Structural Acoustics
Vibro-Acoustics, NVH & Acoustic FEM

Great NVH problem. The approach: you mesh both the structural panels (body-in-white shell elements) and the air cavity inside the cabin as a 3D volume of acoustic elements. Structural elements have displacement DOF (u, v, w); acoustic elements have a single DOF — pressure p. At the interface between the panel and the air, the two are coupled: the normal acceleration of the panel surface drives the acoustic pressure gradient, and (in fully coupled mode) the acoustic pressure acts as a normal traction on the panel. For air-structure coupling in a car cabin, the acoustic mass loading on the panels is usually small — you can treat it as one-way coupling (structural vibration drives acoustics, acoustic back-pressure ignored) and get good results efficiently.

The acoustic wave equation for pressure p in a homogeneous, inviscid fluid is:

where c is the speed of sound (343 m/s in air at 20°C; 1500 m/s in water). In FEM, this is discretised using acoustic finite elements with pressure as the nodal DOF. The resulting finite element equation for harmonic excitation at frequency ω is:

For the fully coupled structural-acoustic system, structural and acoustic DOF are assembled into one block system. The coupling matrices link normal structural acceleration to acoustic pressure gradient at the fluid-structure interface.

Two main strategies for exterior domains. The first is perfectly matched layers (PML) or absorbing boundary conditions — artificial damping layers that simulate a non-reflecting boundary at a finite mesh truncation surface. PML is highly effective for regular geometries. The second and more elegant solution is infinite elements (IFEs) — special elements attached to the outer boundary of the finite acoustic mesh that analytically represent the outgoing wave behaviour (\(p \sim e^{ikr}/r\) in 3D). Abaqus, ANSYS Acoustics, and LMS Virtual.Lab all support infinite elements. They accurately model far-field radiation with no artificial reflections, at modest computational overhead. The mesh truncation surface should be at least λ/2 away from the vibrating structure.

An acoustic transfer function (ATF) maps an input (typically a structural force at a specific point — engine mount, wheel hub) to an output (sound pressure level at a specific receiver — driver's ear position). Run a harmonic frequency sweep (say 20–500 Hz for NVH) and you get a frequency-response function in Pa/N or dB re 20 μPa/N. Peaks in the transfer function indicate cavity resonances (acoustic modes of the cabin air volume) or structural-acoustic coupling resonances. The sound pressure level (SPL) is computed as:

A booming cabin resonance typically appears as a 10–20 dB spike in the SPL spectrum. The FEM-predicted peak frequency and shape tell you where to add damping material, modify panel thickness, or add acoustic foam to control it.

Quite a broad field. In consumer electronics: speaker cone design, headphone acoustic cavity tuning, microphone capsule response — FEM is now standard for audio product development. In defence: submarine acoustic signature prediction — the hull vibration from machinery radiates sound that sonars detect; minimising this drives the entire drivetrain isolation design. In aerospace: acoustic fatigue of aircraft skin panels exposed to jet engine noise — 150+ dB SPL levels cause high-cycle fatigue in thin aluminium panels. In industrial machinery: transformer hum (120 Hz hum from electromagnetic forces), HVAC duct noise, gearbox whine in electric vehicles (EVs have lost the masking effect of combustion noise, making gear mesh frequencies newly audible and important).