rosenbrock type methods applied to finite element computations within finite strain viscoelasticity abstract in this article time adaptive high order rosenbrock type methods are applied to the system of differential algebraic equations which results from the space discretization using finite elements based on a constitutive model of finite strain viscoelasticity it is shown that in this smooth problem more efficient finite element computations result in comparison to classical finite element approaches since the time integration on the basis of rosenbrock type methods does not lead to a system of non linear equations in other words all aspects of implicit finite elements as local iterations on gauss point level and global equilibrium iterations do not occur the first introduction to this approach proposed by hartmann and wensch 22 is extended here to the case of finite strain applications where the geometrical non linear deformation has an essential contribution to the non linearities additionally a clear decomposition into local element or gauss point work and global computational work using the schur complement is introduced to exploit the classical finite element character moreover the extension to the reaction force computation which is different to the classical approach and the influence to mixed element formulations here the three field formulation for displacements pressure and dilatation are discussed the performance of various rosenbrock type methods is investigated and shows that for low accuracy requirements as in order one methods the proposal yields a drastic reduction of the computational time

mathematical analysis and numerical solution of models with dynamic preisach hysteresis abstract hysteresis is a phenomenon that is observed in a great variety of physical systems which leads to a nonlinear and multivalued behavior making their modeling and control difficult even though the analysis and mathematical properties of several rate independent and rate dependent hysteresis models are known this is not the case for the bertotti s rate dependent extension of the classical preisach model the so called dynamic preisach model for which current approaches lack a proper functional analytic framework which is essential to formulate optimization problems and develop stable numerics both being crucial in practice this paper deals with the description and mathematical analysis of the dynamic preisach hysteresis model toward that end we complete a widely accepted definition of the dynamic model commonly used to describe the constitutive relation between the magnetic field h and the magnetic induction b in which the values of b not only depends on the present values of h but also on the past history and its velocity we first analyze mathematically some important properties of the model and compare them with known results for the static preisach model then we consider a parabolic problem with dynamic hysteresis motivated by electromagnetic field equations under suitable assumptions we prove the existence of a solution of a weak formulation of the problem and solve it numerically finally we report a numerical test in order to assess the order of convergence and to illustrate the behavior of the numerical solution for different configurations of the dynamic preisach model

stress computation and consistent tangent operator using non linear kinematic hardening models one possibility of formulating the tinite element method is founded on the principle of virtual displacement in which we want to include a rate independent elastoplastic constitutive model based on the assumption of a yield surface the constitutive equations result from the assumptions of small deformations hooke s law for the elastic domain the normality rule for the evolution of plastic strains the von mises yield condition and a special kind of kinematic hardening due to armstrong and frederick 1 in which linear kinematic hardening is generalized with a saturation term we show that it is not generally recommendable to propose large load steps to this end we investigate the influences of the non linear kinematic hardening model on the stress computation and the resulting consistent elastoplastic tangent operator the main topics of this paper are 1 development of a problem optimized backward euler method with regard to the kinematic hardening model 2 study of the influence of the saturation term on the numerical accuracy through isoerror maps and 3 computation of the consistent elastoplastic tangent operator

an efficient stress algorithm with applications in viscoplasticity and plasticity this paper deals with two main topics the first one concerns the equivalence of stress algorithms based on a backward euler step applied on viscoplastic models of chaboche type and their elastoplastic counterpart generally the stress algorithm yields a system of non linear algebraic equations and the corresponding consistent tangent operator occurring in the principle of virtual displacements leads to a system of linear equations this procedure can be obtained utilizing only numerical methods the second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of armstrong frederick terms which is equivalent to a multi surface plasticity model applying this model a so called problem adapted stress algorithm is derived where only one non linear equation must be solved this result is independent of the number of terms in the hardening model furthermore only the viscoplastic algorithm must be implemented since it includes the elastoplastic constitutive model as a special case 1997 by john wiley sons ltd

finite strain viscoplasticity with nonlinear kinematic hardening phenomenological modeling and time integration abstract this article deals with a viscoplastic material model of overstress type the model is based on a multiplicative decomposition of the deformation gradient into elastic and inelastic part an additional multiplicative decomposition of inelastic part is used to describe a nonlinear kinematic hardening of armstrong frederick type two implicit time stepping methods are adopted for numerical integration of evolution equations such that the plastic incompressibility constraint is exactly satisfied the first method is based on the tensor exponential the second method is a modified euler backward method special numerical tests show that both approaches yield similar results even for finite inelastic increments the basic features of the material response predicted by the material model are illustrated with a series of numerical simulations

