# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 2 Materials and Methods
### Proposal for the Location of the CHIEF Points  
Although, in principle, the proposed strategy is very simple to implement (and may be very efficient), defining where to place CHIEF additional points is not a trivial task, especially if complex geometries are regarded.  
To help define the location of these points, it is important to understand that non-unique responses of the system occur due to its ill-conditioning. An indicator of this behavior is the occurrence of small singular values (or a large relationship between the largest and the smallest singular values) or a large condition number of the matrix. Taking this into account, a strategy based on the addition of CHIEF points within \(\Omega^{-}\), which allows for this ill-conditioning to be reduced, can be developed by optimizing the location of these points based on a parameter that allows this behavior to be represented.  
The proposed strategy consists of first computing the system matrix using the standard BEM formulation, without any CHIEF points. The possibility of ill-conditioning is then evaluated by means of the evaluation of the smallest singular value (SV) of the matrix and comparing it with a given limit. If this smallest SV is below the specified limit, a CHIEF point is added at a random location, and after this an optimization routine is applied, finding the position of the CHIEF point which maximizes the smallest SV. It should be noted that inside the optimization algorithm only an update of a very limited number of lines (equal to the number of CHIEF points) in the system matrix is required, and the rest of the matrix remains unchanged. After optimization, if the SV limit has been reached, the procedure stops; otherwise, optimization is performed once again with one additional point, and the process goes on until either the SV limit has been reached or a maximum number of CHIEF points has been added. Figure 2 presents a flowchart illustrating this procedure.  
In the present work, the implementation of the whole process is performed using MATLAB R2022b, and the optimization algorithm makes use of the \"fmincon\" function from the Optimization Toolbox, choosing the \"interior point\" with a maximum of 100 iterations. Two different implementations of the algorithm have been tested:  
ALG1--For each new set of CHIEF points, all the points are optimized simultaneously (e.g., if a 2 nd point needs to be added, a new optimization is performed for both points together);  
ALG2--After each optimization, the new optimized point position is fixed and kept for the next step (e.g., if a 2 nd point needs to be added, only the new point's position is optimized, and the first point is kept at the previously determined position).  
It should be noted that, for both algorithms, and according to Figure 2, the inclusion of CHIEF points only occurs when the smallest SV is below a given threshold, which allows for the original BEM response (i.e., with no CHIEF points) to be kept in all other situations. In addition, the number of CHIEF points is increased by one at a time, thus keeping the number of additional constraints to a minimum level. It will be shown later in this work that the smallest SV of the matrix has a relationship with the error of the calculated solution, and the higher its value, the lower the global error.