orthogonalized generalized iso geometric analysis ogiga and its applications to problems of fracture mechanics abstract ill conditioning of system matrices is one of the drawbacks of generalized extended isogeometric analysis giga xiga which results in increase in computational cost in the case of iterative solvers or erroneous results in the case of direct solvers linear dependency between the basis functions which can be measured in terms of smallest angle among all the bases is one of the attributes of ill conditioning in xiga in the present study a simplified method to find the smallest angle among all the bases is introduced also to reduce the problem of ill conditioning orthogonalized generalized isogeometric analysis ogiga is proposed in which the standard and the corresponding enriched basis functions are orthogonalized in order to study the performance of present method various fracture mechanics problems like 2 d homogenous cracks bi material interface cracks inclined cracks and 3 d crack problems are chosen the effect of crack position and the radius of enrichment zone on accuracy scaled condition number and smallest angle among all the bases is studied the results obtained from the proposed ogiga method are compared with results obtained from standard xiga and xiga with enrichment function orthogonalization oe xiga agathos et al 2019 in all the cases considered the conditioning of system matrices is significantly improved by employing present methodology and it can be easily extended to wide variety of problems

inexact schwarz algebraic multigrid preconditioners for crack problems modeled by extended finite element methods summary traditional algebraic multigrid amg preconditioners are not well suited for crack problems modeled by extended finite element methods xfem this is mainly because of the unique xfem formulations which embed discontinuous fields in the linear system by addition of special degrees of freedom these degrees of freedom are not properly handled by the amg coarsening process and lead to slow convergence in this paper we proposed a simple domain decomposition approach that retains the amg advantages on well behaved domains by avoiding the coarsening of enriched degrees of freedom the idea was to employ a multiplicative schwarz preconditioner where the physical domain was partitioned into healthy or unfractured and cracked subdomains first the healthy subdomain containing only standard degrees of freedom was solved approximately by one amg v cycle followed by concurrent direct solves of cracked subdomains this strategy alleviated the need to redesign special amg coarsening strategies that can handle xfem discretizations numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force amg preconditioner applied to the linear system copyright 2011 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

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

analysis and computation of the pressure field of preatherosclerotic transport in the artery wall prerequisite to atherogenesis is low density lipoprotein ldl cholesterol transport from the blood into the artery wall across its endothelial cell ec lining whereas most ec junctions are tight enough to preclude ldl passage some approx 1 2000 6000 rare ecs have temporarily widened junctions typically approx 10 5 of the problem s radial scale that allow passage of large molecular aggregates like ldl in a purely diffusive transport model laplace s equation describes the steady tracer profile throughout the two major subendothelial wall layers connected by matching conditions in a diffusive and transmural pressure driven advective transport model laplace s equation describes the steady pressure distribution in the wall whose gradient is proportional to the advecting velocity both problems yield the same boundary value problem that has mixed dirichlet robin boundary conditions we construct a linear system of integral equations that approximates the series expansion coefficients o

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

a stable generalized finite element method sgfem of degree two for interface problems abstract generalized or extended finite element methods gfem xfem of degree 1 for interface problems have been reported in the literature they i yield optimal order of convergence i e o h h is a discretization parameter ii are stable in a sense that conditioning is not worse than that of the standard fem and iii are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements however higher order gfem xfem with the properties i iii have not been successfully addressed yet various enrichment schemes for gfem xfem based on d or d p k d is a distance function or the absolute value of level set function and p k is the polynomial basis of degree k have been reported recently to obtain higher order convergence but they are not stable or robust in general in fact they even may not yield the optimal orders of convergence in this paper we propose a stable gfem xfem of degree 2 sgfem2 for the interface problems where we used the enrichment scheme based on d 1 x y instead of d or d 1 x y x 2 x y y 2 in the literature we prove that the sgfem2 yields the optimal order of convergence i e o h 2 for the interface problems with curved smooth interfaces a local principal component analysis technique has been proposed which ensures that the sgfem2 is stable and robust numerical experiments for straight and curved interfaces have been presented to illuminate these properties

