# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 3 Numerical Experiments
### Case 2--Radiating Irregular Body
As a matter of fact, the results show that with just one CHIEF point, the irregular character of the solution curve is still very clear if a random CHIEF point is considered, with some spikes still being observable in many of the frequencies previously identified for the BEM solution, although with smaller amplitudes (for the frequency of 385 Hz, this spike is still quite pronounced, and almost no difference can be seen in relation to that of the classical BEM solution). As for the smallest SV curve, pronounced dips are  
Figure 13: Response computed for the second test problem, considering either (**a1–a3**) fixed or (**b1–b3**) optimized CHIEF point positions, for one (**a1,b1**), two (**a2,b2**) or three (**a3,b3**) points. For the optimized case, this number of points is the maximum considered in the optimization process. still visible at many of those frequencies, further indicating that the addition of a randomly located CHIEF point may not be very effective. On the other hand, when the position of the CHIEF point is optimized, a smoother curve is obtained, and most of the spikes appear to be filtered by the optimization of that point's location. Nevertheless, at high frequencies some irregularity is still observed and can be associated with lower values of the smallest SV, as seen in the same plot. By adding an additional point, the behavior seems to improve both for the optimized and for the non-optimized case, although it is clear that the optimization of these positions leads to an even smoother curve than before, with the smallest SV curve being shifted up at higher frequencies as well. Finally, by optimizing considering a maximum of three points, a very good response is registered. Figure 14 shows the number of points used at each frequency, revealing that indeed three points were required just at higher frequencies to control the system's conditioning to a desired level. When no optimization is performed, even considering three randomly distributed CHIEF points, significant and visible irregularities may still be evident in the curve, mostly at high frequencies. Indeed, this second test case clearly shows that it can be quite challenging to choose good positions for the CHIEF points when a complex geometry is analyzed. For some additional insights regarding the effect of the random and optimized approaches, Figure 15 presents a detailed view of the response around one of the irregular frequencies (i.e., 255 Hz), considering the classical BEM, a random CHIEF point generation and the proposed optimization approach. It is evident that the classical BEM result has a dip and a peak around 253 Hz, diverging from the smooth trend of the response. Using one or two random CHIEF points may help to reduce this behavior, but it may still be not sufficient to provide appropriate results. Optimizing the position of these points, on the other hand, turns the curve quite smooth, correcting the non-uniqueness of the response. Figure 14: Number of CHIEF points considered for each frequency, when the optimization was performed for a maximum of three CHIEF points. Figure 15: Detail of the response at frequencies around 255 Hz, considering the BEM, one fixed CHIEF point, two fixed CHIEF points and two optimized CHIEF points. Finally, Figure 16 presents the response at the frequency of 253 Hz, where the higher error peak was observed in Figure 15, computed using both ALG1 and with random point generation. The real and imaginary parts of the response are shown for different numbers of CHIEF points, ranging from 0 to 10. In both parts of the response, the results computed using ALG1 are clearly more stable, and the algorithm seems to be able to select a good position of the CHIEF points that minimizes oscillations of the response. This is particularly true for the real part of the response, which seems to stabilize after one point is introduced. Figure 16c1-c4 shows the distribution of optimal points for one, two, three and six CHIEF points, for the same frequency of 253 Hz.