a mesh based partition of unity method for discontinuity modeling this study explores a numerical analysis scheme the manifold method within the framework of the partition of unity method the manifold method is rooted in the discrete element method and has been used in solving discrete continuum interaction problems at its core two meshes are employed in an analysis the mathematical mesh provides the nodes for building a finite covering of the solution domain and the partition of unity functions while the physical mesh provides the domain of integration after providing a geometric interpretation and giving a construct overview the working of the method is illustrated with a number of examples

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

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

an adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses in this paper a first application of an adaptive generalized finite element method to free longitudinal vibration analysis of straight bars and trusses is presented the generalized finite element method is developed by enriching the standard finite element method space whose basis performs a partition of unity with knowledge about the differential equation being solved the enrichment functions used are dependent on the geometric and mechanical properties of the element the proposed approach converges very fast and is able to approximate the frequency related to any vibration mode the variational problem of free vibration is formulated and the main aspects of the adaptive generalized finite element method are presented and discussed the efficiency and convergence of the proposed method in vibration analysis of uniform and non uniform straight bars are checked the application of this technique in a truss is also presented the frequencies obtained by the adaptive generalized finite element method are compared with those obtained by the analytical solution the composite element method and the h version of finite element method

variational multiscale element free galerkin method for 2d burgers equation a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for 2d burgers equation with various values of re the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh recreation are involved meanwhile compared with the variational multiscale finite element method a strong assumption that is the fine scale vanishes identically over the element boundaries although non zero within the elements is not needed subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method finally a 2d burgers equation having large re is solved and the results have also been compared with the ones computed by two other methods the numerical results show that the proposed method can indeed obtain accurate numerical results for 2d burgers equation having large re which does not refer to the choice of a proper stabilization parameter

orbital hp clouds for solving schrodinger equation in quantum mechanics solving schrodinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions in addition to the highly coupled nature between wave functions this work introduces orbital and polynomial enrichment functions to the partition of unity for solution of schrodinger equation under the framework of hp clouds an intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed due to the employment of higher order basis functions a higher order stabilized conforming nodal integration is developed the proposed methods are implemented using the density functional theory for solution of schrodinger equation analysis of several single and multi electron nucleus structures demonstrates the effectiveness of the proposed method

the generalized method of moments for electromagnetic boundary integral equations the generalized method of moments gmm is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations past work on gmm for electromagnetics was confined to geometries modeled by piecewise flat tessellations and suffered from spurious internal line charges in the present article we redesign the gmm scheme and demonstrate its ability to model scattering from pec scatterers composed of mixtures of smooth and non smooth geometrical features furthermore we demonstrate that because the partition of unity provides both functional and effective geometrical continuity between patches gmm permits mixtures of local geometry descriptions and approximation function spaces with significantly more freedom than traditional moment methods

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

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

mesh independent p orthotropic enrichment using the generalized finite element method this paper is aimed at presenting a simple yet effective procedure to implement a mesh independent p orthotropic enrichment in the generalized finite element method the procedure is based on the observation that shape functions used in the gfem can be constructed from polynomials defined in any co ordinate system regardless of the underlying mesh or type of element used numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p orthotropic enrichment copyright 2002 john wiley sons ltd

derivatives of maximum entropy basis functions on the boundary theory and computations summary in this paper we obtain explicit expressions to evaluate the derivatives of maximum entropy max ent basis function on the boundary of a convex domain in the max ent formulation the basis functions are obtained by maximizing a concave functional subjected to linear constraints reproducing conditions in doing so it is found that the lagrange multipliers blow up when x and the expressions for the derivatives of the max ent basis functions in are of an indeterminate form for points on we appeal to l hopital s rule to derive expressions to determine the derivatives of the basis functions we consider the shannon entropy functional and the relative entropy functional with different choices of the prior weight function the first order derivatives of all basis functions are bounded in contrast on an irregular grid with a certain nodal spacing some of the second derivatives of the basis functions are unbounded on the boundary necessary and sufficient conditions on the priors to obtain bounded lagrange multipliers are established optimal convergence rates for fourth order problems are demonstrated for a galerkin approach with a quadratically complete partition of unity enriched max ent approximation copyright 2013 john wiley sons ltd

