exponential convergence estimates for a single neuron system of neutral type the future behavior of a dynamical system is determined by its initial state or initial function nontrivial neuron system involving adaptive learning corresponds to the memorization of initial information in this paper exponential estimates and sufficient conditions for the exponential stability of a single neuron system of neutral type are studied of particular importance is the fact that exponential convergence guarantees that this system is capable of memorizing initial functions furthermore this system is also capable of conveying much more information with respect to the initial functions memorized by neuron system with time delay the proofs follow some new results on nonhomogeneous difference equations evolving in continuous time combined with the lyapunov krasovskii functional and the descriptor system approach the exponential stability conditions are expressed in terms of a linear matrix inequality which lead to less restrictive and less conservative exponential estimates

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

global format for energy momentum based time integration in nonlinear dynamics summary a global format is developed for momentum and energy consistent time integration of second order dynamic systems with general nonlinear stiffness the algorithm is formulated by integrating the state space equations of motion over the time increment the internal force is first represented in fourth order form consisting of the end point mean value plus a term containing the stiffness matrix increment this form gives energy conservation for systems with internal energy as a quartic function of the displacement components this representation is then extended to general energy conservation via a discrete gradient representation the present procedure works directly with the internal force and the stiffness matrix at the time integration interval end points and in contrast to previous energy conserving algorithms it does not require any special form of the energy function nor use of mean value products at the element level or explicit use of a geometric stiffness matrix an optional monotonic algorithmic damping increasing with response frequency is developed in terms of a single damping parameter in the solution procedure the velocity is eliminated and the nonlinear iterations are based on the displacement components alone the procedure represents an energy consistent alternative to available collocation methods with an equally simple implementation copyright 2014 john wiley sons ltd

a note on convergence under dynamical thresholds with delays we complement the study of the asymptotic behaviour of the dynamical threshold neuron model with delay introduced by gopalsamy and leung 1997 by providing a description of the dynamics of the system in the remaining parameters range we characterize the regions of harmless delays and those in which delay induced oscillations appear

bifurcation analysis on a two neuron system with distributed delays abstract a general two neuron model with distributed delays is studied in this paper its local linear stability is analyzed by using the routh hurwitz criterion if the mean delay is used as a bifurcation parameter we prove that hopf bifurcation occurs for a weak kernel this means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value the direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical analysis are also given

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

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed

stability of bifurcating periodic solutions for van der pol equation with continuous distributed delay the van der pol equation with continuous distributed time delay is given its linear stability is investigated by employing the routh hurwitz criteria moreover the local asymptotic stability conditions are also derived by using the mean time delay as a bifurcation parameter the model is found to undergo a sequence of hopf bifurcations the direction and the stability criteria of the bifurcating periodic solutions are obtained by applying the norm form theory and the center manifold theorem some numerical simulation examples for justifying the theoretical results are also given

formulation and analysis of two energy consistent methods for nonlinear elastodynamic frictional contact problems energy conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability furthermore some physical phenomena such as friction can generate dissipation then in this work we present and analyse two energy consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon the first approach permits one to enforce respectively the kuhn tucker and persistency conditions during each time step by combining an adapted continuation of the newton method and a lagrangean formulation in addition the second method which is based on the work in p hauret p le tallec energy controlling time integration methods for nonlinear elastodynamics and low velocity impact comput methods appl mech eng 195 2006 4890 4916 represents a specific penalisation of the unilateral contact conditions some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods

stability analyses of cellular neural networks with continuous time delay the stability of cellular neural networks cnns with continuous time delay is investigated in the present paper a main theorem together with a few corollaries related to the existence and boundedness of the solution for this kind of cnns as well as its global asymptotic stability is derived some numerical simulation examples aimed at justifying the theoretical results are also given

a neural network for storing individual patterns in limit cycles a neural network model in which individual memories are stored in limit cycles is studied analytically and numerically in this model there are two kinds of interactions a hopfield like term that tends to stabilize the system in a memorized state and a second term with a time delay that acts to induce transitions between a memorized state and its complement state for a proper choice of the values of the parameters this model exhibits limit cycle behaviour in which the overlap with a target pattern oscillates in time an asymmetrically diluted version of the model is studied analytically in the limit of extreme dilution we find that the model with cycles performs better than a similarly diluted version of the hopfield model the performance of the fully connected model is studied by numerical simulations we find a behaviour qualitatively similar to that of the dilute model the model with cycles is found to perform better than the hopfield model as a pattern classifier if the memory loading level and the degree of corruption of the input patterns are high

