Iterative solution methods for large-scale finite element models in structural acoustics.

摘要

Finite-element discretizations of time-harmonic structural acoustics problems in exterior domains result in large, sparse systems of linear equations with complex symmetric coefficient matrices. In many situations, these matrix problems need to be solved repeatedly for different right-hand sides, with the same coefficient matrix. This work is concerned with the design and implementation of efficient Krylov-subspace iterative methods for solving these matrix problems.; The discretization of exterior problems, which involve an infinite fluid domain, is achieved by introducing a truncation boundary, and posing a Dirichlet-to-Neumann (DtN) map on it which incorporates the asymptotic behavior of the solution. The DtN map involves non-local spatial integrals whose discretization destroys the sparsity of the global coefficient matrix. We describe a novel discretization of the DtN map that allows the evaluation of matrix-vector products, used in iterative methods, without storage penalties related to its non-local nature.; The study of acoustic radiation and scattering problems involves solution of linear systems with the same coefficient matrix, but different right-hand sides. More generally, linear systems with multiple right-hand sides are also encountered in other areas of engineering analysis and design. We describe a block quasi-minimal residual (BL-QMR) algorithm for the simultaneous solution of non-Hermitian linear systems with multiple right-hand sides. The BL-QMR algorithm is a block Krylov-subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences.; A J-symmetric variant of the BL-QMR method is also introduced to exploit the symmetry of coefficient matrices such as those arising in acoustics. Extensive numerical tests on acoustics problems show that, instead of solving each of the multiple linear systems individually, it is always more efficient to employ BL-QMR. The importance of deflations in finite-precision arithmetic and their effect on convergence is also clearly illustrated.; A general approach for implementation of unstructured grid computations on distributed-memory multi-processor computers is also described. Based on the proposed discretization of DtN condition, as well as a suitable preconditioner, an iterative solution approach is described that does not require the assembly of any global matrix, and its performance is illustrated on the Connection Machine CM-5 system.