numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games we introduce a near linear complexity geometric and meshless algebraic multigrid multiresolution method for pdes with rough l infty coefficients with rigorous a priori accuracy and performance estimates the method is discovered through a decision game theory formulation of the problems of 1 identifying restriction and interpolation operators 2 recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator and 3 gambling on the value of the solution of the pde based on a hierarchy of nested measurements of its solution or source term the resulting elementary gambles form a hierarchy of deterministic basis functions of h 1 0 omega gamblets that 1 are orthogonal across subscales subbands with respect to the scalar product induced by the energy norm of the pde 2 enable sparse compression of the solution space in h 1 0 omega and 3 induce an orthogonal multiresolution operator decomposition the operating diagram of the multig

absolute continuity results for superprocesses with some applications let x1 and x2 be instances of a measure valued dawsonwatanabe 4 super process where the underlying spatial motions are given by a borel right process x and where the branching mechanism has finite variance a necessary and sufficient condition on x0 and x2 is found for the law of xi 2 to be absolutely continuous with respect to the law of x2 the conditions are the natural absolute continuity conditions on s but some care must be taken with the set of times s t being considered the result is used to study the closed support of super brownian motion and give sufficient conditions for the existence of a nontrivial collision measure for a pair of independent superlevy processes or more generally for a super levy process and a fixed measure the collision measure gauges the extent of overlap of the two measures as a final application we give an elementary proof of the instantaneous propagation of a super levy process to all points to which the underlying levy process can jump this result is then extended to a much larger class of superprocesses using different techniques

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

the enriched space time finite element method est for simultaneous solution of fluid structure interaction the paper introduces a weighted residual based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures the presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple flow immersed solid objects the fluid flow is described by the incompressible navier stokes equations the current configuration of the thin structure of linear elastic material with non linear kinematics is mapped to the flow using the zero iso contour of an updated level set function the formulation of fluid structure and coupling conditions uniformly uses velocities as unknowns the integration of the weak form is performed on a space time finite element discretization of the domain interfacial constraints of the multi field problem are ensured by distributed lagrange multipliers the proposed formulation and discretization techniques lead to a monolithic algebraic system well suited for strongly coupled fluid structure systems embedding a thin structure into a flow results in non smooth fields for the fluid based on the concept of the extended finite element method the space time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions this leads to the present enriched space time est method numerical examples of fluid structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems the test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail copyright 2007 john wiley sons ltd

continuum discontinuum modelling of shear bands a methodology to model shear bands as strong discontinuities within a continuum mechanics context is presented the loss of hyperbolicity of the ibvp is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction separation model acting at the discontinuity surface the extended finite element method xfem is employed for the spatial discretization of the governing equations this enables the shear bands to be arbitrarily positioned within the mesh examples that study the shear band progression within a rate independent material are presented copyright 2005 john wiley sons ltd

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

the generalized finite element method for helmholtz equation part ii effect of choice of handbook functions error due to absorbing boundary conditions and its assessment abstract in part i t strouboulis i babuska r hidajat the generalized finite element method for helmholtz equation theory computation and open problems comput methods appl mech engrg 195 2006 4711 4731 we introduced the q version of the generalized finite element method gfem for the helmholtz equation and we addressed its a pollution error due to the wave number b exponential q convergence c robustness to perturbations of the mesh the roundoff and numerical quadrature errors and d a posteriori error estimation here we continue the development of the gfem for helmholtz and we address the effects of a alternative handbook functions and mesh types b the error due to the artificial truncation boundary conditions and its assessment the conclusions are 1 the employment of plane wave wave band and vekua handbook functions lead to equivalent results and 2 for high q the most significant component of error may be the one due to the artificial truncation boundary conditions a rather straightforward approach for assessing this error is proposed

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

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

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a discontinuous enrichment method for three dimensional multiscale harmonic wave propagation problems in multi fluid and fluid solid media an evanescent wave occurs when a propagating incident wave impinges on an interface between two fluid solid or fluid solid media at a subcritical angle mathematical properties of such a wave make it difficult to capture with standard finite element discretization schemes for this reason the discontinuous enrichment method dem developed in comput methods appl mech eng 2001 190 6455 6479 comput methods appl mech eng 2003 192 1389 1419 comput methods appl mech eng 2003 192 3195 3210 int j numer meth engng 2004 61 1938 1956 wave motion 2004 39 4 307 317 int j numer meth engng 2006 66 2086 2114 int j numer meth engng 2006 66 796 815 is extended here to the solution of a class of three dimensional evanescent wave problems in the frequency domain to this effect new dem elements for three dimensional elastodynamic problems are first proposed then these and other dem elements previously developed for the efficient solution of the helmholtz problem are further enriched with free space solutions of model evanescent wave problems in order to achieve high accuracy at practical mesh resolution for fluid fluid and fluid solid applications the performance of the extended dem elements is reported to be better than that of its basic helmholtz and navier counterparts and superior to that achieved by the classical high order polynomial finite element method copyright 2008 john wiley sons ltd

upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations using non local multi continuum method nlmc abstract in this paper we present an upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects we will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations with the aim of handling non homogeneous boundary conditions on perforations we start with describing our method for the laplace equation and then extending the framework for the elasticity problem and parabolic equations the resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid we will present numerical results 1 for steady and unsteady problems 2 for laplace and elastic operators and 3 for neumann and robin non homogeneous boundary conditions on perforations numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedoms

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

residual free bubbles for embedded dirichlet problems abstract we examine a stabilized method employing residual free bubbles for enforcing dirichlet constraints on embedded finite element interfaces in particular we focus attention on problems where the underlying mesh is not explicitly fitted to the geometry of the interface the residual free bubble problem is derived for a simple case and extensions are discussed we show that under certain conditions stabilization only requires knowledge of the residual free bubble on the interface we then examine methods to approximate the residual free bubble a series of benchmark tests are used to demonstrate the accuracy and robustness of the method comparisons are made to nitsche s method employing an eigenvalue estimate for the global stability parameter particular emphasis is placed on the accuracy of flux evaluations on the interface finally we employ the method to simulate an evolving interface problem motivated by resin transfer molding

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

can a finite element method perform arbitrarily badly in this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm and show that adaptive procedures cannot improve this slow convergence we also show that the l 2 norm and the nodal point errors converge arbitrarily slowly with the l 2 norm two cases need to be distinguished and the usual duality principle does not characterize the error completely the constructed elliptic problems are one dimensional

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

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

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

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

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

shear band capturing using a multiscale extended digital image correlation technique abstract finite elements have been used recently to solve the optical flow conservation principle invoked to determine displacement fields by digital image correlation inspired by these recent advances and by the computational effort that has been accomplished during the past 10 years for the simulation of discontinuities by the extended finite element method xfem an extended correlation technique is introduced for capturing shear band like discontinuities from images of real mechanical tests because of the specific information at hand i e gray level images this extended finite element approach is included in a non linear multi grid solver the performances of the proposed approach are discussed on two examples namely the analysis of a bolted assembly and the formation of piobert luders bands

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

a contact algorithm for frictional crack propagation with the extended finite element method we present an incremental quasi static contact algorithm for path dependent frictional crack propagation in the framework of the extended finite element fe method the discrete formulation allows for the modeling of frictional contact independent of the fe mesh standard coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity the contact constraint is borrowed from non linear contact mechanics and embedded within a localized element by penalty method newton raphson iteration with consistent linearization is used to advance the solution we show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called latin for frictional crack propagation numerical examples include simulation of crack initiation and propagation in 2d plane strain with and without bulk plasticity in the presence of bulk plasticity the problem is also solved using an augmented lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution copyright 2008 john wiley sons ltd