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

convergence under dynamical thresholds with delays necessary and sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium of a class of delay differential equations modeling the action of a neuron with dynamical threshold effects

conservation of generalized momentum maps in mechanical optimal control problems with symmetry summary in this paper a structure preserving direct method for the optimal control of mechanical systems is developed the new method accommodates a large class of one step integrators for the underlying state equations the state equations under consideration govern the motion of affine hamiltonian control systems if the optimal control problem has symmetry associated generalized momentum maps are conserved along an optimal path this is in accordance with an extension of noether s theorem to the realm of optimal control problems in the present work we focus on optimal control problems with rotational symmetries the newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem this is true for a variety of one step integrators used for the discretization of the state equations examples are the one step theta method a partitioned variant of the theta method and energy momentum em consistent integrators numerical investigations confirm the theoretical findings copyright 2016 john wiley sons ltd

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

extended state space time integration with high frequency energy dissipation the introduction of algorithmic dissipation into conservative time integration algorithms is addressed in two steps the problem is formulated in state space and it is demonstrated that the use of two balanced dissipation terms one in the equation of motion and the other in the relation between the state space variables can produce dissipation that increases monotonically with the modal frequency this formulation is only partially satisfactory because it leads to dissipation also in the low frequency range a substantially improved form of algorithmic dissipation is then developed by representing the two balanced dissipation terms via auxiliary state space variables related to the displacement and velocity increments by a first order filter the filter technique retains the high frequency behavior of the original balanced algorithm but reduces the low frequency dissipation from first order in the frequency to third order energy balance equations are derived and the behavior of the two algorithms is described by detailed spectral analysis with explicit asymptotic results directly in terms of the damping parameter of the algorithms both algorithms satisfy full symmetry of displacement and normalized velocity variables for the free algorithmically damped vibration of any single mode and therefore exhibit monotonic transients without overshoot effects it is also demonstrated how the algorithm can be arranged in single solve format where only the displacement update requires a matrix equation solution while the velocity and the auxiliary state space variables are updated subsequently in each time step by simple vector relations with scalar coefficients copyright 2007 john wiley sons ltd

hopf bifurcation and stability of periodic solutions for van der pol equation with time delay abstract in this paper the van der pol equation with a time delay is considered where the time delay is regarded as a parameter it is found that hopf bifurcation occurs when this delay passes through a sequence of critical value a formula for determining the direction of the hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem

formulation and analysis of conserving algorithms for frictionless dynamic contact impact problems this paper presents the formulation of conserving time stepping algorithms for frictionless dynamic contact of solids a new class of finite element methods is proposed for the solution of these problems that exhibit the same conservation laws as the underlying continuum dynamical system the proposed methods are based on a penalty regularization of the constrained contact problem and lead to full conservation of the total energy of the system including the regularization penalty potential during persistent contact and restoration of the original energy upon release both linear and angular momenta are conserved by the scheme furthermore the newly developed methods have the ability to enforce the associated constraints in the velocity besides the impenetrability constraint in the displacements while preserving the conservation restoration properties of the final numerical scheme a modification of these schemes is described that assures positive energy dissipation if desired even in the highly nonlinear setting of contact impact problems leading to contact schemes with high frequency energy dissipation representative numerical simulations are presented illustrating the performance of the proposed numerical schemes

