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

Abstract

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 noisebarrier top element. Keywords: acoustics, boundary element method, diffraction, fast multipole method, helmholtz equation, noise barriers, periodicity, scattering. INTRODUCTIONThe boundary element method (BEM) [1] is a widely used engineering tool to simulate the radiation, scattering, and diffraction of acoustic waves for a bounding geometry, i.e., the obstacle. Applications include the design of exhaust pipes, loudspeaker waveguides, virtual acoustics as well as noise barriers [2]. For instance, the BEM was used to investigate the acoustic properties of various noise-barrier materials and noise-barrier shapes, e.g., the general noise-barrier shape [3] or the design of noise-barrier top edges [4].The major disadvantage of the three-dimensional boundary element method (3D-BEM) is its computational cost, i.e., the required amount of physical memory (RAM) and the computation time. The computational cost increases with the size of the obstacle and the simulated wave number. For instance, the calculation of the standardized sound diffraction index (defined in the European standard EN 16272-4) of a noise barrier's top edge stipulates the simulation of a 10 m long and 4 m high noise barrier in third-octave bands ranging from 100 Hz to 5,000 Hz resulting in a numerical problem of approximately two million unknowns (when considering eight constant rectangular elements per wavelength in the simulation [5]). However, the available amount of RAM limits the number of unknowns in the numerical calculation and CPU speed defines the computation ...

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