3d finite difference modeling of elastic wave propagation in the laplace fourier domain abstractwith the recent interest in the laplace fourier domain full waveform inversion we have developed new heterogeneous 3d fourth and second order staggered grid finite difference schemes for modeling seismic wave propagation in the laplace fourier domain our approach is based on the integro interpolation technique for the velocity stress formulation in the cartesian coordinate system five averaging elastic coefficients and three averaging densities are necessary to describe the heterogeneous medium with harmonic averaging of the bulk and shear moduli and arithmetic averaging of density in the fourth order approximation we improved the accuracy of the scheme using a combination of integral identities for two elementary volumes small and large around spatial gridpoints where the wave variables are defined two solution approaches are provided both of which are solved with iterative krylov methods in the first approach the stress variables are eliminated and a linear system for the veloc

tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

a mass redistributed finite element method mr fem for acoustic problems using triangular mesh the accuracy of numerical results using standard finite element method fem in acoustic problems will deteriorate with increasing frequency due to the dispersion error such dispersion error depends on the balance between the stiffness and mass of discretization equation systems this paper reports an improved finite element method fem for solving acoustic problems by re distributing the mass in the mass matrix to tune the balance aiming to minimize the dispersion errors this is done by shifting the integration point locations when computing the entries of the mass matrix while ensuring the mass conservation the new method is verified through the detailed numerical error analysis and a strategy is also proposed for the best mass redistribution in terms of minimizing dispersion error the relative dispersion error of present mass redistributed finite element method mr fem is found to be much smaller than the fem solution in both theoretical prediction and numerical examination the present mr fem works well by using the linear triangular elements that can be generated automatically which enables automation in computation and saving computational cost in mesh generation numerical examples demonstrate the advantages of mr fem in comparison with the standard fem using the same triangular meshes and quadrilateral meshes

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

a parallel element by element solution scheme a parallel element by element scheme is developed for finite element solution of elliptic boundary value problems it is shown that for a broad class of computational grids the solution algorithm is totally parallelizable moreover the conversion of an existing serial ebe code to a parallel code is seen to be quite simple the method is implemented on the alliant fx 8 and sequent balance parallel computers and speedup performance studies are conducted for a representative elliptic pde in two dimensions the present procedure can be applied quite generally to other finite element applications and the associated programs can be modified in a straightforward way to implement the method

high convergence order finite elements with lumped mass matrix a method for deriving hexahedral finite elements with lumped mass matrices for 3 d problems is presented these elements meet the theoretical conditions for high order convergence and two numerical examples based on the 3 d scalar wave equation show that this is also the case in practice and that their accuracy is comparable to elements with consistent mass matrices

on the use of nurbs based discretizations in the scaled boundary finite element method for wave propagation problems abstract we discuss the application of non uniform rational b splines nurbs in the scaled boundary finite element method sbfem for the solution of wave propagation problems at rather high frequencies we focus on the propagation of guided waves along prismatic structures of constant cross section comparisons are made between nurbs based discretizations and high order spectral elements in terms of the achievable convergence rates we find that for the same order of shape functions nurbs can lead to significantly smaller errors compared with lagrange polynomials the difference becomes particularly important at very high frequencies where spectral elements are prone to instabilities furthermore we analyze the behavior of nurbs for the discretization of curved boundaries where the benefit of exact geometry representation becomes crucial even in the low frequency range

bidirectional approximate reasoning for rule based systems using interval valued fuzzy sets abstract in this paper we extend the work of chen et al fuzzy sets and systems 91 1997 339 353 to present a new method to deal with bidirectional approximate reasoning for rule based systems based on the direction of matching between interval valued fuzzy sets we also use some examples to illustrate the bidirectional approximate reasoning process because the proposed method can perform bidirectional approximate reasoning based on the direction of matching between interval valued fuzzy sets it is more reasonable and more powerful than the one presented in chen et al fuzzy sets and systems 91 1997 339 353 the proposed method can provide a useful way to deal with bidirectional approximate reasoning for rule based systems using interval valued fuzzy sets