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

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

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

conservative fourth order time integration of non linear dynamic systems abstract an energy conserving time integration algorithm with fourth order accuracy is developed for dynamic systems with nonlinear stiffness the discrete formulation is derived by integrating the differential state space equations of motion over the integration time increment and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts this process introduces the time derivatives of the state space variables and these are then substituted from the original state space differential equations the resulting discrete form of the state space equations is a direct fourth order accurate representation of the original differential equations this fourth order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components energy conservation for a force potential of general form is obtained by addition of a higher order secant type correction term the formulation leads to a consistent representation of the motion within a time increment corresponding to cubic hermite interpolation in time this in turn leads to excellent phase representation with only a small fourth order error permitting integration of oscillatory systems with only a few integration points per period three numerical examples demonstrate the high accuracy of the algorithm

transient three dimensional domain decomposition problems frame indifferent mortar constraints and conserving integration the present work deals with transient large deformation domain decomposition problems the tying of dissimilar meshed grids is performed by applying the mortar method in this connection a reformulation of the original linear mortar constraints is proposed which retains frame indifference for arbitrary discretizations of the interface furthermore a specific coordinate augmentation technique is proposed to make possible the design of an energy momentum scheme numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy momentum scheme for three dimensional problems copyright 2009 john wiley sons ltd

global robust stability of neural networks with time varying delays in this paper we study the global robust stability of neural networks with time varying delays using the lyapunov functional method and matrix inequality technique several sufficient conditions are presented to show the existence of equilibrium and global robust stability of neural networks which is easy to apply then we give a simulation to justify the obtained results

on the determination of the limit cycles using the harmonic balance method several methods have been developed to search for limit cycles or bifurcation curves of nonlinear differential equations recently some non perturbative methods have been proposed however the majority of these methods are based on the so called harmonic balance method in this paper we will see how to use the harmonic balance method to obtain information of the number period and form of the limit cycles of certain nonlinear equations the method also gives a sequence of algebraic approximations of the limit cycles we will also compare our results with other non perturbative methods the results can be generalized to other nonlinear differential equations

a finite element formulation of non smooth contact based on oriented volumes for quadrilateral and hexahedral elements this paper concerns the finite element modeling of contact between solid bodies with a special emphasis on the treatment of non smooth conditions a new approach for the formulation of the contact constraint that allows for a simple and unified treatment of all potential contact scenarios in the presence of large deformations is presented based on the calculation of an oriented volume this approach possesses the ability to resolve complicated contact scenarios efficiently while being simple to implement and more widely applicable than the contact formulations available to date we show that the formulation boils down to a node to surface gap function that works effectively for non smooth contact situations the numerical implementation using the midpoint rule shows the need to guarantee the conservation of the total energy during impact for which a lagrange multiplier method is used the result is a robust contact detection and resolution algorithm

hopf bifurcation analysis of a bam neural network with multiple time delays and diffusion in this paper a bam neural network with multiple time delays and diffusion under homogeneous neumann boundary conditions is investigated by analyzing the corresponding characteristic equations the local stability of the trivial uniform steady state and the existence of hopf bifurcation under two different cases are established respectively by using the normal form theory and the center manifold reduction of partial functional differential equations pfdes explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions numerical simulations are carried out to illustrate the main results

hybrid state space time integration of rotating beams abstract an efficient time integration algorithm for the dynamic equations of flexible beams in a rotating frame of reference is presented the equations of motion are formulated in a hybrid state space format in terms of local displacements and local components of the absolute velocity with inspiration from hamiltonian mechanics where displacement and momentum have similar roles both sets of state space variables are interpolated between nodes by the same shape functions leading to a general format where all inertia effects are represented via the classic constant mass matrix while effects of the system rotation enter via global operations with the angular velocity vector the algorithm is based on an integrated form of the equations of motion with energy and momentum conserving properties if a kinematically consistent non linear formulation is used a consistent monotonic scheme for algorithmic energy dissipation in terms of local displacements and velocities typical of structural vibrations is developed and implemented in the form of forward weighting of appropriate mean value terms in the algorithm the algorithm is implemented for a beam theory with consistent quadratic non linear kinematics valid for moderate finite rotations the equations of this non linear beam theory are generated in explicit form by extension of the constitutive stiffness and the geometric stiffness of a linear beam theory at the element level the performance of the algorithm is illustrated by numerical examples

