numerical modelling of crack propagation automatic remeshing and comparison of different criteria modelling of a crack propagating through a finite element mesh under mixed mode conditions is of prime importance in fracture mechanics three different crack growth criteria and the respective crack paths prediction for several test cases are compared the maximal circumferential stress criterion the strain energy density fracture criterion and the criterion of the maximal strain energy release rate are implemented using advanced finite element techniques a fully automatic remesher enables to deal with multiple boundaries and multiple materials the propagation of the crack is calculated with both remeshing and nodal relaxation several examples are presented to check for the robustness of the numerical techniques and to study specific features of each criterion

on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

quadrilateral mesh revisited several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent the effect of the bi section condition on the degenerate mesh conditions is also checked

all electron kohn sham density functional theory on hierarchic finite element spaces in this work a real space formulation of the kohn sham equations is developed making use of the hierarchy of finite element spaces from different polynomial order the focus is laid on all electron calculations having the highest requirement onto the basis set which must be able to represent the orthogonal eigenfunctions as well as the electrostatic potential a careful numerical analysis is performed which points out the numerical intricacies originating from the singularity of the nuclei and the necessity for approximations in the numerical setting with the ambition to enable solutions within a predefined accuracy in this context the influence of counter charges in the poisson equation the requirement of a finite domain size numerical quadratures and the mesh refinement are examined as well as the representation of the electrostatic potential in a high order finite element space the performance and accuracy of the method is demonstrated in computations on noble gases in addition the finite element basis proves its flexibility in the calculation of the bond length as well as the dipole moment of the carbon monoxide molecule

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

the interpolation theorem for narrow quadrilateral isoparametric finite elements the interpolation theorem for convex quadrilateral isoparametric finite elements is proved in the case when the condition varrho k h k ge varrho 0 0 is not satisfied where h k is the diameter of the element k and varrho k is the radius of an inscribed circle in k the interpolation error is o h k 2 in the l 2 k norm and o h k in the h 1 k norm provided that the interpolated function belongs to h 2 k in the case when the long sides of the quadrilateral k are parallel the constants appearing in the estimates are evaluated

on the strong discontinuity approach in finite deformation settings taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena this work extends previous results in infinitesimal strain settings to finite deformation scenarios by means of the strong discontinuity analysis and taking isotropic damage models as target continuum stress strain constitutive equation projected discrete tractions displacement jumps constitutive models are derived together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime a variable bandwidth model to automatically induce those strong discontinuity conditions and a discontinuous bifurcation procedure to determine the initiation and propagation of the discontinuity are briefly sketched the large strain counterpart of a non symmetric finite element with embedded discontinuities frequently considered in the strong discontinuity approach for infinitesimal strains is then presented finally some numerical experiments display the theoretical issues and emphasize the role of the large strain kinematics in the obtained results copyright 2003 john wiley sons ltd

layer adapted meshes for convection diffusion problems abstract in the present paper a classification and a survey are given of layer adapted meshes for stationary convection dominated convection diffusion problems for a number of standard numerical schemes theoretical results are presented that demonstrate that the use of properly layer adapted meshes yields robust methods i e methods that perform equally well no matter how dominant the convection we review a number of techniques used in the convergence analysis of these methods

modeling three dimensional crack propagation a comparison of crack path tracking strategies the development of a new finite element technique for the simulation of discontinuous failure phenomena in three dimensions is the key objective of this study in contrast to the widely used extended finite element technique we apply a purely deformation based strategy based on an independent interpolation of the deformation field on both sides of the discontinuity this method has been applied successfully for two dimensional crack propagation problems in the past however when it comes to three dimensional failure phenomena it faces the same difficulties as the extended finite element method unlike in two dimensions the characterization for the three dimensional failure surface is non unique and the tracking of the discrete crack can be performed in several conceptually different ways in this work we review the four most common three dimensional crack tracking strategies we perform a systematic comparison in terms of standard algorithmic quality measures such as mesh independency efficiency robustness stability and computational cost moreover we discuss more specific issues such as crack path continuity and integratability in commercial finite element packages the features of the suggested crack tracking algorithms will be elaborated by means of characteristic benchmark problems in failure analysis copyright 2008 john wiley sons ltd

piecewise polynomial interpolation in muckenhoupt weighted sobolev spaces and applications we develop a constructive piecewise polynomial approximation theory in weighted sobolev spaces with muckenhoupt weights for any polynomial degree the main ingredients to derive optimal error estimates for an averaged taylor polynomial are a suitable weighted poincare inequality a cancellation property and a simple induction argument we also construct a quasi interpolation operator built on local averages over stars which is well defined for functions in l 1 l1 we derive optimal error estimates for any polynomial degree on simplicial shape regular meshes on rectangular meshes these estimates are valid under the condition that neighboring elements have comparable size which yields optimal anisotropic error estimates over n n rectangular domains the interpolation theory extends to cases when the error and function regularity require different weights we conclude with three applications nonuniform elliptic boundary value problems elliptic problems with singular sources and fractional powers of elliptic operators

new finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids this paper presents new finite elements with embedded strong discontinuities in the form of jumps in the mechanical displacement field as well as jumps in the electric potential for the modeling of failure in electromechanical coupled solids such materials which are commonly used in smart systems in the form of actuators or sensors are prone to defects in the form of cracks represented by such strong discontinuities in this work a discrete account of the individual jumps is made with the possibility of their propagation through the individual finite elements without the need of remeshing or refinement strategies originated for purely mechanical based materials and commonly referred as the strong discontinuity approach an extension to account for the electromechanical coupling and the additional jumps in the electric potential is made in this work the decomposition of that methodology into a global problem representing here the electromechanical boundary value problem and a local problem through which the strong discontinuities are introduced serves convenient when developing the corresponding finite elements based on the requirement to avoid stress locking phenomena this idea is extended in this work to develop new finite elements based on the incorporation of electrical separation modes directly into the discrete formulation keeping its local and computational efficient property with regard to the storage and possible static condensation of the parameters used to describe the mechanical and electrical discontinuities on a constitutive level the mechanical traction separation law is extended to account for a softening type behavior in the relation between the normal component of the electric displacement field and the surface charge density this is accomplished through the derivation of a localized electromechanical damage model based on the principle of maximum damage dissipation the performance of the resulting new finite elements is illustrated with several numerical simulations including a compact tension test and a three point bending test of piezoelectric ceramics and compared with experimental results available in the literature

finite element methods for the multi scale modeling of softening hinge lines in plates at failure abstract this paper presents a multi scale framework for the modeling of softening hinge lines in inelastic plates and its finite element implementation the proposed framework considers the reissner mindlin problem of thick plates as the large scale problem with the hinge introduced locally in small neighborhoods modeling the small scales of the material response the hinges are defined as strong discontinuities of the generalized displacements describing the plate deformations the consideration of the large scale limit of vanishing small scales leads to a large scale model of these localized failures of plates involving the solution of a problem in terms of those large scale generalized displacements with the standard regularity conditions enhanced finite elements define the natural framework for the numerical implementation of these considerations the new finite elements are enhanced with the singular strain fields associated with the discontinuities or hinge lines which given their local character are statically condensed at the element level we present a general strategy for the definition of these local strain fields so the kinematics of the hinge lines are captured correctly allowing in particular their fully softened state with no stresses and hence avoiding the so called stress locking we present several particular finite elements formulated in this framework including triangular and quadrilateral finite elements several numerical simulations are presented illustrating their performance

perturbing uniformly stable nonlinear scalar delay differential equations on considere l equation differentielle a retard perturbee x t f t x t g t x t ou g o s r s x r x on donne des conditions sur g qui assurent que la solution zero est uniformement stable sous certaines conditions

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

an anisotropic pressure stabilized finite element method for incompressible flow problems we consider a pressure stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios

asymptotic theory for a class of nonautonomous delay differential equations this paper deals with asymptotic behavior of solutions of the nonlinear nonautonomous delay differential equation x t t r t t f t x s d t s su where xf t x 0 f t 0 0 t r t nondecreasing t s is nondecreasing and of bounded variation general sufficient conditions which are easy to verify are obtained for the solutions to be bounded and asymptotically stable locally and globally these results improve many existing ones principally by allowing i r t to be unbounded ii both discrete and distributed delays and iii the equation to be strongly nonlinear and nonautonomous various examples are given in the form of corollaries with a highly flexible integrand

anisotropic error bounds of lagrange interpolation with any order in two and three dimensions abstract in this paper using the newton s formula of lagrange interpolation we present a new proof of the anisotropic error bounds for lagrange interpolation of any order on the triangle rectangle tetrahedron and cube in a unified way

hierarchical decomposition of domains with fractures we consider the efficient and robust numerical solution of elliptic problems with jumping coefficients occuring on a network of fractures these thin geometric structures are resolved by anisotropic trapezoidal elements we present an iterative solution concept based on a hierarchical separation of the fractures and the surrounding rock matrix upper estimates for the convergence rates are independent of the the jump of coefficients and of the width of the fractures and depend only polynomially on the number of refinement steps the theoretical results are illustrated by numerical experiments

a finite element method for the computational modelling of cohesive cracks the present contribution is concerned with the computational modelling of cohesive cracks in quasi brittle materials whereby the discontinuity is not limited to interelement boundaries but is allowed to propagate freely through the elements in the elements which are intersected by the discontinuity additional displacement degrees of freedom are introduced at the existing nodes therefore two independent copies of the standard basis functions are used one set is put to zero on one side of the discontinuity while it takes its usual values on the opposite side and vice versa for the other set to model inelastic material behaviour a discrete damage type constitutive model is applied formulated in terms of displacements and tractions at the surface some details on the numerical implementation are given concerning the failure criterion the determination of the direction of the discontinuity and the integration scheme finally numerical examples show the performance of the method

theoretical and computational issues in modelling material failure in strong discontinuity scenarios the paper deals with several aspects related to numerical modelling of material failure in strong discontinuity settings a the onset and development of local material failure in terms of continuum constitutive models equipped with strain softening closed forms formulas for the solutions of the discontinuous material bifurcation problem are given for a class of those models b finite elements with embedded discontinuities nodal and elemental enrichments families are formulated in the continuum strong discontinuity approach csda c instability treatment a discrete viscous perturbation method at the failure surfaces is presented as a way to substantially improve the robustness of the numerical simulations

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

uniform error estimates for certain narrow lagrange finite elements error estimates of dupont and scott are used to derive uniform error estimates for lagrange finite elements in jrn n 2 under the following conditions 1 the elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis and 2 the space of approximating functions ft in each element t must include the space of polynomials of degree 1 n 2 if n is odd this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded for example if n 3 the usual requirement with shape regularity is that each ft contain all first degree polynomials the result of this paper requires that each ft contain all quadratic polynomials and consequently does not apply to linear courant elements in tetrahedrons or trilinear tensor elements in rectangular boxes counterexamples in these two cases are included

anisotropic interpolation error estimate for arbitrary quadrilateral isoparametric elements the aim of this paper is to show that for any p in 1 infty the w 1 p anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition mac and the property of comparable lengths for opposite sides clos i e on all those quadrilaterals with interior angles uniformly bounded away from pi and with both pairs of opposite sides having comparable lengths for rectangular elements our interpolation error estimate agrees with the usual one whereas for perturbations of rectangles the most general quadrilateral elements previously considered as far as we know our result has some advantages with respect to the pre existing ones the interpolation error estimate that we proved is written by using two neighboring sides of the element instead of the sides of the unknown perturbed rectangle and on the other hand conditions mac and clos are much simpler requirements to verify and with a clearer geometrical sense than those involved in the definition of perturbations of a rectangle

interpolation error estimates in w 1 p for degenerate q 1 isoparametric elements optimal order error estimates in h 1 for the q 1 isoparametric interpolation were obtained in acosta and duran siam j numer anal37 18 36 1999 for a very general class of degenerate convex quadrilateral elements in this work we show that the same conlusions are valid in w 1 p for 1 p 3 and we give a counterexample for the case p 3 showing that the result cannot be generalized for more regular functions despite this fact we show that optimal order error estimates are valid for any p 1 keeping the interior angles of the element bounded away from 0 and independently of the aspect ratio we also show that the restriction on the maximum angle is sharp for p 3

an anisotropic superconvergent nonconforming plate finite element the classical finite element convergence analysis relies on the following regularity condition there exists a constant c independent of the element k and the mesh such that h k r k

on the interpolation error estimates for q_1 quadrilateral finite elements in this paper we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over frac 1 j p 1 for 1 leq p leq infty on the reference element where j is the jacobian of the nonaffine mapping this allows us to introduce weak geometric conditions depending on p leading to interpolation error estimates in the w 1 p norm for any p in 1 infty which can be regarded as a generalization of the regular decomposition property rdp condition introduced in g acosta and r g duran siam j numer anal 38 2000 pp 1073 1088 for p 2 and new rdp conditions nrdp for p neq2 we avoid the use of the reference family elements which allows us to extend the results to a larger class of elements and to introduce the nrdp condition in a more unified way as far as we know the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 leq p leq infty

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

on friedrichs poincare type inequalities friedrichs and poincare type inequalities are important and widely used in the area of partial differential equations and numerical analysis most of their proofs appearing in references are the argument of reduction to absurdity in this paper we give direct proofs of friedrichs type inequalities in h 1 and poincare type inequalities in some subspaces of w 1 p the dependencies of the inequality coefficients on the domain and some sub domains are illustrated explicitly

global stability for a class of nonlinear nonautonomous delay equations it is easy to imagine that more realistic models for single species growths should take into account both the changing of the environment and the effects of time delay thus it is neces sary and interesting to study the following general nonlinear nonautonomous delay equation x t l x t i s

a computational framework of three dimensional configurational force driven brittle crack propagation abstract we consider a variational formulation of quasi static brittle fracture and develop a new finite element based computational framework for propagation of cracks in three dimensional bodies we outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global clausius planck inequality in the sense of coleman s method consequently the crack propagation direction associated with the classical griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front the variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces therefore the constitutive setting of crack propagation in the space discretized finite element context is naturally related to discrete nodes of a typical finite element mesh in a consistent way with the node based setting the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh the crucial step for the success of this procedure is its embedding into an r adaptive crack facet reorientation procedure based on configurational force based indicators in conjunction with crack front constraints we propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness providing a robust algorithmic setting in the postcritical range the predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

global attractivity in scalar delayed differential equations with applications to population models for the scalar delay differential equation x t 1 x t f t xt sufficient conditions for the global stability of the zero solution are given the behavior of global solutions on r is also addressed several applications to population models which improve known results are presented

extended finite element method on polygonal and quadtree meshes in this paper we present mesh independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes this approach falls within the class of extended and generalized flnite element methods where the partition of unity framework is used to introduce additional enrichment functions within the classical displacement based flnite element approximation for crack modeling a discontinuous function and the two dimensional asymptotic crack tip flelds are used as enrichment functions linearly complete partition of unity approximations are adopted on polygonal convex and nonconvex elements and quadtree meshes excellent agreement with reference solution results is obtained for mixed mode stress intensity factors on benchmark crack problems and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics

dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment summary a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity the approach is limited to rate independent materials so that the transition occurs on a set of measure zero the discrete discontinuity is treated by the extendednite element method xfem whereby arbitrary discontinuities can be incorporated in the model without remeshing loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model a new method was developed for the case when the discontinuity ends within an element it facilitates the modelling of crack tips that occur within an element in a dynamic setting the method is applied to several dynamic crack growth problems including the branching of cracks copyright 2003 john wiley sons ltd

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

three dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range summary this paper presents new three dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range the goal is to model localized surfaces of failure in solids such as cracks at fracture through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors in this way such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface unknown a priori an important issue is also the consideration of general finite element formulations in the developments e g basic displacement based mixed or enhanced assumed strain finite element formulations as needed to optimally resolve the continuum problem in the bulk the actual modeling of the discontinuity effects including the incorporation of the cohesive law defining the discontinuity constitutive response is carried out at the element level with the proper enhancement of the discrete strain field of the element the added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom in fact this global local structure of the finite element methods developed in this work arises naturally from a multi scale characterization of these localized solutions with the discontinuities understood to appear in the small scales thus leading directly to these computationally efficient numerical methods for their numerical resolution easily incorporated to an existing finite element code the focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three dimensional setting these interpolations are shown to be necessary for hexahedral elements to avoid the so called stress locking that occurs with simpler constant approximations of the jumps namely a spurious transfer of stresses across the discontinuity not allowing its full release and hence resulting in an overstiff or locked numerical solution the design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element rather than from an assumed discontinuous interpolation of the displacements assuring with this approach their locking free response by design an additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three dimensional setting of interest here the paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements copyright 2012 john wiley sons ltd

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

interpolation error estimates of a modified 8 node serendipity finite element interpolation error estimates for a modified 8 node serendipity finite element are derived in both regular and degenerate cases the latter of which includes the case when the element is of triangular shape for u in w 3 p k defined over a quadrilateral k the error for the interpolant pi k u is estimated as u pi k u w alpha p k le ch 3 alpha k u w 3 p k alpha 0 1 where 1 le p le infty in the regular case and 1 le p 3 in the degenerate case respectively thus the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for 1 le p 3 results for some related elements are also given

new finite elements with embedded strong discontinuities in the finite deformation range this paper develops new finite elements with embedded strong discontinuities for the modeling of the failure of solids in the finite deformation range the cases of interest include cracks shear bands delamination in composites and similar failures modeled by a propagating discontinuity in the deformation mapping that incorporates a cohesive law in terms of the associated displacement jumps the new elements accommodate the discontinuity in their interiors with a linear interpolation of the displacement jumps in contrast to the existing elements with constant jumps a general methodology is proposed for the development of such finite elements but the final focus is on quadrilateral elements for plane problems where a constant jump interpolation leads to a spurious transfer of stresses through the discontinuity or stress locking in certain common configurations the key aspect of the new elements is the proper enhancement of the deformation gradient that can represent correctly the kinematics of linearly separating discontinuities including a relative finite rotation across the discontinuity and in a frame indifferent manner the enhanced parameters modeling the displacement jumps are introduced independently for each element allowing for their static condensation at the element level and hence to arrive at a global system of equations with the same number and connectivity of the original global degrees of freedom for the underlying finite element mesh this situation leads to a particularly simple implementation of the proposed approach in an existing finite element code the performance of the new elements is illustrated with several numerical simulations including delamination in composite materials and propagating shear bands in a finitely deforming elastoplastic solid

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

3d crack propagation in unreinforced concrete a two step algorithm for tracking 3d crack paths tensile failure of unreinforced concrete involves progressive micro cracking and the related strain softening can coalesce into geometrical discontinuities which separate the material advanced mechanical theories and numerical schemes are required to efficiently and adequately represent crack propagation in 3d in this paper we use the concept of strong discontinuities to model concrete failure we introduce a cohesive fracture process zone which is characterized by a transversely isotropic traction separation law we combine the cohesive crack concept with the partition of unity finite element method where the finite element space is enhanced by the heaviside function the concept is implemented for tetrahedral elements and the failure initialization is based on the simple non local rankine criterion for each element we assume the embedded discontinuity to be flat in the reference configuration which leads to a non smooth crack surfaces approximation in 3d in general different concepts for tracking non planar cracks in 3d are reviewed in addition we propose a two step algorithm for tracking the crack path where a predictor step defines discontinuities according to the non local failure criterion and a corrector step draws in non local information of the existing discontinuities in order to predict a closed 3d crack surface implementation details are provided the proposed framework is used to analyze the predictability of concrete failure by two benchmark examples i e the nooru moharned test and the brokenshire test we compare our numerical results which are mesh independent with experimental data and numerical results adopted from the literature

applicability of the bramble hilbert lemma in interpolation problems of narrow quadrilateral isoparametric finite elements two approaches to deriving the interpolation theorem for narrow quadrilateral isoparametric finite elements not satisfying the condition rho k h k greater than or equal to rho 0 0 rho k and h k being the radius of inscribed circle and the diameter of the quadrilateral k respectively are presented the first one using the bramble hilbert lemma is successful only in deriving the l 2 k estimate the nonapplicability of the standard approach via the bramble hilbert lemma in the case of h 1 k estimate is presented and a fully efficient method giving the optimum rate of convergence o h in the h 1 k norm is described in the end the dependence of the interpolation error on the geometry of a quadrilateral is demonstrated by an example

a new generalized finite element method for two scale simulations of propagating cohesive fractures in 3 d summary this paper presents a novel numerical framework based on the generalized finite element method with global local enrichments gfemgl for two scale simulations of propagating fractures in three dimensions a non linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters the cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized extended finite element method and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface the proposed method employs fine scale solutions of non linear local boundary value problems extracted from the original global problem in order to not only construct scale bridging enrichment functions but also to identify damaged states in the global problem thus enabling accurate global solutions on coarse meshes this is in contrast with the available gfemgl in which the local solution field contributes only to the kinematic description of global solutions the robustness efficiency and accuracy of this approach are demonstrated by results obtained from representative numerical examples copyright 2015 john wiley sons ltd

error estimates for 3 d narrow finite elements we obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three dimensional narrow elements first for the lagrange interpolation we prove optimal error estimates both in order and regularity in l p for p 2 for p 2 it is known that this result is not true applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than h 2 these results are valid both for tetrahedral and rectangular elements second for the case of rectangular elements we obtain optimal in order and regularity error estimates for an average interpolation valid for functions in w 1 s p with 1 p and 0 s 1

an eigenerosion approach to brittle fracture the present work is concerned with the verification and validation of a variant of the eigenfracture scheme of schmidt et al 2009 based on element erosion which we refer to as eigenerosion eigenerosion is derived from the general eigenfracture scheme by restricting the eigendeformations in a binary sense they can be either zero in which case the local behavior is elastic or they can be equal to the local displacement gradient in which case the corresponding material neighborhood is failed or eroded when combined with a finite element approximation this scheme gives rise to element erosion i e the elements can be either intact in which case their behavior is elastic or be completly failed or eroded and have no load bearing capacity we verify the eigenerosion scheme through comparisons with analytical solutions and through convergence studies for mode i fracture propagation both in two and three dimensions and for structured and random meshes finally by way of validation we apply the eigenerosion scheme to the simulation of mixed modes i iii experiments in poly methyl methacrylate plates

