Time-harmonic Elastic Wave Equation Research Articles

The monotonicity method for inclusion detection and the time harmonic elastic wave equation

We consider the problem of reconstructing inhomogeneities in an isotropic elastic body using time harmonic waves. Here we extend the so called monotonicity method for inclusion detection and show how to determine certain types of inhomogeneities in the Lamé parameters and the density. We also included some numerical tests of the method.

A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

In this paper, we propose and analyse a numerical method to solve 2D Dirichlet time-harmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure (P-) and Shear (S-) waves via a suitable Helmholtz–Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the L2-norm for the displacement field, in which the contributions to the error associated with the P- and S- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the P- and S- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented.

Unique continuation for the Lamé system using stabilized finite element methods

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.

Determination of piecewise homogeneous sources for elastic and electromagnetic waves

This paper is concerned with inverse source problems for the time-harmonic elastic wave equations and Maxwell's equations with a single boundary measurement at a fixed frequency. We show the uniqueness and a Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains.

El-WaveHoltz: A time-domain iterative solver for time-harmonic elastic waves

We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.

Direct imaging for the moment tensor point sources of elastic waves

We investigate an inverse source problem of the time-harmonic elastic wave equation. Some novel sampling-type numerical schemes are proposed to identify the moment tensor point sources in the Lamé system from near-field measurements. Rigorous theoretical justifications are provided to show that the locations and moment tensors of the elastic sources can be uniquely determined from the multi-frequency displacement data. Several numerical examples are also presented to illustrate the validity and robustness of the proposed method.

Inverse Elastic Scattering for a Random Source

Consider the inverse random source scattering problem for the two-dimensional time-harmonic elastic wave equation with an inhomogeneous, anisotropic mass density. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudo-differential operator. The goal is to recover the principle symbol of the covariance operator from the displacement measured in a domain away from the source. For such a distributional source, we show that the direct problem has a unique solution by introducing an equivalent Lippmann--Schwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principle symbol of the covariance operator can be uniquely determined by the amplitude of the displacement averaged over the frequency band, generated by a single realization of the random source. The analysis employs the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators.

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

A source transfer domain decomposition method for time-harmonic elastic wave equations

We extend the source transfer domain decomposition method (STDDM) proposed by Chen et al., in order to solve the time-harmonic elastic wave equation that is discretized using the spectral element method. Some numerical examples are presented, demonstrating that STDDM can be applied as an efficient preconditioner in the preconditioned GMRES method for solving the PML equation of the time-harmonic elastic wave equation, with constant and heterogeneous wave numbers. Regarding the problem with constant wave number, the preconditioned GMRES method converges within just a small number of iterations when the discretization error is reduced by high order spectral elements.

Multigrid-based ‘shifted-Laplacian’ preconditioning for the time-harmonic elastic wave equation

We investigate the numerical performance of an iterative solver for a frequency-domain finite-difference discretization of the isotropic elastic wave equation. The solver is based on the 'shifted-Laplacian' preconditioner, originally designed for the acoustic wave equation. This preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration. However, the application of multigrid to the elastic case is not straightforward because standard methods, such as point-Jacobi, fail to smooth the S-wave wavenumber components of the error when high P-to-S velocity ratios are present. We consider line smoothers as an alternative and apply local-mode analysis to evaluate the performance of the various components of the multigrid preconditioner. Numerical examples in 2-D demonstrate the efficacy of our method.

Sweeping preconditioners for elastic wave propagation with spectral element methods

We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LD L T factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.

Wavelet based ILU preconditioners for the numerical solution by PUFEM of high frequency elastic wave scattering

Daubechies family of wavelets combined to the Incomplete Lower–Upper (ILU) factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the Partition of Unity Finite Element Method (PUFEM). After applying the discrete wavelet transform (DWT) to each dense block in the final matrix and the known right-hand side, due to the local enrichment by pressure ( P) and shear ( S) plane waves, the resulting linear system is solved by the restarted Generalized Minimum RESidual method (GMRES) with ILU preconditioners allowing fill-in elements in the L and U matrix factors. A reordering algorithm of the vertices based on Gibbs method is also introduced. It leads to a significant reduction of the bandwidth of the wave based Finite Element (FE) matrix and enables the ILU preconditioners to be more effective. To study the performance of the proposed preconditioners, a problem of a horizontal S plane wave scattered by a rigid circular body in an infinite elastic medium is considered. The numerical tests show the good performance of the DWT based ILU preconditioners in improving the rate of convergence of GMRES, for high numbers of approximating P and S plane waves, on coarse mesh grids containing multi-wavelength sized elements. Moreover, the Haar DWT enhances the scalability with respect to the problem size, when the number of the nodal points increases, of the ILU preconditioner which uses the threshold strategy in the control of the fill-in elements. Despite the high level of the conditioning, a good accuracy may be achieved for a discretization level of around 1.9 degrees of freedom per S wavelength, which is far below the rule of thumb of 10 nodal points per wavelength, adopted for the conventional FE.

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

AbstractThe purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77:1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries.The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical analysis of two plane wave finite element schemes based on the partition of unity method for elastic wave scattering

This work deals with a numerical model based on the partition of unity finite element method (PUFEM), for the two-dimensional time harmonic elastic wave equations. This approach consists in enriching the polynomial finite element spaces locally by superimposed shear (S) and pressure (P) plane waves. Two plane wave Finite elements schemes are particularly investigated here and compared from the point of view of levels of accuracy and conditioning. The problem considered is a horizontal S plane wave scattered by a rigid circular body, in an infinite elastic medium. A finite square domain in the vicinity of the scatterer is considered with the analytical solution of the problem imposed on its boundary through a Robin type boundary condition. An error analysis with respect to the mesh size and the plane wave enrichment is carried out and some numerical aspects, related to the conditioning and its behaviour as a function of the frequency and the number of approximating plane waves, are outlined. The aim is to gain a better practical understanding of the conditioning behaviour of the PUFEM in order to improve its reliability. The results show, for the same number of degrees of freedom per wavelength, higher order elements lead to better accuracy than low order elements. However, this is achieved in detriment of the conditioning especially when the number of degrees of freedom per wavelength is too large.

Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method

AbstractIn this paper, we investigate a numerical approach based on the partition of unity finite element method, for the time‐harmonic elastic wave equations. The aim of the proposed work is to accurately model two‐dimensional elastic wave problems with fewer elements, capable of containing many wavelengths per nodal spacing, and without refining the mesh at each frequency. The approximation of the displacement field is performed via the standard finite element shape functions, enriched by superimposing pressure and shear plane wave basis, which incorporate knowledge of the wave propagation. A variational framework able to handle mixed boundary conditions is described. Numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented. The results show the performance of the proposed method in both accuracy and efficiency. Copyright © 2008 John Wiley & Sons, Ltd.

Time-harmonic elasticity with controllability and higher-order discretization methods

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss–Lobatto quadrature points of the elements, and the Gauss–Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge–Kutta time discretization. As a consequence of these choices, only matrix–vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.