Investigation on behaviors of acoustoelastic cavities using a novel reduced finite element–dual reciprocity boundary element formulation

This paper presents the acoustic and structure characteristics of chambers with elastic/rigid baffles and elastic walls using a novel formulation. The acoustic fluid is assumed to be compressible and inviscid, and the effect of mean flow is neglected. The acoustic field is formulated based on the dual reciprocity boundary element method (DRBEM), and the finite element method (FEM) is used to model the structural dynamics of elastic walls. The reduced-order model (ROM) of the coupled system is derived based on modal analysis of the acoustic field with rigid boundaries and the structure in a vacuum. Numerical examples are presented, and the acoustic modes and frequencies of the chambers are determined by analysis of eigenvalues of the coupled governing equations. The accuracy of the introduced model, which is based on the reduced-order DRBEM-FEM formulation, is examined by comparing the results with analytical and numerical predictions reported in the literature, and good agreements are achieved. Besides, the effects of baffles are studied for different configurations and the elasticity of a wall in a combustion chamber is investigated.


Introduction
The interaction between acoustic waves and solid structures is a field of interest in fluid-structure interaction (FSI) dynamics (L. Chen et al., 2020;Ghalandari, Ziamolki, et al., 2019;Zeinalzadeh & Pakatchian, 2021). These fields of FSI problems have been challenging areas in many engineering applications, such as high-frequency instability and humming of combustion chambers in fluid fuel engines (Anderson & Yang, 1995). Reviews of numerical solutions in low-frequency structure-acoustic problems, with an emphasis on the most popular state-of-the-art methods, have been presented in several research studies (Atalla & Bernhard, 1994;Li et al., 2021;Shu et al., 2012;D. Zhao & Li, 2015). Acoustic resonance in the chamber domain is the main cause of instabilities. Taciturnity of flame, blowout (Hussin et al., 2021), humming and hydrodynamic acoustic forces exerted on the walls of the chambers, which may damage the entire structures of combustion, are considered to account for 90% of destructive results of the CONTACT Yedan Shen yedanshentrina@sina.com; Mohammad Ghalandari Ghalandary.13518@gmail.com; Shahab S. Band shamshirbands@yuntech.edu.tw acoustic problem (Ar'kov et al., 1972;Iurashev et al., 2017;Sui et al., 2021). Radial baffles, which are used at the interfaces of the injector, are considered as common solution techniques for controlling acoustic resonance, especially in modified premixed combustion chambers (Hambric et al., 2016;Han et al., 2021). Dissipation of acoustic energy and increasing damping effects by 30% can decrease the intensity of acoustic resonance in combustion chambers (Baer & Mitchell, 1977) by about 40%. The characteristic effects of these components on the nozzle of a combustion chamber were studied by Farshchi et al. (2009). They showed that the converged nozzle had a great influence on longitudinal and tangential modes. Wieber (1966) experimentally studied the effects of a redial baffle on a cold acoustic combustion chamber. Later, Webster et al. (2017) experimentally conducted both cold and hot firing tests and reported the consequences of the configuration of a redial baffle on damping under both tangential and longitudinal modes.
Investigations on the dynamic properties of such applied problems by the valuable numerical method have become prevalent in recent years. The finite difference, finite volume, finite element (FEM) and boundary element (BEM) methods  have been commonly utilized by many researchers. Among these methods, the BEM, by representing its inherent mesh features, is efficient and convenient for acoustic problems. Indeed, meshing the boundary instead of the entire domain is the principal advantage of BEM, as it decreases the size of equations and subsequently saves computational cost and time (Firouz-Abadi et al., 2008). Several BEMs to investigate the analysis of eigenvalues in Helmholtz problems were presented by Kamiya et al. (1993) and Ali et al. (1995). The multiple reciprocity method (MRM) and dual reciprocity method (DRM) (Fahmy, 2018;Yu et al., 2021) have provided an interesting application of BEMs, especially in the simulation of Helmholtz equations. Kontoni et al. (1991), Sladek et al. (1993) and W. Zhao et al. (2019) used these techniques in the eigenvalue analysis of acoustic problems in two-and three-dimensional fluid domains. The isogeometric BEM is a new and thriving approach for tackling acoustic problems with the same basic traditional BEM phenomena (L. L. Chen et al., 2019;Dölz et al., 2018;Simpson et al., 2014).
The FEM model is another useful robust method which was used for acoustic applications in the late 1960s by Gladwell and Zimmermann (1966). In contrast to the symmetric and sparse features of FEM matrices, the BEM formulation suffers from fully populated and asymmetric matrices. Indeed, the FEM formulation was derived based on the variational technique for a specific acoustic problem. Later, Craggs (1972) studied the natural properties and mode shape of acoustic inclusions using an FEM model. Kim et al. (2015) assessed the stability of a combustion chamber using the FEM and showed a maximum of 1% error difference between their results and the analytical solution.
To take the advantage of both the FEM and BEM in fluid-structure problems, some researchers (Gaul & Wenzel, 2002;Junge et al., 2011) have been motivated to use a combination of both methods simultaneously. Despite this, there remains a need to reduce both the size of equations and the computational effort, and methods based on reduced-order techniques have received a lot of attention recently (Gonzalez-Flesca et al., 2018). Ohayon (2001) and Ohayon and Soize (2012) proposed several variational formulations for the modal analysis of bounded fluid-structure systems to investigate internal vibration-acoustic and hydroelastic systems. The application of modal synthesis for combined structure-acoustic systems was used by Wolf (1976) to determine the low-frequency free and forced interior acoustic properties of passenger compartments. An investigation on vibrational behavior of the acoustic-structure problem using an isogeometric BEM was conducted using the directed coupling method and symmetric-iterative coupling method (Wu et al., 2020). This approach was shown to have low accuracy near the natural frequency of the structure.
Although there are some limitations in the modeling and solution of fluid by the FEM and structures using the BEM, a novel efficient reduced-order FEM-BEM model for the treatment of the acoustic instability of chambers using elastic baffles and walls is presented. To derive the governing equation of the acoustic medium, the fluid is assumed to be homogeneous, isotropic, inviscid and compressible, and the effect of mean flow is neglected. Using the dual reciprocity boundary element method (DRBEM), the equation of the acoustic domain is introduced to the elastic structure. The reduced-order model of elastic structure is obtained via modal analysis using the FEM technique. The reduction of computational cost by the BEM and the robustness of the FEM approach are the main advantages of this formulation model, which is a novel technique for the solution of real problems with the lowest computational cost for this type of problem (Gonzalez-Flesca et al., 2018;W. Zhao et al., 2019). Indeed, a challenging part of this study that arises from the BEM in the solution of rigid and elastic baffled structures is solved by the zoning method, which enables the BEM to approximate the acoustic characteristics of cavities with baffles. The minimum differences between the results and those from the literature prove the capability of the proposed model to solve interactive coupled problems in different forms of cavities.

