discontinuous displacement approximation for capturing plastic localization it is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries interfaces of the finite element subdivision the regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept as a by product of the regularization an intrinsic band width is introduced as a constitutive property rather than a geometric feature of the finite element mesh in this way the spurious mesh sensitivity which is obtained when the displacement approximation is continuous can be avoided another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material an interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface up to within a scalar factor an analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a shear band for a general plasticity model the significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress it is shown that a unique structural post peak response in accordance with non linear fracture mechanics can be achieved when the plastic softening modulus is properly related to the bandwidth the paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope

efficiency and accuracy of numerical solutions to the time dependent schrodinger equation recent significant improvements of the numerical solutions of the time dependent schrodinger equation beg the question as to whether these recent methods are comparable in efficacy in terms of accuracy and computational time to the current method of choice i e the chebyshev expansion of the time evolution operator and the fast fourier transform method of determining the kinetic energy in this paper we review the methods in question and by studying the time development of a coherent wave packet in an oscillator well we are able to assess the effectiveness of the various methods it turns out that the new generalizations come close to within an order of magnitude to being able to generate solutions as precisely and efficiently as the chebyshev fast fourier transform method the strict unitarity of the generalized methods may be an advantage we also show that the fast fourier transform approach to calculating the kinetic energy can be replaced by straightforward numerical differentiation to obtain the same precision

efficient and accurate numerical methods for the klein gordon schrodinger equations in this paper we present efficient unconditionally stable and accurate numerical methods for approximations of the klein gordon schrodinger kgs equations with without damping terms the key features of our methods are based on i the application of a time splitting spectral discretization for a schrodinger type equation in kgs ii the utilization of fourier pseudospectral discretization for spatial derivatives in the klein gordon equation in kgs iii the adoption of solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the numerical methods are either explicit or implicit but can be solved explicitly unconditionally stable and of spectral accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping terms in kgs conserve or keep the same decay rate of the wave energy as that in kgs without or with a linear damping term keep the same dynamics of the mean value of the meson field and give exact results for the plane wave solution extensive numerical tests are presented to confirm the above properties of our numerical methods for kgs finally the methods are applied to study solitary wave collisions in one dimension 1d as well as dynamics of a 2d problem in kgs

from the conserved kuramoto sivashinsky equation to a coalescing particles model abstract the conserved kuramoto sivashinsky cks equation t u x x u u x x u x 2 has recently been derived in the context of crystal growth and it is also strictly related to a similar equation appearing e g in sand ripple dynamics we show that this equation can be mapped into the motion of a system of particles with attractive interactions decaying as the inverse of their distance particles represent vanishing regions of diverging curvature joined by arcs of a single parabola and coalesce upon encounter the coalescing particles model is easier to simulate than the original cks equation the growing interparticle distance l represents coarsening of the system and we are able to establish firmly the scaling l t t we obtain its probability distribution function g l numerically and study it analytically within the hypothesis of uncorrelated intervals finding an overestimate at large distances finally we introduce a method based on coalescence waves which might be useful to gain better analytical insights into the model

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

construction of high order force gradient algorithms for integration of motion in classical and quantum systems a consequent approach is proposed to construct symplectic force gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders contrary to the scheme proposed by chin and kidwell phys rev e 62 8746 2000 where high order algorithms are introduced by standard iterations of a force gradient integrator of order four the present method allows one to reduce the total number of expensive force and its gradient evaluations to a minimum at the same time the precision of the integration increases significantly especially with increasing the order of the generated schemes the algorithms are tested in molecular dynamics and celestial mechanics simulations it is shown in particular that the efficiency of the advanced fourth order based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12 respectively the results corresponding to sixth and eighth order based composition schemes are also presented up to the sixteenth order for orders 14 and 16 such highly precise schemes at considerably smaller computational costs allow to reduce unphysical deviations in the total energy up in 100000 times with respect to those of the standard fourth order based iteration approach

