New Formulations of Fast Boundary Element Methods and Acoustic Applications

Acoustics, functioning either positively or negatively, is considered an important aspect of products in automotive, aerospace, smart electronics, home appliances, etc. With the rapid development of computer-aided engineering (CAE), acoustics simulations have become essential for product design, analysis, and optimization. A robust and efficient acoustics simulation software can significantly reduce the time-to-market thus improving the competitiveness of the products. Among all kinds of available numerical techniques, the boundary element method (BEM) has emerged as a mature and robust tool for industrial needs over the years.

Operating merely on the boundary surface of the problem, the BEM can significantly reduce the time and effort of model preparation for CAE analysis. It can conveniently solve unbounded problems as the Sommerfeld radiation condition is inherently satisfied. Indirect BEM considers the surface potentials as the boundary variables making it feasible to handle open boundary problems and interior-exterior combined problems. However, the high computational cost hinders its application to large-scale problems in practice. The conventional BEM is very much limited to small problems as the computational complexity grows with $\mathcal{O}(N^2)$ in assembly and $\mathcal{O}(N^3)$ in dense solvers, where $N$ is the number of unknowns of the model.

This dissertation introduces novel ideas and develops new solutions to further enhance the acoustics BEM for wide industrial applications. The research work leads to contributions on both numerical and application aspects. The first part of the dissertation focuses on the development of fast BEM solvers. As a well-known acceleration technology, the fast multipole method (FMM) is considered and further investigated to accelerate the BEM for single-frequency calculations. Several new ideas are investigated including a new flexible partition algorithm, efficient hypersingular integral evaluation, and explicit fast multipole far-field calculations. A novel multipole accelerated BEM (MABEM) is formulated by combining the FMM and robust direct solvers with efficient parallelizations. For small to medium size industrial problems, MABEM outperforms the existing BEM and FMBEM solvers.

For very large problems, a new flexible multi-level FMBEM (f-FMBEM) is formulated by using efficient iterative schemes. Various iterative solvers and preconditioning strategies are implemented and evaluated to improve the convergence and robustness of the f-FMBEM. Through various numerical studies, it is observed that the new f-FMBEM significantly outperforms the existing BEM and FMBEM solvers.

Furthermore, a wideband frequency simulation is usually needed for industrial applications. Existing model order reduction (MOR) techniques exhibit difficulties to cope with fast BEM solvers such as $\mathscr{H}$-matrices or FMBEM, making their applications rather limited. In this research, we propose to apply a matrix-free MOR technology based on the Loewner framework to address this issue. The performance of accelerating multiple-frequency BEM simulations has been studied in various scenarios including complex geometries and boundary conditions. Together with the new MABEM and f-FMBEM, a holistic fast BEM solution is formulated to meet industrial needs.

The proposed holistic fast BEM solution offers new capabilities and possibilities to address challenges that were difficult or even unfeasible in the past. In the context of virtual acoustics and auralization, accurate modeling and simulation of room acoustic is fundamental to meeting the challenge. Considering the inadequate modeling of sound waves from many geometric acoustics solvers, wave-based solvers such as BEM become more interesting, especially in complex room scenarios. The proposed fast BEM solution has shown advantages in handling such cases and can be considered a valuable numerical tool for practical room acoustics simulations.  

In addition to robust and fast numerical methods, accurate acoustics simulations also rely on proper inputs, of which material properties are one of the keys. However, such properties are not readily available in practice. Standardized measurement techniques have certain limitations. It poses additional challenges when the target surface has smooth perturbations. Such rough surfaces can be found in water surfaces, absorptive foams with thickness variations, etc. To estimate the rough surface properties in terms of surface shape and surface impedance, a parametric characterization framework is introduced, which benefits from the proposed fast indirect BEM solution. With sufficient quality inputs, the new framework can estimate the surface shape and surface impedance both independently and simultaneously.