Structure domain
In this paper, the FEM is used to model the structural dynamics of the system. The flexible structure is modeled by beam or shell elements. Neglecting damping, the FEM formulation of structural dynamics can be expressed as: where M s and K s represent the mass and stiffness matrices, respectively. d is the vector of nodal displacements and f denotes the external nodal force vector. Reduced-order models (ROMs) of the structure domain can be constructed using the modal analysis technique. The motion of the structure can be described by an expansion of the structural displacement as a reduced set of basis vectors, expressed as where N s is the number of modes, is a matrix with mass-normalized mode shapes in the columns and ξ is a vector containing the corresponding modal coordinates. By substituting Equation (2) into Equation (1), pre-multiplying by T and using the orthogonality of mode shapes, the governing equations are reduced to a set of uncoupled equations, as where I is an identity matrix and s is a diagonal matrix whose elements are squares of natural frequencies. A few modes can be used to construct the ROM of the structure by suppressing higher frequency modes. Thus, the number of mode shapes used to describe the vibration of the structure is much smaller than the degrees of freedom of the complete system. Using the ROM technique, the size of the equations decreases considerably, and computational cost and time are significantly improved (Gonzalez-Flesca et al., 2018). Vector R denotes generalized forces equivalent to work done by external forces at each mode shape. In the absence of body and other external forces, the pressure of acoustic waves (p) is the prominent source of excitation. Thus, the resulting generalized forces can be expressed as where ψ n i is the normal displacement of the ith mode shape and can be computed by projecting mode shapes along the normal direction outward from the surface (S).

