Analytic Preconditioners for the Boundary Integral Solution of the Scattering of Acoustic Waves by Open Surfaces (original) (raw)
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Frequency-robust preconditioning of boundary integral equations for acoustic transmission
Journal of Computational Physics
The scattering and transmission of harmonic acoustic waves at a penetrable material are commonly modelled by a set of Helmholtz equations. This system of partial differential equations can be rewritten into boundary integral equations defined at the surface of the objects and solved with the boundary element method (BEM). High frequencies or geometrical details require a fine surface mesh, which increases the number of degrees of freedom in the weak formulation. Then, matrix compression techniques need to be combined with iterative linear solvers to limit the computational footprint. Moreover, the convergence of the iterative linear solvers often depends on the frequency of the wave field and the objects' characteristic size. Here, the robust PMCHWT formulation is used to solve the acoustic transmission problem. An operator preconditioner based on on-surface radiation conditions (OSRC) is designed that yields frequencyrobust convergence characteristics. Computational benchmarks compare the performance of this novel preconditioned formulation with other preconditioners and boundary integral formulations. The OSRC preconditioned PMCHWT formulation effectively simulates large-scale problems of engineering interest, such as focused ultrasound treatment of osteoid osteoma.
Journal of Computational Acoustics, 1999
E cient iterative methods for the numerical solution of three-dimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a non-re ecting boundary condition on the arti cial boundary. The nite element discretization of the approximate boundary value problem is performed using locally tted meshes, and algebraic ctitious domain methods with separable preconditioners are applied to the solution of the arising mesh equations. These methods are based on imbedding the original domain into a larger one with a simple geometry (for example, a sphere or a parallelepiped). The iterative solution method is realized in a low-dimensional subspace, and partial solution methods are applied to the linear systems with the preconditioner. Results of numerical experiments demonstrate the e ciency and accuracy of the approach.
A direct boundary integral equation method for the acoustic scattering problem
Engineering Analysis with Boundary Elements, 1993
This paper presents a direct boundary integral equation method for solving the exterior Neumann problem of the Helmholtz equation which is a mathematical formulation of the acoustic scattering problem at a perfectly hard body. It is proved that the integral equation obtained from the Helmholtz representation is equivalent to the original boundary value problem and it has a unique solution in a suitable Sobolev space in the framework of pseuo-differential operators. Moreover, the numerical treatment and error estimate are given by using a Galerkin approximation.
2015
Frequency-domain waveform modeling in the acoustic and elastic approximations requires the solution of large illconditioned linear systems. In the context of frequencydomain full waveform inversion, the solutions of these systems are required for a large number of sources (i.e. righthand sides). Because of their tremendous memory requirements, direct solvers are not yet adapted to the solution of 3D elastodynamics equations. We are thus interested in the use of efficient iterative solvers adapted to the solution of these systems. The CGMN method has shown robust convergence properties for 2D and 3D elastic problems in highly heterogeneous media, compared to standard Krylov methods, but still requires a large number of iterations to reach sufficient accuracy. In this study, the design of an efficient preconditioning strategy adapted to this method is investigated. This preconditioner is computed as a sparse approximate inverse of a heavily damped wave propagation operator. In addition, the single seed method is used to increase the efficiency of the solver for multiple right-hand sides. The efficiency of these two combined strategies is evaluated on the 2D BP2004 model in the visco-acoustic approximation, up to 40 Hz. An overall time speed-up equal to 3 and a reduction of the number of iterations by a factor 10 are observed.
An approach to iterative solution of 3D acoustic wave equation in the frequency domain is introduced, justified and verified numerically. It is based on special one- and two-level preconditioners, which are constructed by means of inverse operator for complex damped Helmholtz equation with a depth dependent coefficient. An essential element of the process is computing how these preconditioners acts on a 3D vector. This computation is achieved by performing 2D Fast Fourier Transform along lateral coordinates, followed by solving a number of ordinary differential equations with respect to depth. Both of these operations are effectively parallelized, thus allowing efficient computation. For media with strong lateral variations, such preconditioner can be applied in two stages to increase the rate of convergence. Results of numerical experiments demonstrate good accuracy and acceptable computation times.
