# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 1 Introduction  
The boundary-element method (BEM) can be used to analyze and solve dynamic acoustic problems both in the frequency domain (FD) and in the time domain (TD). Both the BEM-FD and the BEM-TD suffer from numerical instabilities related to the (numerical) resolution of both the Kirchhoff-Helmholtz integral boundary equation and its normal derivative [1]. Investigations into the cause of such instabilities can be found in [2, 3]. In general, it can be understood as an oscillation generated by the numerical solution procedure, where the coupling between the incident field and the internal resonance modes does not entirely vanish [4]. In this case (i.e., by understanding it as an eigenvalue boundary problem), fictitious eigen-frequencies arise due to the problem's resonant frequencies, resulting in spurious oscillations that generate unreliable results. Many publications have addressed the described problem, and some recent works mention different alternatives to circumvent the problem, for example the recent review paper by [PERSON] [5] or the overview presented by [PERSON] [6], which identify the relevant problem of irregular frequencies in external BEM models and refer to the main alternatives to address and solve that issue. One of the first approaches to deal with this problem in the frequency domain was presented by [PERSON] [5], who proposed the combined Helmholtz integral equation formulation (CHIEF), consisting of using the Helmholtz integral equation in the boundary combined with additional compatible equations, formulated from the integral equation forsome internal points, external to the infinite domain and conveniently located to improve the conditioning of the final matrix. A popular alternative strategy is the [PERSON] and Miller approach, initially formulated in [7], which combines the classical BEM integral equation and its normal derivative and, when coupled together by a specific coefficient, allows the definition of stable solutions at the fictitious frequencies. Although this is a well-accepted method, there are still discussions regarding its application, such as in [6] and [8]. Some authors [6] even mention that half of the references on the method are incorrect due to an erroneous coupling coefficient. In other works [9], complementary strategies have been devised to improve the behavior of the method. Alternative works can be found in the literature addressing the stability of the BEM for external problems using other techniques. [PERSON] and [PERSON] [10], for instance, formulated and analyzed a modified boundary-element method for Dirichlet conditions, showing that it was stable for all wave numbers, thus avoiding the unstable behavior at irregular frequencies. In contrast to other existing approaches, the proposed method relies on the use of boundary integral operators which are already available in standard boundary-element methods. Therefore, the additional effort in the implementation is neglectable. [PERSON] et al. [11] addressed the problem using the so-called Closed Virtual Impedance Surface Method (CVIS), and stated that the technique allowed the virtual elimination of the effects of these fictitious frequencies. [PERSON] et al. [12] proposed the use of a modified Green's function and presented results showing that it can indeed remove certain fictitious frequencies. However, the authors indicated that their results were an initial proof of concept for the method, requiring further developments. In spite of existing alternatives, due to its simplicity, the CHIEF method is still one of the most widely used techniques to circumvent non-uniqueness problems at irregular frequencies of BEM models and also has the potential to be applied to other techniques. Indeed, recent works confirm its use and effectiveness. [PERSON] et al. [13] used CHIEF points to stabilize a BEM formulation with Krylov subspace recycling to improve efficiency, indicating a rule of thumb for using a number of CHIEF points which is about 5% of the total number of nodes.