Acoustic domain
In this section, the BEM formulation of the acoustic domain is presented and then the ROM of the equations is introduced based on the modal analysis technique.
To derive the governing equations of the acoustic field, the fluid is assumed to be homogeneous, isotropic and compressible. In addition, the fluid is considered to be irrotational and inviscid. Based on these considerations and assuming infinitesimal fluctuation of pressure, temperature, velocity and density distributions, the continuity and momentum equations in combination with isentropic relationships are reduced to the well-known linear acoustic wave equation. The isentropic equation relating sound pressure and density perturbation is a constitutive equation in the acoustic field. Thus, the linear governing equation of acoustic waves in terms of the velocity potential field ( ) can be expressed as: where ∇ 2 is the Laplace operator and c is the speed of sound in the media. In general, there are three types of boundary conditions in the acoustic governing equation and these can be classified as follows.
The velocity potential or pressure: The velocity: The impedance: where µ is the absorbent coefficient of the surface, q is the normal derivative of the velocity potential, which is the normal velocity, and n is the outward normal direction on the boundary. In this paper, the application of the second type of boundary condition, known as the Neumann boundary condition, is involved.

Boundary element formulation
Apart from the acoustic wave equation (Equation 5), there exist many other engineering applications in which the governing equations contain Laplacian and other terms: The term b can be a function of coordinates, the primary variable ( ) and its time or spatial derivatives.
Using the fundamental solution and by implementation of Green's second identity, one can convert the boundary-value problem (Equation 9), and the corresponding boundary conditions (Equations 6-8) into the boundary integral equations. The basic solution * for potential problems must satisfy the following Poisson equation: where δ(R − R p ) denotes the Diarc delta function, which represents a unit source at the point with the coordinate of R p = x pî + y pĵ + z pk , and * explains the response at the field point R = xî + yĵ + zk due to that source. The basic solution * for three-dimensional problems and its normal derivative are given by: * = 1 4πr (11) where n is the outward normal vector on the boundary and r is the vector of the distance between the source point and field point, and its absolute value is denoted by r. Applying Green's second identity to the governing equation (Equation 9), the conventional form of the boundary integral formulation is derived as in which χ is a coefficient denoting the internal angle of the source point in radians. If the source point is located on a smooth or flat boundary, c p is equal to 0.5, and for source points into the field regions, it is equal to 1. It can be seen that the boundary element formulation of the Poisson equation leads to the conventional form of the boundary integral equation (Equation 13), which consists of two integral parts: one part is the boundary integral and the other is the domain integral part. Such domain integral terms have been a major difficulty in the development of the BEM, which weakens the main advantage of this method. As mentioned in Section 1, dealing with boundary integrals instead of domain integrals and meshing the boundary instead of the internal domain are the main benefits of the BEM compared to other numerical methods such as the FEM. The need to handle domain integrals has motivated many researchers to find solution methods which are general, simple and easy to use. Many solution methods, such as the internal cell method (ICM), particular solution method (PSM), MRM and, more recently, the DRM have been proposed as the result of considerable research efforts. The following subsection explains the outstanding performance of the DRM in addressing the solution of domain integrals as well as boundary integrals.

