An interpolation-based fast-multipole accelerated boundary integral equation method for the three-dimensional wave equation
Introduction
The application of the fast multipole method (FMM) [1], [2], [3], [4] to accelerate the time-domain boundary integral equation method (TDBIEM) still remains a challenge, whereas its frequency-domain counterpart (including the static limit) has been investigated extensively [5].
In particular, for wave problems, the plane-wave time-domain (PWTD) algorithm is known as the time-domain version of the FMM [6]. The core of the PWTD algorithm is the plane-wave expansion of the fundamental solution (or the retarded Greenʼs function for free space) of the wave equation. This expansion corresponds to the multipole expansion (in ordinary FMMs) that allows a given kernel to be re-expressed as a degenerate form such as , where x and y represent the positions of the target and source, respectively [7]. Using the plane-wave expansion, one can develop a fast method to evaluate the retarded potential or time-dependent pairwise interactions among targets and sources by systematically grouping them in space–time. The computational cost of the conventional integral equation (IE) solver (that adopts an ordinary marching-on-in-time (MOT) scheme) is , whereas that of the multilevel PWTD-enhanced IE solver scales as , where and denote the spatial and temporal degrees of freedom, respectively. The value of n is two for the complete spherical expansion and one for finite-cone representations [8]. Thus far, the PWTD algorithm has been successfully applied to accelerate the time-domain IE solvers for electromagnetics [6], [8], [9], [10], [11], [12], acoustics [13], [14], and elastodynamics [15], [16]. Concurrently, the accelerated Cartesian expansion (ACE) algorithm is proposed to resolve the breakdown of the PWTD algorithm in the low-frequency regime [17]. The ACE algorithm uses the Taylor expansion instead of the plane-wave expansion and attains a computational complexity of in that regime.
The proposed algorithm is similar to the PWTD algorithm; in particular, the mechanism of time-gating, which allows to a set of elapsed time-steps to be managed together at the current time-step (see Section 4), belongs to the PWTD algorithm. Specifically, because the low-frequency regime is the prime target, the proposed algorithm can be identified as an alternative for the ACE algorithm rather than that for the PWTD algorithm. However, the proposed algorithm uses neither the plane-wave expansion nor the Taylor expansion. Instead, it interpolates the fundamental solution with respect to the spatial and temporal variables for the target and source in order to reduce the fundamental solution to a degenerate form. Using this approach, one can develop another fast algorithm with a computational complexity of (where or 1/2; see Section 4.4). This algorithm is slightly slower than the PWTD and ACE algorithms but is faster than the conventional algorithm.
The class of FMMs that exploit interpolation, as in this study, is termed the interpolation-based FMM. In general, an FMM in this class is kernel-independent, i.e. it can manage a certain class of kernels (or pairwise interactions) in a single formulation. (Note that interpolation is not the only method to construct kernel-independent FMMs; see [18] for example.) In addition, broadly speaking, the mathematical formulation and numerical implementation of the interpolation-based FMMs are simpler than those of the classical FMMs specialised for individual kernels (e.g. [1], [2], [3], [4], [6]).
Presently, the class of FMMs includes the one-dimensional FMM [19], the black-box FMM [20], the directional FMM [21], and the FMM for low-frequency three-dimensional (3D) electromagnetics [22]. These FMMs correspond to the static or frequency-domain problems. Meanwhile, the concept of the interpolation-based FMM was extended to the transient heat problem in [23], where a time-domain FMM for the heat kernel was proposed to reduce the computational cost from to in order to evaluate the heat potential.
From the viewpoint of kernel independency, the proposed FMM for the wave equation can be regarded as a variant of the time-domain interpolation-based FMM for the heat equation in [23]. In fact, these FMMs have some similarities in their formulations. The difference essentially originates in the properties of the kernels in relevant boundary integral equations. To be specific, the heat kernel is very smooth and decays exponentially when the distance between the target and source increases, i.e. it decays as , where r and T denote distances in space and time, respectively. Hence, one can ignore pairwise interactions when r is sufficiently large. In contrast, the kernel for the wave equation is always nondifferentiable on the wavefront and decays more slowly, i.e. . Therefore, one needs to consider a formulation that is different from the heat equation, especially in the multipole-to-local (M2L) operation.
The remainder of this paper is organised as follows: Section 2 summarises the basics of the conventional MOT-based TDBIEM for the wave equation. Section 3 highlights the mathematical aspect of the proposed interpolation-based FMM used to accelerate the TDBIEM. In Section 4, from the formulae derived in Section 3, the proposed algorithm is developed in a similar manner of the multilevel PWTD algorithm. In Section 5, the constructed fast TDBIEM is numerically investigated to validate its computational accuracy and time in comparison with the conventional TDBIEM.
Section snippets
Time-domain boundary integral equation method for the wave equation
This section summarises the conventional MOT-based TDBIEM (e.g. [24]), which is accelerated using the fast algorithm that will be described in Sections 3 and 4. Following the accurate and memory-efficient TDBIEM for elastodynamics [25], [26], this study considers a TDBIEM (for the wave equation) that is based on the collocation method that uses the piecewise-constant and piecewise-linear bases for space and time, respectively. The selection of these bases is not absolutely necessary, but the
Formulation of time-domain interpolation-based FMM
Let us develop a fast method to evaluate the RHS of (4). To this end, this study proposes an interpolation-based FMM, which is based on approximating the integral kernel(s) of interest through interpolation. This section derives the formulae to construct the FMM discussed in Section 4.
Algorithm
Using the formulae derived in the previous section, one can construct a fast algorithm to solve the BIE in (2) with the computational complexity of , where δ is 1/3 or 1/2 according to the assumption of the spatial distribution of boundary elements. The details of the algorithm are described below along with four pseudo codes.
Numerical experiments
This section addresses the numerical aspects of the proposed interpolation-based TDBIEM. First, the interpolation of the kernels U and W in (5a), (5b) is examined in Section 5.1. Next, in Section 5.2, the accuracy and performance of the proposed TDBIEM are validated by performing analysis for an external scattering problem. Finally, in Section 5.3, a simulation regarding architectural acoustics is demonstrated to observe the feasibility of the proposed method to solve practical problems. All
Conclusion
A novel fast time-domain boundary integral equation method (TDBIEM) to solve initial–boundary value problems for the three-dimensional wave equation in the low-frequency regime is proposed in this paper, based on the studies of the plane-wave time-domain (PWTD) algorithm for wave problems [6], [14] and the interpolation-based fast multipole method (FMM) particularly for the transient heat problem [23]. The cubic Hermite interpolation, which uses a finite difference for the first-order
Acknowledgements
The author would like to thank Dr. H. Yoshikawa of Kyoto University for his useful discussions especially on the conventional TDBIEM. This work was supported by JSPS KAKENHI (Grand-in-Aid for Challenging Exploratory Research) Grant Number 24656072.
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