotropic plate since its thickness is not neg-ligible compared to its lateral dimensions. Therefore,a 'third order shear deformation model' is used, i.e. amodel where the displacement eld over the heightof the plate is decomposed into a third order polyno-mial, in terms of the distance to the neutral plane ofthe plate, as a shape function. This model has beenpresented by Garg et al. [18] and is shortly describedin what follows.The displacement pattern over the height of the

plate with respect to the neutral plane is assumedto be a third order polynomial and is further decom-posed in the x1- and x2-coordinate directions, denedas the in-plane macro-scale coordinates as in Fig. 1(a),into a combination of sine and cosine waves as shapefunctions that satisfy the boundary conditions. Thebasis functions which appear in Eq. (1) are taken tobe the mass normalized mode shapes of the decoupled,simply-supported, thick orthotropic plate:

φk(x) = akχk(z) ψk(x, y) (11)

χk(z) =3∑

l=0

vk,lzl (12)

ψk(x, y) =

cos (αkx) sin (βky)sin (αkx) cos (βky)sin (αkx) sin (βky)

(13)

αk =mkπ

Lxand βk =

nkπ

Ly(14)

where is the element wise multiplication of two vec-tors, Lx and Ly denote the planar dimensions of theleaf in respectively the x1- and x2-coordinate direc-tions, mk ∈ N0 and nk ∈ N0 denote the number ofhalf wavelengths in those directions, specic for modenumber k, ak is a mass normalization constant, vk,l

a scaling vector and z is the distance to the neutralplane of the plate. The modes are numbered accordingto increasing natural frequency.Using Hamilton's principle [19], the equations of

motion for free vibration of an orthotropic plate canbe obtained. By substituting the shape functions inthe equations of motion, a 12-dimensional eigenvalueproblem is obtained for each combination of mk and

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nk in terms of the scaling vector vk :=[vT

k,0,vTk,1,v

Tk,2,v

Tk,3]T :

[Xk − λkMk] vk = 0 (15)

with the elements of Xk and Mk as described by Garget al. [18]. This eigenvalue problem results in 12 eigen-values λk and eigenvectors vk. The eigenvalues, λk,correspond to the square of the natural frequenciesωk of the plate. The eigenvectors, vk, correspond tothe scaling vectors as in Eq. (12). The normalizationconstant satises

ak =

√4

ρLxLy

(Lzvk,0 · vk,0 +

L7z

448vk,3 · vk,3

+L3z

12(2vk,0 · vk,2 + vk,1 · vk,1)

+L5z

80(2vk,1 · vk,3 + vk,2 · vk,2)

)−1/2

(16)

where Ωl denotes the volume of the wall leaf, ρ is itsequivalent density and Lz its thickness.Since the basis functions have been chosen to be the

mass-normalized mode shapes of the leaf, the corre-sponding generalized coordinates are the modal am-plitudes of the leaf. Its equation of motion thereforereads

Ddq = f (17)

where Dd is a diagonal matrix with entries

Dd,kk = −ω2 + ω2k(1 + iηk), (18)

i :=√−1 is the imaginary unit and ηk denotes the

damping loss factor of mode k.

3.2. Transmission suite model

In the previous section, the deterministic model of thehomogenized wall was described in full detail. In thepresent section, this wall model is rigorously coupledto the sound elds in the adjoining rooms.The sound elds are modelled as diuse. Although

a diuse sound eld represents a random ensemble ofrooms, it can also be used to model the sound eld inone specic room at high frequencies when the modaldensity of the room is typically larger than unity. Thisis because at high frequencies, the variance of the totalsound energy across the random ensemble represent-ing the diuse eld is low, and therefore the meantotal energy of the ensemble will be close to the to-tal energy in any particular member of the randomensemble [20]. This is not the case at low frequencies.When a diuse eld model is employed at low frequen-cies as in the present paper, it therefore representsan ensemble of sound elds rather than one particu-lar sound eld. This can cause considerable variationwhen comparing the mean total sound energy of the

diuse model with the total sound energy in one par-ticular sound eld.

A perfectly diuse sound eld requires diusely re-ecting boundaries, i.e., boundaries that reect an in-cident plane wave in a random direction that is sta-tistically independent of the direction of incidence.When boundary reections are specular rather thandiuse, the mean energy density will not be uniformanymore because of interference between incomingand reected sound waves, but this eect can be ac-counted for [21]. A harmonic linear relationship be-tween the mean total energy in the diuse eld andthe mean squared amplitude of the blocked reverber-ant forces at the locations where the loading that gen-erates the diuse eld can be established. This rela-tionship is termed the diuse eld reciprocity rela-tionship [3]. It forms the basis of a hybrid methodof analysis, in which a system consisting of compo-nents that are modelled as deterministic and com-ponents that are modelled as diuse can be rigor-ously analyzed. This method is termed the hybriddeterministic-statistical (DET-SEA) energy analysismethod [4, 22].

Within the hybrid DET-SEA framework, a trans-mission suite (room-wall-room) model has been devel-oped by Reynders et al. [5]. In this model, which isadopted in the present work, the rooms are taken tocarry a diuse eld, while the wall is modeled deter-ministically.

In the room-wall-room setup, the hybrid model con-tains two diuse (SEA) subsystems - the sending andreceiving rooms - and in the context of a sound trans-mission analysis, the quantity of interest is the so-called coupling loss factor between both rooms. If in astationary situation, the sound power ow from room1 to room 2 (through the wall) is denoted as P12, thenthe coupling loss factor η12 is dened as [20]

η12 :=ωn1P12

(E1

n1− E2

n2

)(19)

wshere the total acoustic energy of room k is denotedas Ek and its modal density (i.e., the expected num-ber of modes per unit radial bandwidth) as nk. Al-though the coupling loss factor is a random quantity(because the sound elds in the rooms are random,diuse elds), only its mean value will be of inter-est in the present analysis as the intention is to pre-dict the mean sound transmission loss of the wall.The coupling loss factor relates directly to the soundtransmission coecient, which is dened as the ratiobetween the power ow P12 from room 1 to room 2,and the incident sound power on the wall in room 1.The relationship reads [23]

τ =4V1ω

LxLycη12 (20)

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Figure 3. Perforation pattern of the hollow core brick un-der consideration (ly1 = 0.495m, ly2 = 0.235m, ly3 =0.135m and m = 12.78 kg).

where V1 denotes the volume of the sending room andc the speed of sound in air. The sound transmissionloss (or airborne sound insulation) of the wall thenimmediately follows from

R := 10 log1

τ= 10 log

η12LxLyc

4V1ω(21)

By considering the interaction between the walland the direct eld response of the rooms at theirinterfaces, the coupling loss factor η12 can be rigor-ously evaluated within the hybrid DET-SEA frame-work [4, 5, 24]. The direct eld response of a room isthe sound eld that would occur if the room wouldbe of innite extent, i.e. , if the room would behaveas an acoustic half-space as seen from the room-wallinterface when that interface is embedded in an in-nite planar bae. The related acoustic dynamic sti-ness matrix is then termed the direct eld dynamicstiness matrix of the room. For room 1 for example,this matrix describes the relationship between the dis-placements and forces at the interface with the wallleaf:

Ddir1q = fdir1 (22)

The forces, fdir1, act on the (generalized) degrees offreedom of the wall leaf q due to the pressure eldin the acoustic half-space. The direct eld dynamicstiness matrix Ddir1 is obtained by numerically eval-uating the Rayleigh integral, e.g. using a wavelet dis-cretization; this approach has been described by Lan-gley [25].Once the direct eld acoustic dynamic stiness ma-

trices of both rooms have been computed, the cou-pling loss factor is obtained from [4]

η12 =2

πωn1

∑rs

Im(Ddir2,rs

) (D−1

totIm (Ddir1)D−Htot

)rs

(23)

where

Dtot := Dd + Ddir1 + Ddir2, (24)

Dd denotes the dynamic stiness matrix of the wall asdened in Eq. (17), and Ddir1 and Ddir2 are the directeld acoustic dynamic stiness matrices in terms ofthe (modal) wall coordinates q.

Figure 4. Cross-sectional view of the simplied model ofthe perforated brick used for the computation of the ho-mogenized elasticity tensor CH

ijkl. The handling holes arestraightened and the tongue and groove connection is sim-plied to give at edges.

4. Results and discussion

The model that was presented in the previous sectionis employed in the present section for predicting theairborne sound insulation of a hollow core brick wall.In order to assess the accuracy of the sound insula-tion prediction, it is compared with the results frommeasurements. The brick is shown in Fig. 3. The per-foration pattern of the brick is slightly simplied; therounded edges of the handling holes are straightened.The tongue of the tongue and groove connection ismodelled inside the groove, this approach is valid sincestatic periodic boundary conditions are applied at thebrick faces. A cross-section of this simplied model isshown in Fig. 4.

4.1. Experimental setup

The test setup for the measurement of the airbornesound insulation of walls consists of two adjacentreverberation rooms separated by the test speci-men. The sound insulation has been measured in theKULeuven Laboratory for Acoustics. The test open-ing has a width of 3.25 m and a height of 2.95 m.The volumes of the transmission rooms are 87 m3.The simplied perforation pattern of the consideredbrick is shown in Fig. 4. The airborne sound insulationis modelled for 320 frequency lines in the frequencyrange of 50-3150 Hz. These frequencies therefore cor-respond to the 1/48 band center frequencies. The har-monic transmission losses are band-averaged over 1/3octave bands for comparison with the experimentaldata.

4.2. Comparison with measurements

The wall is constructed using hollow bricks with anaverage density of 813 kg/m3 and a width of 13.5cm. The Young's modulus is taken to be 10 GPa inthe x1- and x3-directions with corresponding Poissoncoecients of 0.2. The Young's modulus of the ter-racotta in the x2-direction, along the perforations,is 3.5 GPa and is signicantly lower than the other

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Figure 5. Comparison of the sound insulation curves of thehybrid DET-SEA model and the experimental results forthe hollow core brick shown in Fig. 4.

two directions, as in [9]. This modulus is computedto make sure that the measured rst thickness res-onance at 2484 Hz matches the eigenfrequency ofthe 1-1-thickness-mode of the homogenized brick wall.The damping loss factor η of the brick wall is basedon the estimation of Craick for brick and concretewalls/oors [26]

η ≈ 2.5√ω

+ 0.01. (25)

A plaster layer of 1 cm with a density of 800 kg/m3,a Young's modulus of 5 GPa and a Poisson ration of0.2 is applied on one side.The result of the model is presented in Fig. 5 both

at the 1/48 band center frequencies and for the aver-aged 1/3 octave bands. The 1/3 octave band valuesare compared to the experimental data. The analy-sis is performed on a single personal computer with a2.7 GHz Intel Core i7 processor and 16 GB RAM. Allcomputations are performed in Matlab. The compu-tation time is around 15 minutes.

4.3. Physical interpretation and comparison

with experiments

An important resonance phenomenon for thick wallsare the so-called thickness resonances, i.e. the fre-quency where the two interfaces of the leaf vibratewith a 90 shift in phase angle. The eigenfrequencycorresponding to the 1-1-thickness resonance-modecan be computed by evaluating the eigenvalue prob-lem of Eq. (15). This gives an eigenfrequency of2484 Hz. As can be observed in Figs. 5, the predic-tion of the rst thickness resonance frequency agreeswell with the experimental ndings. This comes to nosurprise since one of the Young's moduli is computedso that the eigenfrequency matches the experimentalresults.A dip in the sound insulation is observed at 339 Hz.

This is the so-called coincidence dip. It occurs atthe critical frequency of the leaf, i.e., the lowest fre-quency at which the free bending wavelength of an

Figure 6. Comparison of the sound insulation curves for ahomogeneous brick and the perforated brick

innite wall matches the projected free wavelength inair. The theoretical value of the critical frequency is248 Hz in the x2-coordinate direction and 310 Hz inthe x1-coordinate direction for the rst setup. Thepredicted coincidence dip matches the theoretical val-ues well. At low frequencies, the modal behavior of theleaf becomes important. Both the resonance and anti-resonance dips in the frequency range of 100-300 Hzare predicted accurately. From Fig. 5, it can be ob-served that the predicted sound insulation agrees wellwith the experimental ndings.

An important input parameter determining thesound insulation is the perforation pattern of thebrick. The sound insulation of hollow bricks has ex-perimentally been shown to be lower than for a ma-terial with the same surface mass and leaf thickness.The sound insulation of a homogeneous brick is com-puted and compared with the results of Fig. 5. Theleaf thickness and surface mass is taken to be equalto the previously described brick. The homogenizedmaterial parameters are calculated for a homogeneousbrick where the base cell material parameters are sim-ply interpolated, between the material parameters ofterracotta and a void, based on the volume fraction ofthe brick. The results are displayed in Fig. 6. From therst coincidence frequency of the leaf, the sound insu-lation of the homogeneous brick is consistently higherthan for the hollow core brick. This can be explainedby the dierence in the critical frequencies and rstthickness resonance frequency of the two bricks. Thetwo critical frequencies in the x1- and x2-coordinatedirections for the homogeneous brick are lower thanthe second critical frequency of 310 Hz of the perfo-rated brick. The sound insulation therefore increasesby 7.5 dB per octave band, as described in the clas-sical models for thin single leaf walls, from a lowerfrequency on. The sound insulation of the homoge-neous brick follows this increase of 7.5 dB/oct up toa frequency of 2000 Hz when the curve starts show-ing a distinct dip. This dip can be explained by thepresence of the rst thickness resonance at 5344 Hz.

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5. Conclusion

In this paper, an accurate and computationally ef-cient model is presented for the prediction of theairborne sound insulation of hollow core brick wallswithin the hybrid DET-SEA framework. The eigen-modes are used to approximate the vibration eld ofthe wall leaf. A semi-analytical model of a thick, sim-ply supported orthotropic plate for the wall leaf isused to reduce computational cost. The sound insu-lation is computed at the 1/48 octave band centerfrequencies for a frequency range of 50 - 3150 Hz for320 frequency lines to include the oscillatory behaviorand is averaged over 1/3 octave bands for compari-son with the experimental results. The computationtime for the setup presented here is 15 minutes. Themodel has been validated against experimental dataobtained at the KU Leuven Laboratory of Acoustics.

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