representation of singular fields without asymptotic enrichment in the extended finite element method summary in this paper we replace the asymptotic enrichments around the crack tip in the extended finite element method xfem with the semi analytical solution obtained by the scaled boundary finite element method sbfem the proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the xfem to model cracks in homogeneous and or heterogeneous materials without a priori knowledge of the asymptotic solutions a heaviside enrichment is used to represent the jump across the discontinuity surface we call the method as the extended sbfem numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number a simple code is annexed to compute the terms in the stiffness matrix which can easily be integrated in any existing fem xfem code copyright 2013 john wiley sons ltd

local adaptive q enrichments and generalized finite elements for transient heat diffusion problems abstract a residual a posteriori error estimate is used in conjunction with a q adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the generalized finite element method gfem the error estimate allows to assess local as well as global errors of the gfem solutions and provides a tool to adaptively enrich the approximation space with the proposed adaptive algorithm the total number of degrees of freedom dofs is significantly reduced in comparison to uniform enrichment for a comparable accuracy a reduction of up to 80 in dofs is achieved moreover the error estimate captures the deterioration of the conditioning of the system matrix typically encountered in enrichment based methods and enables the possibility of keeping the condition number within an acceptable limit

stable generalized finite element method and associated iterative schemes application to interface problems abstract the generalized finite element method gfem is an extension of the finite element method fem where the standard finite element space is augmented with a space of non polynomial functions called the enrichment space the functions in the enrichment space mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully applied to a wide range of problems however it often suffers from bad conditioning i e its conditioning may not be robust with respect to the mesh and in fact the conditioning could be much worse than that of the standard fem in this paper we present a numerical study that shows that if the angle between the finite element space and the enrichment space is bounded away from 0 uniformly with respect to the mesh then the gfem is stable i e the conditioning of gfem is not worse than that of the standard fem a gfem with this property is called a stable gfem sgfem the last part of the paper is devoted to the derivation of a robust iterative solver exploiting this angle condition it is shown that the required wall clock time is greatly reduced compared to popular gfems used in the literature

general iterative methods for a one parameter nonexpansive semigroup in hilbert space abstract let h be a hilbert space and f a fixed contractive mapping with coefficient 0 1 a a strongly positive linear bounded operator with coefficient 0 consider two iterative methods that generate the sequences x n y n by the algorithm respectively i x n i n a 1 t n 0 t n t s x n d s n f x n ii y n 1 i n a 1 t n 0 t n t s y n d s n f y n where n and t n are two sequences satisfying certain conditions and t s s 0 is a one parameter nonexpansive semigroup on h it is proved that the sequences x n y n generated by the iterative method i and ii respectively converge strongly to a common fixed point x f which solves the variational inequality a f x x z 0 z f

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

strongly stable generalized finite element method application to interface problems abstract in this paper we present the theoretical justification for the stable generalized finite element method sgfem when applied to smooth interface problems we prove that a generalized finite element method gfem is stable i e its conditioning is a not worse than that of the finite element method and b robust with respect to the mesh if the enrichment space of the gfem satisfies two axioms we provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied we show that the enrichment space of the gfem used to address the interface problems in 2d satisfies the sufficient conditions and thus the two axioms yielding an sgfem the idea of a strongly stable gfem associated with one of the two axioms has been introduced strong stability of the gfem is important for designing efficient iterative methods to solve the underlying linear system a proof of the optimal convergence of the gfem for the interface problem has also been derived in this paper the work in this paper is the continuation of kergrene et al 2016 where the stability of the gfem when applied to interface problems was established through numerical experiments the numerical results in kergrene et al 2016 indicated that the gfem is indeed strongly stable

