The Three-Dimensional Quasi-Periodic Boundary Element Method: Implementation, Evaluation, and Use Cases

The boundary element method (BEM) is a widely used engineering tool in acoustics. The major disadvantage of the three-dimensional boundary element method (3D-BEM) is its computational cost, which increases with the size of the simulated obstacle and the simulated wave number. Thus, the geometrical details of the obstacle and the simulated frequency range are limited by computer speed and memory. The computational cost for simulating large obstacles like noise barriers is often reduced by applying the two-dimensional boundary element method (2D-BEM) on three-dimensional obstacles. However, the 2D-BEM limits the geometry of the boundary to obstacles with a one-dimensionally constant profile. An interesting compromise solution between the 2D-BEM and the 3D-BEM is the quasi-periodic boundary element method (QP-BEM). The QP-BEM allows the simulation of periodically repetitive complex three-dimensional structures and periodic sound fields while keeping the computational cost at a reasonable level. In this study, first, the QP-BEM was implemented and coupled with the fast multipole method. Second, the QP-BEM was used to simulate the sound field radiated by a simple geometric object, i.e., a uniformly vibrating cylinder. Results were compared to an analytic solution, for the evaluation of the numerical accuracy of our QP-BEM implementation. For the demonstration of some use cases, third, the QP-BEM was used to simulate the sound field scattered by a sonic crystal noise barrier and a noise- barrier top element.