modified taylor reproducing formulas and h p clouds we study two different approximations of a multivariate function f by operators of the form n i 1 r f x i wi x wi x where w i is an m reproducing partition of unity and t f x i x are modified taylor polynomials of degree r expanded at x i the first approximation was introduced by xuli 2003 in the univariate case and generalized for convex domains by guessab et al 2005 the second one was introduced by duarte 1995 and proved in the univariate case in this paper we first relax the guessab s convexity assumption and we prove duarte s reproduction formula in the multivariate case then we introduce two related reproducing quasi interpolation operators in sobolev spaces a weighted error estimate and jackson s type inequalities for h p cloud function spaces are obtained last numerical examples are analyzed to show the approximative power of the method

anomalous behavior of bilinear quadrilateral finite elements for modeling cohesive cracks with xfem gfem summary the performance of partition of unity based methods such as the generalized finite element method or the extended finite element method is studied for the simulation of cohesive cracking the focus of investigation is on the performance of bilinear quadrilateral finite elements using these methods in particular the approximation of the displacement jump field representing cohesive cracks by extended finite element method generalized finite element method and its effect on the overall behavior at element and structural level is investigated a single element test is performed with two different integration schemes namely the newton cotes lobatto and the gauss integration schemes for the cracked interface contribution it was found that cohesive crack segments subjected to a nonuniform opening in unstructured meshes or an inclined crack in a structured finite element mesh result in an unrealistic crack opening the reasons for such behavior and its effect on the response at element level are discussed furthermore a mesh refinement study is performed to analyze the overall response of a cohesively cracked body in a finite element analysis copyright 2013 john wiley sons ltd

a partition of unity based fe meshfree quad4 element with radial polynomial basis functions for static analyses abstract recently a partition of unity pu based finite element called fe meshfree quad4 element has been published that element employed polynomial basis functions for the local approximation la in the present paper a new fe meshfree quad4 element employing hybrid radial polynomial basis functions for the la is proposed an advantage of radial polynomial basis is the freedom from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions another advantage is the improved accuracy of finite element solution the new element has been applied to several linear and nonlinear test problems the results demonstrate that the new element gives much better performance as compared to the previous fe meshfree quad4 element with pure polynomial basis even with a lower order basis viz with just three or four polynomial terms the present element is capable of giving more accurate solution than the previous element which employs a six term polynomial basis the new element also exhibits a much higher degree of tolerance to mesh distortion than the known quadratic elements moreover the linear dependence problem otherwise associated with many of the partition of unity pu based elements is completely eliminated from the present element

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

a generalized smoothed particle hydrodynamics method for nonlinear dynamic problems abstract a generalized smoothed particle hydrodynamics sph method was developed for the numerical simulation of nonlinear dynamic problems the performance of the proposed method was demonstrated by solving burgers equations over a wide range of the viscosity parameters in one and two space dimensions it was shown that the present results agree very well with other existing solutions as compared to other meshless methods this algorithm is quite straightforward and simple and can be coded easily

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

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

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

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

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

moving particle finite element method with superconvergence nodal integration formulation and applications abstract a new approach of moving particle finite element method has been developed which is capable to gain a global superconvergence through solving particle kernel function to satisfy high order consistencies the nodal based moving particle finite element method inconjunction with the proposed superconvergence approach provides an optimized combination in numerical accuracy and computation efficiency the three dimensional engineering scale simulations demonstrate that this scheme is robust and capable to handle high speed penetration and dynamic crack propagation with intersonic and supersonic speeds

modeling quasi static crack growth with the extended finite element method part i computer implementation abstract the extended finite element method x fem is a numerical method for modeling strong displacement as well as weak strain discontinuities within a standard finite element framework in the x fem special functions are added to the finite element approximation using the framework of partition of unity for crack modeling in isotropic linear elasticity a discontinuous function and the two dimensional asymptotic crack tip displacement fields are used to account for the crack this enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces and hence quasi static crack propagation simulations can be carried out without remeshing in this paper we discuss some of the key issues in the x fem and describe its implementation within a general purpose finite element code the finite element program dynaflow is considered in this study and the implementation for modeling 2 d cracks in isotropic and bimaterial media is described in particular the array allocation for enriched degrees of freedom use of geometric based queries for carrying out nodal enrichment and mesh partitioning and the assembly procedure for the discrete equations are presented we place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework

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

shell finite cell method a high order fictitious domain approach for thin walled structures abstract this article presents a generalization of the recently proposed finite cell method to thin walled structures this approach uses a combination of well known fictitious domain methods with high order hierarchical ansatz spaces known from the p version of the finite element method whereas the original concept embeds a three dimensional structure in a simple domain being meshed into a grid of cube shaped cells the extension presented in this paper applies the fictitious domain approach to a two dimensional master domain defined in the parameter plane of the geometric model implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach it is also remarked that the present approach can easily be carried over to isogeometric analysis opening an attractive possibility to simulate trimmed nurbs surfaces

a 3d interface enriched generalized finite element method for weakly discontinuous problems with complex internal geometries an interface enriched generalized finite element method gfem is introduced for 3d problems with discontinuous gradient fields the proposed method differs from conventional gfem by assigning the generalized degrees of freedom to the interface nodes i e nodes generated along the interface when creating integration subdomains instead of the nodes of the original mesh a linear combination of the lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field this approach provides a great flexibility for evaluating the enrichment functions including for cases where elements are intersected with multiple interfaces we show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional gfem the potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber reinforced composites and in actively cooled microvascular materials

two dimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm abstract we propose a methodology for the two dimensional simulation and optimization of material distribution of functionally graded materials for thermomechanical processes the proposed approach focuses on metal ceramic functionally graded materials which offer great promise in applications where the operating conditions are severe the two dimensional quasi static heat conduction and thermoelasticity problems are analyzed using the element free galerkin method the spatial distribution of ceramic volume fraction is obtained by piecewise bicubic interpolation of volume fractions defined at a finite number of grid points the effective material properties at a point in the domain are estimated from the local volume fractions of the material constituents using the mori tanaka and self consistent homogenization schemes the volume fraction distribution is optimized using a real coded genetic algorithm the element free galerkin code is validated by comparing the results with an exact thermoelasticity solution for a simply supported plate subsequently we use the genetic algorithm and element free galerkin analyses to optimize the material composition for two model problems in the first problem we minimize the peak residual stress when the functionally graded component is cooled from a high fabrication temperature the goal of the second model problem is to minimize the mass of a graded component with constraints on the peak effective stress and maximum temperature experienced by the metal

a simple first order well conditioned and optimally convergent generalized extended fem for two and three dimensional linear elastic fracture mechanics abstract this paper proposes a first order generalized extended finite element method g xfem for 2 d and 3 d linear elastic fracture mechanics problems the conditioning of the method is of the same order as the standard fem and it is robust with respect to the position of the mesh relative to 2 d or 3 d fractures the method achieves an optimal rate of convergence in energy norm even when the solution of the problem in the neighborhood of the fracture front is not contained in the spaces spanned by the adopted singular enrichments control of conditioning associated with heaviside enrichment functions is achieved by shifting them by their nodal values combined with a diagonal pre conditioner in the case of singular enrichment functions a simple extension of the enrichment shifting concept denoted as discontinuous shifting is adopted these enrichment modifications are space preserving the solution space spanned by the modified enrichments is the same as the space spanned by the unmodified ones this is often not the case for conditioning control strategies proposed in the literature the computational cost of these enrichment modifications is negligible and their implementation in existing g xfem software is straightforward the optimal convergence well conditioning and robustness of the method are numerically illustrated with the aid of representative 2 d and 3 d problems including the case of a non planar fracture with a curved fracture front

a two scale generalized finite element approach for modeling localized thermoplasticity summary predicting localized nonlinear thermoplastic behavior and residual stresses and deformations in structures subjected to intense heating is a prevalent challenge in a range of modern engineering applications the authors present a generalized finite element method targeted at this class of problems involving the solution of intrinsically parallelizable local boundary value problems to capture localized time dependent thermo elasto plastic behavior which is embedded in the coarse structural scale approximation via enrichment functions the method accommodates approximation spaces that evolve in between time or load steps while maintaining a fixed global mesh which avoids the need to map solutions and state variables on changing meshes typical of traditional adaptive approaches representative three dimensional examples exhibiting localized transient nonlinear thermal and thermomechanical effects are presented to demonstrate the advantages of the method with respect to available approaches especially in terms of its flexibility and potential for realistic future applications in this area parallelism of the approach is also discussed copyright 2016 john wiley sons ltd

generalized finite element analysis of three dimensional heat transfer problems exhibiting sharp thermal gradients abstract in this paper heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods the effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated convergence studies show that low order linear and quadratic elements require strongly refined meshes for acceptable accuracy we propose a generalized fem with global local enrichments for the class of problems investigated in the paper in this procedure a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized fem with solutions of local boundary value problems the local problems are defined using the same procedure as in the global local fem where boundary conditions are provided by a coarse scale global solution coarse uniform global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence the use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem the two way information transfer provided by the proposed generalized fem is appealing to several classes of problems especially those involving multiple spatial scales the proposed methodology brings the benefits of generalized fem to problems where limited or no information about the solution is known a priori

effects of element distortions on the performance of isoparametric elements we discuss the effects of element distortions on the performance of displacement based isoparametric quadrilateral finite elements suitable example problems for both the lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions the effects of angular and curved edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields we then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution the conclusion is that the lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use

moving particle finite element method with global smoothness we describe a new version of the moving particle finite element method mpfem that provides solutions within a c 0 finite element framework the finite elements determine the weighting for the moving partition of unity a concept of general shape function is proposed which extends regular finite element shape functions to a larger domain these are combined with shepard functions to obtain a smooth approximation the moving particle finite element method combines desirable features of finite element and meshfree methods the proposed approach in fact can be interpreted as a moving partition of unity finite element method or moving kernel finite element method this method possesses the robustness and efficiency of the c 0 finite element method while providing at least c 1 continuity

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 discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid solid problems abstract an evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub critical angle the sub scale nature of such a wave makes it difficult to be captured computationally in this paper the discontinuous enrichment method dem developed in c farhat i harari l p franca the discontinuous enrichment method comput methods appl mech engrg 190 2001 6455 6479 c farhat i harari u hetmaniuk a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime comput methods appl mech engrg 192 2003 1389 1419 c farhat i harari u hetmaniuk the discontinuous enrichment method for multiscale analysis comput methods appl mech engrg 192 2003 3195 3210 c farhat r tezaur p weidemann goiran higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems int j numer methods engrg 61 2004 1938 1956 c farhat p wiedemann goiran r tezaur a discontinuous galerkin method with plane waves and lagrange multipliers for the solution of short wave exterior helmholtz problems on unstructured meshes new computational methods for wave propagation wave motion 39 4 2004 307 317 l zhang r tezaur c farhat the discontinuous enrichment method for elastic wave propagation in the medium frequency regime int j numer methods engrg 66 2006 2086 2114 r tezaur c farhat three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems int j numer methods engrg 66 2006 796 815 is extended to a class of evanescent wave problems new dem elements are constructed by enriching polynomial approximations with free space evanescent solutions in order to achieve high accuracy for problems with fluid fluid or fluid solid interfaces on which evanescent waves may occur the new dem elements are applied to the solution of various two dimensional model problems with evanescent waves their performance is observed to be better than that of the basic helmholtz type dem elements and superior to that of the classical higher order polynomial finite element method

a new formulation for imposing dirichlet boundary conditions on non matching meshes generating matching meshes for problems with complex boundaries is often an intricate process and the use of non matching meshes appears as an appealing solution yet enforcing boundary conditions on non matching meshes is not a straightforward process especially when prescribing those of dirichlet type by combining a type of generalized finite element method gfem with the lagrange multiplier method a new technique for the treatment of essential boundary conditions on non matching meshes is introduced in this manuscript the new formulation yields a symmetric stiffness matrix and is straightforward to implement as a result the methodology makes possible the analysis of problems with the use of simple structured meshes irrespective of the problem domain boundary through the solution of linear elastic problems we show that the optimal rate of convergence is preserved for piecewise linear finite elements yet the formulation is general and thus it can be extended to other elliptic boundary value problems

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

generalized finite element enrichment functions for discontinuous gradient fields a general gfem xfem formulation is presented to solve two dimensional problems characterized by c0 continuity with gradient jumps along discrete lines such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media the new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks we show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines the use of the proposed enrichments in both linear and quadratic approximations is investigated as well as their combination with interface enrichment functions available in the literature through a detailed convergence study we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems in the linear case the effectiveness of correction functions proposed in the literature is also investigated copyright 2009 john wiley sons ltd

a partition of unity based fe meshfree quad4 element for geometric non linear analysis the recently published fe meshfree quad4 element is extended to geometrical non linear analysis the shape functions for this element are obtained by combining meshfree and finite element shape functions the concept of partition of unity pu is employed for the purpose the new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh distortion tolerant compatibility properties from the finite element fe shape functions updated lagrangian formulation is adopted for the non linear solution several numerical example problems are solved and the performance of the element is compared with that of the well known q4 qm6 and q8 elements the results show that for regular meshes the performance of the element is comparable to that of qm6 and q8 elements and superior to that of q4 element for distorted meshes the present element has better mesh distortion tolerance than q4 qm6 and q8 elements copyright 2009 john wiley sons ltd

a unified enrichment approach addressing blending and conditioning issues in enriched finite elements abstract we present a combination of techniques to improve the convergence and conditioning properties of partition of unity pu enriched finite element methods by applying these techniques to different types of enrichment functions namely polynomial discontinuous and singular higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements

a fully smoothed xfem for analysis of axisymmetric problems with weak discontinuities in this paper we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities the salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique this is accomplished by combining the gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique which is used to transform the domain integral into boundary integral the proposed technique completely eliminates the need for isoparametric mapping and the computing of jacobian matrix even for the mass matrix when employed over the enriched elements the proposed technique does not require sub triangulation for the purpose of numerical integration the accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities it can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion

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

on the realization of the wolfe conditions in reduced quasi newton methods for equality constrained optimization this paper describes a reduced quasi newton method for solving equality constrained optimization problems a major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced hessian of the lagrangian a new approach is proposed in this paper the idea is to search for the next iterate along a piecewise linear path the path is designed so that some generalized wolfe conditions can be satisfied these conditions allow the algorithm to sustain the positive definiteness of the matrices from iteration to iteration by a mechanism that has turned out to be efficient in unconstrained optimization

the meshless standard and hypersingular boundary node methods applications to error estimation and adaptivity in three dimensional problems the standard singular boundary node method bnm and the novel hypersingular boundary node method hbnm are employed for the usual and adaptive solutions of three dimensional potential and elasticity problems these methods couple boundary integral equations with moving least squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter the hypersingular residuals developed for error estimation in the mesh based collocation boundary element method bem and symmetric galerkin bem by paulino et al are extended to the meshless bnm setting a simple a posteriori error estimation and an effective adaptive refinement procedure are presented the implementation of all the techniques involved in this work are discussed which includes aspects regarding parallel implementation of the bnm and hbnm codes several numerical examples are given and discussed in detail conclusions are inferred and relevant extensions of the methodology introduced in this work are provided copyright 2001 john wiley sons ltd

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

a stable generalized extended p hierarchical fem for three dimensional linear elastic fracture mechanics abstract in this paper the quadratic stable generalized finite element method sgfem proposed in sanchez rivadeneira and duarte 2019 is extended to 3 d fracture problems with non planar crack surfaces and curved crack fronts this sgfem is based on p hierarchical fem enrichments and its approximation space is the same as in its gfem counterpart singular enrichments are modified using a discontinuous finite element interpolant for conditioning control this leads with the proper choice of enrichments to stiffness matrices with a scaled condition number of o h 2 the same order as in the standard fem the robustness of the method with respect to the position of a crack relative to the mesh is demonstrated a scalar implementation of singular enrichments that can exactly reproduce the first term of the mode i ii and iii of the asymptotic expansion of the elasticity solution is presented improved approximations of signed distance functions based on integration sub elements are also presented convergence studies of a fully 3 d fracture problem with a non planar crack surface and a curved crack front are presented they show that the convergence rate of the proposed method on a sequence of uniform meshes is three times higher than quadratic quarter point finite elements

mesh independent modelling of cracks by using higher order shape functions a mesh independent crack modelling approach based on displacement approximation with higher order shape functions is proposed the heaviside step function based local enrichment method known as extended finite element method is modified by replacing the step function with a higher order shape functions approximation polynomial b spline approximation functions are used in the present paper an advantage of the proposed method is that its implementation only involves integration of the products of original shape functions and their derivatives and does not require modification of the integration domains a volume integral based expression is proposed to calculate the effective surface area of the crack modelled by using an approximate step function it is shown to give the actual crack surface area in the limit of the approximate step function approaching the heaviside function the convergence and accuracy of the method is illustrated in examples of transverse and oblique with respect to loading direction crack problems in rectangular plates uniaxial tension of a unidirectional composite with an open hole is considered hoop stress relaxation due to longitudinal splitting is successfully modelled by the method proposed and compared to direct modelling by using ansys software published in 2002 by john wiley sons ltd

a hierarchical finite element method based on the partition of unity in this paper we consider the application of hierarchical functions to base approximations which are a partition of unity the particular hierarchical functions used are added to base finite element interpolations which for c0 approximations are a particular case of the partition of unity we also show how the functions may be constructed to preserve the interpolation property of the base finite element functions an application to linear elasticity is used to illustrate the properties and stability of the approximation

a new hybrid boundary node method based on taylor expansion and the shepard interpolation method summary a novel meshless method based on the shepard and taylor interpolation method stim and the hybrid boundary node method hbnm is proposed based on the shepard interpolation method and taylor expansion the stim is developed to construct the shape function of the hbnm in the stim the shepard shape function is used as the basic function which is the zero level shape function and the high power basic functions are constructed through taylor expansion four advantages of the stim are the interpolation property the arbitrarily high order consistency the absence of inversion for the whole process of shape function construction and the low computational expense these properties are desirable in the implementation of meshless methods by combining the stim and the hbnm a much more effective meshless method is proposed to solve the elasticity problems compared with the traditional hbnm the stim can improve accuracy because of the use of high power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process numerical examples are given to demonstrate the accuracy and efficiency of the proposed method copyright 2015 john wiley sons ltd

modelling and computing rn sn policies for inventory systems with non stationary stochastic demand abstract this paper addresses the single item non stationary stochastic demand inventory control problem under the non stationary r s policy in non stationary r s policies two sets of control parameters the review intervals which are not necessarily equal and the order up to levels for replenishment periods are fixed at the beginning of the planning horizon to minimize the expected total cost it is assumed that the total cost is comprised of fixed ordering costs and proportional direct item inventory holding and shortage costs with the common assumption that the actual demand per period is a normally distributed random variable about some forecast value a certainty equivalent mixed integer linear programming model is developed for computing policy parameters the model is obtained by means of a piecewise linear approximation to the non linear terms in the cost function numerical examples are provided

investigation of linear dependence problem of three dimensional partition of unity based finite element methods abstract a known problem of partition of unity based generalized finite element methods is the linear dependence of the approximation space which leads to singular stiffness matrix up to now the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available in our previous paper prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes comput methods appl mech engrg 200 2011 665 674 the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two dimensional partition of unity based generalized finite element methods this paper extends the previous work to three dimensional cases the linear dependence problem is first investigated at an element level and then extended to the whole mesh derivation of general formulations in a three dimensional setting is undoubtedly more challenging than a two dimensional setting because of the complicated element topology the principle of the increase of rank deficiency is once more applied the methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three dimensional cases this work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity based finite element methods

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 john wiley sons ltd

robust and direct evaluation of j2 in linear elastic fracture mechanics with the x fem numerical simulation of fracture mechanics is of major importance in the design of structures such as planes or dams for such complex structures robustness and accuracy of the simulations are necessary the finite element method is classically considered for the simulation of such problems however the crack must be explicitly meshed in order to incorporate the discontinuity of the displacement field across crack faces this process has to be handled at each step of propagation the main drawback of this approach is the computational and human cost of remeshing the x fem method was proposed as a response to this issue 1 2 it uses the concept of partition of unity 3 in two ways first to take into account the displacement jump across crack faces far away from the crack tip and also to enrich the approximation close to the tip with some analytical asymptotic fields another important issue in fracture mechanics is the prediction of crack growth direction several criteria have been proposed one can cite for example the maximum hoop stress criterion 4 the maximum tangential stress criterion or the symmetry principle 5 the maximum energy release rate criterion 6 also called vectorial j integral criterion 7 is an alternative which was initially proposed by hellen and blackburn 7 the main difference in evaluating this criterion with respect to the classical j integral rely on the difficulties in the evaluation of the second component j2 of the energy release rate vector j it is well known that j1 which is basically the j integral is path independent concerning j2 herrmann and herrmann 8 have shown that it is path independent only in a modified manner i e by adding an extra contribution on the crack faces however this new contribution requires an accurate determination of the mechanical fields near the crack tip it is well known that the quality of these fields near a singularity is not ensured in the finite element context this is why some authors proposed specific strategies to avoid the integration near the tip for an accurate numerical evaluation of j2 9 10 for enriched finite