Reduced basis methods for numerical room acoustic simulations with parametrized boundaries

TL;DR

This work tackles the computational bottleneck of wave-based room acoustic simulations under parametrized boundary conditions by introducing a reduced basis method (RBM) embedded in a Laplace-domain spectral element model. An offline stage builds a POD-based reduced basis via a cotangent-lift symplectic projection from high-fidelity FOM solutions, enabling a compact ROM that solves a reduced system for new boundary parameters with $N_{rb} \ll N$. Boundary conditions are treated in both frequency-independent and frequency-dependent forms, the latter using ADE and Weeks-based time reconstruction to recover impulse responses in the time domain. Numerical results demonstrate large online speedups (up to $\sim 10^3$) in 3D with modest accuracy losses, alongside manageable storage costs, highlighting the method’s potential for rapid parametric studies in building design and optimization. The study thus establishes a practical, stable RBM pipeline for high-order SEM room acoustics, enabling efficient exploration of absorption and scattering at boundaries across large parameter spaces.

Abstract

The use of model-based numerical simulation of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively and this is computationally expensive. We present a reduced basis methods (RBM) to achieve a computational cost reduction relative to a traditional full order model (FOM), for wave-based room acoustic simulations with parametrized boundary conditions. In this study, the FOM solver is based on the spectral element method, however other numerical methods could be applied. The RBM reduces the computational burden by solving the problem in a low-dimensional subspace for parametrized frequency-independent and frequency-dependent boundary conditions. The problem is formulated and solved in the Laplace domain, which ensures the stability of the reduced order model based on the RBM approach. We study the potential of the proposed RBM framework in terms of computational efficiency, accuracy and storage requirements and we show that the RBM leads to 100-fold speed-ups for a 2D case with an upper frequency of 2kHz and around 1000-fold speed-ups for an analogous 3D case with an upper frequency of 1kHz. While the FOM simulations needed to construct the ROM are expensive, we demonstrate that despite this cost, the ROM has a potential of three orders of magnitude faster than the FOM when four different boundary conditions are simulated per room surface. Moreover, results show that the storage model for the ROM is relatively high but affordable for the presented 2D and 3D cases.

Figures (10)

• Figure 1: FOM impulse response simulations for different parameter values and absorption coefficients. The source location is $(s_x,s_y) = (1, 1)$ m and the receiver location is $(r_x,r_y)=(0.2, 0.2)$ m. Simulations were carried out with $P=4$, $\Delta t = 5.9\times 10^{-6}$ s, $N_e=15$ and $N_e=20$, $N_s= 3000$ and $(\sigma, b)=(10, 1000)$.
• Figure 2: Contour plot of the error obtained using 11 equally spaced points for $\sigma \in[0.1, 2000]$ and 11 equally spaced points for $b \in[0.1,90]$ for three different boundary conditions to obtain (\ref{['optimization']}).
• Figure 3: Simulated pressure using the 2D FOM and ROM for different parameter values $Z_s$ and $d_{mat}$.
• Figure 4: Error and cpu time of the 2D ROM for different basis functions. Simulations were carried out with a fixed $PPW=13$ for frequency-independent boundaries and $PPW=10$ for frequency-independent at $1000$ Hz, $P=4$, $Z_s = 5000$ kgs$^{-1}$m$^{-2}$ and $d_{mat}=0.15$ m.
• Figure 5: Speedup for 2D and 3D. (a) Speedup as function of error, (b) Speedup as function of $N_{rb}$.