# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 3 Numerical Experiments
### Case 2--Radiating Irregular Body  
A second test case has been set up with the geometry depicted in Figure 11. For this case, an irregular boundary has been defined, following Equation (17):  
\[r(\theta)=1+\frac{\cos(5\theta)}{3}+\frac{\sin(4\theta)}{7}, \tag{17}\]  
where (\(r\), \(\theta\)) refers to the coordinates of the boundary points in cylindrical coordinates. Along this boundary, a unit particle velocity is imposed. The system is discretized using 100 constant boundary elements. In Figure 12a, the computed solution using the standard BEM formulation shows the calculated absolute acoustic pressure, together with the smallest SV of the BEM matrix. As in the previous test case, the pressure curve exhibits a very irregular shape, with sharp peaks and dips that indicate an irregular behavior of the BEM that may be associated with non-unique solutions of the system of equations. The smallest SV curve also exhibits pronounced dips at almost all of those frequencies, a behavior indicating that, once more, a relationship between the occurrence of very low SVs and non-unique solutions of the BEM system exists, even in a stronger form than in the previous example. Figure 11: Geometry of the second test problem, considering a flower-shaped domain with unit particle velocity along its boundary. Figure 10: Final position of the CHIEF points after the optimization procedure considering min (SV) > 0.3: (**a**) ALG1, requiring six points; (**b**) ALG2, requiring 10 points. To assess the method's behavior with the addition of a CHIEF point at some of these irregular frequencies, calculations were performed to observe the smallest SV of the system as a function of the position of this CHIEF point. Figure 12b-f illustrate this behavior for frequencies of 165, 255, 295, 315 and 385 Hz (for which an irregular behavior is observed), in the form of color maps, where lighter (yellow) shades indicate positions for which the existence of a CHIEF point will lead to higher values of the smallest SV (i.e., better conditioning), while blue shades indicate the opposite behavior. It is clear from these figures that there is a very strong dependence of the smallest SV from the position of the CHIEF point, a fact that has also been observed in the first test case but that is even more evident here. Indeed, for each analyzed frequency, there are significant parts of the analyzed space for which the collocation of a CHIEF point will have a very limited effect, an observation that indicates that the use of an optimization algorithm to assist the selection of the position of this point may be of great interest. At some specific positions, the effect of the introduction of an internal CHIEF point may be almost negligible, as is the case of the darker shades of blue in all the five figures. In Figure 13, results are presented for this test case considering different situations. In the left column, results considering one, two or three randomly positioned points are illustrated, and, in the right column, the corresponding results considering the proposed algorithm are presented. In this last case, the number of used CHIEF points may vary between 0 and the maximum number allowed during the optimization process, which  
Figure 12: (**a**) BEM response for the second test case, and (**b–f**) color maps of the minimum singular value of the system matrix when one CHIEF point is considered for different frequencies. can be one (Figure 13b1), two (Figure 13b2) or three (Figure 13b3). To avoid a lengthy presentation of results, only ALG1 will be addressed here, but similar findings are also seen for ALG2. By observing the presented results, it becomes quite clear that, in all cases, the use of the proposed optimization algorithm leads to a smoother solution curve than the use of random points.