Engineering Analysis with Boundary Elements 104 (2019) 170–182 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Fictitious eigenfrequencies in the BEM for interior acoustic problems Chang-Jun Zheng a,∗ , Chuan-Xing Bi a , Chuanzeng Zhang b , Yong-Bin Zhang a , Hai-Bo Chen c a Institute of Sound and Vibration Research, Hefei University of Technology, Hefei, Anhui, 230009, PR China b Department of Civil Engineering, University of Siegen, Siegen, D-57068, Germany c CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230027, PR China a r t i c l e i n f o Keywords: Boundary element method Fictitious eigenfrequencies Interior acoustic problems Burton–Miller formulation Nonlinear eigenvalue problems a b s t r a c t It is widely known that the boundary element method (BEM) without any special treatment suffers from the ficti- tious eigenfrequency problem for the numerical solutions of exterior acoustic problems. This problem has drawn much attention and been extensively studied over the last several decades. However, this paper is concerned with the existence and influence of the fictitious eigenfrequencies when using the BEM for the numerical solutions of interior acoustic problems. To this end, an eigenvalue analysis technique is developed for the acoustic BEM. The nonlinear eigenvalue problem caused by the acoustic BEM is converted into an ordinary linear one by using a contour integral method. Therefore, the conversion is fulfilled by solving a series of BEM systems of equations without any special or complicated treatment of the governing equations or the linear systems. Three interior acoustic examples including two with simply connected domains and one with a multiply connected domain are used to reveal the existence and influence of the fictitious eigenfrequencies. Furthermore, the Burton–Miller formulation with a variable coupling parameter is found to be able to remove such fictitious eigenfrequencies, and the optimal choice of the coupling parameter is investigated. 1. Introduction The boundary element method (BEM), sometimes also called the boundary integral equation method (BIEM), is one of the major numerical techniques for solving partial differential equations of a variety of physical problems. In this method, the partial differential equation (PDE) is first transformed into a boundary integral equation (BIE) defined over the surfaces enclosing the problem domain. The BIE is then solved by discretizing the surfaces into small patches, or more exactly, boundary elements. In comparison with other numerical techniques based on the domain discretization, e.g., the finite element method (FEM), the BEM has some special features [1,2]. One of the most remarkable features is that it can deal with unbounded domains rigorously without any approximation or truncation of the domains. This unique feature enables us to solve the wave propagation problems [3–10], e.g., acoustic [4–7], elastodynamic [8,9] and electromagnetic problems [10], in unbounded domains with high accuracy. However, some inherent shortcomings also exist. A well-known one is that, although the solution to the original PDE with adequate boundary con- ditions is unique, the transformed BIE fails to yield a unique solution at certain frequencies for exterior wave propagation problems [11,12]. These frequencies correspond to the resonant frequencies of the associ- ∗ Corresponding author. E-mail addresses: cjzheng@hfut.edu.cn (C.-J. Zheng), cxbi@hfut.edu.cn (C.-X. Bi), c.zhang@uni-siegen.de (C. Zhang), ybzhang@hfut.edu.cn (Y.-B. Zhang), hbchen@ustc.edu.cn (H.-B. Chen). ated interior problems and are often called fictitious eigenfrequencies since they have no physical meaning, but just arise from the drawback of the BIE to solving exterior wave propagation problems. The fictitious eigenfrequency phenomenon of the Kirchhof– Helmholtz integral equation when solving exterior acoustic problems has been extensively studied in the past decades, see for example [13–15] for frequency-domain problems and [16] for time-domain problems. Two methods appropriate for practical applications have been widely used to circumvent the fictitious eigenfrequency problem. One is the combined Helmholtz integral equation formulation (CHIEF) [11] which can tackle the problem by adding additional constraints at some internal points. This method is very simple to implement but the determination of appropriate number and positions of the internal points may become troublesome as the frequency increases [15,17]. The other approach to circumvent the fictitious eigenfrequency problem is the Burton–Miller formulation [12] which uses a linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. It has been proved mathematically that such a linear combination pro- vides unique solutions at all frequencies, provided that the imaginary part of the coupling parameter is nonzero [12]. The advantage of this method is that there is no necessity to make the annoying choice of internal points. The main difficulty is that the normal derivative BIE https://doi.org/10.1016/j.enganabound.2019.03.042 Received 6 November 2018; Received in revised form 22 February 2019; Accepted 31 March 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.