## 2 An Iterating Boundary Element Method with a Relaxation Technique
### The Relaxation Technique
 \tag{8}\]  
\[A_{I}(Q)=\omega_{I}\rho V_{I}(Q) \tag{9}\]  
\[\omega_{I}^{*}=\omega_{I}\frac{t_{n}^{*}}{t_{n}}=\omega_{J}\frac{(t_{n}+F\; \Delta t)}{t_{n}} \tag{10}\]  
where \(n\) is the time step index. Equation (10) can be reformulated as  
\[\omega_{I}^{*}=\omega_{I}\frac{t_{n}^{*}}{t_{n}}=\omega_{J}\frac{(t_{n}+F \Delta t)}{t}=\omega_{J}\bigg{(}1+\frac{F}{n-1}\bigg{)} \tag{11}\]  
Equation (11) helps make clear that the frequency shifting is time-step-dependent and could be viewed as an iterative, or relaxation, technique for the solution of the problem. The method introduced in this paper avoids numerical problems due to the internal resonance frequency by initializing the iteration with each frequency scaled to \(\omega_{I}\Big{(}1+\frac{F}{n-1}\Big{)}\). By rescaling the frequency at successive time steps, at the end of the iterating process (i.e., the last time step) the shifted frequency is  
\[\omega_{I}^{*}=\omega_{I}\bigg{(}1+\frac{F}{N-1}\bigg{)}\approx\omega_{I} \tag{12}\]  
In other words, the solution frequencies are approximately equal to the true frequencies. Clearly, by taking enough time steps, \(N\) is large enough to make \(\omega_{I}^{*}\) as close as necessary to \(\omega_{I}\), and the solution is valid. This method works because the calculated solution coefficients change by a small amount between time steps, so that a small change in the frequency does not disturb the solution and allows for smooth convergence to the correct answer; when \(\omega_{J}^{*}\approx\omega_{J}\), there is only a small number of time steps required, so instability in the solution does not grow before the end of the time steps. \(A_{J}(Q)\) does not change the numerical value used in the computation and will not cause numerical instability at the eigenfrequency of the interior. The term \(A_{part}\) in Equation (6) can also be reformulated:  
\[ \begin{array}{l}A_{part}(t_{n}+F\ \Delta t)=A_{part}(t_{n}^{*})\\ =\sum\limits_{m=1}^{M}\left\{\left[P_{r}(Q,t_{n}^{*}-r/c)\ \left(\frac{-1}{r^{2}} \right)-\frac{\partial P_{r}(Q,t_{n}^{*}-r/c)}{\partial t}\ \frac{1}{c}\right]\left(\
abla^{\rightarrow}\overrightarrow{r}. \overrightarrow{n}\right)\right\}_{m}\frac{\Delta S_{m}}{4\pi}.\end{array}  \tag{13}\]  
The interpolated term (\(P_{r}(Q,t^{*}-r/c)\)) and the differential term (\(\frac{\partial P_{r}(Q,t^{*}-r/c)}{\partial t}\)) in Equation (13) can be evaluated by the linear finite difference method (FDM) in Burden and Faires [23]. The reason the linear FDM is used is because linear FDM is common and easy for interpolation. Substituting Equations (8) and (13) into Equation (6) gives Equation (14). \[ \begin{array}{l}C(H_{m})\ P_{r}(H_{m},t_{n})\ =\\ \sum\limits_{m=1}^{M}\left\{\left[\begin{array}{l}P_{r}(Q,t_{n}+F\Delta t-r/ c)\ \left(\frac{-1}{r^{2}}\right)-\\ \frac{\partial P_{r}(Q,t_{n}+F\Delta t-r/c)}{\partial t}\ \frac{1}{c}\end{array}\right] \left(\
abla^{\rightarrow}\overrightarrow{r}.\overrightarrow{n}\right) \right\}_{m}\frac{\Delta S_{m}}{4\pi}+\\ \sum\limits_{m=1}^{M}-\left\{\left[\begin{array}{l}\sum\limits_{j=1}^{ \infty}A_{J}(Q)\ \sin\left(\omega_{j}^{*}t_{n}-\frac{\omega_{J}r}{c}\right) \end{array}\right]\frac{1}{r}\right\}_{m}\frac{\Delta S_{m}}{4\pi}\end{array}  \tag{14}\]  
Equation (14) verifies that the pulsating frequency, \(\omega_{J}\), is shifted to \(\omega_{J}^{*}\) for the radiating pressure at the \(n\)th time step (time \(t_{n}\)), \(P_{r}(H_{m},t_{n})\). It is not necessary to determine the pressures at the shifted time instants. The radiating pressure of the field point is calculated by using Equation (14), and the iterating process is controlled by three outer loops, as illustrated in Figure 3. In the outermost loop, \(I\) refers to the \(I\)th iteration. In the middle loop, \(n\) refers to different time steps. In the innermost loop, \(m\) refers to different elements on the pulsating object. Each iteration is performed from the first time step to the last time step (i.e., \(n=1\)\(\sim\)\(N\)).