energy conserving explicit local time stepping for second order wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena to overcome that stability restriction local time stepping methods are developed which allow arbitrarily small time steps precisely where small elements in the mesh are located when combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix the resulting discrete numerical scheme is explicit is inherently parallel and exactly conserves a discrete energy starting from the standard second order leap frog scheme time stepping methods of arbitrary order of accuracy are derived numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory

mixed perfectly matched layers for direct transient analysis in 2d elastic heterogeneous media abstract we are concerned with elastic waves arising in plane strain problems in an elastic semi infinite arbitrarily heterogeneous medium specifically we discuss the development of a new mixed displacement stress formulation for forward elastic wave simulations in perfectly matched layer pml truncated heterogeneous media to date most pml formulations split the displacement and stress fields resulting in non physical components for each field in this work we favor unsplit schemes primarily for the relative ease by which the resulting forms can be incorporated into existing codes the ease by which the resulting semi discrete forms can be integrated in time and the ease by which they can be used in adjoint formulations arising in inverse problems contrary to most past and current developments we start by following classical lines and apply complex coordinate stretching to the governing equations in the frequency domain while retaining both displacements and stress quantities as unknowns with the aid of auxiliary variables the resulting mixed form is rendered second order in time thereby allowing the use of standard time integration schemes we report on numerical simulations demonstrating the stability and efficacy of the approach

mixed spectral finite elements for the linear elasticity system in unbounded domains in this paper we present a mixed formulation of a spectral element approximation of the linear elasticity system after studying the main features of this approach we construct perfectly matched layers pmls for modeling unbounded domains then algorithmic issues are discussed and numerical results are given

optimized higher order time discretization of second order hyperbolic problems construction and numerical study we investigate explicit higher order time discretizations of linear second order hyperbolic problems we study the even order 2m schemes obtained by the modified equation method we show that the corresponding cfl upper bound for the time step remains bounded when the order of the scheme increases we propose variants of these schemes constructed to optimize the cfl condition the corresponding optimization problem is analyzed in detail the optimal schemes are validated through various numerical results

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

efficient computation of the sound fields above a layered porous ground an efficient computation of sound fields due to a monopole source placed above a porous layer is presented this paper examines an improved scheme whereby the steepest descent path is selected for the numerical evaluation of the sommerfeld integral along the steepest descent path a standard gaussian hermite quadrature can be used to calculate the sound fields effectively the suggested numerical scheme is accurate at all frequencies except in the very near field the proposed method is more numerically efficient than other computational schemes especially at long ranges and high source frequencies

on the long range propagation of sound over irregular terrain abstract the theory of sound propagation over randomly irregular nominally plane terrain of finite impedance is discussed the analysis is an extension of the theory of coherent scatter originally proposed by biot 1 for an irregular rigid surface it combines biot s approach wherein the surface irregularities are modeled by a homogeneous distribution of hemispherical bosses with more conventional analyses in which the ground is modeled as a smooth plane of finite impedance at sufficiently low frequencies it is shown that the interaction of the surface irregularities with the near field of a ground based source leads to the production of surface waves which are effective in penetrating the ground shadow zone predicted for a smooth surface of the same impedance

sound propagation above an impedance boundary an asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method numerical calculation indicates that the relatively simple solution is more accurate than the thomasson asymptotic solution and the chien soroka solution furthermore it is shown that another solution simplified still more is also in good agreement with the exact solution

a hybrid method combining the edge source integral equation and the boundary element method for scattering problems a hybrid method for acoustic scattering problems is studied in this paper the boundary element method is combined with a recently developed edge diffraction based method j acoust soc am 133 pp 3681 3691 2013 although the edge diffraction method has been shown to provide accurate results for convex rigid objects at a very attractive computational cost it has some numerical challenges for certain radiation directions the hybrid method suggested here has the same structure as the boundary element method bem a first step where the sound pressure is calculated on the surface of the scattering object and a second step where the scattered sound is obtained at any external receiver point in this method the edge diffraction based method is used for the first step and then the calculation of the scattered sound is calculated a la bem by means of the kirchhoff helmholtz integral equation several benchmark cases are studied and the results are compared to different methods

a conjugated infinite element method for half space acoustic problems many acoustic problems especially in environmental acoustics involve half space domains bounded by a plane subjected to normal admittance boundary conditions in the low frequency domain the numerical treatment of such problems usually relies on boundary element methods based on a particular green s function suited for the half admittance plane in the present paper an alternative hybrid finite infinite element scheme is proposed the method relies on a direct treatment of nonhomogeneous boundary conditions along infinite element edges or faces the procedure is validated through comparisons with an available reference solution

