dynamic shakedown analysis and bounds for elastoplastic structures with nonassociative internal variable constitutive laws abstract reference is made to discrete finite element structural models formulated in terms of generalized variables the constitutive laws adopted are elastic plastic nonlinear hardening with internal variables and generally nonassociative the notion of reduced domain is employed as a generalization of a similar concept introduced in earlier developments of shakedown theory on this basis a unified theory is presented which encompasses necessary and separately sufficient conditions for shakedown by a static approach as a further generalization of classical melan s theorem bounds on various post shakedown quantities sufficient and necessary shakedown criteria by a kinematic approach as a further extension of neal symonds koiter theorem by suitable specializations and relaxations of the achievable results the criteria and bounding inequalities established here are formulated as mathematical programming problems in view of numerical applications

localization and regularization behavior of mixed finite elements for 2d structural problems with damaging material abstract a class of lagrangian mixed finite elements is presented for applications to 2d structural problems based on a damage constitutive model attention is focused on localization and regularization issues as compared with the correspondent behavior of lagrangian displacement based elements a non local regularization procedure of integral type is adopted a predictor corrector technique is used to solve the evolution problem of the damage variable the proposed elements show superior performances for typical structural applications

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

mixed finite element formulations and related limitation principles a general treatment abstract the complete set of variational principles for an elastic structural model with non homogeneous boundary conditions and inelastic strains is presented a general methodology for deriving the approximate mixed formulation and the relevant matrix form is also shown and applied the limitation principles for a structural model with a linear elastic behaviour are derived and it is shown that each principle relating two different approximate formulations depends upon the state variable interpolated in one formulation and not in the other

variationally consistent modeling of finite strain plasticity theory with non linear kinematic hardening abstract variational constitutive updates provide a physically and mathematically sound framework for the numerical implementation of material models in contrast to conventional schemes such as the return mapping algorithm they are directly and naturally based on the underlying variational principle hence the resulting numerical scheme inherits all properties of that principle in the present paper focus is on a certain class of those variational methods which relies on energy minimization consequently the algorithmic formulation is governed by energy minimization as well accordingly standard optimization algorithms can be applied to solve the numerical problem a further advantage compared to conventional approaches is the existence of a natural distance semi metric induced by the minimization principle such a distance is the foundation for error estimation and as a result for adaptive finite elements methods though variational constitutive updates are relatively well developed for so called standard dissipative solids i e solids characterized by the normality rule the more general case i e generalized standard materials is far from being understood more precisely int j sol struct 2009 46 1676 1684 represents the first step towards this goal in the present paper a variational constitutive update suitable for a class of nonlinear kinematic hardening models at finite strains is presented two different prototypes of armstrong frederick type are re formulated into the aforementioned variationally consistent framework numerical tests demonstrate the consistency of the resulting implementation

numerical investigation on the stability of singular driven cavity flow by applying the singularity subtraction technique to the unsteady driven cavity problem the stability of the impulsively started flow is investigated without smoothing the corner singularity a second order spectral projection method allows localization of the critical reynolds number for the first hopf bifurcation in the interval 8017 6 8018 8

accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

the ivp for a nonlocal perturbation of the benjamin ono equation in classical and weighted sobolev spaces abstract we prove that the initial value problem associated to a nonlocal perturbation of the benjamin ono equation is locally and globally well posed in sobolev spaces h s r for any s 3 2 and we establish that our result is sharp in the sense that the flow map of this equation fails to be c 2 in h s r for s 3 2 finally we study persistence properties of the solution flow in the weighted sobolev spaces z s r h s r l 2 x 2 r d x for s r 0 we also prove some unique continuation properties of the solution flow in these spaces

a strain gradient isotropic elastoplastic damage model with j3 dependence abstract a plastic strain gradient version of an isotropic elastoplastic damage model that depends on the third invariant j3 of the stress deviator is developed the model is based on the non local equivalent plastic strain ep defined by peerlings et al 2001 and engelen et al 2003 and introduces a material length l to the constitutive equations it is shown that the non local equivalent plastic strain ep at a material point p can be identified with the average value of the local von mises equivalent plastic strain e p over a sphere centered at p and of radius approximately equal to 3 l a methodology for the numerical integration of the constitutive equations is presented the algorithm is appropriate for rate independent as well as rate dependent viscoplastic models the model is implemented in the abaqus general purpose finite element program and both quasi static and dynamic problems are solved two possible abaqus implementations are discussed first user elements are developed which can be used for the solution of both quasi static and dynamic problems reduced 1 point gauss integration is discussed in 8 node hexahedral elements and the physical stabilization method of puso 2000 is used to remove the resulting numerical singularities hourglass control second the implementation of the model via user material subroutines is discussed quasi static problems can be solved with abaqus standard using a coupled temperature displacement steady state analysis together with user subroutine umat in which temperature is identified with the non local equivalent plastic strain ep the solution of dynamic problems requires use of abaqus explicit together with a dynamic temperature displacement analysis option and user subroutines vumat and dflux several example problems are solved

nonlocal models for damage and fracture comparison of approaches abstract the paper analyzes nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials a number of nonlocal formulations found in the literature are classified according to the type of variable subjected to nonlocal averaging analytical and numerical solutions of a simple one dimensional localization problem are presented it is shown that some of the formulations inevitably lead to residual stresses even at very late stages of the deformation process and consequently they are not capable of modeling complete separation in a widely open macroscopic crack the mechanisms leading to this specific type of stress locking are explained based on a theoretical analysis of the nonlocal constitutive equations it is also pointed out that the nonlocal approach distorts the shape of the stress strain diagram which has to be taken into account when designing an appropriate local softening law

basis functions for triangular and quadrilateral high order elements in this paper we present a unified description of new spectral bases suitable for high order hp finite element discretizations on hybrid two dimensional meshes consisting of triangles and quadrilaterals all bases presented are for c0 continuous discretizations and are described both as modal and as mixed modal nodal expansions general jacobi polynomials of mixed weights are employed that accommodate automatic exact numerical quadratures generalized tensor products and variable expansion order in each element the approximation properties of the bases are analyzed in the context of the projection linear advection and diffusion operators

the kinematic laplacian equation method a novel procedure for the navier stokes equations in the vorticity velocity formulation is presented the time evolution of the vorticity is solved as an ode problem on each node of the spatial discretization using at each step of the time discretization the spatial solution for the velocity field provided by a new pde expression called the kinematic laplacian equation kle this complete decoupling of the two variables in a vorticity in time velocity in space split algorithm reduces the number of unknowns to solve in the time integration process and also favors the use of advanced ode algorithms enhancing the efficiency and robustness of time integration the issue of the imposition of vorticity boundary conditions is addressed as well as the details of the implementation of the kle by isoparametric finite element discretization we shall see some validation results of the kle method applied to the classical case of a circular cylinder in impulsive started pure translational steady motion at several reynolds numbers in the range 5

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a locking free finite element method for naghdi shells abstract in this paper a locking free method using mixed finite elements is introduced to approximate the solution of naghdi shell problems with small parameter t the thickness of the shell the approach of arnold and brezzi 1997 is employed with some important changes instead of proving the discrete inf sup condition for arbitrary geometric coefficients which does not seem possible we prove a weaker stability condition for smooth enough geometrically dependent coefficients which is sufficient for deducing the optimal error estimate as long as h 2 t is uniformly bounded for extremely small t we can relax this condition either using a larger bubble space or stabilizing the problem by replacing t2 by t2 h4 in either case an optimal error estimate still holds

non smooth multisurface plasticity and viscoplasticity loading unloading conditions and numerical algorithms rate independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non smooth fashion are considered in detail it is shown that the standard kuhn tucker optimality conditions lead to the only computationally useful characterization of plastic loading on the computational side an unconditionally convergent return mapping algorithm is developed which places no restrictions aside from convexity on the functional forms of the yield condition flow rule and hardening law the proposed general purpose procedure is amenable to exact linearization leading to a closed form expression of the so called consistent algorithmic tangent moduli for viscoplasticity a closed form algorithm is developed based on the rate independent solution the methodology is applied to structural elements in which the elastic domain possesses a non smooth boundary numerical simulations are presented that illustrate the excellent performance of the algorithm

spectral element methods for elliptic problems in nonsmooth domains engineering applications frequently require the numerical solution of elliptic boundary value problems in irregularly shaped domains for smooth problems spectral element methods have proved very successful since they can accommodate fairly complicated geometries while retaining a rapid rate of convergence geometric singularities however often give rise to singular solutions the accuracy of the spectral element methods is then degraded and they offer no apparent advantage over low order finite element methods in many cases however the singular structure of the solution is known and its form may be exploited by the spectral element method among the various ways of doing so through supplementary basis functions eigenfunction expansions and graded meshes the method of auxiliary mapping proves to be particularly effective for certain simple cases the problem is transformed to a coordinate system in which the solution is analytic and exponential convergence is recovered even when this is not possible the singularity is usually much weaker after mapping so that other treatments are more effective in the new coordinate system in this paper we study different ways of treating singularities and in particular the method of auxiliary mapping coupled with the use of supplementary basis functions error estimates are presented explaining why the combined approach is more effective and these estimates are confirmed through a number of numerical experiments for the laplace poisson and helmholtz equations

shell deformation states and the finite element method a benchmark study of cylindrical shells abstract we demonstrate some of the basic mathematical characteristics of shell deformation states and shell models from the point of view of finite element approximation as benchmark cases we analyze in detail three simple cylindrical shell problems these problems are seemingly similar but turn out to be quite different we 1 compute the exact displacement and strain fields in the model problems both numerically and symbolically using two variants of classical shell models 2 for comparison compute numerically the displacements and strains corresponding to certain higher order dimension reduction models derived from the energy principle 3 demonstrate the dissimilarity of the three benchmark problems as to their asymptotic behaviour in the limit of zero thickness and 4 show both theoretically and experimentally how this diversity is reflected in the accuracy of standard finite element approximations of both low and high order

extremum properties of finite step solutions in elastoplasticity with nonlinear mixed hardening abstract the class of elastic plastic constitutive laws assumed herein can be described as follows we envisage y 1 yield criteria or modes which define the current yield surface each yield function consists of two addends an effective stress and a yield limit y 1 y the former is an order one positively homogeneous function of the difference between the stresses ii and reference stresses ii which are generally nonlinear functional of the plastic strain history so that kinematic hardening is accounted for the yield limit y is generally a nonlinear function of y nondecreasing internal variables so that another hardening mechanism which may reduce to isotropic hardening is accommodated in the model the rates of these variables play the role of plastic multipliers in the flow rule a finite step in the geometrically linear evolutive analysis of such solids is defined according to the backward difference or stepwise holonomic strategy for approximate time integration for the finite step boundary value problem thus arising various extremum characterizations of solutions are established and the underlying constitutive restrictions are pointed out and discussed

duality preserving discretization of the large time increment methods abstract the large time increment method is a well established iterative computational method for time dependent non linear structural analyses which has the peculiarity to produce at each iteration an approximation of the complete structural response over the whole considered history of loading the method establishes a converging iterative sequence by exploiting the specific structure of the equations governing the continuum problem the relatively large body of literature on the subject has been mostly concerned with assessing conceptual and practical properties of the method by referring to the continuum problem so far computer implementations where based on heuristic considerations and standard finite element technology in the present paper a different approach is followed a space and time discretization which preserves the duality structure of the continuum problem is introduced first then the method is reformulated for the discrete problem it is shown how the so called generalized variable modelling preserves the fundamental duality structure of the continuum problem a proof of convergence of the iterative scheme to the solution of the discrete problem is also outlined

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

a displacement like finite element model for j2 elastoplasticity variational formulation and finite step solution abstract the displacement like finite element formulation for finite step j2 elastoplasticity is revisited in this paper the classical computational strategy according to which plastic loading is tested at the gauss points of each element and an independent return mapping algorithm is performed for given incremental displacements is consistently derived from a suitably discretized version of a min max variational principle the sequence of solution phases to be performed within each load step adopting a full newton s method is illustrated in detail and the importance of a correct update of the plastic strains is emphasized it is further shown that in order to increase the rate of convergence and the stability properties of the newton s method the consistent elastoplastic tangent operator must be exploited even at the first iteration of each load step subsequent to the first yielding of the structural model this is in contrast with the traditional implementation according to which the elastic operator is used at the first iteration of each load step the effectiveness of the present approach is shown by a set of numerical examples referred to plane strain problems

a high performance element for the analysis of 2d elastoplastic continua abstract the paper presents a four node element for the analysis of 2d continua described by elastic perfectly plastic von mises laws its formulation is based on the assumed stress approach which has been widely tested in the elastic field the elastoplastic formulation of the element is obtained by enriching its variational basis with the weak enforcement of the plastic loading unloading conditions in this way the elastoplastic problem is formulated with respect to the stress parameters of the element consistently with the weak statement of the compatibility condition

shakedown analysis in poroplasticity by linear programming static lower bound and kinematic upper bound theorems for shakedown analysis and limit analysis as special case are presented for two phase fully saturated poroplastic solids in a context characterized by the following features quasi static non associative non softening perfect or hardening poroplasticity piecewise linearized yield loci and hardening constant permeability linear fluid transport law small deformations non traditional galerkin finite element discretization with reference to cyclic periodic external actions numerical examples centred on linear programming evidence the computational potentialities and limitations of direct non evolutive methods of integrity assessment according to limit state design concepts in poroplasticity copyright 2000 john wiley sons ltd

scale resolution locking and high order finite element modelling of shells abstract we demonstrate both by theory and experiment the benefits of using standard finite elements of relatively high degree in shell problems the difficulty of shell problems lies in their asymptotic diversity at zero thickness and in the multiple scale character of the deformation field when the thickness is small due to these characteristics standard finite elements of low degree often resolve the length scales of practical shell deformation very poorly unless extremely fine meshes are used however with standard finite elements of sufficiently high degree say cubic or quartic at least the quality of numerical scale resolution improves remarkably we show by simple error analysis that this effect is not problem specific but rather robust among the diversity of shell problems as numerical test cases we analyze two challenging problems in cylindrical shell geometry

on the implementation of rate independent standard dissipative solids at finite strain variational constitutive updates this paper is concerned with an efficient variationally consistent implementation for rate independent dissipative solids at finite strain more precisely focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the helmholtz energy and the yield function or equivalently a dissipation functional finite strain plasticity is recast into an equivalent minimization problem in contrast to previous models the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions based on this approach a novel numerical implementation representing the main contribution of the paper is given in sharp contrast to by now classical approaches such as the return mapping algorithm and analogously to the theoretical part the numerical formulation is variationally consistent i e all unknown variables follow naturally from minimizing the energy of the considered system consequently several different numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem extending previously published works on variational constitutive updates the advocated model does not rely on any material symmetry and therefore it can be applied to a broad range of different plasticity theories as two examples an anisotropic hill type and a barlat type model are implemented numerical examples demonstrate the applicability and the performance of the proposed implementation

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

locking free finite element methods for shells we propose a new family of finite element methods for the naghdi shell model one method associated with each nonnegative integer k the methods are based on a nonstandard mixed formulation and the kth method employs triangular lagrange finite elements of degree k 2 augmented by bubble functions of degree k 3 for both the displacement and rotation variables and discontinuous piecewise polynomials of degree k for the shear and membrane stresses this method can be implemented in terms of the displacement and rotation variables alone as the minimization of an altered energy functional over the space mentioned the alteration consists of the introduction of a weighted local projection into part but not all of the shear and membrane energy terms of the usual naghdi energy the relative error in the method measured in a norm which combines the h i norm of the displacement and ro tation fields and an appropriate norm of the shear and membrane stress fields converges to zero with order k 1 uniformly with respect to the shell thickness for smooth solutions at least under the assumption that certain geometrical coefficients in the nagdhi model are replaced by piecewise constants

an internal variable variational formulation of viscoplasticity in the paper evolutive processes and variational principles in viscoplasticity are examined in detail the treatment is faced by the recourse to an internal variable formulation and by taking advantage of the appropriate tools in order to deal with multivalued operators this general framework enables the development of a consistent variational formulation for viscoplastic models for this purpose the problem at hand is set in operational form by encompassing into a unique expression the field equations and the constitutive equations the viscoplastic structural operator in all variables is accordingly integrated a family of viscoplastic potentials is deduced and the relevant sequence of extremum properties in one or more unknown variables is derived

treatment of internal constraints by mixed finite element methods unification of concepts a general method to generate assumed stress and strain fields within the context of mixed finite element methods is presented the assumed fields are constructed in such a way that internal constraints are satisfied a priori consequently the locking behaviour commonly observed in finite element solutions of problems with internal constraints is avoided to this end the assumed stress and strain fields are constructed to satisfy a priori the homogeneous part of the equilibrium equations thus avoiding fraeijs de veubeke s limitation principle results obtained using the proposed methodology on a nearly incompressible plane strain problem and thin plate application using a shear deformable theory are indicated

variationally consistent state determination algorithms for nonlinear mixed beam finite elements abstract this paper presents and evaluates several variationally consistent state determination algorithms for geometric and material nonlinear beam finite elements based on the hellinger reissner hr assumed stress variational principle the specifically targeted element type is derived from the two field displacement and generalized stress form of this principle momentum balance and strain compatibility are expressed in a weak form and the constitutive equations are satisfied directly at each integration point i e section along the element length a specific formulation of one element of this type is presented in a companion paper interelement compatibility is not enforced in the dual stress resultant variables for this element and thus various state determination algorithms can be formulated at the local i e element and section levels four state determination algorithms referred to as the l l l n n l and n n procedures are considered the first symbol in these names indicates the element level and the second denotes the section level furthermore the symbol n indicates that local nonlinear iteration is performed prior to returning to the higher level whereas the symbol l indicates that only the linearized equations are satisfied at the corresponding local level the l l algorithm corresponds to conventional newton iteration on both the element and section levels whereas the n n algorithm entails the iterative satisfaction of the nonlinear element compatibility and section constitutive equations for each global iteration the performance of these algorithms is evaluated and compared it is found that the algorithms in which the local nonlinear strain displacement compatibility conditions are solved iteratively at the element level for every global iteration lead to fewer global iterations at the expense of iterations at the element level in particular the n n algorithm is overall the most efficient of the four algorithms for the problems tested also the n n algorithm has substantially smaller element storage requirements the n n algorithm is expected to be particularly advantageous for cases in which a large number of elements remain elastic and inelasticity is concentrated in a few elements

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

the well posedness of the dissipative korteweg de vries equations with low regularity data abstract we study the cauchy problem of a dissipative version of the kdv equation with rough initial data by working in a bourgain type space we prove the local and global well posedness results for sobolev spaces of negative order and the order number is lower than the well known value 3 4 in some sense this paper is intended to show how the bourgain type space is applicable to the study of semilinear equations with a linear part which contain both dissipative and dispersive terms

tensor based finite element formulation for geometrically nonlinear analysis of shell structures in the present study a finite element computational model for the nonlinear analysis of shell structures is presented a tensor based finite element formulation is presented to describe the mathematical model of a shell in a natural and simple way by using curvilinear coordinates in addition a family of high order elements with lagrangian interpolations is used to avoid membrane and shear locking and no mixed interpolations are employed a first order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the lagrangian description this approach takes into account thickness stretching and therefore 3d constitutive equations are utilized numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites as well as functionally graded shells are found to be excellent and show the validity of the developed finite element model moreover the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation of shells

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

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 variational theory for finite step elasto plastic problems abstract an extended version ofgeneralized standard elasto plastic material is considered in the framework of an internal variable theory of associated plasticity according to a backward difference scheme for time integration of the flow rule a finite step structural problem is formulated in a geometrically linear range convex analysis and a brand new potential theory for monotone multivalued operators are shown to provide the natural mathematical setting for the derivation of the related variational formulation a general stationarity principle is obtained and then specialized to obtain a minimum principle in terms of displacements plastic strains and internal variables a critical comparison with an analogous minimum principle recently proposed in literature is performed showing the inadequacy of classical procedures in deriving non smooth variational formulations

development and testing of stable invariant isoparametric curvilinear 2 and 3 d hybrid stress elements abstract linear and quadratic serendipity hybrid stress elements are examined in respect of stability coordinate invariance and optimality a formulation based upon symmetry group theory successfully addresses these issues in undistorted geometries and is fully detailed for plane elements the resulting least order stable invariant stress polynomials can be applied as astute approximations in distorted cases through a variety of tensor components and variational principles a distortion sensitivity study for two and three dimensional elements provides favourable numerical comparisons with the assumed displacement method

a variational constitutive update algorithm for a set of isotropic hyperelastic viscoplastic material models abstract in three dimensional formulations of plasticity the rate of plastic deformation is usually decomposed in direction and amplitude in the most simple cases of von mises type flows this allows to assume a known radial flow direction and computations are reduced to the determination of the amplitude of the plastification in the context of a variational formulation of finite plasticity it is verified that this decomposition between constitutive and kinematic aspects is accomplished by the choice of a quadratic elastic potential hencky model the aim of this paper is to show that a wide set of simple hyperelastic plastic isotropic models not restricted to quadratic elastic behavior can be constructed by relaxing the classical decomposition amplitude direction by the sum of spectral quantities in addition this approach allows for a unified and efficient numerical implementation in which a particular hyperelastic viscoplastic behavior is obtained by a specific choice of the elastic plastic and dissipative pseudo potentials

least squares continuous sensitivity shape optimization for structural elasticity applications a general first order formulation of the continuous sensitivity equations is introduced that improves the accuracy of the sensitivity boundary conditions in the continuous sensitivity method design or shape parameter gradients are computed from a continuous system of partial differential equations instead of the discretized system which avoids the problematic mesh sensitivity calculations of discrete methods for shape variation problems the first order formulation for both the underlying elasticity and sensitivity equations is amenable to solution by a high order polynomial least squares finite element model the continuous sensitivity boundary value problem which can be posed in local or total derivative form is generally simpler in local sensitivity form for shape variation problems although the local form is not unique and total material sensitivities are generally desired for structural elasticity problems the local sensitivity solution is easily transformed to a material derivative the first order formulation and the transformation to material derivatives are demonstrated for two elasticity example problems the least squares continuous sensitivity solutions are presented and compared with analytic sensitivities and finite difference results and should prove convenient validation benchmarks for other continuous sensitivity applications

on the well posedness of higher order viscous burgers equations abstract we consider higher order viscous burgers equations with generalized nonlinearity and study the associated initial value problems for given data in the l 2 based sobolev spaces we introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well posedness for data with sobolev regularity below l 2 we also prove ill posedness for this type of models and show that the local well posedness results are sharp in some particular cases viz when the orders of dissipation p and nonlinearity k 1 satisfy a relation p 2 k 1

finite elements with embedded displacement discontinuity a generalized variable formulation this paper is intended to bring a contribution towards a satisfactory simulation of those fracture phenomena which result in the appearance and development of discrete cracks to this purpose a general mixed finite element formulation is proposed based on the concept of generalized variables in prager s sense the displacement field inside an element is modelled by the sum of two contributions a regular continuous part which is governed by standard shape functions and a possibly discontinuous one which is introduced soon after a suitable criterion is satisfied the formulation is first specialized to a one dimensional case then a triangular element for two dimensional problems is described in detail analytical and numerical examples are presented in order to clarify the formulation and to point out the essential role of inter element conformity copyright 2000 john wiley sons ltd

strain localization and bifurcation in a nonlocal continuum abstract the conditions for localization and wave propagation in a strain softening material described by a nonlocal damage based constitutive relation are derived in closed form localization is understood as a bifurcation into a harmonic mode the criterion for bifurcation is reduced to the classical form of singularity of a pseudo acoustic tensor this tensor is not a material property as it involves the wavelength of the bifurcation mode through the fourier transform of the weight function used in the definition of the nonlocal damage a geometrical solution is provided to analyse localization the conditions for the onset of bifurcation are found to coincide in the nonlocal and in the corresponding local cases in the nonlocal continuum the wavelength of the localization mode is constrained to remain below a threshold which is proportional to the characteristic length of the continuum the analysis in dynamics exhibits the well known property of wave dispersion in some instances i e for large wavelength modes wave celerities become imaginary but waves with a sufficiently short wavelength are found to propagate during softening in all the situations

a unified approach for displacement equilibrium and hybrid finite element models in elasto plasticity abstract finite element models for elasto plastic incremental analysis are derived from a three field variational principle the newton raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle the iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models follow directly in the displacement model the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process while the equilibrium of the stresses is restored only in the mean after convergence in the equilibrium model the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process when convergence is achieved the stresses are locally in equilibrium with the applied external loads in both cases a tangential stiffness matrix can be constructed even for perfectly plastic materials and it allows one to obtain always very good convergence properties examples are shown for plane stress and axisymmetric cases

accurate ω ψ spectral solution of the singular driven cavity problem this article provides accurate spectral solutions of the driven cavity problem calculated in the vorticity stream function representation without smoothing the corner singularities a prima facie impossible task as in a recent benchmark spectral calculation by primitive variables of botella and peyret closed form contributions of the singular solution for both zero and finite reynolds numbers are subtracted from the unknown of the problem tackled here numerically in biharmonic form the method employed is based on a split approach to the vorticity and stream function equations a galerkin legendre approximation of the problem for the perturbation and an evaluation of the nonlinear terms by gauss legendre numerical integration results computed for re 0 100 and 1000 compare well with the benchmark steady solutions provided by the aforementioned collocation chebyshev projection method the validity of the proposed singularity subtraction scheme for computing time dependent solutions is also established

a discrete formulation for elastic solids with damaging interfaces abstract an elastic solid with embedded interfaces loci of possible displacement discontinuities is considered here decohesion and quasi brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps a discrete formulation in terms of interface variables only is obtained the space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented some examples are shown in order to clarify the formulation

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

the b bar method and the limitation principles abstract the generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity plasticity and unilateral models the modified forms of the hu washizu and hellinger reissner principles and the displacement based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables the related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided it is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the b bar method lead to a limitation principle accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement pressure variational principle a second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two field mixed formulation and the displacement based method numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations in elasticity or elastoplasticity under the validity of the limitation principles

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

non local yield limit degradation presented is a new type of a non local continuum model which avoids problems of convergence at mesh refinement and spurious mesh sensitivity in a softening continuum characterized by degradation of the yield limit the key idea which has recently been proposed in a general context and has already been applied to softening damage due to stiffness degradation is to apply the non local concept only to those parameters which cause the degradation while keeping the definition of the strains local compared to the previously advanced fully non local continuum formulation the new approach has the advantage that the stresses are subjected to the standard differential equations of equilibrium and standard boundary or interface conditions the new formulation exhibits no zero energy periodic modes imbrication of finite elements is unnecessary and finite elements with standard continuity requirements are sufficient two dimensional finite element solutions with up to 3248 degrees of freedom are presented to document convergence and efficacy the formulation is applied to tunnel excavation in a soil stabilized by cement grouting with the objective of preventing cave in burst of the tunnel sides due to compression softening

on uniqueness of the dynamic finite step problem in gradient dependent softening plasticity the dynamic evolution of an elastoplastic softening solid is considered a material model including in the yield function the laplacian of the plastic multiplier is used to regularize the problem the dynamic finite step problem is formulated according to a generalized mid point integration scheme space discretization is carried out by a mixed finite element technique based on generalized variables a sufficient uniqueness condition of the finite step solution is proved for a one dimensional problem also a necessary and sufficient condition is presented a simple numerical test shows the regularizing properties mesh independence of the proposed model and the positive influence of the gradient term also on the time step amplitude ensuring uniqueness of solution

a benchmark study of reduced strain shell finite elements quadratic schemes we study experimentally the accuracy and reliability of some low order shell finite element schemes based on modifying the standard displacement formulation by reduced strain expressions we focus on quadrilateral elements with a quadratic displacement approximation three benchmark problems with different asymptotic behaviour in the limit of zero shell thickness is used in the experiments following the error analysis of a reduced strain scheme we study two components of the total error the approximation error and the consistency error we demonstrate that the performance of the methods is both case and mesh dependent when a bending dominated problem is solved none of the methods studied can avoid the usual worst case locking effect of the approximation error on general meshes for a membrane dominated problem the total error is typically dominated by the consistency error which often convergences slowly copyright 2000 john wiley sons ltd

a plastic nonlocal damage model abstract in the present paper a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials a new isotropic damage model which is able to describe the behavior of a wide class of cementitious materials is presented a regularization technique based on the introduction of the damage laplacian in the damage limit function is adopted to overcome the analytical and computational problems induced by the softening constitutive law a drucker prager type of plastic limit function is proposed considering isotropic hardening a numerical procedure based on an implicit backward euler technique for the time integration of the plastic and damage evolution equations is presented to solve each nonlinear step a predictor corrector iterative method is developed within the splitting method in particular the damage evolution is determined solving a constrained minimization problem of a convex functional the proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

a symmetric galerkin boundary domain element method for finite elastic deformations abstract the symmetric galerkin boundary element method sgbem is reformulated for problems of finite elasticity with hyperelastic material and incompressibility using fundamental solutions related to a fictitious homogeneous isotropic and compressible linear elastic material the proposed formulation contains besides the standard boundary integrals domain integrals which account for the problem s nonlinearities through some fictitious initial strain and stress fields required to satisfy appropriate consistency equations the boundary domain integral equation problem so obtained is shown to admit a stationarity principle a consequence of the hu washizu one which covers a number of particular cases including the compressible case as well as the fully linear case either incompressible and compressible so recovering known results of the literature the continuum rate problem is also addressed and suitably discretized by boundary and domain elements the related algebraic solving equation system is shown to exhibit symmetry and some sign definiteness features it is formally solved in terms of the node values of the displacement gradient the idrostatic pressure the initial strain and stress fields a long with the unknown boundary displacements and tractions details regarding the fundamental solutions and the stationarity principle are reported in three appendices at the end of the paper

galerkin spectral method for the vorticity and stream function equations a galerkin legendre spectral method for the solution of the vorticity and stream function equations in uncoupled form under no slip conditions in a square domain is presented which fully exploits the separation of variables in the two elliptic problems benefits from a nonsingular influence matrix and is able to solve the singular driven cavity problem modulo gibbs phenomenon even without regularizing the boundary condition at the corners

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

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

on the well posedness for the generalized ostrovsky stepanyams and tsimring equation abstract in this work we prove that the initial value problem associated with equation u t u xxx h u x h u xxx u k u x 0 x r t 0 where 0 and h denotes the usual hilbert transform is locally well posed in sobolev spaces h s r for s 0 and k 1 2 3 and globally well posed in l 2 r

a unified approach for variationally consistent finite elements in elastoplasticity abstract a general variationally consistent approach for elastoplastic finite element analysis is presented a weak formulation of the finite step i e in terms of finite increments elastoplastic boundary value problem is obtained by enforcing the stationarity of a mixed functional of hu washizu type a peculiarity of the present formulation is that the constitutive equations are expressed in weak form also all fields are then independently discretized in terms of so called generalized variables the result is a set of non linear algebraic equations a subset of which can be interpreted as the backward difference time integrated constitutive law of the finite element an iterative procedure where all calculations are at the element and not at the gauss point level resting on newton raphson scheme is proposed for the solution of the governing equations it is shown how the customary displacement approach and also some other mixed formulations in elastoplasticity can be regarded as special cases of the present formulation

a regularized force based beam element with a damage plastic section constitutive law a new beam finite element is presented with a generalized section constitutive law based on damage mechanics and plasticity to analyse the cyclic structural response of plane frames both displacement based and force based fb approaches are used and compared to demonstrate the significant advantages of the fb formulation in the presence of material non linearity in order to overcome the analytical problems and the pathological mesh dependency of the numerical response in the presence of strain softening post peak behaviour a classical non local regularization procedure is adopted first based on the integral definition of the associated variable governing the damaging evolution process subsequently for the fb element a new simple regularization technique is proposed based on a selected integration procedure along the element length which predefines the location of the gauss points in the beam region where the localization phenomena take place as for the other computational aspects an iterative element state determination is adopted for the fb formulation and a local predictor corrector algorithm is used to solve the incremental evolution problems of the damage and plastic internal variables finally some examples are shown on simple beams and frames subjected to monotonically increasing and cyclic loading conditions copyright 2006 john wiley sons ltd

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

equilibrium based iterative solutions for the non linear beam problem summary the paper describes a procedure for the non linear analysis of structures which are an assemblage of beams with material non linearities of general type the approach uses the equilibrium integrals and a consistent iterative formulation at the element level within the general framework of the displacement method for the solution of the global structural problem the application of dierent approaches to the non linear beam problem is presented and discussed including the traditional stiness and exibility approach and some mixed formulations the proposed equilibrium based approach is shown to be more accurate and more robust than the traditional compatibility based approach on which most of the non linear beam elements available today are based similar advantages are also found with respect to an approach based on the three eld mixed assumed strain method the paper presents a new approach to the numerical static or dynamic analysis of structures made up of an assemblage of non linear beam elements with spread material non linearities an essential step in the numerical incremental solution of such a problem within the usual framework of the displacement method at the global level consists in nding for each individual element the end forces which correspond to given end displacements together with the associated generalized strains and stresses at the dierent sections along the element this step is often referred to as element state determination for simplicity in the following the beam element will be considered as partially constrained at the ends so that no rigid body displacement is allowed and as a consequence we shall refer to end deformations q rather than to end displacements and to the associated end forces or generalized stresses p the standard stiness and exibility approaches will be briey reviewed before going into the presentation of the proposed algorithms after that a review of some mixed approaches is carried out showing the application of two and three eld formulation to the non linear beam problem pointing out dierences and similarities with the proposed algorithms finally numerical examples are presented so as to compare the behaviour of the various approaches when implemented in a non linear nite element beam used in a standard displacement based finite element fe code

a two scale finite element relaxation analysis of shear bands in non convex inelastic solids small strain theory for standard dissipative materials abstract we propose a fundamentally new approach to the treatment of shear band localizations in strain softening elastic plastic solids at small strains based on energy minimization principles associated with micro structure developments the point of departure is a general internal variable formulation that determines the inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasihyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments in finite elasticity furthermore localization phenomena are interpreted as micro structure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these micro structures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem we develop based on an approximated rank one convexification of the incremental stress potential a computational two scale procedure for a mesh objective treatment of localization problems it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the micro shearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which illustrate the mesh independence of the results

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

a consistent approach to continuum and discrete rate elastoplastic structural problems abstract the rate elasto plastic structural problem with hardening is presented and is cast within the general theory of structural models with convex constraints a consistent derivation of the constitutive variational principles is performed and the equivalence with the elastic predictor plastic corrector scheme of computational plasticity is proved following general concepts of convex analysis and of potential theory the more general variational formulation is derived the space discretization is achieved by the finite element approach the definition of a global yield function instead of a local one leads to a unique scalar plastic multiplier instead of a field of plastic multipliers and avoids their discretization mixed variational principles can thus be derived from the continuous ones and related computational schemes are also presented

fe formulation of a nonlocal plasticity theory a nonlocal continuum plasticity theory is presented the nonlocal field introduced here is defined as a certain weighted average of the corresponding local field taken over all the material points in the body hereby a quantity with the dimension of length occurs as a material parameter when this so called internal length is equal to zero the local classical plasticity theory is regained in the present model the yield function will depend on a nonlocal field the consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers the integral equations are classified as fredholm equations of the second kind and the existence of a solution will be commented upon after discretization a matrix equation is obtained and an algorithm for finding the solution is proposed for a generalized von mises material a plane boundary value problem is solved with a fe method since the nonlocal quantities are integrals c0 continuous elements are sufficient the solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations in the numerical simulations shear band formation is analysed and the results display mesh insensitivity

least squares finite element formulation for shear deformable shells abstract we present a least squares based finite element formulation for the numerical analysis of shear deformable shell structures the variational problem is obtained by minimizing the least squares functional defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of hilbert spaces the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise i e stability requirements such as inf sup conditions never arise the proposed formulation retains the generalized displacements and stress resultants as independent variables and in view of the nature of the variational setting upon which the finite element model is built allows for equal order interpolation a p type hierarchical basis is used to construct the discrete finite element model based on the least squares formulation exponentially fast decay of the least squares functional is verified for increasing order of the modal expansions several well established benchmark problems are solved to demonstrate the predictive capability of the least squares based shell elements shell elements based on this formulation are shown to be effective in both membrane and bending dominated states

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

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

on the low regularity of the modified korteweg de vries equation with a dissipative term abstract we consider a dissipative version of the modified korteweg de vries equation u t u x x x u x x u 3 x 0 we prove global well posedness results on the associated cauchy problem in the sobolev spaces h s r for s 1 4 while for s 1 2 we prove some ill posedness issues

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

plane isoparametric hybrid stress elements invariance and optimal sampling formulation and applications of the hybrid stress finite element model to plane elasticity problems are examined conditions for invariance of the element stiffness are established for two dimensional problems the results of which are easily extended to three dimensional cases next the hybrid stress functional for a 3 d continuum is manipulated into a more convenient form in which the location of optimal stress strain sampling points can be identified to illustrate these concepts 4 and 8 node plane isoparametric hybrid stress elements which are invariant and of correct rank are developed and compared with existing hybrid stress elements for a 4 node element lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation alternative 8 node elements are considered and the best invariant element is shown to be one in which the stress compatibility equations are invoked results are also presented which demonstrate the validity of the optimal sampling points the effects of reduced orders of numerical integration and the behaviour of the elements for nearly incompressible materials

on the membrane locking of h p finite elements in a cylindrical shell problem we study the performance of high order h p finite clement schemes in approximating the nearly inexlensional deformation state of a thin cylindrical shell roof with both ends free severe locking is demonstrated for both quadratic and cubic elements and even for higher order if the grid is not aligned with the axis of the cylinder carefully designed selective reduced integration is shown to improve the accuracy of the finite element approximation at low and moderately high orders however so far this technique seems to work well only on aligned grids

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

a transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular in this paper we propose an implicit high order compact hoc finite difference scheme for solving the two dimensional 2d unsteady navier stokes n s equations on irregular geometries on orthogonal grids our scheme is second order accurate in time and fourth order accurate in space it is used to solve three pertinent fluid flow problems namely the flow decayed by viscosity the lid driven square cavity and the flow in a constricted channel it is seen to efficiently capture both transient and steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions apart from including the good features of hoc schemes our formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations detailed comparison data produced by the scheme for all the three test cases are provided and compared with analytical as well as established numerical results excellent comparison is obtained in all the cases