Fast and accurate boundary element methods for large-scale computational acoustics

Abstract

The boundary element method (BEM) is a powerful algorithm to solve the Helmholtz equation for harmonic acoustic waves. The explicit use of Green’s functions avoids domain truncation of unbounded regions and accurately models wave propagation through homogeneous materials. Furthermore, fast multipole and hierarchical compression techniques provide efficient computations for dense matrix multiplications. However, the convergence of the iterative linear solvers deteriorates significantly when frequencies are high or materials have large contrasts in density or speed of sound. This talk presents several algorithmic improvements of the BEM. First, a preconditioner based on on-surface radiation conditions drastically reduces the iteration count of linear solvers at high frequencies. Second, anovel boundary integral formulation remains well-conditioned for high-contrast transmission problems. We used our fast and accurate BEM implementation to simulate focused ultrasound propagation in the human body, which can be translated to important biomedical applications such as the non-invasive treatment of liver cancer and neuromodulation of the brain. We validated the methodology within the benchmarking exercise of the International Transcranial Ultrasonic Stimulation Safety and Standards (ITRUSST) consortium. As a second application, we simulated the collective resonances of water-entrained arrays of air bubbles. Finally, we implemented all functionality in our open-source Python library, OptimUS.