a parameter free perfectly matched layer formulation for the finite element based solution of the helmholtz equation this paper presents a parameter free perfectly matched layer pml method for the finite element based solution of the helmholtz equation we employ one of bermudez et al s unbounded absorbing functions for the complex coordinate mapping underlying the pml with this choice the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost characterised by the number of elements in the bulk and the pml regions is the thickness of the perfectly matched layer pml we show that for the case of planar waves the absorbing function performs best for pmls whose thickness is much smaller than the wavelength we then perform extensive numerical experiments to explore its performance for non planar waves considering domain shapes with smooth and polygonal boundaries different solution types smooth and singular and a wide range of wavenumbers k to identify an optimal range for the normalised pml thickness k pml such that within this range the error introduced by the presence of the pml is consistently small and insensitive to change this implies that if the pml thickness is chosen from within this range no further pml optimisation is required i e the method is parameter free we characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh bermudez et al s type a pml damping function performs best for k pml 1 extensive numerical experiments reveal an optimal range of pml thicknesses i opt for k pml i opt the error is small and insensitive to change choosing k pml from anywhere within i opt yields a parameter free pml method

explicit finite element perfectly matched layer for transient three dimensional elastic waves the use of a perfectly matched layer pml model is an efficient approach toward the bounded domain modelling of wave propagation on unbounded domains this paper formulates a three dimensional pml for elastic waves by building upon previous work by the author and implements it in a displacement based finite element setting the novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration thus making it practicable for use in large scale three dimensional dynamic analyses an efficient method of calculating the strain terms in the pml is developed in order to take advantage of the lack of the overhead of solving equations at each time step the pml formulation is studied and validated first for a semi infinite bar and then for the classical soil structure interaction problems of a square flexible footing on a i half space ii layer on a half space and iii layer on a rigid base numerical results for these problems demonstrate that the pml models produce highly accurate results with small bounded domains and at low computational cost and that these models are long time stable with critical time step sizes similar to those of corresponding fully elastic models copyright 2008 john wiley sons ltd

a new family of fourth order locally one dimensional schemes for the three dimensional wave equation in this paper we present a new family of locally one dimensional lod schemes with fourth order accuracy in both space and time for the three dimensional 3d acoustic wave equation it is well known that high order explicit schemes offer efficient time steppings but with restrictive cfl conditions implicit discretization gives unconditional stable schemes but with inefficient time steppings our lod schemes can be seen as a compromise between explicit and implicit schemes in the way that our schemes have more relax cfl conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations which is more efficient than implicit schemes moreover the new scheme is four layer in time and three layer in space so that boundary conditions can be imposed in the classical way the computations of the initial conditions for the three intermediate time layers are explicitly constructed furthermore the stability condition is derived and the restriction for the time step is given explicitly finally numerical examples are completed to show the performance of our new method

remark on essential selfadjointness of dirac operators with coulomb potentials on considere des operateurs de dirac a j 1 3 j d j p x p x j 1 n p j x a j d j i j on demontre que toutes les informations sur le caractere autoadjoint et l invariance du spectre essentiel sont contenues dans l inegalite r 1 u c j d j p u

generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media it is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs however the computational cost of conventional numerical modeling methods such as finite difference method and finite element method becomes prohibitively expensive when applied to very large models we propose a generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element the application of multiscale basis functions can capture the fine scale medium property variations and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite element method for wave equation while restricting the error to low values we formulate the continuous galerkin and discontinuous galerkin formulation of the multiscale method both of which have pros and cons applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

on the strong unique continuation principle for inequalities of maxwell type our goal of this paper is to prove carleman type inequalities for solutions a of maxwell type inequalities curl a div a cre 1 a assuming ij ec 0 1 in a neighborhood of 0 such inequalities also arise by nonlinear maxwell equations with lipschitz nonlinearity f a provided that a is locally bounded our investigation was inspired by the study of the corresponding problem for the elliptic dirac system

scattering theory for the dirac operator with a long range electromagnetic potential abstract we consider the dirac operator with a long range potential v x scalar pseudo scalar and vector components of v x may have arbitrary power like decay at infinity we introduce wave operators with time independent modifiers these modifiers are pseudo differential operators whose symbols are roughly speaking constructed in terms of approximate eigenfunctions of the stationary problem we derive and solve eikonal and transport equations for the corresponding phase and amplitude functions from an analytical point of view our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper this allows us to fit the long range scattering theory for the dirac operator into the framework of smooth perturbations finally we find the asymptotics for large times t of solutions u x t of the time dependent dirac equation

