A comparison of numerical approaches to quantify sound insulation of lightweight wooden floor structures

Mohamad Bader Eddin 1

Department of Applied Sciences, University of Quebec at Chicoutimi 555 Bd de lÚniversité, Chicoutimi, QC G7H 2B1, Canada

Jonathan M. Broyles 2

The Pennsylvania State University State College, PA 16801, United States

Sylvain M´ e nard 3

Department of Applied Sciences, University of Quebec at Chicoutimi 555 Bd de lÚniversité, Chicoutimi, QC G7H 2B1, Canada

Delphine Bard 4

Engineering Acoustics, Lund University John Ericssons v¨ a g 1, 223 63 Lund, Sweden

Jean-Luc Kouyoumji 5

Technological Institute FCBA All. de Boutaut, 33000 Bordeaux, France

ABSTRACT Quantifying air-borne and structure-borne sound insulation is an important design consideration for the indoor comfort in a building. Although sound insulation performance is commonly measured experimentally, numerical methods can have time-saving and economic benefits. Further, numerical methods can be incorporated within building simulations to provide an estimate of the acoustic environment. In response, this paper evaluates three di ff erent computational approaches for quantifying sound insulation in one-third octave bands (50 Hz -5 kHz) of a lightweight floor including: an analytical (theoretical) model, a finite element model (FEM), and an artificial neural network (ANN) model. The three numerical methods are tested on the sound insulation of a cross laminated timber (CLT) floor. The results of this study show that the ANN model is able to accurately predict the air-borne and impact sound insulation performance at frequencies above

1 Mohamad.Bader-Eddin1@uqac.ca

2 j.broyles@psu.edu

3 Sylvain_Menard@uqac.ca

4 Delphine.Bard@construction.lth.se

5 Jean-Luc.Kouyoumji@fcba.fr

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

250 Hz, but over-predicts the air-borne performance and under-predicts the impact performance at low frequencies. However, the analytical and FEM strategies provide acceptable estimations, useful during the conceptual design stage, but with higher deviations than ANN model across all frequencies. While no model is able to accurately represent acoustic behavior across all frequencies, this work highlights the advantages and disadvantages when applied to predicting the sound insulation of a CLT floor. Keywords: sound insulation, artificial neural networks, numerical analysis, building acoustics, floor structures

1. INTRODUCTION

In recent years, mass timber and engineering wood products (EWP), especially cross laminated timber (CLT), have been used widely in construction engineering due to its availability in nature and ease of use, and environmental benefit [1–3]. In the 20 th century, wood-framed structures grew rapidly to be the dominant building construction system in housing to accommodate the growing population in North America [4, 5]. Although this type of structure is ideal for sustainable buildings, solves the problem of durability, and reduces construction time and cost, a potential consequence is that the subjective sound insulation quality is considered to be lower than that of a heavy structure that has the same sound insulation data [6,7]. To meet adequate acoustic requirements, it is essential to use prediction tools and acoustic products with expected performance to provide suitable indoor acoustic quality (IAQ) in buildings. Many prediction tools such analytical expressions, finite element models (FEMs), and other statistical analyses are introduced to estimate indoor acoustic behavior inside buildings [8–11]. Those methods can reduce the cost and e ff orts of experimental acoustical tests. However, certain prediction tools showed significant deviations in forecasting the acoustic performance of structural elements [11,12]. The simplest approach of a prediction method is to use theory-based analytical expressions incorporating the mass, sti ff ness and damping of a structure [13]. This is reasonable for a single leaf, rectilinear structure, but it is not the case for multi-layered and lightweight constructions. An accurate estimation of sound insulation for double structures has been and remains to be a challenge [13]. Some construction details, such as: mechanical connections between di ff erent materials, can hardly be considered in modeling, especially using analytical methods [14]. Additionally, the diversity of construction materials makes the forecasting more di ffi cult. Moreover, the standard method for estimating the acoustic performance of building elements as indicated in ISO 12354 Part 1 and 2 [15, 16], are presently not suitable for multi-layered complex and lightweight structures [17]. The standardized methods are widely used and were developed based on data from heavy monolithic constructions [17]. The finite element method (FEM) has been utilized extensively in acoustic analysis to predict both air-borne and impact sound insulation [10,18–20]. However, the results of FEM are highly sensitive to many factors, including the influence of uncertainty and variation in the material properties, especially in wood species because of its heterogeneity and the lack of the material properties database [10, 21, 22]. Another challenge in FEM is the structural links between layers, which is critical for the e ff ective forecasting of sound insulation for double-leaf structures [23]. In a comparison study of existing prediction models [24], it was found that only five of the seventeen acoustic prediction models considered the structural connections, and just two of models considered rigidity in the modeling. Recently, the applications of machine learning (ML) have been widely used for solving complex problems in various fields, such as speech recognition, image classification, and building acoustic domain [8, 25–27]. ML can be defined as a branch of statistics in which a model can learn based on provided data [28]. An advantage of machine learning approaches, especially artificial neural networks (ANN), is that they perform best when a huge dataset is provided, thus improving analytic, predictive power and increasing the accuracy of the results.

In building acoustics, ANN models were performed for estimation of air-borne sound insulation curves of masonry walls [8]. 34 laboratory measurements of monolithic brick walls were used in the database with di ff erent geometric and physical parameters. Another use of ANN was to forecast the sound insulation of sandwich partition panels and to predict the weighted air-borne sound reduction index R w and sound transmission class STC values [9]. In another study [29], ANN was utilized to predict the acoustic spectrum of di ff erent lightweight wooden floor structures in one-third frequency bands (50 Hz − 5 kHz ). The study was limited to 67 measurements and used the thickness and the installation order of the materials as the only parameters in the model. A complementary study was carried out [30] but used 252 standardized laboratory measurements and more construction parameters, such as: material type, material thickness and density, area of the floor structure, etc. The study then conducted a sensitivity analysis to estimate the most important parameters in forecasting sound insulation curves. Although many previous studies have demonstrated the usefulness of prediction of the sound insulation performance using numerical methods, there is a lack of studies comparing the methods directly to one another. To address this gap, the goal of this study is to make a comparison between theoretical approaches, finite element model, and a developed prediction model based on artificial neural networks to predict air-borne and impact sound insulation curves for a single-board CLT panel of 140 mm . Since a realistic baseline for comparison the acoustic performance of a certain structure is experimental measurements, the latter will be used as a reference to compare other prediction tools. The comparison aims to focus on strengths and limitations of each approach in sound insulation estimations.

2. METHODS

2.1. Theoretical methods Since the first half of the 20 th century, the computation of sound transmission through a solid structure has attracted interest in building acoustics [11]. Several analytical and empirical formulations have been developed for simple monolithic homogeneous isotropic thin partitions and double leaf structures [31–34]. However, those tools showed remarkable limitations [11, 12, 35]. Building partitions involve di ff erent technological solutions that can hardly be taken into consideration. Therefore, more advanced models are required. Sound transmission through a single leaf homogeneous partition was proposed by Cremer [32, 35, 36]. In the case of a simple homogeneous panel, the most important property to be considered is the mass per unit area of the panel. The well-known mass law (Equation 1) gives a very simple prediction to a transmission loss [31]:

R = 20 log ( mf ) − 47 , (1)

where m is the mass per unit area ( kg / m 2 ) and f the frequency ( Hz ). Although the simplicity is advantageous for homogeneous, rectilinear structures, the law commonly overestimates the sound insulation. Further, previous studies have found that mass law is insu ffi cient for di ff erentiating air-borne transmission performance for structures with the same mass density, but di ff erent geometric characteristics [37]. A more elaborate, theory-based analytical approach includes infinite panel theory. Using this theory, the sti ff ness and damping are also accounted for in the estimation of the transmission coe ffi cient, as shown in Equation 2:

τ ( φ, ω ) = (2 ρ 0 c 0 / sin φ ) 2

(2 ρ 0 c 0 / sin φ ) + ( η ( D /ω )( k 0 sin φ ) 4 ) 2 + ( ω m − ( D /ω )( k 0 sin φ ) 4 ) 2 . (2)

Once the transmission coe ffi cient is obtained, the air-borne transmission loss for di ff erent

frequencies can be obtained (Equation 3),

TL = 10 log ( 1

τ ) , (3)

with the transmission loss binned into one-third octave bands.

2.2. Finite element method The finite element method, although first introduced in the mid-20 th century, has not been extensively used for sound insulation predictions. Part of this hesitation is due to the computational complexity to create a finite element model and the expertise to interpret the results. Additionally, FEM makes many assumptions that directly influence the estimated sound insulation performance, such as material properties and damping. Despite these limitations, FEM has been e ff ective for predicting acoustic transmission loss as proven through experimental validations. In this paper, an FEM, using Ansys 2021 v1, is created to predict the structure-borne sound insulation of a CLT floor. Impact sound is estimated by taking the ratio of inputted sound power to radiated sound power. The CLT floor was modeled using hexahedral elements with a uniform element width of 0.1 m. The material properties used to model the CLT floor are shown in Table 1, which were used from [10] as material properties. After preprocessing, the modal response was used to obtain the direct frequency response. The input power is generated by a 1 N force, representing footfall, located on the top of the floor. Three locations are evaluated on the CLT floor, as shown in Figure 1. The force creates a displacement and corresponding velocities, on the floor, which radiates on the underside of the floor. An air hemisphere, representing the far field, is modeled to obtain the acoustic pressure at the boundary of the hemisphere to obtain the radiated sound power. The impact performance was averaged for the three forcing locations. A damping coe ffi cient of 4% was used, along with density of 500 kg / m 3 .

Figure 1: Locations of the forces on the floor.

Table 1: Parameters used in the FEM of the CLT floor.

Modulus of Elasticity Shear Modulus Poisson’s Ratio

E 1 = 9,000 MPa G 1 , 2 = 900 MPa ν 1 , 2 = 0.3

E 2 = 4,000 MPa G 1 , 3 = 90 MPa ν 1 , 3 = 0.3

E 3 = 4,000 MPa G 2 , 3 = 63 MPa ν 2 , 3 = 0.4

2.3. Artificial neural networks The idea of artificial neural networks approach was motivated by the structure of a real brain [38]. It utilizes very simple calculations (multiplication, additions and fundamental logic elements) to solve complex mathematical problems [39,40]. The architecture of ANN consists of layers, and each layer contains computation units called neurons [41]. Those units are linked to one another through weights which are used to scale each input to a neuron. ANN computes input values using activation function and propagates them to the output neuron(s) using the weights as intermediate features [42]. The training data contains a set of input-output pairs of the function to be learned. It gives feedback on the correctness of the weights, depending on the accuracy of the predictions (output values). Errors that are made by ANN are viewed to evaluate the values of weights between neurons. Therefore, the weights are adjusted in a network. Features of ANN (weights and bias) are tuned during the learning phase. Since an ANN model is typically arranged in layers and each layer is an array of neurons, a data set flows through each neuron in an input-output manner. A neuron can be expressed as a mathematical function that computes one or more input values and outputs as a single value. The neuron is identified as follows,

y = f ( X ( w i x i + b )) , (4)

where y , w i , x i and b represent output, weight value, input and bias of a specific neuron respectively. Output values of neurons are defined as activation values, which are utilized as an input to the activation function.

2.3.1 Database collection

The database is developed based on 252 standardized laboratory measurements received from Lund University in Sweden, FCBA in France, FPinnovations and CNRC [44] in Canada. The measurements consist of air-borne and impact sound insulation tests performed on 142 di ff erent floor structures in the frequency range of 50 Hz to 5 kHz . The air-borne sound reduction index measurements were carried out according to ISO 140-3 (1995) [45] or the latest ISO 10140-2 (2010) [46] and ASTM E90-09 (2016) [47]. The impact sound pressure level data were measured following ISO 140-6 (1998) [48] or the latest ISO 10140-3 (2010) [49] and ASTM E492-09 (2016) [50]. Measurement data based on ASTM standards were converted to comply with ISO standards descriptors, the weighted air- borne sound R w and the weighted normalized impact sound pressure level L n , w in order to have a total agreement with the acoustic descriptors. The entire database is split into three subsets: training, validation and test set. The training set is employed to initiate the ANN features. The validation set is used for optimizing the architecture of the model, while the test set is used for estimating the predictive capabilities of the chosen model.

