on the modelling of incompressibility in linear and non linear elasticity with the master slave approach this is the pre peer reviewed version of the following article munoz jose j on the modelling of incompressibility in linear and non linear elasticity with the master slave approach international journal for numerical methods in engineering abril 2008 vol 74 num 2 p 269 293 which has been published in final form at http www3 interscience wiley com journal 116323442 abstract

on the sharpness of l 2 error estimates of h 1 0 projections onto subspaces of piecewise high order polynomials in a plane polygonal domain consider a poisson problem u f with homogeneous dirichlet boundary condition and the p version finite element solutions of this we give various upper and lower bounds for the error measured in l 2 in the case of a single element i e a convex domain we reduce the question of sharpness of these estimates to the behavior of a certain inf sup constant which is numerically determined and a likely sharp estimate is then conjectured this is confirmed during a series of numerical experiments also for the case of a reentrant comer for a one dimensional analogue problem of rotational symmetry sharp l 2 error estimates are proven directly and via an extension of the classical duality argument here we give sharp l error estimates in some weighted and unweighted norms also

an estimate of the babuska brezzi inf sup discrete stability constant for general linear petrov galerkin finite element formulations an estimate of the babuska brezzi stability constant an a priori lower bound estimate of the babuska brezzi bb inf sup discrete stability constant is given for general linear petrov galerkin formulations the estimate is based on reasonable assumptions regarding a specific auxiliary adjoint problem the similarities and discrepancies of the presented method compared to the well known fortin s criterion are also noted the estimate may be used to obtain numerical verification of the bb condition under the assumption that the finite element basis functions are almost orthogonal

finite element analysis of anisotropic non linear incompressible elastic solids by a mixed model a mixed finite element method is presented for geometrically and materially non linear analysis of anisotropic incompressible hyperelastic materials an incremental iteractive total lagrangian formulation is adopted the nodal displacements and the hydrostatic pressure are independently interpolated leading to a mixed system of equations with characteristic zero diagonal terms computations are carried out using a three dimensional linear displacement constant pressure element a mixed penalty approximation is then employed to eliminate the pressure variables at the element level the anisotropic material handling capability of the formulation is tested through a number of transversely isotropic problems and the results compared to analytical solutions to demonstrate the applicability of this formulation to model complex anisotropic problems the inflation of a cut toroidal tube with helical fibre orientation is analysed

fractional randomness and the brownian bridge abstract this paper introduces a statistical approach to fractional randomness based on the central limit theorem we show under general conditions that fractional noise randomness defined relative to a uniform distribution implies as well a fractional brownian bridge randomness rather than a fractional brownian motion we analyze further their fractional properties namely their variance and covariance and obtain specific results for particular distributions including the fractional uniform distribution and an exponential distribution the results we obtain have both practical and theoretical implications to the many applications of fractional calculus and in particular when they are applied to modeling statistical problems where time scaling and randomness prime this is the case in finance insurance and risk models as well as in other areas of interest

stabilized low order finite elements for failure and localization problems in undrained soils and foundations abstract geomaterials in general and soils in particular are highly nonlinear materials presenting a very strong coupling between solid skeleton and intersticial water in the limit of zero compressibility of water and soil grains and zero permeability which correspond to the classical undrained assumption of soil mechanics the functions used to interpolate displacements and pressures must fulfill either the babuska brezzi conditions or the much simpler patch test proposed by zienkiewicz and taylor these requirements exclude the use of elements with equal order interpolation for pressures and displacements for which spurious oscillations may appear the simplest elements with continuous pressures which can be used in 2d are the quadratic triangle and quadrilateral with linear and bilinear pressures respectively the purpose of this paper is to present a stabilization technique allowing the use of both linear triangles for displacements and pressures t3p3 and bilinear quadrilaterals q4p4 the proposed element will be applied to obtain limit loads and failure surfaces in simple boundary value problems for which analytical solutions exist

on the stability of bilinear constant velocity pressure finite elements an analysis of the babuska stability of bilinear constant finite element pairs for viscous flow calculations is given an unstable mode not of the checkerboard type is given for which the stability constant turns out to beo h thus the indicated spaces are not stable in general for numerical calculation

a modified streamline diffusion method for solving the stationary navier stokes equation recently hughes et al 11 12 proposed new finite element schemes of petrov galerkin type for solving the stokes problem which do not require the discrete version of the ladyshenskaya babuska brezzi condition lbb condition in this paper we derive a conforming finite element method for solving the stationary navier stokes equations which combines the advantages of arbitrary finite element spaces for velocity pressure with the favourable properties of the streamline diffusion method in the case of moderate and high reynolds number

the patch test for mixed formulations this paper presents a simple extension of the patch test to mixed formulations to provide the necessary and sufficient conditions for convergence the general algebraic conditions of babuska and brezzi are given a simple form and a conceptual application of the patch tests serves to point out the instability of several well known formulations for incompressible problems the test is also applied to mixed displacement strain formulations of elasticity and limiting formulations are noted

the finite element patch test revisited a computer test for convergence validation and error estimates abstract the patch test was introduced in the mid 1960s first as a check of satisfactory performance for elements violating continuity requirements and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors in its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct later it was established that the test could also be used to find the convergence order of any finite element type while consistency is one of the necessary requirements for convergence the second is of course that of stability in the original early work only displacement type structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms in a 1986 reformulation of the patch test this stability or solvability requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms 1 2 although some misinterpretation of the patch test has occurred in literature 3 4 the authors believe that with certain restrictions on regularity of the problem dealt with the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical babuska brezzi criteria 5 6 in recent years another important use of the patch test has been found by babuska and his coworkers 7 9 this work tests the asymptotic effectivity of various error estimating procedures used with various element patterns

a computational strategy for thermo poroelastic structures with a time space interface coupling this paper deals with a computational strategy suitable for the simulation of multiphysics problems based on the large time increment latin method the simulation of such problems must encounter the possible different time and space scales which usually arise with the different physics herein we focus on using different time and space discretizations for each physics by introducing an interface with its own discretization the feasibility of both a time and space couplings is exemplified on a nonlinear 3 physics strongly coupled problem the thermal fluid structure interaction in a thermo poroelastic structure

finite element methods principles for their selection abstract principles for the selection of a finite element method for a particular problem are discussed these principles are stated in terms of the notion of approximability optimality and stability several examples are discussed in detail as illustrations conclusions regarding the selection of finite element methods are summarized in the final section of the paper

comparison of monolithic and splitting solution schemes for dynamic porous media problems 5 summary proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents we compare the monolithic and splitting solution of the different multi field formulations feasible in porous media dynamics because of the inherent solid fluid momentum 9 interactions one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint here the 11 resulting set of differential algebraic equations dae is solved by the finite element method fem following two different strategies 1 an implicit monolithic approach where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators 2 a semi explicit implicit splitting scheme in the sense of a fractional step method 15 where the dae are first discretized in time split using intermediate variables and then discretized in space using linear equal order approximations for all primary unknowns finally a one and a two dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies therefore several test cases differing in the used multi field formulation the 19 monolithic time stepping method and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter in the end we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem copyright q 2009 john wiley sons ltd 23

stable element free galerkin solution procedures for the coupled soil pore fluid problem it is always difficult to solve the coupled soil pore fluid problem when the soil is saturated and impermeable because this situation often results in intensive oscillations of the solutions this topic has been discussed widely in the field of the finite element method but rarely by meshless methods the element free galerkin efg method has outstanding advantages of solving this problem based on the fact that its interpolation function can be constructed flexibly by the nodes in the compact domain in this study an efg numerical model together with a stabilization technique is proposed to obtain stable solutions especially for the saturated and impermeable soil close agreement between computational results and analytical solutions for one and two dimensional examples shows that the proposed numerical model not only provides highly accurate solutions for the saturated soil with relatively high permeability but also eliminates the oscillations of the solutions very effectively for the saturated and impermeable soil furthermore the influences of the hydraulic anisotropy a typical property of two dimensional problems on the proposed stabilization technique are discussed it is suggested that the optimal distribution of the nodes of essential variables can be designed according to the relative importance of the permeability in different directions copyright c 2010 john wiley sons ltd

a novel unsymmetric 8 node plane element immune to mesh distortion under a quadratic displacement field an 8 node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions viz the compatibility enforcing shape functions and completeness enforcing shape functions the former are chosen to satisfy exactly the minimum inter as well as intra element displacement continuity requirements while the latter are chosen to satisfy all the linear and higher order completeness requirements so as to reproduce exactly a quadratic displacement field numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions in this respect the proposed 8 node unsymmetric element emerges to be better than the existing symmetric quad8 quad8 9 quad9 quad12 and quad16 elements and matches the performance of the quartic element quad25 for test problems involving a cubic or higher order displacement field the proposed element yields a solution accuracy that is comparable to or better than that of quad8 quad8 9 and quad9 elements furthermore the element maintains a good accuracy even with the reduced 2 2 numerical integration copyright 2003 john wiley sons ltd

an extension of the feti domain decomposition method for incompressible and nearly incompressible problems incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill conditioning that prevents accuracy in pressure estimation and can lead to a lack of convergence for iterative solution algorithms a multilevel dual domain decomposition method is then chosen as an iterative algorithm the original feti and feti dp methods are extended to deal with such problems when the discretization of the pressure field is discontinuous throughout the elements a dedicated augmentation of the algorithms is proposed and the different methods are compared with several preconditioners for bidimensional test cases the resulting approaches are both optimal and numerically scalable and their costs are estimated with a complexity analysis

taming chaotic systems with dithers abstract a simple approach is proposed to tame chaos by injecting high frequency signals commonly called dithers into the chaotic systems based on the relaxed method an appropriate dither is introduced to suppress chaotic motion if the frequency of dither is high enough the trajectory described by the dithered chaotic system and the trajectory of its corresponding mathematical model the relaxed system would be made as close as desired this fact enables us to get a rigorous prediction of the dithered chaotic system s behavior by obtaining the behavior of the relaxed system finally an example with simulations is given to illustrate the concepts discussed throughout this paper

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

p version of mixed finite element methods for stokes like problems abstract we investigate various sample extremum saddle point and minimization problems from the point of view of stability of increasing order mixed finite element methods problems discussed include the stokes poisson and linear elasticity problems

petrov galerkin solutions of the incompressible navier stokes equations in primitive variables with adaptive remeshing the solution of the incompressible navier stokes equations in primitive variables is addressed in this paper a petrov galerkin formulation which automatically introduces streamline upwinding and allows equal order interpolation for all flow variables is presented the solution algorithm is segregated involving a series of updatings of the velocity and pressure fields error estimation and adaptive remeshing strategies for both steady and unsteady problems are demonstrated in some representative numerical examples

a new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems abstract in order to accurately model the behaviour of geostructures it is usually not possible to neglect the interaction between the soil skeleton and the pore fluid classical finite element models taking into account this interaction are formulated in terms of the displacement and pore pressure fields and are based on the assumption that the fluid acceleration relative to the soil skeleton is negligible this type of mixed problems is similar to others found in solid and fluid mechanics and might give rise to numerical instabilities unless certain requirements are met there are two classical approaches to this problem the first is usually known as the zienkiewicz taylor patch test for mixed formulations as a consequence of this test the interpolation degree of the displacement field is required to be higher than the corresponding one of the pressure field mathematically speaking this is a necessary condition for stability the second approach mathematically more involved is usually known as the bab ska brezzi inf sup condition and constitutes a sufficient condition for stability however it is possible to obtain stable formulations circumventing the interpolation degree requirement through the so called stabilization techniques these techniques were initially applied in the context of fluid mechanics and later extended to solid mechanics this article presents a new formulation in which stabilization is achieved through an approach based on the simo rifai enhanced strain element

research of combined hybrid method applied in the reissner mindlin plate model this paper has introduced two kinds of combined hybrid variational formulations for plate bending finite elements based upon the reissner mindlin theory according to whether assumed constant moment stress is introduced when assumed constant shear stress has been introduced due to two options for incompatible displacement modes four new types of combined hybrid elements are proposed a set of standard tests for both thin and thick plates show that new elements perform well

variational space time elements for large scale systems abstract in this paper we introduce a new galerkin based formulation for transient continuum problems governed by partial differential equations in space and time therefore we aim at a direct finite element discretization of the space time suitable for massive parallel analysis of the arising large scale problem the proposed formulation is applied to thermal mechanical and fluid systems as well as to a kuramoto sivashinsky problem representing the general class of higher order formulations in material science using nurbs based shape functions we verify whenever possible the conservation properties of the formulation finally a series of examples demonstrate the applicability to all systems presented in this paper

a simple and efficient hellinger reissner type mixed finite element for nearly incompressible elasticity abstract the mixed finite element method is a promising approach in order to overcome locking phenomena of classical displacement based finite elements in the nearly incompressible regime for the elasticity problem in this work we present a novel element based on hellinger reissner s principle for linear elasticity essential for the construction of the element is a restriction of the solution space for the stresses resulting in a formulation with displacements in h 1 b and stresses in h div b the symmetry of the stresses is achieved in a weak sense without the necessity of additional degrees of freedom this formulation leads for the case of lowest order interpolation to a very efficient and robust finite element even satisfying the numerical inf sup test in two and three dimensions

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

a petrov galerkin formulation for the incompressible navier stokes equations using equal order interpolation for velocity and pressure a new petrov galerkin method for the incompressible navier stokes equations is presented the use of the so called optimal upwind parameter in multidimensions is justified by a time scale analysis of the relevant physical processes the resulting procedure circumvents the babuska brezzi condition and allows equal order interpolation for velocity and pressure to be used

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

error estimates and convergence rates for various incompressible elements this note reports numerical experiments on the efficiency of simple error estimates derived earlier1 applied to incompressible mixed or related penalty type formulations the rate of convergence and performance of various mixed elements is compared numerical results from a driven cavity and an incompressible elastic problem demonstrate that the t6b1 3d and t6 3c elements give a faster rate of convergence than the t6 1d element however in the case of a plane extrusion analysis stronger singularity the rate of convergence for the t6b1 3d element drops and is inferior to that of the t6 1d while the t6 3c element still proves superior to the other two elements

a general methodology for deriving shear constrained reissner mindlin plate elements in this paper the necessary requirements for the good behaviour of shear constrained reissner mindlin plate elements for thick and thin plate situations are re interpreted and a simple explicit form of the substitute shear strain matrix is obtained this extends the previous work of the authors presented in references 18 and 31 the general methodology is applied to the re formulation of some well known quadrilateral plate elements and some new triangular and quadrilateral plate elements which show promising features some examples of the good behaviour of these elements are given

an iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics abstract the discretization of the u p model for saturated porous media results in the semi discrete system of mixed type in displacements and pressures the babuska brezzi condition or the simpler patch test proposed by zienkiewicz and taylor precludes the use of elements with the equal low order of interpolation for u and p unless special stabilization techniques such as the fractional step algorithms are used the purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones the method is based on introducing an iteration algorithm the numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm

on computing the pressure by thep version of the finite element method for stokes problem this paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving stokes problem using thep version of the finite element method when one uses a localh 1 projection we show that optimal rates of convergence for the pressure approximation is achieved when the pressure is not inh 1 or the value of the pressure is only needed at a few points one may extract the pressure pointwise using e g a single layer potential recovery negative zero and higher norm estimates for the stokes velocity are derived within the framework of thep version of the f e m

an analysis of dual formulations for the finite element solution of two body contact problems abstract this paper examines the convergence properties of dual finite element formulations of the two dimensional frictionless two body contact problem under the assumption of infinitesimal kinematics the centerpiece of the proposed analysis is the well known babuska brezzi condition suitably adapted to the present problem it is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the babuska brezzi condition thus producing increasingly oscillatory solutions under mesh refinement alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

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

the babuska brezzi condition and the patch test an example abstract this paper discusses an example of a mixed finite element method in one dimension which satisfies the engineering patch test but does not satisfy a uniform inf sup condition the inf sup constant for this method is shown to be of order h2 the reasons for good results from benchmark problems for this method are addressed by characterizing the nature of input data for which the method performs well numerical results illustrating the effect of the non uniform inf sup condition are presented

wave propagation and localization problems in saturated viscoplastic geomaterials this paper presents an improved algorithm to deal with wave propagation and localization problems in saturated viscoplastic geomaterials it consists of a mixed formulation in terms of effective stress velocity and pore pressure that uses a fractional step algorithm allowing the use of equal order of interpolation for the three variables and the simplest element such as the linear triangle the viscoplastic model used is of modified cam clay type viscoplasticity results in a strong source term that deteriorates the accuracy of the two step taylor galerkin algorithm therefore a runge kutta splitting scheme has been used to deal with the source terms resulting in a better accuracy of the method copyright 2006 john wiley sons ltd

controlling a complex dynamical network to attain an inhomogeneous equilibrium abstract many networks are made up of a few groups with nodes in the same group having the same kind of function in this work the problem of controlling a complex dynamical network to attain an inhomogeneous equilibrium point is investigated which means that nodes in the same group achieve the same equilibrium point as an isolated node while different groups correspond to different equilibrium points an open loop constant control approach is first proposed to obtain the inhomogeneous equilibrium point of the network then the feedback pinning control approach is applied to make the inhomogeneous equilibrium point asymptotically stable

iterative method for constrained and mixed approximation an inexpensive improvement of f e m performance a general iterative procedure for solving mixed lagrangian finite element equations is introduced the method allows a dramatic improvement in accuracy of simple irreducible formulations to be achieved in a few resolutions applications to incompressible flow stress analysis and metal forming problems are given the procedures can be easily implemented into standard programs

a quadrilateral mixed finite element with two enhanced strain modes summary an improved plane strain stress element is derived using a hu washizu variational formulation with bilinear displacement interpolation seven strain and stress terms and two enhanced strain modes the number of unknowns of the four node element is increased from eight to ten degrees of freedom for linear and non linear applications the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element which is denoted by qe2 the excellent performance of the proposed element is demonstrated using several linear and non linear examples

global chaos control of non autonomous systems this study describes a global approach of controlling chaos to reduce tedious waiting time caused by using conventional local controllers with euler s method a non autonomous system is approximated by a non linear difference system and then an approximate global poincare map function is derived from the difference system by iterating one or more periods of a periodic excitation based on the map function unstable periodic orbits embedded in a chaotic motion can be detected and a global controller for a targeted unstable periodic orbit is designed the global controller makes all the unstable periodic orbits vanish except a targeted periodic orbit furthermore a lyapunov s direct method is applied to confirm that the global controller can asymptotically stabilize the unique periodic orbit for practical applications system models are usually unknown to obtain a mathematical model non linear system identification based on the harmonic balance principle is applied to an unknown chaotic system of a noisy environment simulation results demonstrate that the global controller successfully regularizes a chaotic motion even if the chaotic trajectory is far from the targeted periodic orbit

consistent mesh tying methods for topologically distinct discretized surfaces in non linear solid mechanics this work presents a new technique for joining dissimilar meshes in non linear finite element analysis with the novel result that the adjoining surfaces do not need to be spatially coincident for patch tests to be passed the work builds in some sense on results previously presented by dohrmann et al international journal for numerical methods in engineering 1991 40 1205 but considerably generalizes upon the results presented there most importantly it is shown that consistency of the tied finite element formulation may be achieved such that implementation of element formulations is not affected through the use of a three field variational principle applied over the contacting region a key conclusion of the work is that although the approach advocated bears some resemblance to mortar element schemes for contact analysis see for example mcdevitt and laursen international journal for numerical methods in engineering 2000 48 1525 a straightforward application of a mortaring scheme is not sufficient to guarantee patch test passage when dissimilar meshing of curved surfaces leads to local open volumes or overlaps between tied surfaces in their reference configurations the proposed algorithm is demonstrated through a number of numerical examples performed within a matrix free non linear equation solving framework published in 2003 by john wiley sons ltd

a stabilised mixed meshfree method for incompressible media application to linear elasticity and stokes flow abstract this paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation the aims of this paper are twofold to avoid the volumetric locking that occurs at the incompressible limit and to circumvent the inf sup condition while approximating both displacement and pressure with the same set of nodes and shape functions we employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes where the gradient is corrected as proposed by duan et al 2014 the gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed a threat that could undermine the stabilisation effort the resulting novel method passes the mixed patch tests and is demonstrated to be effective and optimally accurate in linear elasticity and stokes flow test cases

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

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

incompressibility without tears how to avoid restrictions of mixed formulation the severe restrictions of the babuska brezzi stability criteria permit only a limited number of interpolations for velocities displacements and pressure to be used for incompressible behaviour these restrictions preclude the use of many useful elements we show in this paper how such difficulties can be side stepped by seeking the steady state solution through the use of various time marching schemes this permits a simple iterative approach to incompressible or nearly incompressible problems of fluid mechanics or solid mechanics and provides a stimulus for the use of such procedures in metal forming flow etc

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

nonlinear feedback for controlling the lorenz equation this work presents a nonlinear feedback approach for controlling the lorenz equation the derivation of the feedback is based on linearizing an input output dynamic of the system which leads to large regions of asymptotic stability here the input signal to the lorenz equation is the applied heat via the rayleigh number the performance of the nonlinear feedback is tested via the stabilization of equilibrium points and periodic orbits

an eulerian mixed formulation for viscoplastic materials abstract this paper presents an eulerian mixed formulation for viscoplastic materials the algorithm is formulated in terms of deviatoric stress velocity and hydrostatic pressure using a fractional step algorithm that allows the use of equal order of interpolation for the three variables the constitutive law is based on the viscoplasticity theory of perzyna with the von mises yield criterion the jaumann rate of cauchy stress tensor is used to ensure the objectivity of the constitutive law a splitting scheme based on a two step taylor galerkin algorithm for the advective terms plus a fourth order runge kutta scheme for the sources has been used to improve the accuracy of the method

numerical analysis of strain localization in metal powder forming processes it is well known that strain localization and indeed displacement discontinuity can arise in materials exhibiting plastic behaviour indeed such localization is almost certain to occur if strain softening or non associated behaviour exists though it can be triggered even when ideal plasticity is assumed this study is concerned mainly with the manner in which the numerical discretization process has to be devised so as to capture the localization phenomenon in this paper a method is presented for applying the mixed formulation to study the prediction of localization phenomenon in powder forming processes this study is focused on the performance of mixed u triangular elements to study their efficiency in indicating localization for various mesh refinements an adaptive analysis using element elongation is applied in the modelling of strain localization the numerical results are obtained for a von mises yield criterion applied to a multi level component at the final stage of compaction it is shown that if a correct approximation is used then both the uniform and non uniform mesh refinements will converge to the correct answer and clearly indicate the localization phenomenon it is also observed that an adaptive analysis using element elongation can be effective in the modelling of such phenomena copyright 2001 john wiley sons ltd