extremum properties of finite step solutions in elastoplasticity with nonlinear mixed hardening abstract the class of elastic plastic constitutive laws assumed herein can be described as follows we envisage y 1 yield criteria or modes which define the current yield surface each yield function consists of two addends an effective stress and a yield limit y 1 y the former is an order one positively homogeneous function of the difference between the stresses ii and reference stresses ii which are generally nonlinear functional of the plastic strain history so that kinematic hardening is accounted for the yield limit y is generally a nonlinear function of y nondecreasing internal variables so that another hardening mechanism which may reduce to isotropic hardening is accommodated in the model the rates of these variables play the role of plastic multipliers in the flow rule a finite step in the geometrically linear evolutive analysis of such solids is defined according to the backward difference or stepwise holonomic strategy for approximate time integration for the finite step boundary value problem thus arising various extremum characterizations of solutions are established and the underlying constitutive restrictions are pointed out and discussed

coarsening dynamics in a simplified dnls model we investigate the coarsening evolution occurring in a simplified stochastic model of the discrete nonlinear schrodinger dnls equation in the so called negative temperature region we provide an explanation of the coarsening exponent n 1 3 by invoking an analogy with a suitable exclusion process in spite of the equivalence with the exponent observed in other known universality classes this model is certainly different in that it refers to a dynamics with two conservation laws

nonlinear dynamics in one dimension a criterion for coarsening and its temporal law we develop a general criterion about coarsening for some classes of nonlinear evolution equations describing one dimensional pattern forming systems this criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time an intermediate scenario may occur namely interrupted coarsening the power of the criterion on which a brief account has been given politi and misbah phys rev lett 92 090601 2004 and which we extend here to more general equations lies in the fact that the statement about the occurrence of coarsening or selection of a length scale can be made by only inspecting the behavior of the branch of steady state periodic solutions the criterion states that coarsening occurs if ensuremath lambda ensuremath a g0 while a length scale selection prevails if ensuremath lambda ensuremath a l0 where ensuremath lambda is the wavelength of the pattern and a is the amplitude of the profile prime refers to differentiation this criterion is established thanks to the analysis of the phase diffusion equation of the pattern we connect the phase diffusion coefficient d ensuremath lambda which carries a kinetic information to ensuremath lambda ensuremath a which refers to a pure steady state property the relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent our calculation based on the idea that ensuremath mid d ensuremath lambda ensuremath mid ensuremath sim ensuremath lambda 2 t is exemplified on several nonlinear equations showing that the exact exponent is captured we are not aware of another method that so systematically provides the coarsening exponent contrary to many situations where the one dimensional character has proven essential for the derivation of the coarsening exponent this idea can be used in principle at any dimension some speculations about the extension of the present results are outlined

a variational theory for finite step elasto plastic problems abstract an extended version ofgeneralized standard elasto plastic material is considered in the framework of an internal variable theory of associated plasticity according to a backward difference scheme for time integration of the flow rule a finite step structural problem is formulated in a geometrically linear range convex analysis and a brand new potential theory for monotone multivalued operators are shown to provide the natural mathematical setting for the derivation of the related variational formulation a general stationarity principle is obtained and then specialized to obtain a minimum principle in terms of displacements plastic strains and internal variables a critical comparison with an analogous minimum principle recently proposed in literature is performed showing the inadequacy of classical procedures in deriving non smooth variational formulations

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

solving the time dependent schrodinger equation numerically we introduce an explicit scheme to solve the time dependent schr odinger equation the scheme is a straightforward extension of the second order differencing scheme to the fourth sixth and higher order accuracy the accuracy is remarkably improved with minor changes in the second order differencing program this method is conditionally stable there is a trade off between the higher order accuracy and the condition of stability the performance is evaluated and compared to the standard methods such as the crank nicholson scheme cn and the chebyshev scheme ch the new scheme is much more accurate than cn almost equal to ch

a fourier method solution for the time dependent schrodinger equation as a tool in molecular dynamics a new method is presented for the solution of the time dependent schrbdinger equation in its application to physical and chemical molecular phenomena the method is based on discretizing space and time on a grid and using the fourier method to produce both spatial derivatives and second order differencing for time derivatives the method conserves norm and energy and preserves quantum mechanical commutation relations one and twodimensional examples where a comparison to analytic results is possible are investigated many phenomena of physics and chemistry have a common dynamical evolution pattern this pattern begins with an initial state which under the influence of the potentials evolves through time to produce a final asymptotic state among such phenomena are chemical reactions photodissociation unimolecular breakdown surface scattering and desorption the present study presents a numerical solution based on quantum mechanics for the common pattern or for the time dependent evolution of the state of the system the importance of exact numerical solutions is twofold first these solutions give a quantitative description of physical problems and also are of use as a bench mark to check approximation methods and to define their range of validity

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

a cocr method for solving complex symmetric linear systems the conjugate orthogonal conjugate gradient cocg method has been recognized as an attractive lanczos type krylov subspace method for solving complex symmetric linear systems however it sometimes shows irregular convergence behavior in practical applications in the present paper we propose a conjugate a orthogonal conjugate residual cocr method which can be regarded as an extension of the conjugate residual cr method numerical examples show that cocr often gives smoother convergence behavior than cocg

finite elements with embedded displacement discontinuity a generalized variable formulation this paper is intended to bring a contribution towards a satisfactory simulation of those fracture phenomena which result in the appearance and development of discrete cracks to this purpose a general mixed finite element formulation is proposed based on the concept of generalized variables in prager s sense the displacement field inside an element is modelled by the sum of two contributions a regular continuous part which is governed by standard shape functions and a possibly discontinuous one which is introduced soon after a suitable criterion is satisfied the formulation is first specialized to a one dimensional case then a triangular element for two dimensional problems is described in detail analytical and numerical examples are presented in order to clarify the formulation and to point out the essential role of inter element conformity copyright 2000 john wiley sons ltd

coarsening dynamics in one dimension the phase diffusion equation and its numerical implementation many nonlinear partial differential equations pdes display a coarsening dynamics i e an emerging pattern whose typical length scale l increases with time the so called coarsening exponent n characterizes the time dependence of the scale of the pattern l t t n and coarsening dynamics can be described by a diffusion equation for the phase of the pattern by means of a multiscale analysis we are able to find the analytical expression of such diffusion equations here we propose a recipe to implement numerically the determination of d the phase diffusion coefficient as a function of the wavelength of the base steady state u 0 x d carries all information about coarsening dynamics and through the relation d l l 2 t it allows us to determine the coarsening exponent the main conceptual message is that the coarsening exponent is determined without solving a time dependent equation but only by inspecting the periodic steady state solutions this provides a much faster strategy than a orward time dependent calculation we discuss our method for several different pdes both conserved and not conserved

parameter identification of a cohesive crack model by kalman filter abstract the kalman filter kf methodology is apt to solve parameter identification inverse problems in a statistical context through a sequence of estimations which starts from an a priori estimation by an expert bayesian approach and exploits a time stepping flow of experimental data until convergence is empirically ascertained such methodology is here adopted for the identification of the material parameters together with their uncertainties in a mode i cohesive crack model on the basis of experimental data generated by wedge splitting tests on concrete specimens the simulation of the experiments is based on the assumptions of a piecewise linear cohesive model with four parameters to identify on the crack path and of linear elasticity elsewhere in view of regularly progressive fracture processes the discrete crack model and consequently the overall finite element analysis are formulated as linear complementarity problems this mathematical construct is exploited to obtain the sensitivity matrix key ingredient of the kf extended to nonlinear inverse problems in a computationally convenient closed form various issues peculiar of kf identification in the present mechanical context are critically discussed in the light of the numerical solutions achieved

a discrete formulation for elastic solids with damaging interfaces abstract an elastic solid with embedded interfaces loci of possible displacement discontinuities is considered here decohesion and quasi brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps a discrete formulation in terms of interface variables only is obtained the space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented some examples are shown in order to clarify the formulation

phase instability and coarsening in two dimensions instabilities and pattern formation is the rule in nonequilibrium systems selection of a persistent length scale or coarsening increase in the length scale with time are the two major alternatives when and under which conditions one dynamics prevails over the other is a long standing problem particularly beyond one dimension it is shown that the challenge can be defied in two dimensions using the concept of phase diffusion equation we find that coarsening is related to the ensuremath lambda dependence of a suitable phase diffusion coefficient d 11 ensuremath lambda depending on lattice symmetry and conservation laws these results are exemplified analytically on prototypical nonlinear equations

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

on uniqueness of the dynamic finite step problem in gradient dependent softening plasticity the dynamic evolution of an elastoplastic softening solid is considered a material model including in the yield function the laplacian of the plastic multiplier is used to regularize the problem the dynamic finite step problem is formulated according to a generalized mid point integration scheme space discretization is carried out by a mixed finite element technique based on generalized variables a sufficient uniqueness condition of the finite step solution is proved for a one dimensional problem also a necessary and sufficient condition is presented a simple numerical test shows the regularizing properties mesh independence of the proposed model and the positive influence of the gradient term also on the time step amplitude ensuring uniqueness of solution

finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions a new finite element formulation aimed at the solution of problems involving strain localization is presented the proposed formulation incorporates displacement interpolated embedded localization lines results are shown to converge to an exact solution when the mesh is refined and also to be quite insensitive to mesh distortions

fast and stable method for simulating quantum electron dynamics a fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time dependent schrodinger equation this method is a real space real time evolution method implemented by several computational techniques such as suzuki s exponential product cayley s form the finite differential method and an operator named adhesive operator this method conserves the norm of the wave function manages periodic conditions and adaptive mesh refinement technique and is suitable for vector and parallel type supercomputers applying this method to some simple electron dynamics we confirmed the efficiency and accuracy of the method for simulating fast time dependent quantum phenomena

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

canonical transforms and the efficient integration of quantum mechanical wave equations the integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry the wave function s energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations we demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude our approach includes the so called kramers henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency

parallel solver for the time dependent linear and nonlinear schrodinger equation a solution of the time dependent schrodinger equation is required in a variety of problems in physics and chemistry these include atoms and molecules in time dependent electromagnetic fields time dependent approaches to atomic collision problems and describing the behavior of materials subjected to internal and external forces we describe an approach in which the finite element discrete variable representation fedvr is combined with the real space product rsp algorithm to generate an efficient and highly accurate method for the solution of the time dependent linear and nonlinear schrodinger equation the fedvr provides a highly accurate spatial representation using a minimum number of grid points while the rsp algorithm propagates the wave function in operations per time step parallelization of the method is transparent and is implemented here by distributing one or two spatial dimensions across the available processors within the message passing interface scheme the complete formalism and a number of three dimensional examples are given its high accuracy and efficacy are illustrated by a comparison with the usual finite difference method

analysis of a leap frog pseudospectral scheme for the schrodinger equation the numerical properties of a leap frog pseudospectral scheme for the schrodinger equation are analyzed stability second order accuracy in time and spectral accuracy in space are discussed considering the linear schrodinger equation with potential in a periodic setting further issues regarding phase error gauge invariance conservation properties and commutation relations are addressed results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known crank nicholson finite difference scheme

stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach abstract robustness and stability of the continuum strong discontinuity approach csda to material failure are addressed after identification of lack of symmetry of the finite element formulation and material softening in the constitutive model as possible causes of loss of robustness two remedies are proposed 1 the use of an specific symmetric version of the elementary enriched e fem finite element with embedded discontinuities and 2 a new implicit explicit integration of the internal variable in the constitutive model which renders the tangent constitutive algorithmic operator positive definite and constant the combination of both developments leads to finite element formulations with constant in the time step and non singular tangent structural stiffness allowing dramatic improvements in terms of robustness and computational costs after assessing the convergence and stability properties of the new strategies three dimensional numerical simulations of failure problems illustrate the performance of the proposed procedures

mathematical programming approaches for the safety assessment of semirigid elastoplastic frames abstract this paper presents two complementary mathematical programming based approaches for the accurate safety assessment of semirigid elastoplastic frames under quasistatic loads the inelastic behavior of the flexible connections and material plasticity are accommodated through piecewise linearized nonlinear yield surfaces as is necessary for this class of structures geometric nonlinearity is taken into account moreover only a 2nd order geometric approximation is included as this is sufficiently accurate for practical structures the work described has a twofold contribution first we develop an algorithm that can robustly and efficiently process the complete path dependent nonholonomic response of the structure in a stepwise path independent holonomic fashion the governing formulation is cast in mixed static kinematic variables and leads naturally to what is known in the mathematical programming literature as a mixed complementarity problem mcp the novelty of the proposed algorithm is that it processes the mcp directly without using some iterative and often cumbersome predictor corrector procedure second in the spirit of simplified analyses the classical limit analysis approach is extended to compute the limit load multiplier under the simultaneous influence of joint flexibility material and geometric nonlinearities and limited ductility our formulation is an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints mpec various nonlinear programming based algorithms are proposed to solve the mpec finally four numerical examples concerning practical structures and benchmark cases are provided to illustrate application of the analyses as well as to validate the accuracy and robustness of the proposed schemes

numerical validation of the quantum lattice boltzmann scheme in two and three dimensions the multidimensional formulation of the quantum lattice boltzmann qlb scheme is validated against analytical solutions of the time dependent nonrelativistic schrodinger equation in two and three spatial dimensions as a result we demonstrate the viability of the quantum lattice boltzmann scheme for the numerical solution of the time dependent schrodinger equation in multiple spatial dimensions in addition the links between the qlb scheme and sequential splitting methods for partial differential equations are also clarified

accurate numerical solutions of the time dependent schrodinger equation we present a generalization of the often used crank nicolson cn method of obtaining numerical solutions of the time dependent schr odinger equation the generalization yields numerical solutions accurate to order ensuremath delta x 2r ensuremath 1 in space and ensuremath delta t 2m in time for any positive integers r and m while cn employ r m 1 we note dramatic improvement in the attainable precision circa ten or greater orders of magnitude along with several orders of magnitude reduction of computational time the improved method is shown to lead to feasible studies of coherent state oscillations with additional short range interactions wave packet scattering and long time studies of decaying systems

a unified approach for variationally consistent finite elements in elastoplasticity abstract a general variationally consistent approach for elastoplastic finite element analysis is presented a weak formulation of the finite step i e in terms of finite increments elastoplastic boundary value problem is obtained by enforcing the stationarity of a mixed functional of hu washizu type a peculiarity of the present formulation is that the constitutive equations are expressed in weak form also all fields are then independently discretized in terms of so called generalized variables the result is a set of non linear algebraic equations a subset of which can be interpreted as the backward difference time integrated constitutive law of the finite element an iterative procedure where all calculations are at the element and not at the gauss point level resting on newton raphson scheme is proposed for the solution of the governing equations it is shown how the customary displacement approach and also some other mixed formulations in elastoplasticity can be regarded as special cases of the present formulation

higher order splitting algorithms for solving the nonlinear schrodinger equation and their instabilities since the kinetic and potential energy terms of the real time nonlinear schroedinger equation can each be solved exactly the entire equation can be solved to any order via splitting algorithms we verified the fourth order convergence of some well known algorithms by solving the gross pitaevskii equation numerically all such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved a detail error analysis reveals that the noise or elementary excitations of the nonlinear schroedinger equation obeys the bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process for a continuum wave function this instability is unavoidable no matter how small the time step for a discrete wave function the instability can be avoided only for delta tk sub max sup 2 or approx 2 pi where k sub max pi delta x