analysis of three dimensional fracture mechanics problems a two scale approach using coarse generalized fem meshes this paper presents a generalized finite element method gfem based on the solution of interdependent global structural and local crack scale problems the local problems focus on the resolution of fine scale features of the solution in the vicinity of three dimensional cracks while the global problem addresses the macro scale structural behavior the local solutions are embedded into the solution space for the global problem using the partition of unity method the local problems are accurately solved using an hp gfem and thus the proposed method does not rely on analytical solutions the proposed methodology enables accurate modeling of three dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts the boundary conditions for the local problems are provided by the coarse global mesh solution and can be of dirichlet neumann or cauchy type the effect of the type of local boundary conditions on the performance of the proposed gfem is analyzed several three dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance both in terms of problem size and cpu time are presented copyright 2009 john wiley sons ltd

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

level set topology optimization of structural problems with interface cohesion summary this paper presents a finite element topology optimization framework for the design of two phase structural systems considering contact and cohesion phenomena along the interface the geometry of the material interface is described by an explicit level set method and the structural response is predicted by the extended finite element method in this work the interface condition is described by a bilinear cohesive zone model on the basis of the traction separation constitutive relation the non penetration condition in the presence of compressive interface forces is enforced by a stabilized lagrange multiplier method the mechanical model assumes a linear elastic isotropic material infinitesimal strain theory and a quasi static response the optimization problem is solved by a nonlinear programming method and the design sensitivities are computed by the adjoint method the performance of the presented method is evaluated by 2d and 3d numerical examples the results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered in addition 3d examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality the optimization framework presented is limited to two phase structural systems where the material interface is coincident in the undeformed configuration and to structural responses that remain valid considering small strain kinematics copyright 2017 john wiley sons ltd

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

finite cover method for linear and non linear analyses of heterogeneous solids we introduce the finite cover method fcm as a generalization of the finite element method fem and extend it to analyse the linear and non linear mechanical behaviour of heterogeneous solids and structures the name fcm is actually an alias for the manifold method mm and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials after reviewing the concept of physical and mathematical covers for approximating functions in the fcm we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces the problem essentially involves the discontinuities in strains and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces we simulate such non linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the fcm and those of the fem copyright 2003 john wiley sons ltd

limit theorems for coupled continuous time random walks scaling limits of continuous time random walks are used in physics to model anomalous diffusion in which a cloud of particles spreads at a different rate than the classical brownian motion governing equations for these limit processes generalize the classical diffusion equation in this article we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent this leads to an efficient method of computing the limit and a surprising connection to fractional derivatives

extra dof free and linearly independent enrichments in gfem abstract a gfem without extra dof is developed without the extra dof the resulting linear system becomes independent in size of the order of local approximation the resulting pu approximation becomes free of linear dependence and the resulting stiffness matrix becomes good conditioned numerical studies show the new gfem s excellent convergence properties and excellent stability

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

an integrated procedure for three dimensional structural analysis with the finite cover method in this paper an integrated procedure for three dimensional 3d structural analyses with the finite cover method fcm is introduced in the pre process of this procedure the geometry of a structure is modelled by 3d cad followed by digitization to have the corresponding voxel model and then the structure is covered by a union of mathematical covers namely a mathematical mesh independently generated for approximation purposes since the mesh topology in the fcm does not need to conform to the physical boundaries of the structure the mesh can be regular and structured thus the numerical analysis procedure is free from the difficulties mesh generation typically poses and in this sense enables us to realize the mesh free analysis after formulating the fcm with interface elements for the static equilibrium state of a structure we detail the procedure of the finite cover modelling including the geometry modelling with 3d cad and the identification of the geometry covered by a regular mesh for numerical integration prior to full 3d modelling and analysis we present a simple numerical example to confirm the equivalence of the performance of the fcm and that of the standard finite element method fem finally representative numerical examples are presented to demonstrate the capabilities of the proposed analysis procedure

