Non-uniform grid accelerated local–global boundary condition (NG-LGBC) for acoustic scattering

Abstract

A global absorbing boundary condition utilizing accelerated integration scheme based on the non-uniform grid (NG) approach is hybridized with a local boundary condition, producing an accurate and efficient mesh truncation scheme for finite method analysis of open-region acoustic scattering problems. The method affords arbitrarily shaped, in particular non-convex boundaries that conform to essentially concave obstacles of dimensions large compared with the wavelength.

Introduction

In solving unbounded problems by the finite element method (FEM), finite difference time domain (FDTD) or finite difference frequency domain (FDFD) methods, one often faces the need to choose between local and global absorbing boundary conditions (ABCs). Local ABCs are usually considered more efficient because they utilize a finite stencil, therefore they have become relatively popular in recent years. One can categorize two groups of local ABCs: The first group is based on local approximations of the one way wave equation or the radiation condition. Early (late 70s) local ABCs used pseudo-differential operators as generalizations of the one-way wave equations. Such ABCs were developed by Engquist and Majda [1], [2]. These conditions were implemented in the context of Maxwell’s equations and the Navier–Stokes equations by Mur [3], Rudy and Strikwerda [4], respectively. Bayliss and Turkel [5], [6], [7], [8] developed an asymptotic expansion that can be considered as a generalization of the Sommerfeld radiation condition and applied it to both time- and frequency-domain calculation. Many other local ABCs, extending the ones mentioned above, such as [9], are available in the literature. Numerically derived local ABCs have been also proposed [10]. The main drawback of these methods is in their limited accuracy and essential dependence on the angle of arrival. The second group of local ABCs overcomes this limitation by employing absorption rather than radiation-based condition. The most notable method in this class is the perfectly matched layer (PML) introduced by Berenger [11]. It appears to yield a major improvement in the reduction of boundary reflections compared to any ABC proposed previously and as such has attracted considerable attention. The PML has been formulated in terms of both Berenger’s original non-Maxwellian equations and Maxwellian forms [12]. The main disadvantage of the PML is the need to surround the computational domain with a thick layer that may add a significant number of unknowns to the problem.

In addition to the aforementioned issues, a major drawback common to all local ABC formulations is the requirement for the boundary to be a convex, typically separable surface such as a box or sphere, which may not conform very well to the shape of the scattering obstacle. The convexity requirement translates into a sizeable “white space”¹ when treating an essentially concave geometry and hence implies a significant additional computational cost.

In contrast to local ABCs, global boundary conditions (GBCs) involve integration over the entire boundary of the computational domain. These formulations can be based on one of the following: (1) Kirchoff-like integration of equivalent sources on a given surface, resulting in the field on an adjacent external surface; (2) construction of an impedance-like operator (viz., a Green’s function) transforming the magnetic (electric) field into the electric (magnetic) field over the surface or, equivalently, building an integral equation for a surface problem. The latter formulation is then discretized via the boundary elements method (BEM) and then incorporated as a dense block into the FEM total matrix (this procedure can be viewed as a FEM–BEM hybridization, see below). Global BCs require no approximations, thus having the potential for providing better accuracy. The boundary can trace the surface of the scattering object, avoiding the convexity requirement, thereby reducing the number of mesh points in the “white space”. The main disadvantage associated with GBCs, however, is the creation of dense rather than sparse matrix blocks, rendering these methods relatively expensive in terms of computer resources. GBCs were originally developed in the early 70s in the context of the Laplace and Helmholz equations [13]. The unimoment method [15] was introduced as generalized impedance relationships in the context of individual cylindrical and spherical wave functions. The early 80s saw the global lookback scheme [14] that enabled the computational domain to be truncated near the sources. The field on the boundary was generated from those field values at retarded times on an interior surface one cell away from the boundary via an integral representation of the field. Non-local boundary conditions, in particular the Dirichlet-to-Neumann (DtN) operator [16], [17], offer the advantage of being exact while allowing fast evaluation. However, these methods require a separable boundary, similarly to the unimoment method. Removal of the requirement of specific boundary shapes has been offered by the difference potential method (DPM) [18] and by the Green’s function method (GFM) [19], [20]. These methods provide a compromise between the efficiency and accuracy of the local and global ABCs, respectively. The GFM is the result of a purely discrete formulation of the Green’s function tailored to a given arbitrary surface, rather than using an outright discretization of the continuous Green’s function. Extensive survey of the ABC literature can be found in [21], [22], [23].

Hybridization methods for GBCs in the context of the FEM–BEM, referred to as the FEM-boundary integral (FEM-BI) approach, are reviewed in [24, Chapters 13.3 and 13.4]. This hybridization scheme possesses the drawbacks in terms of computational cost mentioned above in the context of GBCs, unless an efficient integration method is employed. In order to improve efficiency, discretized integration can be performed by using the fast multipole method (FMM) [25], [26], [27], [28]. This method is known to reduce the computational complexity of each multiplication operation from $\mathcal{O}(N^2)$ to $\mathcal{O}(N^{1\text{.}5})$, where N is proportional to the number of boundary unknowns. Multilevel fast multipole algorithm (MLFMA) can further reduce computational complexity to $\mathcal{O}(N\log N)$ [29], [30], [31], [32], [33], [34]. An alternative fast GBC method, proposed in [35], employs a two level non-uniform grid (NG) algorithm [36] for accelerating the pertinent surface integrations.

For any GBC, artificial resonances resulting from the closed surface integral formulation, may result in a high condition number for the FEM matrix, which in turns causes iterative solution algorithms to slow down or stagnate at times. In particular, if the closed surface has concave portions, that are of primary interest here, additional resonance artifacts may occur due to internal reflections within the concave inlet.

Blending local and global ABCs is a remedy to the aforementioned difficulty. Recently suggested novel GBCs combining the local–global blend with fast integration algorithms are the FEM/adaptive absorbing boundary condition method [37] and its time-domain counterpart using the plane wave time domain (PWTD) method [38].

In this work, it is proposed to alleviate the aforementioned problems by hybridizing the fast scheme of [35] with a conventional local ABC, such as Mur’s in order to achieve enhanced accuracy and improved convergence for problems of the concave type as defined in Section 2. The hybridization scheme, first proposed in [39], is introduced in Section 3. The details of the fast integration scheme are reviewed in Section 4, and numerical tests in Section 5 show that this algorithm is indeed capable of reducing the computational cost of evaluating the boundary integrals from $\mathcal{O}(N^2)$ to $\mathcal{O}(N^{1\text{.}5})$. A multilevel NG algorithm, to be developed in the next phase of this work, will achieve an asymptotic complexity of $\mathcal{O}(N\log N)$. Improved convergence rate of the iterative algorithm, thanks to the proposed hybridization of the boundary conditions, is also demonstrated in Section 5. Conclusions are duly drawn in Section 6.