Frequency domain iterative solver for elasticity with semi‐analytical preconditioner (original) (raw)
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A damping preconditioner for time-harmonic wave equations in fluid and elastic material
Journal of Computational Physics, 2009
A physical damping is considered as a preconditioning technique for acoustic and elastic wave scattering. The earlier preconditioners for the Helmholtz equation are generalized for elastic materials and three-dimensional domains. An algebraic multigrid method is used in approximating the inverse of damped operators. Several numerical experiments demonstrate the behavior of the method in complicated two-dimensional and three-dimensional domains.
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.
Seg Technical Program Expanded Abstracts, 1999
An iterative numerical method for solving the wave equation in an inhomogeneous medium with constant density is presented. The method is based on a Krylov iterative method and enhanced by a powerful preconditioner. For the preconditioner, a complex Shifted-Laplace operator is proposed, designed specifically for the wave equation. A multigrid method is used to approximately compute the inverse of the preconditioner. Numerical examples on 2D problems show that the combined method is robust and applicable for a wide range of frequencies. Extension to 3D is straightforward.
Applied Numerical Mathematics, 2010
The analysis of electromagnetic scattering by electrically large one-side open-ended cavities remains a challenge. Finite element discretisation of the vector wave equation to solve for the electric field inside the cavity leads to ill-conditioned indefinite linear systems of large dimension, a result of the requirement of a fine nearly uniform discrete sampling in the computational domain. Direct methods based on frontal solution techniques to solve the resulting linear system have been used because efficient iterative methods were not available. With the arrival of the shifted-Laplace preconditioner of Erlangga, iterative solution of the indefinite system becomes tractable. This paper discusses the modifications required for application of the shifted-Laplace preconditioner to cavity scattering and some preliminary results of this approach. The shifted-Laplace preconditioner is shown to be very effective for improving the convergence rate of the iterative solution algorithm. However, to be able to handle problems with larger number of degrees of freedom, it is necessary to include a multigrid algorithm to solve the preconditioner system, as this will allow the use of short recurrence Krylov subspace methods, as opposed to the currently employed long-recurrence method, for which the storage requirements of the Krylov basis become unpractically large.
International Journal for Numerical Methods in Engineering, 2010
We present new iterative solvers for large-scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second-order accuracy can be obtained at very small overcost with respect to first-order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2×2 block symmetric indefinite linear system arising from mixed (displacement-pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods.
An Ecient Preconditioner for Iterative Solvers
The method of moments solution of the Maxwell's equations leads to a dense system of complex equations. Direct solution of these equations using LU factorization becomes unwieldy as the size of the scatterer increase in terms of wavelength. Iterative solvers, such as those based on Krylov projection methods, offer an alternative approach for solving large system of equations. Most often, the iterative methods are used in combination with some kind of preconditioning to improve the condition number of the system matrix A in order to acheive accelerated convergence [1-2]. This paper discusses the application of Multi-Frontal Preconditioners (MFPs) for the Krylov projection methods for an efficient solution of the dense system of linear equations. The MFP uses combined unifrontal/multi-frontal approach to handle arbitrary sparsity patterns and enables a general fill-in reduction[3]. The paper specifically focuses on the efficient solution of complex general systems, without making any assumptions regarding the positive definitness of the operators. Performances of several popular Krylov projection methods are presented to demostrate the computational efficiency of the present method, using the MFP.
Preconditioned Mixed Spectral Element Methods for Elasticity and Stokes Problems
SIAM Journal on Scientific Computing, 1998
Preconditioned iterative methods for the indenite systems obtained by discretizing the linear elasticity and Stokes problems with mixed spectral elements in three dimensions are introduced and analyzed. The resulting stiness matrices have the structure of saddle point problems with a penalty term, which is associated with the Poisson ratio for elasticity problems or with stabilization techniques for Stokes problems. The main results of this paper show that the convergence rate of the resulting algorithms is independent of the penalty parameter, the number of spectral elements N and mildly dependent on the spectral degree n via the inf-sup constant. The preconditioners proposed for the whole indenite system are block-diagonal and block-triangular. Numerical experiments presented in the nal section show that these algorithms are a practical and ecient strategy for the iterative solution of the indenite problems arising from mixed spectral element discretizations of elliptic systems.
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
A new iterative solver for the time-harmonic wave equation
GEOPHYSICS, 2006
The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another...