parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

multi scale second order computational homogenization of multi phase materials a nested finite element solution strategy this paper presents the detailed implementation and computational aspects of a novel second order computational homogenization procedure which is suitable for a multi scale modelling of macroscopic localization and size effects the second order scheme is an extension of the classical first order computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro micro scale transition from the microstructural analysis the macroscopic stress and higher order stress tensors are obtained thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum the higher order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix for the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed as an example the second order computational homogenization approach is applied for the multi scale analysis of simple shear of a constrained heterogeneous strip where a pronounced boundary size effect appears

implicit algorithm for finite deformation hypoelastic viscoplasticity in fcc metals an implicit objective stress update algorithm is proposed for a hypoelastic viscoplastic model a thermal dynamic yield function which is derived based on the thermal activation analysis and dislocation interaction mechanisms is used along with the consistency approach and the framework of additive viscoplasticity in deriving the proposed model for fcc metals the corotational formulation approach is utilized in developing the proposed model in the finite deformation field for the case of the newton raphson iteration method a new expression for the consistent algorithmic tangent stiffness matrix of rate dependent metals is derived by direct linearization of the stress update algorithm finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program abaqus numerical implementation for a simple tensile problem is used for validating the material parameters of the ofhc copper under low and high strain rates and temperatures the numerical results of the adiabatic true stress true strain curves compare very well with the experimental data the effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh independent results copyright 2006 john wiley sons ltd

spectral dimension of a wire network by a rigorous analysis for diffusion on a wire network the spectral dimension of a fractal is shown to be independent of the local structure it is discussed that diffusion does not occur when the lattice spacing tends to zero on such a fractal that the spectral dimension is less than the hausdorff dimension if a free field on the fractal exhibits a certain long distance behaviour for a sierpinski carpet the spectral dimension is evaluated within bond moving approximation migdal kadanov renormalisation as a result the author obtains a value smaller than the hausdorff dimension

efficient iteration schemes for anisotropic hyperelasto plasticity numerical issues arising when integrating hyperelasto plastic constitutive equations with elastic anisotropy stemming from anisotropic damage and plastic anisotropy as represented by kinematic hardening are discussed in particular solution algorithms for the corresponding non linear system of equations due to implicit integration algorithms are addressed it is shown that algorithms like staggered iteration and quasi newton techniques are superior to a pure newton technique when the cpu time is compared however the drawback of the staggered and the quasi newton technique is that rather small time steps must be taken to ensure convergence which can be of importance when applying complex constitutive models in a finite element programme copyright 2005 john wiley sons ltd

on the spatial formulation of anisotropic multiplicative elasto plasticity abstract this contribution aims at a thermodynamically consistent and modular formulation of anisotropic multiplicative elasto plasticity embedded in the well established framework of non standard dissipative materials we adopt as a key ingredient the introduction of additional symmetric arguments typically structural tensors into the relevant scalar valued isotropic tensor functions on this basis the fundamental covariance relation leads us to a formulation which allows itself to be set up directly in terms of spatial arguments and thus guarantees a convenient implementation within a finite element setting

higher order and higher floating point precision numerical approximations of finite strain elasticity moduli two real domain numerical approximation methods for accurate computation of finite strain elasticity moduli are developed and their accuracy and computational efficiency investigated with reference to hyperelastic constitutive models with known analytical solutions the methods are higher order and higher floating point precision numerical approximation the latter being novel in this context a general formula for higher order approximation finite difference schemes is derived and a new procedure is proposed to implement increased floating point precision the accuracy of the approximated elasticity moduli is investigated numerically using higher order approximations in standard double precision and increased quadruple precision it is found that as the order of the approximation increases the elasticity moduli tend towards the analytical solution using higher floating point precision the approximated elasticity moduli for all orders of approximation are found to be more accurate than the standard double precision evaluation of the analytical moduli application of the techniques to a finite element problem shows that the numerically approximated methods obtain convergence equivalent to the analytical method but require greater computational effort it is concluded that numerical approximation of elasticity moduli is a powerful and effective means of implementing advanced constitutive models in the finite element method without prior derivation of difficult analytical solutions