pore scale modeling and continuous time random walk analysis of dispersion in porous media 1 we provide a physically based explanation for the complex macroscopic behavior of dispersion in porous media as a function of peclet number pe using a pore scale network model that accurately predicts the experimental dependence of the longitudinal dispersion coefficient dl on pe the asymptotic dispersion coefficient is only reached after the solute has traveled through a large number of pores at high pe this implies that preasymptotic dispersion is the norm even in experiments in statistically homogeneous media interpreting transport as a continuous time random walk we show that 1 the power law dispersion regime is controlled by the variation in average velocity between throats the distribution of local pe giving dl pe with 3 1 2 where is an exponent characterizing the distribution of transit times between pores 2 the crossover to a linear regime dl pe for pe pecrit 400 is due to a transition from a diffusion controlled late time cutoff to transport governed by advective movement and 3 the transverse dispersion coefficient dt pe for all pe 1

a high order finite element technique with automatic treatment of stress singularities by semi analytical enrichment abstract this paper presents an approach to the automatic enrichment of finite elements in the vicinity of a stress singularity the enrichment consists of semi analytical singular modes constructed using the scaled boundary finite element method sbfem in contrast to analytical methods the sbfem provides modes for inhomogeneous and anisotropic materials without additional effort the finite element basis can be of arbitrary order and remains unaltered by the enrichment the approach requires enrichment in only one layer of elements around a node due to the compatibility of sbfem with fem there is no need for transitional elements and there are no parasitic terms the approach is tested for several benchmark problems the stress intensity factors are computed based on techniques inspired by the sbfem the proposed procedure is compared to a standard finite element implementation and shows a significant improvement in the error of the displacement field for problems involving singular stresses

prophlex an hp adaptive finite element kernel for solving coupled systems of partial differential equations in computational mechanics this paper presents an overview of the basic design and architecture of the prophlex hp adaptive finite element kernel prophlex was designed to be a commercial robust implementation of hp adaptivity driven by residual error estimation with the primary goal of being physics independent and computationally efficient on a wide array of computer hardware platforms prophlex can solve virtually any class of engineering problems which be may be mathematically formulated as a system of linear or nonlinear second order partial differential equations and associated boundary conditions it has been applied to compressible and incompressible fluid dynamics linear and nonlinear solid mechanics heat flow problems as well as semiconductor device simulation examples of prophlex customization for linear and nonlinear solid mechanics are presented

generalized nodes and high performance elements the paper concerns the development of robust and high accuracy finite elements with only corner nodes using a partition of unity based finite element approximation construction of the partition of unity based approximation is accomplished by a physically defined local function of displacements a 4 node quadratic tetrahedral element and a 3 node quadratic triangular element are developed eigenvalue analysis shows that linear dependencies in the partition of unity based finite element approximation constructed for the new elements are eliminable numerical calculations demonstrate that the new elements are robust insensitive to mesh distortion and offer quadratic accuracy while also keeping mesh generation extremely simple copyright 2005 john wiley sons ltd

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

a multiscale extended finite element method for crack propagation in this paper we propose a multiscale strategy for crack propagation which enables one to use a refined mesh only in the crack s vicinity where it is required two techniques are used in synergy a multiscale strategy based on a domain decomposition method to account for the crack s global and local effects efficiently and a local enrichment technique the x fem to describe the geometry of the crack independently of the mesh the focus of this study is the avoidance of meshing difficulties and the choice of an appropriate scale separation to make the strategy efficient we show that the introduction of the crack s discontinuity both on the microscale and on the macroscale is essential for the numerical scalability of the domain decomposition method to remain unaffected by the presence of a crack thus the convergence rate of the iterative solver is the same throughout the crack s propagation

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 fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

the length of a shortest geodesic loop at a point in this paper we prove that given a point p m n where m n is a closed riemannian manifold of dimension n the length of a shortest geodesic loop lp m n at this point is bounded above by 2nd where d is the diameter of m n moreover we show that on a closed simply connected riemannian manifold m n with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of m n or the length of a shortest closed geodesic on m n is bounded from above by 4d

treatment of material discontinuity in the element free galerkin method abstract a method is presented for application of the element free galerkin method efg to solid mechanics problems containing material discontinuities by using the efg method the trial and test functions for the weak form are constructed with moving least square interpolants in each material domain additional constraints are imposed at the variational level to ensure satisfaction of interface conditions as a result only nodal data no elemental connectivity is needed given c 1 continuity of the interpolant weighting function the dependent variable and its gradient are continuous in each material domain two linear elastic bi material examples are presented to illustrate the effectiveness of the method

