Acoustic Analysis Using the Boundary Element Method (BEM)
Acoustic Analysis Using the Boundary Element: Theoretical Foundations
What is Acoustic BEM?
Professor, what is acoustic BEM?
A method that transforms the Helmholtz equation into a Boundary Integral Equation and calculates the sound field using only surface meshes. Its greatest advantage is that it naturally handles external sound fields (infinite domains) because volume meshes are unnecessary.
Governing Equation
Helmholtz equation for the acoustic field:
$k = \omega/c$: wave number. This is transformed into a boundary integral form using Green's theorem:
$G$: free-space Green's function:
$c(\mathbf{x})$: $1/2$ on the boundary, $1$ inside the domain, $0$ outside the domain.
What are its advantages compared to FEM?
Summary
BEM's Origins are in 1960s Elasticity Theory
The mathematical foundation of the Boundary Element Method was laid by Jaswon (1963) and Sympson from Stanford University. They published a method for discretizing integral equations for elasticity problems, but at the time, applications to acoustics were not envisioned. The adaptation to acoustic BEM came about a decade later, with Schenck's CHEFS method published in 1968 as an IBM technical report being the pioneer, influencing later NASTRAN and commercial solvers.
Computational Methods for Acoustic Analysis Using the Boundary Element
Discretization of Acoustic BEM
How do you solve the boundary integral equation numerically?
Discretize the surface into triangular or quadrilateral elements. Approximate sound pressure $p$ and normal velocity $v_n$ with nodal values.
$[H]$, $[G]$: Influence matrices. Integral values of the Green's function between each element pair.
Handling Singular Integrals
The most technically challenging part of BEM implementation is handling singular integrals (where the Green's function diverges as $r \to 0$):
- Weak singularity ($1/r$): Regularize using polar coordinate transformation
- Strong singularity ($1/r^2$): Define via Cauchy principal value
- Hypersingular integral: Hadamard finite-part integral
Non-Uniqueness Problem
In external BEM, the solution becomes non-unique at the internal eigenfrequencies of the closed boundary. Countermeasures:
- CHIEF method: Place additional equations at interior points (overdetermined system)
- Burton-Miller method: Linear combination of the standard BIE and its normal derivative BIE. Theoretically complete.
The Burton-Miller method is standard. It requires handling hypersingular integrals but reliably eliminates non-uniqueness.
FMM (Fast Multipole Method)
You mentioned that the dense matrix makes large-scale problems unsolvable...
Solved by FMM (Fast Multipole Method). Approximates calculations for groups of distant elements collectively:
- Memory: $O(N^2) \to O(N)$
- Computational cost: $O(N^2) \to O(N\log N)$
BEM with 1 million elements has become practical. Implemented in Actran, COMSOL, etc.
Summary
FMM Reduces BEM Computational Cost from O(N²) to O(N log N)
Traditional BEM required O(N²) memory and computational cost relative to the number of nodes N, making large-scale models practically impossible. The Fast Multipole Method (FMM) published by Greengard and Rokhlin in 1987 brought about a revolution, making acoustic analysis of automotive body scale (hundreds of thousands of nodes) realistic. Nuances' VirtualLab Acoustics and LMS Sysnoise 5.x are known as products that implemented FMM-BEM early on.
Acoustic Analysis Using the Boundary Element in Practice
Acoustic BEM in Practice
Typical applications include engine radiated noise, tire noise, transformer noise, and acoustic radiation from exhaust pipes.
Analysis Flow
1. Structural Vibration Analysis — Obtain surface vibration velocity $v_n$ via FEM
2. Create BEM Surface Mesh — Extract from FEM mesh or create independently
3. Set Boundary Conditions — Set $v_n$ (Neumann condition)
4. BEM Solution — Calculate surface sound pressure $p$
5. Sound Field Evaluation — Calculate sound pressure, radiated power at arbitrary observation points
Practical Checklist
BEM Mesh Guidelines
| Maximum Frequency [Hz] | Wavelength at 340m/s [m] | Maximum Element Size [mm] |
|---|---|---|
| 500 | 0.68 | 113 |
| 1000 | 0.34 | 57 |
| 2000 | 0.17 | 28 |
| 5000 | 0.068 | 11 |
| 10000 | 0.034 | 5.7 |
The mesh gets quite fine above 5kHz.
Exactly. Mesh density is the main challenge for high-frequency analysis. Automatic mesh adaptation methods are under research.