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

algorithms for coupled problems that preserve symmetries and the laws of thermodynamics part i monolithic integrators and their application to finite strain thermoelasticity abstract we propose discretizations for a general class of nonlinear coupled thermomechanical problems of evolution the most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model to formulate such methods we exploit the geometric structure afforded by the generic formalism of non equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the generic structure as an illustration a complete discretization of finite strain thermoelasticity is presented using finite elements in space and a monolithic integrator in time simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

multi conformal symplectic pdes and discretizations past work on integration methods that preserve a conformal symplectic structure focuses on hamiltonian systems with weak linear damping in this work systems of pdes that have conformal symplectic structure in time and space are considered meaning conformal symplecticity is fully generalized for pdes using multiple examples it is shown that pdes with this particular structure have interesting applications what it means to preserve a multi conformal symplectic conservation law numerically is explained along with presentation of two numerical methods that preserve such properties then the advantages of the methods are briefly explored through applications to linear equations consideration of momentum and energy dissipation and backward error analysis numerical simulations for two pdes illustrate the properties of the methods as well as the advantages over other standard methods

nonperturbative but approximate method for solving differential equations and finding limit cycles a nonperturbative method for solving differential equations and for finding limit cycles is proposed and is illustrated on the anharmonic oscillator and on the van der pol equation it is shown to give the amplitude period and equation of the limit cycle with a better accuracy than any perturbative results so far obtained

an energy momentum conserving algorithm using the variational formulation of visco plastic updates in this paper we use the variational formulation of elasto plastic updates proposed by ortiz and stainier comput methods appl mech eng 1999 171 419 444 in the context of consistent time integration schemes we show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms moreover we present numerical examples that illustrate the efficiency of our developments copyright 2005 john wiley sons ltd

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

event driven integration of linear structural dynamics models under unilateral elastic constraints abstract this paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints i e constraints with a linear force displacement characteristic whenever active the presented procedure relies on an event driven strategy for the handling of the contact constraints in combination with one step schemes dedicated to the time integration of the second order equations of motion efficiency of the procedure follows from the use of cubic hermite interpolation to continuously extend the normal gap functions that reflect the openings of the contact interfaces robustness follows from the proper handling of complex numerical situations e g numerical grazing or discontinuity sticking through appropriate algorithm structure and numerical implementation and integration stability is guaranteed by the very nature of the algorithm and that of the one step integration scheme following a detailed coverage of the integration procedure and the countermeasures to the expected numerical difficulties three application examples are treated for illustration purposes a matlab implementation of the procedure is provided online download and usage information are given in appendix a

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

modified newmark family for non linear dynamic analysis a simple modification of the newmark scheme is suggested which keeps the total energy of mechanical systems constant and thereby enhances the unconditional stability in non linear dynamic analysis numerical damping which is formulated on an energy basis is also introduced and the dissipative character of the algorithm is guaranteed in the non linear regime representative examples in support of such a modified newmark family are presented copyright 2004 john wiley sons ltd

computational schemes for non linear elasto dynamics this paper deals with the development of computational schemes for the dynamic analysis of non linear elastic systems the focus of the investigation is on the derivation of unconditionally stable time integration schemes presenting high frequency numerical dissipation for these types of problem at first schemes based on galerkin and time discontinuous galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed though useful these schemes require casting the equations of motion in the symmetric hyperbolic form which is not always possible furthermore this approaches to unacceptably high computational costs next unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form both energy preserving and energy decaying schemes are derived numerical examples are presented to demonstrate the accuracy and efficiency of the proposed schemes copyright 1999 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 time integration scheme for geometrically exact beam an energy conserving finite element formulation for the dynamic analysis of geometrically non linear beam like structures undergoing large overall motions has been developed the formulation uses classical displacement based planar beam finite elements described in an inertial frame it takes into account finite axial bending and shear strains a theoretically consistent approach is used to derive a novel and simple energy conserving scheme using the unconventional incremental strain update rather than the standard strong form numerical examples demonstrate perfect energy and momenta conservation stability and robustness of the scheme and good convergence properties in terms of both the newton raphson method and time step size c 2006 elsevier b v all rights reserved