2.3.2 Prediction model configuration

The ANN model is based on a multilayer perceptron algorithm which consists of two hidden layers with 40 and 30 neurons in each layer respectively. The LeakyReLU (Leaky Rectified Linear Unit)

function [43, 51, 52] was utilized as an activation function for both layers. This activation function is based on the ReLU function, which has a small slope for negative values. It helps to overcome the vanishing gradient issues which usually face normal activation functions such as tan and sigmoid functions by giving negative gradients instead of zeros [53]. In the training phase, the Adam optimizer [54] was used, which is one of several optimization algorithms used to optimize neural networks [55]. The cost function is a mathematical expression which is used to evaluate the performance of the model. In this case, the prediction deals with continuous values because the target is to predict the acoustic curve. Therefore, the root-mean-squared error (RMSE) can be used as a cost function,

v t

n X

1 n

RMS E =

i = 1 (ˆ y i − y i ) 2 , (5)

where n is the total number of measurements used as a training set, ˆ y i and y i are the predicted and measured values respectively.

3. RESULTS AND DISCUSSION

3.1. Prediction of air-borne sound insulation Figure 2 illustrates the sound reduction index of a 140 mm CLT panel based on experimental test, mass law (Equation 1), analytical expression (Equation 3), and ANN model values. It can be seen that the transmission loss curve based on mass law gives higher values than experimental ones across all frequencies. By contrast, the analytical expression values are lower and closer to measured ones, especially at frequencies above 250 Hz . However, the prediction model using artificial neural networks tends to over-predict sound reduction index values in the low frequency bands (50-250 Hz ), while a better agreement is achieved, the higher frequency bands. Moreover, the used approaches show a limitation in estimation of the air-borne sound reduction index values below 250 Hz , where the fundamental resonance (first eigenfrequencies) usually occur [13]. Table 2 depicts the calculated single number quantities (SNQs), R w , and correction terms ( C 100 − 3150 and C 50 − 5000 ) for sound insulation curves based on measured and di ff erent prediction tools. The highest error variation in the weighted reduction index R w is 15 dB that is given by the mass law. However, a better accuracy is obtained using the analytical expression and ANN model with a deviation of 2 dB . Regarding correction terms, the di ff erences did not exceed 1 dB for all approaches. Overall the ANN model shows a satisfied predictive capability in the estimation of SNQs for a 140 mm bare CLT panel. Table 3 represents a comparison between measured and predicted air-borne sound insulation curves split into three di ff erent zones: low (50-200 Hz ), middle (250 Hz − 1 kHz ), and high frequencies (1.25-5 kHz ). The mass law gives the highest deviations among all other methods. However, the analytical expression shows better accuracy at middle and high frequencies. Results in Table 3 reveals that the ANN model demonstrated the best accuracy across all frequencies with an overall deviation of 2.05 dB .

3.2. Prediction of impact sound insulation Two prediction methods using FEM and ANN models are used to estimate the impact sound insulation curve of a 140 mm bare CLT floor. It is observed that both models showed a good correlation in the estimation of impact sound pressure values (Figure 3). However, The FEM demonstrates non-negligible deviations at middle and high frequencies, while the ANN values are closer to measured ones. Similar to the prediction of air-borne sound insulation curves at low frequencies, the fundamental resonance area is still a challenge to achieve good estimations. Concerning SNQs and correction terms, Table 4 reveals that the FEM reveals a better accuracy

Figure 2: A comparison between air-borne sound insulation data of a 140 mm bare CLT floor

Table 2: Comparison between measured and predicted weighted sound reduction indices R w of a 140 mm CLT panel.

Transmission loss curve R w ( dB ) C 100 − 3150 C 50 − 5000

Experimental values 35 0 0

Mass law values 50 -1 -1

Analytical expression values 33 -1 -1

ANN model 37 -1 0

Table 3: Comparison between measured and predicted air-borne sound reduction indices clustered in low, middle, and high frequency ranges using the absolute mean error values.