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

upscaling levy motions in porous media with long range correlations bacterial motility has been modeled by levy motions which were upscaled in porous media with drift processes such as levy processes and time changed brownian motion via central limit theorems it was possible to prove the limit theorems and upscale the processes because of the independence of their increments therefore it has not been applied to processes with correlated increments such as fractional brownian motions in this paper the upscaling approach is generalized to porous media with long range correlated processes the processes are modeled by a fractional brownian velocity process and p diffusive position processes that were defined and used to classify diffusion processes by o malley and cushman a renormalization group classification of nonstationary and or infinite second moment diffusive processes j stat phys 146 989 1000 2012 10 1007 s10955 012 0448 3 a few examples of p diffusive processes are discussed by computing the values of the parameter p

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

fractional cauchy problems on bounded domains fractional cauchy problems replace the usual first order time derivative by a fractional derivative this paper develops classical solutions and stochastic analogues for fractional cauchy problems in a bounded domain dr d with dirichlet boundary conditions stochastic solutions are constructed via an inverse stable subordi nator whose scaling index corresponds to the order of the fractional time derivative dirichlet problems corresponding to iterated brow nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half derivative in time 1 introduction in this paper we extend the approach of meerschaert and scheffler 23 and meerschaert et al 24 to fractional cauchy problems on bounded domains our methods involve eigenfunction expansions killed markov processes and inverse stable subordinators in a recent related paper 7 we establish a connection between fractional cauchy problems with index 1 2 on an unbounded domain and iterated brownian motion ibm defined as zt b yt where b is a brownian motion with values in r d and y is an independent one dimensional brownian motion since ibm is also the stochastic solution to a cauchy problem involving a fourth order derivative in space 2 14 that paper also establishes a connection between certain higher order cauchy problems and their time fractional analogues more generally baeumer meerschaert and nane 7 shows a connection between fractional cauchy problems with 1 2 and higher order cauchy problems that involve the square of the generator in the present paper we

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

symplectic embeddings of polydisks if p and p are symplectic polydisks of radii r 1 r n and r 1 r n respectively then we prove that p symplectically embeds in p provided that c n r 1 r 1 and c n r 1 r n r 1 r n up to a constant factor these conditions are optimal

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 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

symplectic capacity and short periodic billiard trajectory we prove that a bounded domain in mathbb r n with smooth boundary has a periodic billiard trajectory with at most n 1 bounce times and of length less than c n r where c n is a positive constant which depends only on n and r is the supremum of radius of balls in this result improves the result by c viterbo which asserts that has a periodic billiard trajectory of length less than c n rm vol omega 1 n to prove this result we study symplectic capacity of liouville domains which is defined via symplectic homology

linear dependence problems of partition of unity based generalized fems a known problem of partition of unity based generalized finite element methods referred to as gfem is the linear dependence problem which leads to singular global stiffness matrices thus far attempts to eliminate the linear dependence problem have been unsuccessful numerical experiments are carried out among several gfems to investigate the problem based on the numerical experiments simple but effective approaches to eliminating the linear dependence problem are suggested

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

on the topological properties of symplectic maps in this paper we show that symplectic maps have surprising topological properties in particular we construct an interesting metric for the symplectic diffeomorphism groups which is related but not obviously to the topological properties of symplectic maps and phase space geometry we also prove a certain number of generalised symplectic fixed point theorems and give an application to a hamiltonian system

