Abstract
Based on the governing equations of energy flow analysis in plate and acoustic domain, an unstructured zeroorder energy finite element method (uEFEM^{0}) formulation is presented to simulate the highfrequency behavior of plate structures in contact with acoustic cavities. The new formulation is derived using EFEM^{0}, in which the energy primary variable is conserved in each element. The bending, longitudinal, and shear wave fields in the plates are all included. By meshing plates with triangular grids and cavities with tetrahedral grids, this new formulation can be easily used for modeling structures with arbitrary shape. The formulation is validated by comparing the results obtained from uEFEM^{0} with those from statistical energy analysis and literature. Good correlations are observed, and the advantages of the uEFEM^{0} formulation are identified.
1. Introduction
Vibration analysis in highfrequency range is an important part in the design phase to reduce the vibration level of the structures. The finite element method (FEM) and boundary element method (BEM) are rational choices to predict the vibration level of structures in lowfrequency range. However, the wavelength is relatively short and the data, such as material properties and joint behavior, are highly uncertain for highfrequency problems [1]. Thus, a large number of elements are demanded, and FEM and BEM at high frequency require unreasonable computation time and storage, and the analysis result would be highly uncertain.
Averaging is considered a feasible way to overcome the frequency limitations of traditional deterministic methods. Statistical energy analysis (SEA), developed by Lyon [2], is widely accepted as an analysis tool in practice because of its easytounderstand nature. In SEA, a builtup structure is divided into a number of subsystems, and a singleaveraged lumped energy with respect to time and space in each subsystem is provided at the end. Therefore, SEA possesses difficulties in determining the spatial variation of energy density of the domain of interest.
This disadvantage is overcome by introducing energy flow analysis (EFA), which is based on a differential equation of energy density [1, 38]. EFA is an effective method for predicting distribution of the time and space averaged energy density and the power flow path. It is based on a set of assumptions [9], enumerated as follows: (1) linear, elastic, dissipative and isotropic systems; (2) small hysteretic damping loss factor ($\eta \ll \text{1}$); (3) steadystate conditions with harmonic excitation of frequency $\omega $; (4) far from singularities, evanescent waves are neglected; and (5) interferences between propagative waves are not considered. In 1989, Nefske and Sung [6] applied EFA on highfrequency vibration in beamlike structure and developed the energy finite element method (EFEM). This approach was extended within other basic types of structures, such as rod [1], plate [4, 10] and acoustic domain [8]. Zhang et al. [11] presented an alternate approach, and derived the EFEM governing equations for plate or an acoustic space, in which unify governing equation is obtained. Kong et al. [12] pointed out that the highfrequency vibration in structures can be recognized as a linear superposition of the direct field and the reverberant field, and the results obtained from EFA is the reverberant field part. Several researchers tried to modify the governing equations in specific structures, in order to extent the validity region of EFA [1316]. In 2011, a random EFA was proposed by You et al. [17], which can be employed to predict the highfrequency vibration level of structures under a random loading.
When applying EFEM in complex structures, a problem is that the energy density is discontinuous at the junctions [6], so that the elements at the junctions must be physically disconnected by duplicate nodes at the joints. In 2000, Wang [5] used a piecewise constant approximation to the energy differential equations and developed a zeroorder EFEM (EFEM^{0}), in which the variables are stored in the centroid of the control volumes so that duplicate nodes are not necessary. Thus, the existing FEM mesh can be more easily used to perform an EFEM^{0} analysis. However, in his research, the EFEM^{0} is based on structured grids, and only beam and plate structures are considered, the vibration motion of acoustic domain is not included.
In this paper, a new formulation of unstructured EFEM^{0} (uEFEM^{0}) that considers both outplane and inplane waves is presented, in which plates and acoustic domains are meshed by triangular and tetrahedral grids, respectively. By applying the analytical method to evaluate the power transfer coefficients between structuralstructural [18] and structuralacoustic [8] junctions, this new formulation can be employed to solve highfrequency structuralacoustic problems of builtup structures with arbitrary shape. Two numerical examples are given to verify the new formulation.
2. Overview of FEA
The procedures of the EFA are initially reviewed to develop the uEFEM^{0} formulation including a brief review on the power transfer through the joints.