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

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

numerical integration of the stiff dynamics of geometrically exact shells an energy dissipative momentum conserving scheme this paper presents a new family of time stepping algorithms for the integration of the dynamics of non linear shells we consider the geometrically exact shell theory involving an inextensible director field the so called five parameter shell model the main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid in this context we develop time stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system and allow the introduction of a controllable non negative energy dissipation to handle the high numerical stiffness characteristic of these problems a series of algorithmic parameters for the different components of the deformation of the shell i e membrane bending and transverse shear fully control this numerical dissipation recovering existing energy momentum schemes as a particular choice of these algorithmic parameters we present rigorous proofs of the numerical properties of the resulting algorithms in the full non linear range furthermore it is argued that the numerical dissipation is introduced in the high frequency range by considering the proposed algorithm in the context of a linear problem the finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation dissipation properties we present several representative numerical simulations illustrating the performance of the newly proposed methods the robustness gained over existing methods in these stiff problems is confirmed in particular copyright 2002 john wiley sons ltd

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

generalized energy momentum method for non linear adaptive shell dynamics in the present study a generalization of the energy momentum method denoted by generalized energy momentum method applied to the non linear dynamics of shells will be developed within the framework of the generalized a method this algorithmic environment contains the unconditionally stable energy momentum method and its numerically damped version as well as the classical newmark and a methods as special cases in order to control the size of the time steps of the integration scheme with respect to accuracy and eaciency an adaptive time stepping procedure based on local a posteriori error estimation will be improved for non linear dynamical systems and applied to the proposed class of algorithms the spatial discretization is realized by an eight noded finite shell element of reissner mindlin type including an extensible shell director field permitting the application of three dimensional material laws the original formulation of this finite element will be developed for non linear dynamic analysis and adapted for the employment within the introduced energy conserving decaying time integration scheme 1999 elsevier science s a all rights reserved

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

stability and hopf bifurcation of a general delayed recurrent neural network in this paper stability and bifurcation of a general recurrent neural network with multiple time delays is considered where all the variables of the network can be regarded as bifurcation parameters it is found that hopf bifurcation occurs when these parameters pass through some critical values where the conditions for local asymptotical stability of the equilibrium are not satisfied by analyzing the characteristic equation and using the frequency domain method the existence of hopf bifurcation is proved the stability of bifurcating periodic solutions is determined by the harmonic balance approach nyquist criterion and graphic hopf bifurcation theorem moreover a critical condition is derived under which the stability is not guaranteed thus a necessary and sufficient condition for ensuring the local asymptotical stability is well understood and from which the essential dynamics of the delayed neural network are revealed finally numerical results are given to verify the theoretical analysis and some interesting phenomena are observed and reported

asymptotic stability of a class of neutral delay neuron system in a critical case in this brief the asymptotic stability properties of a neutral delay neuron system are studied mainly in a critical case when the exponential stability is not possible if a critical value of the coefficient in the neutral delay neuron system is considered then the difficulty for our investigation is caused by the fact that the spectrum of the linear operator is asymptotically approximated to the imaginary axis it is obvious that in such a case the equation is not exponentially stable and one needs more subtle methods in order to characterize this type of asymptotic stability hence first the local asymptotic stability for the neutral delay neuron system is studied and the main tools involved are the asymptotic expansions of characteristic roots laplace transforms and function series and a complete analysis of the stability diagram is also presented then based on the energy method the globally asymptotic stability results for the neutral delay neuron system are derived