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

bidirectional approximate reasoning based on interval valued fuzzy sets abstract this paper proposes a bidirectional approximate reasoning method based on interval valued fuzzy sets where fuzzy production rules are used for knowledge representation and the fuzzy terms appearing in the fuzzy production rules of a rule based system are represented by interval valued fuzzy sets the proposed method is more flexible than the one presented in the paper by bien and chun ieee trans fuzzy systems 2 1994 177 due to the fact that it allows the fuzzy terms appearing in the fuzzy production rules of a rule based system to be represented by interval valued fuzzy sets rather than general fuzzy sets furthermore because the proposed method requires only simple arithmetic operations and because it allows bidirectional approximate reasoning it can be executed much faster and more flexible than the single input single output approximate reasoning scheme presented in the paper by gorzalczany fuzzy sets and systems 21 1981 10

extended fuzzy logic sets and systems the concepts of sets and approximate reasoning within extended fuzzy logic fle provide a systematic procedure for transforming unprecisiated knowledge into a nonlinear mapping over what we define here as f sets an f set differs from a fuzzy set in that it is associated with the restriction of validity in addition to that of possibility therefore by f set we can simultaneously deal with two different types of uncertainties one that is related to ill known objects represented by incomplete information information with its one or more aspects being imprecise vague partial nonspecific undetermined and another that is related to truth values considering gradualness here we define new concepts of vartheta cuts and alpha vartheta cuts introduce the f extension principle and consider arithmetic computations within fle we then address other aspects of the proposed fle system such as fuzzification and validification operations in input processing stage set conversion and defuzzification in output processing stage and inferencing in fact in this paper we intend to develop fle theoretically and practically from the stands of sets and systems to extend the concept of approximate reasoning as a consequence of this development we assert that considering the validity degree of methods and information can lead to more reasonable and trustworthy results through capturing more uncertainty

convergence of a finite volume scheme for coagulation fragmentation equations this paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation a time explicit finite volume scheme is developed based on a conservative formulation of the equation it is shown to converge under a stability condition on the time step while a first order rate of convergence is established and an explicit error estimate is given finally several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution

non conforming curved finite element schemes for time dependent elastic acoustic coupled problems high order numerical methods for solving time dependent acoustic elastic coupled problems are introduced these methods based on finite element techniques allow for a flexible coupling between the fluid and the solid domain by using non conforming meshes and curved elements since characteristic waves travel at different speeds through different media specific levels of granularity for the mesh discretization are required on each domain making impractical a possible conforming coupling in between advantageously physical domains may be independently discretized in our framework due to the non conforming feature consequently an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques namely by reducing the total number of degrees of freedom differently from other non conforming approaches proposed so far our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage so that no extra computations are necessary during the time evolution of the simulation on the other hand as an advantage of using curvilinear elements the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used in other words higher order schemes are in consonance with higher order approximations of the geometry concerning the time discretization we analyze both explicit and implicit schemes these schemes are energy conserving and for the explicit case the stability is guaranteed by a cfl condition in order to illustrate the accuracy and convergence of these methods a set of representative numerical tests are presented

a bayesian approach to estimate uncertainty for full waveform inversion using a priori information from depth migration abstractfull waveform inversion fwi enables us to obtain high resolution subsurface images however estimating model uncertainties associated with this technique is still a challenging problem we have used a bayesian inference framework to estimate model uncertainties associated with fwi the uncertainties were assessed based on an a posteriori covariance operator evaluated at the maximum a posteriori model for the prior distribution we have used a spatially nonstationary covariance operator based on a plane wave construction with local dips measured from migrated images preconditioned frequency domain fwi was used to estimate the maximum a posteriori model efficient manipulation of the posterior covariance was based on a low rank approximation of the data misfit hessian preconditioned by the prior covariance operator the strong decay of the singular values indicated that data were mostly informative about a low dimensional subspace of model parameters to reduce computational cost of the random

computationally efficient reasoning using approximated fuzzy intervals in this paper we emphasize the use of sigmoid like membership functions which take values in the open unit interval and propose the membership driven inference mdi reasoning scheme with sigmoid like membership functions one can avoid the so called indetermination part of the conclusion which occur in reasoning with the compositional rule of inference cri moreover the mdi and the min and product based cri are closed under such membership functions the axiomatic properties of the mdi reasoning scheme are shown including not only the generalized modus ponens but also the generalized modus tollens the generalized chain rule and more as a special sigmoid like function we present the so called squashing function by which piecewise linear fuzzy intervals can be arbitrarily approximated we show that by utilizing approximated fuzzy intervals in rules and premises the mdi reasoning scheme can be efficiently calculated only by the parameters that define the fuzzy sets in the rule and the premise

