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

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

convolutional perfectly matched layer for elastic second order wave equation in this work a method is presented to extend the convolutional perfectly matched layer c pml to simulate acoustic wave propagation in elastic media with a second order equation formulation this non physical layer is used at the computational edge of a finite element method algorithm in frequency domain and a pseudo spectral algorithm in time domain as an absorbing boundary condition abc to truncate unbounded media numerical results show that the c pml abc attenuates the outgoing surface waves more effectively than classical pml abc as proposed by berenger j comput phys 114 195 200 1994 for electromagnetic waves in the case of oblique incidence where the pml method suffers from large spurious reflections moreover a modification of the proposed c pml formulation is also discussed in order to stabilize the absorbing layer in anisotropic solids where numerical instabilities can appear

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

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

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

3d finite difference synthetic acoustic log in cylindrical coordinates a finite difference method of numerical simulation of sonic logging has been developed and implemented the very general statement is considered the surrounding medium is allowed to be 3d heterogeneous and a source can be located at any point inside or outside the well to provide a maximally precise description of the sharpest interface of the problem the interface between the well and surrounding formations we use cylindrical coordinates with the axis directed along the well in order to avoid an excessive azimuth inflation of grid cells with an increase of radius we perform periodical refinement of the grid step in the azimuth direction in order to truncate the area of computations perfectly matched layer pml for cylindrical coordinates is developed and implemented its main advantage in comparison with other approaches is an extremely low level of artificial reflections and the absence of necessity to perform splitting of variables in the azimuth direction based on this numerical method a software oriented for the use of parallel computations is developed and implemented under message passing interface library results of numerical experiments for a well with completion embedded within layered elastic background with a vertical planar crack are presented and discussed

the inverse medium problem in heterogeneous pml truncated domains using scalar probing waves abstract we discuss the inverse medium problem associated with semi infinite domains in particular we attempt to image the spatial variability of shear moduli or shear wave velocities from scant surficial measurements of an arbitrarily heterogeneous semi infinite domain s response to prescribed dynamic excitations we use a full waveform approach to drive the inversion process within a pde constrained optimization framework due to the semi infinite extent of the targeted domains we introduce perfectly matched layers pmls to arrive at finite computational domains the numerical implementation is based on a mixed finite element method that is used to resolve the ensuing state and adjoint boundary value problems both of which are pml endowed to alleviate the inherent solution multiplicity we use tikhonov and total variation tv regularization schemes in conjunction with a regularization factor continuation scheme to further improve the optimizer s chances to converge we also discuss a source frequency continuation scheme we report on two dimensional numerical experiments using synthetic data included are layered profiles and profiles involving inclined layers and inclusions we also report on our methodology s reconstruction of the highly heterogeneous marmousi benchmark velocity model

attenuation compensation in anisotropic least squares reverse time migration abstractanisotropic and attenuating properties of subsurface media cause amplitude loss and waveform distortion in seismic wave propagation resulting in negative influence on seismic imaging to correct the anisotropy effect and compensate amplitude attenuation a compensated amplitude vertical transverse isotropic vti least squares reverse time migration lsrtm method is adopted in this method the attenuation term of an attenuated acoustic wave equation is extended to a vti quasi differential wave equation which takes care of effects from anisotropy and attenuation the finite difference method is used to solve the equation in which attenuation terms are solved in the wavenumber domain and other terms are solved in the space or wavenumber domain stable regularization operators are derived and introduced to the equations to eliminate severe numerical noise in high frequency components during backward propagation meanwhile a demigration operator migration operator and gradient formula for att

analysis of a finite pml approximation to the three dimensional elastic wave scattering problem we consider the application of a perfectly matched layer pml technique to approximate solutions to the elastic wave scattering problem in the frequency domain the pml is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain the complement of the scatterer the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem we prove existence and uniqueness of the solutions to the infinite domain and truncated domain pml equations provided that the truncated domain is sufficiently large we also show exponential convergence of the solution of the truncated pml problem to the solution of the original scattering problem in the region of interest we then analyze a galerkin numerical approximation to the truncated pml problem and prove that it is well posed provided that the pml damping parameter and mesh size are small enough finally computational results illustrating the efficiency of the finite element pml approximation are presented

stability of perfectly matched layer regions in generalized finite difference method for wave problems the generalized finite difference method gfdm has been proved to be a good procedure to solve the problem of wave propagation as other numerical methods when using a finite model to study an unbounded domain a way of avoiding wave reflection at the boundaries is needed perfectly matched layers pml absorbing boundary conditions are commonly used this paper aims to obtain a perfectly matched layer formulation for the gfdm for which the stability is guaranteed

a new family of mixed finite elements for the linear elastodynamic problem we construct and analyze a new family of quadrangular in two dimensions or cubic in three dimensions mixed finite elements for the approximation of elastic wave equations our elements lead to explicit schemes via mass lumping after time discretization including in the case of anisotropic media error estimates are given for these new elements

spectral element method for elastic and acoustic waves in frequency domain numerical techniques in time domain are widespread in seismic and acoustic modeling in some applications however frequency domain techniques can be advantageous over the time domain approach when narrow band results are desired especially if multiple sources can be handled more conveniently in the frequency domain moreover the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain in this paper we present a spectral element method sem in frequency domain to simulate elastic and acoustic waves in anisotropic heterogeneous and lossy media the sem is based upon the finite element framework and has exponential convergence because of the use of gll basis functions the anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems compared with the conventional finite element method the number of unknowns in the sem is significantly reduced and higher order accuracy is obtained due to its spectral accuracy to account for the acoustic solid interaction the domain decomposition method ddm based upon the discontinuous galerkin spectral element method is proposed numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain

a staggered grid convolutional differentiator for elastic wave modelling the computation of derivatives in governing partial differential equations is one of the most investigated subjects in the numerical simulation of physical wave propagation an analytical staggered grid convolutional differentiator cd for first order velocity stress elastic wave equations is derived in this paper by inverse fourier transformation of the band limited spectrum of a first derivative operator a taper window function is used to truncate the infinite staggered grid cd stencil the truncated cd operator is almost as accurate as the analytical solution and as efficient as the finite difference fd method the selection of window functions will influence the accuracy of the cd operator in wave simulation we search for the optimal gaussian windows for different order cds by minimizing the spectral error of the derivative and comparing the windows with the normal hanning window function for tapering the cd operators it is found that the optimal gaussian window appears to be similar to the hanning window function for tapering the same cd operator we investigate the accuracy of the windowed cd operator and the staggered grid fd method with different orders compared to the conventional staggered grid fd method a short staggered grid cd operator achieves an accuracy equivalent to that of a long fd operator with lower computational costs for example an 8th order staggered grid cd operator can achieve the same accuracy of a 16th order staggered grid fd algorithm but with half of the computational resources and time required numerical examples from a homogeneous model and a crustal waveguide model are used to illustrate the superiority of the cd operators over the conventional staggered grid fd operators for the simulation of wave propagations

a perfectly matched layer for fluid solid problems application to ocean acoustics simulations with solid ocean bottoms a time domain legendre spectral element method is described for full wave simulation of ocean acoustics models i e coupled fluid solid problems in unbounded or semi infinite domains taking into account shear wave propagation in the ocean bottom the technique can accommodate range dependent and depth dependent wave speed and density as well as steep ocean floor topography for truncation of the infinite domain to efficiently absorb outgoing waves a fluid solid complex frequency shifted unsplit perfectly matched layer is introduced based on the complex coordinate stretching technique the complex stretching is rigorously taken into account in the derivation of the fluid solid matching condition inside the absorbing layer which has never been done before in the time domain two implementations are designed a convolutional formulation and an auxiliary differential equation formulation because the latter allows for implementation of high order time schemes leading to reduced numerical dispersion and dissipation a topic of importance in particular in long range ocean acoustics simulations the method is validated for a two dimensional fluid solid pekeris waveguide and for a three dimensional seamount model which shows that the technique is accurate and numerically long time stable compared with widely used paraxial absorbing boundary conditions the perfectly matched layer is significantly more efficient at absorbing both body waves and interface waves

a note on conditioning stability and collocation matrices the concept of a well conditioned bvp as considered in recent literature essentially corresponds to the concept of a uniformly stable ivp but this terminology suggests a shift from absolute to relative quantities which makes it more difficult to understand the conditioning of numerical discretizations especially for higher order differential operators in this note we wish to make it clear that the notion of well conditioning of bvps is not always a direct analogue of the u the condition number of a discretization of a well conditioned bvp generally grows unboundedly when higher accuracy is desired because it contains a discretization of a differential operator this condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted for collocation an implementation which uses a local runge kutta solution representation produces a much better conditioning because the repeated numerical differentiation is avoided instead the highest derivative is approximated locally by a discontinuous piece wise polynomial and local integrations yield lower solution derivatives upon imposing the collocation equations the highest derivative approximation can be locally eliminated resulting in a sophisticated discretization of the corresponding first order system

multiple steady state solutions in a multijunction semiconductor device multiple steady state solutions are constructed for a multijunction semiconductor device using a combination of asymptotic and numerical techniques the multijunction device is modeled by the drift diffusion equations of semiconductor physics in the drift diffusion model three sign alterations of the doping are assumed and a shockley read hall recombination is allowed for in the parameter range for which the current voltage relation is monotonic the quasi neutral approximation to the drift diffusion model is found to be valid and solutions are constructed asymptotically in the limit of large doping densities in contrast in the parameter range for which the current voltage relation is s shaped the quasi neutral approximation is found to poorly approximate solutions to the full model on the middle and lower branches a suitable modification to the quasi neutral approximation is then introduced to describe the middle and lower branches of the current voltage relation under the assumption of no recombi

a well posed shooting method for transferable dae s consider a tpbvp for transferable nonlinear dae s in general the shooting equation has a singular jacobian a multiple shooting method which has a nonsingular jacobian and also produces consistent initial values for the integration is presented the estimation of the condition of the jacobian shows the well posedness of the method some illustrative examples are given

local time space mesh refinement for simulation of elastic wave propagation in multi scale media this paper presents an original approach to local time space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro heterogeneities the main features of the algorithm are the application of temporal and spatial refinement on two different surfaces the use of the embedded stencil technique for the refinement of grid step with respect to time the use of the fast fourier transform fft based interpolation to couple variables for spatial mesh refinement the latter makes it possible to perform filtration of high spatial frequencies which provides stability in the proposed finite difference schemes in the present work the technique is implemented for the finite difference simulation of seismic wave propagation and the interaction of such waves with fluid filled fractures and cavities of carbonate reservoirs however this approach is easy to adapt and or combine with other numerical techniques such as finite elements discontinuous galerkin method or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations

an efficient complex frequency shifted perfectly matched layer for second order acoustic wave equation summary in the context of simulations of wave propagations in unbounded domain absorbing boundary conditions are often used to truncate the simulation domain to a finite space perfectly matched layer pml has proven to be an excellent absorbing boundary conditions however as this technique was primarily designed for the first order equation system it cannot be applied to the second order equation system directly in this paper based on a complex coordinate stretching technique we developed a novel efficient auxiliary differential equation form of the complex frequency shifted pml for the second order equation system this facilitates the use of complex frequency shifted pml in acoustic simulations based upon wave equations of second order form compared with previous state of the art methods the proposed one has the advantage of simpler implementation it is an unsplit field scheme that can be extended to higher order discretization schemes conveniently numerical results from both homogeneous and heterogeneous computational domains are provided to illustrate the validity of the method copyright 2013 john wiley sons ltd

conditioning of the steady state semiconductor device problem when solving numerically the steady state semiconductor device problem using appropriate discretizations extremely large condition numbers are often encountered for the linearized discrete device problem these condition numbers are so large that if they represented a sharp bound on the amplification of input errors or even of roundoff errors then the obtained numerical solution would be meaningless as it turns out one major reason for these large numbers is due to poor row and column scaling which is essentially harmless and or can be fixed but another reason could be an ill conditioned device which yields a true loss of significant digits in the numerical calculation in this paper we carry out a conditioning analysis for the steady state device problem we consider various quasilinearizations as well as gummel type iterations and obtain stability bounds which may indeed allow ill conditioning in general these bounds are exponential in the potential variation and are sharp e g for a thyristor but for devices where each smooth subdomain has an ohmic contact e g a pn diode moderate bounds guaranteeing well conditioning are obtained moreover the analysis suggests how various row and column scalings should be applied in order for the measured condition numbers to correspond more realistically to the true loss of significant digits in the calculations

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

irregular perfectly matched layers for 3d elastic wave modeling we have developed a perfectly matched layer pml absorbing boundary condition that can be imposed along an arbitrary geometric boundary in 3d elastic wave modeling the scheme is developed by using the local coordinate system based pml splitting equations and integral approach of the pml equations under a discretization of tetrahedral grids however no explicit coordinate transformations arise the local coordinate system based pml splitting equations make it possible to decay incident waves around the direction normal to the irregular geometric boundaries instead of a coordinate axis direction based on the resulting 3d irregular pml model we can flexibly construct the computational domain with smaller nodes by cutting uninterested zones this results in significant reductions in computational cost and memory requirements for 3d simulations by building a smooth artificial boundary the irregular pml model can avoid the respective treatments to the edges and corners of the artificial boundaries also the irregular pml model may reduce the grazing incidence that makes the pml model less efficient by changing the shapes of the artificial boundaries numerical testing was used to demonstrate the performance of the irregular pml model

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

perfectly matched transmission problem with absorbing layers application to anisotropic acoustics in convex polygonal domains this paper presents an original approach to design perfectly matched layers pml for anisotropic scalar wave equations this approach is based first on the introduction of a modified wave equation and second on the formulation of general perfectly matched transmission conditions for this equation the stability of the transmission problem is discussed by way of the adaptation of a high frequency stability necessary condition and we apply our approach to construct pml suited for any convex domain with straight boundaries a new variational formulation of the problem including a lagrange multiplier at the interface between the physical and the absorbing domains is then set and numerical results are presented in 2d and 3d these results show the efficiency of our approach when using constant damping coefficients combined with high order elements

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

stable p wave modeling for reverse time migration in tilted ti media we present an approach for p wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis tti media suitable for anisotropic reverse time migration the proposed approach is based on wave equations derived from first principles the equations of motion and hooke s law under the acoustic ti approximation consequently no assumptions are made about the spatial variation of medium parameters a rotation of the stress and strain tensors to a local coordinate system aligned with the ti symmetry axis makes it possible to benefit from the simple and sparse form of the ti elastic tensor in that system the resulting wave equations can be formulated either as a set of five first order or as a set of two second order partial differential equations for the constant density case the second order tti wave equations involve mixed and nonmixed second order spatial derivatives with respect to global nonrotated coordinates we propose a numerical implementation of these equations using high order centered finite differences to minimize modeling artifacts related to the use of centered first derivative operators we use discrete second derivative operators for the nonmixed second order spatial derivatives and repeated discrete first derivative operators for the mixed derivatives such a combination of finite difference operators leads to a stable wave propagator provided that the operators are designed properly in practice stability is achieved by slightly weighting down terms that contain mixed derivatives this has a minor practically negligible effect on the kinematics of wave propagation the stability of the presented scheme in inhomogeneous tti models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples

application of the multiaxial perfectly matched layer m pml to near surface seismic modeling with rayleigh waves perfectly matched layer pml absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling when modeling rayleigh waves with the existence of the free surface the classical pml algorithm becomes unstable when the poisson s ratio of the medium is high numerical errors can accumulate exponentially and terminate the simulation due to computational overflows numerical tests show that the divergence speed of the classical pml has a nonlinear relationship with the poisson s ratio generally the higher the poisson s ratio the faster the classical pml diverges the multiaxial pml m pml attenuates the waves in pmls using different damping profiles that are proportional to each other in orthogonal directions the proportion coefficients of the damping profiles usually vary with the specific model settings if they are set appropriately the m pml algorithm is stable for high poisson s ratio earth models through numerical tests of 40 models with poisson s ratios that varied from 0 10 to 0 49 we found that a constant proportion coefficient of 1 0 for the x and z directional damping profiles is sufficient to stabilize the m pml for all 2d isotropic elastic cases wavefield simulations indicate that the instability of the classical pml is strongly related to the wave phenomena near the free surface when applying the multiaxial technique only in the corners of the pml near the free surface the original m pml technique can be simplified without losing its stability the simplified m pml works efficiently for homogeneous and heterogeneous earth models with high poisson s ratios the analysis in this paper is based on 2d finite difference modeling in the time domain that can easily be extended into the 3d domain with other numerical methods

perfectly matched layers for transient elastodynamics of unbounded domains one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies in a recent work computer methods in applied mechanics and engineering 2003 192 1337 1375 the authors presented inter alia time harmonic governing equations of pmls for anti plane and for plane strain motion of visco elastic media this paper presents a corresponding time domain displacement based governing equations of these pmls and b displacement based finite element implementations of these equations suitable for direct transient analysis the finite element implementation of the anti plane pml is found to be symmetric whereas that of the plane strain pml is not numerical results are presented for the anti plane motion of a semi infinite layer on a rigid base and for the classical soil structure interaction problems of a rigid strip footing on i a half plane ii a layer on a half plane and iii a layer on a rigid base these results demonstrate the high accuracy achievable by pml models even with small bounded domains

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

a discretization scheme for an extended drift diffusion model including trap assisted phenomena an extended drift diffusion model is considered to account for the kinetics of electrons trapped in defect states within a semiconductor material a discretization scheme based on newton krylov iterations and mixed finite volumes is then proposed and applied to the model even in the presence of schottky contacts i e robin type boundary conditions numerical results concerning the simulation of an electro optical device in several working conditions are presented last

time varying boundary conditions in simulation of seismic wave propagation we propose two new boundary conditions to regulate coherent reflections from the model boundaries in numerical solutions of wave equations both boundary conditions have the common feature that the boundary condition is varied with respect to time the first boundary condition expands or contracts the computational model during a modeling simulation the effect is to cause a doppler shift in the reflected wavefield that can be used to shift energy outside a frequency band of interest in addition when the computational domain is expanding the range of possible incidence angles on the boundary is restricted this can be used to increase the effectiveness of many existing absorbing boundary conditions that are more effective for incidence angles close to normal the second boundary condition is an extension of random boundaries by carefully changing the realization of a random boundary over time a more diffusive wavefield can be simulated we show results with 2d numerical simulations of the scalar wave equation for both these boundary conditions the first boundary condition has application to modeling but both these boundary conditions have potential application within algorithms that rely upon modeling kernels such as reverse time migration and full waveform inversion

a gauss newton full waveform inversion in pml truncated domains using scalar probing waves abstract this study considers the characterization of subsurface shear wave velocity profiles in semi infinite media using scalar waves using surficial responses caused by probing waves a reconstruction of the material profile is sought using a gauss newton full waveform inversion method in a two dimensional domain truncated by perfectly matched layer pml wave absorbing boundaries the pml is introduced to limit the semi infinite extent of the half space and to prevent reflections from the truncated boundaries a hybrid unsplit field pml is formulated in the inversion framework to enable more efficient wave simulations than with a fully mixed pml the full waveform inversion method is based on a constrained optimization framework that is implemented using karush kuhn tucker kkt optimality conditions to minimize the objective functional augmented by pml endowed wave equations via lagrange multipliers the kkt conditions consist of state adjoint and control problems and are solved iteratively to update the shear wave velocity profile of the pml truncated domain numerical examples show that the developed gauss newton inversion method is accurate enough and more efficient than another inversion method the algorithm s performance is demonstrated by the numerical examples including the case of noisy measurement responses and the case of reduced number of sources and receivers

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

frequency domain acoustic wave modeling with hybrid absorbing boundary conditions abstractacoustic waves are modeled using a 2d frequency domain finite difference scheme with three different artificial damping boundaries the absorbing boundary condition abc perfectly matched layer pml and a hybrid strategy combining pml and abc we evaluated the features of each simulation for a given artificial boundary condition for homogeneous and heterogeneous media for shorter pml layers the hybrid absorbing scheme gave better results when compared with the traditional pml technique with a significant reduction of the spurious reflections and a shorter computational run time

decomposed element free galerkin method compared with finite difference method for elastic wave propagation the decomposed element free galerkin defg method is a modified scheme to resolve shortcomings of memory use in element free galerkin efg methods defg solves elastic wave equation problems by alternately updating the stress strain relations and the equations of motion as in the staggered grid finite difference fd method defg requires at most twice the memory space a size comparable to that used by the fd method in addition defg can adopt perfectly matched layer pml absorbing boundary conditions as in the fd case to confirm that defg performs as well as fd a 2d defg under pml boundary conditions was compared with an fd with fourth order spatial accuracy fd4 using an exact analytical solution of ps reflection waves the defg results are as accurate as those obtained by fd4 in a comparison using lamb s problem with eight nodal spaces for the shortest s wavelength defg provides a remarkably accurate rayleigh waveform over a distance of at least 50 wavelengths compared with 10 wavelengths for

a conservative numerical scheme for modeling nonlinear acoustic propagation in thermoviscous homogeneous media abstract a nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented this system constitutes a full nonlinear wave model that has been formulated in the conservation form initially this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the lax wendroff theorem and the scheme can match the solutions of the westervelt and burgers equations numerically here high order numerical descriptions of strongly nonlinear wave propagations become of great interest for that matter we consider finite difference formulations of the weighted essentially non oscillatory weno schemes associated with explicit strong stability preserving runge kutta ssp rk time integration methods although this strategy is known to be computationally demanding it is found to be effective when implemented to be solved in graphical processing units gpus as we consider wave propagations in unbounded domains perfectly matching layers pml have been also considered in this work the proposed system model is validated and illustrated by using one and two dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media

a singularly perturbed boundary value problem modelling a semiconductor device this paper is concerned with the static one dimensional modelling of a semiconductor device namely the pn junction when a bias is applied the governing equations are the well known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts because of the different orders of magnitude of the solution components at the boundaries we scale the components individually and obtain a singular perturbation problem we analyse the equilibrium case zero bias applied and set up approximate models posed as singularly perturbed second order equations by neglecting the hole and electron current densities this makes sense for small forward bias and for moderate reverse bias for the full problem we prove an a priori estimate on the number of electron hole carrier pairs and derive asymptotic expansions as the perturbation parameter tends to zero by setting up the reduced system and the boundary

dispersive waves and pml performance for large stencil differencing in the parabolic equation the discrete reflection absorption performance of perfectly matched layer pml boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the pml region most previous work analyzing pml reflection performance has assumed use of low order second or fourth small stencil three or five point spatial discretizations whose dispersion is poor outside a narrow low wavenumber band large stencil differencing schemes regardless of order are easily created with physically realistic wide band dispersive characteristics but the effect their adoption has on pml performance has not been analyzed previously in the context of a semi discrete approximation to the parabolic paraxial equation this work derives the analytic form of the discrete pml reflection coefficient for an arbitrary order arbitrary stencil interior differencing scheme coupled to an arbitrary order arbitrary stencil pml region this result is then used to show that the absorption performance of the pml regions is largely insensitive to changes in interior differencing as a consequence the benefits of superior dispersion characteristics provided by use of large stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the pml

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

a higher order perfectly matched layer formulation for finite difference time domain seismic wave modeling we have developed a higher order perfectly matched layer pml formulation to improve the absorption performance for finite difference time domain seismic modeling first we outlined a new unsplit correction approach which allowed for traditional first order pmls to be added directly to existing codes in a straightforward manner then using this framework we constructed a pml formulation that can be used to construct higher order pmls of arbitrary order the greater number of degrees of freedom associated with the higher order pml allow for enhanced flexibility of the pml stretching functions thus potentially facilitating enhanced absorption performance we found that the new approach can offer increased elastodynamic absorption particularly for evanescent waves we also discovered that the extra degrees of freedom associated with the higher order pml required careful optimization if enhanced absorption was to be achieved furthermore these extra degrees of freedom increased the computational requirements in comparison with first order schemes we reached our formulations using one compact equation thus increasing the ease of implementation additionally the formulations are based on a recursive integration approach that reduce pml memory requirements and do not require special consideration for corner regions we tested the new formulations to determine their ability to absorb body waves and surface waves we also tested standard staggered grid stencils and rotated staggered grid stencils

an accelerated monotone iterative method for the quantum corrected energy transport model a non stationary monotone iterative method is proposed and analyzed for the quantum corrected energy transport model in nanoscale semiconductor device simulation for the density gradient equations it is analytically and numerically shown that the convergence rate of the method is optimal in the sense of gummel s decoupling iteration this is a globally convergent method in the sense that the initial guess can be taken as a lower or an upper solution which is independent of applied voltages the method integrates the monotone parameters grid sizes and scharfetter gummel fitting in an adaptive and automatic way to treat the singularly perturbed nature of the model that incurs boundary junction and quantum potential layers in the device

an approximate newton method for the solution of the basic semiconductor device equations an iterative method for the solution of the steady state fundamental semiconductor device equations is presented and analyzed it is shown that while requiring the same amount of operations per iteration as gummel cycle iterations the new method has local convergence properties similar to those of newton s method

an unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media the perfectly matched layer pml absorbing technique has become popular in numerical modeling in elastic or poroelastic media because of its efficiency in absorbing waves at nongrazing incidence however after numerical discretization at grazing incidence large spurious oscillations are sent back from the pml into the main domain the pml then becomes less efficient when sources are located close to the edge of the truncated physical domain under study for thin slices or for receivers located at a large offset we develop a pml improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first order system in velocity and stress we show its efficiency for both nondissipative and dissipative biot porous models based on a fourth order staggered finite difference method used in a thin mesh slice the results obtained are improved significantly compared with those obtained with the classical pml