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

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

robust numerical calculation of tangent moduli at finite strains based on complex step derivative approximation and its application to localization analysis abstract an extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard fem software this method is based on the complex step derivative approximation csda approach the csda is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e g finite difference approximation therefore the csda can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision implementation details of the robust numerical approximation of tangent moduli from stress calculations using the csda are given and their performance is illustrated through representative examples involving finite deformations in addition to that we focus on the determination of material instabilities therefore an accompanying localization analysis is performed where the acoustic tensor is directly computed from the approximation of the moduli it is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation on the other hand the csda approach provides high accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely

consistent tangent operator for cutting plane algorithm of elasto plasticity abstract the paper presents a derivation of the consistent tangent operator cto for the cutting plane algorithm cpa for a class of plasticity models that are suitable to be integrated using cpa an explicit recursive expression is analytically derived and is updated in each iteration of the cpa integration procedure to yield the final value of the cto when the cpa is converged

efficient and modular algorithms in modeling finite inelastic deformations objective integration parameter identification and sub stepping techniques this paper presents a general algorithmic formulation for constitutive equations describing inelastic material deformation the algorithmic representation focuses on the generality and modularity which is a key aspect within the development phase of constitutive equations it is shown explicitly how to solve the incremental constitutive equations by use of objective algorithms and an implicit integration scheme applying numerical as well as analytical derivatives although it is shown that using numerical gradients is significantly slower their application is easier especially in the development phase it is also demonstrated that having the constitutive equations modeled in the general context introduced the issue of parameter identification can be tackled by the same implementation using a driver algorithm besides parameter identification the general structure is also applied to two different fe simulations first the objectivity of different integration methods will be presented considering a simple shear loading superposed by a rigid rotation using a strain rate independent material model as example in the second example it is shown for a strain rate dependent material model how the simulation of the wire drawing process can be accelerated crucially by using sub stepping techniques

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

efficient computation of the far field of parabolic reflectors by pseudo sampling algorithm the newly developed pseudo sampling representation is applied for computing the far field of an offset parabolic reflector with a cluster feed illumination a new comb type fast fourier transform fft algorithm is used in a computer program both precision and computational time are analyzed demonstrating the excellent performance of the method

asymptotic form of the spectral dimension of the sierpinski gasket type of fractals the authors have studied the spectral dimension d48t of an infinite class of fractals the first member b 2 of the class is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the ordinary triangular lattice by studying the electric resistance of the fractals they have been able to calculate exact values of d for the first 200 members of the class an analysis of the obtained data reveals that for large b the spectral dimension should approach the upper limit of 2 according to the formula d approximately 2 constant ln b beta where beta is not larger than one this result implies among other things that the scaling exponents of the resistivity and diffusion constant should logarithmically vanish at the fractal lattice crossover

an integration scheme for prandtl reuss elastoplastic constitutive equations we investigate the generalized midpoint rule for the time integration of elastoplastic constitutive equations for pressure independent yield criteria the incremental equations are divided into one scalar hydrostatic pressure dilation rate equation and a stress deviator strain rate deviator tensorial equation this formulation is applied to two classical problems bulging of a tube under internal pressure and tension test on a notched specimen

operator split methods for the numerical solution of the elastoplastic dynamic problem the elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques the additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition in the context of a finite element discretization these product algorithms entail for every time step the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity the proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions with or without corners the stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem

a return mapping algorithm for elastoplastic and ductile damage constitutive equations using the subloading surface method summary in this paper is presented a return mapping algorithm for an elastoplastic damage model that couples the constitutive equations of an unconventional plasticity model with the phenomenological description of ductile damage in the continuum damage mechanics framework this approach combines the advantages of describing the mechanical property degradation by using the lemaitre model with the realistic accumulation of plastic deformation in cyclic mobility problems from the subloading surface model most of the damage models for ductile failure focus on monotonic loading conditions whereas the present model aims to extend the investigation to cyclic loading and low cycles fatigue problems a cutting plane algorithm is adopted to describe the material behavior that consider both isotropic and kinematic hardening laws two simple numerical studies show that the elastoplastic and damage model is implemented correctly displaying a quadratic rate of convergence for local equilibria and precise solutions even for large prescribed strain increments

