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A solution methodology is introduced for the efficient computation of the acoustic field over restricted domains and for a frequency window. Typically, such partial field solutions include, for example, surfaces enclosing the radiating structure or even single points in the computational domain. The multiple-frequency partial-field (MFPF) method starts out by reformulating the finite element matrix system into a suitable shifted form. The DtN map is used as a radiation boundary condition and is interpreted as a low rank update of the matrix problem. The shifted standard form is then approximated by a rational matrix-valued Padé approximant and solved simultaneously over a frequency range. To obtain the Padé approximation, a banded unsymmetric Lanczos process is applied on the standard shifted form exploiting the matrix Padé-via-Lanczos connection. Numerical examples show the feasibility of the outlined procedure.
This paper addresses the efficient solution of acoustic problems in which the primary interest is obtaining the solution only on restricted portions of the domain but over a wide range of frequencies. The exterior acoustics boundary value problem is approximated using the finite element method in combination with the Dirichlet-to-Neumann (DtN) map. The restriction domain problem is formally posed in transfer function form based on the finite element solution. In order to obtain the solution over a range of frequencies, a matrix-valued Padé approximation of the transfer function is employed, using a two-sided block Lanczos algorithm. This approach provides a stable and efficient representation of the Padé approximation. In order to apply the algorithm, it is necessary to reformulate the transfer function due to the frequency dependency in the nonreflecting boundary condition. This is illustrated for the case of the DtN boundary condition, but there is no restriction on the approach which can also be applied to other radiation boundary conditions. Numerical tests confirm that the approach offers significant computational speed-up.
A Krylov subspace projection method which provides simultaneous solutions of the Helmholtz equation at multiple frequencies in one solution step is presented. The projector is obtained with an unsymmetric block Lanczos algorithm applied to a transfer function derived from a finite element discretization. This approach is equivalent to a matrix-valued Padé approximation of the transfer function. The proposed method is an extension of the formulation presented in [J. Comput. Acoust. 8 (2000) 223] to unsymmetric systems and allows the treatment of a much wider range of practical problems, including near-field and fluid–structure interaction computations