higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

local linearization filters for non linear continuous discrete state space models with multiplicative noise in this paper the local linearization method for the approximate computation of the prediction and filtering estimates of continuous discrete state space models is extended to the general case of non linear non autonomous models with multiplicative noise the approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a local linearization of both the non linear state equation and the non linear measurement equation in addition the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the ll filters is illustrated by mean of numerical experiments

numerical solution of differential equations with colored noise using the general theory of numerical integration of stochastic differential equations a constructive approach to numerical methods for a system with colored noise is proposed efficient methods up to the 5 2 strong order and up to the third weak order including runge kutta and implicit schemes are presented the algorithms are tested on the kubo oscillator

the perimeter generating function of punctured staircase polygons using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons staircase polygons with a single internal staircase hole we find that all the terms in the generating function can be reproduced from a linear fuchsian differential equation of order 8 we perform an analysis of the properties of the differential equation

variance reduced weak monte carlo simulations of stochastically driven oscillators of engineering interest abstract analytical solutions of nonlinear and higher dimensional stochastically driven oscillators are rarely possible and this leaves the direct monte carlo simulation of the governing stochastic differential equations sdes as the only tool to obtain the required numerical solution engineers in particular are mostly interested in weak numerical solutions which provide a faster and simpler computational framework to obtain the statistical expectations moments of the response functions the numerical integration tools considered in this study are weak versions of stochastic euler and stochastic newmark methods a well known limitation of a monte carlo approach is however the requirement of a large ensemble size in order to arrive at convergent estimates of the statistical quantities of interest presently a simple form of a variance reduction strategy is proposed such that the ensemble size may be significantly reduced without affecting the accuracy of the predicted expectations of any function of the response vector process provided that the function can be adequately represented through a power series approximation the basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the stochastic forcing function which is typically a filtered white noise process through a set of variance reduced functions in the process the additional computational cost due to system augmentation is far smaller than the accrued advantages due to a drastically reduced ensemble size indeed we show that the proposed method appears to work satisfactorily even in the special case of the ensemble size being just 1 the variance reduction scheme is first illustrated through applications to a nonlinear duffing equation driven by additive and multiplicative white noise processes a problem for which exact stationary solutions are known this is followed up with applications of the strategy to a few higher dimensional systems i e 2 and 3 dof nonlinear oscillators under additive white noises

estimations des erreurs de meilleure approximation polynomiale et d interpolation de lagrange dans les espaces de sobolev d ordre non entier on etablit des majorations explicites de i erreur de meilleure approximation polynomiale ainsi que des majorations explicites et nonexplicites de i erreur d interpolation de lagrange lorsque la fonction consideree appartient a un espace de sobolev d ordre non entier defini sur un ouvert borne de n les resultats obtenus generalisent les resultats connus dans le cas des espaces de sobolev d ordre entier explicit bounds for the best polynomial approximation error explicit and non explicit bounds for the lagrange interpolation error are derived for functions belonging to fractional order sobolev spaces defined over a bounded open set in n thus the classical results of the integer order sobolev spaces are extended

ising model susceptibility the fuchsian differential equation for χ 4 and its factorization properties we give the fuchsian linear differential equation satisfied by x 4 the four particle contribution to the susceptibility of the isotropic square lattice ising model this fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to x 4 the corresponding order ten linear differentia operator exhibits a large set of factorization properties among these factorizations one is highly remarkable it corresponds to the fact that the two particle contribution x 2 is actually a solution of this order ten linear differential operator this result together with a similar one for the order seven differential operator corresponding to the three particle contribution x 3 leads us to a conjecture on the structure of all the n particle contributions x n

dual methods for optimizing finite element flexural systems abstract modern numerical methods for the optimization of large discretized systems are now well developed and highly efficient in the case of thin walled elastic structures modeled by finite elements however this is not yet true for structures whose components are subject simultaneously to bending and extension loads in this paper the idea of generalized optimality criterion goc set forth in previous papers for bar membrane and pure bending elements is extended to deal with general beam and flat shell elements the modifications brought to the goc result in explicit approximations for the behavior constraints that are correct up to the first order but that exhibit a more complex algebraic form indeed these explicit expressions are no longer merely linear in the reciprocal design variables however they continue to be additively separable and therefore dual methods remain fully applicable just as in the original statement of the goc approach numerical examples will be offered to demonstrate the efficiency of the method presented