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

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

a higher order time integration method for viscoplasticity time integration in viscoplasticity is discussed from both engineering and mathematical points of view a second order integration scheme is presented and combined with an adaptive time stepping algorithm which utilizes local error estimates for the choice of time steps and a closed form expression for the consistent tangent operator is derived the algorithm is applied to the viscoplastic model by choi and krempl

renormalisation on symmetric fractals the authors introduce and investigate new classes of sierpinski type fractals they determine their fractal and spectral dimensions using renormalisation procedures and for particular classes they give these dimensions in closed form the spectral dimensions densely fill the interval allowing them to flexibly choose models for applications

coupled plastic damaged model abstract a constitutive model that couples plasticity and damage is presented the model is thermodynamically consistent and comes from a generalization of classical plasticity theory and isotropic damage theory of kachanov coupling between plasticity and damage is achieved through a simultaneous solution of the plastic and the damage problem after a description of the model a numerical algorithm for the integration of the resulting constitutive equations is presented it is an euler backward type of algorithm that is particularly suitable to solve plain stress non linear problems with a 2d finite element program the consistent stiffness matrix is also derived the paper is completed with some application examples that show that the model presented accurately reproduces the behaviour of elastic plastic damaged materials

exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals this paper presents a new stress update algorithm for large strain rate independent single crystal plasticity the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors the key contribution of this paper is an algorithmic formulation of the exponential map exp sl 3 sl 3 for updating the special linear group sl 3 of unimodular plastic deformation maps this update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration the resulting fully implicit stress update algorithm treats the possibly redundant constraints of single crystal plasticity by means of an active set search it exploits intrinsically the simple representation of the schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of eulerian vectors updates the performance of the proposed algorithm is demonstrated by means of a representative numerical example

determination of the tangent stiffness tensor in materials modeling in case of large deformations by calculation of a directed strain perturbation abstract the finite element method in the field of materials modeling is often relying to the tangent stiffness tensor of the constitutive law this so called jacobian matrix is required at each time increment and describes the local material behavior it assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state for increasingly sophisticated material models the tangent stiffness tensor can be derived analytically only with great effort if at all numerical methods like the forward difference the central difference and the complex step derivative approximation approach are widely used for its calculation for each of these methods it is necessary to generate a specific strain perturbation however in large strain formulations it is not possible to perturb the strain directly but one can only modify the deformation gradient we present our methods to generate a directed strain perturbation for the green lagrange euler almansi and logarithmic strain measures as a function of the deformation gradient and compare them with other commonly used methods an increase in accuracy and rate of convergence can be achieved with the proposed procedures

a highly accurate 1st and 2nd order differentiation scheme for hyperelastic material models based on hyper dual numbers abstract in this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models which are derived in terms of the first and second derivatives of a strain energy function this numerical scheme provides a compact model independent framework which means that once the framework is implemented any other hyperelastic material model can be incorporated by solely modifying the energy function the method is based on the numerical calculation of strain energy derivatives using hyper dual numbers and thus referred to as hyper dual step derivative hdsd the hdsd does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values furthermore it enables the calculation of derivatives of arbitrary order this is a great advantage compared to other numerical approaches as e g the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values another alternative the complex step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values but it only provides first derivatives and is thus not able to calculate stresses and moduli at once in this paper representative numerical examples using an anisotropic model are provided showing the performance of the proposed method in detail an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme furthermore examples demonstrate the robustness and simple implementation of the hdsd scheme in finite element software it turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time