Dual reciprocity method (DRM)
The DRM is a treatment for domain integrals which was first introduced for solving acoustic-structure problems. It is an innovative approach to transforming domain integrals into boundary integrals. Based on the PSM, the solution of Equation (13) is the summation of a homogeneous solution with a particular solutionˆ , which satisfies the following equation: In general, it is difficult to find a particular solution for any type of Poisson equation, especially when the term b is nonlinear and time dependent. However, it is possible to approximate b as a set of basic functions which have particular solutions as: where α j denotes undetermined coefficients and f j is known as the approximating functions, which are geometry dependent . N B and N I are the number of boundary and internal points, respectively. The relationship between the approximating functions and their corresponding particular solutions is as follows: Using Equations (15)-(17) and substituting the results into Equation (13) yields: By applying Green's second identity for the term on the right-hand side of Equation (18), one obtains: whereq j is analogous to q = ∂ ∂n and n is the outward normal direction on the boundary. It is observed that there is no domain integral in Equation (19). This boundary integral equation can be solved by discretizing the boundary into small elements. The shape of these elements can be triangular or quadrilateral. By integrating over each boundary element and adding the contribution of all elements, Equation (19) can be written as: where k is the number of boundary elements. The variation of the unknown variable and q over the elements can be approximated in terms of their nodal values by introducing appropriate interpolation functions as: where k and q k are vectors consisting of nodal values of acoustic potential and normal velocity, respectively, and N k is a matrix of element interpolation functions. It should be noted that for simplicity and to increase the efficiency of the methods,ˆ andq, which are known functions, can also be approximate over boundary elements, as well as unknown variables. Inserting Equations (21) and (22) into Equation (20), the boundary integral equation can be expressed as: in which By representing the above integrals in the local coordinate systems of elements, they can be evaluated numerically by the Gauss quadrature method. Employing Equation (23) for all boundary and internal points by using point collocation techniques, a set of linear algebraic equations is obtained: The terms c i are incorporated into the diagonal terms of the matrix H. The unknown coefficient vector α can be obtained by evaluating the term b at N B + N I points, which may be written in matrix form as: Each column of the matrix F consists of approximate functions f j at N B + N I collocation points. By obtaining α from the previous equation and substituting the result into Equation (26), one derives: where and q are vectors containing nodal values of acoustic potential and normal velocity at the walls, entire boundary and internal collocation points.
In the acoustic wave equation, b is equal to a timedependent inertia term. Finally, by replacing the inertia term in Equation (28), the DRM-based BEM formulation of the acoustic domain with baffles can be written as: where M f and K f are the acoustic mass and stiffness matrices, respectively. They are composed of submatrices corresponding to the walls, boundary and internal points, and have a size of N B + N I .Û s andQ are matrices whose columns are a vector of particular solutions and their normal derivative at N B + N I points, respectively. Since the value ofq at internal points is meaningless, the corresponding value ofq in matrixQ is zero. According to the mass conservation law, q values at the inlet and outlet of the interface are equal. Besides, the potential values at the interface and the entire domain are equal.
One of the most important considerations influencing the accuracy of DRM results is the choice of appropriate approximate functions f . Many types of approximate functions have been proposed, but radial-based functions  are the simplest and most accurate.
The corresponding particular solutions and normal derivatives become: According to the results reported in the literature, choosing the first two terms of f and the resultingˆ andq functions will provide the most accurate alternatives. The accuracy of the DRM results also depends on the number of internal points and, consequently, the number of approximating functions that are used for the expansion of b. Although there exist rare cases in which the solutions are sensitive to the number of internal points, the convergence of results in terms of the number of internal points is accomplished in this paper as well.

Reduced order model
To derive the ROM of the acoustic equation, the velocity potential is expressed in terms of acoustic mode shapes in a closed cavity with rigid walls. The solution of the acoustic equation is assumed to be harmonic and is given as: Substituting Equation (35) into Equation (29) and applying the boundary conditions (q = 0) yields standard eigenvalue problems, which can be used to determine acoustic natural frequencies and the corresponding mode shapes. The right and left eigenvalue problems are as follows: whereλ i = ω 2 i is the ith eigenvalue, and¯ r i and¯ l i are, respectively, the ith right and left eigenvectors, which are bi-orthogonal with respect to the mass and stiffness matrices. By considering all eigenvectors, one can obtain: where f is a matrix with the square of natural frequencies in the diagonal. Matrices X and Y contain columns of the right and left eigenvectors, respectively. The solution of Equation (29) is assumed to be given by a linear combination of the right eigenvectors as: where N f is the number of acoustic mode shapes and η i denotes the generalized modal coordinates. By substituting Equation (39) into Equation (29), pre-multiplying with Y T and using bi-orthogonality relationships, the equations are reduced to:

Coupled domain
The reduced model of the coupled system can be obtained based on the ROM of each structure or fluid subdomain. This aim is accomplished by solving the structural problem in vacuo (in the absence of acoustic fluid) and the acoustic subdomain with rigid boundaries.
In the structure subdomain, generalized structural forces due to acoustic pressure can be computed with other boundary integrals over the entire surface. Therefore, by approximating the pressure p = N k p k within each boundary element and performing the integration over the boundary mesh, one obtains: in which By substituting Equation (41) into Equation (3) and using the linearized unsteady Bernoulli's equation (p = −ρΦ), the equation for the structure can be expressed as: Substituting the acoustic potential expansion (Equation 39) into this equation yields: In the interaction of acoustic fluid and structure at the boundary, fluid particles and structure move together, which is known as the continuity of displacement field at the interface. Using the continuity condition, the acoustic fluid velocity vector in the normal direction (q) can be computed by taking the time derivative of the structure normal displacement (d n ), namely: By substituting Equation (45) into the acoustic equation (Equation 40), one obtains: where T is a matrix whose columns are structure mode shapes projected in the normal direction. Finally, by combining Equations (44) and (46), the ROM of the coupled system can be written as: in which:

