A Boundary Element Method (BEM) Solver for Low Frequency Room Modes

Abstract

Room modes are known to be problematic in small critical listening environments. They degrade the acoustic quality at low frequencies, producing peaks and nulls in the frequency domain and ringing in the time domain. The Finite Element Method (FEM) is currently the easiest way to predict such resonances for arbitrarily shaped rooms. This solves for mode frequencies and shapes, as well as Q-factors and decay rates. Such 'eigenfrequency' solvers are commonplace in FEM, but FEM has the disadvantage of needing to mesh the entire air volume in the room. The Boundary Element Method (BEM) avoids this and only requires a simple boundary mesh, but solution of its eigenfrequency problems is much more challenging and appears in only a few academic papers. Here we transfer those approaches to Room Acoustics. We implement the block Sakurai-Sugiura method, which uses a contour integral in the complex frequency plane to convert the BEM eigenfrequency problem, which is usually non-linear in wavenumber, into a standard linear eigenfrequency problem that is straightforward to solve. The method is demonstrated through application to a cuboid room and an irregularly shaped room, both with impedance boundary conditions. Results are validated against FEM and discussed.

Analysis and Control of Acoustic Modes in Cylindrical Cavities with application to Direct Field Acoustic Noise (DFAN) Testing.

“…Another useful metric is Modal Decay Time 𝑀𝑇 60 , the time in seconds it takes a mode to decay by 60dB. Both of these can be found from the FEM eigenfrequencies, which are complex for a damped problem [24]. Q-factor is equal to real(𝑓 𝑛,𝑚,𝑙 ) 2 × imag(𝑓 𝑛,𝑚,𝑙 ) ⁄ and 𝑀𝑇 60,𝑛,𝑚.𝑙 = 3 ln(10) 2π × imag(𝑓 𝑛,𝑚,𝑙 ) ⁄ .…”

“…This matching process is non-trivial since modes from the two models are not identical, and the FEM solver finds many extra highly damped modes that are of little physical importance. Modes were matched using the Modal Assurance Criterion [24] and similarity of eigenfrequency. Matches are typically clear cut for important high-Q modes but may be ambiguous or not possible for highly damped modes.…”

“…Since the acoustic domain is unbounded a BEM solver might be advantageous. Historically these have only been able to compute frequency response results but eigenfrequency solvers for BEM are emerging [24]. This would avoid the need to choose PML parameters, which were found to affect the Q-factor of the lowest frequency modes.…”