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

robustness in stable generalized finite element methods sgfem applied to poisson problems with crack singularities abstract in this paper we study the performance of the generalized finite element method gfem applied to the poisson problem with crack singularities recently a gfem with modified heaviside enrichments was proposed in gupta et al 2013 to approximate the solution of a 2d elasticity problem with a crack it was shown that the gfem is indeed a stable gfem sgfem i e it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method fem however the robustness of the conditioning of the gfem with respect to the position of the mesh relative to the crack was not addressed in gupta et al 2013 in this paper we observed that the conditioning of the gfem with the enrichments used in gupta et al 2013 is not robust when applied to approximate the solution of the poisson problem moreover the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces we proposed using additional singular enrichments at the nodes close to the crack near the crack tip we proved that the gfem with enrichments proposed in this paper yields optimal order of convergence irrespective of the position of the mesh moreover with a local orthogonalization procedure we have shown through numerical experiments that the conditioning of this gfem is not worse than that of the standard fem and the conditioning is robust with respect to the position of the mesh thus the gfem with the enrichments suggested in this paper is indeed an sgfem when applied to a poisson problem with the crack singularities

stable generalized iso geometric analysis sgiga for problems with discontinuities and singularities abstract numerical analysis of physical mathematical problems based on generalized extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks the presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed in order to alleviate these drawbacks three different stable generalized isogeometric analysis sgiga methods are proposed in this paper in sgiga the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function problems with weak and strong discontinuities singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed sgiga methods from the results it is observed that sgiga yields optimal convergence rates as well as better conditioning of system matrices the results obtained from the proposed sgiga methods are also compared with the results from the established methods stable generalized finite element method sgfem and extended isogeometric analysis xiga to study the relative performances with respect to accuracy and conditioning

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

harmonic solutions of a mixed boundary problem arising in the modeling of macromolecular transport into vessel walls the earliest events leading to atherosclerosis involve the transport of lpw density lipooprotein ldl cholesterol from the blood across endothelial cells that line the artery wall laplace s equation describes the steady state diffusion profile of a tracer through the vessel wall this gives rise to a boundary value problem with mixed dirichlet and robin conditions we construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution we prove the existence of the solution to this problem analytically by using gershgorin s theorem on the location of the eigenvalues of the corresponding matrix we give a uniqueness proof using miranda s theorem c miranda p d e of elliptic type springer verlag berlin 1970 the analytical construction method forms the basis for a numerical calculation algorithm we apply our results to the transport problem above and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time

a robust preconditioning technique for the extended finite element method the extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions but the resulting stiffness matrices can become ill conditioned in that case iterative solvers need a large number of iterations to obtain an acceptable solution in this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments a domain decomposition is employed and the algorithm is very well suited for parallel computations the method was tested in numerical experiments to show its effectiveness the experiments have been conducted for structures containing cracks and material interfaces we show that the corresponding enrichments can result in arbitrarily ill conditioned matrices the method proposed here however provides well conditioned matrices and can be applied to any sort of enrichment the complexity of this approach and its relation to the domain decomposition is discussed computation times have been measured for a structure containing multiple cracks for this structure the computation times could be decreased by a factor of 2

improved implementation and robustness study of the x fem for stress analysis around cracks numerical crack propagation schemes were augmented in an elegant manner by the x fem method the use of special tip enrichment functions as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh which is of industrial interest especially when a numerical model for crack propagation is desired this paper improves the implementation of the x fem method for stress analysis around cracks in three ways first the enrichment strategy is revisited the conventional approach uses a topological enrichment only the elements touching the front are enriched we suggest a geometrical enrichment in which a given domain size is enriched the improvements obtained with this enrichment are discussed second the conditioning of the x fem both for topological and geometrical enrichments is studied a preconditioner is introduced so that off the shelf iterative solver packages can be used and perform as well on x fem matrices as on standard fem matrices the preconditioner uses a local nodal cholesky based decomposition third the numerical integration scheme to build the x fem stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme a 2d benchmark problem is designed to show the improvements and the robustness

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

strongly stable generalized finite element method ssgfem for a non smooth interface problem ii a simplified algorithm abstract the solution of interface problems with non smooth interfaces having corners connected by straight lines exhibit weak singularity along the interface and may have singularities at the corners in an earlier paper zhang et al 2019 a strongly stable generalized finite element method ssgfem was proposed to approximate solution of such problems the method required two separate enrichment functions at o h 1 number of nodes h is the mesh parameter and a local orthogonalization technique requiring additional operations was used to ensure its stability and robustness in this paper we present a simplified ssgfem that does not require two enrichment functions at any of the nodes as a result we obtain an optimally converging gfem that is stable i e its scaled condition number is of the same order that of a standard fem and robust i e the conditioning is independent of the position of meshes relative to the interface these properties are illuminated by well designed experiments the simplified ssgfem does not require lot and requires less degrees of freedom and is easier to implement in contrast to the ssgfem proposed in zhang et al 2019 we also proved the optimal convergence for the simplified ssgfem in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner

a coupled eulerian lagrangian extended finite element formulation for simulating large deformations in hyperelastic media with moving free boundaries we present a coupled eulerian lagrangian cel formulation aimed at modeling the moving interface of hyperelastic materials undergoing large to extreme deformations this formulation is based on an eulerian description of kinematics of deformable bodies together with an updated lagrangian formulation for the transport of the deformation gradient tensor the extended finite element method xfem is used to discretize the mechanical equilibrium and deformation gradient transport equations in a two phase domain a mixed interpolation scheme biquadratic for the velocity and bilinear for the deformation gradient is adopted to improve the accuracy of the numerical formulation the interface describing the deformed shape of the body is represented by the level set function and is evolved using the grid based particle method the performance of the scheme is explored in two dimensions in the compressible regime for an adequate spatial and temporal discretization our numerical results are in good agreement with theory and with numerical results from the traditional lagrangian formulation in abaqus the advantage of the proposed formulation is that material motion is not coupled with that of the mesh this eliminates the issues of mesh distortion and the need for remeshing associated with lagrangian formulations when bodies undergo very large distortions it is therefore well adapted to describe the motion of complex fluids and soft matter whose physical properties are intermediate between conventional liquids and solids

an analytic series method for laplacian problems with mixed boundary conditions mixed boundary value problems are characterised by a combination of dirichlet and neumann conditions along at least one boundary historically only a very small subset of these problems could be solved using analytic series methods analytic is taken here to mean a series whose terms are analytic in the complex plane in the past series solutions were obtained by using an appropriate choice of axes or a co ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure dirichlet or neumann form in this paper i will consider the more general problem where the mixed boundary conditions cannot be resolved by a co ordinate transformation that is a dirichlet condition applies on part of the boundary and a neumann condition applies along the remaining section i will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa in addition i will extend this technique to the general mixed boundary value problem defined on an arbitrary boundary where the boundary is not parallel to the abscissa i will demonstrate the efficacy of the method on a well known seepage problem

an algebraic multigrid approach to solve extended finite element method based fracture problems summary this article proposes an algebraic multigrid amg approach to solve linear systems arising from applications where strong discontinuities are modeled by the extended finite element method the application of amg methods promises optimal scalability for solving large linear systems however the straightforward or black box use of existing amg techniques for extended finite element method problems is often problematic in this paper we highlight the reasons for this behavior and propose a relatively simple adaptation that allows one to leverage existing amg software mostly unchanged numerical tests demonstrate that optimal iterative convergence rates can be attained that are comparable with amg convergence rates associated with linear systems for standard finite element approximations without discontinuities published 2012 this article is a us government work and is in the public domain in the usa

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

an extended finite element method with algebraic constraints xfem ac for problems with weak discontinuities abstract we present a new extended finite element method with algebraic constraints xfem ac for recovering weakly discontinuous solutions across internal element interfaces if necessary cut elements are further partitioned by a local secondary cut into body fitting subelements each resulting subelement contributes an enrichment of the parent element the enriched solutions are then tied using algebraic constraints which enforce c 0 continuity across both cuts these constraints impose equivalence of the enriched and body fitted finite element solutions and are the key differentiating feature of the xfem ac in so doing a stable mixed formulation is obtained without having to explicitly construct a compatible lagrange multiplier space and prove a formal inf sup condition likewise convergence of the xfem ac solution follows from its equivalence to the interface fitted finite element solution this relationship is further exploited to improve the numerical solution of the resulting xfem ac linear system examples are shown demonstrating the new approach for both steady state and transient diffusion problems

convergence of iterates of nonexpansive mappings and orbits of nonexpansive semigroups abstract let x be a uniformly convex banach space and let c be a bounded closed and convex subset of x assume that c has nonempty interior and is locally uniformly rotund let t be a nonexpansive self mapping of c if t has no fixed point in the interior of c then there exists a unique point x on the boundary of c such that each sequence of iterates of t converges in norm to x we also establish an analogous result for nonexpansive semigroups

an extended finite element method for a diffuse interface model this work presents an extended finite element method xfem for a diffuse interface model which describes interfacial phenomena of multi phase flow the diffuse interface has a non zero thickness over which the phase field variable changes continuously the diffuse interface thickness is typically very small compared to the observed domain size resulting in a high gradient solution in this work the finite element approximation of the phase field variable is locally enriched with a tangent hyperbolic function which characterizes a high gradient solution of the diffuse interface model we study a one dimensional advection and two diffusion problems and demonstrate the remarkable improvement of the solution by the local enrichment

a well conditioned and optimally convergent xfem for 3d linear elastic fracture summary a variation of the extended finite element method for three dimensional fracture mechanics is proposed it utilizes a novel form of enrichment and point wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two dimensional and three dimensional crack problems a bespoke benchmark problem is introduced to determine the method s accuracy in the general three dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40 for moderately refined meshes and topological enrichment moreover when a fixed enrichment volume is used the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method diminishing the number of iterations by almost one order of magnitude copyright 2015 john wiley sons ltd

robust and scalable domain decomposition solvers for unfitted finite element methods abstract unfitted finite element methods e g extended finite element techniques or the so called finite cell method have a great potential for large scale simulations since they avoid the generation of body fitted meshes and the use of graph partitioning techniques two main bottlenecks for problems with non trivial geometries however the linear systems that arise from these discretizations can be much more ill conditioned due to the so called small cut cell problem the state of the art approach is to rely on sparse direct methods which have quadratic complexity and are thus not well suited for large scale simulations in order to solve this situation in this work we investigate the use of domain decomposition preconditioners balancing domain decomposition by constraints for unfitted methods we observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior as a result we propose a customization of the classical bddc methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner these changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids a complete set of complex 3d numerical experiments shows the good performance of the proposed preconditioners

stable generalized finite element method sgfem for parabolic interface problems abstract we study a stable generalized finite element method sgfem for parabolic interface problems with smooth interfaces the standard finite element fe space based on a simple mesh is enriched by a one side distance function to simulate gradient jumps of exact solutions on the interfaces the mesh is quasi uniform and independent of the interfaces so that mesh matching or refinement along the interfaces is not needed the main idea to derive the sgfem is to modify the one side distance function by subtracting its fe interpolant the sgfem approximates the parabolic interface problem with an optimal convergence rate and is stable in that its scaled condition number is of the same order as the fem o h 2 h is a mesh size parameter and does not depend on the relative positions of the mesh and interfaces the proposed sgfem is a conforming method and is free from any penalty parameters or stability schemes moreover the components of the approximation functions of the sgfem are independent when extended to vector valued interface problems the optimal convergence rates of sgfem for curved interfaces are proven theoretically for the backward euler and crank nicolson formulations the convergence and conditioning are tested by numerical experiments

computation of limit and shakedown loads using a node based smoothed finite element method summary this paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node based smoothed fem in combination with a primal dual algorithm an associated primal dual form based on the von mises yield criterion is adopted the primal dual algorithm together with a newton like iteration are then used to solve this associated primal dual form to determine simultaneously both approximate upper and quasi lower bounds of the plastic collapse limit and the shakedown limit the present formulation uses only linear approximations and its implementation into finite element programs is quite simple several numerical examples are given to show the reliability accuracy and generality of the present formulation compared with other available methods copyright 2011 john wiley sons ltd

substructuring fe xfe approaches applied to three dimensional crack propagation two substructuring methods are investigated in order to allow for the use of the extended finite element method x fem within commercial finite element fe codes without need for modifying their kernel the global fe problem is decomposed into two subdomains the safe domain and the cracked domain based on the value of the level sets representing the crack the safe domain is treated by the host fe software while the cracked domain is treated by an independent xfe code the first substructuring method consists of calculating the schur matrix of a cracked super element with the xfe code the second technique introduces the finite element tearing and interconnecting method feti which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains the stiffness matrices and nodal forces are provided by the xfe and fe codes for the cracked and safe subdomains respectively the solutions obtained with these two techniques are rigorously equivalent to those computed with the stand alone xfe code first the computational efficiency of the two approaches is demonstrated second a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3d cracked geometries finally this contribution presents an application of the fe xfe feti method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure

a fourier method for initial value problems with mixed boundary conditions abstract works concerning solutions of initial problems with mixed boundary conditions are very rare in the literature and essentially depend on the geometry of some individual problems k b ranger for example considered the cooling of a finite needle and strip with prescribed cooling temperature in a medium bounded by two parallel planes perpendicular to the axis and maintained at zero temperature 1 because of the axial symmetries of these problems ranger could employ integral operators which map 2 d harmonics into the solution of the reduced wave equation and into that of the heat equation thereby in this work a systematic method for solving initial plane problems with mixed boundary conditions is proposed it can be considered as a natural generalization of the fourier method that deals with the uniform initial problems on the one hand and of that started by ju i cersi 2 dealing with steady mixed boundary problems on the other by gradual modifications the latter method has become of wide application to solutions of problems in several branches of mathematical physics this has recently been expounded 3 we shall start with presenting the method by means of a concrete example that geometrically as well as physically is rather simplified the heat conductivity in the unit circle with dirichlet neumann boundary conditions the method is applicable to problems with different configurations further it can be extended to deal with more complicated physical situations of mixed character such as dynamical problems of the theory of elasticity with the typical stress strain boundary conditions

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 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

strongly stable generalized finite element method ssgfem for a non smooth interface problem abstract in this paper we propose a strongly stable generalized finite element method ssgfem for a non smooth interface problem where the interface has a corner the ssgfem employs enrichments of 2 types the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions both types of enrichments have been modified by a simple local procedure of subtracting the interpolant a simple local orthogonalization technique lot also has been used at the nodes enriched with both enrichment functions we prove that the ssgfem yields the optimal order of convergence the numerical experiments presented in this paper indicate that the conditioning of the ssgfem is not worse than that of the standard finite element method and the conditioning is robust with respect to the position of the mesh relative to the interface

weak convergence and non linear ergodic theorems for reversible semigroups of non lipschitzian mappings abstract letgbe a semitopological semigroup letcbe a closed convex subset of a uniformly convex banach spaceewith afrechetdifferentiable norm and t tt t g be a continuous representation ofgas asymptotically nonexpansive type mappings ofcinto itself such that the common fixed point setf t of t incis nonempty we prove in this paper that ifgis right reversible then for every almost orbitu of t formula consists of at most one point further formula is nonempty for eachx cif and only if there exists a nonexpansive retractionpofcontof t such thatpts tsp pfor alls gandp x is in the closed convex hull of tsx s g x c this result is applied to study the problem of weak convergence of the net u t t g to a common fixed point of t

nonexpansive retractions and nonlinear ergodic theorems in banach spaces on donne des theoremes ergodiques non lineaires pour des semigroupes commutatifs d applications non dilatation sans utiliser le concept de moyenne compacte

extended finite element method with global enrichment a variation of the extended finite element method for 3d fracture mechanics is proposed it utilizes global enrichment and point wise as well as integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two and three preprint submitted to elsevier science 29 january 2015 page 1 of 81 http mc manuscriptcentral com nme international journal for numerical methods in engineering

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

higher order stable generalized finite element method the generalized finite element method gfem is a galerkin method where the trial space is obtained by augmenting the trial space of the standard finite element method fem by non polynomial functions called enrichments that mimic the local behavior of the unknown solution of the underlying variational problem the gfem has excellent approximation properties but its conditioning could be much worse than that of the fem however if the enrichments satisfy certain properties then the conditioning of the gfem is not worse than that of the standard fem and the gfem is referred to as the stable gfem sgfem in this paper we address the higher order sgfem that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning yielding a robust implementation of the higher order sgfem

a stable generalized extended fem with discontinuous interpolants for fracture mechanics abstract this paper presents numerical studies with three classes of quadratic generalized fem gfem approximations and shows that all of them lead to errors that are orders of magnitude smaller than the fem with quarter point elements provided that appropriate enrichments are selected however all of them lead to severely ill conditioned systems of equations with a condition number up to o h 10 in the case of the gfem based on a quadratic partition of unity enrichment modifications able to address the ill conditioning of quadratic gfem approximations while preserving their optimal convergence are proposed a robust enrichment modification strategy based on a discontinuous fe interpolant is proposed to control the conditioning of branch function enrichment the discontinuous fe interpolant is a generalization of the continuous one used with the stable gfem sgfem we show that sgfem spaces based on p hierarchical fem enrichments are the same as their gfem counterparts this guarantees that both gfem and sgfem spaces will lead to the same solution this is not the case for the other classes of second order spaces the robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated

asymptotic behavior for almost orbits of a reversible semigroup of non lipschitzian mappings in a metric space let m be a metric space and a hausdorff topology on m such that m is compact let s be a right reversible semitopological semigroup and t s s s a representation of s as asymptotically nonexpansive type self mappings of m and u a bounded almost orbit of we study the convergence of the net u s s s in m when the triplet m satisfies various types of opial conditions our results extend and unify many previously known results 2002 elsevier science usa all rights reserved

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem

isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth abstract we present a novel numerical method to simulate crack growth in 3d directly from the computer aided design cad geometry of the component without any mesh generation the method is an isogeometric boundary element method igabem based on non uniform rational b splines nurbs nurbs basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations a stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements convergence studies in the crack opening displacement is performed for a penny shaped crack and an elliptical crack two approaches to extract stress intensity factors sifs the contour m integral and the virtual crack closure integral are compared using dual integral equations the results show remarkable accuracy in the computed sifs leading to smooth crack paths and reliable fatigue lives without requiring the generation of any mesh from the cad model of the component under consideration

weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems the first ergodic theorems for semigroups of nonlinear operators date back to work done by j b baillon cl 23 in 1975 1976 where he proved in 2 that i i lim l t j t s i x dt exists for an odd nonlinear contraction semigroup s t a 0 in real hilbert space in subsequent years various authors have extended his results to motions and almost orbits of nonlinear strongly continuous semigroups in uniformly convex banach spaces e g see 4 10 22 24 26 27 30 31 33 351 as well as 4 10 241 for yet further references these results apply to strong and integral solutions of the quasi autonomous nonlinear cauchy problem