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

strong and weak arbitrary discontinuities in spectral finite elements methods for constructing arbitrary discontinuities within spectral finite elements are described and studied we use the concept of the extended finite element method xfem which introduces the discontinuity through a local partition of unity so there is no requirement for the mesh to be aligned with the discontinuities a key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity we found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements for the quadrature of the galerkin weak form since the integrand is discontinuous we use a strategy of subdividing the discontinuous elements into 6 and 10 node triangles the order of the element depends on the order of the spectral method for curved discontinuities several numerical examples are solved to examine the accuracy of the methods for straight discontinuities we achieved the optimal convergence rate of the spectral element for the curved discontinuity the convergence rate in the energy norm error is suboptimal we attribute the suboptimality to the approximations in the quadrature scheme we also found that modification of the adjacent elements is only needed for lower order spectral elements copyright 2005 john wiley sons ltd

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

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

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

three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems recently a discontinuous galerkin finite element method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of helmholtz problems in the mid frequency regime in this paper this method is extended to three dimensions and several new elements are proposed computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher order galerkin method

nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media abstract our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media we consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures to construct our upscaled model we will apply the nonlocal multicontinua nlmc upscaling technique the upscaled coefficients are obtained by using some multiscale basis functions which are solutions of local problems defined on oversampled regions for each continuum within a target coarse element we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise the resulting multiscale basis functions have been shown to have good approximation properties to illustrate the idea of our approach we will consider a dual continua background model consisting of discrete fractures in two space dimensions that is we consider a system with three continua we will present several numerical examples and they show that our method is able to capture the interaction between matrix continua and discrete fractures on the coarse grid efficiently

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

on time integration in the xfem the extended finite element method xfem is often used in applications that involve moving interfaces examples are the propagation of cracks or the movement of interfaces in two phase problems this work focuses on time integration in the xfem the performance of the discontinuous galerkin method in time space time finite elements fes and time stepping schemes are analyzed by convergence studies for different model problems it is shown that space time fe achieve optimal convergence rates special care is required for time stepping in the xfem due to the time dependence of the enrichment functions in each time step the enrichment functions have to be evaluated at different time levels this has important consequences in the quadrature used for the integration of the weak form a time stepping scheme that leads to optimal or only slightly sub optimal convergence rates is systematically constructed in this work copyright 2009 john wiley sons ltd

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

a posteriori error estimation for generalized finite element methods in this paper we address the problem of a posteriori error estimation for generalized finite element methods based on the partition of unity method the computational results focus on the question of the reliability of the error estimators and its assessment

dislocations by partition of unity a new finite element method for accurately modelling the displacement and stress fields produced by a dislocation is proposed the methodology is based on a local enrichment of the finite element space by closed form solutions for dislocations in infinite media via local partitions of unity this allows the treatment of both arbitrary boundary conditions and interfaces between materials the method can readily be extended to arrays of dislocations 3d problems large strains and non linear constitutive models results are given for an edge dislocation in a hollow cylinder and in an infinite medium for the cases of a glide plane intersecting a rigid obstacle and an interface between two materials copyright 2005 john wiley sons ltd

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

a local space time discontinuous finite element method we present a locally enriched space time finite element method for solving hyperbolic problems with discontinuities the space time formulation is coupled to a standard semidiscrete finite element method the discontinuities are captured by a space time version of the extended finite element method which treats arbitrary moving discontinuities since the discontinuities are local the enriched space time method is only needed around the discontinuities which provides significant computational savings the coupling is implemented through a weak enforcement of the continuity of the flux between the space time and semidiscrete domains in a manner similar to discontinuous galerkin methods the method is illustrated through one dimensional problems it displays remarkable accuracy in capturing moving discontinuities both in their amplitude and speed

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

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

fast tree based redistancing for level set computations level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface such as the signed distance to the interface this paper presents efficient algorithms for this redistancing problem the algorithms use quadtrees and triangulation to compute global approximate signed distance functions a quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants given a polygonal interface with n elements our algorithms run in o n space and o n log n time two dimensional numerical results show they are highly efficient in practice

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

concurrently coupled atomistic and xfem models for dislocations and cracks a method for the modeling of dislocations and cracks by atomistic continuum models is described the methodology combines the extended finite element method with the bridging domain method bdm the former is used to model crack surfaces and slip planes in the continuum whereas the bdm is used to link the atomistic models with the continuum the bdm is an overlapping domain decomposition method in which the atomistic and continuum energies are blended so that their contributions decay to their boundaries on the overlapping subdomain compatibility between the continua and atomistic domains is enforced by a continuous lagrange multiplier field the methodology allows for simulations with atomistic resolution near crack fronts and dislocation cores while retaining a continuum model in the remaining part of the domain and so a large reduction in the number of atoms is possible it is applied to the modeling of cracks and dislocations in graphene sheets energies and energy distributions compare very well with direct numerical simulations by strictly atomistic models copyright 2008 john wiley sons ltd

on error estimator and p adaptivity in the generalized finite element method this paper addresses the issue of a p adaptive version of the generalized finite element method gfem the technique adopted here is the equilibrated element residual method but presented under the gfem approach i e by taking into account the typical nodal enrichment scheme of the method such scheme consists of multiplying the partition of unity functions by a set of enrichment functions these functions in the case of the element residual method are monomials and can be used to build the polynomial space one degree higher than the one of the solution in which the error functions is approximated global and local measures are defined and used as error estimator and indicators respectively the error indicators calculated on the element patches that surrounds each node are used to control a refinement procedure numerical examples in plane elasticity are presented outlining in particular the effectivity index of the error estimator proposed finally the p adaptive procedure is described and its good performance is illustrated by the last numerical example

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

arbitrary discontinuities in space time finite elements by level sets and x fem an enriched finite element method with arbitrary discontinuities in space time is presented the discontinuities are treated by the extended finite element method x fem which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper surface which is described by level sets a space time weak form for conservation laws is developed where the rankine hugoniot jump conditions are natural conditions of the weak form the method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non linear burgers equations by capturing the discontinuity in time as well as space results are improved over capturing the discontinuity in space alone and the method is remarkably accurate implications to standard semi discretization x fem formulations are also discussed copyright 2004 john wiley sons ltd

two scale shear band evolution by local partition of unity we introduce a methodology to model shear band evolution in the quasi static regime using the extended finite element method we enrich the finite element polynomial displacement field with a fine scale function which models the high displacement gradient in the shear band for this purpose we use a local partition of unity and a parameterized displacement enrichment based on closed form solutions for one dimensional shear bands a stabilized consistent penalty method is used to circumvent locking in the regularized elasto viscoplastic plane strain regime and to guarantee element stability the loss of stability of the boundary value problem is used as an indicator of shear band initiation point and direction shear band development examples are shown illustrating the capabilities of the method to track shear band evolution and strains as high as 1000 copyright 2005 john wiley sons ltd

p version of the generalized fem using mesh based handbooks with applications to multiscale problems in this paper we analyse the p convergence of a new version of the generalized finite element method generalized fem or gfem which employs mesh based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the partition of unity method pum we show that the p version of our gfem is capable of achieving very high accuracy for multiscale problems which may be impossible to solve using the standard fem we analyse the effect of the main factors affecting the accuracy of the method namely a the data and the buffer included in the handbook domains and b the accuracy of the numerical construction of the handbook functions we illustrate the robustness of the method by employing as model problem the laplacian in a domain with a large number of closely spaced voids similar robustness can be expected for problems of heat conduction and elasticity set in domains with a large number of closely spaced voids cracks inclusions etc copyright 2004 john wiley sons ltd

a regularized xfem framework for embedded cohesive interfaces this work focuses on xfem modeling of elastodamaging interfaces characterized by regularized kinematics after regularization of the displacement discontinuity a process zone of finite width with continuous strain fields emerges the stress fields are independently energy deduced modeled via specific constitutive laws and subsequently introduced within the virtual work principle the methodology can address cohesive interfaces with vanishing and finite thickness in a unified way two dimensional results for a strain localization problem and several delamination tests are discussed in detail and compared with previous numerical experimental results