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

the rate of convergence of gmres on a tridiagonal toeplitz linear system the generalized minimal residual method gmres is often used to solve a nonsymmetric linear system ax b but its convergence analysis is a rather difficult task in general a commonly used approach is to diagonalize a x x 1 and then separate the study of gmres convergence behavior into optimizing the condition number of x and a polynomial minimization problem over a s spectrum this artificial separation could greatly overestimate gmres residuals and likely yields error bounds that are too far from the actual ones on the other hand considering the effects of both a s spectrum and the conditioning of x at the same time poses a difficult challenge perhaps impossible to deal with in general but only possible for certain particular linear systems this paper will do so for a nonsymmetric tridiagonal toeplitz system sharp error bounds on and sometimes exact expressions for residuals are obtained these expressions and or bounds are in terms of the three parameters that define a and chebyshev polynomials of the first kind

an implicit integration algorithm with a projection method for viscoplastic constitutive equations robinson s viscoplastic model a representative of the so called overstress models is integrated by use of the generalized midpoint rule the solution of the non linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with newton s method consistent tangent moduli are calculated and the quadratic convergence of the global newton equilibrium iteration is shown the time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration various numerical examples are considered to demonstrate the efficiency of the methods

numerical aspects associated with the implementation of a finite strain elasto viscoelastic viscoplastic constitutive theory in principal stretches this paper treats the numerical implementation of a finite strain elasto viscoelastic viscoplastic constitutive model for semi crystalline polymers written in principal stretches a parallel configuration of the three model elements is used that enables the decoupled algorithmic treatment of each response within a stress update numerical scheme the numerical aspects associated with the use of principal stretch constitutive expressions in a tensor space numerical environment are initially developed for the general cases of any elastic or inelastic constitutive element included is the formulation of the closed form consistent tangential modulus tensor the principal space algorithmic treatments of the elastic viscoelastic and viscoplastic elements are then used as specific examples of particular importance is the development of a principal space closest point projection return mapping algorithm for viscoplasticity including isotropic strain hardening preliminary numerical examples are presented to illustrate the versatility of the model

efficient implicit integration for finite strain viscoplasticity with a nested multiplicative split abstract an efficient and reliable stress computation algorithm is presented which is based on implicit integration of the local evolution equations of multiplicative finite strain plasticity viscoplasticity the algorithm is illustrated by an example involving a combined nonlinear isotropic kinematic hardening numerous backstress tensors are employed for a better description of the material behavior the considered material model exhibits the so called weak invariance under arbitrary isochoric changes of the reference configuration and the presented algorithm retains this useful property even more the weak invariance serves as a guide in constructing this algorithm the constraint of inelastic incompressibility is exactly preserved as well the proposed method is first order accurate concerning the accuracy of the stress computation the new algorithm is comparable to the euler backward method with a subsequent correction of incompressibility ebmsc and the classical exponential method em regarding the computational efficiency the new algorithm is superior to the ebmsc and em some accuracy tests are presented using parameters of the aluminum alloy 5754 o and the 42crmo4 steel fem solutions of two boundary value problems using msc marc are presented to show the correctness of the numerical implementation

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

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

on the numerical treatment of finite deformations in elastoviscoplasticity abstract this paper deals with the generalization of a geometric linear viscoplastic model to finite strains and its numerical application we owe the original formulation of the applied model to perzyna and chaboche it includes nonlinear isotropic and kinematic hardening as well as a nonlinear rate dependence the constitutive equations are integrated numerically in the context of a finite element formulation from theoretical considerations it is known that in the case of vanishing viscosity or slow processes rate independent plasticity arises as an asymptotic limit accordingly the numerical formulation includes this property in fact the stress algorithm corresponding to viscoplasticity is reduced to the asymptotic limit in a most simple way namely by setting the viscosity parameter equal to zero furthermore it is shown that the numerical integration of the constitutive model involves the solution of only one nonlinear equation for one scalar unknown this even applies to a sum of armstrong frederick terms the algorithm incorporates the inelastic incompressibility on the level of the gau points numerical computations of examples taken from metal forming technology show the physical significance of the model and the reliability of the numerical algorithm these calculations have been carried out by means of the finite element program psu

remarks on the interpretation of current non linear finite element analyses as differential algebraic equations for the numerical solution of materially non linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point wise defined evolution equations characterizing the material behaviour the interpretation of such systems as differential algebraic equations dae allows modern day integration algorithms from numerical mathematics to be efficiently applied especially the application of diagonally implicit runge kutta methods dirk together with a multilevel newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward euler schemes for the local time integration moreover the notion of the consistent tangent operator becomes more obvious in this context the quadratical order of convergence of the multilevel newton algorithm is usually validated by numerical studies however an analytical proof of this second order convergence has already been given by authors in the field of non linear electrical networks we show that this proof can be applied in the current context based on the dae interpretation mentioned above we finally compare the proposed procedure to several well known stress algorithms and show that the inclusion of a step size control based on local error estimations merely requires a small extra time investment copyright 2001 john wiley sons ltd

from potential theory to matrix iterations in six steps the theory of the convergence of krylov subspace iterations for linear systems of equations conjugate gradients biconjugate gradients gmres qmr bi cgstab and so on is reviewed for a computation of this kind an estimated asymptotic convergence factor rho le 1 can be derived by solving a problem of potential theory or conformal mapping six approximations are involved in relating the actual computation to this scalar estimate these six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming

micromechanical modeling of coupled viscoelastic viscoplastic composites based on an incrementally affine formulation this study proposes a micromechanical modeling of inclusion reinforced viscoelastic viscoplastic composites based on mean field approaches for this we have generalized the so called incrementally affine linearization method which was proposed by doghri et al 2010a for elasto viscoplastic materials the proposal provides an affine relation between stress and strain increments via an algorithmic tangent operator in order to find the incrementally affine expression we start by the linearization of evolution equations at the beginning of a time step around the end time of the step next a numerical integration of the linearized equations is required using a fully implicit backward euler scheme the obtained algebraic equations lead to an incrementally affine formulation which is form similar to linear thermoelasticity therefore known homogenization models for linear thermoelastic composites can be applied the proposed method can deal with general viscoelastic viscoplastic constitutive models with an arbitrary number of internal variables the semi analytical predictions are validated against finite element simulations and experimental results

least squares residuals and minimal residual methods we study krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step minimal residual mr methods mr methods are often formulated in terms of a sequence of least squares ls problems of increasing dimension we present several basic identities and bounds for the ls residual these results are interesting in the general context of solving ls problems when applied to mr methods they show that the size of the mr residual is strongly related to the conditioning of different bases of the same krylov subspace using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of mr methods different bases also lead to different implementations which are mathematically equivalent but can differ numerically our theoretical results are used for a finite precision analysis of implementations of the gmres method y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 we explain that the choice of the basis is fundamental for the numerical stability of the implementation as demonstrated in the case of simpler gmres h f walker and l zhou numer linear algebra appl 1 1994 pp 571 581 the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis in particular we prove that simpler gmres is inherently less numerically stable than the classical gmres implementation due to saad and schultz siam j sci statist comput 7 1986 pp 856 869

a time integration method for the viscoelastic viscoplastic analyses of polymers and finite element implementation the present study introduces a time integration algorithm for solving a non linear viscoelastic viscoplastic ve vp constitutive equation of isotropic polymers the material parameters in the constitutive models are stress dependent the algorithm is derived based on an implicit time integration method computational inelasticity springer new york 1998 within a general displacement based finite element fe analysis and suitable for small deformation gradient problems schapery s integral model is used for the ve responses while the vp component follows the perzyna model having an overstress function a recursive iterative method int j numer meth engng 2004 59 25 45 is employed and modified to solve the ve vp constitutive equation an iterative procedure with predictor corrector steps is added to the recursive integration method a residual vector is defined for the incremental total strain and the magnitude of the incremental vp strain a consistent tangent stiffness matrix as previously discussed in ju j eng mech 1990 116 1764 1779 and simo and hughes computational inelasticity springer new york 1998 is also formulated to improve convergence and avoid divergence available experimental data on time dependent and inelastic responses of high density polyethylene are used to verify the current numerical algorithm the time integration scheme is examined in terms of its computational efficiency and accuracy numerical fe analyses of microstructural responses of polyethylene reinforced with elastic particle are also presented copyright 2009 john wiley sons ltd

sharpness in rates of convergence for the symmetric lanczos method the lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem there is a well established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs these bounds suggest at least linear convergence in terms of the number of lanczos steps assuming there are gaps between individual eigenvalues in practice often superlinear convergence is observed the question is do the existing bounds tell the correct convergence rate in general an affirmative answer is given here for the two extreme eigenvalues by examples whose lanczos approximations have errors comparable to the error bounds for all lanczos steps

gmres convergence analysis for a convection diffusion model problem when gmres y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 is applied to streamline upwind petrov galerkin supg discretized convection diffusion problems it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norm several approaches were made to understand this behavior however the existing analyses are solely based on the matrix of the discretized system and they do not take into account any influence of the right hand side determined by the boundary conditions and or source term in the pde therefore they cannot explain the length of the initial period of slow convergence which is right hand side dependent we concentrate on a frequently used model problem with dirichlet boundary conditions and with a constant velocity field parallel to one of the axes instead of the eigendecomposition of the system matrix which is ill conditioned we use its orthogonal transformation into a block diagonal matrix with nonsymmetric tridiagonal toeplitz blocks and offer an explanation of gmres convergence we show how the initial period of slow convergence is related to the boundary conditions and address the question why the convergence in the second stage accelerates

distribution of the discretization and algebraic error in numerical solution of partial differential equations abstract in the adaptive numerical solution of partial differential equations local mesh refinement is used together with a posteriori error analysis in order to equilibrate the discretization error distribution over the domain since the discretized algebraic problems are not solved exactly a natural question is whether the spatial distribution of the algebraic error is analogous to the spatial distribution of the discretization error the main goal of this paper is to illustrate using standard boundary value model problems that this may not hold on the contrary the algebraic error can have large local components which can significantly dominate the total error in some parts of the domain the illustrated phenomenon is of general significance and it is not restricted to some particular problems or dimensions to our knowledge the discrepancy between the spatial distribution of the discretization and algebraic errors has not been studied in detail elsewhere

computational homogenisation of composite plates consideration of the thickness change with a modified projection strategy in the present paper a method for the modelling of the mechanical behaviour of composite plates especially hybrid laminates and sandwich plates is proposed the chosen method is the computational homogenisation or the so called fe 2 for plates the principle is to split the considered problem into two scales on the one hand the two dimensional fe computation of a plate is performed on the macroscale on the other hand a three dimensional fe problem is computed on the mesoscale which enables the discretisation of the layers organisation from each integration point of the macroscale the deformations are projected to the representative volume element rve on the mesoscale where a dirichlet boundary value problem is solved finally the homogenisation of the stresses is conducted to define the stress resultants of the macroscale the proposed method presents the advantage to take into account any material behaviour without any transformation of the constitutive laws even non linear ones furthermore the new projection strategy presented in this work considers the thickness change of the plate which enables a resolution of the poisson locking the method is illustrated by the simulation of the mechanical behaviour of both sandwich plate and of hybrid laminate

coupled viscoelastic viscoplastic modeling of homogeneous and isotropic polymers numerical algorithm and analytical solutions a coupled viscoelastic viscoplastic ve vp model is implemented and studied the total strain is the sum of ve and vp parts and the cauchy stress is given by a linear ve model as a boltzmann integral of the history of ve strains the proposed computational algorithm features fully implicit integration return mapping based on a two step ve predictor vp corrector strategy and a consistent tangent operator the algorithm is applied to j2 vp coupled with ve very compact expressions are obtained which are form identical to classical elasto viscoplasticity evp provided that the constant linear elastic shear and bulk moduli are replaced with incremental relaxation moduli which are appropriate functions of the time increment two different integration methods to obtain the incremental moduli are proposed and assessed closed form solutions for uniaxial tension and simple shear are developed based on an original solution method for integro differential equations the analytical results enable to illustrate the constitutive model and provide unambiguous benchmarks for numerical algorithms model predictions are compared with experimental data and reasonable correlation is obtained

on meinardus examples for the conjugate gradient method the conjugate gradient cg method is widely used to solve a positive definite linear system ax b of order n it is well known that the relative residual of the kth approximate solution by cg with the initial approximation x 0 0 is bounded above by 2 k k k k 1 with k k 1 k 1 where k k a a 2 a 1 2 is a s spectral condition number in 1963 meinardus numer math 5 1963 pp 14 23 gave an example to achieve this bound for k n 1 but without saying anything about all other 1 k n 1 this very example can be used to show that the bound is sharp for any given k by constructing examples to attain the bound but such examples depend on k and for them the k 1 th residual is exactly zero therefore it would be interesting to know if there is any example on which the cg relative residuals are comparable to the bound for all 1 k n 1 there are two contributions in this paper 1 a closed formula for the cg residuals for all 1 k n 1 on meinardus example is obtained and in particular it implies that the bound is always within a factor of 2 of the actual residuals 2 a complete characterization of extreme positive linear systems for which the kth cg residual achieves the bound is also presented

a note on conjugate gradient convergence part ii summary continuing our previous analysis we derive the exact number of conjugate gradient iterations needed to achieve a given tolerance for the one dimensional discrete poisson equation on a uniform grid and a particularly smooth solution vector

the rate of convergence of gmres on a tridiagonal toeplitz linear system ii abstract this paper continues the recent work of the authors r c li w zhang the rate of convergence of gmres on a tridiagonal toeplitz linear system numer math 112 2009 267 293 electronically published on 19 december 2008 on the rate of convergence of gmres for a tridiagonal toeplitz linear system ax b much simpler formulas than the earlier ones for gmres residuals when b is the first or the last column of the identity matrix are established and these formulas allow us to confirm the rate of convergence that was conjectured but only partially proven earlier simpler and sharper bounds than earlier ones when all b s entries except its first and last ones are zeros are also obtained

fully implicit integration and consistent tangent modulus in elasto plasticity this paper deals with the numerical integration of a class of rate independent elasto plastic models the backward euler scheme is used to integrate the rate constitutive relations the non linear equations obtained are solved by the newton method the consistent tangent modulus is obtained by exact linearization of the algorithm in the case of j 2 elastoplasticity with non linear isotropic hardening and non linear kinematic hardening chaboche marquis model explicit formulas are derived without any approximations

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

computation in finite strain viscoelasticity finite elements based on the interpretation as differential algebraic equations abstract in this article the interpretation of current non linear finite element calculations applied to constitutive equations of evolutionary type as a system of differential algebraic equations is continued p ellsiepen s hartmann int j numer methods engrg 51 2001 679 this procedure is applied to a model of finite strain viscoelasticity which incorporates a particular model of non linear rate dependence three specific topics are studied firstly we investigate a possible treatment of the interpretation in view of mixed finite elements secondly the application of stiffly accurate diagonally implicit runge kutta methods which yield on element level the same structure as a backward euler based integration step is analysed it is shown that the resulting stress algorithm is reducible to the solution of only three non linear equations at each gauss point one maxwell element lastly the effect with respect to expense and achievable accuracy of a time adaptive procedure is focused which is necessary in the case of different time scales such as relaxation or creep dominated processes

formulation and numerical solution of non smooth elasto visco plasticity models abstract in this work we describe in detail a model for small strain elasto visco plasticity with convex non smooth yield functions and associative nonlinear kinetic laws restricted to linear hardening using concepts of non smooth convex geometry numerical methods are developed to integrate the evolution equations of the model these algorithms are analyzed and shown to inherit a discrete dissipation inequality irrespective of the smoothness of the yield function the results are applied to models based on tresca s and drucker prager s yield criteria which include all possible types of non smoothness numerical examples are shown to illustrate the performance of the methods

on a numerical method for 2d magnetic field computations in a lamination with enforced total flux the paper deals with a numerical method for the evaluation of the electromagnetic loss in a lamination of an electric machine based upon the maxwell equations the underlying problem consists in the computation of the unidirectional magnetic field in a cross section of the laminate orthogonal to the enforced flux the method starts from a suitable variational formulation of the governing parabolic problem with a nonlocal neumann boundary condition accompanied by a dirichlet side condition involving nonlinear and hysteresis effects through the differential permeability coefficient the variational problem is solved numerically by a finite element method combined with a finite difference technique which is deviced so as to take into account the nonlinear and hysteresis behaviour of the material the numerical method is found to be effective and reliable

a return mapping algorithm for cyclic viscoplastic constitutive models abstract this paper presents a return mapping algorithm for cyclic viscoplastic constitutive models that include material memory effects the constitutive model is based on multi component forms of kinematic and isotropic hardening variables in conjunction with von mises yield criterion armstrong frederick a f type rules are used to describe the nonlinear evolution of each of the multi component kinematic hardening variables a saturation type exponential rule is used to describe the nonlinear evolution of each of the isotropic hardening variables the concept of memory surface is used to describe the strain range dependent material memory effects that are induced by the prior strain histories in this paper the above class of cyclic viscoplastic constitutive models is formulated within a consistent thermodynamic framework that encompasses the standard generalized materials framework furthermore a complete algorithmic treatment of the above rate dependent constitutive model is also presented for any desired stress or strain constrained configuration subspace a generalized midpoint algorithm is used to integrate the rate constitutive equations the consistent tangent operator is obtained by linearizing the return mapping algorithm and is found to be unsymmetric due to the presence of nonlinear evolution rules for the kinematic hardening variables several numerical examples representing the cyclic hardening and softening behavior transient and stabilized hysteresis behavior and the non fading memory effects of the material are presented these examples demonstrate the accuracy and robustness of the present algorithmic framework for modeling the cyclic viscoplastic behavior of the material

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

complementary 2 d finite element procedures for the magnetic field analysis using a vector hysteresis model the main purpose of this paper is to incorporate a rened hysteresis model viz a vector preisach model in 2 d magnetic eld computations two complementary formulations based either on the scalar or on the vector potential are considered the governing maxwell equations are rewritten in a suitable way that allows to take into account the proper magnetic material parameters and moreover to pass to a variational formulation the variational problems are solved numerically by a finite element approximation using a quadratic mesh followed by a time discretization method based upon a modied cranck nicholson algorithm finally the eectiveness of the presented mathematical tools have been conrmed by several numerical experiments comparing the complementarity of the two procedures 1998 john wiley sons ltd

a note on conjugate gradient convergence the one dimensional discrete poisson equation on a uniform grid with n points produces a linear system of equations with a symmetric positive definite coefficient matrix hence the conjugate gradient method can be used and standard analysis gives an upper bound of o n on the number of iterations required for convergence this paper introduces a systematically defined set of solutions dependent on a parameter beta and for several values of beta presents exact analytic expressions for the number of steps k beta tau n needed to achieve accuracy tau the asymptotic behavior of these expressions has the form o n alpha as n rightarrow infty and o tau gamma as tau rightarrow 0 in particular two choices of beta corresponding to nonsmooth solutions give alpha 0 i e iteration counts independent of n this is in contrast to the standard bounds the standard asymptotic convergence behavior alpha 1 is seen for a relatively smooth solution numerical examples illustrate and supplement the analysis