Engineering Analysis With Boundary Elements, 2003
Boundary element discretization of the Kirchhoff-Helmholtz integral equation gives rise to a linear system of equations. This system may be solved directly or iteratively. Application of direct solvers is quite common but turns out to be inefficient for large scale problem with 10,000 unknowns and more. These systems can be solved on behalf of iterative methods. This paper is dedicated to testing performance of four iterative solvers being the Restarted Bi-Conjugate Gradient Stabilized algorithm, the Conjugate Gradient method applied to the normal equations (CGNR), the Generalized Minimal Residual (GMRes) and the Transpose Free Quasi Minimal Residual. For that, we distinguish between internal and external problems. Performance of iterative solvers with respect to problem size, polynomial degree of interpolation, wave-number, wave-number over problem size, absorption at surface, and smoothness of the surface is investigated. Furthermore, the effect of diagonal preconditioning is illuminated. All examples consist of different meshes of up to more than 100,000 elements. In general, the methods perform well for the internal problems, a duct problem, a sedan cabin compartment and a fictitious small concert hall. GMRes proves to solve the problems most efficiently. External problems appear more challenging due to the hypersingular operator of the Burton and Miller formulation. Scattering of a plane wave at a sphere and at a cat's eye are investigated as well as a tire noise problem. The first two are remarkably efficiently solved in the medium and high frequency range by CGNR whereas the tire noise example is only solved by GMRes. In all examples, at least one or two solution methods turn out to require less operations than a direct solver. The effect of diagonal preconditioning is marginal especially for higher frequencies. q
GEOPHYSICS, 2013
We tested a biconjugate gradient stabilized (BiCGSTAB) solver using a multigrid-based preconditioner for solving the acoustic wave (Helmholtz) equation in the frequency domain. The perfectly matched layer (PML) method was used as the radiation boundary condition (RBC). The equation was discretized using either a second-or fourth-order finite-difference (FD) scheme. The convergence of an iterative solver depended strongly on the RBC used because the spectrum of the discretized equation also depends on it. We used a geometric multigrid approach to construct a preconditioner for our FD frequency-domain (FDFD) forward solver equipped with the PML boundary condition. For efficiency, this preconditioner was only constructed using a second-order FD scheme with negligible attenuation inside the PML domain. The preconditioner was used for accelerating the convergence rate of the FDFD forward solver for cases when the discretization grids were oversampled (i.e., when the number of discretization points per minimum wavelength was greater than 10). The number of multigrid levels was also chosen adaptively depending on the number of discretization grids. We found that the multigrid preconditioner can speed up the total computational time of the BiCGSTAB solver for oversampled cases or at low frequencies. We also observed that the BiCGSTAB solver using an accurate PML boundary condition converged for realistic SEG benchmark models at high frequencies.
Lecture Notes in Computational Science and Engineering, 2005
The construction of accurate generalized impedance boundary conditions for the three-dimensional acoustic scattering problem by a homogeneous dissipative medium is analyzed. The technique relies on an explicit computation of the symbolic asymptotic expansion of the exact impedance operator in the interior domain. An efficient pseudolocalization of this operator based on Padé approximants is then proposed. The condition can be easily integrated in an iterative finite element solver without modifying its performances since the pseudolocal implementation preserves the sparse structure of the linear system. Numerical results are given to illustrate the method.
International Journal for Numerical Methods in Engineering, 2005
The present text deals with the numerical solution of two-dimensional high-frequency acoustic scattering problems using a new high-order and asymptotic Padé-type artificial boundary condition. The Padé-type condition is easy-to-implement in a Galerkin least-squares (iterative) finite element solver for arbitrarily convex-shaped boundaries. The method accuracy is investigated for different model problems and for the scattering problem by a submarine-shaped scatterer. As a result, relatively small computational domains, optimized according to the shape of the scatterer, can be considered while yielding accurate computations for high-frequencies.
An Incomplete Lu Preconditioner for Problems in Acoustics
Journal of Computational Acoustics, 2005
We present an incomplete LU preconditioner for solving discretized Helmholtz problems. The preconditioner is based on an analytic factorization of the Helmholtz operator. This allows us to take the physical properties of the acoustics problem modeled by the Helmholtz equation into account in the preconditioner. We show how the parameters in the preconditioner can be chosen in order to make it effective. Numerical experiments show that the new preconditioner leads to convergent iterative methods even for large wave numbers, and it outperforms classical ILU preconditioners by a large margin.