## 4 Spatial Acoustic Scattering
However, a slightly higher energy lobe can be discerned at \(\theta=20^{\circ}\) in the FEM case. This is likely due to the finite size of the surface sample in the simulation, resulting in a decrease of the scattered sound energy at grazing angles, thus further relatively enhancing the remaining scattered energy and slightly reducing the intrinsic autocorrelation value. Despite such differences, the FEM model can be estimated to be in good agreement with the theoretical prediction. The missing scattering distribution under Figure 4c is at the core of this work's rationale, as it would be an unnecessarily complex task to simulate the fine features of the metadiffuser in a large 3D volume. The complex physics and the small geometry of the QRM not only would require an extremely fine meshing grid, and thus a very high computational load for conducting the same simulations, but viscothermal losses would have to be taken into account as well. This is why the aforementioned equivalent surface impedance as a fitted RLC circuit within an FDTD model is proposed to bypass the numerical limitations of large-scale multiphysics simulations, while faithfully reproducing the intended scattering of the metadiffuser embedded in a larger scene. It is worth noting that viscothermal losses inside the metasurface are implicitly encoded by the RLC approach in the FDTD scheme. However, in order to strengthen the forthcoming analysis between FEM and FDTD, scattering comparisons of a flat surface and a much simpler acoustic diffuser (e.g., a Quadratic Residue Diffuser) can be found in the Supplementary Materials, where excellent agreement between the different numerical methods can be observed. Figure 4d shows the scattering distribution of the equivalent surface impedance simulated through FEM where the \(N=5\) impedance patches were modelled using impedance boundary conditions with input values identical to the analytical (TMM) surface impedance of the QRM. Again, it can be observed that the main dip at \(\theta\approx-20^{\circ}\) is correctly reproduced, and that the main axial lobes are also in good agreement with the theory, leading to a normalised diffusion coefficient \(\delta_{n,0}=0.56\) close to that of Figure 4b. The major changes that can be distinguished compared to the FEM simulation of the QRM are the energy distribution of the lateral lobes and the smoothing of the \(\theta\approx 20^{\circ}\) energy lobe. The former seems to resemble that of a flat panel scattering. Perhaps this is due to the disappearance of the slit cavities within each impedance patch which may cause variations from the estimated slit impedance values as these are dependent on the free air radiation correction of the slits. More investigation on that matter is needed. Figure 4e similarly represents the scattered sound distribution of the ZEQ in the FDTD solver. Results are in excellent agreement with the ZEQ FEM data, with a very similar normalised diffusion coefficient \(\delta_{n,0}=0.57\). A minor increase in scattered sound energy can however be perceived between FEM and FDTD ZEQ models which also appears in other cross-numerical comparisons that were conducted for traditional sound diffusers and flat surfaces (see Supplementary Material). This slight energy increase in FDTD RLC modelling may then be attributed to energy propagation modelling in each numerical environment (FEM/FDTD), but remains nonetheless almost negligible with a difference in diffusion coefficient of 1%. Although the equivalent surface impedance method proposed here results in a major simplification of the more intricate geometry of the metamaterial being studied, it can be seen that it is quite efficient for replicating an approximation of its scattered field into the surrounding space. Additionally, it has been previously shown that these FDTD simulations with such RLC circuit boundary conditions are amenable to parallel acceleration [29].