a nitsche stabilized finite element method for frictional sliding on embedded interfaces part ii intersecting interfaces abstract we extend the weighted nitsche s method proposed in the first part of this study to include multiple intersecting embedded interfaces these intersections arise either inside a computational domain where two internal interfaces intersect or on the boundary of the computational domain where an internal interface intersects with the external boundary we propose a variational treatment of both the interfacial kinematics and the external dirichlet constraints within nitsche s framework we modify the numerical analysis to account for these intersections and provide an explicit expression for the weights and the method parameters that arise in the nitsche variational form in the presence of junctions finally we demonstrate the performance of the method for both perfectly tied interfaces and perfectly plastic sliding interfaces through several benchmark examples

a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

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

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

a symmetric method for weakly imposing dirichlet boundary conditions in embedded finite element meshes summary in this paper we propose a way to weakly prescribe dirichlet boundary conditions in embedded finite element meshes the key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation relatively small and can be fixed a priori moreover the formulation is symmetric for symmetric problems an additional element discontinuous stress field is used to enforce the boundary conditions in the poisson problem additional terms are required in order to guarantee stability in the convection diffusion equation and the stokes problem the proposed method is then easily extended to the transient navier stokes equations copyright 2012 john wiley sons ltd

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

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

robust imposition of dirichlet boundary conditions on embedded surfaces summary we develop both stable and stabilized methods for imposing dirichlet constraints on embedded three dimensional surfaces in finite elements the stable method makes use of the vital vertex algorithm of bechet et al bechet et al 2009 int j numer meth engng 78 8 931 954 albeit with a modified set of discontinuous basis functions for the lagrange multiplier field on the embedded surface the stabilized method extends the work of dolbow and harari dolbow and harari 2009 int j numer meth engng 78 2 229 252 to three dimensional surfaces algorithmic and implementational details of both methods are provided several three dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches the results indicate that both methods yield optimal rates of convergence in various quantities of interest with the primary differences being in the surface flux the utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques published by john wiley sons ltd copyright c 2000 john wiley sons ltd

asymptotic smoothing and attractors for the generalized benjamin bona mahony equation on r 3 abstract we study the asymptotic behavior of the solutions of the benjamin bona mahony equation defined on r 3 we first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain which involves the littlewood paley projection operators we then prove the existence of an attractor for the benjamin bona mahony equation in the phase space h 1 r 3 by showing the solutions are point dissipative and asymptotic compact finally we establish the regularity of the attractor and show that the attractor is bounded in h 2 r 3

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

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

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

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

modeling of failure in composites by x fem and level sets within a multiscale framework abstract composites or multi phase materials are characterized by a distinct heterogeneous microstructure the failure modes of these materials are governed by several micromechanical effects like debonding phenomena and matrix cracks the overall mechanical behavior of composites in the linear as well as the nonlinear regime is not only governed by the material properties of the components and their bonds but also by the material layout in the present contribution the material structure is resolved and modeled on a small scale allowing to deal with these effects for the numerical simulation we apply a combination of the extended finite element method x fem n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer methods engrg 46 1999 131 150 and the level set method lsm s osher j sethian fronts propagating with curvature dependent speed algorithms based on hamilton jacobi formulations j comput phys 79 1988 12 49 in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity the geometry of material interfaces and cracks is described by the lsm the combination of both x fem and lsm turns out to be very natural since the enrichment can be described and even constructed in terms of level set functions in order to project the material behavior modeled on a small scale onto the large or structural scale we employ the variational multiscale method vmm t hughes g feijoo l mazzei j b quiney the variational multiscale method a paradigm for computational mechanics comput methods appl mech engrg 166 1998 3 24 this concept is based on an additive split of the displacement field into large and small scale parts for an efficient solution of the discrete problem we postulate that the small scale displacements are locally supported in order to achieve this objective one has to assume appropriate constraint conditions it can be shown that the applied numerical model allows a considerable flexibility concerning the variation of the material design and consequently of the mechanical behavior of a composite

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

on methods for stabilizing constraints over enriched interfaces in elasticity summary enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non smooth problems internal features such as boundaries are represented in such methods by using discontinuous enrichment of the standard finite element basis within such frameworks however imposition of interface constraints and or constitutive relations can cause unexpected difficulties depending upon how relevant fields are interpolated on un gridded interfaces this work address the stabilized treatment of constraints in an enriched finite element context both the lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields we demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner in a bubble stabilized approach fine scale degrees of freedom are added over elements supporting the interface the variational form can be shown to have a similar form to a second approach we consider nitsche s method with the exception that the stabilization terms follow directly from the bubble functions in this work we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two dimensional elasticity we examine several benchmark problems in elasticity and show that only nitsche s method and the bubble stabilization approach produce stable traction fields over internal boundaries we also demonstrate a novel difference between the penalty method and nitsche s method in that the latter passes the patch test exactly regardless of the stabilization parameter s magnitude results for more complicated geometries and triple interface junctions are also presented copyright 2008 john wiley sons ltd

discrete fast algorithms for two dimensional linear prediction on a polar raster new generalized split levinson and schur algorithms for the two dimensional linear least squares prediction problem on a polar raster are derived the algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk the covariance function of the random field is assumed to have a toeplitz plus hankel structure for both its radial part and its transverse angular part this assumption is valid for some types of random fields such as isotropic random fields the algorithms generalize the split levinson and schur algorithms in two ways 1 to two dimensions and 2 to toeplitz plus hankel covariances

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

interface material failure modeled by the extended finite element method and level sets multi phase materials are characterized by the fact that they possess a specific heterogeneous microstructure this feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models furthermore such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted in multiscale analyses usually the microstructure of a material point is resolved directly thus the material behavior at the material point can be modeled in an elegant natural and simple way the present contribution aims at a detailed geometric modeling of multi phase materials as well as at a local mechanical modeling of material interfaces and interfacial failure the geometry of the material distribution within the microstructure is described through level set functions the mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite element method x fem the combination of both the level set method and the x fem allows one to model such internal features of the microstructure without the adaptation of the mesh thus expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity remarkably the level set method cannot only be applied to describe the geometry of the material interfaces and cracks respectively but also to support and even to develop the corresponding enrichment function

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

finite deformation formulation for embedded frictional crack with the extended finite element method we reformulate an extended finite element fe framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation including finite stretching and rotation for the fe representation we consider a galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration contact and frictional constraints employ two kuhn tucker conditions a contact separation constraint nesting over a stick slip constraint for the case when the crack faces are in frictional sliding mode we integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics we then present plane strain simulations demonstrating various aspects of the extended fe solutions the mechanisms considered include combined opening and frictional sliding in initially straight curved and s shaped cracks with and without bulk plasticity to gain further insight into the extended fe solutions we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks including the distribution of the normal component of traction on the crack faces copyright 2009 john wiley sons ltd

a nitsche based method for unilateral contact problems numerical analysis we introduce a nitsche based formulation for the finite element discretization of the unilateral contact problem in linear elasticity it features a weak treatment of the non linear contact conditions through a consistent penalty term without any additional assumption on the contact set we can prove theoretically its fully optimal convergence rate in the h1 norm for linear finite elements in two dimensions which is o h 1 2 when the solution lies in h 3 2 0 1 2 an interest of the formulation is that conversely to lagrange multiplier based methods no other unknown is introduced and no discrete inf sup condition needs to be satisfied

stabilized low order finite elements for frictional contact with the extended finite element method abstract contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity in which the normal contact pressure exhibits spurious oscillation this oscillation does not go away with mesh refinement and in some cases it even gets worse as the mesh is refined using a lagrange multipliers formulation we trace this problem to non satisfaction of the lbb condition associated with equal order interpolation of slip and normal component of traction in this paper we employ a stabilized finite element formulation based on the polynomial pressure projection ppp technique which was used successfully for stokes equation and for coupled solid deformation fluid diffusion using low order mixed finite elements for the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method we use low order linear triangular elements tetrahedral elements for 3d for both slip and normal pressure degrees of freedom and show the efficacy of the stabilized formulation on a variety of plane strain plane stress and three dimensional problems

a stable lagrange multiplier space for stiff interface conditions within the extended finite element method this paper introduces a new algorithm to define a stable lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method in contrast to earlier approaches we do not work with an interior penalty formulation as e g for nitsche techniques but impose the constraints weakly in terms of lagrange multipliers roughly speaking a stable and optimal discrete lagrange multiplier space has to satisfy two criteria a best approximation property and a uniform inf sup condition owing to the fact that the interface does not match the edges of the mesh the choice of a good discrete lagrange multiplier space is not trivial here we propose a new algorithm for the local construction of the lagrange multiplier space and show that a uniform inf sup condition is satisfied a counterexample is also presented i e the inf sup constant depends on the mesh size and degenerates as it tends to zero numerical results in two dimensional confirm the theoretical ones copyright

a mixed augmented lagrangian extended finite element method for modelling elastic plastic fatigue crack growth with unilateral contact the complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods a new technique for the finite element modelling of elastic plastic fatigue crack growth with unilateral contact on the crack faces is presented the extended finite element method x fem is used to discretize the equations allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh this paper presents an augmented lagrangian formulation in the x fem framework that is able to deal with elastic plastic crack growth with treatment of contact an original formulation which takes advantages of two powerful numerical methods is presented next the numerical issues such as contact treatment and numerical integration are addressed and finally numerical examples are shown to validate the method copyright 2007 john wiley sons ltd