A domain decomposition solver for acoustic scattering by elastic objects in layered media

Abstract

A finite element solution procedure is presented for accurately computing time-harmonic acoustic scattering by elastic targets buried in sediment. An improved finite element discretization based on trilinear basis functions leading to fourth-order phase accuracy is described. For sufficiently accurate discretizations 100 million to 1 billion unknowns are required. The resulting systems of linear equations are solved iteratively using the GMRES method with a domain decomposition preconditioner employing a fast direct solver. Due to the construction of the discretization and preconditioner, iterations can be reduced onto a sparse subspace associated with the interfaces. Numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths.

Introduction

We develop an efficient numerical method for computing time-harmonic acoustic wave scattering by an elastic object in three-dimensional layered media. It is capable of modeling the scattering of sonar signals by undersea targets located in or near the seabed in littoral environments with plane or rippled interface between water and sediment. One application for such problems is the detection of hazardous or/and lost objects buried in sediment. For this purpose it is essential to have a numerical approximation which can accurately predict the scattered field by such targets. Our model problem in this paper is the scattering by a solid aluminum cylinder buried in sediment. A similar experimental test setup was used in [30] to perform measurements. This is an exterior problem which has to be truncated into a bounded domain for a finite element discretization; see Fig. 1.

We consider scattering problems in a frequency range in which the size of the computational domain is of the order of 100 wavelengths. The discretization has to have sufficiently many nodes per wavelength. In the considered frequency range the phase (pollution) error of the solution with a second-order accurate discretization is large unless tens of nodes are used per wavelength [20]. This would lead to huge systems of linear equations which are computationally intractable. Many ways have been proposed to reduce the phase error; see [20], for example. We employ a generalization of the approach proposed in [13] in which standard bilinear finite elements are used for two-dimensional Helmholtz problems together with a modified quadrature rule. This leads to fourth-order phase accuracy and a higher order approximation on orthogonal uniform meshes for a constant speed of sound. Actually it coincides with the nine-point fourth-order compact finite difference scheme [29] in the homogeneous medium. A related fourth-order finite difference method for the Helmholtz equation was considered in [12]. The resulting matrix has exactly the same structure and complexity as the one obtained by using the standard Gauss quadrature rule. As shown in Section 5.1 it drastically increases the performance and capability of our finite element modeling. The scheme with eight points per wavelength sufficiently well achieves the resolution we need and thus leads to a linear system with much smaller numbers of unknowns.

The resulting system of linear equations is too large to be solved using a direct method. We propose an iterative solution procedure with a decomposition preconditioner based on a domain embedding approach [15], [16], [17], [21], [22], [24], [26]. In our solution procedure the computational domain is decomposed into the near field subdomain which encloses an elastic target (the interior box in Fig. 1) and the far-field subdomain (the rest of the rectangle computational domain Π). In the far-field subdomain the discretization is based on an orthogonal uniform mesh which is locally adapted to the interface between the water and sediment. We use a multiplicative domain decomposition, i.e. we develop a preconditioning method for the resulting linear equation that partitions the matrix into the decoupled blocks. We will use a separable preconditioner based on the perfectly vertically layered media in our solution procedure for the far-field subdomain. The far-field preconditioner is based on a fast direct solver [16], [33]. Since the media is vertically layered with the wavy interface, our preconditioner coincides with the system matrix except the rows corresponding to unknowns near the interfaces. Thus, we can reduce iterations on a small sparse subspace as has been shown in [22], [24], for example, and Section 4.3. This procedure reduces GMRES iterates onto a sparse subspace which leads to much reduced storage and computational requirements. The GMRES iteration is evaluated by the partial solution method based on a cyclic reduction type fast direct solver. This reduction makes our iterative method extremely efficient and enables us to solve systems with billions of unknowns as our numerical examples demonstrate. With multigrid preconditioners like in [1], [9], [10] and with nonorthogonal grids in [28] such reduction cannot be made and the solution of the model problem would require much more memory and computation. With the separation-of-variables preconditioner in [31] the reduction of the iterations onto a sparse subspace can be made with the model problem.

A similar domain decomposition approach to the one considered here was used for two-dimensional problems in [4], [5], [23]. Specifically, the approach in [23] is extended to the three-dimensional case. Such extension is nontrivial and involves the use of higher order discretization in the layered media and efficient direct solver for the 3D separable preconditioner. Another preconditioning technique for scattering problems in layered media without an object has been considered in [1], [10], [28], [31], for example, and with an object in [19].

The outline of the paper is the following. We begin by defining a model problem in Section 2. For the finite element discretization a weak formulation, meshes, a domain decomposition, the forming of linear systems and a modified quadrature rule are described in Section 3. The iteration with a domain decomposition preconditioner is consider in Section 4. Also, the solution procedures for the far-field and near-field preconditioners are given and how iterations can be performed in a sparse subspace is described. Numerical experiments are presented in Section 5. They include an accuracy study for the discretizations and an efficiency study of the iterative solution. The paper ends with the conclusions in Section 6.