generalized bremmer series with rational approximation for the scattering of waves in inhomogeneous media the bremmer series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper sparse matrix representations of these pseudodifferential operators are derived the authors focus on designing sparse matrices keeping the accuracy high at the cost of ignoring any critical scattering angle phenomena such matrix representations follow from rational approximations of the vertical slowness and the transverse laplace operator symbols and of the vertical derivative as they appear in the parabolic equation method sparse matrix representations lead to a fast algorithm an optimization procedure is followed to minimize the errors in the high frequency limit for a given discretization rate the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counterpropagating constituents each of these steps is represented by a sparse matrix equation the resulting algorithm provides an improvement of the parabolic equation method in particular for transient wave phenomena and extends the latter method systematically for backscattered waves

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

reducing two way splitting error of ffd method in dual domains the fourier finite difference ffd method is very popular in seismic depth migration but its straightforward 3d extension creates two way splitting error due to ignoring the cross terms of spatial partial derivatives traditional correction schemes either in the spatial domain by the implicit finite difference method or in the wavenumber domain by phase compensation lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts we propose compensating the two way splitting error in dual domains alternately in the spatial and wavenumber domains via fourier transform first we organize the expanded square root operator in terms of two way splitting ffd plus the usually ignored cross terms second we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions finally we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator in implementation the compensation terms are similar to the high order corrections of the generalized screen method but their functions are to compensate the two way splitting error rather than the expansion error numerical experiments show that optimized one term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1 error for all azimuths is improved up to 60 from 35 compared with traditional single domain methods our scheme can handle lateral velocity variations even for strong velocity contrasts much more easily with only one additional fourier transform based on the two way splitting ffd method which helps retain the computational efficiency

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

a critical analysis of quadratic beam finite elements a new methodology of evaluation of c0 beam elements is presented it is shown that knowing the stiffness matrix of an arbitrary type of element it is possible to create equivalent equilibrium conditions expressed in the form of one difference equation for a regular beam discretized by these elements the study of the convergence of one difference equation gives an interpretation of the source of troubles occurring in low order bending elements which is more convincing than the usually applied consideration of the conditioning of element stiffness matrices a careful examination of quadratic mindlin elements provides a very clear explanation of the shear locking essence in the timoshenko beam the presented method enables one to identify errors that appear also in the reduced integrated or constrained elements for each type of analysed quadratic element an adequate difference equation is derived and compared with the exact one based on this comparison a simple method of corrections is proposed that completely eliminates the errors associated with the application of c0 bending elements

predicting and avoiding shear locking in beam vibration problems using the b spline field approximation method the b spline field approximation method is used to construct discrete models of straight timoshenko beams the traditional timoshenko beam discretization treats deflection and rotation as the independent field variables and permits the use of c0 discretizations but can sometimes exhibit unwanted shear locking behaviour in this paper a new explanation of shear locking is presented and is used in selecting a variety of b spline interpolation fields that can be predicted not to suffer from shear locking it is also exploited in an adaptation of a timoshenko model where deflection and shear strain are independently discretized field variables this requires a c1 discretization of the deflection but avoids shear locking in every case and allows for a more economical solution than previous implementations of this formulation as a means of illustrating the performance characteristics of the formulation presented in the paper the vibration characteristics of these models are compared with traditional finite element models for this problem the b spline method outperforms the finite element method and has the advantage of an easy predictability of the locking behaviour that can be expected

2 d depth migration in transversely isotropic media using explicit operators stable explicit depth extrapolation filters can be used to propagate plane waves corresponding to the qp and qsv quasi p and quasi sv propagation modes for transversely isotropic ti media here i discuss and compare results of two different methods for obtaining the filters for ti media with a vertical axis of symmetry vti the first a modified taylor series method is used to calculate the n coefficients of a finite length filter such that the taylor expansion around vertical propagation matches the spatial fourier transform of the downward continuation operator for vti media second a least squares method is used to calculate the filter coefficients such that the amplitude and phase departures from the ideal response of the downward continuation operator for vti media are minimized over a range of frequencies and propagation angles in both methods the amplitude response of the filter is forced to be less than unity in the evanescent region to achieve stability in general as exemplified in

splitting of operators alternate directions and paraxial approximations for the three dimensional wave equation we design a family of new three dimensional 3 d paraxial equations well adapted to numerical solution by the alternate directions method these equations do not suffer from bad anisotropic effects and the cost for their numerical integration remains cheap in comparison with classical paraxial equations

penalty resolution of the babuska circle paradox abstract if the solution for a simply supported circular plate is approximated by a sequence of finite element solutions on successively refined polygonal domains the finite element approximations converge to the solution of a different problem in the present study we present a finite element formulation with boundary penalty that produces valid approximations to the simply supported plate the penalty term is shown to require the use of reduced integration the dependence of the penalty parameter on mesh size h is also examined numerical experiments confirm the validity of the method and determine rates of convergence a second approach involving a modified corner condition is also considered and error estimates determined this scheme is implemented also using a discrete penalty technique the results and rates are compared with the boundary penalty method and their relative merits discussed

optimized finite difference operator for broadband seismic wave modeling high resolution image and waveform inversion of smallscale targets requires the handling of high frequency seismic wavefields however conventional finite difference fd methods have strong numerical dispersions in the presence of high frequency components to reduce these numerical dispersions we optimized the constant coefficients of the fd operator by maximizing the wavenumber coverage within a given error limitation we set up three general criteria to enhance the convergence of the algorithm and reduce the optimization effort we selected the error limitation to be 0 0001 this being the smallest in the literature which led to perfect agreement between theoretical analyses and numerical experiments the accuracy of our optimized fd methods can even reach that of much higher order unoptimized fd methods which means great savings of computational efforts and memory demand these advantages become even more apparent with 3d modeling especially for saving memory demand

assumed strain formulation for triangular c0 plate elements based on a weak form of the kirchhoff constraints application of a weak form of the kirchhoff constraints in the formulation of triangular assumed shear strain c0 plate element is described it is shown that compared to a similar formulation of the quadrilateral elements formulation of the triangles requires some additional considerations which lead to additional terms in the energy expression with those terms performance of the assumed shear strain triangular elements is comparable to that of the quadrilateral elements

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

fourier finite difference migration many existing migration schemes cannot simultaneously handle the two most important problems of migration imaging of steep dips and imaging in media with arbitrary velocity variations in all directions for example phase shift k migration is accurate for nearly all dips but it is limited to very simple velocity functions on the other hand finite difference schemes based on one way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events we propose a new hybrid migration method named fourier finite difference migration wherein the downward continuation operator is split into two downward continuation operators one operator is a phase shift operator for a chosen constant background velocity and the other operator is an optimized finite difference operator for the varying component of the velocity function if there is no variation of velocity then only a phase shift operator will be applied automatically on the other hand if there is a strong variatio

a stabilization procedure for the quadrilateral plate element with one point quadrature a stabilization procedure is developed for controlling the kinematic modes of the four node bilinear quadrilateral element when single point quadrature is used these kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness in this stabilization procedure additional generalized strains are defined which are activated by the kinematic modes these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral by using a scaling law developed in an earlier paper the stabilization parameters are defined so they do not adversely affect the element s performance several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems

a fractal patch test element consistency is generally checked using the patch test on an element patch of finite size this condition may in certain cases be too restrictive and disqualifies elements that appear to be convergent a method termed fractal patch test is presented in which the patch size is maintained constant while the distorted mesh is refined examples are given for four node quadrilateral elements used in plane stress and strain analysis and for plate bending elements

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

3 d implicit finite difference migration by multiway splitting we show that 3 d implicit finite difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome the basic idea is as follows we approximate the 3 d square root operator by a sequence of 2 d operators in three four or six directions to solve the azimuth symmetry problem each 2 d square root operator is then approximated by a sequence of implicit 2 d operators to improve steep dip accuracy this sequence contains some unknown coefficients which are calculated by a taylor expansion technique or by an optimization technique in the taylor expansion method the square root and its approximation are expanded into power series by comparing the terms the unknown coefficients are calculated the more 2 d finite difference operators for cascading are taken and the more directions for downward continuation are chosen the more terms from power series can be compared to obtain a higher degree migration operator with better

physical shape functions in finite element analysis of moderately thick plates a new rectangular finite element for moderately thick plates is presented the element is based on the concept of physical shape functions i e functions which contain in themselves physical and geometrical properties of the element the analytic formulae of stiffness geometric stiffness and mass matrices are presented for isotropic material the element is free from locking and zero energy modes not corresponding to rigid body motions several examples are presented for static initial stability and free vibration problems

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

the embedded discontinuous galerkin method application to linear shell problems a new discontinuous galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement based galerkin method is presented those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach this makes the present formulation easily implementable within the existing commercial computer codes the proposed approach is named the embedded discontinuous galerkin method it is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity an application of the method to linear shell problems is then outlined and numerical results are presented

an acoustic wave equation for anisotropic media a wave equation derived using the acoustic medium assumption for p waves in transversely isotropic ti media with a vertical symmetry axis vti media yields a good kinematic approximation to the familiar elastic wave equation for vti media the wavefield solutions obtained using this vti acoustic wave equation are free of shear waves which significantly reduces the computation time compared to the elastic wavefield solutions for exploding reflector type applications from this vti acoustic wave equation the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a ti medium are derived these equations based on the acoustic assumption shear wave velocity 0 are much simpler than their elastic counterparts yet they yield an accurate description of traveltimes and geometrical amplitudes numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex ti models

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

a lock free material finite element for non linear oscillations of laminated plates abstract the objective of the present paper is to propose an efficient accurate and robust four node shear flexible composite plate element with six degrees of freedom per node to investigate the non linear oscillatory behavior of unsymmetrical laminated plates the degrees of freedom considered are three displacement u v w along x y and z axis two rotations x y abouty and x axis and twist xy the elementc employs coupled displacement field which is derived using moment shear equilibrium and in plane equilibrium of composite strips along the x andy axis the displacement field so derived not only depend on the element co ordinates but are a function of extensional bending extensional bending and transverse shear stiffness coefficients as well a bi cubic polynomial distribution with 16 generalized undetermined coefficients for the transverse displacement is assumed the element stiffness and mass matrices are computed numerically by employing 3 3 gauss legendre product rules the element is found to be free of shear locking and does not exhibit any spurious modes the element is found to be free of shear locking and does not exhibit any spurious modes in order to compute the non linear frequencies linear mode shape corresponding to fundamental frequency is assumed as the spatial distribution and non linear finite element equations are reduced to a single non linear second order ordinary differential equation this equation is solved by employing direct numerical integration method a series of numerical examples is solved to demonstrate the efficacy of the proposed material finite element

a study of stabilization and projection in the 4 node mindlin plate element in this paper a mindlin plate element is formulated based on the hellinger reissner principle and the technique the stiffness consists of a constant stress one point quadrature matrix and a stabilization matrix the stabilization matrix is compared with those previously proposed in addition the element uses a projection to modify the nodal displacements so that the patch test is satisfied the projection matrix is based on a mode decomposition several numerical cases are presented and it is shown that the mode decomposition projection is necessary both for satisfaction of the patch test and convergence

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

a stable one way wave propagator for vti media for the purpose of one way wave equation imaging a pseudoscreen propagator is developed for transversely isotropic media with vertical axes of symmetry the phase shift for propagation through a depth slice is decomposed into three terms a gazdag phase shift for propagation in a laterally homogeneous reference medium a correction for lateral variability of vertical propagation and a remaining wide angle term for oblique directions of propagation based on rational function approximation for this remaining wide angle term a fourier finite difference ffd approach with four way splitting is applied fourth order pade approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction therefore a method of coefficient optimization is developed in conjunction with a method of choosing an adequate homogeneous reference medium in a depth slice by symmetrizing the finite difference operators and because of the choice of the optimized coefficients the propagator is stable in the sense that the least squares norm of the wavefield measured for a frequency depth slice does not grow with increasing depth of propagation a small amount of artificial damping is applied to suppress artifacts that appear at the critical angle defined by the velocities in the reference medium and the actual medium synthetic examples confirm that good kinematic accuracy is achieved for a wide range of propagation angles typically up to 60

scalar generalized screen algorithms in transversely isotropic media with a vertical symmetry axis the scalar generalized screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis vti although wave propagation in a transversely isotropic medium is essentially elastic we use an equivalent acoustic system of equations for the qp waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of mild anisotropy the enhanced accuracy of the generalized screen method as compared to the split step fourier methods allows the extension to vti media the generalized screen expansion of the one way propagator follows closely the method used in the isotropic case the medium is defined in terms of a background and a perturbation the generalized screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation we cast the theory into numerical algorithms and assess the accuracy of the generalized screen method in a partic

optimized chebyshev fourier migration a wide angle dual domain method for media with strong velocity contrasts a wide angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one way waveequation migration we have developed a dual domain one way propagator using truncated chebyshev polynomials and a globally optimized scheme our method increases the accuracy of the expanded square root operator by adding two high order terms to the traditional split step fourier propagator first we approximate the square root operator using taylor expansion around the reference background velocity then we apply the first kindchebyshevpolynomialstoeconomizetheresultsofthe taylor expansion finally we optimize the constant coefficients using the globally optimized scheme which are fixed and feasible for arbitrary velocity models theoretical analysis and numericalexperimentshavedemonstratedthatthemethodhasvery highaccuracyandexceedstheunoptimizedfourierfinite difference propagator for the entire range of practical velocity contrasts the accurate propagation angle of the method is always about 60 under the relative error of 1 for complex media with weak moderate and even strong lateral velocity contrasts the method allows us to handle wide angle propagations and strong lateral velocity contrast simultaneously by using fourier transform alone only four 2d fourier transforms are required for each step of 3d wavefield extrapolation and the computing cost is similar to that of the fourier finite difference method compared with the finite difference method our method has no twowaysplittingerrori e azimuthal anisotropyerrorfor3dcases and almost no numerical dispersion for coarse grids in addition it has strong potential to be accelerated when an enhanced fast fouriertransformalgorithmemerges

anisotropic complex pade hybrid finite difference depth migration standard real valued finite difference fd and fourier finite difference ffd migrations cannot handle evanescent waves correctly which can lead to numerical instabilities in the presence of strong velocity variations a possible solution to these problems is the complex pade approximation which avoids problems with evanescent waves by rotating the branch cut of the complex square root we have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable fd and hybrid fd ffd migrations for such media our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60 for fd migration as confirmed by numerical impulse responses and work with synthetic data

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

a kirchhoff mode method for c0 bilinear and serendipity plate elements abstract a new formulation for the c0 quadrilateral and serendipity plate elements is presented it is an extension of the mode decomposition approach that has been successfully applied in evaluating the c0 triangular linear plate element in contrast to the triangular element the concept of an equivalent discrete kirchhoff configuration is utilized the elements are applied to several examples and the performance of these elements is shown to be quite good

directional illumination analysis using the local exponential frame we have developed an efficient method of directional illuminationanalysisinthelocalangledomainusinglocalexponential frame beamlets the space domain wavefields with different shot receiver geometries are decomposed into the local angle domain by using the local exponential beamlets which form a tight frame with the redundancy ratio two and are implemented by a linear combination of local cosine and localsinetransforms becauseofthefastalgorithmsofthelocal cosine sine transforms this method is much more efficient than the previously used decomposition methods in directional illumination analysis such as the local slant stackingmethodandthegabor daubechiesframemethod theresults of directional illumination di maps and the acquisitiondipresponsesadrforthe2dseg eagesaltmodel and the 45 shot 3d seg eage model demonstrated the validity and feasibility of our method compared with the illumination results using local slant stacking decomposition thenewmethodproducesilluminationmapsofsimilarquality but it does so a few times faster furthermore because of its high computational efficiency and saving in memory usage the new method makes the 3d directional illumination analysisreadilyapplicableintheindustry

derivation and application of extended parabolic wave theories ii path integral representations the n dimensional reduced scalar helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through 1 the n dimensional extended parabolic weyl pseudodifferential equation and 2 an imbedding in an n 1 dimensional parabolic schrodinger equation the first relationship provides the basis for the parabolic based hamiltonian phase space path integral representation of the half space propagator the second relationship provides the basis for the elliptic based path integral representations associated with feynman and fradkin feynman and garrod and feynman and dewitt morette exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow and wide angle extended parabolic wave theories the path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

a mixed formulation of c0 linear triangular plate shell element the role of edge shear constraints a simple and effective linear c0 triangular element for plates and shells is developed on the basis of the hellinger reissner mixed variational principle with independently assumed stress and displacement fields two main features are emphasized in this development concerning the assumed transverse shear stress field firstly this assumption results in edge type penalty constraints in the thin plate shell regimes secondly the element is used in the form of a four triangle cross diagonal macroelement mesh as a result the element is shown to be free from shear locking in thin plate shell applications and it satisfied all the appropriate patch tests required for kirchhoff plate models in addition the element exhibits good overall convergence properties in a variety of test problems for plates and shells finally from the computational standpoint the element is very efficient since all stiffness component matrices are derived in explicit forms

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

optimized implicit finite difference and fourier finite difference migration for vti media propagation velocity of seismic waves in heterogeneous vti media depends not only on spatial location but also on their propagation direction which leads to a much more complex dispersion relation than in isotropic media as a result designing implicit finite difference fd schemes for wavefield extrapolation in anisotropic media through analytic taylor series expansion is more difficult implicit fd and fourier finite difference ffd schemes are developed for vertical transversely isotropic vti media based on function fitting the dispersion relation of vti media is approximated with a rational function and its coefficients are estimated by weighted least squares optimization because these coefficients are functions of thomsen anisotropy parameters e and and vary laterally in heterogeneous vti media they are calculated before wavefield extrapolation and stored in a table implicit fd and ffd schemes for vti media are almost the same as for isotropic media except that coefficients are looke

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

globally optimal finite difference schemes based on least squares abstractspatial finite difference fd coefficients are usually determined by the taylor series expansion te or optimization methods the former can provide high accuracy on a smaller wavenumber or frequency zone and the latter can give moderate accuracy on a larger zone present optimization methods applied to calculate fd coefficients are generally gradient like or global optimization like algorithms and thus iterations are involved they are more computationally expensive and sometimes the global solution may not be found i examined second order spatial derivatives and computed the optimized spatial fd coefficients over the given wavenumber range using the least squares ls method the results indicated that the fd accuracy increased with increasing operator length and decreasing wavenumber range therefore for the given error and operator length globally optimal spatial fd coefficients can be easily obtained some optimal fd coefficients were given i developed schemes to obtain optimized ls

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

global optimization of generalized nonhyperbolic moveout approximation for long offset normal moveout the approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media the generalized nonhyperbolic moveout approximation gma brings a considerable improvement in accuracy compared to known analytical approximations however this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters which would degrade the inversion accuracy due to error accumulation in velocity analysis we optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset to depth ratio within practical ranges theoretical analyses and numerical experiments show that the traditional optimization scheme using 2 norm solved by the least squares method could not provide an error constrained result in addition a direct optimization without constant coefficient extension would not lead to a satisfactory accuracy improvement we construct the objective function using the maximum norm and so

a c0 triangular plate element with one point quadrature on developpe un element de plaque triangulaire du type de mindlin qui utilise des champs lineaires pour les rotations et les deflexions transversales l element ne necessite qu un seul point de quadrature ce qui est tres interessant pour l analyse non lineaire

a hermite reproducing kernel approximation for thin plate analysis with sub domain stabilized conforming integration a hermite reproducing kernel rk approximation and a sub domain stabilized conforming integration ssci are proposed for solving thin plate problems in which second order differentiation is involved in the weak form although the standard rk approximation can be constructed with an arbitrary order of continuity the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems by imposing the kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously it is demonstrated that the minimum normalized support size coverage of kernel functions can be significantly reduced with this proposed approximation the galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived subsequently an ssci method is developed to achieve the exact pure bending solution as well as to maintain spatial stability numerical examples demonstrate that the proposed formulation offers superior convergence rates accuracy and efficiency compared with those based on higher order gauss quadrature rule copyright 2007 john wiley sons ltd

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas

design of one way wavefield extrapolation operators using smooth functions in wlsq optimization many depth migration methods use one way frequency space depth extrapolation methods these methods are generally considered to be expensive so it is important to find the most efficient way of implementing them this usually means making spatial convolution operators that are as short as possible applying the extrapolation operators in a recursive way using small depth steps also demands that the operators do not amplify the wavefield at every depth step weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm the extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the remez exchange method and nonlinear optimization in this paper the weighted least squares technique is refined by using different model functions by smoothing the phase and amplitude transition at the evanescent c

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

reduced integration and the shear flexible beam element the timoshenko beam element is studied in an investigation of shear locking in the development of c 0 continuous elements using shear flexible or penalty function type formulations a penalty type argument is used to degenerate thick elements to thin elements and the various approximations of the shear related energy terms act as different types of constraints depending on the formulation two types of constraints may emerge which are classified as true or spurious formulations that ensure only true constraints in the extreme penalty limit cases display superior performance in the thick element situation the argument is extended to a shallow curved timoshenko beam and another mechanism inplane locking arises if all energy terms corresponding to the membrane energy are exactly integrated