Noise Prediction Simulation — From FEM-BEM Coupling to SEA

Roughly speaking, it's a two-stage process. First, perform FEM vibration analysis of the structure to obtain the surface velocity distribution. Next, use BEM (Boundary Element Method) to calculate the sound pressure distribution radiated from that vibrating surface.

The most straightforward example is automotive road noise. Vibration travels from the tire → suspension → body panels, and sound is radiated into the cabin. Tracing this "structure → acoustics" chain is a classic application of FEM-BEM coupling. Other examples include compressor noise, motor noise in home appliances, and environmental noise from construction machinery, all predicted using this method.

The concept is simple, but you need to choose the method based on the frequency band. For low frequencies (~500 Hz), use FEM-BEM; for mid-to-high frequencies (500 Hz~), use SEA (Statistical Energy Analysis); for aerodynamic noise, use the FW-H method. There's a division of tools according to purpose. Understanding this overall picture first will make the later discussion easier to grasp.

The starting point is the wave equation. For sound pressure $p(\mathbf{x}, t)$:

Here, $c$ is the speed of sound (approximately 343 m/s in air). Transforming this into the frequency domain yields the Helmholtz equation for a steady-state sound field vibrating at angular frequency $\omega$:

Exactly. The rule of thumb is at least 6 elements per wavelength (or 3 elements for quadratic elements). For example, at 1000 Hz, $\lambda \approx 0.34$ m, so the element size needs to be about 57 mm or smaller. As frequency increases, the mesh explodes—this is the fate of noise prediction.

For acoustic radiation from a vibrating surface embedded in an infinite rigid wall (baffled surface), the Rayleigh integral can be used. It directly calculates the sound pressure at any point $\mathbf{x}$ from the surface normal velocity $v_n(\mathbf{y})$:

Here, $r = |\mathbf{x} - \mathbf{y}|$ is the distance from the source point to the receiver point, and $\rho_0$ is the air density. $e^{-jkr}/r$ is the Green's function, representing a spherical wave from a point source. This is sufficient for calculations like speaker radiation patterns.

That's where the Kirchhoff-Helmholtz integral equation and BEM come in. It's a more general formulation that can handle radiation from arbitrary closed surfaces.

Applying Green's theorem to the Helmholtz equation yields the Kirchhoff-Helmholtz integral equation. For the exterior domain (radiation problem):

$G(\mathbf{x},\mathbf{y}) = \dfrac{e^{-jk|\mathbf{x}-\mathbf{y}|}}{4\pi|\mathbf{x}-\mathbf{y}|}$ is the 3D free-space Green's function. $c(\mathbf{x})$ is a coefficient that is $1/2$ on the boundary and $1$ inside the exterior domain. Since $\partial p/\partial n = j\omega\rho_0 v_n$, substituting the surface velocity $v_n$ obtained from structural FEM gives the sound pressure $p$.

Yes. For exterior radiation problems, FEM requires meshing a huge air domain for sound to propagate far away, plus setting up non-reflective boundaries (like PML). With BEM, only the surface mesh is needed, and the radiation condition (Sommerfeld condition) is automatically satisfied. BEM is overwhelmingly advantageous for external radiation problems like automotive pass-by noise.

There are two main metrics. First, Sound Pressure Level (SPL) compares the sound pressure at a receiver point to a reference value:

Next, Sound Power Level ($L_W$) is an indicator of the radiated energy of the sound source itself and does not depend on the observation position:

The sound power $W$ is obtained by integrating the acoustic intensity $I$ over the vibrating surface:

Correct. In terms of standards, environmental noise regulations are usually specified in SPL, but $L_W$ is used for comparing the performance of noise sources. Measurement methods for sound power are defined in ISO 3744 and ISO 3745, so comparisons with simulation results are often done within this framework.

The basics are the same, but there are points specific to noise prediction. The goal is to output the surface normal velocity $v_n(\omega)$ from a frequency response analysis of forced vibration. The FEM equation of motion is:

Once displacement $\mathbf{u}$ is obtained, the surface velocity is $v_n = j\omega\, \mathbf{u} \cdot \mathbf{n}$. In practice, the choice of damping model is crucial. Rayleigh damping ($\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$) is easy but can cause the damping ratio to vary unphysically over a wide frequency range. For automotive NVH analysis, it's common to set measured modal damping values frequency by frequency.

Exactly. If the damping ratio is twice the actual value, the SPL at a resonance peak can change by as much as 6 dB. That's why, at the structural FEM stage, thoroughly verifying correlation with experimental modal analysis (MAC value) is what determines the reliability of noise prediction.

Discretize the integral equation by dividing the surface $S$ into $N$ boundary elements. In matrix form:

$\mathbf{p}$ is the nodal sound pressure vector, $\mathbf{q} = \partial p/\partial n = j\omega\rho_0 v_n$ is the normal sound pressure gradient. Since $v_n$ is known from structural FEM, $\mathbf{q}$ is known, and solving this system of equations yields the surface sound pressure $\mathbf{p}$. Once the surface pressure is known, the sound pressure at any external point (field point) can also be calculated via integration.

Good question. BEM matrices become dense matrices (full matrices). That's because every node interacts with every other node. For $N$ nodes, it's an $N \times N$ full matrix, requiring $O(N^2)$ memory, and solving with direct methods is $O(N^3)$. This is the bottleneck for large-scale BEM problems, which is solved by FMM-BEM, discussed later.

One-way coupling (weak