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

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

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

an immersed boundary method for fluids using the xfem and the hydrodynamic boltzmann transport equation abstract this paper presents a stabilized finite element formulation of the hydrodynamic boltzmann transport equation hbte to predict nearly incompressible fluid flow the hbte is discretized with hermite polynomials in the velocity variable and a streamline upwind petrov galerkin formulation is used to discretize the spatial variable a nonlinear stabilization scheme is presented from which a simple linear stabilization scheme is constructed in contrast to the navier stokes ns equations the hbte is a first order equation and allows for conveniently enforcing dirichlet conditions along immersed boundaries a simple and efficient formulation for enforcing dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method xfem numerical experiments indicate that both the linear and non linear stabilization methods are sufficiently accurate and stable but the linear formulation reduces the computational cost significantly the accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints but may degrade the accuracy of the global solution comparisons with results of a single relaxation time lattice boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement

bounds for decoupled design and analysis discretizations in topology optimization summary topology optimization formulations using multiple design variables per finite element have been proposed to improve the design resolution this paper discusses the relation between the number of design variables per element and the order of the elements used for analysis we derive that beyond a maximum number of design variables certain sets of material distributions cannot be discriminated based on the corresponding analysis results this makes the design description inefficient and the solution of the optimization problem non unique to prevent this we establish bounds for the maximum number of design variables that can be used to describe the material distribution for any given finite element scheme without introducing non uniqueness copyright 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

harmonic enrichment functions a unified treatment of multiple intersecting and branched cracks in the extended finite element method a unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented within each element that is intersected by a crack the enrichment function for the crack is obtained via the solution of the laplace equation with dirichlet and vanishing neumann boundary conditions a single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks without recourse to special cases which provides flexibility over the existing approaches in which each case is treated separately numerical integration is rendered to be simple there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique copyright 2010 john wiley sons ltd

on upscaling operator stable levy motions in fractal porous media the dynamics of motile particles such as microbes in random porous media are modeled with a hierarchical set of stochastic differential equations which correspond to micro meso and macro scales on the microscale the motile particle is modeled as an operator stable levy process with stationary ergodic markov drift the micro to meso and meso to macro scale homogenization is handled with generalized central limit theorems on the mesoscale the lagrangian drift or the lagrangian acceleration is assumed levy to account for the fractal character of many natural porous systems diffusion on the mesoscale is a result of the microscale asymptotics while diffusion on the macroscale results from the mesoscale asymptotics renormalized fokker planck equations with time dependent dispersion tensors and fractional derivatives are presented at the macro scale

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

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

a new way to treat material discontinuities in the numerical manifold method abstract a new technique to describe the material discontinuities within the numerical manifold method framework is presented a new concept of weak discontinuous physical covers is introduced special local approximations which are continuous in the approximation but discontinuous in its first derivative are tailored for weak discontinuous physical covers this new method takes full advantage of the partition of unity several typical examples are presented results have shown the accuracy and convenience of the proposed method in modeling material interfaces especially composite materials with multiple holes inclusions

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes in the partition of unity pu based methods the global approximation is built by multiplying a partition of unity by local approximations within this framework high order approximations are achieved by directly adopting high order polynomials as local approximations and therefore nodes along sides or inside elements which are usually adopted in the conventional finite element methods are no more required however the pu based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom in this paper the origin of the rank deficiency in the pu based approximation space is first dissected at an element level and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency finally examples are investigated to validate the present approach the current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix it can be resolved in its own manner

characterization of long range correlations in complex distributions and profiles characterizing long range correlations in complex distributions such as the porosity logs of field scale porous media and profiles such as the fracture surfaces of rock and materials is an important problem we carry out an extensive analysis of such distributions represented by synthetic and real data to determine which method provides the most efficient and accurate tool for characterizing them the synthetic data and profiles are generated by a fractional brownian motion fbm and the real data analyzed are a porosity log of an oil reservoir and time variations of the pressure fluctuations in three phase flow in a fluidized bed the fbm is generated by three different numerical methods and the data are analyzed by seven different techniques our analysis indicates that the size of the data array greatly influences the accuracy of characterization of its long range correlations we also find that if the size of the data array is large enough the commonly used rescaled range r s method of analyzing fbm series fails to provide accurate estimates of the hurst exponent although it can provide a reasonably accurate analysis of a data array that is generated by a fractional gaussian noise in contrast the maximum entropy and wavelet decomposition methods offer highly accurate and efficient tools of characterizing long range correlations in complex distributions and profiles new methods that are somewhat similar to the r s method are also suggested

finite time approach to equilibrium in a fractional brownian velocity field we consider the solution of the equation r t w r t r 0 r0 0 where w is a fractional brownian motion f b m with the hurst exponent 0 1 we show that for almost all realizations of w the trajectory reaches in finite time the nearest equilibrium point i e zero of the f b m either to the right or to the left of r0 depending on whether w r0 is positive or not after reaching the equilibrium the trajectory stays in it forever the problem is motivated by studying the separation between two particles in a gaussian velocity field which satisfies a local self similarity hypothesis in contrast to the case when the forcing term is a brownian motion then an analogous statement is a consequence of the markov property of the process we show our result using as the principal tools the properties of time reversibility of the law of the f b m see lemma 2 4 below and the small ball estimate of molchan commun math phys 205 1999 97 111

