Frequency-dependent absorption and transmission boundary for the finite-difference time-domain method

In frequency-domain analyses, absorption and transmission characteristics are often modeled as complex surface impedance and complex transfer impedance, respectively. It is however difficult to take the frequency-dependent characteristics into account directly in the finite-difference time-domain (FDTD) method. In this study, a locally reactive boundary using mechanical mass-damper-spring (MDS) systems, which is herein called an MDS boundary, is formulated for sound absorption and transmission analyses by the FDTD method. In addition, the stability conditions of the MDS boundary are discussed. One-dimensional numerical examples show that the MDS boundary can approximate various simple absorption and transmission frequency-dependent characteristics by tuning the parameters of masses, damping constants, and spring constants. Some of them also show that the stability condition of the mechanical MDS system itself is not sufficient and the stability condition considering the effects of the adjacent cells to the MDS boundary can offer the stable calculations. Furthermore, the procedure in a situation where the MDS boundary is not located in parallel with the cell grids is verified by a three-dimensional numerical example.