Abstract
A novel hybrid deterministicstatistical approach named ESFESEA method specially used to predict the sound Transmission loss of panels in midfrequency is proposed in this paper. The proposed hybrid methods takes the best advantages of edgedbased smoothing FEM (ESFEM) and statistical energy analysis (SEA) to further improve the accuracy of midfrequency transmission loss predictions. The application of ESFEM will “soften” the wellknown “overlystiff” 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 modela 2D dash panel in a car. The proposed ESFESEA is verified by various numerical examples.
1. Introduction
The predictions of sound Transmission loss in midfrequency 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 abovementioned approaches.
When the frequency of interest shifted to the midfrequency domain, the airborne sound transmission prediction through a panel become more difficult. The midfrequency 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 midfrequency transmission loss problem properly. Recently, the hybrid FESEA developed by Robin S. Langley becomes the mainstream method meant for midfrequency problems [6]. However, the use of conventional FEM in conventional hybrid methods entails some inherent drawbacks, which are closely related to the wellknown “overlystiff” feature of FEM and its sensitivity to numerical pollution errors [7, 8]. In this paper, a novel hybrid edgebased smoothed FEM coupled with statistical energy analysis (ESFESEA) method is proposed, and can in principle handle all these restrictions lies in a single deterministic or statistical method and conventional hybrid methods. The edgebased smoothed technique will help to extend the application of conventional FEM to midfrequency with higher accuracy and efficiency while SEA will deal with all the uncertainty in the transmission system. The embedding of the ESFEM in hybrid ESFESEA framework presents a promising combination for solving midfrequency transmission loss problems. This is also the essential motivation of this work.
2. Theory summary
The hybrid ESFESEA takes the best advantages of two well established techniques: SEA and ESFEM. These two methods are briefly reviewed in the following.
2.1. The basic formulations of ESFEM
In the conventional FEM, the calculation of stiffness matrix can be written as follows [9]:
in which ${\mathbf{B}}_{b}$ and ${\mathbf{B}}_{s}$ are straindeflection matrix for bending and shearing, respectively. ${\text{D}}_{\text{b}}$ is the bending stiffness constitutive coefficients, and ${\mathbf{D}}_{s}$ is transverse shear stiffness constitutive.
In ESFEM, strains calculation is based on the edgebased smoothed domain rather than the 3node triangles element domain. The smoothing operation is firstly applied to the bending (inplane) strain and the shear (offplane) stain of the plate over each of the edgebased smoothing domains:
in which ${N}_{k}^{e}$ is the number of the subdomain of edge $k$ and ${A}_{e}^{i}$ is the area of $i$th subdomain in a triangle element. ${A}_{k}$ is the area of the smoothing domain ${\mathrm{\Omega}}_{k}$, which can be calculated as follows:
Using the smoothed strain matrix ${\stackrel{}{\mathbf{B}}}_{s}$ and ${\stackrel{}{\mathbf{B}}}_{b}$ to replace ${\mathbf{B}}_{s}$ and ${\mathbf{B}}_{b}$, the smoothed stiffness matrix $\overline{\mathbf{K}}$ based on the smoothing domain can be evaluated as:
Finally the edgedbased discretized system equations for ReissnerMinlin plate elements can be rewritten as:
where $\mathbf{M}$ is the mass matrix, $\mathbf{u}$ is total unknown variables, $\mathbf{F}$ is referred as vector of nodal forces.
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:
where ${P}_{i}$ and ${P}_{j}$ are the external source excitations which are applied directly to the subsystem. ${\eta}_{ij}$ and ${\eta}_{ji}$ are the coupling loss factors of two subsystems. $\omega $ is the angular frequency. ${\eta}_{id}$ and ${\eta}_{jd}$ are the damping loss factors. ${E}_{i}$ and ${E}_{j}$ are the average vibrational energy of two subsystems. The power balance equations can be easily extend to the a big system with $n$ numbers of subsystems. Detailed principles of statistical energy analysis (SEA) can refer to Ref. [4].