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

on the smoothed finite element method recently liu et al proposed the smoothed finite element method by using the non mapped shape functions and then introducing the strain smoothing operator when evaluating the element stiffness in the framework of the finite element method however the theories and examples by liu et al are not sufficient for general quadrilateral elements this paper shows that the non mapped shape functions used in the smoothed finite element have disadvantages in existence linearity non negativity and patch test copyright 2008 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

allman s triangle rotational dof and partition of unity a simplification of the 1984 allman triangle one of historically most important elements with rotational dofs is presented it is found that this old element takes a typical form of the partition of unity approximation the allman s rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications the stiffness matrix of the 1984 allman triangle which is originally computed from the linear strain triangular element the 6 nodes quadratic triangle can be obtained instead in a cheaper way from that of the constant strain triangular element the constraint of the rotational terms during essential boundary treatment which remains equivocal and ambiguous is understood to be mandatory the partition of unity notion enables a straightforward extension of the allman s rotational dof to meshfree approximations in numerical examples we discuss suppression of spurious zero energy modes and patch tests standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs copyright 2006 john wiley sons ltd

two scale diffusion deformation coupling model for material deterioration involving micro crack propagation in this paper we introduce a two scale diffusion deformation coupled model that represents the aging material deterioration of two phase materials involving micro crack propagations the mathematical homogenization method is applied to relate the micro and macroscopic field variables and a weak coupling solution method is employed to solve the two way coupling phenomena between the diffusion of scalar fields and the deformation of quasi brittle solids the macroscopic mechanical behavior represented by the derived two scale two way coupled model reveals material nonlinearity due to micro scale cracking induced by the scalar field induced deformation which can be simulated by the finite cover method after verifying the fundamental validity of the proposed model and the analysis method we perform a simple numerical example to demonstrate their ability to predict aging material deterioration copyright 2010 john wiley sons ltd

random walks with infinite spatial and temporal moments the continuous time random walk of montroll and weiss has been modified by scher and lax to include a coupled spatial temporal memory we treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial and temporal moments of the memory are infinite the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated the resulting behavior includes localized diffusive wavelike and levy s stable laws for the appropriate scaled variable we show that an infinite mean waiting time can lead to long time diffusive behavior while a finite mean waiting time is not sufficient to ensure the same

critical assessment of different mass lumping schemes for higher order serendipity finite elements abstract an efficient and robust finite element based transient analysis of structures is important in many engineering applications in this context a diagonal or lumped mass matrix is an essential prerequisite in the last decades several methods to construct such a mass matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches is needed in the present article we consequently investigate established mass lumping schemes such as the row sum method and the diagonal scaling method as well as the recently proposed manifold based method furthermore the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions note that the manifold based method and the diagonal scaling method are very general approaches in the sense that they are applicable to arbitrary nodal based shape functions with arbitrary polynomial orders while guaranteeing the positive definiteness of the lumped mass matrix with respect to the row sum method the positivity is not ensured and negative or zero diagonal components might be computed consequently a detailed analysis of the influence and performance of these mass lumping schemes on the numerical results is conducted to this end several dynamic benchmark models including modal harmonic and transient analyses are selected showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal mass matrix are employed

development of contact algorithm for three dimensional numerical manifold method summary this paper customizes a contact detection and enforcing scheme to fit the three dimensional 3 d numerical manifold method nmm a hierarchical contact system is established for efficient contact detection the mathematical mesh a unique component in the nmm is utilized for global searching of possible contact blocks and elements followed by the local searching to identify primitive hierarchies all the potential contact pairs are then transformed into one of the two essential entrance modes point to plane and crossing lines modes among which real contact pairs are detected through a unified formula the penalty method is selected to enforce the contact constraints and a general contact solution procedure in the 3 d nmm is established because of the implicit framework an open close iteration is performed within each time step to determine the correct number of contact pairs among multi bodies and to achieve complete convergence of imposed contact force at corresponding position the proposed contact algorithm extensively utilizes most of the original components of the nmm namely the mathematical mesh cells and the manifold elements as well as the external components associated with contacts such as the contact body the contact facet and the contact vertex in particular the utilization of two mutually approaching mathematical cells is efficient in detecting contacting territory which makes this method particularly effective for both convex and non convex bodies the validity and accuracy of the proposed contact algorithm are verified and demonstrated through three benchmark problems copyright 2013 john wiley sons ltd

numerical manifold space of hermitian form and application to kirchhoff s thin plate problems summary for second order problems where the behavior is described by second order partial differential equations the numerical manifold method nmm has gained great success because of difficulties in the construction of the h 2 regular lagrangian partition of unity subordinate to the finite element cover however few applications of the nmm have been found to fourth order problems such as kirchhoff s thin plate problems parallel to the finite element methods this study constructs the numerical manifold space of the hermitian form to solve fourth order problems from the minimum potential principle meanwhile the mixed primal formulation and the penalized formulation fitted to the nmm for kirchhoff s thin plate problems are derived the typical examples indicate that by the proposed procedures even those earliest developed elements in the finite element history such as zienkiewicz s plate element regain their vigor copyright 2013 john wiley sons ltd

a renormalization group classification of nonstationary and or infinite second moment diffusive processes anomalous diffusion processes are often classified by their mean square displacement if the mean square displacement grows linearly in time the process is considered classical if it grows like t with 1 the process is considered subdiffusive or superdiffusive respectively processes with infinite mean square displacement are considered superdiffusive we begin by examining the ways in which power law mean square displacements can arise namely via non zero drift nonstationary increments and correlated increments subsequently we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present finally we introduce an alternative classification scheme based on renormalization groups this scheme classifies processes with stationary increments such as brownian motion and fractional brownian motion in the same groups as the mean square displacement scheme but does a better job of classifying processes with nonstationary increments and or processes with infinite second moments such as stable levy motion a numerical approach to analyzing data based on the renormalization group classification is also presented

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

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

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

incremental secant modulus iteration scheme and stress recovery for simulating cracking process in quasi brittle materials using xfem in this paper an incremental secant modulus iteration scheme using the extended generalized finite element method xfem is proposed for the simulation of cracking process in quasi brittle materials described by cohesive crack models whose softening law is composed of linear segments the leading term of the displacement asymptotic field at the tip of a cohesive crack which ensures a displacement discontinuity normal to the cohesive crack face is used as the enrichment function in the xfem the opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation a statically admissible stress recovery sar technique is extended to cohesive cracks with special treatment of non homogeneous boundary tractions the application of locally normalized co ordinates to eliminate possible ill conditioning of sar and the influence of different weight functions on sar are also studied several mode i cracking problems in quasi brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme as well as the characteristics of global responses and local fields obtained numerically by the xfem

accurate recovery based upper error bounds for the extended finite element framework this paper introduces a recovery type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method xfem the paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a fem framework the upper bound property requires the recovered solution to be equilibrated and continuous the proposed technique consists of using a recovery technique especially adapted to the xfem framework that yields equilibrium at a local level patch by patch then a postprocess based on the partition of unity concept is used to obtain continuity the result is a very accurate but only nearly statically admissible recovered stress field with small equilibrium defaults introduced by the postprocess sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults as demonstrated by the numerical tests

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

displacement energy of unit disk cotangent bundles we give an upper bound of a hamiltonian displacement energy of a unit disk cotangent bundle d m in a cotangent bundle t m when the base manifold m is an open riemannian manifold our main result is that the displacement energy is not greater than c r m where r m is the inner radius of m and c is a dimensional constant as an immediate application we study symplectic embedding problems of unit disk cotangent bundles moreover combined with results in symplectic geometry our main result shows the existence of short periodic billiard trajectories and short geodesic loops