Non-modal Acceleration Methods for Broadband Acoustic Simulation (Niet-modale versnellingsmethoden voor breedbandige akoestische simulatie)
Non-Modal Acceleration Methods for Broadband Acoustic Simulation
Lenzi, Marcos; R0253219
Numerical modelling has become an indispensable tool in engineering design and analysis. In acoustics, many numerical methods are available for a given class of problems and/or frequency ranges of analysis. At low frequencies, deterministic methods such as the Finite Element Method (FEM) and Boundary Element Method (BEM) are often applied. In this family of methods, in order to fully characterize a system, the solution of the Helmholtz equation is commonly required over a broad frequency range. The simplest approach which consists in solving the system of equations (obtained from the discretization) for each frequency, becomes computationally prohibitive for fine frequency increments, particularly when dealing with large systems, like those encountered when addressing mid-frequency problems. This thesis proposes the development of computational algorithms that lead to Fast Frequency Sweep (FFS) approaches for the FEM and BEM, where reduced-order models are built via Padé approximations. In the fast frequency sweep for FEM, many well-established acceleration techniques are available for systems exhibiting polynomial frequency dependency of second-order kind. This thesis treats systems of more complicated wavenumber dependency, likely to be encountered when applying frequency dependent boundary conditions and/or loadings. The Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) is selected as the method of choice. The method is benchmarked first against the Second-Order Arnoldi (SOAR) algorithm on a simple second-order system. Then it is applied to realistic large-scale interior and exterior Helmholtz problems exhibiting high-order polynomial or rational frequency behaviour. In either case, the proposed methodology is shown to reduce the computational time of the frequency sweep by an order of magnitude when compared to the direct approach. In the BEM, the development of an efficient approach for computing frequency sweeps is more involved. The matrices arising from Boundary Element (BE) discretizations are fully populated and their assembly is computationally demanding. Therefore, the two operations performed when computing the response at one frequency, namely assembling the system matrix and solving the associated linear system, may be of the same order of magnitude in terms of the required computational time. To provide a reasonable speedup for the frequency sweep problem, an efficient algorithm must accelerate both operations. Hence, the proposed approach possesses two distinct features. First, it bypasses forming the system matrices for each individual frequency by assembling them only at a few master frequencies. The matrices at the rest of the frequencies are obtained by interpolation. Moreover, it avoids solving a dense linear system at each frequency by extrapolating the response around some expansion frequencies. This is achieved by constructing Padé approximants via the WCAWE algorithm. The method is applied on two numerical examples: one exterior academic problem consisting of a sphere with a vibrating cap radiating in free field; and a vehicle interior/exterior acoustics problem. In both cases, very good accuracy levels can be maintained while providing significant speed-up factors.