## 2 An Iterating Boundary Element Method with a Relaxation Technique
### The Relaxation Technique  
In Kao and [PERSON] [13], Equations (1)-(3) are applied in solving the radiating waves. The right-hand side of Equation (1) is calculated by Equations (2) and (3). The calculated value is then used to update the left-hand side of Equation (1). However, the algorithm is expected to have numerical instability for the eigenfrequency of the interior problem (i.e., interior resonance frequency). Numerical instability also appears for the frequency-domain BEM at eigenfrequencies corresponding to an internal problem. The pulsating frequency, \(\omega_{J}\), appears in Equation (3), \(B_{part}\). In order to overcome these problems at the interior resonance frequency, \(B_{part}\) should be reformulated in this paper. The main concept of the proposed boundary element method is shifting the frequency by shifting the field time in every time step, in order to avoid numerical divergence due to the eigenfrequency of the interior problem. The idea process can be illustrated by Figure 2, which shows that the radiating pressure for the \(n\)th time step at time \(t_{n}\) is \(P_{r}(t_{n})\). The time interval of each step is \(\Delta t\). In this paper, each time interval is treated as a time element. For the \(n\)th time step, the time-shifted point is at time (\(t_{n}+F\) At). \[t_{n}^{*}=t_{n}+F\Delta t,\hskip 14.226378 pt(0<F<1) \tag{4}\]  
The value of \(F\) is between 1 and 0 (i.e., 0 \(<F<1\)). In this paper, the value of \(F\) is recommended to be a multiple of 0.35 (i.e., 0.35 or 0.7) for quick convergence. The recommended \(F\) value (0.35) was obtained by the numerical tests carried out in Section 3.1. According to the numerical test, a larger pulsating frequency needs a larger \(F\) (i.e., \(F=0.7\)) to effectively stabilize the iterating process. Figure 1: An arbitrary radiator in the half-space domain (Label 1–4: the square faces of the cubic, Label 5: the pyramid face). Figure 2: The illustration of the boundary element method with a relaxation technique. In every time step, \(P_{r}(t_{n})\) is approximated by \(P_{r}(t_{n}+F\Delta t)\). In other words, the radiating pressure of the \(n\)th time step at time \(t_{n}\), \(P_{r}(t_{n})\), is calculated by the time-shifted point of the \(n\)th time element:  
\[P_{r}(t_{n})\approx P_{r}(t_{n}+F\;\Delta t),\;\;\;\;\;(0<F<1). \tag{5}\]  
The concept illustrated above was applied to the equations (Equations (1)-(3)) derived by [PERSON] and [PERSON] [13], and the boundary integral equation was then discretized as Equation (6). \[ \begin{array}{ll}C(H_{m})\;P_{r}(H_{m},t_{n})&\approx\sum\limits_{L=1}^{M} \big{[}A_{part}(t_{n}+F\;\Delta t)+B_{part}(t_{n}+F\;\Delta t)\big{]}\\ &=\sum\limits_{L=1}^{M}\big{[}A_{part}(t_{n}^{*})+B_{part}(t_{n}^{*})\big{]} \end{array}  \tag{6}\]  
The time period includes \(N\)-many time steps (\(n=1\)-\(N\)). \(C(H_{m})\) is the solid angle of the \(m\)th field point. Because the field point is located on the smooth element of the radiator in this paper, \(C(H_{m})\) has the value 0.5. The time of the field point at the \(n\)th time step (\(t_{n}\)) is presented as follows:  
\[t_{n}=\frac{2\pi}{\omega_{I}}\;\frac{(n-1)}{(N-1)}=\Delta t(n-1),\;\;\;\;\;\; \;(n=1,2,\ldots\ldots,N). \tag{7}\]  
If a larger \(N\) is selected, the iterations can continue for longer. Because the present method is a time-domain scheme, \(B_{part}\) in Equation (6) can be reorganized as follows:  
\[ \begin{array}{ll}&B_{part}(t_{n}+F\;\Delta\;t)=B_{part}(t_{n}^{*})\\ &=\sum\limits_{m=1}^{M}-\left\{\left[\sum\limits_{l=1}^{\infty}A_{I}(Q)\sin \left(\omega_{I}t_{n}^{*}-\frac{\omega_{I}r}{c}\right)\right]\frac{1}{r} \right\}_{m}\frac{\Delta S_{m}}{4\pi}\\ &=\sum\limits_{m=1}^{M}-\left\{\left[\sum\limits_{l=1}^{\infty}A_{I}(Q)\sin \left(\omega_{I}^{*}t_{n}-\frac{\omega_{I}r}{c}\right)\right]\frac{1}{r} \right\}_{m}\frac{\Delta S_{m}}{4\pi}\end{array}