# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 3 Numerical Experiments  
### Case 1--Radiating Circular Cavity  
The first test case analyzed here (which is illustrated in Figure 3) corresponds to a circular cavity with a unit radius, embedded within a homogeneous acoustic medium. Figure 3: Schematic representation of the first test case, with a circular cavity subject to a unit particle velocity throughout its boundary. Figure 2: Flowchart of the proposed algorithm. The host medium allows a propagation velocity of 340 m/s and has a density of 1.21 kg/m\({}^{3}\). Along the boundary, a unit particle velocity is imposed as \(v_{n}=-\frac{1}{\eta\omega}\frac{\partial p}{\partial n(\xi)}=1\), and for this case an analytical solution can be derived, assuming the form:  
\[p(r)=-\frac{\mathrm{i}}{4}A_{0}[J_{0}(kr)-iY_{0}(kr)], \tag{15}\]  
where  
\[A_{0}=\frac{-4\rho cv_{n}(a)}{J_{1}(ka)-\mathrm{i}\,Y_{1}(ka)}, \tag{16}\]  
and \(v_{n}(a)=1\) is the unitary particle velocity imposed at \(r=a\), with \(a=1\) being the cylinder radius. In addition, \(J_{0}\) and \(J_{1}\) stand for the first kind Bessel function of zero and first order, respectively, whereas \(Y_{0}\) and \(Y_{1}\) are the second kind Bessel function of zero and first order, respectively, and \(r\) stands for the distance of the point of analysis to the center of the cavity. In this case, the acoustic response is computed at a distance of 1.8 m from the center of the cavity, for frequencies between 5 and 500 Hz, and its real and imaginary parts are depicted in Figure 4b. In Figure 4c, the absolute pressure value, analytically calculated and using the BEM, is displayed, using a discretization with 63 boundary elements for the mesh (see Figure 4a for this adopted discretization). From these results, it is clear that a good match is obtained throughout the referred frequency range, except for some specific frequencies, in which it is possible to observe large deviations. Figure 5a illustrates the relative errors of the BEM results, together with the smallest SV of the BEM matrix, and Figure 5b shows the reciprocal of the matrix condition number (computed using MATLAB's \"rcond\" command). Observing these plots, the three error  
Figure 4: Test case with a circular cavity with prescribed particle velocity: (**a**) BEM discretization with 63 constant elements; (**b**) real and imaginary parts of the analytical solution; (**c**) absolute value of the pressure solution considering analytically and numerically (with the BEM) computed responses. peaks are quite evident, occurring at frequencies around 130, 300 and 470 Hz, for which relative errors reaching 10% can be seen. In both Figure 5a,b, it can be seen that these frequencies coincide with those describing dips in the smallest SV and \"rcond\" curves, indicating that these errors are associated with the ill-conditioning of the system. However, it should be noted that such dips also occur at other frequencies, for which no error increase is registered. The conventional approach of adding CHIEF points to stabilize the response that has been initially used by introducing one, two or three CHIEF points located randomly inside the cavity. Results for those cases are represented in Figure 6a-c, in which it is possible to observe that a significant improvement of the error is attained at the previously mentioned frequencies. The error improvement is, however, non-uniform throughout the frequency domain, and significant oscillations occur from frequency to frequency. Indeed, at some points the CHIEF points seem to originate a small degradation of the response, even when three points are used. The smallest SV is also represented in each plot, and it can be seen that the addition of the first CHIEF point significantly improves the first dip in that curve but only slightly affects the remaining ones. In particular, the error peak at 470 Hz still occurs, although slightly attenuated. As more points are added, the smallest SV curve improves, but dips are still clearly visible, indicating that a more stable matrix is obtained, in particular at higher frequencies.