Results and discussion
Based on the presented formulation (DRBEM-FEM), some test cases (Farshchi et al., 2009;Laudien et al., 1995;Wieber, 1966) are studied in this section and are compared with FEM simulation and analytical results.  Besides, the formulation is examined for cylindrical geometry and for a nozzle combustion chamber filled with air (ρ f = 1.21 kg/m 3 , c = 343 m/s) by determining their acoustic modal parameters. The investigations are followed by rigid baffled cavities for both cylindrical and nozzle geometries. The effects of flexibility of the baffles on the acoustic characteristics of nozzle combustion are also presented. Furthermore, an axisymmetric cavity filled by another fluid (ρ f = 565 kg/m 3 and c = 730 m/s) is adopted to study the effect of elasticity of steel walls on the acoustic domain by the proposed formulation.

Cylindrical combustion chamber
The first test case is a rigid cylindrical chamber ( Figure 1) with a length (L C ) of 53 cm and a diameter (D C ) of 34 cm. The domain is discretized by 20 shell elements in the longitudinal direction, seven shell elements in the radial direction and 40 shell elements in the circumferential direction. A sensitivity analysis of results with respect to the number of internal collocation points chosen in the DRM is established to ensure the accuracy and efficiency of the method. The accuracy of the introduced formulation is evaluated for chambers with closed-end boundary conditions. The results of a mesh study on the first 18 natural frequencies of the cavity for the corresponding mesh, with respect to the variation of the number of internal points to boundary points (N I /N B ), are presented in Table 1.
Based on the mesh sensitivity analysis of a cylinder in the literature (Farshchi et al., 2009;Laudien et al., 1995;Wieber, 1966), the accuracy of the numerical results is enhanced with the increase in the number of internal points with uniform distribution, and is acceptable for A comparison of the results for the first 18 modes with the FEM and the literature (Farshchi et al., 2009; Laudien et al., 1995;Wieber, 1966) is presented in Table 2. It should be noted that there is a slight difference, of less than 1.5%, between the present model, the FEM and the literature for the corresponding modes, and 3% L 2 error norm percentage, which confirms the current formulation and the code developed based on it.
The results of a cylindrical chamber with an open-end condition are given in Table 3 and are compared with those of previous studies. The computations show a maximum of 1% difference for the corresponding modes and overall 1.5 L 2 error norm percentages between FEM and BEM in open cylinders. Besides, it can be evaluated that there is a maximum variation of 50% between open and closed cylinders, which can be detected for the first mode. An overall L 2 error norm of 59.61% between open and closed cylinders is computed.
The corresponding acoustic mode shapes of open and closed chambers are shown in Figure 2 for the first six frequencies. In the mode shape figures, red and blue colors show qualitatively the maximum and minimum displacements, respectively. Variations in longitudinal modes are more significant than those in other modes, and these

Cylindrical chamber with radial baffles
A typical configuration of a cylindrical chamber with two baffles is shown in Figure 3.
To study the acoustic behavior of chambers with baffles, a second test case is considered as a cylindrical chamber with different radial baffle configurations. The frequencies of the acoustics with three, four and five baffles mounted radially at the inlet with 660 shell elements and 1320 nodes are given in Table 4. Letters T and L refer to modes in the tangential and longitudinal directions, respectively, and their index refers to the mode number. The baffles have a length of 30 mm. Results are extracted for an L/D ratio of 0.233 and show a 7.7-10.7% reduction in the first frequency for arrangements with four to six baffles, respectively.
Based on a mesh sensitivity analysis of a cylinder cavity with radial baffles and the literature (Farshchi et al., 2009;Laudien et al., 1995;Wieber, 1966), the accuracy of the numerical results is acceptable for N I N B = 0.22. The slight difference between the current results and those in the literature can be inferred from the thickness of the baffles. The baffles significantly affect the circumferential and longitudinal-circumferential modes. It can be also concluded that the frequencies corresponding to L1T1 mode tend to disappear with an increasing number of baffles.
In addition, the influence of the baffle length is more prominent than the number of baffles, especially on the first circumferential mode. The computations for a chamber with five baffles, with an L/D ratio of approximately 0.333, show a 24% reduction in the frequency for T1 mode, whereas the reduction for the corresponding frequency reported in Laudien et al. (1995) is about 23%. The effects of baffle angle in comparison to FEM soft- ware on the non-dimensional frequencies are indicated in Figures 4-6. It can be observed that the frequency differences between the two circumferential modes are more tangible at 180°. The length of the baffle is 53 cm, c is the speed of sound and . Modes T1, L1T1 and L2T1 at both 180°and 360°configurations have reductions of about 9.7%, 3.7% and 5.3%, respectively. The mode shapes of baffled and unbaffled cylinders are illustrated in Figure 7.

Combustion chamber
The third case is a combustion chamber with a converging nozzle. The chamber consists of two parts, namely a cylindrical section and a converging section. The length (L c ) of the cylinder section is 11 cm and the diameter (D c ) is 34 cm. The nozzle section has a length (L n ) of 42 cm and a 12 cm outlet diameter (D n ). The grid of the domain is constructed by shell elements (20 elements in the longitudinal direction, seven elements in the radial direction and 40 elements in the circumferential direction) (Figure 8). The boundary conditions considered for the walls and opening are the same as those in the previous test case. Employing Newman boundary conditions at the inlet and outlet indicates differences in the pressure contours at the inlet and converged areas. Indeed, mode shapes of the nozzle depict a concentration in the pressure gradient at the throat area (Figure 9). A mesh study of the nozzle cavity is conducted for the first six frequencies  using results from the literature (Farshchi et al., 2009;Laudien et al., 1995;Wieber, 1966) and the FEM, and the accuracy of numerical results is obtained for N I N B = 0.21. An evaluation of natural frequencies for the closed-end nozzle shows negligible differences between the current model and the literature (Table 5). Moreover, the frequency of a nozzle chamber has a higher value than that of a cylindrical one. This can be inferred from the effective length and volume reduction in nozzle topology.
The first four natural frequencies for the closed-end combustion chamber reveal a significant decrease in longitudinal mode compared to those for the open-end chamber (Table 6).
Maximum variations of 73% for the corresponding modes, an overall error norm (L 2 ) of 83% between the open and closed chambers in the longitudinal direction, and near equality of both frequencies in the circumferential mode are found.

Elastic converging nozzle with elastic baffle
The effects of nozzle flexibility in the combustion chamber on acoustic frequencies are investigated in this example. The chamber is considered as a cantilever body with aluminum of 2024 material properties (ρ s = 2700 kg/m 3 , υ = 0.3 and E = 71 GPa) at environmental temperature. Modal characteristics of the coupled system are evaluated using an efficient DRBEM/FEM based on the ROM formulation with 5700 shell elements and 22,400 nodes. The first 10 natural frequencies of the coupled domain are extracted using the lowest four mode shapes of individual acoustic and structure subdomains. Table 7 presents the coupled acoustoelastic frequencies for both 2 and 5 mm thicknesses. Frequencies of the structure at lower wall stiffness have a greater effect on the coupled domain. As a result of the coupling effect, structure frequencies decrease by 0.3% and acoustic frequencies increase by 0.4% for every mode. Besides, based on the results of the baffled cavity, the elasticity of the baffles highlights their coupling effects in circumferential and longitudinal-circumferential modes. Figure 10 illustrates the effect of the flexibility parameter ( ω a c L) versus the non-dimensional thickness ( t D ) for acoustic dominant frequencies, where t is the thickness, c is the speed of sound, L is the length and D is the diameter of the cavity. It can be deduced that there is a coupling between acoustics and structure in high structural flexibility. In low non-dimensional thickness of structures, the results  show that the first mode becomes stiff owing to coupling effects. The results also demonstrate a remarkable coupling system in L1T1 with the second mode of structures. Decreases in the structural and acoustic frequencies of 2.7% and 2.6%, respectively, are approximated.