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

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

a discontinuous collocated grid implementation for high order finite difference modeling abstractthe discontinuous grid method can greatly reduce the storage requirements and computational costs of finite difference fd modeling for large velocity contrast models however traditional discontinuous grid methods have to use interpolation when refining the wavefield in transition zones and would cause apparent artifacts we have developed a new discontinuous collocated grid scheme for high order fd modeling we refined the wavefield on a rotated coordinate system where the interpolation is not required again the horizontal and vertical spatial derivatives can be accurately converted into diagonal derivatives within the rotated coordinate system thus our scheme would be free of artifacts caused by improper interpolation the ratio from coarse to fine grid spacing is restricted to 2n for our scheme where n is a positive integer numerical experiments demonstrate that the proposed discontinuous collocated grid scheme reduces the artificial reflections by about two orders of magnitude compar

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

an improved vacuum formulation for 2d finite difference modeling of rayleigh waves including surface topography and internal discontinuities rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography modeling of rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics for simulation of rayleigh waves the traction free boundary condition needs to be satisfied on the free surface avacuum formulation naturally incorporates surface topographyinfinite difference fd modeling by treating the surface grid nodes as the internal grid nodes however theconventionalvacuumformulationdoesnotcompletely fulfill the free surface boundary condition and becomes unstableformodelingusinghigh orderfdoperators wedeveloped a stable vacuum formulation that fully satisfies the free surface boundary condition by choosing an appropriate combination of the staggered grid form and a parameter averaging scheme the elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes the improved vacuum formulation can accurately and stably simulate rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high poisson s ratios 0 4 this method requires fewer grid points per wavelength than the stress image based methods internal discontinuities in a model can be handled without modification of the algorithm onlyminorchangesarerequiredtoimplementtheimproved vacuum formulation in existing 2d fd modeling codes

love wave waveform inversion in time domain for shallow shear wave velocity abstracthigh frequency surface wave techniques are widely used to estimate s wave velocity of near surface materials surface wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered model assumption during calculation of dispersion curves waveform inversion directly fits the waveform of observed data and it can be applied to any kinds of earth models we have used the love wave waveform inversion in the time domain to estimate near surface s wave velocity we used the finite difference method as the forward modeling method the source effect was removed by the deconvolution technique which made our method independent of the source wavelet we defined the difference between the deconvolved observed and calculated waveform as the misfit function we divided the model into different sizes of blocks depending on the resolution of the love waves and we updated the s wave velocity of each blo

an optimal absorbing boundary condition for elastic wave modeling absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries lindman 1975 clayton and engquist 1977 reynolds 1978 liao et al 1984 cerjan et al 1985 randall 1988 higdon 1991 we are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage this is practical for 3 d simulation of elastic wave propagation by a finite difference method peng and toksoz 1994 developed a method to design a class of optimal absorbing boundary conditions for a given operator length in this short note we give a brief introduction to this technique and we compare the optimal absorbing boundary conditions against those by reynolds 1978 and higdon 1991 using examples of 3 d elastic finite difference modeling on an ncube 2 parallel computer in the appendix we also give explicit formulas for computing coefficients of the optimal abso

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

3d frequency domain finite difference viscoelastic wave modeling using weighted average 27 point operators with optimal coefficients abstractexperimental data suggest that the viscoelastic behavior of rocks is more easily and accurately described in the frequency domain than in the time domain supporting the idea of simulating seismic wave propagation in the frequency domain we evaluated weighted averaged 27 point finite difference operators for 3d viscoelastic wave modeling in the frequency domain within the proposed framework we developed general equations for normalized phase velocities that can be used with arbitrary finite difference operators three sets of weighting coefficients for second order central finite difference operators that minimize the numerical dispersion for up to five grid points per wavelength were found using a damped least squares ls criterion as well as a global optimization scheme based on l1 and l2 norm criteria the three sets produced very similar dispersion curves and improvement provided by global optimization appeared marginal in this respect we also evaluated a discrete form for the heterogen

wave propagation in heterogeneous media effects of fine scale heterogeneity we study the multiple scattering effects caused by fine scale heterogeneity for this purpose it is unreasonable to rely on a linearization of the dependency of the wavefield in the parameters that describe the medium therefore the only tools that correctly model wave propagation are based on the numerical solution of the two way wave equation by finite differences or finite elements a fine grid provides a straightforward approach to account for fine scale heterogeneity in this situation there is no need for high order schemes variational methods allow us to exhibit a numerical scheme that accounts for heterogeneity and that is by construction stable provided a stability condition is fulfilled this condition is a sufficient stability condition contrary to the classical necessary stability conditions in addition general mathematical results prove the finite difference solution is close to the solution of the wave equation when the grid is fine enough the multiple scattering effects caused by fine scale heterogeneity are very important in particular we observe that imaging the so computed synthetic data by standard migration techniques that assume a linearization of the above mentioned dependency shows a strongly noise corrupted image this illustrates the importance of preprocessing data to remove the effect of multiple scattering we try to improve the signal to noise ratio by removing multiples related to the free surface although significant noise reduction is achieved even more sophisticated preprocessing is required to obtain a clear image of the subsurface

staggered grid spectral element methods for elastic wave simulations in this paper we develop and analyze a new class of spectral element methods for the simulations of elastic wave propagation the major components of the method are the spatial discretization and the choice of interpolation nodes the spatial discretization is based on piecewise polynomial approximation defined on staggered grids the resulting method combines the advantages of both staggered grid based methods and classical non staggered grid based spectral element methods our new method is energy conserving and does not require the use of any numerical flux because of the staggered local continuity of the basis functions our new method also uses radau points as interpolation nodes and the resulting mass matrix is diagonal thus time marching is explicit and is very efficient moreover we give a rigorous proof for the optimal convergence of the method in terms of dispersion we present a numerical study for the numerical dispersion and show that this error is of very high order finally some numerical convergence tests and applications to unbounded domain problems with perfectly matched layer are shown

an average derivative optimal scheme for modeling of the frequency domain 3d elastic wave equation because of its high computational cost we needed to develop an efficient numerical scheme for the frequency domain 3d elastic wave equation in addition the numerical scheme should be applicable to media with a liquid solid interface to address these two issues we have developed a new average derivative optimal 27 point scheme with arbitrary directional grid intervals and a corresponding numerical dispersion analysis for the frequency domain 3d elastic wave equation the novelty of this scheme is that its optimal coefficients depend on the ratio of the directional grid intervals and poisson s ratio in this way this scheme can be applied to media with a liquid solid interface and a computational grid with arbitrary directional grid intervals for media with a variable poisson s ratio we have developed an effective and stable interpolation method for optimization coefficients compared with the classic 19 point scheme this new scheme reduces the required number of grid points per wavelength for equal and unequal directional grid intervals the reduction of the number of grid points increases as the poisson s ratio becomes larger in particular the numerical s wave phase velocity of this new scheme becomes zero whereas the classic 19 point scheme produces a spurious numerical s wave phase velocity as poisson s ratio reaches 0 5 we have performed numerical examples to develop the theoretical analysis

a time domain discontinuous galerkin method for mechanical resonator quality factor computations numerical simulations are becoming increasingly important in the design of micromechanical resonators in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses frequency based approaches have been shown to predict these accurately however they require the solution of eigenvalue problems or the inversion of helmholtz type operators which are known to be very difficult for large scale iterative solvers we propose using a time domain approach instead where a broadband input signal is propagated through the system with a local explicit time stepper this is achieved using a new high order discontinuous galerkin dg discretization for the linear elasticity equations in particular a second order formulation with compact dg fluxes and a runge kutta time integrator where the block diagonal mass matrices allow for efficient stable and accurate time stepping our solver scales well on distributed parallel computers even in three spatial dimension and for large problem sizes the resulting output signal is analyzed using a well known filter diagonalization method which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data we validate the properties of our scheme on model problems and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well

dispersion reducing finite elements for transient acoustics this paper focuses on increasing the accuracy of low order four node quadrilateral finite elements for the transient analysis of wave propagation modified integration rules originally proposed for time harmonic problems provide the basis for the proposed technique the modified integration rules shift the integration points to locations away from the conventional gauss or gauss lobatto integration points with the goal of reducing the discretization errors specifically the dispersion error presented here is an extension of the idea to time dependent analysis using implicit as well as explicit time stepping schemes the locations of the stiffness integration points remain unchanged from those in time harmonic case on the other hand the locations of the integration points for the mass matrix depend on the time stepping scheme and the step size furthermore the central difference method needs to be modified from its conventional form to facilitate fully explicit computation the superior performance of the proposed algorithms is illustrated with the help of several numerical examples

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

improved pml for the fdtd solution of wave structure interaction problems in a previous paper an optimum perfectly matched layer has been designed for the finite difference time domain fdtd solution of wave structure interaction problems the present paper shows the results of subsequent investigations that were done with the intention of reducing the thickness of this optimum pml four improvements to the pml are presented the resulting reductions of the thickness are discussed

absorbing boundary conditions for acoustic and elastic waves in stratified media abstract absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains prior progress on this problem for acoustic and elastic waves has generally been concerned with waves propagating through uniform media the present paper is concerned with waves in stratified media which are of interest for example in geophysical problems suppose that the medium consists of homogeneous layers separated by parallel horizontal interfaces and suppose that absorbing boundary conditions are needed along a vertical computational boundary the boundary conditions that are described in this paper are based on a quantity known as the ray parameter according to snell s law this parameter remains the same when a plane wave propagates through a stratified medium and undergoes reflection refraction and in the case of elastic waves conversion once can therefore use the same absorbing boundary conditions in all layers for acoustic waves the absorption properties are the same in all layers for elastic waves the absorption properties vary somewhat from one layer to another however one still obtains good absorption in all layers even in the presence of strong contrasts between layers the boundary conditions are also effective in absorbing rayleigh waves which propagate along free surfaces of elastic media the boundary formulas developed here can be applied without modification to problems in both two and three dimensions

vector based elastic reverse time migration based on scalar imaging condition abstractthe scalar images pp ps sp and ss of elastic reverse time migration ertm can be generated by applying an imaging condition as crosscorrelation of pure wave modes in conventional ertm helmholtz decomposition is commonly applied in wavefield separation which leads to a polarity reversal problem in converted wave images because of the opposite polarity distributions of the s wavefields polarity reversal of the converted wave image will cause destructive interference when stacking over multiple shots besides in the 3d case the curl calculation generates a vector s wave which makes it impossible to produce scalar ps sp and ss images with the crosscorrelation imaging condition we evaluate a vector based ertm vb ertm method to address these problems in vb ertm an amplitude preserved wavefield separation method based on decoupled elastic wave equation is exploited to obtain the pure wave modes the output separated wavefields are both vectorial to obtain the scalar images the scala

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

a simple multi directional absorbing layer method to simulate elastic wave propagation in unbounded domains the numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries this point is critical for the analysis of wave propagation in heterogeneous or layered solids various techniques such as absorbing boundary conditions infinite elements or absorbing boundary layers e g perfectly matched layers lead to an important reduction of such spurious reflections in this paper a simple absorbing layer method is proposed it is based on a rayleigh caughey damping formulation which is often already available in existing finite element softwares the principle of the caughey absorbing layer method is first presented including a rheological interpretation the efficiency of the method is then shown through 1d finite element simulations considering homogeneous and heterogeneous damping in the absorbing layer 2d models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences a comparison with the pml method is first performed for pure p waves and the method is shown to be reliable in a more complex 2d case involving various wave types and incidences it may thus be used for various types of problems involving elastic waves e g machine vibrations seismic waves etc copyright 2010 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

reproducing kernel banach spaces with the l 1 norm ii error analysis for regularized least square regression a typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error the hypothesis error and the regularization error using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically following this direction we illustrate how reproducing kernel banach spaces with the 1 norm can be applied to improve the learning rate estimate of 1 regularization in machine learning

a comparative study of finite element and spectral element methods in seismic wavefield modeling the computational accuracy and efficiency of finite element method and spectral element method sem are investigated thoroughly in time domain elastic wavefield modeling the diagonal mass matrices of the fem and sem free from matrix inversion are compared comprehensively by making full use of the mass lumped technique and quadrature rules we investigate the fem and sem based respectively on quadrilateral with the polynomials of degrees one and two and on triangular grids with the polynomials of degrees one and three generally the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used the fem has a comparable accuracy to the sem with the same number of interpolant points in view of the triangular and quadrilateral sems the former suffers from larger computational costs and relatively lower accuracy compared with the latter furthermore the convergence study proves that the triangular sem produces consistently larger errors than the quadrilateral sem for any order and element sizes however the triangular sem can adapt to arbitrary complex geometries effectively in terms of efficiency the fem has an efficiency comparable with the sem on condition that the order of interpolation polynomials is identical in addition a perfectly matched layer pml boundary condition in variational form is deduced by introducing four intermediate variables in frequency domain the pml avoids convolution calculation and obtains an exact solution through inverse fourier transform in time domain the numerical examples verify the validity and effectiveness of the pml

elastic least squares reverse time migration with hybrid l1 l2 misfit function abstract we have developed the theory and synthetic tests of elastic least squares reverse time migration elsrtm in this method a least squares reverse time migration algorithm is used to image multicomponent seismic data based on the first order elastic velocity stress wave equation in which the linearized elastic modeling equations are used for forward modeling and its adjoint equations are derived based on the adjoint state method for back propagating the data residuals also we have developed another elsrtm scheme based on the wavefield separation technique in which the p wave image is obtained using p wave forward and adjoint wavefields and the s wave image is obtained using p wave forward and s wave adjoint wavefields in this way the crosstalk artifacts can be minimized to a significant extent in general seismic data inevitably contain noise we apply the hybrid l1 l2 misfit function to the elsrtm algorithm to improve the robustness of our elsrtm to noise numerical tests on synthetic dat

least square regression with lp coefficient regularization the selection of the penalty functional is critical for the performance of a regularized learning algorithm and thus it deserves special attention in this article we present a least square regression algorithm based on lp coefficient regularization comparing with the classical regularized least square regression the new algorithm is different in the regularization term our primary focus is on the error analysis of the algorithm an explicit learning rate is derived under some ordinary assumptions

stochastic joint inversion of 2d seismic and seismoelectric signals in linear poroelastic materials a numerical investigation the interpretation of seismoelectrical signals is a difficult task because coseismic and seismoelectric converted signals are recorded simultaneously and the seismoelectric conversions are typically several orders of magnitude smaller than the coseismic electrical signals the seismic and seismoelectric signals are modeled using a finite element code with perfectly matched layer boundary conditions assuming a linear poroelastic body we present a stochastic joint inversion of the seismic and seismoelectrical data based on the adaptive metropolis algorithm to obtain the posterior probability density functions of the material properties of each geologic unit this includes the permeability porosity electrical conductivity bulk modulus of the dry porous frame bulk modulus of the fluid bulk modulus of the solid phase and shear modulus of the formations a test of this approach is performed with a synthetic model comprising two horizontal layers and a reservoir partially saturated with oil which is embedded in the second layer the result of the joint inversion shows that we can invert the permeability of the reservoir and its mechanical properties