recursion for the smallest eigenvalue density of β wishart laguerre ensemble the statistics of the smallest eigenvalue of wishart laguerre ensemble is important from several perspectives the smallest eigenvalue density is typically expressible in terms of determinants or pfaffians these results are of utmost significance in understanding the spectral behavior of wishart laguerre ensembles and among other things unveil the underlying universality aspects in the asymptotic limits however obtaining exact and explicit expressions by expanding determinants or pfaffians becomes impractical if large dimension matrices are involved for the real matrices beta 1 edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices very recently an analogous recurrence scheme has been obtained for the complex matrices beta 2 in the present work we extend this to beta wishart laguerre ensembles for the case when exponent alpha in the associated laguerre weight function lambda alpha e beta lambda 2 is a non negative integer while beta is positive real this also gives access to the smallest eigenvalue density of fixed trace beta wishart laguerre ensemble as well as moments for both cases moreover comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly exact evaluations for large values of n the matrix dimension and alpha also enable us to compare with tracy widom density and large deviation results of katzav and castillo we also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the wigner smith matrix and are relevant to the problem of quantum chaotic scattering

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

modelling of cohesive crack growth in concrete structures with the extended finite element method in this paper the extended finite element method xfem is used for a discrete crack simulation of concrete using an adaptive crack growth algorithm different criteria for predicting the direction of the extension of a cohesive crack are investigated in the context of the xfem the robustness of these criteria is analyzed with reference to three examples and the results are compared to experiments furthermore the efficient application of the quadtree data structure for an iterative mesh refinement in combination with the xfem is shown considerably to reduce the computational effort

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

nonparametric filtering for stochastic nonlinear oscillations this paper proposes a method of nonparametric filtering for stochastic nonlinear oscillations with particular interest in their derivative estimation based on a second order ordinary differential equation a stochastic oscillation is modeled by a two variate stochastic differential equation without specifying the function form of the drift function where the first variable is assumed to be observable but not the other given the discrete time series with observation error the proposed method enables us to estimate the values of the drift function and its derivatives including those of the unobservable variable according to the results of numerical experiments to compare estimation accuracy with a parametric method the proposed method shows better performance in the estimation of nonlinear models

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

random matrix theory and entanglement in quantum spin chains we compute the entropy of entanglement in the ground states of a general class of quantum spin chain hamiltonians those that are related to quadratic forms of fermi operators between the first n spins and the rest of the system in the limit of infinite total chain length we show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given hamiltonian and those characterizing the haar measure of the associated group these averages are either toeplitz determinants or determinants of combinations of toeplitz and hankel matrices recent generalizations of the fisher hartwig conjecture are used to compute the leading order asymptotics of the entropy as n this is shown to grow logarithmically with n the constant of proportionality is determined explicitly as is the next constant term in the asymptotic expansion the logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal field theoretic calculations in these calculations the constant of proportionality was determined in terms of the central charge of the virasoro algebra our results therefore lead to an explicit formula for this charge we also show that the entropy is related to solutions of ordinary differential equations of painleve type in some cases these solutions can be evaluated to all orders using recurrence relations

a finite element method with mesh separation based approximation technique and its application in modeling crack propagation with adaptive mesh refinement summary this paper presents a fem with mesh separation based approximation technique that separates a standard element into three geometrically independent elements a dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation the novel technique makes it very easy for mesh generation of problems with complex or solution dependent varying geometry it offers a flexible way to construct displacement approximations and provides a unified framework for the fem to enjoy some of the key advantages of the hansbo and hansbo method the meshfree methods the semi analytical fems and the smoothed fem for problems with evolving discontinuities the method enables the devising of an efficient crack tip adaptive mesh refinement strategy to improve the accuracy of crack tip fields both the discontinuities due to intra element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level the effectiveness and robustness of the present method are benchmarked with several numerical examples the numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test and it is essential to apply local adaptive mesh refinement for low fracture energy problems copyright 2014 john wiley sons ltd