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

how delays affect neural dynamics and learning we investigate the effects of delays on the dynamics and in particular on the oscillatory properties of simple neural network models we extend previously known results regarding the effects of delays on stability and convergence properties we treat in detail the case of ring networks for which we derive simple conditions for oscillating behavior and several formulas to predict the regions of bifurcation the periods of the limit cycles and the phases of the different neurons these results in turn can readily be applied to more complex and more biologically motivated architectures such as layered networks in general the main result is that delays tend to increase the period of oscillations and broaden the spectrum of possible frequencies in a quantifiable way simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior adaptable delays are then proposed as one additional mechanism through which neural systems could tailor their own dynamics accordingly we derive recurrent backpropagation learning formulas for the adjustment of delays and other parameters in networks with delayed interactions and discuss some possible applications

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

an energy consistent material point method for dynamic finite deformation plasticity energy consistency for the material point method mpm is examined for thermodynamically consistent hyperelastic plastic materials it is shown that mpm can be formulated with implicit three field variational finite element algorithms which dissipate energy and conserve momentum for that class of material models with a consistent mass matrix the resulting overall numerical method inherits the energy dissipative and momentum conserving properties of the mesh solution thus the proposed mpm algorithm satisfies by construction a time discrete form of the second law of thermodynamics properties of the method are illustrated in numerical examples

energy dissipative time finite elements for classical mechanics abstract based upon the temporal discretisation of hamilton s canonical equations by means of the continuous galerkin method a petrov galerkin time finite element formulation is presented to analyse non linear mechanical systems for long time integration with the presence of dissipation the time stepping scheme features a correct estimation of physical dissipative energy for mechanical systems with weak dissipation this enables to introduce a moderate numerical dissipation to damp out undesired spurious high frequency modes in a controllable manner numerical examples are performed to demonstrate that the proposed scheme allows us to obtain a correct amount by which the energy changes over the integration run

asymptotic stability criteria for a two neuron network with different time delays in this paper the asymptotic stability of a two neuron system with different time delays has been investigated some criteria for determining the global asymptotically stability of equilibrium are derived from the theory of monotonic dynamical system and the approach of lyapunov functional for local asymptotic stability some elegant criteria are also obtained by the nyquist criteria we find that one of them depends on the length of delays while the other ones do not in the latter case the delays are sometimes called harmless delays the results obtained have leading significance in the study of neural networks composed of a large number of neurons with different time delays

thermodynamically consistent time stepping algorithms for non linear thermomechanical systems we present the basic theory for developing novel monolithic and staggered time stepping algorithms for general non linear coupled thermomechanical problems the proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics for isolated systems they preserve the total energy and the entropy never decreases furthermore if the governing equations of the problem have symmetries the proposed integrators preserve them too the formulation of such methods is based on two ideas expressing the evolution equation in the so called general equations for non equilibrium reversible irreversible coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields the new methods can be considered as an extension of the energy momentum integration algorithms to coupled thermomechanical problems to which they reduce in the purely hamiltonian case in the article the new ideas are applied to a simple coupled problem a double thermoelastic pendulum with symmetry numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability which stems precisely from their ability to preserve the structure of the evolution equations they discretize copyright 2009 john wiley sons ltd

energy controlling time integration methods for nonlinear elastodynamics and low velocity impact it is now well established that discrete energy conservation dissipation plays a key role for the unconditional stability of time integration schemes in nonlinear elastodynamics in this paper from a rigorous conservation analysis of the hilber hughes taylor time integration scheme h hilber t hughes r taylor improved numerical dissipation for time integration algorithms in structural dynamics earthquake engrg struct dynam 5 1977 283 292 we propose an original way of introducing a controllable energy dissipation while conserving momenta in conservative strategies like j simo n tarnow the discrete energy momentum method conserving algorithms for nonlinear elastodynamics z angew math phys 43 1992 757 792 moreover we extend the technique proposed in o gonzalez exact energy and momentum conserving algorithms for general models in nonlinear elasticity comput methods appl mech engrg 190 13 14 2000 1763 1783 to provide energy controlling time integration schemes for frictionless contact problems enforcing the standard kuhn tucker conditions at time discretization points we also extend this technique to viscoelastic models numerical tests involving the impact of incompressible elastic or viscoelastic bodies in large deformation are proposed to confirm the theoretical analysis

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions