velocity attenuation and quality factor in anisotropic viscoelastic media a perturbation approach velocity attenuation and the quality q factor of waves propagating in homogeneous media of arbitrary anisotropy and attenuation strength are calculated in high frequency asymptotics using a stationary slowness vector the vector evaluated at the stationary point of the slowness surface this vector is generally complex valued and inhomogeneous meaning that the real and imaginary parts of the vector have different directions the slowness vector can be determined by solving three coupled polynomial equations of the sixth order or by a nonlinear inversion the procedure is simplified if perturbation theory is applied the elastic medium is viewed as a background medium and the attenuation effects are incorporated as perturbations in the first order approximation the phase and ray velocities and their directions remain unchanged being the same as those in the background elastic medium the perturbation of the slowness vector is calculated by solving a system of three linear equations the phase att

a unified theory for formulation of hybrid stress membrane elements a theory is proposed in this paper to explain in a unified manner the three rational approaches using the incompatible displacements u in the hybrid element formulation limited to the plane stress case the theory suggests that the introduction of u is to constrain the total value of the invariant i2 x y of the assumed stresses over the element to zero in the plane stress case this is the well known condition for the stress distributions to be independent of the elastic constants of the material the theory proposed is also used to explain the closeness of the performances of the four 4 node hybrid stress membrane elements 5 i 5 ii 5 a and 5 c that have been developed previously the differences among them are shown to be of the second degree in the distortion measures of the element shape when they are compared on the basis of the non vanishing total value of i2 of their respectively adopted assumed stress field several numerical examples are used to verify and to illustrate the proposed theory

minimum traveltime calculation in 3 d graph theory traveltime calculation is a crucial part of seismic migration schemes especially prestack migration there are many different ways to compute traveltimes these methods can be divided into three categories 1 ray tracing julian and gubbins 1977 cerven et al 1977 these treat the problem as a initial value problem by shooting rays from the source to the receivers or they can also treat the problem as a two point boundary value problem an initial raypath is bent using perturbation theory until fermat s principle is satisfied nichols 1994 also computed traveltimes with the amplitude information attached to it in two dimensions 2 finite difference methods reshel and kosloff 1986 vidale 1988 van trier and symes 1991 these solve the eikonal equation directly by using different numerical schemes such as the runge kutta method wavefront expansion or upwind finite difference 3 graph theory moser 1991 fisher and lees 1993 meng et al 1994 this method recasts the traveltime prob

shortest path calculation of seismic rays like the traveling salesman who wants to find the shortest route from one city to another in order to minimize his time wasted on traveling one can find seismic raypaths by calculating the shortest traveltime paths through a network that represents the earth the network consists of points that are connected with neighboring points by connections as long as the traveltime of a seismic wave along it the shortest traveltime path from one point to another is an approximation to the seismic ray between them by fermat s principle the shortest path method is an efficient and flexible way to calculate the raypaths and traveltimes of first arrivals to all points in the earth simultaneously there are no restrictions of classical ray theory diffracted raypaths and paths to shadow zones are found correctly there are also no restrictions to the complexity or the dimensionality of the velocity model furthermore there are no problems with convergence of trial raypaths toward a specified receiver nor with ray

on numerically solving the complex eikonal equation using real ray tracing methods a comparison with the exact analytical solution the exact analytical solution of the complex eikonal equation describing p and s waves radiated by a point source situated in a simple type of isotropic viscoelastic medium was ascertained the velocity attenuation model is smoothly inhomogeneous with a constant gradient of the square of the complex slowness the resultant traveltime is complex its real part describes the wave propagation and its imaginary part describes the attenuation effects the solution was further used as a reference solution for numerical tests of the accuracy and robustness of two approximate ray tracing approaches solving the complex eikonal equation real elastic ray tracing and real viscoelastic ray tracing numerical modeling revealed that the real viscoelastic ray tracing method is unequivocally preferable to elastic ray tracing it is more accurate and works even in situations when the elastic ray tracing fails also the ray fields calculated by the real viscoelastic ray tracing are excellently reproduced even in the case when the elastic ray tracing yields completely distorted results compared with complex ray tracing which is limited to simple types of media the real viscoelastic ray tracing offers a fast and computationally straightforward procedure for calculating complex traveltimes in complicated 3d inhomogeneous attenuating structures

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

associated stieltjes carlitz polynomials and a generalization of heun s differential equation abstract the generating function of stieltjes carlitz polynomials is a solution of heun s differential equation and using this relation carlitz was the first to get exact closed forms for some heun functions similarly the associated stieltjes carlitz polynomials lead to a new differential equation which we call associated heun thanks to the link with orthogonal polynomials we are able to deduce two integral relations connecting associated heun functions with different parameters and to exhibit the set of associated heun functions which generalize carlitz s part of these results were used by the author to derive the stieltjes transform of the measure of orthogonality for the associated stieltjes carlitz polynomials using asymptotic analysis here we present a new derivation of this result

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

a rational approach for choosing stress terms for hybrid finite element formulations a new approach for choosing the stress terms for hybrid stress elements is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacements examples using 4 node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting element will provide the most accurate solutions

on the incompressible constraint of the 4 node quadrilateral element the standard plane 4 node element is written as the summation of a constant gradient matrix usually obtained from underintegration and a stabilization matrix the split is based on a taylor series expansion of element basis functions in the incompressible limit the locking effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher order term in the taylor series the incompressibility condition is formulated in a weak sense so that the element displacement field is divergence free when integrated over the element volume the resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first order terms of the taylor series the corresponding eigenvalue enforces incompressibility implicitly by means of a penalty constraint analytical expressions for that constant dilatation eigenpair are derived for arbitrary element geometries it is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner for several classical and recent elements the eigensystems are analysed numerically it is shown that most of the formulations reflect the incompressible constraint identically in the incompressible limit the numerical accuracies of the elements are compared

development and testing of stable invariant isoparametric curvilinear 2 and 3 d hybrid stress elements abstract linear and quadratic serendipity hybrid stress elements are examined in respect of stability coordinate invariance and optimality a formulation based upon symmetry group theory successfully addresses these issues in undistorted geometries and is fully detailed for plane elements the resulting least order stable invariant stress polynomials can be applied as astute approximations in distorted cases through a variety of tensor components and variational principles a distortion sensitivity study for two and three dimensional elements provides favourable numerical comparisons with the assumed displacement method

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

a second order discontinuous galerkin fast sweeping method for eikonal equations in this paper we construct a second order fast sweeping method with a discontinuous galerkin dg local solver for computing viscosity solutions of a class of static hamilton jacobi equations namely the eikonal equations our piecewise linear dg local solver is built on a dg method developed recently y cheng c w shu a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations journal of computational physics 223 2007 398 415 for the time dependent hamilton jacobi equations the causality property of eikonal equations is incorporated into the design of this solver the resulting local nonlinear system in the gauss seidel iterations is a simple quadratic system and can be solved explicitly the compactness of the dg method and the fast sweeping strategy lead to fast convergence of the new scheme for eikonal equations extensive numerical examples verify efficiency convergence and second order accuracy of the proposed method

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

extension of field consistence approach into developing plane stress elements in this research paper the possibilities for extending the field consistence approach l yunhua on shear locking in finite elements licentiate thesis stockholm 1997 l yunhua explanation and elimination of shear locking and membrane locking with field consistence approach comput methods appl mech engrg in press starting from different variational principles to plane stress elements are investigated in the extension two main difficulties are explicitly solving a set of coupled partial differential equations and satisfying inter element compatibility the first one is alleviated by constructing element interpolations from a set of quasi general solutions rather than the real general solutions to the euler lagrangian equations the second one is solved by combining the field consistence approach with the iso parametric interpolation technique the traditional assumed stress method is improved and an efficient plane stress element is obtained it seems that the relations between the three commonly used variational principles can be more reasonably established in the framework of the field consistence approach

the generating function method for quadratic asymptotically symmetric birth and death processes the general form of the generating function associated with quadratic asymptotically symmetric birth and death processes is given with no restriction but the positivity of the transition rates the explicit form of the moments is obtained when their behaviour is explosive for large time finally the relationship between the generating function method and the karlin mcgregor approach based on orthogonal polynomial theory is established

computation of uniform wave forms using complex rays complex rays and polynomial phase functions are used to numerically solve the helmholtz equation in a realistic two dimensional smoothly varying heterogeneous velocity model with multiple adjacent cusp caustics together these two methods allow the determination of global uniformly asymptotic solutions in the presence of arbitrarily many caustics two algorithms are introduced to this end a two point ray tracing algorithm for complex rays and a perturbation method for constructing polynomial phase functions model representation in complex space is performed via discrete cosine transform analysis geometrical and uniformly asymptotic solutions are computed for a linear layer test model as well as a velocity model from yucca mountain

solving the birth and death processes with quadratic asymptotically symmetric transition rates birth and death processes with quadratic transition rates are considered with the constraint lim n to infty lambda n mu n 1 the partial differential equation governing the generating function is solved quite generally and gives an integral representation for it in the process a generalized mehler fork transform is defined and inverted from the generating function we extract the transition probabilities which are shown to be the solution of the chapman kolmogorov equations various approximation schemes are then devised to deal with large populations and to extract the large time behavior for the transition probabilities

spectral properties of birth death polynomials we consider sequences of polynomials that are defined by a three terms recurrence relation and orthogonal with respect to a positive measure on the nonnegative axis by a famous result of karlin and mcgregor such sequences are instrumental in the analysis of birth death processes inspired by problems and results in this stochastic setting we present necessary and sufficient conditions in terms of the parameters in the recurrence relation for the smallest or second smallest point in the support of the orthogonalizing measure to be larger than zero and for the support to be discrete with no finite limit point

short note o n implementation of the fast marching algorithm in this note we present an implementation of the fast marching algorithm for solving eikonal equations that in practice reduces the original run time from o nlogn to linear this lower run time cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm this improvement is achieved introducing the straight forward untidy priority queue obtained via a quantization of the priorities in the marching computation we present the underlying framework estimations on the error and examples showing the usefulness of the proposed approach

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials we study a birth and death process with quartic transition rates for which the transition probabilities but for the honest ones are not unique for their related polynomials we exhibit infinitely many different orthogonality measures among which stems the honest one these results are obtained firstly by reducing the kolmogorov equations to linear partial differential equations and secondly by mapping them to partial differential equations with constant coefficients which can be solved analytically

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

new strategy for assumed stresses for 4 node hybrid stress membrane element an oblique skew co ordinate system defined at the origin of the isoparametric co ordinates is introduced as the reference co ordinate system for assumed stresses in the formulation of a 4 node hybrid membrane element the initially uncoupled stresses in these reference co ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of pian and wu7 when suitable incompatible displacements are introduced such strategy allows the element stiffness to be formulated by using either the hellinger reissner or the complementary energy functional numerical results show that the performances of three elements constructed by using different sets of incompatible displacements in this paper are practically identical and are all of slightly higher accuracy than that of the pian sumihara element6

relations between incompatible displacement model and hybrid stress model an examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by irons in 1972 but the basic differences between the two are also identified for 8 node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by irons purely from his physical insight