# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 1 Introduction
[PERSON] and [PERSON] [14] used a fast multipole BEM for aero-acoustic problems, using the CHIEF method to avoid irregular frequencies. In their proposal, the CHIEF point selection for 3D problems is automatized while still random, being linked to a multi-level octree decomposition, together with a point-in-polyhedron algorithm; the thumb rule of 5% of CHIEF points is also mentioned in this work. In a very recent paper, [PERSON] et al. [15] also showed the applicability of the CHIEF method to recent meshless techniques, particularly to the singular boundary method (SBM). In this work, the so-called modified SBM is indeed an extension of the method by incorporating CHIEF points to avoid unstable results at specific frequencies, using a so-called self-regularization technique. These authors also compared their technique with a Burton-Miller SBM and indicated that the CHIEF method can lead to more accurate results in the tested problem. Following the given references, the classic CHIEF method is not only a relevant technique for the BEM itself but is also useful as a complement in more recent methods. For the latter, the principle of application is exactly the same as for the BEM, and so the same exact problems are expected to occur, in particular the definition of convenient location and number of CHIEF points for each problem. This so-called \"convenient location\" of CHIEF points is not easily established, and a bad selection of these point locations may cause the procedure to fail or lead to less accurate results. [PERSON] et al. [16] proposed a CHIEF-block idea to enforce constraints in a weighted residual sense over a small region. However, if the CHIEF point is located over or near the nodal line of its corresponding mode, it may not provide a valid constraint [17; 18]. To overcome this problem, [PERSON] et al. [8; 9] presented an analytical study to select a valid number of CHIEF points and their positions for circular cavities by using circulants. However, their discussions were limited to circular configurations, and its extension to general geometrical formats is not trivial. After that, [PERSON] et al. [19] proposed a criterion in selecting the minimum number of CHIEF points and their positions by testing the orthogonal condition between the influence row vector and the right unitary vector, proving a further advancement on the topic. On the other hand, [PERSON] et al. [20] suggested that the best efficiency was achieved by the random selection of CHIEF points. In this paper, a method based on an optimization process, which maximizes the smallest singular value of the system matrix, is considered to formulate a criterion for selecting the number of CHIEF points as well as their locations. The method is here applied to 2D problems, although its extension to higher dimensional problems should not pose any difficulties. The paper is organized as follows: first, the general formulation of the BEM is given for acoustic problems; there follows a brief description of the CHIEF method; next, the algorithms that are proposed and analyzed in this paper are described; and, finally, numerical examples are presented and discussed.