Acoustic wave analysis is required in various applications involving parametric analyses across different settings. The acoustic BEM is a numerical method based on solving the discretized boundary integral equation (BIE) corresponding to the Helmholtz equation. It has unique advantage in reducing problem dimensionality and handling exterior problems without domain truncation. However, the computational efficiency of the BEM solutions due to its nonsymmetric and fully-populated matrices is a critical issue. Even with iterative solver based accelerated BEM, large-scale problems and those with high frequencies often necessitate numerous iterations, hampered by ill-conditioned system matrices. A fast direct BEM for 3D acoustic problems is proposed in this thesis, which is more suitable fo...[ Read more ]

Acoustic wave analysis is required in various applications involving parametric analyses across different settings. The acoustic BEM is a numerical method based on solving the discretized boundary integral equation (BIE) corresponding to the Helmholtz equation. It has unique advantage in reducing problem dimensionality and handling exterior problems without domain truncation. However, the computational efficiency of the BEM solutions due to its nonsymmetric and fully-populated matrices is a critical issue. Even with iterative solver based accelerated BEM, large-scale problems and those with high frequencies often necessitate numerous iterations, hampered by ill-conditioned system matrices. A fast direct BEM for 3D acoustic problems is proposed in this thesis, which is more suitable for broadband acoustic simulation of complex structures. The main idea of the fast direct solver is based on the hierarchical matrix, low-rank decomposition and fast matrix inversion formula. The numerical results show that the same level of accuracy and higher computational efficiency can be achieved by using this fast direct BEM compared with the conventional direct BEM. The fast direct solver avoids the convergence problem of the iterative solver, but still requires repeated calculations over the range of parameter variations. To address these challenges, we propose a novel data-driven surrogate modelling approach called B-FNO, which combines BEM and Fourier neural operator (FNO) for wave analysis in varying domains and frequencies. This approach utilizes FNO to map problem boundaries and other parameters to boundary solutions of BIEs. The B-FNO approach significantly improves accuracy compared to well-known neural network surrogate models. Moreover, compared to accelerated BEM, the B-FNO approach requires a much smaller number of iterations. We validate the effectiveness of our method through numerical experiments on a series of 2D and 3D benchmark problems. To address the limitations in efficiency and memory usage of B-FNO when applied to large-scale 3D problems, we further introduce a decomposed B-FNO. By leveraging tensor decomposition and the separable property of fast Fourier transform, the time complexity is significantly reduced from O(N²log N) to O(N²) . The complex boundary conditions and large-scale geometries are considered to demonstrate the potential of the decomposed B-FNO for broad application.