modelling of wave propagation in composite plates using the time domain spectral element method abstract this paper presents results of numerical simulation of the propagation of transverse elastic waves corresponding to the a0 mode of lamb waves in a composite plate the problem is solved by the spectral element method spectral plate finite elements with 36 nodes defined at gauss lobatto legendre points are used as a consequence of the selection of lagrange polynomials for element shape functions discrete orthogonality is obtained leading to the diagonal form of the element mass matrix this results in a crucial reduction of numerical operations required for the solution of the equation of motion by time integration numerical calculations have been carried out for various orientations and relative volume fractions of reinforcing fibres within the plate the paper shows how the velocities of transverse elastic waves in composite materials depend on the orientation and the relative volume fraction of the reinforcement

inference with a single fuzzy conditional proposition abstract a topic in the area of approximate reasoning that has been extensively researched is the following given a single fuzzy conditional proposition if x then y is b with a and b being fuzzy sets defining particular values of the linguistic variables x and y respectively and given some a a then what can be inferred about the value of y when a us replaced by a in the works of l zadeh j baldwin e hisdal r yager m mizumoto et al different solutions to this problem were proposed though different they all share one common feature they ignore the fact that a single fuzzy conditional proposition represents only one specific manifestation of the causal influence exerted by x on y which is being described by binding the fuzzy set b to the fuzzy set a thus the general pattern of the causal relatinship between the two linguistic variables if forgotten as a result the inference made when the original value of x is replaced by a new one might in some cases contradict the general pattern of the causal relationship the present paper makes an attempt to specify these types of inference with a single fuzzy conditional proposition which do not depend on the general pattern of the causal relationship between x and y it also discusses some associated problems

applying finite element analysis to the memory variable formulation of wave propagation in anelastic media finite element methods are applied to solution of seismic wave motion in linear viscoelastic media using the memory variable formalism the displacements are represented as a superposition of a set of basis functions it is shown that if memory variables are represented using the spatial derivatives of those basis functions rather than the basis functions themselves the equations to be solved are simpler and require less computer memory using this formulation results for sh waves in one and two dimensions are calculated using a simple explicit finite element in space finite difference in time scheme these results agree with those found with a method of lines solution results in a homogeneous medium also agree with the frequency domain solutions of kjartansson s constant q method

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

compact high order stencils with optimal accuracy for numerical solutions of 2 d time independent elasticity equations abstract a new approach for the increase in the order of accuracy of high order numerical techniques used for time independent elasticity is suggested on uniform square and rectangular meshes it is based on the optimization of the coefficients of the corresponding discrete stencil equations with respect to the local truncation error it is shown that the second order of accuracy of 2 d linear finite elements with 9 point stencils is optimal and cannot be improved however the order of accuracy of 25 point stencils similar to those for quadratic finite and isogeometric elements can be significantly improved we have developed new 25 point stencils for 2 d elastic problems with optimal 10th order of accuracy the numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for high order finite elements at the same number of degrees of freedom the new approach yields significantly more accurate results than those obtained by high order up to the tenth order finite elements the numerical experiments also show that the new approach with 25 point stencils yields very accurate results for nearly incompressible materials with poisson s ratio 0 4999

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

accurate and efficient seismic modeling in random media the optimum method for seismic modeling in random media must 1 be highly accurate to be sensitive to subtle effects of wave propagation 2 allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths this is necessary to obtain statistically meaningful overall attributes of wavefields high order staggered grid finite difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling investigations for random media reveal that both methods lead to nearly identical wavefields the small differences can be attributed mainly to differences in the numerical dispersion this result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous the differentiator can be longer than the medium scale leng

applicability of the fuzzy operators in the design of fuzzy logic controllers a study of the different roles played by the fuzzy operators in fuzzy control is developed in this paper the behavior of a very large amount of fuzzy operators is analyzed and a comparison of the accuracy of many fuzzy logic controllers designed by means of these operators is carried out in order to do that a comparison methodology is defined and two fuzzy control applications are selected the inverted pendulum problem and the fuzzy modeling of the real curve y x

source independent full waveform inversion of seismic data a set of seismic trace data is collected in an input data set that is first fourier transformed in its entirety into the frequency domain a normalized wavefield is obtained for each trace of the input data set in the frequency domain normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data the normalized wavefield is source independent complex and dimensionless the normalized wavefield is shown to be uniquely defined as the normalized impulse response provided that a certain condition is met for the source this property allows construction of the inversion algorithm disclosed herein without any source or source coupling information the algorithm minimizes the error between data normalized wavefield and the model normalized wavefield the methodology is applicable to any 3 d seismic problem and damping may be easily included in the process

numerical solution of the smoluchowski kinetic equation and asymptotics of the distribution function we obtain the numerical solution of the smoluchowski kinetic equation for model kernels u m1 m2 varies as m1 m2 lambda and u m1 m2 varies as m1m2 lambda 2 0 lambda or 2 we show that the behaviour of the solution for the kernel u varies as m1 m2 lambda at 0 lambda or 1 and u varies as m1m2 lambda 2 at 0 lambda or 2 becomes self similar after some time the shape of the scaling function is analysed in particular a simple approximate expression for it at u varies as m1 m2 lambda is found an interesting result is obtained for u varies as m1m2 lambda 2 0 lambda 1 the asymptotic behaviour of the scaling function proved to be non power we develop the procedure for determining tcr the critical indices and the exponent of the power law asymptotics of the smoluchowski equation solution the concrete values of these quantities for the model kernels are obtained the stability conditions of the algorithm for numerical solving are analysed the possibilities afforded by numerical solution for investigation of the smoluchowski equation are discussed

handling the nonlinearity of a fuzzy logic controller at the transition between rules abstract a fuzzy logic controller approximates a desired control surface by using the outputs of its fuzzy control rules the shape of this control surface is mainly influenced by the linguistic variables of the rule base the basic operators of the fuzzy sets the implication the inference and the defuzzification method the linearity nonlinearity of this control surface is discussed in many studies however none of those studies has explicitly considered the linearity nonlinearity at the transition between fuzzy logic rules in this paper a simple approach to control the linearity nonlinearity at this transition by a modified center of gravity defuzzification method is proposed namely by introducing the so called defuzzification weights to the overlapping areas of the consequent those defuzzification weights can be determined heuristically or automatically both cases will be demonstrated for the latter a feedforward neural network is employed

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

alternatives to the rayleigh quotient for the quadratic eigenvalue problem we consider the quadratic eigenvalue problem lambda 2 ax lambda bx cx 0 suppose that u is an approximation to an eigenvector x for instance obtained by a subspace method and that we want to determine an approximation to the corresponding eigenvalue lambda the usual approach is to impose the galerkin condition r theta u theta 2 a theta b c u perp u from which it follows that theta must be one of the two solutions to the quadratic equation u au theta 2 u bu theta u cu 0 an unnatural aspect is that if u x the second solution has in general no meaning when u is not very accurate it may not be clear which solution is the best moreover when the discriminant of the equation is small the solutions may be very sensitive to perturbations in u in this paper we therefore examine alternative approximations to lambda we compare the approaches theoretically and by numerical experiments the methods are extended to approximations from subspaces and to the polynomial eigenvalue problem

full wave directional illumination analysis in the frequency domain directional illumination analysis based on one way wave equations has been studied extensively however its inherent limitations e g one way propagation wide angle error and amplitude inaccuracy can severely hinder its applications for accurate survey design and true reflection imaging corrections in complex media we have analyzed the illumination in the frequency domain using full two way wave propagators considering the extensive computation and huge storage required for time domain methods and the fact that the illumination is frequency dependent this full wave analysis can provide frequency dependent full angle true amplitude illumination not only for the downgoing waves but also for the upgoing waves including turning waves and reflected waves two methods were considered to decompose the full wavefield into the local angle domain a direct full dimensional decomposition and more efficient split step decomposition composed of three lower dimensional decompositions the results of illumination analysis demonstrated the advantages of this method the two decomposition methods produced similar results

analytical solutions of smoluschowski s coagulation equation it is proven that smoluchowski s coagulation equation with a kernel kij which satisfies kij or 0

some considerations about a least squares model for fuzzy rules of inference abstract in this paper the method of least squares is applied to the fuzzy inference rules we begin studying the conditions in which from a fuzzy set we can build another through the method of least squares then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens we present different theorems and examples that demonstrate the fundamental advantages of the method studied we conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective and

a recursive green s function technique for acoustic scattering from heterogeneous objects a fast efficient algorithm for computing acoustic fields scattered by inhomogeneous objects in an otherwise homogeneous space is presented the algorithm called the recursive green s function method rgfm constructs the domain green s function by recursively combining known green s functions from smaller subdomains the fields on the scatterer surface are then computed using a boundary integral formulation proper implementation of the rgfm results in a storage requirement of o n and computational costs of o n3 2 and o n2 for two and three dimensional problems respectively where n is the total number of discrete points in the inhomogeneous region results are compared with those obtained from exact solutions to show the accuracy of the method

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

3 d large scale wave propagation modeling by spectral element method on cray t3e multiprocessor highly efficient algorithms and massively parallel computers are needed for acoustic full wave modeling in large scale 3 d realistic unbounded media in this work a spectral element method in conjunction with a new iterative solution technique is presented the very high spatial accuracy of the algorithm allows for increased computational efficiency when compared to standard finite element method in addition the use of an iterative solver based on a local element by element formulation and a matrix vector product which is factored and converted to a faster matrix matrix product allows for a significant reduction both in storage requirements and in the computational complexity the resulting algorithm can easily be implemented on high performance vector and or parallel processors the numerical scheme has been tested on a cray t3e supercomputer our experimental results show that efficiency is very high and that good load balancing can be obtained at run time because relevant blocks of computations are performed at the element level and are totally independent

new 25 point stencils with optimal accuracy for 2 d heat transfer problems comparison with the quadratic isogeometric elements abstract a new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent poisson equation has been suggested on uniform square and rectangular meshes it is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error by a simple modification of the coefficients of 25 point stencils the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the poisson equation despite the significant increase in accuracy the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh the numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented

lumped mass matrix in explicit finite element method for transient dynamics of elasticity abstract the mass matrix of structural dynamics is discussed for the approximations by linear elements with constant jacobians the a priori error estimate is derived for the explicit finite element method using diagonal mass matrix with smooth solutions it is shown that by using diagonal mass matrix or consistent mass matrix the displacement velocity and the energies have the same convergence rates in fact the difference in numerical errors by these two methods is of order h2 the same conclusion is valid for a certain type of reissner mindlin plate element for transient dynamics

fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions in this article three methods are presented to perform the center of gravity cog defuzzification method in the context of linguistic fuzzy models with t norm based inference one well known method the discretisation method and two new methods the slope based method and the modified transformation function method the methods are worked out for trapezoidal membership functions forming a fuzzy partition in the sense of ruspini experimental results show that the newly introduced methods exhibit excellent accuracy at an extremely low computational cost compared to the widely applied discretisation method

the effects of membership function on fuzzy reasoning abstract fuzzy modus ponens has been considered a generalization of the classic modus ponens logic inference rule it has been expected to provide a more powerful and useful inference capability in the process of fuzzy reasoning the definitions of membership functions of fuzzy sets used in the reasoning process may however adversely affect the reasoning results if the insignificant parts of the membership functions are incompatible this paper explains these undesirable effects and proposes a method to resolve this problem in addition an algorithm for the compositional rule of inference is developed to perform the fuzzy modus ponens syllogism test results shown in this paper indicate that the algorithm and the underlying proposed methods are reasonable approaches to a robust fuzzy logic reasoning system

a finite version of smoluchowski s coagulation equation the authors formulate a finite version of smoluchowski s coagulation equation in which molecules of size n react in the normal way whereas molecules of size n to 2n can be produced but cannot react they prove that the solution converges to the solution of the infinite system of equations in the limit n to infinity

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

least squares legendre spectral element solutions to sound propagation problems this paper presents a novel algorithm and numerical results of sound wave propagation the method is based on a least squares legendre spectral element approach for spatial discretization and the crank nicolson proc cambridge philos soc 43 50 67 1947 and adams bashforth d gottlieb and s a orszag numerical analysis of spectral methods theory and applications cbms nsf monograph siam 1977 schemes for temporal discretization to solve the linearized acoustic field equations for sound propagation two types of nasa computational aeroacoustics caa workshop benchmark problems icase larc workshop on benchmark problems in computational aeroacoustics edited by j c hardin j r ristorcelli and c k w tam nasa conference publication 3300 1995a are considered a narrow gaussian sound wave propagating in a one dimensional space without flows and the reflection of a two dimensional acoustic pulse off a rigid wall in the presence of a uniform flow of mach 0 5 in a semi infinite space the f

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

iterative finite difference solution analysis of acoustic wave equation in the laplace fourier domain we have investigated numerical characteristics of iterative solutions to the acoustic wave equation in the laplacefourier lf domain we transformed the time domain acoustic wave equation into the lf domain the transformed equation was discretized with finite differences and was solved with iterative methods finite difference modeling experiments demonstrate that iterative methods require an infinitesimal stopping tolerance to accurately compute the pressure field especially at long offsets to understand the requirement for such infinitesimal tolerance values we analyzed the evolution of intermediate solution vectors residual vectors and search direction vectors during the iteration the analysis showed that the requirement arises from the fact that in the solution space the amplitude of the pressure field varies more than sixty orders of magnitude on the common log scale accordingly we propose a rule of thumb for choosing a proper stopping tolerance value we also examined numerical dispersion errors in terms of the grid sampling resolutions per skin depth and wavelength we found that despite the similarity of the form of the acoustic wave and electromagnetic diffusion equations the former is different from the latter due to the fact that in the lf domain the skin depth of the acoustic wave equation is decoupled from its wavelength this aspect requires that in the lf domain its grid size be determined by considering the minimum grid sampling resolutions based not only the wavelength but also the skin depth

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

time space domain mesh free finite difference based on least squares for 2d acoustic wave modeling abstractfinite difference fd methods approximate derivatives through a weighted summation of function values from neighboring nodes traditionally these neighboring nodes are assumed to be distributed regularly such as in square or rectangular lattices to improve geometric flexibility one option is to develop fd in a mesh free discretization in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra etc these mesh free fds have had successful applications especially in computational geoscience however they are all space domain fd schemes in which fd coefficients are derived by approximating spatial derivatives individually in the space domain for acoustic wave modeling it has been proven that space domain fd methods normally have higher dispersion error than time space domain fd methods which determine fd coefficients by approximating the time space domain disp

optimal reduction of numerical dispersion for wave propagation problems part 1 application to 1 d isogeometric elements abstract a numerical technique with the optimal coefficients of the stencil equation has been suggested based on this approach new high order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1 d case by the modification of the mass and stiffness matrices the order of the dispersion error is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations it was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high order isogeometric elements the analysis of the dispersion error of the high order isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix the second order of the dispersion error cannot be improved with the conventional stiffness matrix however the dispersion error for the lumped mass matrix can be improved from the second order to order 2 p by the modification of the stiffness matrix the numerical examples confirm the computational efficiency of the new high order isogeometric elements with reduced dispersion we have also showed that numerical results obtained by the new and conventional high order isogeometric elements may include spurious oscillations due to the dispersion error these oscillations can be quantified and filtered by the two stage time integration technique recently developed in our papers the approach developed in the paper can be directly applied to other space discretization techniques with similar stencil equations

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

numerical simulation of interface waves by high order spectral modeling techniques few problems in elastodynamics have a closed form analytical solution the others can be investigated with semianalytical methods but in general one is not sure whether these methods give reliable solutions the same happens with numerical techniques for instance finite difference methods solve in principle any complex problem including those with arbitrary inhomogeneities and boundary conditions however there is no way to verify the quantitative correctness of the solutions the major problems are stability with respect to material properties numerical dispersion and the treatment of boundary conditions in practice these problems may produce inaccurate solutions in this paper the study of complex problems with two different numerical grid techniques in order to cross check the solutions is proposed interface waves in particular are emphasized since they pose the major difficulties due to the need to implement boundary conditions the first method is based on global differential operators where the solution is expanded in terms of the fourier basis and chebyshev polynomials while the second is the spectral element method an extension of the finite element method that uses chebyshev polynomials as interpolating functions both methods have spectral accuracy up to approximately the nyquist wave number of the grid moreover both methods implement the boundary conditions in a natural way particularly the spectral element algorithm we first solve lamb s problem and compare numerical and analytical solutions then the problem of dispersed rayleigh waves and finally the two quarter space problem we show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist the combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

the influence of the mass matrix on the dispersive nature of the semi discrete second order wave equation abstract the application of discrete solution methods to the second order wave equation can yield a dispersive representation of the non dispersive wave propagation problem resulting in a phase speed that depends not only upon the wavelength of the signal being propagated but also upon the direction of propagation in this work the dependence of the dispersive errors on the wave propagation direction mesh aspect ratio and wave number is investigated with the goal of understanding and hopefully reducing the phase and group errors associated with the two dimensional bilinear finite element an analysis of the dispersive effects associated with the consistent row sum lumped and higher order mass matrices has led to a reduced coupling penta diagonal mass matrix that yields improved phase and group errors with respect to wavelength and propagation direction the influence of row sum lumping the finite element mass matrix is demonstrated to always introduce lagging phase and group error while a linear combination of the lumped and consistent mass matrices i e a higher order mass matrix is shown to improve the dispersion characteristics of both the reduced and full integration element using a 2 5 phase error metric the higher order mass matrix can reduce the mesh resolution requirements relative to the lumped mass matrix by nearly a factor of two in one dimension a factor of 4 in two dimensions similarly the reduced coupling mass matrix can reduce the grid requirements by a factor of approximately four relative to the lumped mass matrix a factor of 16 in two dimensions although the reduced coupling mass matrix requires additional storage relative to the lumped mass the factor of 16 reduction in grid requirements promises tremendous overall savings for computing a fixed wavelength spectrum or alternatively for resolving shorter wavelengths with a fixed grid resolution

