# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 3 Numerical Experiments
### Case 1--Radiating Circular Cavity
To better understand the influence of the CHIEF point position, the colormaps in Figure 7 illustrate the smallest SV computed for the matrix as a function of the position of one CHIEF point. Results for six frequencies are illustrated, including those where error peaks have been identified and other frequencies for which a dip in the smallest SV curve occurs but is not associated with any error. It is interesting to note that for the first three presented plots (see Figure 7a-c), there is a strong dependence of the smallest SV on the position of the CHIEF point. Moreover, for the two higher frequencies (i.e., 300 and 470 Hz), there are regions where the collocation of one CHIEF point can be totally ineffective (keeping a very small value of the SV). It is also interesting to note that for the other three illustrated frequencies (see Figure 7d-f), a completely different behavior occurs, and there is almost no dependence of the smallest SV on the position of the CHIEF point. Indeed, these results are in line with the expected behavior, with the first three frequencies corresponding to fictitious eigenvalues of the problem, for which non-unique responses are obtained, and the other three just being associated with a more regular behavior of the BEM and not specifically with that same type of non-unique response. Figure 5: BEM relative error and (**a**) smallest singular value of the BEM matrix; (**b**) reciprocal of the condition number of the BEM matrix. Figure 6: BEM relative error with one (**a**), two (**b**) and three (**c**) CHIEF points. The smallest singular value for each case is also plotted. Figure 7: Smallest singular value of the BEM matrix as a function of the position of one CHIEF point: (**a–c**) correspond to frequencies where simultaneously an error peak in the BEM model and a minimum in the singular value curve occur (see Figure 4a); (**d–f**) correspond to frequencies where no error peak occurs in the BEM model, but a minimum in the singular value curve occurs. The proposed optimization algorithms have been applied to this test case, and the corresponding results are depicted in Figure 8a-d, considering a threshold of min(SV) \(>\) 0.1 for both algorithms. Figure 8a,c illustrate the relative error and the smallest SV curve registered for ALG1 and ALG2, while Figure 8b,d show the number of CHIEF points added during the optimization process. In both cases, min(SV) \(>\) 0.1 was attained with a small number of points, and the smallest SV curves are clearly more regular than when random points are used, without evident dips. The global error level is well controlled, particularly at higher frequencies, and no relevant peaks are observed. It is important to emphasize that CHIEF points are only used in a very limited number of frequencies (only when min(SV) \(>\) 0.1), and the used number of points depends on the specific frequency being analyzed. For the lower frequencies, typically one or two points are sufficient for both algorithms, while some of the higher frequencies require three or even four optimized CHIEF points. ALG2 seems to have a trend of requiring more points than ALG1 at the higher frequencies, which is in accordance with the expected behavior for these algorithms. Indeed, ALG2 independently optimizes one point at a time (point 2 is optimized only after point 1 is fixed, and so on), and as so its final point distribution is sub-optimal by nature. ALG1, on the other hand, optimizes the position of all points simultaneously, allowing a more effective search for the optimal distribution. However, ALG1 typically has a larger number of variables involved in the optimization, rendering a slower process. For example, considering the frequency of 470 Hz, both ALG1 and ALG2 require three CHIEF points for reaching the target smallest SV, but the computational effort time required by ALG1 is around 3.5 times higher. Even for 415 Hz, for which ALG1 only needs two points and ALG2 requires four points, ALG2 is 2 times faster.