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

variational formulation using integral equations to solve sound scattering above an absorbing plane the boundary element method is used to model two dimensional acoustic radiation and scattering from a body of arbitrary shape above an infinite plane of flat surface and homogeneous impedance the particularity of the study is the use of an indirect integral representation of the solution given in terms of the jumps of pressure and its normal derivative through the boundaries a variational formulation is associated with the boundary indirect integral equations modeling our problem the major difficulty in the formulation is the infinite feature of the plane which is avoided by introducing an appropriate green s function numerical results of the attenuation of sound by noise barriers are presented they show good agreement with other results in the literature

initial value problem for the dynamic system of anisotropic elasticity the main object of this paper is the cauchy problem for the dynamic system of anisotropic elasticity existence and uniqueness theorems of weak and smooth solutions of this problem are established by the reduction of the original elasticity system into a symmetric hyperbolic system of the first order the numerical method of the cauchy problem solving for anisotropic elastic system with polynomial data is obtained and its correctness is established the simulations of the numerical solutions are presented

sound diffraction by a many sided barrier or pillar abstract the sound field produced when a spherical sound wave is incident on a rigid many sided barrier or pillar is calculated on the basis of keller s geometrical theory of diffraction kouyoumjian and pathak s asymptotic solution for a single wedge and pierce s method for a doubly diffracted wave from a comparison between calculated results and experimental results it is shown that the method described is applicable to the prediction of the propagation of noise around a thick barrier a building a number of these obstacles or a number of thin barriers with an extremely high accuracy

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

sound propagation over a large wedge a comparison between the geometrical theory of diffraction and the parabolic equation low frequency propagation over wedges hills and barriers is an important problem with many applications two different propagation models are used to compare the sound field generated by a steady single frequency source in front of over and behind a large rigid v shaped wedge predictions along the boundary made with both models at several frequencies are compared and found to be in excellent agreement the results provide additional evidence that the parabolic approximation is able to account accurately for propagation effects caused by terrain irregularities

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

a two scale solution algorithm for the elastic wave equation operator based upscaling is a two scale algorithm that speeds up the solution of the wave equation by producing a coarse grid solution which incorporates much of the local fine scale solution information we present the first implementation of operator upscaling for the elastic wave equation by using the velocity displacement formulation of the three dimensional elastic wave equation basis functions that are linear in all three directions and applying mass lumping the subgrid solve first stage of the two step algorithm reduces to solving explicit difference equations at the second stage of the algorithm we upscale both velocity and displacement by using local subgrid information to formulate the coarse grid problem the coarse grid system matrix is independent of time sparse and banded this paper explores both serial and parallel implementations of the method the main simplifying assumption of the method special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm leads to an easily parallelizable algorithm because very little communication is required between processors in fact for this upscaling implementation calculation of the load vector for the coarse solve dominates the cost of a time step we show that for a homogeneous medium convergence is second order in space and time so long as both the coarse and fine grids are simultaneously refined a series of heterogeneous medium numerical experiments demonstrate that the upscaled solution captures the fine scale fluctuations in the input parameters accurately most notable for use in a seismic inversion algorithm the upscaling algorithm accurately locates the depth of reflectors interface changes

a novel boundary integral formulation for three dimensional analysis of thin acoustic barriers over an impedance plane this article presents a three dimensional formulation for the analysis of acoustic barriers over an impedance plane as infinitely thin structures the barriers are therefore modeled as simple surfaces rather than volumetric structures using this approach the problems caused by near singular integrations and near degenerate systems of equations are averted and mesh generation is made easier a dual boundary element method is used in the analysis involving the simultaneous solution of standard and hypersingular boundary integral equations an optimization procedure is used to speed up the assembling of the system of equations increasing the applicability of the method to a wider range of frequencies

an efficient time domain perfectly matched layers formulation for elastodynamics on spherical domains summary many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain one widely used approach for truncating the infinite domain is the so called method of perfectly matched layers pmls most existing pml formulations are developed for finite difference methods based on the first order velocity stress form of the elasticity equations and they are not straight forward to implement using standard finite element methods fems on unstructured meshes some of the problems with these formulations include the application of boundary conditions in half space problems and in the treatment of edges and or corners for time domain problems several pml formulations which do work with fems have been proposed although most of them still have some of these problems and or they require a large number of auxiliary nodal history memory variables in this work we develop a new pml formulation for time domain elastodynamics on a spherical domain which reduces to a two dimensional formulation under the assumption of axisymmetry our formulation is well suited for implementation using fems where it requires lower memory than existing formulations and it allows for natural application of boundary conditions we solve example problems on two dimensional and three dimensional domains using a high order discontinuous galerkin dg discretization on unstructured meshes and explicit time stepping we also study an approach for stabilization of the discrete equations and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation copyright 2014 john wiley sons ltd

an efficient finite element time domain formulation for the elastic second order wave equation a non split complex frequency shifted convolutional pml the perfectly matched layer pml technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first order system in velocity and stress in attenuating non grazing bulk and surface waves this paper develops a novel convolutional pml formulation based on the second order wave equation with displacements as the only unknowns to annihilate spurious reflections from near grazing waves the derived variational form allows for the use of e g finite element and the spectral element methods as spatial discretization schemes a recursive convolution update scheme of second order accuracy is employed such that highly stable effective time integration with the newmark beta implicit and explicit with mass lumping method is achieved the implementation requires minor modifications of existing displacement based finite element software and the stability and efficiency of the proposed formulation is verified by relevant two dimensional benchmarks that accommodate bulk and surface waves copyright 2011 john wiley sons ltd

absolutely continuous spectrum of dirac operators for long range potentials abstract using the resolvent method and the technique of weighted l2 estimates we deduce the non existence of the singular continuous spectrum of the dirac operator p x on the interval 1 we assume that the hermitian matrix potential p x p1 x p2 x is divided into a long range part p 1 x o x e r r p 1 x o x e and a short range part p 2 x o x 1 e x with local singularities p x o x a j 1 of coulomb type for n nuclei a1 an r 3

mixed higher order spectral finite elements for reissner mindlin equations we construct a class of high order numerical approximations for the reissner mindlin plate model in the time domain based on mixed spectral finite elements with mass lumping in this way we obtain explicit time stepping schemes we first compare the reissner mindlin model to three dimensional 3d solutions to validate our method then we show the advantages of the schemes in terms of accuracy and computational time

an analytic secondary source model of edge diffraction impulse responses a new impulse response model for the edge diffraction from finite rigid or soft wedges is presented which is based on the exact biot tolstoy solution the new model is an extension of the work by medwin et al h medwin et al j acoust soc am 72 1005 1013 1982 in that the concept of secondary edge sources is used it is shown that analytical directivity functions for such edge sources can be derived and that they give the correct solution for the infinite wedge these functions support the assumption for the first order diffraction model suggested by medwin et al that the contributions to the impulse response from the two sides around the apex point are exactly identical the analytical functions also indicate that medwin s second order diffraction model contains approximations which however might be of minor importance for most geometries access to analytical directivity functions makes it possible to derive explicit expressions for the first and even second order diffraction for certain geometries an example of this is axisymmetric scattering from a thin circular rigid or soft disc for which the new model gives first order diffraction results within 0 20 db of published reference frequency domain results and the second order diffraction results also agree well with the reference results scattering from a rectangular plate is studied as well and comparisons with published numerical results show that the new model gives accurate results it is shown that the directivity functions can lead to efficient and accurate numerical implementations for first and second order diffraction

an optimization based approach for solving a time harmonic multiphysical wave problem with higher order schemes this study considers developing numerical solution techniques for the computer simulations of time harmonic fluid structure interaction between acoustic and elastic waves the focus is on the efficiency of an iterative solution method based on a controllability approach and spectral elements we concentrate on the model in which the acoustic waves in the fluid domain are modeled by using the velocity potential and the elastic waves in the structure domain are modeled by using displacement traditionally the complex valued time harmonic equations are used for solving the time harmonic problems instead of that we focus on finding periodic solutions without solving the time harmonic problems directly the time dependent equations can be simulated with respect to time until a time harmonic solution is reached but the approach suffers from poor convergence to overcome this challenge we follow the approach first suggested and developed for the acoustic wave equations by bristeau glowinski and periaux thus we accelerate the convergence rate by employing a controllability method the problem is formulated as a least squares optimization problem which is solved with the conjugate gradient cg algorithm computation of the gradient of the functional is done directly for the discretized problem a graph based multigrid method is used for preconditioning the cg algorithm

space time mesh refinement for elastodynamics numerical results we propose here a generalization to the elastodynamic equations of the space time mesh refinement technique intro duced in raffinement de maillage spatio temporel pour les equations de maxwell ph d thesis universitede dau phine paris 2000 for the maxwells equations this method uses a discrete energy conservation to ensure the stability the method is presented in a variational way applicable to other type of hyperbolic systems several numerical experiments are provided to show the efficiency of this approach 2004 elsevier b v all rights reserved

near field computation for sound propagation above ground using complex image theory in this paper a very simple method is introduced for the efficient computation of the sound field in the near region above an impedance ground or an extended reaction ground the procedure is based on complex image theory using prony s method the slowly varying factor in the integrand of the sommerfeld integral is approximated by a short series in terms of complex exponential functions the result can be exactly evaluated by the sommerfeld identity a new closed form approximate expression of the sound field is obtained which is represented as the original source its quasistatic image and several images with complex amplitudes and positions avoiding numerical evaluation of the sommerfeld integral makes these closed form expressions computationally very efficient the representation of the field with complex images makes the formula useful in measurement of ground impedance or in other cases requiring repeated mapping of the sound field numerical results show that these closed form green s functions are very accurate for the computation of the near field