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

finite element modeling of linear elastodynamics problems with explicit time integration methods and linear elements with the reduced dispersion error abstract we have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time integration methods these techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements the analytical study of numerical dispersion for the new techniques is carried out in the 1 d 2 d and 3 d cases the numerical study of the efficiency of the dispersion reduction techniques includes the two stage time integration approach with the filtering stage developed in our previous papers that quantifies and removes spurious high frequency oscillations from numerical results we have found that in contrast to the standard linear elements with explicit time integration methods and the lumped mass matrix the finite element techniques with reduced dispersion yield more accurate results at small time increments smaller than the stability limit in the 2 d and 3 d cases the recommendations for the selection of the size of time increments are suggested the new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations

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

general templates for n noded bar elements based on reduced representations and numerical dispersion reduction by optimized finite elements abstract various kinds of finite elements are proposed and employed for one dimensional wave field simulations each of which has their own capabilities and disadvantages this paper deals with structural similarities and differences that arise at the element level by the use of reduced forms of mass and stiffness matrices general parameterized forms are then developed for mass and stiffness matrices of n noded elements elements with n nodes with arbitrary arrangement optimization for the free parameters will deliver elements with better performance and less numerical dispersion error two of such elements are developed and compared for their ability in dispersion error remission

dissipation preserving spectral element method for damped seismic wave equations this article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method the conformal symplectic method is a variation of conventional symplectic methods to treat non conservative time evolution problems which has superior behaviors in long time stability and dissipation preservation to reveal the intrinsic dissipative properties of the model equations we first reformulate the original systems in their equivalent conformal multi symplectic structures and derive the corresponding conformal symplectic conservation laws we thereafter separate each system into a conservative hamiltonian system and a purely dissipative ordinary differential equation system based on the splitting methodology we solve the two subsystems respectively the dissipative one is cheaply solved by its analytic solution while for the conservative system we combine a fourth order symplectic nystrm method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free surface topography the strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method a relative larger courant number than that of the traditional newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength a benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate the classical lamb problem is used to demonstrate the ability of modeling rayleigh wave in elastic wave propagation more comprehensive numerical experiments are presented to investigate the long time simulation low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating homogeneous and heterogeneous media

numerical analysis of a narrow angle one way elastic wave equation and extension to curvilinear coordinates in this paper we review the finite difference implementation of a narrow angle one way vector wave equation for elastic 3d media extrapolation is performed in the frequency domain where the second order accurate lateral spatial difference operators are sufficiently accurate for narrow angle propagation we perform a numerical analysis of the finite difference scheme to highlight the stability and dispersion characteristics the von neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths source pulse spectral content and angular range of forward propagation of interest although the algorithm is reasonable its computational efficiency is limited by the second order accurate extrapolation step therefore the extrapolation scheme can be improved we extend the cartesian narrow angle formulation to curvilinear coordinates where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the cartesian propagator an example of curvilinear extrapolation for a simple model consisting of a high velocity sphere within a homogeneous background velocity structure shows that the narrow angle propagator is capable of modeling frequency dependent geometric spreading and diffraction effects in curvilinear coordinates

simulation of anisotropic wave propagation based upon a spectral element method we introduce a numerical approach for modeling elastic wave propagation in 2 d and 3 d fully anisotropic media based upon a spectral element method the technique solves a weak formulation of the wave equation which is discretized using a high order polynomial representation on a finite element mesh for isotropic media the spectral element method is known for its high degree of accuracy its ability to handle complex model geometries and its low computational cost we show that the method can be extended to fully anisotropic media the mass matrix obtained is diagonal by construction which leads to a very efficient fully explicit solver we demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2 d transversely isotropic test case and by comparing its predictions against those based upon a finite difference method for a 2 d heterogeneous anisotropic medium we show its generality and its flexibility by modeling wave propagation in a 3 d transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid

post gelation solutions to smoluchowski s coagulation equation it is proven how the post gelation behaviour originally suggested by flory 1993 can be obtained as a result of a limiting process passing from a finite to an infinite system in a previous paper by the authors it was shown how the post gelation behaviour first suggested by stockmayer 1943 can be obtained by passing to the limit of an infinite system in a different way it is thus demonstrated that different post gelation solutions of smoluchowski s coagulation equation can be obtained by different limiting processes