a state space model of the hemodynamic approach nonlinear filtering of bold signals in this paper a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent bold responses the proposed method constitutes an alternative to the recently proposed friston neuroimage 16 2002 513 method and has some advantages over it the procedure is based on recent groundbreaking time series analysis techniques that have been in this case adopted to characterize hemodynamic responses in functional magnetic resonance imaging fmri this work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons first our model includes physiological noise previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation secondly by using the innovation method and the local linearization filter not only the parameters but also the underlying states of the system generating responses can be estimated these states can include things like a flow inducing signal triggered by neuronal activation de oxyhemoglobine cerebral blood flow and volume finally radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach which could be essential to understanding those brain areas indirectly related to the stimulus hence thirdly by inferring about the radial basis parameters we are able to perform a blind deconvolution which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also to implicitly reconstruct the underlying synaptic dynamics induced experimentally which caused these states variations from this study we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps spm nonlinear bold phenomena and unspecific input temporal sequences must be included in the fmri analysis

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

theoretical aspects of the smoothed finite element method sfem this paper examines the theoretical bases for the smoothed finite element method sfem which was formulated by incorporating cell wise strain smoothing operation into standard compatible finite element method fem the weak form of sfem can be derived from the hu washizu three field variational principle for elastic problems it is proved that 1d linear element and 2d linear triangle element in sfem are identical to their counterparts in fem while 2d bilinear quadrilateral elements in sfem are different from that of fem when the number of smoothing cells scs of the elements equals 1 the sfem solution is proved to be variationally consistent and has the same properties with those of fem using reduced integration when sc approaches infinity the sfem solution will approach the solution of the standard displacement compatible fem model when sc is a finite number larger than 1 the sfem solutions are not variationally consistent but energy consistent and will change monotonously from the solution of sfem sc 1 to that of sfem sc it is suggested that there exists an optimal number of sc such that the sfem solution is closest to the exact solution the properties of sfem are confirmed by numerical examples copyright 2006 john wiley sons ltd

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

high order fuchsian equations for the square lattice ising model χ 6 this paper deals with the six particle contribution to the magnetic susceptibility of the square lattice ising model we have generated modulo a prime series coefficients for the length of the series is sufficient to produce the corresponding fuchsian linear differential equation modulo a prime we obtain the fuchsian linear differential equation that annihilates the depleted series the factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper the depleted differential operator is shown to have a structure similar to the corresponding operator for it splits into factors of smaller orders with the left most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral e the right most factor has a direct sum structure and using series calculated modulo several primes all the factors in the direct sum have been reconstructed in exact arithmetics

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

extended finite element method with edge based strain smoothing esm xfem for linear elastic crack growth abstract this paper presents a strain smoothing procedure for the extended finite element method xfem the resulting edge based smoothed extended finite element method esm xfem is tailored to linear elastic fracture mechanics and in this context to outperform the standard xfem in the xfem the displacement based approximation is enriched by the heaviside and asymptotic crack tip functions using the framework of partition of unity this eliminates the need for the mesh alignment with the crack and re meshing as the crack evolves edge based smoothing es relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes and produces a softening effect leading to a close to exact stiffness super convergence and ultra accurate solutions the present method takes advantage of both the es fem and the xfem thanks to the use of strain smoothing the subdivision of elements intersected by discontinuities and of integrating the singular derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction

dynamic modeling of neuronal responses in fmri using cubature kalman filtering this paper presents a new approach to inverting fitting models of coupled dynamical systems based on state of the art cubature kalman filtering crucially this inversion furnishes posterior estimates of both the hidden states and parameters of a system including any unknown exogenous input because the underlying generative model is formulated in continuous time with a discrete observation process it can be applied to a wide variety of models specified with either ordinary or stochastic differential equations these are an important class of models that are particularly appropriate for biological time series where the underlying system is specified in terms of kinetics or dynamics i e dynamic causal models we provide comparative evaluations with generalized bayesian filtering dynamic expectation maximization and demonstrate marked improvements in accuracy and computational efficiency we compare the schemes using a series of difficult nonlinear toy examples and conclude with a special focus on hemodynamic models of evoked brain responses in fmri our scheme promises to provide a significant advance in characterizing the functional architectures of distributed neuronal systems even in the absence of known exogenous experimental input e g resting state fmri studies and spontaneous fluctuations in electrophysiological studies importantly unlike current bayesian filters e g dem our scheme provides estimates of time varying parameters which we will exploit in future work on the adaptation and enabling of connections in the brain

