A novel hybrid deterministic-statistical approach named ES-FE-SEA method specially used to predict the sound Transmission loss of panels in mid-frequency is proposed in this paper. The proposed hybrid methods takes the best advantages of edged-based smoothing FEM (ES-FEM) and statistical energy analysis (SEA) to further improve the accuracy of mid-frequency transmission loss predictions. The application of ES-FEM will “soften” the well-known “overly-stiff” behavior in the standard FEM solution and reduce the inherent numerical dispersion error. While the SEA approach will deal with the physical uncertainty in the relatively higher frequency range. Two different types of subsystems will be coupled based on “reciprocity relationship” theorem. The proposed was firstly applied to a standard simple numerical example, and excellent agreement with reference results was achieved. Thus the method is then applied to a more complicated model-a 2D dash panel in a car. The proposed ES-FE-SEA is verified by various numerical examples

Aiguo Cheng

Fei Wu

Guangyao Li

Gui Rong Liu

Zhicheng He

English

Vibroengineering PROCEDIA

 34 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
Mid-frequency prediction of transmission loss using a 
novel hybrid deterministic and statistical method 
Fei Wu1, Gui Rong Liu2, Guangyao Li3, Aiguo Cheng4, Zhicheng He5 
1, 3, 4, 5State Key Lab of Advanced Technology for Vehicle Body Design and Manufacture,  
Hunan University, Changsha, 410082, P. R. China 
2Department of Aerospace Engineering and Engineering Mechanics,  
University of Cincinnati 2851 Woodside Dr, Cincinnati, OH 45221, USA 
1Corresponding author 
E-mail: 1wufeifrank@gmail.com, 2liugr@ucmail.uc.edu, 3gyli@hnu.cn, 4cheng-aiguo@163.com, 
5hezhicheng815@gmail.com 
(Accepted 15 July 2014) 
Abstract. A novel hybrid deterministic-statistical approach named ES-FE-SEA method specially 
used to predict the sound Transmission loss of panels in mid-frequency is proposed in this paper. 
The proposed hybrid methods takes the best advantages of edged-based smoothing FEM 
(ES-FEM) and statistical energy analysis (SEA) to further improve the accuracy of mid-frequency 
transmission loss predictions. The application of ES-FEM will “soften” the well-known 
“overly-stiff” behavior in the standard FEM solution and reduce the inherent numerical dispersion 
error. While the SEA approach will deal with the physical uncertainty in the relatively higher 
frequency range. Two different types of subsystems will be coupled based on “reciprocity 
relationship” theorem. The proposed was firstly applied to a standard simple numerical example, 
and excellent agreement with reference results was achieved. Thus the method is then applied to 
a more complicated model-a 2D dash panel in a car. The proposed ES-FE-SEA is verified by 
various numerical examples. 
Keywords: transmission loss, edge-based smoothed finite element (ES-FEM), ES-FE-SEA, 
mid-frequency. 
1. Introduction 
The predictions of sound Transmission loss in mid-frequency regime have been one of 
hotspots and difficulties in academy and engineering. There are many contributed efforts to 
improve the modeling techniques to solve such transmission loss problems. Some variational 
approaches [1] and wave approaches [2] attempted to solve the transmission loss based on the 
exact analytical solutions. Because of the basic simplifications and assumptions, these approaches, 
however, are only suitable for special cases and provide just rough estimates. Numerical modeling 
approaches such as the finite element method (FEM) [3] can be applied without restrictions for 
different geometries of cavities and plates but often have limited applicability to mid and high 
frequency range responses. The statistical energy analysis (SEA) [4] is another possible approach 
for solving the transmission loss problems. SEA approach can deal with high frequency 
transmission loss problems, but often cannot be applied in the lower frequency range All in all 
there are still a lot of problems to calculate the transmission loss using the above-mentioned 
approaches. 
When the frequency of interest shifted to the mid-frequency domain, the airborne sound 
transmission prediction through a panel become more difficult. The mid-frequency response of 
transmission loss systems often exhibit the mixed behaviors (either deterministic or statistic 
behaviors) [5]. A single deterministic method like FEM or statistic method as SEA can hardly 
solve mid-frequency transmission loss problem properly. Recently, the hybrid FE-SEA developed 
by Robin S. Langley becomes the mainstream method meant for mid-frequency problems [6]. 
However, the use of conventional FEM in conventional hybrid methods entails some inherent 
drawbacks, which are closely related to the well-known “overly-stiff” feature of FEM and its 
sensitivity to numerical pollution errors [7, 8]. In this paper, a novel hybrid edge-based smoothed 
MID-FREQUENCY PREDICTION OF TRANSMISSION LOSS USING A NOVEL HYBRID DETERMINISTIC AND STATISTICAL METHOD.  
FEI WU, GUI RONG LIU, GUANGYAO LI, AIGUO CHENG, ZHICHENG HE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 35 
FEM coupled with statistical energy analysis (ES-FE-SEA) method is proposed, and can in 
principle handle all these restrictions lies in a single deterministic or statistical method and 
conventional hybrid methods. The edge-based smoothed technique will help to extend the 
application of conventional FEM to mid-frequency with higher accuracy and efficiency while 
SEA will deal with all the uncertainty in the transmission system. The embedding of the ES-FEM 
in hybrid ES-FE-SEA framework presents a promising combination for solving mid-frequency 
transmission loss problems. This is also the essential motivation of this work. 
2. Theory summary 
The hybrid ES-FE-SEA takes the best advantages of two well established techniques: SEA 
and ES-FEM. These two methods are briefly reviewed in the following. 
2.1. The basic formulations of ES-FEM 
In the conventional FEM, the calculation of stiffness matrix can be written as follows [9]: 
۹ = න ۰௕்۲௕۰௕݀Ω +
 
ஐ
න ۰௦்۲௦۰௦݀Ω
ஐ
, (1) 
in which ۰௕ and ۰௦ are strain-deflection matrix for bending and shearing, respectively. Db is the 
bending stiffness constitutive coefficients, and ۲௦ is transverse shear stiffness constitutive. 
In ES-FEM, strains calculation is based on the edge-based smoothed domain rather than the  
3-node triangles element domain. The smoothing operation is firstly applied to the bending 
(in-plane) strain and the shear (off-plane) stain of the plate over each of the edge-based smoothing 
domains: 
۰ഥ௕௜(ܠ௞) =
1
ܣ௞
෍ 13 ܣ௘
௜ ۰௕௜(ܠ௞)
ேೖ೐
௜ୀଵ
, ۰ഥ௦௜(ܠ௞) =
1
ܣ௞
෍ 13 ܣ௘
௜ ۰௦௜(ܠ௞),
ேೖ೐
௜ୀଵ
(2) 
in which ௞ܰ௘ is the number of the sub-domain of edge ݇ and ܣ௘௜  is the area of ݅th sub-domain in a 
triangle element. ܣ௞ is the area of the smoothing domain Ω௞, which can be calculated as follows: 
ܣ௞ = න ݀Ω
 
ஐೖ
= 13 ෍ ܣ௘
௜ .
ேೖ೐
௜ୀଵ
(3) 
Using the smoothed strain matrix ۰തݏ  and ۰തܾ  to replace ۰ݏ  and ۰ܾ , the smoothed stiffness 
matrix ۹ based on the smoothing domain can be evaluated as: 
۹ = ෍ ܣ௞
ேೞ
௞ୀଵ
۰ഥ௕்۲௕۰ഥ௕ + ෍ ܣ௞
ேೞ
௞ୀଵ
۰ഥ௦்۲௦۰௦. (4) 
Finally the edged-based discretized system equations for Reissner-Minlin plate elements can 
be re-written as: 
ۻܝሷ + ۹ഥܝ = ۴, (5) 
where ۻ is the mass matrix, ܝ is total unknown variables, ۴ is referred as vector of nodal forces. 
MID-FREQUENCY PREDICTION OF TRANSMISSION LOSS USING A NOVEL HYBRID DETERMINISTIC AND STATISTICAL METHOD.  
FEI WU, GUI RONG LIU, GUANGYAO LI, AIGUO CHENG, ZHICHENG HE 
36 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
2.2. The basic formulations of SEA 
The Statistical Energy Analysis (SEA) calculation is based on the principle of conservation of 
energy which flows in and out of a subsystem. Only one single degree of freedom (energy of 
vibration), rather than potentially thousands of degrees of freedom in FEM, is needed to describe 
the vibrational behavior of a subsystem. Therefore, this approach can dramatically save great 
computational time through reducing the numerical model size. The power balance equations for 
two subsystems can be written in a matrix form as below: 
൤݆ܲ݅ܲ൨ = ߱ ቈ
(ߟ݅݀ + ߟ݆݅) −ߟ݆݅
−ߟ݆݅ (ߟ݆݀ + ߟ݆݅)
቉ ൤ܧ݅ܧ݆൨, (6)
where ௜ܲ and ௝ܲ are the external source excitations which are applied directly to the subsystem. 
ߟ௜௝ and ߟ௝௜ are the coupling loss factors of two subsystems. ߱ is the angular frequency. ߟ௜ௗ and 
ߟ௝ௗ are the damping loss factors. ܧ௜ and ܧ௝ are the average vibrational energy of two subsystems. 
The power balance equations can be easily extend to the a big system with ݊  numbers of 
subsystems. Detailed principles of statistical energy analysis (SEA) can refer to Ref. [4]. 
3. The hybrid ES-FEM/SEA equations 
The coupling of an ES-FEM subsystem and two SEA subsystems is described based on the 
“reciprocity relationship” theorem. According to the diffuse field reciprocity relation, the 
cross-spectrum of the forces on the boundary of a statistical subsystem can be written in terms of 
subsystem vibrational energy as follow: 
〈܁௙௙௥௘௩〉 =
4ܧ
ߨ߱݊ Imሼ۲ௗ௜௥ሽ, (7)
where ۲ௗ௜௥ is the direct field dynamic stiffness matrix related to an acoustic half-space on one 
side of the infinite planar baffle. The dynamic stiffness matrix can be calculated using the approach 
of Jinc functions based on some grid of points embedded in the infinite planar baffle [10]. The 
statistical cavity subsystem is discomposed into “direct field” and “reverberant field”. For each 
subsystem, power balance condition leads to the following set of additional equations: 
௜ܲ௡ = ݌௢௨௧௥௘௩ + ݌ௗ௜௦௦, (8)
where ௜ܲ௡ is the input power that is assumed to be characterized and known. ݌௢௨௧௥௘௩ is the power 
leaving the reverberant field. ݌ௗ௜௦௦ is the dissipated power that is assumed to be proportional to 
damping. Full details of the derivation of the above equations can be found in Ref. [6]. 
We assume that the left cavity is the acoustic source cavity and carries a input power ଵܲ. The 
transmitted power ଷܲ of the right cavity can be obtained using Hybrid ES-FE-SEA equations. The 
transmission loss of the plate is defined as [11]: 
܂ۺ = 10logଵ଴ ቆ ଵܲ
ଶ
ଷܲ
ଶቇ − 10logଵ଴ ൤
ܣ
(ܵଷܽଷ)
൨, (9)
where ܣ is the area of the panel and ܵଷ and ܽଷ are the surface area and absorption coefficient of 
the right hand room, respectively. 
MID-FREQUENCY PREDICTION OF TRANSMISSION LOSS USING A NOVEL HYBRID DETERMINISTIC AND STATISTICAL METHOD.  
FEI WU, GUI RONG LIU, GUANGYAO LI, AIGUO CHENG, ZHICHENG HE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 37 
4. Numerical application of the hybrid ES-FE-SEA 
4.1. Validation of ES-FEM/SEA  
In order to validate the proposed ES-FE-SEA, a standard numerical example for benchmarking 
is examined, which was original studied by Robin S. Langley using hybrid FE-SEA [6]. The 
thickness of panel is 0.015 m. The hybrid FE-SEA model and FEM model shows as following 
Fig. 1. 
 
a) Hybrid FE-SEA (VA-One) 
 
b) FEM model 
Fig. 1. Sketch of a transmission loss system 
The convergence study of the proposed hybrid ES-FE-SEA is first carried on a very fine mesh 
with 3,598 triangle elements and 1,884 nodes. The reference results are obtained by VA-One 
software using the embedded hybrid FE-SEA also based on that fine mesh. The simply supported 
boundary conditions around the plate perimeter are chosen to compare the results. Coupling loss 
factors of two cavities and transmission loss factor of the plate are plotted in Fig. 2. 
a) Coupling loss factor of two cavities 
 
b) The transmission loss factor of the free plate 
Fig. 2. The validation of the ES-FE-SEA 
a) Coupling loss factor of two cavities 
 
b) The transmission loss factor of the free plate 
Fig. 3. The comparison between the ES-FE-SEA and conventional FE-SEA 
The results obtained from ES-FE-SEA show excellent agreements with the reference result. 
Therefore, the validation and effectiveness of the present method is confirmed in this example. In 
500 1,000 1,500 2,000 3000 4,00010
-12
10-10
10-8
10-6
10-4
Frequency(Hz)
Co
up
lin
g L
os
s F
ac
tor
s
 
 
Reference
ES-FE-SEA
500 1,000 1,500 2,000 3000 4,0000
20
40
60
80
Frequency(Hz)
Tr
an
sm
iss
ion
 lo
ss
 (d
B)
 
 
Reference
ES-FE-SEA
500 1,000 1,500 2,000 3000 4,00010
-15
10-10
10-5
Frequency(Hz)
Co
up
lin
g L
os
s F
ac
tor
s
 
 
FE-SEA
Reference
ES-FE-SEA
500 1,000 1,500 2,000 3000 4,0000
10
20
30
40
50
60
70
Frequency(Hz)
Tr
an
sm
iss
ion
 lo
ss
 (d
B)
 
 
FE-SEA
ES-FE-SEA
Reference
MID-FREQUENCY PREDICTION OF TRANSMISSION LOSS USING A NOVEL HYBRID DETERMINISTIC AND STATISTICAL METHOD.  
FEI WU, GUI RONG LIU, GUANGYAO LI, AIGUO CHENG, ZHICHENG HE 
38 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
order to compare the accuracy of ES-FE-SEA and conventional FE-SEA, the proposed method is 
applied to a rough mesh with 244 triangle elements with 143 nodes. The reference result is 
obtained on VA-ONE software by using FE-SEA with a very fine mesh (5,908 triangle elements 
and 3050 nodes). The results obtained by different methods are plotted in Fig. 3. 
As shown in the Fig. 3, it is obvious that ES-FE-SEA provides a higher accuracy results 
compared with the FE-SEA in the whole low to mid-frequency regime. Even applied in the model 
with much less DOFs (244 DOFs), ES-FE-SEA is still able to provide the result in almost the 
same accuracy level as the result of FE-SEA using fine mesh (5,908 DOFs). To sum up, the high 
accuracy and computational efficiency of ES-FEM/SEA for mid-frequency solution is illustrated 
in this example. 
4.2. Dash panel model 
In this part, a transmission loss model of complicated dash panel is established using the hybrid 
ES-FE-SEA as illustrated schematically in Fig. 4. The hybrid model consists of two adjacent 
reverberation cavities and dash panel of interest. Two adjacent reverberation cavity are simply 
extracted by the dash panel with the volume of 1.5 m3. The material parameters of dash panel are 
given as: the density ߩ = 7800 kg/m3, Yong’s modulus ܧ = 210 GPa and Poisson’s ratio ߥ = 0.3. 
The dash panel has the thickness of 15 mm. 
 
a) Hybrid FE-SEA (VA-One) 
 
b) FEM model 
Fig. 4. A transmission loss model of dash panel 
The dash panel compartment is first divided using 260 triangle elements with 160 nodes for 
ES-FEM or FEM. The results are compared against the reference result that is calculated by 
VA-ONE software using coupled FE-SEA model, where the dash panel is divided with 1,842 
triangle elements and 1,012 nodes. The transmission loss factor of the dash panel is calculated 
using different methods as shown in Fig. 5. The analysis frequency range was between 0-2000 Hz. 
 
Fig. 5. The transmission loss factor of the front windshield 
The results for this complicated vehicle example re-enforces the finding from the example for 
benchmarking. In the whole mid-frequency domain, the ES-FEM/SEA constantly provides much 
more accurate result in higher frequency range, compared with the FE-SEA model using the same 
0 200 400 600 800 1000 1200 1400 1600 1800 20000
10
20
30
40
50
60
70
Frequency(Hz)
Tr
an
sm
iss
ion
 lo
ss
 (d
B)
 
ES-FE-SEA
FE-SEA
Reference
MID-FREQUENCY PREDICTION OF TRANSMISSION LOSS USING A NOVEL HYBRID DETERMINISTIC AND STATISTICAL METHOD.  
FEI WU, GUI RONG LIU, GUANGYAO LI, AIGUO CHENG, ZHICHENG HE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 39 
mesh. Thus, the method could be used for the more complicated model. 
5. Conclusions 
This paper investigated the transmission loss of a simple panel and complicated dash panel 
using the newly developed hybrid ES-FE-SEA method in mid-frequencies. Compared with 
conventional FE-SEA, the proposed ES-FE-SEA approach takes the best advantages of 
edged-based smoothing techniques to “soften” the stiffness of the deterministic subsystem, thus 
eliminating the numerical error in standard FEM. Therefore, the ES-FE-SEA constantly provides 
much more accurate result in the mid to high frequency range, comparing with the FE-SEA model 
based on the same mesh. 
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Part I: Theoretical results. Acta Acustica united with Acustica, Vol. 26, Issue 1, 1972, p. 1-18. 
[3] Sakumaa Tetsuya, Kenichi Egawab, Yosuke Yasudac Numerical analysis of sound transmission 
loss of glass pane – on the treatment of edge damping. Moment, Vol. 500, 2008. 
[4] Crocker M. J., A. J. Price sound transmission using statistical energy analysis. Journal of Sound and 
Vibration, Vol. 9, Issue 3, 1969, p. 469-486. 
[5] Ladeveze P., et al. A multiscale computational method for medium-frequency vibrations of 
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[6] Shorter Phil J., Langley R. S. Vibro-acoustic analysis of complex systems. Journal of Sound and 
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[7] Liu G. R., Dai K. Y., Nguyen T. T. A smoothed finite element method for mechanics problems. 
Computational Mechanics, Vol. 39, Issue 6, 2007, p. 859-877. 
[8] Wu F., Liu G. R., Li G. Y., Cheng A. G., He Z. C. A new hybrid smoothed FEM for static and free 
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[9] Liu G. R. Meshfree Methods: Moving Beyond the Finite Element Method. 2nd ed., CRC Press, 2009. 
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Mid-frequency prediction of transmission loss using a novel hybrid deterministic and statistical method

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Mid-frequency prediction of transmission loss using a novel hybrid deterministic and statistical method

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