# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 4 Final Remarks  
The present paper revisits the topic of irregular frequencies that occur in exterior acoustic problems using the boundary-element method, proposing a strategy for the iterative definition of the number and position of CHIEF points in an effective manner. The proposed method is based on the analysis of the singular values of the system matrix, selecting the lowest one and choosing the best position for the CHIEF points that maximizes this parameter. It is known, and it is also shown here, that very low singular values are an indicator of ill-conditioning, and thus the authors have described that the maximization of this value is a possible criterion in selecting the number of CHIEF points and their locations. Two algorithms have been proposed, both leading to good results for the test cases that are discussed in this manuscript. By following these algorithms, the number of CHIEF points is defined incrementally, but without solving the system at each incremental step. Indeed, since only a few lines are added to the initial BEM matrix, the method is not computationally demanding, and by design it is simple to implement and allows existing codes to be reused without very significant changes. Additionally, it also removes the necessity of a preliminary study by the user for each specific problem in order to determine the adequate placement and number of CHIEF points. However, the use of the proposed techniques involves an additional computational overhead due to the optimization algorithm, but in the view of the authors the advantages clearly outweigh the disadvantages. The authors believe that this strategy can aid the implementation of more stable BEM codes that can automatically circumvent non-uniqueness problems.  
It is important to note that the devised strategy is here described for the specific case of 2D problems, but its extension to 2.5D problems or 3D problems should not pose any challenges. Even the application of the technique to complement other numerical formulations, such as the singular boundary method, can be envisaged without significant additional complexity. Therefore, the application in more realistic engineering scenarios can be viable and allow for reliable yet compact (minimizing the number of CHIEF points) implementations, with a systematic approach for the definition of the position and number of additional points to be used instead of the empirical rules which are used in most of the publications in the scientific community dealing with this approach.  
Conceptualization, K.d.A.G. and L.G.; methodology, K.d.A.G., D.S.S. and L.G.; software, K.d.A.G. and L.G.; formal analysis, K.d.A.G., D.S.S., D.S., P.A.C. and L.G.; writing--original draft preparation, K.d.A.G. and L.G.; writing--review and editing, D.S. and P.A.C.; funding acquisition: P.A.C. and L.G. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by FCT--Fundacao para a Ciencia e a Tecnologia, I.P., by Base Funding (UIDB/04029/2020) and Programmatic Funding (UIDP/04029/2020) of the research unit \"Institute for sustainability and innovation in structural engineering--ISISE\", and by Base Funding (UIDB/04708/2020) and Programmatic Funding (UIDP/04708/2020) of the research unit \"CONSTRUCT--Instituto de ID em Estruturas e Construcoes\". This work was also funded by FEDER funds through the COMPETE 2020, Portugal 2020, under the projects POCI-01-0247-FEDER-033990 (iNBrazil) and POCI-01-0247-FEDER-046111 (FERROVIA 4.0), and by national funds (PIDDAC) through FCT/MCTES under the project PTDC/ECI-EGC/3352/2021 (IntRAIL). It was also partially funded by \"Conselho Nacional de Desenvolvimento Cientifico e Tecnologico\" (CNPq) and \"Fundacao de Amparo a Pesquisa do Estado de Minas Gerais\" (FAPEMIG).  
Not applicable.  
The authors declare no conflict of interests.