approximation of cahn hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with c1 elements a variational formulation and c1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase separation processes described by the cahn hilliard diffuse interface model of transport in a mixture or alloy the adaptive scheme is guided by a laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth order non linear problem and is implemented in a parallel solution framework it is then applied to resolve complex evolving interfacial solution behavior for 2d and 3d simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase transformation applications of interest simulation results and adaptive performance are discussed the scheme permits efficient robust multiscale resolution and interface characterization copyright 2008 john wiley sons ltd

mixed variational principles and robust finite element implementations of gradient plasticity at small strains summary this work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to plasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type pdes including micro structural boundary conditions this incorporates non local plastic effects based on length scales which reflect properties of the material micro structure we develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity which is outlined for the simple model problem of von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element fe design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust fe implementations of gradient plasticity this includes a rational method for the definition of elastic plastic boundaries in gradient plasticity along with a post processor that defines the plastic variables in the elastic range we discuss alternative mixed fe designs of the coupled problem including a local global solution strategy of short range and long range fields this includes several new aspects such as extended q1p0 type and mini type finite elements for gradient plasticity all methods are derived in a rigorous format from variational principles numerical benchmarks address advantages and disadvantages of alternative fe designs and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity copyright 2013 john wiley sons ltd

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

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

superconvergence analysis of conforming finite element method for nonlinear schrodinger equation the main aim of this paper is to apply the conforming bilinear finite element to solve the nonlinear schrodinger equation nlse firstly the stability and convergence for time discrete scheme are proved secondly through a new estimate approach the optimal order error estimates and superclose properties in h1 norm are obtained with backward euler b e and crank nicolson c n fully discrete schemes the global superconvergence results are deduced with the help of interpolation postprocessing technique finally some numerical examples are provided to verify the theoretical analysis

galerkin alternating direction method for a kind of three dimensional nonlinear hyperbolic problems a kind of three dimensional nonlinear hyperbolic equation is firstly transformed into a first order system of equations then the galerkin alternating direction procedure for the system is derived the error estimates of the procedure in h 1 norm and l 2 norm respectively are obtained by using the theory and techniques of priori estimate of differential equations the numerical experiment is also given to support the theoretical analysis comparing the results of numerical example with the theoretical analysis they are coincided

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

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

linear oblique derivative problems for the uniformly elliptic hamilton jacobi bellman equation on demontre la resolubilite et l unicite des problemes aux valeurs limites generaux et de dirichlet

formulation and numerical exploitation of mixed variational principles for coupled problems of cahn hilliard type and standard diffusion in elastic solids summary this work develops variational principles for the coupled problem of standard and extended cahn hilliard type species diffusion in solids undergoing finite elastic deformations it shows that the coupled problem of diffusion in deforming solids accounting for phenomena like swelling diffusion induced stress generation and possible phase segregation caused by the diffusing species is related to an intrinsic mixed variational principle it determines the rates of deformation and concentration along with the chemical potential where the latter plays the role of a mixed variable the principle characterizes a canonically compact model structure where the three governing equations involved that is the mechanical equilibrium condition the mass balance for the species content and a microforce balance that determines the chemical potential appear as the euler equation of a variational statement the existence of the variational principle underlines an inherent symmetry of the coupled deformation diffusion problem this can be exploited in the numerical implementation by the construction of time discrete and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation diffusion problem based on low order basis functions they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speedup and reduction of data storage when compared with non symmetric implementations this is a strong argument for the use of the developed variational principles in the computational design of deformation diffusion problems copyright 2014 john wiley sons ltd

the analysis of a finite element method with streamline diffusion for the compressible navier stokes equations this paper deals with the rigorous error analysis of finite element solutions of the two and three dimensional nonstationary compressible navier stokes equations the streamline diffusion technique is applied to the continuity equation to obtain a stabilized finite element scheme it is proved that the numerical formulation has a unique solution and the solution is convergent an a priori error estimate uniform in time is obtained

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

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

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

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

comparative computational analysis of the cahn hilliard equation with emphasis on c1 continuous methods the numerical treatment of the fourth order cahn hilliard equation is nonstandard using a galerkin method necessitates for instance piecewise smooth and globally c 1 continuous basis functions or a mixed formulation the latter is obtained introducing an auxiliary field which allows to rephrase the cahn hilliard equation as a set of two coupled second order equations in view of this the formulation in terms of the primal unknown appears to be a more intuitive and natural choice but requires a c 1 continuous interpolation therefore isogeometric analysis using a spline basis and natural element analysis are addressed in the present contribution mixed second order finite element methods introducing the chemical potential or alternatively a nonlocal concentration as auxiliary field serve as references to which both higher order methods are compared in terms of accuracy and efficiency

mean values of subsolutions of elliptic and parabolic equations abstract integral averages of weak subsolutions and supersolutions in rn of quasilinear elliptic and parabolic equations are investigated the important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena harnack type inequalities are established in terms of these integral averages author

an energetic formulation of a gradient damage model for concrete and its numerical implementation abstract the energetic formulation of a rate independent system assumes that the evolution of the system is driven by two scalar valued functions the storage energy functional and the dissipation pseudo potential the evolution of the material system is then characterized by two energetic principles the global stability condition and the energy balance between stored and dissipated energies with the work of external loading rate independent generalized standard materials are endowed with such structure whereas the existence of such energetic structure is not so apparent in materials with non associated flow rule in this paper we consider a rate independent gradient damage model for concrete where the evolution of the damage does not follow the normality rule we will show that such model can be nevertheless derived by a dissipation potential at the expense of having a state variable dependent potential and therefore the energetic formulation can be obtained also in this case after introducing the incremental minimization problem consistent with such formulation we obtain a discrete version of the stability condition and establish lower and upper a priori energy bounds met by the energetic solution these are fundamental results for the analysis of the formulation the actual numerical solution of the incremental minimization problem is realized by considering first a variable splitting in order to treat the gradient of the damage field as an independent variable and then by applying the augmented lagrangian method to tackle with the resulting constrained optimization problem we solve the first order stationarity conditions of the augmented lagrangian functional by a path following newton s method based on the energy dissipation rate control we show that we are able to describe highly non linear responses of the material such as softening branches and snap back responses several numerical tests are performed to verify the objectivity of the formulation and of the proposed numerical method details of the numerical implementation are also given

local c 0 α estimates for viscosity solutions of neumann type boundary value problems abstract in this article we prove the local c 0 regularity and provide c 0 estimates for viscosity solutions of fully nonlinear possibly degenerate elliptic equations associated to linear or nonlinear neumann type boundary conditions the interest of these results comes from the fact that they are indeed regularity results and not only a priori estimates from the generality of the equations and boundary conditions we are able to handle and the possible degeneracy of the equations we are able to take into account in the case of linear boundary conditions

a thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking a new gradient damage model is derived starting from thermodynamic statements nonstandard terms are added to the free energy potential for this purpose in contrast with existing approaches these terms are related to the kinematics of the problem and therefore come into play already in the elastic regime following the argument that the second law of thermodynamics need only be satisfied globally for a nonlocal continuum an energy balance equation is derived which must be solved together with the usual equilibrium equations numerical solutions of this coupled set of equations have been generated for a one dimensional problem in order to study the localisation behaviour of the new model in the limit where the elastic modulus related to the energy balance goes to 0 an earlier gradient damage model is retrieved this simplified model is used to study cracking of brick masonry under the influence of mechanical loading

phase field modeling of ductile fracture at finite strains a robust variational based numerical implementation of a gradient extended theory by micromorphic regularization summary this work provides a robust variational based numerical implementation of a phase field model of ductile fracture in elastic plastic solids undergoing large strains this covers a computationally efficient micromorphic regularization of the coupled gradient plasticity damage formulation the phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics it has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture specific computational structures such as finite element design of crack discontinuities or intricate crack tracking algorithms the proposed gradient extended plasticity damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities this ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post critical ranges the proposed setting is rooted in a canonical variational principle the coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture the latter represents a coupled resistance to plasticity and damage depending on the gradient extended internal variables that enter plastic yield functions and fracture threshold functions with this viewpoint on the generalized internal variables at hand the thermodynamic formulation is outlined for gradient extended dissipative solids with generalized internal variables that are passive in nature it is specified for a conceptual model of von mises type elasto plasticity at finite strains coupled with fracture the canonical theory proposed is shown to be governed by a rate type minimization principle which fully determines the coupled multi field evolution problem this is exploited on the numerical side by a fully symmetric monolithic finite element implementation an important aspect of this work is the regularization towards a micromorphic gradient plasticity damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms this enhances the robustness of the finite element implementation in particular on the side of gradient plasticity the performance of the formulation is demonstrated by means of some representative examples copyright 2016 john wiley sons ltd

the nonlinear oblique derivative problem for quasilinear elliptic equations on considere des problemes aux valeurs limites pour une equation elliptique quasilineaire d ordre 2 qu 0 dans ou est un domaine borne de r n n 2 et q est donne sur c 2 par qu x a ij x u x du x d ij u x a x u x du x

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix a model for phase separation in multicomponent systems is studied in particular the possibility of a concentration dependence of the mobility matrix is taken into account this leads to a system of fourth order degenerate parabolic partial differential equations we derive these equations from balance laws show some properties of the model and prove a global existence result for the degenerate system

numerical analysis and simulation for a nonlinear wave equation in this work we study a nonlinear wave equation depending on different norms of the initial conditions has bounded solution for all t 0 or 0 t t 0 for some t 0 0 we also prove that the solution may blow up at t 0 proofs of some the analytical results listed are sketched or given for approximate numerical solutions we use the finite element method in the spatial variable and the finite difference method in time the nonlinear system for each time step is solved by newton s modified method we present numerical analysis for error estimates and numerical simulations to illustrate the convergence of the theoretical results we present too the singularity points x t where the blow up occurs for different values in a numerical simulation

variational boundary conditions based on the nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled cahn hilliard equation the primal variational formulation of the fourth order cahn hilliard equation requires c1 continuous finite element discretizations e g in the context of isogeometric analysis in this paper we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the cahn hilliard equation in primal variables our formulation is based on the symmetric variant of nitsche s method does not introduce additional degrees of freedom and is shown to be variationally consistent in contrast to strong enforcement the new boundary condition formulation can be naturally applied to any mapped isogeometric parametrization of any polynomial degree in addition it preserves full accuracy including higher order rates of convergence which we illustrate for boundary fitted discretizations of several benchmark tests in one two and three dimensions unfitted cartesian b spline meshes constitute an effective alternative to boundary fitted isogeometric parametrizations for constructing c1 continuous discretizations in particular for complex geometries we combine our variational boundary condition formulation with unfitted cartesian b spline meshes and the finite cell method to simulate chemical phase segregation in a composite electrode this example involving coupling of chemical fields with mechanical stresses on complex domains and coupling of different materials across complex interfaces demonstrates the flexibility of variational boundary conditions in the context of higher order unfitted isogeometric discretizations

a new error analysis of characteristics mixed fems for miscible displacement in porous media the method of characteristics type is especially effective for convection dominated diffusion problems due to the nature of characteristic temporal discretization the method allows one to use a large time step in many practical computations while all previous theoretical analyses always required certain restrictions on the time stepsize here we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in mathbb r d d 2 3 for this purpose we introduce a new characteristic time discrete system we prove that the l 2 error bound the characteristic time discrete systemof the fully discrete method of characteristics to the time discrete system is tau independent and the numerical solution is bounded in w 1 infty norm unconditionally with the boundedness optimal error estimates are established in a traditional manner numerical results confirm our th

a generalized inf sup test for multi field mixed variational methods abstract we present a variational approach towards identifying conditions for the stability of galerkin methods for multi field saddle point problems in continuum mechanics and continuum thermodynamics with free energy functions that have positive second variations at critical points the framework aims to generalize the discrete inf sup theory and its numerical verification in the context of general problems in an arbitrary number of fields for both linear and nonlinear problems in utilizing a linearized second derivative test the proposed scheme is purely based on uniqueness properties of a mixed lagrangian around the solution thus providing a purely variational requirement for the well posedness of galerkin approximations in maintaining the dependence on parameters and in incorporating the notion of multifold inf sup conditions the scheme provides a generalization of the classical lbb theory that allows the study of multi field saddle point problems in the limit of vanishing parameters and their connection to generalized numerical eigenvalue tests the proposed generalization is trusted to provide a helpful tool for the development of mixed methods that arise in many novel engineering problems due to the coupling of multiple irreversible phenomena or the incorporation of lagrange multipliers in advanced methods an emphasis is given with regard to time dependent problems in irreversible thermodynamics that arise from biot type variational formulations if conjugate variables are employed by means of a legendre fenchel transformation examples are given for a two field variational principle in finite deformation poroelasticity in the undrained limit a three field variational principle for elasticity in the incompressible limit and a recent four field variational principle in gradient extended plasticity

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

variational gradient plasticity at finite strains part ii local global updates and mixed finite elements for additive plasticity in the logarithmic strain space the second part of our work on variational inelasticity with long range effects outlines the formulation and finite element implementation of additive finite gradient plasticity in the logarithmic strain space it is considered to be the most simple approach to finite plasticity suitable for the purely phenomenological description of polycrystalline metals or amorphous materials if structures of the geometrically linear theories are defined in the lagrangian logarithmic strain space we start from a mixed saddle point principle for metric type additive plasticity which is specified for the important model problem of isochoric von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust finite element implementations of gradient plasticity it includes a rational method for the definition of elastic plastic boundaries epbs in gradient plasticity along with a postprocessor that defines the plastic variables in the elastic range we discuss alternative mixed finite element designs of the coupled problem including a local global solution strategy of short and long range fields all methods are derived in a rigorous format from variational principles numerical benchmarks demonstrated the excellent performance of the proposed mixed variational approach to gradient plasticity in the logarithmic strain space

theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity the paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains starting from the classical local continuum formulation which fails to produce physically meaningful and numerically converging results within localization computations a thermodynamically motivated gradient plasticity formulation is envisioned the model is based on an assumption for the helmholtz free energy incorporating the gradient of the internal history variable a yield condition and the postulate of maximum dissipation resulting in an associated structure as a result the driving force conjugated to the hardening evolution is identified as the quasi non local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux at the numerical side besides the balance of linear momentum the algorithmic consistency condition has to be solved in weak form thereby the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non linear programming moreover different discretization techniques are proposed in order to compare the fe performance in local plasticity with the advocated gradient formulation both for hardening and softening copyright 2001 john wiley sons ltd

mixed variational potentials and inherent symmetries of the cahn hilliard theory of diffusive phase separation this work shows that the cahn hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential the principle characterizes a canonically compact model structure where the two balances involved for the species content and microforce appear as the euler equations of a variational statement the existence of the variational principle underlines an inherent symmetry in the two field representation of the cahn hilliard theory this can be exploited in the numerical implementation by the construction of time and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the finite element solution of the cahn hilliard equation based on low order basis functions leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speed up and reduction of data storage when compared with non symmetric implementations in this sense the potentials developed are believed to be fundamental ingredients to a deeper understanding of the cahn hilliard theory

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

unconditional superconvergence analysis of a new mixed finite element method for nonlinear sobolev equation in this paper a new mixed finite element scheme is proposed for the nonlinear sobolev equation by employing the finite element pair q11 q01 q10 based on the combination of interpolation and projection skill as well as the mean value technique the independent superclose results of the original variable u in h1 norm and the variable q a u u t b u u in l2 norm are deduced for the semi discrete and linearized fully discrete systems is the temporal partition parameter what s more the new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained unconditionally while previous literature always require certain time step conditions finally some numerical results are provided to confirm our theoretical analysis and show the efficiency of the method

unconditionally optimal error estimates of a crank nicolson galerkin method for the nonlinear thermistor equations this paper focuses on unconditionally optimal error analysis of an uncoupled and linearized crank nicolson galerkin finite element method for the time dependent nonlinear thermistor equations in d dimensional space d 2 3 in our analysis we split the error function into two parts one from the spatial discretization and one from the temporal discretization by introducing a corresponding time discrete elliptic system we present a rigorous analysis for the regularity of the solution of the time discrete system and error estimates of the time discretization with these estimates and the proved regularity optimal error estimates of the fully discrete crank nicolson galerkin method are obtained unconditionally numerical results confirm our analysis and show the efficiency of the method

finite element approximation of a model for phase separation of a multi component alloy with non smooth free energy a model for the phase separation of a multi component alloy with non smooth free energy is considered an error bound is proved for a fully practical piecewise linear finite element approximation using a backward euler time discretization an iterative scheme for solving the resulting nonlinear algebraic system is analysed finally numerical experiments with three components in one and two space dimensions are presented in the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem which we construct explicitly

uniform error estimates of finite difference methods for the nonlinear schrodinger equation with wave operator we establish uniform error estimates of finite difference methods for the nonlinear schrodinger equation nls perturbed by the wave operator nlsw with a perturbation strength described by a dimensionless parameter varepsilon varepsilon in 0 1 when varepsilon to0 nlsw collapses to the standard nls in the small perturbation parameter regime i e 0 varepsilon ll1 the solution of nlsw is perturbed from that of nls with a function oscillating in time with o varepsilon 2 wavelength at o varepsilon 4 and o varepsilon 2 amplitudes for well prepared and ill prepared initial data respectively this high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in varepsilon of the standard finite difference methods for nlsw such as the conservative crank nicolson finite difference cnfd method and the semi implicit finite difference sifd method we obtain error bounds uniformly in varepsilon at the order of o h 2 tau and o h 2 tau 2 3 with time step tau and mesh size h for well prepared and ill prepared initial data respectively for both cnfd and sifd in the l 2 norm and discrete semi h 1 norm our error bounds are valid for general nonlinearity in nlsw and for one two and three dimensions to derive these uniform error bounds we combine varepsilon dependent error estimates of nlsw varepsilon dependent error bounds between the numerical approximate solutions of nlsw and the solution of nls together with error bounds between the solutions of nlsw and nls other key techniques in the analysis include the energy method cut off of the nonlinearity and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi h 1 norm estimate finally numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

characteristic finite element methods for nonlinear sobolev equations1this work is supported by the doctoral fundation of the educational committee of china 1 in this paper the time stepping along characteristic finite element methods for the nonlinear sobolev equation is considered optimal convergence rates in h 1 and l 2 are demonstrated for this scheme numerical results published elsewhere have confirmed that larger time steps are appropriate with these scheme and that the approximations exhibit improved qualitative behavior

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

superconvergence analysis of a two grid method for nonlinear hyperbolic equations abstract in this paper the superconvergence analysis of a second order fully discrete scheme with two grid method tgm is studied for the two dimensional nonlinear hyperbolic equations by the bilinear finite element the existence and uniqueness of the solution for the scheme are proved rigorously by use of the combination technique of the interpolation and ritz projection the superclose results in h 1 norm are obtained and the global superconvergence is derived through the interpolated postprocessing approach finally some numerical results are provided to verify the theoretical predictions and show the efficiency of the proposed tgm

the neumann problem for equations of monge ampere type we prove the existence of a classical solution to a neumann type problem for nonlinear elliptic equations of monge ampere type the methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed gauss curvature we also present applications to asymptotic problems which approach the non unique case and maximum principles for linear equations of aleksandrov bakelman type

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

new conservative difference schemes for a coupled nonlinear schrodinger system abstract in this paper two conservative difference schemes for solving a coupled nonlinear schrodinger cnls system are numerically analyzed firstly a nonlinear implicit two level finite difference scheme for cnls system is studied then a linear three level difference scheme for cnls system is presented an induction argument and the discrete energy method are used to prove the second order convergence and unconditional stability of the linear scheme numerical examples show the efficiency of the new scheme

a minimization principle for deformation diffusion processes in polymeric hydrogels constitutive modeling and fe implementation abstract this paper presents a recently developed innovative minimization principle for coupled deformation diffusion processes applied to hydrogels and compares this new structure with the classical saddle point formulation in both variational foundation and finite element implementation first balance equations and boundary conditions associated with dissipative fluid transport in solids undergoing large deformation are shown to be rooted in a canonical minimization formulation this two field principle determines the deformation rate and the fluid flux constitutively governed by the scalar free energy and the dissipation potential functions it can be used to derive the well known saddle point formulation by a legendre transformation of the dissipation potential next the variational potential is transformed to its incremental counterpart by means of a discretization in time which offers an intuitive and unconstrained discretization within the finite element method to this end vectorial raviart thomas shape functions are chosen for flux degrees of freedom in order to fulfill the required h d i v b conformity the need for this ansatz space can be interpreted as a counterpart to the lbb condition that arises within the saddle point principle and is usually addressed by mixed element formulations however we are able to demonstrate equivalent or superior performance of the minimization principle in several representative boundary value problems phenomena specific to hydrogels like diffusion induced large volume change with instability patterns in the presence of geometrical constraints are successfully modeled the proposed variational framework can thus be validated and its significance complementary to classical approaches is underlined with the inherent symmetry of the coupled problem as a key feature and consequence of the minimization 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 two grid method for finite volume element approximations of second order nonlinear hyperbolic equations the two grid method is studied for solving a two dimensional second order nonlinear hyperbolic equation using finite volume element method the method is based on two different finite element spaces defined on one coarse grid with grid size h and one fine grid with grid size h respectively the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid a prior error estimate in the h 1 norm is proved to be o h h 3 lnh for the two grid semidiscrete finite volume element method with these proposed techniques solving such a large class of second order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation finally a numerical example is presented to validate the usefulness and efficiency of the method

analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits we propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method in particular by focusing on the dependence of the error bounds on the parameter e known as the measure of the interface thickness optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k in particular it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of chen and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect finally as a nontrivial byproduct the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients the sharp interface limits include the classical stefan problem the generalized stefan problems with surface tension and surface kinetics the motion by mean curvature flow and the hele shaw model

on the use of local maximum entropy approximants for cahn hilliard phase field models in 2d domains and on surfaces abstract we apply a local maximum entropy lme approximation scheme to the fourth order phase field model for the traditional cahn hilliard theory the discretization of the cahn hilliard equation by galerkin method requires at least c 1 continuous basis functions this requirement can be fulfilled by using lme shape functions which are c continuous in this case the primal variational formulations of the fourth order partial differential equation are well defined and integrable hence there is no need to split the fourth order partial differential equation into two second order partial differential equations this splitting scheme is a common practice in mixed finite element formulations with c 0 continuous lagrange shape functions furthermore we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization which can be applied to tackle surfaces of complex geometry and topology and to solve cahn hilliard equation on general surfaces finally the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples

an isogeometric analysis approach to gradient dependent plasticity summary gradient dependent plasticity can be used to achieve mesh objective results upon loss of well posedness of the initial boundary value problem due to the introduction of strain softening non associated flow and geometric non linearity a prominent class of gradient plasticity models considers a dependence of the yield strength on the laplacian of the hardening parameter usually an invariant of the plastic strain tensor this inclusion causes the consistency condition to become a partial differential equation in addition to the momentum balance at the internal moving boundary one has to impose appropriate boundary conditions on the hardening parameter or equivalently on the plastic multiplier this internal boundary condition can be enforced without tracking the elastic plastic boundary by requiring c1 continuity with respect to the plastic multiplier in this contribution this continuity has been achieved by using nurbs as shape functions both for the plastic multiplier and for the displacements one advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher making it consistent with the interpolation of the plastic multiplier this is different from previous approaches which have been exploited the regularising effect of gradient plasticity is shown for one and two dimensional boundary value problems this article is protected by copyright all rights reserved

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

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

a quantitative comparison between c0 and c1 elements for solving the cahn hilliard equation the cahn hilliard ch equation is a time dependent fourth order partial differential equation pde when solving the ch equation via the finite element method fem the domain is discretized by c 1 continuous basis functions or the equation is split into a pair of second order pdes and discretized via c 0 continuous basis functions in the current work a quantitative comparison between c 1 hermite and c 0 lagrange elements is carried out using a continuous galerkin fem formulation the different discretizations are evaluated using the method of manufactured solutions solved with newton s method and jacobian free newton krylov it is found that the use of linear lagrange elements provides the fastest computation time for a given number of elements while the use of cubic hermite elements provides the lowest error the results offer a set of benchmarks to consider when choosing basis functions to solve the ch equation in addition an example of microstructure evolution demonstrates the different types of elements for a traditional phase field model

an algorithm for gradient regularized plasticity coupled to damage based on a dual mixed fe formulation abstract a thermodynamically consistent theory of gradient regularized plasticity with coupling to isotropic damage was presented by svedberg and runesson 23 its main purpose was to describe the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure explicit constitutive relations were established for a model problem based on von mises plasticity with isotropic hardening of two types local and gradient enhanced the main purpose of this paper is to describe a new numerical algorithm for the gradient enhanced formulation using the backward euler rule leads to a gradient enhanced format of the classical closest point projection method the plastic multiplier increment is solved via a dual mixed method which is a convenient feature in the present context this leads to an unambiguous representation of the constitutive relations in the equilibrium equation moreover the classical local algebraic kuhn tucker conditions are obtained in a straightforward fashion in the limit case of local theory the developed algorithm was implemented in the educational finite element package 5 numerical results are shown for the extension of a plate in plane stress these results demonstrate the convergence characteristics with mesh refinement as well as the salient features of the adopted damage based constitutive model

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

unconditional superconvergence analysis of a linearized galerkin fem for nonlinear hyperbolic equations abstract unconditional superconvergence analysis for nonlinear hyperbolic equations with bilinear finite element is studied a linearized galerkin finite element method fem is developed and a time discrete system is established to split the error into two parts which are called the temporal error and the spatial error respectively on one hand a rigorous analysis for the regularity of the time discrete system is presented based on the temporal error estimation skillfully on the other hand the numerical solution u h n 0 t u h n 0 is bounded by the spatial error which is deduced with the help of the ritz projection operator the superclose and global superconvergence estimates of u with order o h 2 2 in h 1 norm are derived without any restriction of through the relationship between the interpolation operator and the ritz projection operator at last numerical results are provided to confirm the theoretical analysis here h is the subdivision parameter and the time step

the neumann problem for hessian equations in this paper we prove the existence of a classical solution to a neumann boundary value problem for hessian equations in uniformly convex domain the method depends upon the establishment of a priori derivative estimates up to second order so we give an affirmative answer to a conjecture of n trudinger in 1987