3. The hybrid ESFEM/SEA equations
The coupling of an ESFEM subsystem and two SEA subsystems is described based on the “reciprocity relationship” theorem. According to the diffuse field reciprocity relation, the crossspectrum of the forces on the boundary of a statistical subsystem can be written in terms of subsystem vibrational energy as follow:
where ${\mathbf{D}}_{dir}$ is the direct field dynamic stiffness matrix related to an acoustic halfspace 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:
where ${P}_{in}$ is the input power that is assumed to be characterized and known. ${p}_{out}^{rev}$ is the power leaving the reverberant field. ${p}_{diss}$ 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 ${P}_{1}$. The transmitted power ${P}_{3}$ of the right cavity can be obtained using Hybrid ESFESEA equations. The transmission loss of the plate is defined as [11]:
where $A$ is the area of the panel and ${S}_{3}$ and ${a}_{3}$ are the surface area and absorption coefficient of the right hand room, respectively.
4. Numerical application of the hybrid ESFESEA
4.1. Validation of ESFEM/SEA
In order to validate the proposed ESFESEA, a standard numerical example for benchmarking is examined, which was original studied by Robin S. Langley using hybrid FESEA [6]. The thickness of panel is 0.015 m. The hybrid FESEA model and FEM model shows as following Fig. 1.
Fig. 1Sketch of a transmission loss system
a) Hybrid FESEA (VAOne)
b) FEM model
The convergence study of the proposed hybrid ESFESEA is first carried on a very fine mesh with 3,598 triangle elements and 1,884 nodes. The reference results are obtained by VAOne software using the embedded hybrid FESEA 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.
Fig. 2The validation of the ESFESEA
a) Coupling loss factor of two cavities
b) The transmission loss factor of the free plate
Fig. 3The comparison between the ESFESEA and conventional FESEA
a) Coupling loss factor of two cavities
b) The transmission loss factor of the free plate
The results obtained from ESFESEA show excellent agreements with the reference result. Therefore, the validation and effectiveness of the present method is confirmed in this example. In order to compare the accuracy of ESFESEA and conventional FESEA, the proposed method is applied to a rough mesh with 244 triangle elements with 143 nodes. The reference result is obtained on VAONE software by using FESEA 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 ESFESEA provides a higher accuracy results compared with the FESEA in the whole low to midfrequency regime. Even applied in the model with much less DOFs (244 DOFs), ESFESEA is still able to provide the result in almost the same accuracy level as the result of FESEA using fine mesh (5,908 DOFs). To sum up, the high accuracy and computational efficiency of ESFEM/SEA for midfrequency 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 ESFESEA 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 m^{3}. The material parameters of dash panel are given as: the density $\rho =$7800 kg/m^{3}, Yong’s modulus $E=$210 GPa and Poisson’s ratio $\nu =$0.3. The dash panel has the thickness of 15 mm.
Fig. 4A transmission loss model of dash panel
a) Hybrid FESEA (VAOne)
b) FEM model
The dash panel compartment is first divided using 260 triangle elements with 160 nodes for ESFEM or FEM. The results are compared against the reference result that is calculated by VAONE software using coupled FESEA 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 02000 Hz.
Fig. 5The transmission loss factor of the front windshield
The results for this complicated vehicle example reenforces the finding from the example for benchmarking. In the whole midfrequency domain, the ESFEM/SEA constantly provides much more accurate result in higher frequency range, compared with the FESEA model using the same 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 ESFESEA method in midfrequencies. Compared with conventional FESEA, the proposed ESFESEA approach takes the best advantages of edgedbased smoothing techniques to “soften” the stiffness of the deterministic subsystem, thus eliminating the numerical error in standard FEM. Therefore, the ESFESEA constantly provides much more accurate result in the mid to high frequency range, comparing with the FESEA model based on the same mesh.
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