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

isomonodromy transformations of linear systems of difference equations we introduce and study isomonodromy transformations of the matrix linear difference equation y z 1 a z y z with polynomial a z our main result is construction of an isomonodromy action of zm n 1 1 on the space of coefficients a z here m is the size of matrices and n is the degree of a z the birational action of certain rank n subgroups can be described by difference analogs of the classical schlesinger equations and we prove that for generic initial conditions these difference schlesinger equations have a unique solution we also show that both the classical schlesinger equations and the schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes similarly to the continuous case for m n 2 the difference schlesinger equations and their q analogs yield discrete painlev e equations examples include dpii dpiv dpv and q pvi

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

on the approximation in the smoothed finite element method sfem this letter aims at resolving the issues raised in the recent short communication int j numer meth engng 2008 76 8 1285 1295 doi 10 1002 nme 2460 and answered by int j numer meth engng 2009 doi 10 1002 nme 2587 by proposing a systematic approximation scheme based on non mapped shape functions which both allows to fully exploit the unique advantages of the smoothed finite element method sfem comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 commun numer meth engng 2009 25 1 19 34 doi 10 1002 cnm 1098 int j numer meth engng 2007 71 8 902 930 comput meth appl mech engng 2008 198 2 165 177 doi 10 1016 j cma 2008 05 029 comput meth appl mech engng 2007 submitted int j numer meth engng 2008 74 2 175 208 doi 10 1002 nme 2146 comput meth appl mech engng 2008 197 13 16 1184 1203 doi 10 1016 j cma 2007 10 008 and resolve the existence linearity and positivity deficiencies pointed out in int j numer meth engng 2008 76 8 1285 1295 we show that wachspress interpolants a rational basis for function approximation academic press inc new york 1975 computed in the physical coordinate system are very well suited to the sfem especially when elements are heavily distorted obtuse interior angles the proposed approximation leads to results that are almost identical to those of the sfem initially proposed in comput mech 2007 39 6 859 877 doi 10 1007 s00466 006 0075 4 these results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of sfems copyright 2009 john wiley sons ltd

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

recurrence equations for the computation of correlations in the 1 r 2 quantum many body system in a previous paper the two particle distribution function and one particle density matrix for the quantum many body system with the 1 r2 pair potential have been expressed as limiting cases of selberg correlation integrals recurrence equations are derived which allow rapid evaluation of these multidimensional integrals the exact results for the two particle distribution are compared with the harmonic approximation

singularity dominated strong fluctuations for some random matrix averages the circular and jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval 0 1 of the real line respectively the averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges for 2 1 at points approaching the eigenvalue support using the theory of generalized hypergeometric functions based on jack polynomials the functional form of the leading asymptotic behaviour is established rigorously in the circular ensemble case this confirms a conjecture of berry and keating

weak local linear discretizations for stochastic differential equations convergence and numerical schemes weak local linear wll discretizations are playing an increasing role in the construction of effective numerical integrators and inference methods for stochastic differential equations sdes with additive noise however due to limitations in the existing numerical implementations of wll discretizations the resulting integrators and inference methods have either been restricted to particular classes of autonomous sdes or showed low computational efficiency another limitation is the absence of a systematic theoretical study of the rate of convergence of the wll discretizations and numerical integratos this task is the main purpose of the present paper a second goal is introducing a new wll scheme that overcomes the numerical limitations mentioned above additionally a comparative analysis between the new wll scheme and some conventional weak integrators is also presented

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

adaptive finite element methods on quadrilateral meshes without hanging nodes hanging nodes have some disadvantages in the implementation of adaptive finite element methods on quadrilateral meshes which usually need further techniques to treat them in this paper we present a shape regular local refinement algorithm on quadrilateral meshes without hanging nodes which can be viewed as an extension of the original red green refinement proposed by bank sherman and weiser refinement algorithms and data structures for regular local mesh refinement in scientific computing applications of mathematics and computing to the physical sciences r s stepleman ed north holland amsterdam 1983 pp 3 17 we also prove this quadrilateral red green refinement can hold the shape regularity and conformity of quadrilateral meshes furthermore we have successfully accomplished the adaptive finite element computation on quadrilateral meshes without hanging nodes numerical results show that the adaptive algorithm with this red green refinement is quasi optimal some properties of this local refinement on quadrilateral meshes are also covered in this paper

simulation of stochastic differential equations through the local linearization method a comparative study a new local linearization ll scheme for the numerical integration of nonautonomous multidimensional stochastic differential equations sdes with additive noise is introduced the numerical scheme is based on the local linearization of the sde s drift coefficient by means of a truncated ito taylor expansion a comparative study with the other ll schemes is presented which shows some advantanges of the new scheme over other ones

the implementation of b splines to hashin and shtrikman variational principle based fft method for the homogenization of composite abstract fast fourier transform fft has been successfully used to estimate the effective properties of composites with meso periodic structure for more than two decades numerous improvements have been done to make it adequate for infinite contrast bound elastic energy and composite voxels the hashin and shtrikman variational principle based fft method handles these three problems simultaneously and is very attractive in this paper b splines were adopted to discretize the polarized field the approximate polarized field was substituted into the hashin and shtrikman variational principle to obtain the discretized system the constitutive matrix of a composite voxel was calculated by both the energetically consistent way and the laminate mixing rule numerical example of a square matrix with a circular inclusion shows that the results are more accurate than the remaining cases when the first and second order b splines as well as the laminate mixing constitutive matrix are used for higher order b spline the summation to compute the discrete green operator for periodic boundary conditions converges faster while the discretized polarized field needs more iterations to converge only when the polarized field is sufficiently smooth high order b spline improves the numerical accuracy and efficiency

the exact perimeter generating function for a model of punctured staircase polygons we have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90 angle with respect to the outer staircase polygon in one approach we calculated a long series expansion for the problem and found that all the terms in the generating function can be reproduced from a linear fuchsian differential equation of order 4 we then solved this ode and found a closed form expression for the generating function this is a highly unusual and most fortuitous result since odes of such high order very rarely permit a closed form solution in a second approach we proved the result for the generating function exactly using combinatorial arguments this latter solution allows many generalizations including models with other types of punctures and a model with any fixed number of nested rotated staircase punctures

a family of weak stochastic newmark methods for simplified and efficient monte carlo simulations of oscillators engineers are often more concerned with the computation of statistical moments or mathematical expectations of the response of stochastically driven dynamical systems than with the determination of path wise response histories with this in perspective weak stochastic solutions of dynamical systems modelled as n degrees of freedom dof mechanical oscillators and driven by additive and or multiplicative white noise or filtered white noise processes are considered in this study while weak stochastic solutions are simpler and quicker to compute than strong sample path wise solutions it must be emphasized that sample realizations of weak solutions have no path wise similarity with strong solutions however the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is close to that of the true response if it exists within a certain order of a given time step size computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems oscillators of engineering interest to efficiently generate such weak responses a novel class of weak stochastic newmark methods wsnms based on implicit ito taylor expansions of displacement and velocity fields is proposed the resulting multiple stochastic integrals msis in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities in fact yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher level msis estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy simplicity and higher computational speed of wsnms vis a vis a few other popularly used stochastic integration schemes as well as the path wise versions of the stochastic newmark scheme copyright 2006 john wiley sons ltd

hypergeometric solutions of trigonometric kz equations satisfy dynamical difference equations abstract the trigonometric kz equations associated to a lie algebra g depend on a parameter h where h g is a cartan subalgebra a system of dynamical difference equations with respect to compatible with the kz equations is introduced by v tarasov and the second author 2000 internat math res notices 15 801 829 we prove that the standard hypergeometric solutions of the trigonometric kz equations associated to sl n also satisfy the dynamical difference equations

piecewise polynomial shape functions for hp finite element methods abstract high order piecewise polynomial shape functions for quadrilateral finite elements are proposed the shape functions are tensor products of one dimensional piecewise functions based on legendre polynomials this allows the construction of h 1 conforming approximations on meshes with arbitrary levels of hanging nodes without the necessity for constrained approximation techniques furthermore meshes can be refined simply by inserting nodes on element edges application examples including the l shape domain a plate problem and a tie rod connection plate illustrate the applicability of the method

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

error estimation and adaptivity for discontinuous failure discontinuous failure is simulated via the introduction of a geometrical discontinuity the cohesive zone is modelled via the use of the partition of unity property of the finite element interpolation by this approach a crack can pass through the elements without any restriction to the underlying mesh despite such a feature it has been confirmed that a sufficiently fine mesh discretization still needs to be ensured in order to obtain a correct crack path and mechanical response the p adaptive scheme driven by error in an energy norm measure or a goal oriented measure has been examined due to its implementational simplicity the results have shown that if considering only increasing the polynomial degree the p approach can greatly improve the results copyright 2008 john wiley sons ltd

a new weight function enrichment in meshless methods for multiple cracks in linear elasticity summary a new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions the main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack represented by a segment but continuous all around the crack tips an analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple multiplicative manner the resulting method allows a more straightforward implementation and simulation of the presence of multiple cracks crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility transparency and diffraction and without using additional unknowns and additional equations for the evolution of the level sets as in extrinsic partition of unity based methods stress intensity factors calculated using the j integral demonstrate excellent agreement with analytical solutions for classical fracture mechanics benchmarks copyright 2011 john wiley sons ltd

nonlinear estimation of the bold signal signal variations in functional magnetic resonance imaging experiments essentially reflect the vascular system response to increased demand for oxygen caused by neuronal activity termed the blood oxygenation level dependent bold effect the most comprehensive model to date of the bold signal is formulated as a mixed continuous discrete time system of nonlinear stochastic differential equations previous approaches to the analysis of this system have been based on linearised approximations of the dynamics which are limited in their ability to capture the inherent nonlinearities in the physiological system in this paper we present a nonlinear filtering method for simultaneous estimation of the hidden physiological states and the system parameters based on an iterative coordinate descent framework state estimates of the cerebral blood flow cerebral blood volume and deoxyhaemoglobin content are determined using a particle filter demonstrated via simulation to be accurate robust and efficient in comparison to linearisation based techniques the adaptive state and parameter estimation algorithm generates physiologically reasonable parameter estimates for experimental fmri data it is anticipated that signal processing techniques for modelling and estimation will become increasingly important in fmri analyses as limitations of linear and linearised modelling are reached

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

difference system for selberg correlation integrals the selberg correlation integrals are averages of the products with respect to the selberg density our interest is in the case m 1 1 when this corresponds to the th moment of the corresponding characteristic polynomial we give the explicit form of an n 1 n 1 matrix linear difference system in the variable which determines the average and we give the gauss decomposition of the corresponding n 1 n 1 matrix for a positive integer the difference system can be used to efficiently compute the power series defined by this average

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

integrals over the circular ensembles relating to classical domains using the theory of classical domains and jack polynomials we generalize several integrals over the unitary group due to fyodorov and khoruzhenko to integrals over the circular orthogonal and circular symplectic ensembles while in the unitary case a fundamental role is played by complex matrices it is anti symmetric and symmetric matrices which play the same role in the latter two cases in our workings a generalization of the evaluation of the integral of a jack polynomial times the selberg weight sought by fyodorov and khoruzhenko is identified

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

smooth finite element methods convergence accuracy and properties a stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented the integration of the stiffness matrix is performed on the boundaries of the finite elements a rigorous variational framework based on the hu washizu assumed strain variational form is developed we prove that solutions yielded by the proposed method are in a space bounded by the standard finite element solution infinite number of subcells and a quasi equilibrium finite element solution a single subcell we show elsewhere the equivalence of the one subcell element with a quasi equilibrium finite element leading to a global a posteriori error estimate we apply the method to compressible and incompressible linear elasticity problems the method can always achieve higher accuracy and convergence rates than the standard finite element method especially in the presence of incompressibility singularities or distorted meshes for a slightly smaller computational cost it is shown numerically that the one cell smoothed four noded quadrilateral finite element has a convergence rate of 2 0 in the energy norm for problems with smooth solutions which is remarkable for problems with rough solutions this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme copyright 2007 john wiley sons ltd

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier