A High-frequency Bem for 3d Acoustic Scattering

The Boundary Element Method (BEM) is a powerful method for simulating scattering of acoustic waves which has many advantages, particularly when the problem concerns an object in an unbounded medium. Its applications are however limited in practice because standard schemes have a computational cost which grows extremely quickly as size and frequency is increased. Fundamentally this occurs because the number of degrees of freedom N required to discretise the boundary with elements that are small with respect to wavelength increases with frequency, scaling O(f) in 3D or O(f) in 2D. BEM produces dense matrices relating these elements, resulting in O(N) computation and storage costs, so O(f) in 3D or O(f) in 2D. Accelerated BEM algorithms such as the Fast Multipole Method can reduce this dependency on N to O(N) for small f and O(N log N) for larger f, but the trend of increasing cost with frequency due to the scaling of N with f remains. An alternative strategy toward remedying this is to design discretisation schemes which do not require more degrees of freedom at higher frequencies. This is the approach adopted by the so called ‘High frequency BEM’ (HF-BEM) algorithms, such as Partition-of-Unity BEM (PU-BEM) and Hybrid Numerical Asymptotic BEM (HNA-BEM). These typically represent the pressure on the boundary using basis functions which are products of suitably chosen oscillatory functions, multiplied with standard piecewise-polynomial interpolators defined on a coarse, frequencyindependent mesh. Such approaches have been shown to achieve significant savings, for example reducing the number of degrees of freedom required to O(log f) for polygonal obstacles in 2D. This paper will give an overview of these methods and will demonstrate a new HNA-BEM algorithm for the modelling of rectangular plates in 3D.