discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

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

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

stochastic bifurcation models we study an ordinary differential equation controlled by a stochastic process we present results on existence and uniqueness of solutions on associated local times trotter and ray knight theorems and on time and direction of bifurcation a relationship with lipschitz approximations to brownian paths is also discussed

nature of singularities for analyticity domains of invariant curves we present theoretical arguments based on infinite dimensional bifurcation theory and numerical evidence based on nonperturbative methods that the boundaries of analyticity of invariant curves can be described as an accumulation of branch points which are typically of order 2 we show how this fact would explain previous numerical results of several authors and how it suggests more efficient numerical algorithms which we implement

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

break up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables two area preserving twist maps are associated to a smooth closed convex table the classical billiard map and the dual billiard map when the table is circular these maps are integrable and their phase spaces are foliated by invariant curves the invariant curves with rational rotation numbers are resonant and do not persist under generic perturbations of the circle we present a sufficient condition for the break up of these curves this condition is expressed directly in terms of the fourier coefficients of the perturbation it follows from a standard melnikov argument

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity analytical estimates this is the second of two papers in which we motivate introduce and analyze a new type of strategy for the stabilization of discontinuous galerkin dg methods in nonlinear elasticity problems the foremost goal behind it is to enhance the robustness of the method without deteriorating the accuracy of the resulting solutions its distinctive property is that for nonlinear elastic problems the stabilization term is solution dependent and hence it is termed an adaptive stabilization mechanism the key contribution of this paper is the construction of a stabilization strategy for which the method is perfectly stable since the stabilization parameters can be explicitly computed as part of the numerical solution this is accomplished through the main result of this paper which consists of a theorem that provides lower bounds for the size of the stabilization parameters numerical examples confirm the guaranteed stability of the resulting method however they also show that the computed lower bounds overestimate the minimum amount of stabilization needed negatively affecting the approximation properties of the method these results underscore the fact that a better understanding of the stabilization mechanisms is needed in order to construct a method that is both robust and efficient

singularities of periodic orbits near invariant curves abstract we show numerically for standard like maps how the singularities in the complex parameter of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number furthermore we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits the results are compared with analyses performed by pade approximants greene s method root criterion and the prediction by renormalization theory

ghost circles for twist maps abstract completely ordered invariant circles are found for the gradient of the energy flow in the state space containing the critical sets corresponding to the birkhoff orbits of all rotation number in particular these ghost circles contain the aubry mather sets and map invariant circles as completely critical sets when these exist we give a criterion for a sequence of rational ghost circles to converge to a completely critical one

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

modeling of convection dominated thermoporomechanics problems using incomplete interior penalty galerkin method abstract in order to obtain converged and accurate numerical solutions in general small time steps are required for time dependent problems for popular continuous finite elements however the use of small time steps often results in oscillatory pressures and temperatures for thermoporomechanics problems with low conductivities moreover convection dominated thermal poromechanics problems often have very sharp temperature fronts that propagate over spatial domains when applied to these problems continuous elements present serious deficiencies in that large error induced in coarse mesh zones will propagate and pollute the solutions in fine mesh zones such oscillatory and polluted solutions are nonphysical and must be precluded we propose discontinuous galerkin methods to be alternatives to handle these challenges this paper focuses on the finite element formulation of the incomplete interior penalty galerkin method on thermoporoelasticity problems with heat convection in this formulation we propose discontinuous spaces for all three field variables to deal with the convection effect an upwinding scheme is simply and elegantly implemented through element interfaces which is favored by the discontinuous galerkin framework in addition the inner pressure boundary conditions important to many engineering problems are also taken into account in our discontinuous galerkin formulation excellent performance of this discontinuous galerkin method on precluding temperature oscillations and on blocking error propagation is demonstrated through solving a convection dominated oil well injection problem

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

learning mutation and long run equilibria in games an evolutionary model with a finite number of players and with stochastic mutations is analyzed the expansion and contraction of strategies is linked to their current relative success but mutuation perturbing the system from its deterministic evolution are present as well the focus is on the long run implications of ongoing mutations which drastically reduce the set of equilibria for 2 by 2 symmetric games with two symmetric strict nash equilibria the risk dominant equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected even though there is another strict nash equilibrium copyright 1993 by the econometric society

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

an optimally convergent discontinuous galerkin based extended finite element method for fracture mechanics the extended finite element method xfem enables the representation of cracks in arbitrary locations of a mesh we introduce here a variant of the xfem rendering an optimally convergent scheme its distinguishing features are as follows a the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip the enrichment region b only one and two enrichment functions are added for anti plane shear and planar problems respectively and c the relaxation of the continuity between the enrichment region and the rest of the domain and the adoption of a discontinuous galerkin dg method therein the method is provably stable for any positive value of a stabilization parameter and by weakly enforcing the continuity between the two regions it eliminates blending elements partly responsible for the suboptimal convergence of some early xfems moreover the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined more importantly the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns quadrature strategies required for the optimal convergence are also discussed finally the dg method was modified to retain stability based on an inf sup condition copyright 2009 john wiley sons ltd

bryuno function and the standard map for the standard map the homotopically non trivial invariant curves of rotation number satisfying the bryuno condition are shown to be analytic in the perturbative parameter e provided e is small enough the radius of convergence of the lindstedt series sometimes called critical function of the standard map is studied and the relation with the bryuno function b is derived the quantity log 2 b is proved to be bounded uniformly in

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

rapid non linear finite element analysis of continuous and discontinuous galerkin methods in matlab matlab is adept at the development of concise finite element fe routines however it is commonly perceived to be too inefficient for high fidelity analysis this paper aims to challenge this preconception by presenting two optimised fe codes for both continuous galerkin cg and discontinuous galerkin dg methods although this has previously been achieved for linear elastic problems no such optimised matlab script currently exists which includes the effects of material non linearity to incorporate these elasto plastic effects the externally applied load is split into a discrete number of loadsteps equilibrium is determined at each loadstep between the externally applied load and the arising internal forces using the newton raphson method the optimisation of the scripts is primarily achieved using vectorised blocking algorithms which minimise ram to cache overheads and maximise cache reuse the optimised codes yielded maximum speed gains of 25 7 and 10 1 when compared to the corresponding unoptimised scripts for cg and dg respectively it was identified that with increasing refinement of the mesh the solver time begins to dominate the overall simulation time this bottleneck has a greater disadvantage on the dg code predominantly due the asymmetric nature of the global stiffness matrix the implementation of an efficient solver would see further improvement to the overall run times particularly for large problems

computation of domains of analyticity for some perturbative expansions of mechanics abstract we compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics we use pade approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence we also use non perturbative methods based on direct computation of the invariant circles and in analogy with greene s criterion approximation by circles with rational rotation we find that the domains computed by all the methods agree within the limits of accuracy we also study rigorously the nature of the singularities when the frequency is rational

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

a continuation method to study periodic orbits of the froeschle map abstract the dynamics of many hamiltonian systems with three degrees of freedom is represented by the froeschle map which is symplectic and four dimensional in this paper we study sequences of periodic orbits approaching the invariant tori in order to obtain information about its stability a homotopic method is used to continue the branches of periodic orbits we found that the existence of turning points is related to the linear stability of the periodic orbit and its rotation vector

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

a hybrid high order method for finite elastoplastic deformations within a logarithmic strain framework we devise and evaluate numerically a hybrid high order hho method for finite plasticity within a logarithmic strain framework the hho method uses as discrete unknowns piecewise polynomials of order k 1 on the mesh skeleton together with cell based polynomi als that can be eliminated locally by static condensation the hho method leads to a primal formulation supports polyhedral meshes with non matching interfaces is free of volumetric locking the integration of the behavior law is performed only at cell based quadrature nodes and the tangent matrix in newton s method is symmetric moreover the principle of virtual work is satisfied locally with equilibrated tractions various two and three dimensional benchmarks are presented as well as comparison against known solutions with an industrial software using conforming and mixed finite elements

formulation of implicit finite element methods for multiplicative finite deformation plasticity some constitutive and computational aspects of finite deformation plasticity are discussed attention is restricted to multiplicative theories of plasticity in which the deformation gradients are assumed to be decomposable into elastic and plastic terms it is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non symmetric even for associated plastic flow both explicit i e no equilibrium iteration and implicit finite element formulations are considered an assumed strain formulation is used to accommodate the near incompressibility associated with fully developed isochoric plastic flow as an example of explicit integration the rate tangent modulus method is reviewed in some detail an implicit scheme is derived for which the consistent tangents resulting in quadratic convergence of the equilibrium iterations can be written out in closed form for arbitrary material models all the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method

shape of analyticity domains of lindstedt series the standard map the analyticity domains of the lindstedt series for the standard map are studied numerically using pade approximants to model their natural boundaries we show that if the rotation number is a diophantine number close to a rational value p q then the radius of convergence of the lindstedt series becomes smaller than the critical threshold for the corresponding kol mogorov arnol d moser curve and the natural boundary on the plane of the complexified perturbative parameter acquires a flower like shape with 2q petals

on the algorithmic and implementational aspects of a discontinuous galerkin method at finite strains in this work algorithmic modifications are proposed and analyzed for a recently developed stabilized finite strain discontinuous galerkin dg method the distinguishing feature of the original method referred to as vmdg is a consistently derived expression for the numerical flux and stability tensor that account for evolving material and geometric nonlinearity in the vicinity of the interface herein the proposed modifications involve simplifications to the residual force vector and tangent stiffness matrix of the vmdg method that lead to formulations similar to other existing dg methods but retain the enhanced definition for the stability parameters the primary objective is to reduce the costs associated with implementing the method as well as executing simulations while retaining accuracy and flexibility thereby making the formulation amenable to boarder material classes such as inelasticity each simplification has associated implications on the mathematical and algorithmic properties of the method such as l 2 convergence rate solution accuracy continuity enforcement and stability of the nonlinear equation solver these implications are carefully quantified and assessed through a comprehensive numerical performance study the range of two and three dimensional problems under consideration involves both isotropic and anisotropic materials both triangular and quadrilateral element types are employed along with h and p refinement the ability of the proposed methods to produce stable and accurate results for such a broad class of problems is highlighted

periodic and quasi periodic orbits for the standard map we consider both periodic and quasi periodic solutions for the standard map and we study the corresponding conjugating functions i e the functions conjugating the motions to trivial rotations we compare the invariant curves with rotation numbers satisfying the bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents n p n q n we prove the following results for n 1 for rotation numbers n n we study the radius of convergence of the conjugating functions and we find lower bounds on them which tend to a limit which is a lower bound on the corresponding quantity for 2 the periodic orbits consist of points which are more and more close to the invariant curve with rotation number 3 such orbits lie on analytical curves which tend uniformly to the invariant curve

fast equilibrium selection by rational players living in a changing world we study a coordination game with randomly changing payoffs and small frictions in changing actions using only backwards induction we find that players must coordinate on the risk dominant equilibrium more precisely a continuum of fully rational players are randomly matched to play a symmetric 2 x 2 game the payoff matrix changes according to a random walk players observe these payoffs and the population distribution of actions as they evolve the game has frictions opportunities to change strategies arrive from independent random processes so that the players are locked into their actions for some time as the frictions disappear each player ignores what the others are doing and switches at her first opportunity to the risk dominant action history dependence emerges in some cases when frictions remain positive

breakup of invariant tori for the four dimensional semi standard map abstract we compute the domain of existence of two dimensional invariant tori with fixed frequency vectors for a four dimensional complex symplectic map the map is a generalization of the semi standard map studied by greene and percival it has three parameters a 1 and a 2 representing the strength of the kicks in each degree of freedom and the coupling the domain of existence of a torus in a 1 a 2 is shown to be complete and log convex for fixed k a 1 a 2 explicit bounds on the domain for fixed k are obtained numerical results show that quadratic irrationals can be more robust than the cubic irrational the spiral mean

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

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

discontinuous galerkin methods for non linear elasticity this paper presents the formulation and a partial analysis of a class of discontinuous galerkin methods for quasistatic non linear elasticity problems these methods are endowed with several salient features the equations that define the numerical scheme are the euler lagrange equations of a one field variational principle a trait that provides an elegant and simple derivation of the method in consonance with general discontinuous galerkin formulations it is possible within this framework to choose different numerical fluxes numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted finally a conceivable surprising characteristic is that as demonstrated with numerical examples these methods provide a given accuracy level for a comparable and often lower computational cost than conforming formulations stabilization is occasionally needed for discontinuous galerkin methods in linear elliptic problems in this paper we propose a sufficient condition for the stability of each linearized non linear elastic problem that naturally includes material and geometric parameters the latter needed to account for buckling we then prove that when a similar condition is satisfied by the discrete problem the method provides stable linearized deformed configurations upon the addition of a standard stabilization term we conclude by discussing the complexity of the implementation and propose a computationally efficient approach that avoids looping over both elements and element faces several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous galerkin method within the class copyright 2006 john wiley sons ltd

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

galerkin schemes and inverse boundary value problems in reflexive banach spaces we develop the galerkin method for a recent version of the lax milgram theorem the generation of the corresponding finite dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive banach spaces in question in addition we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

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 general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors we study numerically the complex domains of validity for kam theory in generalized standard mappings we compare methods based on pade approximants and methods based on the study of periodic orbits

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

on the numerical integration of a class of pressure dependent plasticity models an unconditionally stable algorithm for the numerical integration of elastoplastic pressure dependent constitutive relations is analysed in detail in this paper the application of the method to plane stress problems in which the out of plane strain component is not defined kinematically is discussed the tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations the algorithm is applied to gurson s constitutive model some one dimensional problems are solved and comparisons with exact solutions are made the paper closes with a numerical study of the necking of an axi symmetric specimen using gurson s plasticity model to describe the constitutive behaviour of the material

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