Elastic axisymmetric tank
In this example, acoustic-structure characteristics of a symmetric tank are studied. Tanks are used extensively in pressure vessel industries. As can be seen, an axisymmetric boundary condition can be utilized for this case, where a tank is clamped along its central circumference. The geometry and boundary element mesh with 700 shell elements and 2790 nodes are illustrated in Figure 11. Studies with an emphasis on the vibration of a structure in the presence of an acoustic medium are conducted. The tank is constructed of steel alloy (ρ s = 8000 kg/m 3 , υ = 0.3 and E = 210 GPa) with a thickness of 0.05 m and filled with a compressible fluid (ρ f = 565 kg/m 3 and c = 730 m/s).
Following the above mesh study approach, the value N I N B = 0.21 is used for the axisymmetric tank. The first  10 coupled frequencies of the proposed FSI problem are listed in Table 8 and compared with the numerical results obtained from the FEM-based approach. The lowest six modes of the structure without fluid for harmonic index 1 and the first 40 modes of the acoustic cavity are used in the ROM. The results demonstrate close values between the complete FEM-based results and reduced-order formulation results for the corresponding   modes. As can be seen, the structure dominant frequencies decrease owing to the fluid effect. The mode shapes in mode 2 (structural dominant) and in mode 7 (acoustic dominant) are shown in Figure 12. It can be observed that the modes in the coupled system change significantly and are a combination of the corresponding mode shapes before coupling. In addition, because of the fluid, the coupling of the structure and acoustic medium becomes strong and the fluid influences the structure. Therefore, the structure is deflected and the structural mode shapes combine together via coupling, even in acoustic dominant modes. Consequently, it can be concluded that the vibration characteristics and the response of the structure change considerably as a result of fluid-structure interaction.

Elastic submarine with stiffener and bulkhead
The last test case is a submarine with a hull structure (Jin et al., 2018), which consists of a hemispherical truncated canonical-cylindrical steel shell with 8 mm thickness ( Figure 13). The structure is reinforced with eight bulkheads and several ring stiffeners. Evaluations are conducted for steel alloy (ρ s = 7800 kg/m 3 , υ = 0.3 and E = 210 GPa) filled with air (ρ air = 1.29 kg/m 3 , c air = 340 m/s) and submerged in water (ρ water = 1000 kg/m 3 , c water = 1500 m/s).
Here, the number of stiffeners and their effects on the dynamic behavior of the submerged submarine structure are investigated with 6000 shell elements and 23,950 nodes. The first five modes of the structure and the first five acoustic modes with N I N B = 0.2 are applied to utilize the ROM form of the coupled formulation. Comparison of the first five coupled frequencies for zero and 12 stiffeners with those in the literature shows good agreement between the results.
It should be emphasized that there is a slight difference, of less than 1%, between the present model and the literature for the corresponding modes. Furthermore, it can be observed that the rings increase the first three frequencies and decrease for the third and fourth frequencies of the coupled systems. Results on vibrational characteristics reveal that the response of the structure changes considerably owing to fluid-structure interaction in higher modes.

Conclusions
Acoustic-structure interactions in chambers comprising rigid/elastic baffles and elastic walls are investigated with a novel FEM-BEM formulation in this article. The governing equations of the acoustic and structure domains are introduced, based on the BEM and FEM. Afterwards, the reduced-order model of the coupled system with baffled and unbaffled chambers is derived using a modal analysis technique. The proposed formulation is examined by comparing the results with the literature, and the effects of baffles and elasticity of the walls are studied through a comprehensive study.
The results generally confirm the accuracy of the proposed formulation. Comparison of the results with those reported in the literature and the FEM shows that there is good agreement between them. Considering baffles in the chamber construction generally leads to a considerable reduction of acoustic frequencies. The results indicate that an optimum value can be found for acoustic frequencies of a chamber with two radial baffles by changing the angular position of the baffles. It can be inferred that the converging nozzle section of a combustion chamber transforms pure longitudinal modes into the radial direction and pure tangential modes into the longitudinal direction. Furthermore, this study demonstrates that by considering the elasticity of the nozzle section of a combustor, no coupling occurs in the structure and acoustic mode shapes even if the natural frequencies of the structure and the acoustic domains approach each other. Therefore, it can be inferred that the elasticity of a nozzle has a negligible influence on the acoustic characteristics of the combustor. The averaged error of less than 2% proves the capability of the proposed approach in the coupled analysis of acoustic problems. However, the last case study, on an axisymmetric and elastic submarine, shows high coupling between the acoustics and the structure. In this paper, the main limitation is the application of the proposed formulation to linear acoustic-structure cavity problems with rigid and elastic baffles. Nonlinear fluid-structure acoustic problems with complex structures could be developed in the future. The overall CPU time for acoustic mode evaluation based on the FEM computation is about 30% longer than that of the reduced-order BEM-based formulation.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was sponsored by the philosophy and social science planing project of Guangdong province [grant no. gd20ygl12], basic and applied basic project of Guangzhou city [grant no. 202102020629], Philosophy and social science planning project of Guangzhou city [grant no. 2121GZGJ48].