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

consistent micro macro transitions at large objective strains in curvilinear convective coordinates this paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates the main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro macro transitions in curvilinear convective coordinates where different physical spaces are considered at the homogenized macro continuum and at the locally attached representative micro structures the deformation and the coordinate system of the micro structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro continuum for the consistent formulation of micro macro transitions the operations scale up and scale down are introduced considering the rotated representation of tensor variables at the different physical reference frames of micro and macro structure the second goal of this paper is to use objective strain measures like the green lagrange strain tensor for the solution of boundary value problems on the micro and macro scale by providing the required transformations for the work conjugate stress strain and tangent tensors into variables admissible for the considered micro macro transitions and satisfying the averaging theorem copyright 2007 john wiley sons ltd

return mapping for nonsmooth and multiscale elastoplasticity we present a semi implicit return mapping algorithm for integrating generic nonsmooth elastoplastic models the semi implicit nature of the algorithm stems from freezing the plastic internal variables at their previous state followed by implicitly integrating the stresses and plastic multiplier the plastic internal variables are incrementally updated once convergence is achieved a posteriori locally the algorithm behaves as a classic return mapping for perfect plasticity and hence inherits the stability of implicit integrators however it differs from purely implicit integrators by keeping the plastic internal variables locally constant this feature affords the method the ability to integrate nonsmooth c0 evolution laws that may not be integrable using implicit methods as a result we propose and use the algorithm as the backbone of a semi concurrent multiscale framework in which nonsmooth constitutive relationships can be directly extracted from the underlying micromechanical processes and faithfully incorporated into elastoplastic continuum models though accuracy of the proposed algorithm is step size dependent its simplicity and its remarkable ability to handle nonsmooth relations make the method promising and computationally appealing

complex polynomial phase integration a previously published integration algorithm applicable to the numerical computation of integrals with rapidly oscillating integrands is generalized the previous algorithm involved quadratic approximation of the phase function which was assumed to be real the present generalization concerns approximation of a complex phase function by a polynomial of arbitrary degree as before the integrand is then written without approximation as a slowly varying function multiplied by the polynomial phase exponential and the slowly varying factor is approximated by a finite sum of chebyshev polynomials the integral is thus expressed as a sum of constituent integrals which are computed recursively via lu decomposition applied to a system of linear equations with a banded coefficients matrix examples are presented comparing various degree phase approximants

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

entropic thermoelasticity at finite strains aspects of the formulation and numerical implementation abstract this paper presents a formulation of isotropic finite strain thermoelasticity and addresses some aspects of its numerical implementation on the theoretical side a eulerian setting of isotropic thermoelasticity is discussed based on the finger tensor as a strain measure novel aspects are a direct representation of the eulerian thermoelastic moduli in terms of the finger tensor and a rigorous decomposition of the thermoelastic response functions into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts an algorithmic procedure for the computation of the stress and thermoelastic moduli based on a representation of the free energy in terms of eigenvalues of the unimodular part of the finger tensor is developed and applied to model problems of strictly entropic and modified entropic thermoelasticity furthermore two algorithms for the solution of the coupled problem are discussed based on operator splits of the global field equations of thermoelasticity the paper concludes with some representative numerical simulations of thermoelastic processes in rubber like materials

on the numerical integration of a class of pressure dependent plasticity models with mixed hardening radial return algorithms for both three dimensional and plane stress situations are developed for a class of pressure dependent plasticity models formulated in state variables with mixed hardening the consistent tangent matrix has been developed which among other advantages does not require numerical inversion the algorithms for gurson s mixed hardening model are incorporated in a finite element program to solve several simple uniaxial tension problems when compared with numerically integrated solutions the results from the finite element analysis show excellent agreement finally results of the axisymmetric necking problem using gurson s mixed hardening model are presented as an example of application

multisurface thermoplasticity for single crystals at large strains in terms of eulerian vector updates this paper presents a formulation of large strain rate independent multisurface thermoplasticity for single crystals and addresses aspects of its numerical implementation the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a key feature of the present paper is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors this observation is intrinsically exploited on the numerical side by formulating a new fully implicit stress update algorithm and the associated consistent elastoplastic moduli in terms of these eulerian vectors the proposed return mapping algorithm treats the possibly redundant constraints of large strain multisurface plasticity for single crystals by means of an active set search furthermore it satisfies in an algorithmically exact way the plastic incompressibility condition in situations of multislip the performance of the proposed formulation is demonstrated for two representative numerical simulations of thermoplastic deformation processes in single crystals with isotropic taylor type hardening

quadratic phase integration using a chebyshev expansion an integration algorithm is described which is particularly effective in the numerical treatment of integrands having rapidly varying phase and slowly varying amplitude the algorithm involves approximating the phase function by a quadratic polynomial and rewriting the integrand without approximation as a slowly varying function multiplied by this quadratic phase exponential the slowly varying function is then approximated by chebyshev expansion and the desired integral is thus expressed as a sum of constituent integrals with integrands containing a chebyshev polynomial multiplied by the quadratic phase factor these constituent integrals are computed by means of lu decomposition applied to a system of linear equations with a banded coefficients matrix example results are presented indicating that a substantial reduction in computation time may be realized by means of this approach

correction to a new integration algorithm and its application to the analysis of symmetrical cassegrain microwave antennas an efficient numerical algorithm for physical optics po analysis of reflectors is presented cylindrical symmetry of the reflector is not essential but is assumed for purposes of illustration and this leads to the use of fourier representation in the azimuthal variable the algorithm is based on quadratic approximation of the phase and polynomial approximation of the remaining portion of the integrand the necessary integrals are generated recursively and they involve the fresnel integral additional efficiency is achieved by means of an interpolation scheme therein one interpolates all but the rapidly varying edge diffraction contribution to the integral and then adds the explicitly calculated edge contribution at each far field point a few illustrative examples are included

on the computation of the macroscopic tangent for multiscale volumetric homogenization problems in this contribution methods of computing the macroscopic tangent that is required in multiscale volumetric homogenization problems are investigated a condensation procedure that employs the tangent information from the microscale finite element analysis is derived within a special framework where deformation controlled boundary conditions in micromechanical testing are enforced via the penalty method the developed methodology is demonstrated by sample macroscale problems in the context of the multilevel finite element strategy due to the high memory allocation costs that the condensation procedure induces a perturbation procedure is developed based on a minimal number of micromechanical tests results from the condensation and perturbation procedures are compared for sample infinitesimal and finite deformation inelasticity problems and the algorithmic consistency of the macroscopic tangent with the evolution of the macroscopic stress is discussed it is shown that the ability to compute the macroscopic tangent can also be employed to construct a stress controlled micromechanical testing procedure

a multiscale approach for modeling progressive damage of composite materials using fast fourier transforms abstract composite materials possess a highly complex material behavior and thus advanced simulation techniques are necessary to compute their mechanical response in this regard especially modeling failure and progressive damage presents a challenging task conventional macro mechanical methods and even closed form estimates are in many cases not sufficient to predict the appropriate mechanical material response full field simulations must be resorted to but these are known to be very expensive from the computational point of view in this contribution we propose a more efficient multiscale approach similar to fe2 nonlinear material effects caused by progressive damage behavior are captured directly on the discretized material level using simple isotropic continuum damage laws in contrast to conventional fe2 methods which use the finite element method fem to solve both scales numerically the fine scale problem material level is rewritten in an integral form of lippmann schwinger type and solved efficiently using the fast fourier transformation fft the calculation is carried out on a regular voxel grid that can be obtained from 3d images like tomographies the fine scale problem is integrated in a standard finite element framework which is used to solve the macroscopic bvp component level in the work at hand the scale coupling technique and the computation of the macroscopic tangent are described and in some numerical examples the convergence behavior of the macroscopic newton algorithm is investigated thereby the simulations were considered until localization and softening on the material scale occurred it is shown that the proposed method presents an effective way to determine the exact physical macroscopic response considering arbitrary microstructures and loading conditions

accurate stress updating algorithm based on constant strain rate assumption abstract the calculation of the stresses for path dependent materials possesses a delicate problem here the constitutive equations are reformulated using the assumption of constant strain rate within an increment this assumption allows the constitutive laws to be rewritten as ordinary differential equations and thus a large set of tools is available to solve the constitutive equations within this concept the possibility to form an algorithmic stiffness tensor is investigated and it turns out that the true tensor is expensive to calculate but it is of interest that an inexpensive well behaving approximative algorithmic stiffness tensor can be formulated to investigate the properties of the solution method both with respect to stress calculation and convergence properties in a newton raphson method an elasto plastic damage model based on von mises plasticity is considered an interesting observation is that for the usual closest point projection method the damage evolution is very sensitive to the size of load step

an overlapping domain decomposition preconditioning method for monolithic solution of shear bands metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material s load bearing capability a ductile failure phenomenon known as shear banding shear bands characterized as material instabilities are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise this coupled thermo mechanical localization problem can be formulated as a nonlinear system with two balance equations momentum and energy and two constitutive laws for elasticity and plasticity in our formulation mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time where at every time step a newton type method is used to solve the system monolithically to that end a block jacobian matrix is formed analytically using g teaux derivatives the resulting jacobian is sparse nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation consequently posing a significant challenge to iterative solvers to address this issue an overlapping domain decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms is proposed the key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately with selective updates when necessary the preconditioner is formed on the basis of an additive schwartz method that is applied to a schur complement partition of the system the proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off the shelf solvers we study the effect of hh and kk refinement that are obtained from isogeometric discretizations of the system the overlapping strategy and its parallel performance excellent performance is demonstrated in serial and parallel on all benchmark examples

a computing model of pressure distribution from tight underwear based on the theory of membrane of huge deformation in elasticity a computable model is presented to solve the problem of pressure distribution on a human body from tight underwear this model is steady and reliable with the help of this model it is easy to solve the problem of describing wearing dressed effect in integrated garments cad system

pop approximation to the spectral dimension of dual three dimensional sierpinski carpets the spectral dimension d of a network governs the massless singularity of a free field and the asymptotic behaviour of the diffusion on the network approximate values of d for two types of three dimensional generalisations of the dual sierpinski carpet are obtained using the pop approximation method for one of them which is generated by a cube of side length three with one block at the centre taken away the value obtained is d pop 2 log 26 log 884 93 approximately 2 89 for the other one with seven cubes taken away d pop 2 log 20 log 40 3 approximately 2 31 the algorithm of the pop method is explained the results for 2d symmetric dual sierpinski type carpets are also reported

line search methods in general return mapping algorithms with application to porous plasticity the present paper focuses on the application of line search methods in general return mapping algorithms two inexact line search methods and an exact line search method are investigated regarding their convergence properties within an automatic time incrementation in finite element calculations as an example for the assessment of the algorithms an elastic plastic and an elastic viscoplastic version of gurson s porous plasticity model are used in simulations of the necking of a tensile bar it is shown that larger time increments are possible and therefore a smaller number of increments are required when using line search methods the exact line search method shows the best performance concerning the required number of increments but takes more cpu time to complete the simulation the application of the inexact line search methods in general lowers the number of increments along with a reduction in the cpu time as compared with the case when no line search is used copyright 2007 john wiley sons ltd

computational issues in large strain elasto plasticity an algorithm for mixed hardening and plastic spin in this paper an algorithm for large strain elasto plasticity with isotropic hyperelasticity based on the multiplicative decomposition is formulated the algorithm includes a possible constitutive equation for the plastic spin and mixed hardening in which the principal stress and principal backstress directions are not necessarily preserved it is shown that if the principal trial stress directions are preserved during the plastic flow as assumed in some algorithms a plastic spin is inadvertently introduced for the kinematic mixed hardening case if the formulation is performed in the principal stress space a rotation of the backstress is inadvertently introduced as well the consistent linearization of the algorithm is also addressed in detail copyright 2005 john wiley sons ltd

a theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains the article presents new aspects of large strain isotropic elastoplastic analysis of shells on the shell theory side we consider several possible continuum based parametrizations of the shell deformation and investigate their applicability with respect to the implementation of general three dimensional local constitutive models as an aspect of computational shell analysis we then discuss in detail the numerical implementation of a shell based on the parametrization of the top and bottom surface in terms of a new 8 node brick type mixed finite shell element on the basis of assumed strain and enhanced strain variational approaches on the side of plasticity theory we adopt a recently proposed intermediate configuration free formulation of large strain plasticity based on the notion of a plastic metric and consider its representation with respect to the parameter manifold of the shell this frame is formulated in terms of principal strains and principal stresses with respect to dual co and contra variant eigenvector triads associated with a mixed variant elastic strain tensor the new aspect of computational plasticity is the development of a stress update algorithm for the plasticity model mentioned above which includes a general return mapping in the eigenvalue space the algorithm can be applied to general coordinate charts and is particularly convenient when formulated with respect to the space of the curvilinear coordinates of the shell parametrization we demonstrate the performance of the finite element formulation and constitutive algorithms by means of several numerical examples

geometrically non linear anisotropic inelasticity based on fictitious configurations application to the coupling of continuum damage and multiplicative elasto plasticity the objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically non linear inelasticity in addition to the intermediate configuration of multiplicative elasto plasticity we further introduce two microscopic configurations of lagrangian and eulerian type which characterize the so called fictitious undamaged material this kinematical framework enables us to apply two well established postulates based on standard terminology in non linear continuum mechanics concerning the free energy function the postulate of strain energy equivalence is adopted and in view of the plastic dissipation potential the concept of effective stress is a natural outcome of the underlying kinematical assumptions finally we focus on the integration technique for the class of obtained evolution equations and present numerical examples for a prototype model to underline the applicability of the proposed framework copyright 2003 john wiley sons ltd

three dimensional finite elements for large deformation micropolar elasticity abstract in this paper we propose a three dimensional finite element formulation for micropolar elasticity dealing with large displacements and small strains or equivalently small strains and finite rotations a comprehensive outline of the theory s characteristical features is given and we try to elucidate the set up of a possible non linear finite element implementation one focus of the present study is on a sound verification process featuring the construction of an enhanced patch test and the assessment of quadratic asymptotic rates of convergence aspects of performance and validity are discussed at a set of numerical examples we show that the proposed model is able to reproduce the transition between micropolar and classical continua highly accurate finally we present results underlining the implementation s applicability in the realm of finite deformation with arbitrarily large rotations

reflector antenna fields an exact aperture like approach a new computational approach is presented which allows fast analysis of radiation from large reflector antennas for an aperture a fourier transform ft relationship does exist between far field and aperture distribution accordingly the far field can be exactly reconstructed from the knowledge of approximately one sample per lobe shannon whittaker theorem applied at nyquist rate the finite reflector curvature introduces an extra factor in the radiation integral so that the radiation integral is no longer a ft in order to overcome this difficulty a new pseudosampling expansion which explicitly takes into account the extra factor is developed for parabolic reflector the sampling functions are related to the fresnel integrals and the far field can be exactly reconstructed in terms of aperture far field samples which can be computed using the fast fourier transform fft numerical computations and error analysis show the excellent performance of the method which can be generalized to deal with arbitrary reflector surfaces and near field evaluation

a generalized approach to formulate the consistent tangent stiffness in plasticity with application to the gld porous material model abstract it has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global newton iterations in the solution of the incremental problem in this paper we present a general method to formulate the consistent tangent stiffness for plasticity the robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with j2 flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program abaqus the proposed approach is then applied to an anisotropic porous plasticity model the gologanu leblond devaux model performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall newton iterations caused by using the consistent tangent matrix

simplified technique for evaluating the radiation integrals a simplified numerical method is described for computing the radiation integral of a reflector antenna which is associated with either the aperture field method afm or the induced current method icm the method involves linear approximation of the phase function and absorption of the approximation error into amplitude function then the integrand is rewritten as a slowly varying function multiplied by the phase exponential the slowly varying function is expanded in terms of a polynomial for the radial direction and trigonometric functions for the circumferential direction expansion coefficients are determined by matching at points on the aperture the matching points are chosen in a manner similar to that of a two dimensional numerical integration scheme

numerical differentiation for local and global tangent operators in computational plasticity in this paper numerical differentiation is applied to integrate plastic constitutive laws and to compute the corresponding consistent tangent operators the derivatives of the constitutive equations are approximated by means of difference schemes these derivatives are needed to achieve quadratic convergence in the integration at gauss point level and in the solution of the boundary value problem numerical differentiation is shown to be a simple robust and competitive alternative to analytical derivatives quadratic convergence is maintained provided that adequate schemes and stepsizes are chosen this point is illustrated by means of some numerical examples