exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part ii second order methods we present in this paper the formulation of a new high frequency dissipative time stepping algorithm for nonlinear elastodynamics that is second order accurate in time the new scheme exhibits unconditional energy dissipation and momentum conservation and thus the given name of energy dissipative momentum conserving second order scheme edmc 2 leading also to the conservation of the relative equilibria of the underlying physical system the unconditional character of these properties applies not only with respect to the time step size but equally important with respect to the considered elastic potential moreover the dissipation properties are fully controlled through an algorithmic parameter reducing to existing fully conserving schemes if desired the design of the new algorithm is described in detail including a complete analysis of its dissipation conservation properties in the fully nonlinear range of finite elasticity to motivate the different constructions that lead to the dissipative properties of the final scheme the same arguments are used first in the construction of new linear time stepping algorithms for the system of linear elastodynamics including first and second order schemes the new schemes exhibit a rigorous decay of the physical energy with the second order schemes formulated in a general two stage formula accommodating the aforementioned control parameter in the dissipation of the energy a complete spectral analysis of the new schemes is presented in this linear range to evaluate their different numerical properties in particular the dissipative character of the proposed schemes in the high frequency range is fully demonstrated in fact it is shown that the new second order scheme is l stable most remarkably the extension of these ideas to the nonlinear range is accomplished avoiding the use of multi stage formulae given the freedom gained in using general nonlinear relations while preserving the conservation laws of the momenta and the corresponding relative equilibria several representative numerical simulations are presented in the context of nonlinear elastodynamics to evaluate the performance of the newly proposed schemes

an exhaustive search algorithm for checking limit cycle behavior of digital filters the presence of limit cycles that may arise in fixed point arithmetic implementation of a digital filter can significantly impair its performance this paper presents an algorithm to determine the presence absence of such limit cycles for generality the filter is taken to be in its state space formulation the algorithm is applicable independent of filter order type of quantization nonlinearity and whether the accumulator is single or double length it may be utilized to construct limit cycle free regions in filter coefficient space once a filter is determined to be limit cycle free a technique that provides a robustness region in coefficient space where all filters remain limit cycle free is also presented

energy dissipative momentum conserving time stepping algorithms for finite strain multiplicative plasticity abstract this paper presents a new energy dissipative momentum conserving algorithm for multiplicative models of finite strain plasticity the proposed scheme preserves exactly the conservation laws of linear and angular momenta and leads to the exact energy dissipation between the two computed solutions in a given time step in particular the energy is exactly conserved if the step is elastic thus recovering existing energy momentum conserving schemes in the purely elastic range extensions accommodating a strictly non negative numerical energy dissipation in the high frequency to handle the numerical stiffness of the problem are also discussed briefly these conservation dissipation properties are proven rigorously in the very nonlinear setting of multiplicative finite strain plasticity in fact the analysis drives the design of the new return mapping algorithm proposed to integrate the plastic evolution equations so the above dissipation conservation properties hold these analyses account for the spatial discretization of the problem in the context of the finite element method several representative numerical simulations are presented to illustrate the performance of the new algorithm

a computer aided test for the absence of limit cycles in fixed point digital filters the problem of determining the existence of limit cycles in direct form digital filters with quantization nonlinearities is considered a method based on an exhaustive search is introduced it is applicable to all direct form digital filters regardless of the type of quantization arithmetic the number of quantizers and the order of the filter the equivalence between the concept of global asymptotic stability and the exclusion of limit cycles is proved for linearly stable finite wordlength digital filters the robustness of this test with respect to coefficient perturbation in the digital filter is presented the computational aspects of the algorithm are also discussed the algorithm is shown to be capable of always determining the global asymptotic stability of the tested realization

variational time integrators for finite dimensional thermo elasto dynamics without heat conduction this paper focuses on the formulation of variational integrators vis for finite dimensional thermo elastic systems without heat conduction the dynamics of these systems happen to have a hamiltonian structure after thermal displacements are introduced it is then possible to formulate integrators by taking advantage of standard methods in vi the class of integrators we construct have some remarkable features a they are symplectic b they exactly conserve the entropy of the system or in other words they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes c they nearly exactly conserve the value of the energy for very long times and d they exactly conserve linear and angular momentum we first describe how to adapt any vi for the classical mechanical systems to integrate adiabatic thermo elastic ones and then formulate three new types of integrators the first class based on a generalized trapezoidal rule gives rise to two first order explicit integrators and a second order implicit one by composing then the two first order integrators we construct a second order explicit one finally we formulate a fourth order implicit integrator which is a symplectic partitioned runge kutta method the performance of these new algorithms is showcased through numerical examples copyright 2011 john wiley sons ltd

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part i low order methods for two model problems and nonlinear elastodynamics abstract we present in this paper the development of a class of time stepping algorithms for nonlinear elastodynamics that exhibits the controllable numerical dissipation in the high frequency range required for the robust solution of the resulting numerically stiff systems to motivate and illustrate better the developments in this general case we present first the formulation and analysis of these methods for two simple model problems namely we consider a nonlinear elastic spring mass system and a simplified model of thin elastic beams as it is discussed in detail in this paper the conservation by the numerical algorithm of the momenta and corresponding relative equilibria of these characteristic hamiltonian systems with symmetry is of the main importance these conservation properties lead for a fixed and finite time step to a correct qualitative picture of the phase space where the discrete dynamics takes place even in the presence of the desired and controlled numerical dissipation of the energy this situation is contrasted with traditional dissipative numerical schemes which are shown through rigorous analyses to not only lose their dissipative character in the general nonlinear range but also the aforementioned conservation properties thus leading to a qualitatively distorted approximation of the phase dynamics the key for a successful algorithm in this context is the incorporation of the numerical dissipation in the internal modes of the motion while not affecting the group motions of the systems the algorithms presented in this work accomplish these goals the focus in this first part is given to first order methods representative numerical simulations ranging from applications in nonlinear structural dynamics to nonlinear continuum three dimensional elastodynamics are presented in the context of the finite element method to illustrate these ideas and results

a finite element for vibration analysis of twisted blades based on beam theory on presente une methode d elements finis de discretisation en segments de poutre d aubes tournantes pretordues en employant la methode de deplacement de matrice on developpe les proprietes de rigidite et de masse on obtient de maniere systematique la matrice de rigidite structurale et l effet du mouvement de rotation du rotor sur la matrice de rigidite a partir des fonctions potentielles et d energie cinetique

energy consistent algorithms for dynamic finite deformation plasticity abstract this paper addresses the theoretical development and numerical implementation of energy consistent algorithms for dynamic elastoplasticity emphasizing finite strain constitutive formulations so that unconditional stability of the algorithms is assured even in the fully nonlinear regime the key concept behind energy consistency is the requirement that the discretized system obey an a priori stability estimate which may be derived in general using the first and second laws of thermodynamics this approach to computational dynamic plasticity differs from typical application of traditional algorithms such as newmark or hilber hughes taylor methods where local time integration schemes for plasticity laws are developed somewhat independently from the global time integration scheme for the equations of motion without explicit consideration of thermodynamical restrictions two algorithms based on both additive and multiplicative finite deformation plasticity model are formulated within the energy consistent framework both algorithms possess the desirable feature of nonlinear stability of previous energy momentum algorithms for elastodynamics

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

elimination of limit cycles due to two s complement quantization in normal form digital filters normal form digital filters are attractive due to their desirable properties when implemented in finite wordlength arithmetic these filters are free from all overflow limit cycles and quantization limit cycles when magnitude truncation is used however when two s complement truncation tct quantization is used limit cycles can still exist in this paper it is shown that when block structures are used normal form digital filters can be made free of limit cycles due to tct quantization it is shown that this can be done with a small block size an algorithm is also presented to find the minimum block size required for a given filter some examples are given to illustrate the results

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

stability of distributed 3 d systems implemented on grid sensor networks using the givone roesser and the fornasini marchesini state space models for m d systems a method for distributed information processing in rectangular grid based wireless sensor networks is presented it can be used for distributed implementation of any general linear system on a grid sensor network nonlinearities caused by finite precision number representation schemes used for in node computations and inter node communication are modeled stability of the system is analyzed with special consideration given to the influence of inter node communication on system dynamics a necessary and sufficient condition for global asymptotic stability under finite precision arithmetic is established simulation results are presented to illustrate the theoretical results

assumed strain finite element methods for conserving temporal integrations in non linear solid dynamics this paper presents a new class of assumed strain finite elements to use in combination with general energy momentum conserving time stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time the case of interest corresponds to nearly incompressible material responses in the fully non linear finite strain elastic and elastoplastic ranges the new elements consider the classical scaling of the deformation gradient with an assumed jacobian its determinant defined locally through a weighted averaging procedure at the element level the key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or b bar operator the developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta and the conservation dissipation law of energy for elastic and inelastic problems respectively care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration with these developments at hand a new general b bar operator is introduced that satisfies these conditions the new operator not only accounts for the spatial interpolations e g bilinear displacements with piece wise constant volume but also depends on the discrete structure of the equations in time the aforementioned conservation dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes unconditionally of the time step size and the material model elastic or elastoplastic different finite elements are considered in this framework including quadrilateral and triangular elements for plane problems and brick elements for three dimensional problems several representative numerical simulations are presented involving in particular the use of energy dissipating momentum conserving time stepping schemes recently developed by the author and co workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements including these conservation dissipation properties in time and their locking free response in the quasi incompressible case copyright 2007 john wiley sons ltd

volume preserving energy momentum schemes for isochoric multiplicative plasticity abstract this paper presents a new energy dissipative momentum conserving algorithm for multiplicative finite strain plasticity f f e f p that also preserves exactly the plastic volume for isochoric plastic models the new algorithm exhibits exactly the conservation laws of linear and angular momentum of the underlying physical problem as well as its energy evolution a strictly positive energy dissipation in fact the exact energy dissipation is obtained by design during plastic steps while enforcing the plastic consistency i e the yield condition on the final stress appearing in the equations of motion exact energy conservation is attained in particular during elastic steps the aforementioned preservation of the plastic volume is obtained by a new treatment of the geometric structure behind the considered multiplicative models of finite strain plasticity namely we present a new approximation of the reference and elastic metrics whose contractions with the incremental total and elastic strains lead exactly to the increment of the total and elastic natural volumetric strains respectively the elastic metric is defined in the intermediate configuration defined itself by the proper discrete approximation in time of the plastic deformation gradient f p the new algorithm extends to the plastic range existing energy momentum conserving schemes for nonlinear elastic problems but incorporating a new modified elastic stress formula consistent with this new geometric setting the inherited conservation laws of momenta and especially the non negative character of the energy dissipation leads to an improved performance over existing more classical schemes showing numerical instabilities in the considered highly nonlinear geometric setting of large deformations and strains several numerical simulations are presented illustrating these properties

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

a first order energy dissipative momentum conserving scheme for elasto plasticity using the variational updates formulation abstract in a previous paper l noels l stainier j p ponthot an energy momentum conserving algorithm using the variational formulation of visco plastic updates int j numer methods engrg 65 2006 904 942 the authors demonstrated the efficiency of the variational formulation of elasto plastic updates to develop energy momentum conserving time integration algorithms indeed within such a framework the stress tensor always derives from an incremental potential even when plastic behavior is considered therefore the verification of the conservation of energy in the non linear range can easily be demonstrated the sum of the reversible stored energy and irreversible dissipated energy exactly corresponds to the work of the external forces applied to the structure although this formulation was shown to be accurate and robust the introduction of numerical dissipation for high frequency numerical modes can be necessary to simulate complex phenomena in this work we propose a modification of the variational updates framework to introduce this numerical property leading to a new energy dissipative momentum conserving time integration algorithm for elasto plasticity

constraint energy momentum algorithm and its application to non linear dynamics of shells abstract a modified time integration method is developed for non linear structural dynamics combining controllable algorithmic dissipation of higher modes and at the same time conservation of energy as well as linear and angular momentum although these features seem to be actually inconsistent numerical stability with large time steps for dynamic buckling and snap through problems as well as for dynamical systems with smooth solutions is guaranteed arbitrary implicit one step time integration schemes with newmark approximations are suitable for use as basic algorithms for the proposed method the desired conservation attributes are obtained if the basic algorithm is augmented by energy and momentum constraints the potential of the suggested algorithm is demonstrated by selected applications to non linear dynamics of shell structures

energy momentum consistent finite element discretization of dynamic finite viscoelasticity this paper is concerned with energy momentum consistent time discretizations of dynamic finite viscoelasticity energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics the discretization is energy momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations the performed approximation is based on a three field formulation in which the deformation field the velocity field and a strain like viscous internal variable field are treated as independent quantities the new non linear viscous evolution equation satisfies a non negative viscous dissipation not only in the continuous case but also in the fully discretized system the initial boundary value problem is discretized by using finite elements in space and time thereby the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments incremental objectivity although the present approach makes possible to design schemes of arbitrary order the focus is on finite elements relying on linear lagrange polynomials for the sake of clearness the discrete energy momentum consistency is based on the collocation property and an enhanced second piola kirchhoff stress tensor the obtained coupled non linear algebraic equations are consistently linearized the corresponding iterative solution procedure is associated with newly proposed convergence criteria which take the discrete energy consistency into account the iterative solution procedure is therefore not complicated by different scalings in the independent variables since the motion of the element is taken into account for solving the viscous evolution equation representative numerical simulations with various boundary conditions show the superior stability of the new time integration algorithm in comparison with the ordinary midpoint rule both the quasi rigid deformations during a free flight and large deformations arising in a dynamic tensile test are considered copyright 2009 john wiley sons ltd

a co rotational element time integration strategy for non linear dynamics new procedures are proposed for implicit dynamic analysis using the finite element method the main aim is to give stable solutions with significant rigid body motions in particular rotations in contrast to most conventional approaches the time integration strategy is closely linked to the element technologies with the latter involving a form of co rotational procedure for the undamped situation one of the solution procedures leads to an algorithm that exactly conserves energy when constant external forces are applied i e with gravity loading

a consistent time fe method for large strain elasto plasto dynamics in the present paper energy consistent momentum conserving time stepping schemes for geometrically nonlinear multiplicative elasto plasto dynamics referred to as ecmc cg methods are developed in this context the entire discretisation procedure of the global equations of motion is based on galerkin methods in space and time whereby special emphasis is placed on the desired conservation properties related to the approximation of time integrals to guarantee energy consistency also for higher order finite elements in time a thermodynamically motivated enhanced stress tensor which defines a non standard quadrature rule is derived for elasto plastic material behaviour concerning the integration of the local evolution equations in time exemplarily a well established first order accurate exponential update is applied and adapted according to the global time stepping scheme based on linear finite elements in time the performance of the resulting scheme is demonstrated by means of several representative numerical examples whereby it is shown that global energy consistency is guaranteed exactly within the calculation accuracy by applying the proposed concepts

hybrid stress triangular finite element with enhanced performance for statics and dynamics abstract in this paper we develop the mixed variational formulation with independent displacement rotation and stress fields based on the work of hughes and brezzi 1989 however we further suppress rotation field to obtain element superior performance for classical continuum case with hybrid stress interpolation the lowest order whitney s interpolation or raviart thomas vector space is employed to discretize stress field which delivers superior performance by enforcing the continuity of traction field across element boundary we further extend this optimal choice of hybrid stress discrete approximation to dynamic analysis by choosing the appropriate time integration scheme that enforces stability for long term computation several numerical simulations are given to illustrate an enhanced performance of the proposed element and algorithm

two s complement quantization in periodic digital filters the stability of periodically shift variant psv filters are studied when implemented with two s complement truncation tct quantization block form and standard nonblock form implementations are considered and two sufficient conditions are established for stability as a special case second order coupled form psv filters are then investigated under tct quantization stability regions are established within the parameter space for block implementations examples are given to illustrate the results

stability of digital filters implemented with two s complement truncation quantization the stability of digital filters implemented with two s complement truncation quantization is investigated both the direct form and normal form realizations are considered and regions in the parameter plane are obtained where the filters are globally asymptotically stable regions within the stability triangle of a direct form digital filter are also studied where limit cycles of period one and two are possible the results are illustrated with several examples