reducing spurious oscillations in discontinuous wave propagation simulation using high order finite elements in this paper a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves the spatial domain is discretized by the finite element method to obtain accurate results the standard finite element method requires a very fine mesh which is why the computational effort can be very time consuming the use of high order finite element methods such as the spectral element method based on lagrange polynomials through gauss lobatto legendre points or the iso geometric analysis using non uniform rational b splines can reduce the enormous computational costs significantly compared to the standard finite element method however explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave the procedure proposed here is basically similar to the explicit method of noh and bathe but the semi discrete equation of motion is modified by introducing a damping parameter to suit the high order fem analysis of the discontinuous wave propagation the performance due to this modification is tested for one dimensional wave propagation problems using both lumped and consistent mass matrices then the idea of a combination of the consistent and row sum lumped mass matrices is evaluated the proposed method is studied also in two dimensions by considering the lamb problem of wave propagation the results are promising enough to provide a better simulation of the discontinuous wave propagation problem

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

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a posteriori finite element error estimation for second order hyperbolic problems abstract we develop a posteriori finite element discretization error estimates for the wave equation in one dimension we show that the significant part of the spatial finite element error is proportional to a lobatto polynomial and an error estimate is obtained by solving a set of either local elliptic or hyperbolic problems in two dimensions we show that the dichotomy principle of babuska and yu holds for even degree approximations error estimates are computed by solving a set of local elliptic or hyperbolic problems and for odd degree approximations an error estimate is computed using jumps of solution gradients across element boundaries this study also extends known superconvergence results for elliptic and parabolic problems superconvergence in galerkin finite element methods springer verlag new york 1995 to second order hyperbolic problems

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

modeling of wave propagation in plate structures using three dimensional spectral element method for damage detection abstract this paper presents the application of three dimensional 3 d spectral element method sem to wave propagation problems in plate structures for the purpose of damage detection the excellent characteristic of the sem is that the mass matrix is diagonal because of the choice of lagrange interpolation function supported on the gauss lobatto legendre gll points in conjunction with the gll integration rule therefore numerical calculation can be significantly efficient in comparison with the classical finite element method fem by taking advantage of this characteristic a 3 d spectral finite elements sfes based model is developed to simulate the wave propagation in plate structures lamb waves propagating in aluminum plates with and without a crack are investigated responses from 3 d sfes and 2 d sfes based models are compared different lamb wave modes are generated using different excitation approaches and subsequently characteristics of those modes are analyzed the results demonstrate that the proposed model can offer efficient and realistic simulation for lamb wave propagation in plate structures so as to detect damages in those structures

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

the use of the local truncation error for the increase in accuracy of the linear finite elements for heat transfer problems abstract a new approach for the increase in the order of accuracy of the linear finite elements used for the time dependent heat equation and for the time independent laplace equation has been suggested it is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error by a simple modification of the coefficients of the elemental mass and stiffness matrices the accuracy of the linear finite elements is improved by two orders for the heat equation and by four orders for the laplace equation despite the significant increase in accuracy the computational costs of the new technique are the same as those for the conventional linear finite elements on a given mesh 2 d and 3 d numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new linear finite elements are much more accurate than the conventional linear and quadratic finite elements at the same numbers of degrees of freedom

optimally blended spectral finite element scheme for wave propagation and nonstandard reduced integration we study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent finite element mass matrix and lumped spectral element mass matrix for the small wave number limit we find and prove that for the optimum blending the resulting scheme a provides 2p 4 order accuracy for p th order method two orders more accurate compared with finite and spectral element schemes b has an absolute accuracy which is mathcal o p 3 and mathcal o p 2 times better than that of the pure finite and spectral element schemes respectively c tends to exhibit phase lag moreover we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived

a high order numerical approach with cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials abstract recently we have developed a new numerical approach for pdes with constant coefficients on irregular domains and cartesian meshes in this paper we extend it to a much more general case of pdes with variable coefficients that have a lot of applications e g the modeling of functionally graded materials the inhomogeneous materials obtained by 3 d printing and many others here we consider the 2 d wave and heat equations for isotropic and anisotropic inhomogeneous materials the idea of the extension to the case of pdes with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size this leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach however this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique similar to our previous technique the new 2 d approach with compact 9 point stencils uses trivial cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at similar 9 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the third order finite elements with much wider stencils the wave and heat equations are uniformly treated with the new approach

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed