Efficient technique in low-frequency fast multipole boundary element method for plane-symmetric acoustic problems
Introduction
The fast multipole boundary element method (FMBEM), which is a highly efficient boundary element method (BEM) with the use of the fast multipole method (FMM) [1], [2], has been widely studied for the Helmholtz equation [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], as well as for many other equations. The standard BEM requires an operation count even with appropriate iterative solvers along with an memory requirement, where N represents the degrees of freedom (DOF). The FMBEM can reduce both of these to , where and , depending on the number of iterations in the iterative process, the problem geometry, and the implementation.
In the field of acoustics, plane-symmetric problems are often analyzed. Examples are half-space problems on an infinite rigid plane. A case in which plane-symmetric objects such as noise barriers and apartment buildings are on an infinite rigid plane can be a symmetric problem about two or three planes [16], [17]. When a sound field with planes of symmetry perpendicular to each other is analyzed using the standard BEM, where or 3, one can use Green's function that includes the contribution from image regions with respect to the planes of symmetry. This reduces the operation count to with a classical direct solver, or with an iterative solver, along with lowering the memory requirement to , because the calculation boundary is limited to of the whole [18], [19]. The FMBEM cannot directly adopt this technique because Green's functions are expressed by multipole and local expansions, though a couple of efficient techniques in the FMBEM for plane-symmetric acoustic problems have been proposed [8], [14].
One of these techniques has been proposed in the framework of an FMBEM that uses a high-frequency diagonal form proposed by Rokhlin [20] (high-frequency FMBEM: HF-FMBEM) in Ref. [8]. The HF-FMBEM generally uses so-called far-field signature functions, which are functions of a unit vector determining the direction of the plane wave propagation. This technique is based on a symmetrical relation among these functions, and it reduces both the operation count and required memory to about of those using the standard HF-FMBEM. This technique has the advantage that one does not have to calculate or store the far-field signature functions for image regions.
Another efficient technique has been proposed in the framework of an FMBEM for low-frequency problems, which is based on the original multipole expansion theory (low-frequency FMBEM: LF-FMBEM) [14]. Being different from the high-frequency case mentioned above [8], this technique uses the half-space Green's function, and the size of a hierarchical cell structure used in the FMBEM can be limited to a half-region. This technique, however, has the drawback that the operation count and required memory for local expansion coefficients do not decrease, while those for multipole expansion coefficients are halved. Moreover, it is unclear whether or not the limitation of the hierarchical cell size improves the computational efficiency.
In the present paper, we propose another efficient technique in the LF-FMBEM for plane-symmetric acoustic problems, which is based on the symmetries of multipole expansion coefficients produced by monopole or dipole sources. In this technique, neither the multipole nor local expansion coefficients for image regions have to be calculated and stored in RAM, as in the technique for the HF-FMBEM mentioned above [8]. The proposed technique can be extended to symmetrical problems up to , and it is straightforwardly applicable to a variety of formulations for the BEM, such as the hypersingular formulation, the Burton–Miller formulation [21] to avoid fictitious eigenfrequency difficulties, the indirect formulation that can be used for problems with degenerate boundaries [22], [23] and avoiding fictitious eigenfrequency difficulties [15], [24], and their mixed formulations.
The outline of the LF-FMBEM is presented in Section 2. Section 3 derives mathematical symmetries of multipole expansion coefficients. Section 4 presents concrete computational procedures for plane-symmetric acoustic problems, based on these symmetries. In Section 5, numerical experiments using this technique are performed; these show an ideal improvement of computational efficiency, with the proposed technique reducing both the computation time and required memory to about of those using the standard LF-FMBEM. One of the numerical experiments also shows that the computational efficiency for the LF-FMBEM depends very little on the size of a hierarchical cell structure. Section 6 concludes the paper.
Section snippets
Outline of the LF-FMBEM
Here we present the outline of the LF-FMBEM, preceded by those of the BEM and LF-FMM. In the subsection on the BEM, we describe not only the singular formulation but also the indirect one, because the latter is used for calculation in Section 5. Throughout the present paper, the time convention is used.
Symmetries of multipole expansion coefficients
Here we derive mathematical symmetries of multipole expansion coefficients.
An efficient LF-FMBEM for plane-symmetric sound fields
An efficient technique applicable to plane-symmetric problems with one to three planes of symmetry perpendicular to each other is presented here. The computational procedures basically correspond to the six steps of the LF-FMBEM described in Section 2.3 and Ref. [15].
Numerical results
Here we perform numerical experiments to validate the efficient technique just described.
Conclusions
An efficient technique for plane-symmetric acoustic problems has been proposed in the framework of the FMBEM for low-frequency problems. We derived simple mathematical symmetries of multipole expansion coefficients for a plane-symmetric sound field produced by monopole or dipole sources. We presented concrete computational procedures for this technique based on these symmetries, and we discussed its computational efficiency. In the proposed technique, neither multipole nor local expansion
References (29)
- et al.
A multipole Galerkin boundary element method for acoustics
Eng Anal Boundary Elem
(2004) - et al.
A wideband fast multipole method for the Helmholtz equation in three dimensions
J Comput Phys
(2006) - et al.
Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems
Eng Anal Boundary Elem
(2009) Diffraction of sound around barriers: use of the boundary elements technique
J Sound Vib
(1980)Diagonal forms of translation operators for the Helmholtz equation in three dimensions
Appl Comput Harmonic Anal
(1993)On calculation of sound fields around three dimensional objects by integral equation methods
J Sound Vib
(1980)Rapid solution of integral equations of classical potential theory
Comput Phys
(1983)The rapid evaluation of potential fields in particle systems
(1988)- et al.
Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: setup and validation
Acta Acust Acust
(2002) - et al.
Fast multipole boundary element method for large-scale steady-state sound field analysis. Part II: examination of numerical items
Acta Acust Acust
(2003)