2.1. Energy flow equation
The EFA governing differential equation for steady state problem can be developed by considering orthogonal and incoherent waves traveling within the wavebearing domain of interest and expressed as [11]:
where $e$ is the time and spaceaveraged energy density, which characterizes the vibration behavior. $\omega $ is the circular frequency of the analysis, ${c}_{g}$ is the group speed, $\eta $ is the hysteresis damping factor, and ${\pi}_{in}$ is the injected power density. The first term on the left side of Eq. (1) represents the transmitted power, and the second term represents the dissipated power. Thus, Eq. (1) expresses an energy flow balance that involves the power flow, energy loss, and injected power density.
Table 1Form in different vibrating systems
Configuration  Energy equation  Group velocity 
Bending wave  ${c}_{g}^{2}\left(\frac{{\partial}^{2}e}{\partial {x}^{2}}+\frac{{\partial}^{2}e}{\partial {y}^{2}}\right)+\eta \omega e={\pi}_{in}$  ${c}_{g}=2\omega \sqrt[4]{{\rho}_{s}h{\omega}^{2}/D}$ 
Longitudinal wave  ${c}_{g}^{2}\left(\frac{{\partial}^{2}e}{\partial {x}^{2}}+\frac{{\partial}^{2}e}{\partial {y}^{2}}\right)+\eta \omega e={\pi}_{in}$  ${c}_{g}=\sqrt{E/{\rho}_{s}\left(1{\nu}^{2}\right)}$ 
Shear wave  ${c}_{g}^{2}\left(\frac{{\partial}^{2}e}{\partial {x}^{2}}+\frac{{\partial}^{2}e}{\partial {y}^{2}}\right)+\eta \omega e={\pi}_{in}$  ${c}_{g}=\sqrt{E/{2\rho}_{s}\left(1+\nu \right)}$ 
Acoustic wave  ${c}_{g}^{2}\left(\frac{{\partial}^{2}e}{\partial {x}^{2}}+\frac{{\partial}^{2}e}{\partial {y}^{2}}+\frac{{\partial}^{2}e}{\partial {z}^{2}}\right)+\eta \omega e={\pi}_{in}$  ${c}_{g}={c}_{0}$ 
In typical vibrating systems, the form of Eq. (1) is slightly different. As enumerated in Table 1, ${\rho}_{s}$ is the structural density, $E$ and $\nu $ are the Young’s modulus and Poisson ratio, respectively, $D$ is the flexural rigidity, and ${c}_{0}$ is the sound speed. Notably, the energy densities in plates and acoustic domains are recognized as surface and bulk densities, respectively.
2.2. Power flow on coupling boundary
Notably, the governing equations enumerated in Table 1 are valid if the structure and the fluid (air) do not affect each other. However, the primary variable $e$ is discontinuous at the boundaries of connecting elements at the junctions for a structural–acoustic coupled system. These discontinuities can originate from the geometry, from multiple components connected together, from the change of material, or from the different medium interface [7], where ever a reflection/refraction of traveling wave occurs. The structural bending, longitudinal, shear motion, and acoustic motion will not affect each other unless structuralstructural or structuralacoustic interfaces exist [19].
In this work, two kinds of discontinuities are considered, namely, PlatePlate (PP) joint, which represents a junction of plates, and PlateAcoustic (PA) joint, which represents an interface of plate and acoustic domain. As deduced by Dong et al. [19], the relationship between the power flow through the discontinuities and the energy density on the boundary of the discontinuities has the same form in both PP and PA joints, as:
where $\mathbf{I}$ is an identity matrix, ${\mathbf{c}}_{g}$ is a diagonal matrix formed by the group speeds of coupling waves, and $\mathbf{\tau}$ is the power transfer coefficient matrix. The dimensions of the matrices are 2×2 for PA joints, and 3$n$×3$n$ for PP joints, where $n$ is the number of plates which constitute the joint. In this formulation, all the three wave types in plates are considered.
2.3. PA joint relationship
For a structuralacoustic coupling problem, the bending vibration induces the acoustic wave, and the acoustic wave also affects the bending vibration. Zhang et al. [20] pointed out that because of the effect of fluid loading on the structure, the governing equation of bending wave in Table 1 should be modified as follows:
where ${\eta}_{rad}$ is the radiation damping, which is defined as:
In Eq. (4), the parameter $\alpha $ can be calculated by:
where ${f}_{c}$ is the coincidence frequency, at which the structural bending wave number ${k}_{b}$ equals to the acoustic wave number $k$. The bending group speed should be computed with the formulation ${c}_{gB}=2{\left(D{\omega}^{2}/\alpha {\rho}_{s}h\right)}^{1/4}$, where $D$ is the flexural rigidity of the plate.
The radiation efficiency ${\sigma}_{rad}$ quantifies the interaction between structure bending and acoustic waves, the formulation proposed by Leppington [21] is used in this work:
where $r=a/b$$\text{(}r\ge \text{1)}$ is the ratio between the characteristic length $a$ and $b$ of the plate, $\mu ={k}_{sB}/k$ is the wave number ratio, and ${\lambda}_{c}={c}_{0}/{f}_{c}$ is the wavelength of acoustic wave at the coincidence frequency ${f}_{c}$.
By the physical meaning of ${\sigma}_{rad}$, Bitsie [8] derived the power transfer coefficients between bending wave in plate and acoustic wave in air, as:
As the result of conservation law, ${\tau}_{bb}=1{\tau}_{ba}$ and ${\tau}_{aa}=1{\tau}_{ab}$.
Considering the PA joint, the form of the matrices in Eq. (2) can be written as:
2.4. Power flow through the PP joint
Langley and Heron [18] discussed how to compute the power transfer coefficients between an arbitrary number of plates connected to a common beam, using a wave dynamic stiffness matrix method. In this method, the complete set of transmission coefficients is written in the form of ${\tau}_{pr}^{ij}$, where respectively $i$ and $p$ represent the carrier plate and waveform of the incident wave, and $j$ and $r$ represent those of the generated wave. If $i=j$, $\tau $ is recognized as a reflection coefficient. Their published paper has several errors, which are enumerated in Appendix.
If considering a PP joint, which consists of two plates denoted by $i$ and $j$, the matrices in Eq. (2) are in the form of:
3. Implementation of uEFEM^{0}
The discretization procedure of governing equations in Table 1 is realized by EFEM^{0} with unstructured mesh, which indicates that the triangular grids are used for the three types of waves in 2D plates and the tetrahedral grids are used for the acoustic waves in 3D acoustic domains. When the grids are produced, each grid is recognized as a control volume, and the variables are cellcentered, which indicates that they are stored at the centroids of each control volume.
Notice that the governing equation, Eq. (1), is analogous to heat conduction equation, which can be integrated in an elemental control volume ${V}_{i}$, as:
A simple piecewise constant approximation in ${V}_{i}$ is applied for the latter two terms, expressed as:
and by applying the divergence theorem, the first term can be transferred into the power flux through the common boundary of adjacent cells, as:
where ${Q}_{i,j}$ represents the power flux from ${V}_{i}$ to its $j$th adjacent volume ${V}_{j}$, and $m$ is the number of common boundaries. In this paper, different treatments [22, 23] are carried out to calculate ${Q}_{i,j}$ in 2D and 3D domains.
3.1. Calculation of ${\mathit{Q}}_{\mathit{i},\mathit{j}}$ in 2D domain
Fig. 1(a) shows the schematic of two adjacent control volumes of the unstructured grids in a plate, whose centroids are denoted as ${Z}_{i}$ and ${Z}_{j}$, respectively, and their common boundary is $\stackrel{}{AB}$. In Fig. 1(b), ${\mathbf{t}}_{i,j}$ and ${\mathbf{n}}_{i,j}$ are the tangential and normal unit vectors of $\stackrel{}{AB}$, respectively, and ${\mathbf{\tau}}_{i,j}$ and ${\mathbf{\nu}}_{i,j}$ are the unit vectors of the one starting at ${Z}_{i}$ pointing to ${Z}_{j}$ and its vertical one, respectively. $\theta $ is the angle between ${\mathbf{\tau}}_{i,j}$ and ${\mathbf{t}}_{i,j}$. ${d}_{i}$ and ${d}_{j}$ are the distances from each centroid of ${Z}_{i}$ and ${Z}_{j}$ to $\stackrel{}{AB}$ , respectively. $M$ is the intersection of $\stackrel{}{AB}$ and $\overline{{Z}_{i}{Z}_{j}}$.
Fig. 1Two adjacent control volumes and vectors definition on the common boundary in 2D domain
a) Volumes ${Z}_{i}$ and ${Z}_{j}$
b) Vectors definition on $\stackrel{}{AB}$
In 2D plates, the net power flow density ${q}_{i,j}$ is assumed to be uniform on the common boundary $\overline{AB}$; thus, the power flux ${Q}_{i,j}$ can be easily expressed as:
where ${L}_{j}$ is the length of $\overline{AB}$, and $h$ is the thickness of the plate. In this work, the triangular grid is used for the plates; thus, the number of adjacent control volume for each grid, $m$ in Eq. (12), is equal to 3, and ${q}_{i,j}$ is in the form of:
Note that the vectors ${\mathbf{\tau}}_{i,j}$ and ${\mathbf{t}}_{i,j}$ are noncollinear, so ${\mathbf{n}}_{i,j}$ can be decomposed into ${\mathbf{n}}_{i,j}=a{\mathbf{\tau}}_{i,j}+b{\mathbf{t}}_{i,j}$, where:
then the power flux density is resolved into these two directions, and a finite difference is used in each direction for numerical approximation, as in ${V}_{i}$ one can obtain:
and similarly in ${V}_{j}$, the opposite net flow density is:
The symbol $\left\overline{{P}_{1}{P}_{2}}\right$ indicates the distance of the two involved points, ${P}_{1}$ and ${P}_{2}$.
The power flux through the common boundary $\overline{AB}$ is continuous, which yields:
Substituting Eqs. (16), (17) to Eq. (18), ${e}_{M}$ can be easily solved as:
then substituting Eq. (19) into Eq. (16), and notice that:
One can eliminate ${u}_{M}$ and the approximation formula of ${q}_{i,j}$ can be written as:
From Eqs. (13) and (21), the power flux through one boundary of the element ${Z}_{i}$ is solved. This procedure is repeated $m$ times and is superposed. Then, the superposition of all the power flux out of the element ${Z}_{i}$ can be worked out, which indicates that the first term of Eq. (10) is finally solved.
3.2. Calculation of ${\mathit{Q}}_{\mathit{i},\mathit{j}}$ in 3D domain
As shown in Fig. 2, ${Z}_{i}$ and ${Z}_{j}$ denote the centroids of two adjacent tetrahedral grids in 3D domain, whose common boundary is a triangle denoted by ${\u2206}_{BCD}$, and $E$ is the centroid of ${\u2206}_{BCD}$.
First, two functions should be introduced:
${V}_{abcd}$ represents the volume of a tetrahedron whose vertices are $a$, $b$, $c$ and $d$. An area vector, ${\mathbf{S}}_{a,b,c}$, is in the form of:
The use of commas emphasizes that the subscripts $a$, $b$ and $c$ in Eq. (23) are directive, which is different from those in Eq. (22). The direction of vector ${\mathbf{S}}_{a,b,c}$ is determined by right hand rule, starting from $a$ to $b$ and then to $c$.
The governing equation of acoustic domains has the same form as that of plates (listed in Table 1), but the power flow flux from ${Z}_{i}$ to ${Z}_{j}$, ${Q}_{i,j}$, is divided into three parts in 3D solution:
The gradient, $\nabla e\text{,}$ in each part, ${\left(\nabla e\xb7S\right)}_{B,C,E}$ for example, is averaged in a secondary tetrahedron, ${T}_{{Z}_{i}BCE}$:
Otherwise, a similar approximation in ${T}_{{Z}_{j}BCE}$ while calculating ${Q}_{j,i}$ will lead to:
To guarantee the conservativeness of the power flux through the common boundary ${\u2206}_{BCD}$:
Substituting Eqs. (25), (26) into Eq. (27), ${e}_{E}$ can be eliminated, and results in:
where the scalar coefficients can be solved as:
${\varphi}_{B}=\frac{1}{3{V}_{{Z}_{i}BCE}}\left({\mathbf{S}}_{C,{Z}_{i},E}+\frac{{\phi}_{B}}{{\phi}_{E}}{\mathbf{S}}_{C,B,{Z}_{i}}\right)\cdot {\mathbf{S}}_{B,C,E}\mathit{},$
${\varphi}_{C}=\frac{1}{3{V}_{{Z}_{i}BCE}}\left({\mathbf{S}}_{B,E,{Z}_{i}}+\frac{{\phi}_{C}}{{\phi}_{E}}{\mathbf{S}}_{C,B,{Z}_{i}}\right)\cdot {\mathbf{S}}_{B,C,E}\mathit{},$
and:
${\phi}_{B}=\frac{{\mathbf{S}}_{C,E,{Z}_{j}}\cdot {\mathbf{S}}_{B,C,E}}{{V}_{{Z}_{j}BCE}}\frac{{\mathbf{S}}_{C,{Z}_{i},E}\cdot {\mathbf{S}}_{B,C,E}}{{V}_{{Z}_{i}BCE}},{\phi}_{C}=\frac{{\mathbf{S}}_{B,{Z}_{j},E}\cdot {\mathbf{S}}_{B,C,E}}{{V}_{{Z}_{j}BCE}}\frac{{\mathbf{S}}_{B,E,{Z}_{i}}\cdot {\mathbf{S}}_{B,C,E}}{{V}_{{Z}_{i}BCE}}.$
Similar to Eq. (28), the other two parts of power flux, ${\left(\nabla e\xb7S\right)}_{C,D,E}$ and ${\left(\nabla e\xb7S\right)}_{D,B,E}$, can be worked out. Once they are all determined, submit them into Eq. (24), then ${Q}_{i,j}$ in 3D domain can be obtained.
Fig. 2Two adjacent control volumes in 3D domain
Fig. 3Interpolation diagram for computing vertex weighted average e0 from surrounding cellcentered ei
3.3. Node interpolation
When calculating ${Q}_{i,j}$ in both 2D and 3D domains, the variable at the vertex, ${e}_{0}$, is unknown, as shown in Fig. 3; it is evaluated by a weighted average of the surrounding cellcentered values, ${e}_{1}$∼${e}_{N}$, as [24]:
in which the weight functions ${W}_{i}$ are selected to ensure zero Laplacian of the linear function in the shadow region (Fig. 3). $N$ is the number of the surrounding cells.
Aiming at this, Lagrangian multipliers, ${\lambda}_{x}$, ${\lambda}_{y}$ and ${\lambda}_{z}$, are employed, and the weight function is expressed as:
where $\left({x}_{i},{y}_{i},{z}_{i}\right)$ are the spatial locations of the centroids of the cells, and $\left({x}_{0},{y}_{0},{z}_{0}\right)$ is that of the vertex. The Lagrangian multipliers are computed by the local geometric parameters as:
where:
The interpolation coefficients are computed as:
${I}_{xx}=\sum _{i=1}^{N}{\left({x}_{i}{x}_{0}\right)}^{2},{I}_{yy}=\sum _{i=1}^{N}{\left({y}_{i}{y}_{0}\right)}^{2},{I}_{zz}=\sum _{i=1}^{N}{\left({z}_{i}{z}_{0}\right)}^{2},$
${I}_{xy}=\sum _{i=1}^{N}\left({x}_{i}{x}_{0}\right)\left({y}_{i}{y}_{0}\right),{I}_{xz}=\sum _{i=1}^{N}\left({x}_{i}{x}_{0}\right)\left({z}_{i}{z}_{0}\right),{I}_{yz}=\sum _{i=1}^{N}\left({y}_{i}{y}_{0}\right)\left({z}_{i}{z}_{0}\right).$
3.4. Assembling the global equation
First, ${Q}_{i,j}$ is initially computed for each common boundary in 2D and 3D domains, and if the boundary is on any joint, ${Q}_{i,j}$ is computed using Eq. (2) instead. Then, the results are summarized into Eq. (10) and all the element stiffness equations are worked out. Lastly, the corresponding coefficients of each control volume are gathered, and the global equation can be written in a general form as:
where ${N}_{\mathrm{d}\mathrm{o}\mathrm{f}}=3{N}_{P}+{N}_{A}$, ${N}_{P}$ and ${N}_{A}$ are the numbers of control volumes in plates and air cavities, respectively. The notations $B$, $L$, $S$, and $A$, represent the bending, longitudinal, shear, and acoustic waves, respectively. The diagonal matrices BB, LL, and so on, are assembled from ${Q}_{i,j}$ of each wave domain, and the other matrices are from the power transfer of the junctions. $\mathit{e}$ is the energy density vector, and $\mathbf{\Pi}$ is the vector of corresponding power injected into every element.
4. Validation
In this section, two numerical examples are developed to confirm the validity of the proposed uEFEM^{0}. The accuracy of the result obtained from uEFEM^{0} is verified by establishing corresponding models by AutoSEA [25], a commercial software of SEA. The result obtained from uEFEM^{0} is the distribution of energy densities and that of SEA is the lumped energy of each subsystem. For comparison, the results of uEFEM^{0} are integrated into the lumped energy of each subsystem defined by the SEA model as:
4.1. Sevenplate engine foundation model
Several researchers have investigated the 1:2.5 scale model of the diesel propulsion engine foundation and the hull structure of a frigate ship to study the solution of highfrequency structural vibration based on energy flow approach. Lyon [26] divided this model into 7 and 12 substructures and studied the significance of the inplane energy in energy transmission by the SEA method. Vlahopoulos et al. [3] applied EFEM on this sevenplate structure and obtained consistent results with those of Lyon.
Fig. 4Sevenplate engine foundation model
In this paper, the material properties and the dimensions of plates of the sevenplate model are the same as those in Ref. [3]. The uEFEM^{0} model comprises 660 nodes, 1170 triangular control volumes, and 27 PP joints, as shown in Fig. 4.
The analysis is carried out in thirteen 1/3 octaves that cover from 1.25 kHz to 20 kHz, as listed in Table 2. In each octave band, a unit bending power of 1 W is injected at the center of plate 1, whereas it is applied on the top plate in the corresponding SEA model. On all the boundaries of the system, the boundary condition is set as $\mathbf{I}=\left({c}_{g}^{2}/\eta \omega \right)\nabla e\cdot \mathbf{n}=0$.
Table 2Computing frequency range, thirteen 1/3 octaves
Octave ID.  1  2  3  4  5  6  7  8  9  10  11  12  13  
Frequency (kHz)  Center  1.25  1.6  2  2.5  3.15  4  5  6.3  8  10  12.5  16  20 
Lower limit  1.12  1.41  1.78  2.24  2.82  3.55  4.47  5.62  7.08  8.91  11.2  14.1  17.8  
Upper limit  1.41  1.78  2.24  2.82  3.55  4.47  5.62  7.08  8.91  11.2  14.1  17.8  22.4 
Fig. 5 shows the comparison of the total bending energy of plates 1, 4, and 7. The bending energy ratios between plates 7 to 1 and plates 2 to 1 are presented in Figs. 7(a) and 7(b), the results with the legend of ‘EFEA’ were developed by Vlahopoulos [3], and the experimental results were provided by Lyon [26].
It can be demonstrated that:
1) The bending energy results of the incident plate (Plate 1) fit well, whereas those of other plates have slight difference, which is less than 3 dB. This finding is attributed to the plates except the incident plate; the energy is induced by the joints, and different treatments are applied with the joints in uEFEM^{0} and SEA.
2) As observed in Figs. 7(a) and 7(b), the developed uEFEM^{0} shows the best overall correlation to the experimental data.
3) When only the bending mechanism is included in the formulation, power can be transferred from plates 1 to plate 7 only from the bending motion, while in fact an important amount of power is transferred from the inplane motion of plates 2 and 5. Thus, the SEA result with only bending wave included shows a significant error on the energy ratio between plate 7 to plate 1.
4) The uEFEM^{0} formulation is based on a FEM mesh of the structure. Therefore, the spatial distribution of the energy density can be identified easily (Fig. 6).
Fig. 5Bending energy results in plates 1, 4 and 7 (results from uEFEM0, SEA/AutoSEA)
Fig. 6Distribution of bending energy density on the sevenplate model at 1.25 kHz (result from uEFEM0, and the unit: dB, ref = 1×1012 J/m2)
Fig. 7Bending energy ratio between plates (results from uEFEM0, SEA/AutoSEA, and literature)
a) Plates 2 to 1
b) Plates 7 to 1
4.2. Simplified passenger vehicle model
A passenger vehicle is simplified to seven structural plates with an internal air cavity. The plates are made of aluminum, with a uniform thickness $h=$5 mm, and the internal cavity is filled with air. The dimensions and mesh of the vehicle are both shown in Fig. 8. The model comprises 468 structural plate elements and 1220 acoustic air elements, with 79 PP joints and 468 PA joints. The injected powers, 0.25 W each in any octave, are applied at about the positions of four wheels, which are (0.6, 0.3), (0.6, 1.2), (2.5, 0.3), and (2.5, 1.2) in the coordinate system $XOY$ set on the base plate. In the SEA model, a unit power of 1 W is applied at the base plate. The side plates are examples of plates with irregular shape, which would be difficult to deal with EFEM^{0} developed by Wang [5], but easy with this presented uEFEM^{0}.
The material properties are listed in Table 3, and the analysis frequency range is the same as the prior numerical example, which is listed in Table 2.
Table 3Material properties of the vehicle model
Aluminum (plates)  Air (cavity)  
Elastic modulus $E$ (GPa)  Density ${\rho}_{s}$ (kg/m^{3})  Poisson ratio $\nu $  DLF* ${\eta}_{s}$  Density ${\rho}_{0}$ (kg/m^{3})  Sound speed ${c}_{0}$ (m/s^{2})  DLF* ${\eta}_{0}$ 
70  2700  0.33  0.06  1.23  340  0.005 
*DLF: Damping loss factor. The DLFs of all the three wave domains in plates are set the same as ${\eta}_{s}$. 
Fig. 8uEFEM0 model of simplified vehicle
Figs. 9(a) and 9(b) show the comparison of results from uEFEM^{0} and AutoSEA, including the bending energy of plates and sound pressure level (SPL) of the cavity. The spatial variation of the energy density is presented in Fig. 10. From the analysis of the simplified vehicle, the following conclusions can be summarized:
1) Similar to the behaviors of uEFEM^{0} and SEA analysis in the prior numerical example, the total bending energy of the incident plate shows a good agreement, whereas the results in other plates are slightly deviated. The result of SPL fits well for these two methods.
2) As observed in Fig. 10, the bending energy level is highest at the loading points and decreases as the distance of the loading points increases. The inplane energy takes its peak in the side plates, which have the same direction with the loading forces. This phenomenon explains the importance of the inplane energy in the energy transmission in such structures including ‘L’ angles. The SPL shows a significant value at the corners between the base and side plates. This result is due to a number of plate elements with relatively high bending energy that radiate energy into the air cavity at those corners.
3) The simplified passenger vehicle application demonstrates how the uEFEM^{0} formulation can be utilized as a simulation tool for structuralacoustic car applications, thus it may potentially be further developed to be an NVH tool.
Fig. 9Comparison of bending energy in several plates and sound pressure level in air cavity
a) Bending energy
b) Sound pressure level
5. Conclusions
An unstructured zeroorder EFEM formulation is developed for structural–acoustic problem, and both bending and inplane waves are considered in the plate structure. The derived formulation is applied to predict the energy distributions of a sevenplate structure and a simplified vehicle model with an internal air cavity. The formulation is validated by comparing the results obtained from conventional simulation methods and experiments. The bending wave energy decreases as the distance of the location increases from the exciting point. The inplane wave energy has significant values in the plates, which have the same direction with that of the input power, and the highest SPL most likely exists at the corners. Omitting the inplane wave motion results in a significant underestimation in the structures far from the loading points.
Compared with SEA, the developed uEFEM^{0} formulation takes the advantage in determining the energy distribution. Moreover, uEFEM^{0} guarantees the conservativeness of energy in any elemental control volume and addresses the difficulty of traditional EFEM^{0} in dealing with irregular shapes. This work is the first to solve a structuralacoustic problem using a zeroorder EFEM.
Fig. 10Cloud charts of energy density and sound pressure level, at f= 6.3 kHz (The unit of energy density is dB, ref = 1×1012 J/m2, and that of SPL is dB, ref = 2×105 Pa)
a) Bending energy density
b) Longitudinal energy density
c) Shear energy density
d) Sound pressure level
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