finite element methods for the analysis of strong discontinuities in coupled poro plastic media this paper presents the formulation of finite element methods for the numerical resolution of strong discontinuities in poro plastic solids fully coupled infinitesimal conditions are considered these solutions are characterized by a discontinuous displacement field with the associated singular strains and a singular distribution of the fluid content here singular distributions refer to dirac delta functions the singular component of the fluid content distribution models the fluid accumulated per unit area of the discontinuity surface and it is directly related with the dilatancy characterizing singular inelastic strains localized along such a surface it further accounts for a discontinuous fluid flow vector given by darcy s law in terms of a continuous pore pressure field all these considerations are incorporated in the proposed finite element methods through a local enhancement of the finite element interpolations as these discontinuities appear the local character of these interpolations lead after the static condensation of the enhanced fields to a large scale problem exhibiting the same structure as common finite element models of the global poro plastic problem but incorporating now crucially the localized dissipative effects characteristic of the localized failures several numerical simulations are presented to evaluate the performance of the proposed methods

extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids abstract this paper addresses novel extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids on one hand based upon the unified multiscale kinematics of strong discontinuities standard finite elements at the coarse scale are enriched with fine scale kinematics in which non uniform discontinuity modes are incorporated on the other hand the traction continuity condition across the discontinuity is enforced in the statically optimal form as the fine scale statics the admissible discontinuity modes satisfying a priori the traction continuity condition are derived in addition to the relative rigid body motions translations and rotations more general relative stretching modes that induce discontinuous strain stress fields in the bulk are obtained with no limit of a vanishing poisson s ratio the proposed model then particularizes to 2d quadrilateral elements the displacement jumps at the discontinuity nodes are selected as the enrichment parameters and regarded as global variables shared by neighboring elements the resulting model not only is intrinsically stress locking free for a fully softened discontinuity but also automatically guarantees global continuity of the displacement jumps another benefit is that the numerical instability suffered under the unfavorable element discontinuity configuration can be easily circumvented furthermore the vanishing displacement jumps at the discontinuity tip can be enforced trivially as conventional essential boundary conditions and those complex strategies introduced in existing models are avoided together with the failure criterion related to the element average stress and the propagation orientation based on a simplified nonlocal stress a local crack tracking algorithm is adopted to guarantee global continuity of the discontinuity path representative numerical simulations of element and structure benchmark tests are presented to verify the performance of the proposed approach in the modeling of arbitrary discontinuity propagation in solids

stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach abstract robustness and stability of the continuum strong discontinuity approach csda to material failure are addressed after identification of lack of symmetry of the finite element formulation and material softening in the constitutive model as possible causes of loss of robustness two remedies are proposed 1 the use of an specific symmetric version of the elementary enriched e fem finite element with embedded discontinuities and 2 a new implicit explicit integration of the internal variable in the constitutive model which renders the tangent constitutive algorithmic operator positive definite and constant the combination of both developments leads to finite element formulations with constant in the time step and non singular tangent structural stiffness allowing dramatic improvements in terms of robustness and computational costs after assessing the convergence and stability properties of the new strategies three dimensional numerical simulations of failure problems illustrate the performance of the proposed procedures

a multiscale approach for modeling progressive damage of composite materials using fast fourier transforms abstract composite materials possess a highly complex material behavior and thus advanced simulation techniques are necessary to compute their mechanical response in this regard especially modeling failure and progressive damage presents a challenging task conventional macro mechanical methods and even closed form estimates are in many cases not sufficient to predict the appropriate mechanical material response full field simulations must be resorted to but these are known to be very expensive from the computational point of view in this contribution we propose a more efficient multiscale approach similar to fe2 nonlinear material effects caused by progressive damage behavior are captured directly on the discretized material level using simple isotropic continuum damage laws in contrast to conventional fe2 methods which use the finite element method fem to solve both scales numerically the fine scale problem material level is rewritten in an integral form of lippmann schwinger type and solved efficiently using the fast fourier transformation fft the calculation is carried out on a regular voxel grid that can be obtained from 3d images like tomographies the fine scale problem is integrated in a standard finite element framework which is used to solve the macroscopic bvp component level in the work at hand the scale coupling technique and the computation of the macroscopic tangent are described and in some numerical examples the convergence behavior of the macroscopic newton algorithm is investigated thereby the simulations were considered until localization and softening on the material scale occurred it is shown that the proposed method presents an effective way to determine the exact physical macroscopic response considering arbitrary microstructures and loading conditions

a finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation abstract this paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard galerkin approximation strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level the mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting at post localization the stress point integration algorithm along the band is exact and amenable to consistent linearization numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow as well as plane strain compression of dilatant cohesive frictional materials are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

global attractivity results for nonlinear delay differential equations abstract the aim of this paper is to give sufficient conditions for global attractivity of the zero solution of the nonlinear delay differential equationx t p t f x t an example which guarantees that our 3 2 stability condition is the best possible is also given

stability theorems for nonlinear scalar delay differential equations in this paper the stability on nonlinear scalar delay differential equations is studied and improved 3 2 stability theorems are presented

the minimal angle condition for quadrilateral finite elements of arbitrary degree we study w1 p lagrange interpolation error estimates for general quadrilateralqk finite elements with k2 for the most standard case of p 2 it turns out that the constant c involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral moreover the same holds for any p in the range 1p 3 on the other hand for 3p we show that c also depends on the maximal interior angle we provide some counterexamples showing that our results are sharp

locality constraints within multiscale model for non linear material behaviour this paper presents a two scale approach for the mechanical and numerical modelling of materials with microstructure like concrete or fibre reinforced concrete in the non linear regime it addresses applications where the assumption of scale separation as the basis for classical homogenization methods does not hold this occurs when the resolution of micro and macro scale does not differ ab initio or when evolving fluctuations in the macro fields are in the order of the micro scale during the loading progress typical examples are localization phenomena the objective of the present study is to develop an efficient solution method exploiting the physically existing multiscale character of the problem the proposed method belongs to the superposition based methods with local enrichment of the large scale solution by a small scale part u the main focus of the present formulation is to allow for locality of the small scale solution within the large scale elements to achieve an efficient solution strategy at the same time the small scale information exchange over the large scale element boundaries is facilitated while maintaining the accuracy of a refined complete solution thus the emphasis lies on finding appropriate locality constraints for u to illustrate the method the formulation is applied to a damage mechanics based material model for concrete like materials copyright 2006 john wiley sons ltd

a marching cubes based failure surface propagation concept for three dimensional finite elements with non planar embedded strong discontinuities of higher order kinematics summary this paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method when modeling three dimensional 3d solids at failure the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization this results in the need for a propagation of the discrete failure zone through the individual finite elements which is possible for finite elements with embedded strong discontinuities whereas for two dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction more advanced strategies are needed to achieve a smooth failure surface in 3d simulations an example for such method is the global tracking algorithm which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements in particular for a hexahedral eight node finite element 256 such cases exist to capture all those possibilities the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work in addition because many of the possible cases result in non planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces a remedy for such scenarios is proposed various 3d failure propagation simulations outline the performance of the proposed concept copyright 2013 john wiley sons ltd

3d modelling of strong discontinuities in elastoplastic solids fixed and rotating localization formulations this paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials based on the strong discontinuity approach sda computational mechanics 1993 12 277 296 mesh objective 2d and 3d finite element formulations are developed using linear and quadratic 2d elements as well as 8 noded 3d elements in the formulation of the finite element model proposed in the paper the analogy with standard formulations is emphasized the parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique this approach results in linearized constitutive equations formally identical to continuum models therefore the standard return mapping algorithm is used to solve the non linear equations in analogy to concepts used in continuum smeared crack models a rotating formulation of the sda is proposed in addition to the standard concept of fixed discontinuities it is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions the applicability of the proposed sda finite element model as well as its numerical performance is investigated by means of a three dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force copyright 2003 john wiley sons ltd

global attractivity of solutions of nonlinear delay differential equations with a forcing term abstract in this paper we give sufficient conditions for every solution of the delay differential equation x t p t f x t r t to converge to zero our results extend and improve several known results in the literature an application of our main results to an equation derived from mathematical biology is also given

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes