weak singularities may help periodic solutions to exist abstract in a periodically forced semilinear differential equation with a singular nonlinearity a weak force condition enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem the originality of the arguments relies in that contrary to the customary situation in the available references a weak singularity facilitates the arguments of the proofs

some novel developments in finite element procedures for gradient dependent plasticity summary improved algorithms are proposed for a gradient plasticity theory in which the laplacian of an invariant plastic strain measure enters the yield function particular attention is given to the type of finite elements that can be used within the format of gradient dependent plasticity assuming a weak satisfaction of the yield function mixed finite elements are developed in which the invariant plastic strain measure and the displacements are discretized two families of finite elements are developed one in which the invariant plastic strain measure is interpolated using c continuous polynomials and one in which penalty enhanced c continuous interpolants are used the performance of both families of finite elements is assessed numerically in one dimensional and two dimensional boundary value problems the regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated both for mesh refinement and for the directional bias of the grid lines

dispersion analysis and element free galerkin solutions of second and fourth order gradient enhanced damage models gradient dependent damage formulations incorporate higher order derivatives of state variables in the constitutive equations different formulations have been derived for this gradient enhancement comparison of which is difficult in a finite element context due to higher order continuity requirements for certain formulations on the other hand the higher order continuity requirements are met naturally by element free galerkin efg shape functions thus the efg method provides a suitable tool for the assessment of gradient enhanced continuum models dispersion analyses have been carried out to compare different gradient enhanced models with the non local damage model the formulation of the additional boundary conditions is addressed numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second and fourth order derivatives it is shown that with the same underlying internal length scale very different results can be obtained copyright 2000 john wiley sons ltd

a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis a discontinuous galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity the model of gradient plasticity is based on the von mises yield function in which dependence is on the isotropic hardening parameter and its laplacian the problem takes the form of a variational inequality of the second kind the discontinuous galerkin formulation is shown to be consistent and convergent error estimates are obtained for the cases of semi and fully discrete formulations these mimic the error estimates obtained for classical plasticity with the conventional galerkin formulation

two solutions for an elliptic equation with fast increasing weight and concave convex nonlinearities abstract we prove the existence of two nonnegative nontrivial solutions for the equation div k x u a x k x u q 2 u b x k x u p 2 u x r n where n 3 k x exp x 4 2 and the potentials a and b have indefinite sign and satisfy some mild integrability conditions the results hold when a has small norm in a suitable weighted lebesgue space

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

a discontinuous galerkin formulation of a model of gradient plasticity at finite strains abstract the work presented here constitutes an extension to the finite strain regime of a discontinuous galerkin based strain gradient plasticity formulation presented in djoko et al j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis comput methods appl mech engrg 197 2007 1 21 the focus here is on algorithmic and computational aspects of the formulation at finite strains the adoption of a logarithmic hyperelastic plastic formulation preserves the essential features of the infinitesimal formulation this key ingredient allows the predictor corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite strain regime the use of low order elements is essential to contain the computational expense of the formulation but these elements are prone to locking the method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation the form of the consistent tangent modulus is derived for the case of gradient plasticity two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm as well as features of the model of gradient plasticity

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

multiplicity of positive solutions for singular three point boundary value problems at resonance abstract in this paper we study the existence and multiplicity of positive solutions for the three point boundary value problems with singular nonlinear perturbations at resonance the existence results are applicable to both the case of a strong singularity and the case of a weak singularity the proof is based on a nonlinear alternative leray schauder principle and a fixed point theorem in cones for completely continuous operators

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

an adaptive discontinuous finite element method for the transport equation abstract in this paper we introduce a discontinuous finite element method in our approach it is possible to combine the advantages of finite element and finite difference methods the main ingredients are numerical flux approximation and local orthogonal basis functions the scheme is defined on arbitrary triangulations and two different error indicators are derived especially the second one is closely connected to our approach and able to handle arbitrary varying flow directions numerical resuls are given for boundary value problems in two dimensions they demonstrate the performance of the scheme combined with the two error indicators

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

finite element analysis of bifurcation in nonlocal strain softening solids progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise e g ill posedness and multiple bifurcation points this characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters for instance nonlocal damage constitutive relations the issue of possible stable or unstable equilibrium paths multiple localization zones and of the detection of bifurcation points has however never been addressed in the context of nonlocal constitutive laws we extend here the eigenmode analysis and perturbation method proposed by de borst to the study of the bifurcation and post bifurcation response of discrete nonlocal strain softening solids numerical applications on beams show that bifurcation and instability may occur in the post peak regime as opposed to the case of local constitutive relations the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

monotone iterative method for first order differential equations at resonance abstract this paper concerns the application of the monotone iterative technique for first order differential equations involving stieltjes integrals conditions we discuss such problems at resonance when the measure in the stieltjes integral is positive and also when this measure changes the sign sufficient conditions which guarantee the existence of extremal unique and quasi solutions are given three examples illustrate the results

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

a fast convergent rate preserving discontinuous galerkin framework for rate independent plasticity problems abstract in this paper a finite element framework based on the incomplete interior penalty galerkin formulation a non symmetric discontinuous galerkin method is consistently formulated for modeling plasticity problems with small deformation because of its pure displacement based framework this proposed discontinuous galerkin method is possibly able to completely preserve numerical integration algorithms efficiently developed in the traditional continuous galerkin framework besides stresses on element interior quadrature points stresses on element surface quadrature points are also required to return on yielding surfaces in this discontinuous galerkin framework which is able to provide more accurate material yielding profiles than the continuous galerkin framework the performance of the proposed discontinuous galerkin framework has been evaluated in detail for j 2 and pressure dependent plasticities using perfect plasticity plasticity with hardening and associative and non associative material models quadratic convergent rates compatible to the tradition continuous galerkin method for modeling plasticity problems have been achieved within a large penalty range in a nodal based discontinuous galerkin implementation

on the spatial formulation of discontinuous galerkin methods for finite elastoplasticity abstract in this paper we present a consistent spatial formulation for discontinuous galerkin dg methods applied to solid mechanics problems with finite deformation this spatial formulation provides a general accurate and efficient dg finite element computational framework for modeling nonlinear solid mechanics problems to obtain a consistent formulation we employ the incomplete interior penalty galerkin iipg method another requirement for achieving a fast convergence rate for newton s iterations is the consistent formulation of material integrators we show that material integrators that are well developed and tested in continuous galerkin cg methods can be fully exploited for dg methods by additionally performing stress returning on element interfaces finally for problems with pressure or follower loading stiffness contributed from loaded surfaces must also be consistently incorporated in this work we propose the truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems two formulations based on the co rotational and multiplicative decomposition based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity respectively two new terminologies the so called standard surface geometry stiffness and the penalty surface geometry stiffness are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals the performance of our dg formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes fast convergence rates for newton s iterations have been achieved in our iipg implementation

implicit gradient elasticity a new formulation of gradient elasticity is derived and implemented pade approximations are used to introduce an implicit relation between non local strains and displacements as a result the finite element interpolation requires only 0 continuous rather than 1 continuous shape functions the underlying energy functional is presented and it is found that the present formulation is of the mixed type whereby the non local strains act as the primary unknowns and the displacements as the auxiliary unknowns the implications on the interpolation are studied finally the influence of the additional length scale parameter on the global response is assessed copyright 2006 john wiley sons ltd

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

radial symmetry of self similar solutions for semilinear heat equations abstract the symmetry properties of positive solutions of the equation u 1 2 x u 1 p 1 u u p 0 in r n where n 2 p n 2 n was studied it was proved that u must be radially symmetric about the origin provided u x o x 2 p 1 as x and that there exist non radial solutions u satisfying lim sup x x 2 p 1 u x 0

godunov mixed methods for advection diffusion equations in multidimensions time split methods for multidimensional advection diffusion equations are considered in these methods advection is approximated by a godunov type procedure and diffusion is approximated by a low

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

a discontinuous galerkin formulation of non linear kirchhoff love shells discontinuous galerkin dg methods provide a means of weakly enforcing the continuity of the unknown field derivatives and have particular appeal in problems involving high order derivatives this feature has previously been successfully exploited comput methods appl mech eng 2008 197 2901 2929 to develop a formulation of linear kirchhoff love shells considering only the membrane and bending responses in this proposed one field method the displacements are the only unknowns while the displacement field is continuous the continuity in the displacement derivative between two elements is weakly enforced by recourse to a dg formulation it is the purpose of the present paper to extend this formulation to finite deformations and non linear elastic behaviors while the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple stress from the displacement field at the mid surface of the shell the non linear formulation considered implies the evaluation of the general stress tensor across the shell thickness leading to a reformulation of the internal forces of the shell nevertheless since the interface terms resulting from the discontinuous galerkin method involve only the resultant couple stress at the edges of the shells the extension to non linear deformations is straightforward copyright 2008 john wiley sons ltd

cohesive zone models higher order continuum theories and reliability methods for computational failure analysis a concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials the importance of the cohesive zone approach is emphasized and various ways to incorporate the cohesive zone methodology in discretization methods are discussed numerical representations of cohesive zone models suffer from a certain mesh bias for discrete representations this is caused by the initial mesh design while for smeared representations it is rooted in the ill posedness of the rate boundary value problem that arises upon the introduction of decohesion a proper representation of the discrete character of cohesive zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition of unity property of finite element shape functions the effectiveness of the approach is demonstrated for some examples at different scales moreover examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack true discontinuum when adhering to a continuum description of failure higher order continuum models must be used meshless methods are ideally suited to assess the importance of the higher order gradient terms as will be shown finally regularized strain softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes

multiplicity of self similar solutions for a critical equation we consider the equation u 12 x u f u u 2 2u x rn with 0 f superlinear and 2 2n n 2 for n 3 we prove that for each k n there exists k 0 such that the equation has at least k pairs of solutions provided 0 in the proof we use variational methods for the even functional associated to the equation

corner treatment by assigning dual tractions to every node for elastodynamics in td bem in this study a corner treatment on the boundary for time domain boundary element method td bem by assigning dual tractions to every node is proposed the proposed corner treatment on boundary mimics the real mechanical situation on the physical boundary for elastodynamic problem and the computation accuracy is good taking the rectangular boundary as an example the numerical implementation of the proposed corner treatment on boundary by assigning dual tractions to every node is demonstrated two verification examples one dimensional rod in finite medium and two dimensional circular cavity in infinite medium under dynamic loads are conducted in which the results from the td bem based on the proposed corner treatment on boundary are compared with the results from the analytical solutions good agreement between the analytical results and the td bem results based on the proposed corner treatment on boundary for td bem is achieved for the two verification examples

two gradient plasticity theories discretized with the element free galerkin method abstract in this paper the element free galerkin method is exploited to analyze gradient plasticity theories the use of this discretization method has the advantage that higher order continuity which can be required for the plastic multiplier field is provided for in passing we show that the regularization properties of the higher order gradients are necessary since similar to finite element methods a severe discretization sensitivity is encountered otherwise the mesh free discretization method is first applied to an established stress space gradient plasticity theory next a strain space gradient plasticity theory which employs an implicit averaging of an invariant strain measure is proposed and elaborated this theory is insensitive to the order of the interpolation polynomials and as a result the convergence of the newton raphson procedure is better the performance of both gradient plasticity models is examined for the mesh free discretization scheme a comparative study is carried out for a one dimensional bar in tension while the influence of the discretization in biaxial compression is studied next including that of the numerical parameters of the discretization scheme and that of the internal length scale contained in the gradient plasticity theories

a new discontinuous galerkin method for kirchhoff love shells abstract discontinuous galerkin methods dg have particular appeal in problems involving high order derivatives since they provide a means of weakly enforcing the continuity of the unknown field derivatives this paper proposes a new discontinuous galerkin method for kirchhoff love shells considering only the membrane and bending response the proposed one field method utilizes the weak enforcement in such a way that the displacements are the only unknowns while the rotation continuity is weakly enforced this work presents the formulation of the new discontinuous galerkin method for linear elastic shells demonstrates the consistency and stability of the proposed framework and establishes the method s convergence rate after a description of the formulation implementation into a finite element code these properties are demonstrated on numerical applications

positive solutions for a three point boundary value problem at resonance abstract this paper deals with the second order three point boundary value problem x t f t x t t 0 1 x 0 0 x x 1 the existence and multiplicity of positive solutions are investigated by means of the fixed point theorem in cones

upper and lower solutions for the singular p laplacian with sign changing nonlinearities and nonlinear boundary data this paper presents existence results for singular p laplacian equations with nonlinear boundary data our nonlinearity f t u in the differential equation is allowed to change sign and the singularity may occur at u 0 t 0 and t 1

a low order locking free hybrid discontinuous galerkin element formulation for large deformations abstract in this work a hybrid discontinuous galerkin dg quadrilateral element formulation is presented as usual in hybrid formulations the interior of the elements is treated separately from the skeleton which represents the element boundaries these are kinematically decoupled i e displacement jumps can occur between the skeleton and the interior of the elements the global degrees of freedom dofs are defined on the skeleton as the displacements at the corners which allows the implementation into existing finite element codes as usual in hybrid dg formulations the degrees of freedom in the interior are condensed out on the element level leading to the same number of global degrees of freedom as for continuous bilinear elements instead of using conventional shape functions in the interior the deformation gradient f is assumed constant within the element furthermore f is connected to the skeleton degrees of freedom via the weak form this leads to a very simple formulation and implementation the element is tested for several computational examples from the literature special choices of the penalty parameter are investigated which are partially derived analytically it is found that the element is free of volumetric and shear locking moreover the convergence is similar to that of other well known locking free finite element formulations

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

on an equivalence between a discontinuous galerkin method and reduced integration with hourglass stabilization for finite elasticity abstract discontinuous galerkin methods and methods of reduced integration with hourglass stabilization have different points of origin whereas the discontinuous galerkin idea was developed to investigate a wide range of applications among them problems of electrodynamics fluid and solid mechanics the reduced integration technology started out from the observation of the so called locking problem in linear elasticity today the two methodologies are still considered separately from each other although it has already been recognized that discontinuous galerkin methods can be also used to circumvent locking phenomena in elasticity the idea of the present contribution is to find a direct equivalence between a discontinuous galerkin method and a finite element technology based on reduced integration with hourglass stabilization the transfer between the two methods is carried out in the context of finite elasticity thus the derivation is valid for arbitrarily large deformations the main advantage of this analysis is that the choice of the free parameter in nitsche s method often also called stabilization parameter is no longer unclear but prescribed by the connection to the hourglass stabilization further important in this context is the replacement of the scalar parameter by a symmetric 2 2 matrix if two dimensional problems are considered which is here the case the result of the derivation is a discontinuous galerkin formulation which shows very good convergence behaviour not only in the limit of incompressibility but also for bending of thin structures

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

gradient enhanced damage modeling of cracked bodies by reproducing kernel element method abstract in crack modeling by extended finite element method the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics however for the more sophisticated condition of a gradient enhanced damage model firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required in this study the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method they are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost no new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth after describing the formulations several numerical examples are implemented to investigate the ability of the proposed method for both the linear elastic fracture mechanics and the gradient enhanced damage models

multiple positive and sign changing solutions of an elliptic equation with fast increasing weight and critical growth abstract we consider the following equation div k x u k x x u q 2 u q x k x u 2 2 u x r n where n 3 2 q 2 2 n n 2 0 is a parameter k x exp x 4 2 2 2 q 2 2 and 0 q x c r n using variational methods and delicate estimates we establish some existence and multiplicity of positive and sign changing solutions for the problem provided that the maximum of q x is achieved at different points

dispersion and isogeometric analyses of second order and fourth order implicit gradient enhanced plasticity models implicit gradient plasticity models incorporate higher order spatial gradients via an additional helmholtz type equation for the plastic multiplier so far the enrichment has been limited to second order spatial gradients resulting in a formulation that can be discretised using c0 continuous finite elements herein an implicit gradient plasticity model is formulated that includes a fourth order gradient term as well a comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second order gradient plasticity model is made using a dispersion analysis the higher order continuity requirement for the fourth order implicit gradient plasticity model has been met by exploiting the higher order continuity property of isogeometric analysis which uses nonuniform rational b splines as shape functions instead of lagrange polynomials the discretised variables displacements and plastic multiplier may require different orders of interpolation an issue that is also addressed numerical results show that both formulations can be used as a localisation limiter but that quantitative differences occur and a different evolution of the localisation band is obtained for 2 dimensional problems

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

an explicit discontinuous galerkin method for non linear solid dynamics formulation parallel implementation and scalability properties an explicit dynamics spatially discontinuous galerkin dg formulation for non linear solid dynamics is proposed and implemented for parallel computation dg methods have particular appeal in problems involving complex material response e g non local behavior and failure as even in the presence of discontinuities they provide a rigorous means of ensuring both consistency and stability in the proposed method these are guaranteed the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation the semi discrete system of ordinary differential equations is integrated in time using a conventional second order central difference explicit scheme a stability criterion for the time integration algorithm accounting for the influence of the dg discretization stability is derived for the equivalent linearized system this approach naturally lends itself to efficient parallel implementation the resulting dg computational framework is implemented in three dimensions via specialized interface elements the versatility robustness and scalability of the overall computational approach are all demonstrated in problems involving stress wave propagation and large plastic deformations copyright 2007 john wiley sons ltd

spectral analysis of localization in nonlocal and over nonlocal materials with softening plasticity or damage the paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials i e materials for which the yield limits soften as a function of the total strain both plasticity and damage models are considered it is verified analytically in a simple way that the nonlocal integral type model with degrading yield limit depending on the total strain works correctly if and only one adopts an unconventional nonlocal formulation introduced in 1994 by vermeer and brinkgreve and in 1996 by planas and by stromberg and ristinmaa which is here called for the sake of brevity over nonlocal because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is com pensated by assigning to the nonlocal variable weight greater than 1 this is equivalent to a nonlocal variable with a smooth positive weight function of total weight greater than 1 normalized by superposing a negative delta function spike at the center the spectral approach readily confirms that the nonlocal integral type generalization of softening plasticity with an additive format gives correct localization properties only if an over nonlocal formulation is adopted by contrast the nonlocal integral type generalization of softening plasticity with a multiplicative format provides real istic localization behavior just like the nonlocal integral type damage model and thus does not necessitate an over non local formulation the localization behavior of explicit and implicit gradient type models is also analyzed a simple analysis shows that plasticity and damage models with gradient type localization limiter whether explicit or implicit have very different localization behaviors 2005 elsevier ltd all rights reserved

a direct finite element implementation of the gradient dependent theory the enhanced non local gradient dependent theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical theories and the classical local continuum they are successful in explaining the size effects encountered at the micron scale and in preserving the well posedeness of the i bvp governing the solution of material instability triggering strain localization this is due to the incorporation of an intrinsic material length scale parameter in the constitutive description however the numerical implementation of these theories is not a direct task because of the higher order of the governing equations in this paper a direct computational algorithm for the gradient approach is proposed this algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches int j solids struct 1988 24 581 597 int j numer meth engng 1992 35 521 539 eng comput 1993 10 99 121 dissertation 1994 int j numer meth engng 1996 39 2477 2505 int j numer meth engng 1996 39 3731 3755 comput meth appl mech eng 1998 163 11 32 comput meth appl mech eng 1998 163 33 53 euro j mech a solids 1999 18 939 962 int j solids struct 2000 37 7481 7499 a predictor corrector scheme is proposed for the solution of the non linear algebraic problem from the fem the expressions of the continuum and consistent tangent matrices are provided the method is validated by conducting various numerical tests as a result pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered copyright 2005 john wiley sons ltd

conditions for locking free elasto plastic analyses in the element free galerkin method the use of finite elements with linear displacement fields in volume preserving elasto plastic problems can lead to volumetric locking due to the inability of the shape functions to describe certain displacement modes a response of the structure can be obtained which is much too stiff the element free galerkin efg method can provide shape functions that have an intrinsic richer nature and do not exhibit volumetric locking it is shown that the finite element fe method can be conceived as a specialisation of the efg method and that for a specific choice of the efg parameters the fe shape functions are retrieved then volumetric locking behaviour is found again however for other choices of the efg parameters shape functions are obtained that are able to describe isochoric elasto plasticity without volumetric locking

analysis of thin shells by the element free galerkin method a meshless approach to the analysis of arbitrary kirchhoff shells by the element free galerkin efg method is presented the shell theory used is geometrically exact and can be applied to deep shells the method is based on moving least squares approximant the method is meshless which means that the discretization is independent of the geometric subdivision into finite elements the satisfaction of the c1 continuity requirements is easily met by efg since it requires only c1 weights therefore it is not necessary to resort to mindlin reissner theory or to devices such as discrete kirchhoff theory the requirements of consistency are met by the use of a polynomial basis of quadratic or higher order a subdivision similar to finite elements is used to provide a background mesh for numerical integration the essential boundary conditions are enforced by lagrange multipliers membrane locking which is due to different approximation order for transverse and membrane displacements is removed by using larger domains of influence with the quadratic basis and by using quartic polynomial basis which can prevent membrane locking completely it is shown on the obstacle course for shells that the present technique performs well

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

a c0 discontinuous galerkin formulation for kirchhoff plates abstract a particular discontinuous galerkin finite element formulation for the simulation of kirchhoff plates is presented it is rotation free and utilises standard c 0 lagrange finite element basis functions with the required continuity imposed in a weak sense across element boundaries the implications of the scheme in terms of coercivity and convergence of the galerkin problem in various norms are studied with the formulation shown to be stable for any positive value of a penalty parameter a priori error estimates are supported by a range of numerical examples properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples based on the results of the analysis and numerical examples it is concluded that the formulation is robust accurate and relatively simple

a stabilized mixed discontinuous galerkin method for darcy flow abstract a new mixed stabilized discontinuous galerkin formulation for darcy flow is presented the formulation combines several attributes not simultaneously satisfied by other methods it is convergent for any combination of velocity and pressure interpolation higher than first order it exactly satisfies a mass balance on each element and it passes two and three dimensional constant flow patch tests for distorted element geometries the key ingredient in the formulation is a volumetric residual based stabilization term that does not involve any mesh dependent parameters

an implicit gradient model by a reproducing kernel strain regularization in strain localization problems abstract a reproducing kernel strain regularization rksr as a mathematical generalization of gradient theory and non local theory for strain localization problems is presented rksr introduces a correction of the weight function in the non local strain by imposition of gradient reproducing conditions both continuum and discrete forms of rksr are presented and they lead to an implicit representation of gradient models as such rksr provides a gradient type regularization to the localization problem without increasing the order of differentiation in the governing equations hence no additional boundary conditions are required and the need for higher order continuity for the approximation of unknowns in the governing equations is no longer an issue a von neumann spectral analysis is employed to study the spectral properties of rksr of various orders in one dimension it is shown that rksr almost duplicates the spectral properties of second and fourth order gradient theories in summary rksr reproduces the regularization properties of gradient methods without dealing with additional boundary conditions or higher order continuity issues

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

interior penalty discontinuous approximations of convection diffusion problems with parabolic layers a nonsymmetric discontinuous galerkin finite element method with interior penalties is considered for two dimensional convection diffusion problems with regular and parabolic layers on an anisotropic shishkin type mesh with bilinear elements we prove error estimates uniformly in the perturbation parameter in an integral norm associated with this method on different types of interelement edges we derive the values of discontinuity penalization parameters numerical experiments complement the theoretical results

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a characteristics mixed finite element method for advection dominated transport problems we define a new finite element method called the characteristics mixed method for approximating the solution to an advection dominated transport problem the method is based on a space time variational form of the advection diffusion equation our test functions are piecewise constant in space and in time they approximately follow the characteristics of the advective i e hyperbolic part of the equation thus the scheme uses a characteristic approximation to handle advection in time this is combined with a low order mixed finite element spatial approximation of the equation boundary conditions are incorporated in a natural and mass conservative fashion the scheme is completely locally conservative in fact on the discrete level fluid is transported along the approximate characteristics a postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux this has the effect of improving the rate of convergence of

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

on a class of nonlinear elliptic equations with fast increasing weight and critical growth abstract we are concerned with the existence of rapidly decaying solutions for the equation div k x u k x x u q 2 u k x u 2 2 u x r n where n 3 2 q 2 2 n n 2 0 is a parameter k x exp x 4 2 and the number is given by 2 2 q 2 2 we obtain a positive solution if 2 q 2 and a sign changing solution if q 2 the existence results depend on the values of the parameter in the proofs we apply variational methods

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

upper and lower solutions method and a second order three point singular boundary value problem the singular boundary value problem u q t g t u u 0 t 0 1 h 0 1 c 0 1 u 0 0 u 1 cu h is studied in this paper the singularity may appear at t 0 and the function g may be superlinear at u and change sign the existence of solutions is obtained via an upper and lower solutions method

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

a higher order stress based gradient enhanced damage model based on isogeometric analysis abstract the micro damage associated with diffuse fracture processes in quasi brittle materials can be described by continuum damage mechanics in order to overcome the mesh dependence of local damage formulations non local and gradient enhanced approaches are often employed in this manuscript a higher order stress based gradient enhanced formulation is proposed which exploits the higher order continuity of b spline functions in isogeometric analysis iga the proposed formulation does not require the decomposition of the fourth order model into two second order models two numerical examples are presented to demonstrate the performance of the formulation and to compare the obtained solutions with results from conventional gradient enhanced damage formulation

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

on l2 error estimate for nonsymmetric interior penalty galerkin approximation to linear elliptic problems with nonhomogeneous dirichlet data in this paper we present improved a priori error estimates for a nonsymmetric interior penalty galerkin method nipg with super penalty for the problem email protected f in w and u g on email protected using piecewise polynomials of degree less than or equal to r our new l 2 error estimate is of order h r r 1 2 when email protected h r 1 2 email protected and is optimal i e of order h r r 1 when email protected h r 1 email protected where h denotes the mesh size numerical experiments are presented to illustrate the theoretical results

a discontinuous galerkin formulation for a strain gradient dependent damage model abstract the numerical solution of strain gradient dependent continuum problems has been dogged by continuity demands on the basis functions for most commonly accepted models solutions using the finite element method demand c 1 continuity of the shape functions here recent developments in discontinuous galerkin methods are explored and exploited for the solution of a prototype non linear strain gradient dependent continuum model a formulation is developed that allows the rigorous solution of a strain gradient damage model using standard c 0 shape functions the formulation is tested in one dimension for the simplest possible finite element formulation continuous piecewise linear displacement and constant on elements internal variable numerical results are shown to compare excellently with a benchmark solution the results are remarkable given the simplicity of the proposed formulation

a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis this work is the second of a two part investigation into the use of discontinuous galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity part i j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 presented the formulation and analysis of such problems this part focusses on algorithmic and computational aspects of the problem in particular it is shown that the predictor corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity and to discontinuous galerkin formulations conditions for convergence of the algorithms are presented for the elastic secant and consistent tangent predictors the form of the consistent tangent modulus is established for the case of gradient plasticity a selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms as well as features of the model of gradient plasticity adopted here

three dimensional isogeometric solutions to general boundary value problems of toupin s gradient elasticity theory at finite strains abstract we present to the best of our knowledge the first complete three dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity we have chosen for our work toupin s theory toupin 1962 one of the more general formulations of strain gradient elasticity our framework has three crucial ingredients the first is isogeometric analysis hughes et al 2005 which we have adopted for its straightforward and robust representation of c 1 continuity the second is a weak treatment of the higher order dirichlet boundary conditions in the formulation which control the development of strain gradients in the solution the third ingredient is algorithmic automatic differentiation which eliminates the need for linearization by hand of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains we present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions we discuss length scale effects the role of higher order boundary conditions and perhaps most importantly the relevance of the framework to problems with elastic free energy density functions that are non convex in strain space

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

on the c1 continuous discretization of non linear gradient elasticity a comparison of nem and fem based on bernstein bezier patches in gradient elasticity the appearance of strain gradients in the free energy density leads to the need of c 1 continuous discretization m ethods in the present work the performances of c 1 finite elements and the c 1 natural element method nem are compared the triangular argyris and hsieh clough tocher finite elements are reparametrized in terms of the bernstein polynomials the quadrilateral bogner fox schmidt element is used in an isoparametric framework for which a preprocessing algorithm is presented additionally the c 1 nem is applied to non linear gradient elasticity several numerical examples are analyzed to compare the convergence behavior of the different methods it will be illustrated that the isoparametric elements and the nem show a significantly better performance than the triangular elements

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

a discontinuous galerkin method for strain gradient dependent damage study of interpolations and convergence abstract a discontinuous galerkin method has been developed for strain gradient dependent damage the strength of this method lies in the fact that it allows the use of c 0 interpolation functions for continuum theories involving higher order derivatives while in a conventional framework at least c 1 interpolations are required the discontinuous galerkin formulation thereby offers significant potential for engineering computations with strain gradient dependent models when using basis functions with a low degree of continuity jump conditions arise at element edges which are incorporated in the weak form in addition to the formulation itself a detailed study of the convergence properties of the method for various element types is presented and an error analysis is undertaken numerical results of some one and two dimensional problems are presented and discussed