Z.S. Chen, G. Hofstetter, H.A. Mang:
"A 3D Boundary Element Method for Determination of Acoustic Eigenfrequencies Considering Admittance Boundary Conditions";
Journal of Computational Acoustics, 1 (1993), 455 - 468.
In order to evaluate the acoustic properties of car compartments alreadyin the design stage, it is improtant to compute the acoustic eigenfrequencies. In contrast to the finite element method (FEM), which has been employed for determining the acoustic eigenvalues of irregularly shaped cavities [1, 2, 3] for about twenty years, application of the boundary element method (BEM) for the solution of acoustic eigenvalue problems is relatively new [7, 8, 9]. Application of the BEM to the numerical computation of acoustic eigenfrequencies is complicated because the frequency parameter is occuring nonlinearity in the corresponding fundamental solution. Hence, the resulting matrices of the eigenvalue problem depend on the frequency parameter, prohibiting the application of one of the well-known numerical algorithms for the extraction of eigenvalues.
This serious drawback of the BEM can be overcome by means of the so-called "Dual Reciprocity Method" (DRM) [4, 5], or the "Particular Integral Method" (PIM) [6, 7]. The latter is characterized by taking the sound pressure as the sum of a complementary solution and a particular solution, respectively. However, till now the application of this method was restricted to Neumann and Dirichlet boundary conditions.
The aim of this paper is to propose a numerical method for extraction of the acoustic eigenvalues on the basis of a boundary element formulation, which is characterized by a unified treatment of Robin, Dirichlet and Neumann boundary conditions. Hence, admittance boundary conditions can be taken into account. The latter is essential to describe the acoustic properties of linings. The proposed method is an extension of the algorithm, originally proposed by Banerjee et al.  and improved by Coyette and Fyfe  and Ali et al. . In the numerical part of the paper the proposed algorithm is applied to the computation of the smallest eigenfrequencies of a large-scale problem.
 M. Petyt, J. Lea an dG.H. Koopmann, "A finite element method for determining the acoustic modes of irregular shaped cavities", J. Sound Vib., 45 (1976) 495-302.
 P.D. Joppa and I.M. Fyee, " A finite element analysis of the impedance properties of irregular shaped cavities with absorptive boundaries", J. Sound Vib., 56 (1978), 61-69.
 D.J. Nefske, J.A. Wolf and L.J. Howell, "Structural-acoustic finite element analysis of the automobile passenger compartment: A review of current practice" J. Sound Vib., 80 (1982), 247-266.
 D. Nardini and C.A. Brebbia, "A new approach to free vibration analysis using boundary elements", in C.A. Brebbia (ED.), Proc. 4th Int. Conf. on BEM, Springer-Verlag Berlin, 1982, 313-326.
 D. Nardini and C.A. Brebbia, "Transient boundary element elastodynamics using the dual resiprocity an dmodal superposition", in C.A. Brebbia (Ed.), Boundary Elements VIII Conference, Springer-Verlag Berlin, 1986, 435-443.
 S. Ahmad and P.K. Banergee, "Free vibration analysis by BEM using particular integrals", J. Eng. Mech., ASCE, 112 (1986), 682-695.
 P.K. Banerjee, S. Ahmad and H.C. Wang, "A new BEM formulation for the acoustic eigenfrequency analysis", Int. J. Num. Meth. Eng., 26 (1988), 1299-1309.
 J.P. Coyette and K.R. Fyfe, "An Improved Formulation for Acoustic Eigenmode Exraction from Boundary Element Models", J. Vibr. Acoust., 112 (1990), 392-397.
 A. Ali, C. Rajakumrand and S.M. Yunus, "On the formulation of the acoustic boundary element eigenvalue problem", Int. J. Num. Meth. Eng., 31 (1991), 1271-1282.