Absolute Mean Error Values in dB

Frequency range Low Middle High Full range

50 − 200 Hz 250 Hz − 1 kHz 1 . 25 − 5 kHz 50 Hz − 5 kHz

Mass law 7.03 15.01 15.83 12.63

Analytical exp. 7.60 2.40 2.54 4.18

ANN model 6.73 1.49 2.07 2.05

_ ‘Comparing Data for CLT floor = Experimenta Values TS iss tw Values a S naytial Exresion Values 8 x ‘AN Model Values « % 3 2 60 = S 3 @ 3 5 2% ° 50 250 125K 3k Frequency (Hz)

Figure 3: A comparison between impact sound insulation data of a 140 mm bare CLT floor

Table 4: Comparison between measured and predicted normalized impact sound pressure levels L n , w of a 140 mm CLT panel.

impact sound curve L n , w ( dB ) C I , 100 − 2500 C I , 50 − 5000

Experimental values 88 -4 -4

FEM model 89 -9 -9

ANN model 85 -4 -4

Table 5: Comparison between measured and predicted normalized impact sound pressure levels clustered in low, middle, and high frequency ranges using the absolute mean error values.

Absolute Mean Error Values in dB

Frequency range Low Middle High Full range

50 − 200 Hz 250 Hz − 1 kHz 1 . 25 − 5 kHz 50 Hz − 5 kHz

FEM 5.50 7.83 7.10 2.08

ANN model 2.26 2.29 0.95 1.84

than ANN model in predicting the weighted normalized impact sound pressure level ( L n , w ) with variations of 1 and 3 dB respectively. However, ANN model depicts better estimation regarding the calculated correction terms.

Impact sound pressure levels Ln (dB) 100 60 40 20 ‘Comparing Data for CLT floor experimental Values “FEM Model Values A ANN Model values. 50 250 125K Frequency (Hz) 3K

Table 5 presents that the ANN model can estimate impact sound pressure levels better than FEM across all frequencies with errors not exceeding 2 dB . It should be noted that the FEM could be susceptible to certain uncertainties, especially due to the CLT structure and assumptions, that could strongly a ff ect the estimations. In addition, the discrepancies in the FEM results could be described due to the over-simplified modeling method in assumptions of boundary conditions. Advantages and disadvantages of each used method in forecasting the sound insulation curves are presented in Table 6. Although mass law over-predicts the sound insulation curves, it is appropriate for an early approximation in design. The analytical model is satisfactory for more sophisticated geometries, but is still an estimate, suitable for acoustic design exploration. The FEM is appropriate for complex models, in which boundary conditions, and unique material properties can be taken into more consideration (but not complete consideration). The ANN model was the most accurate but relies on existing data - which none of the other methods have. Furthermore, the ANN model requires training, which could be computationally taxing (although a FEM can be computationally costly too, the other approaches not so much). Despite the resealable obtained results from FEM and ANN model, both methods require skills (i.e., modeling and coding skills, respectively) in order to build a suitable prediction tool.

Table 6: Advantages and disadvantages of the di ff erent approaches used in the prediction of sound insulation.

Approach Advantage Disadvantage

-Mass law • suitable for early estimation • gives inflated ratings

• uses one structural parameter (the mass)

-Analytical expression • manipulate di ff erent geometries • high deviations at low frequencies

• acceptable results

-FEM • appropriate for complex models • computationally costly

• reasonable results • broad assumptions are needed

• requires modeling skills

-ANN model • satisfactory accuracy at all frequencies • needs data to be trained

• requires coding skills

4. CONCLUSIONS

The present work demonstrates a comparison of numerical approaches to estimate the acoustic performance of a 140 mm bare CLT floor in one-third octave bands (50 Hz − 5 kHz ). The results show that the ANN model gives reliable estimations to quantify air-borne and impact sound insulation curves. The same applies to analytical method and FEM on prediction of air-borne and impact curves, respectively, but with higher deviations. However, no model is able to estimate the accurate acoustic behavior. In addition, all the approaches illustrate a limitation in the prediction of the acoustic spectrum at low frequency range (50-250 Hz ), where the fundamental frequencies are usually located. The best methods in estimating the sound insulation were the ANN model and FEM. However, these methods depend on the skills of the designer (i.e., coding skills or modeling skills) and the resources available to them (i.e., data to train an ANN, ANSYS licensing, etc.). Further

research would be for more comparison with di ff erent structure configurations investigated.

5. ACKNOWLEDGMENTS

The authors are grateful to Lund university, FCBA, FPInnovations, and CNRC for providing the acoustic measurement data of sound insulation.

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