a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

improving stability by change of representation in time stepping analysis of non linear beams dynamics in this paper we propose a change in the representation of the discrete motion equations in structural non linear dynamics to obtain an improvement in the stability of time numerical integrations in particular natural local state variables are indicated for a finite element approach to beam problems the results relative to newmark approximations for the variations in the displacement and velocity vectors show a significant increase in the range of stability of the time integration process and a reduction in the number of newton iterations required in the time integration steps the proposed method further preserves energy as well as the linear and angular momentum of the dynamical system copyright 2006 john wiley sons ltd

dynamic analysis of structures with unilateral constraints numerical integration and reduction of structural equations structural dynamic equations with unilateral constraints upon the displacements velocities and accelerations are employed in order to represent vibrating elastic structures with normal oblique impact and friction interaction points for obtaining a numerical integration scheme the lagrange multipliers and a minimum work approach are employed at each time step the algorithm is presented as an extension of the generalized newmark scheme it seems to retain the asymptotic features of the original one the reduction of the number of dynamic degrees of freedom of the unilaterally constrained structures is carried out by representing the equations of motion in modal co ordinates of the unconstrained structure and truncating the dynamic contributions of higher modes the presented techniques have been verified by investigating free longitudinal vibroimpact motion laws of an elastic vibroconverter and free longitudinal and bending vibration of a vibroconverter interacting with a moving rigid body by oblique impact and friction forces

surmounting oscillating barriers path integral approach for weak noise we consider the thermally activated escape of an overdamped brownian particle over a potential barrier in the presence of periodic driving a time dependent path integral formalism is developed which allows us to derive asymptotically exact weak noise expressions for both the instantaneous and the time averaged escape rate our results comprise a conceptually different systematic treatment of the rate prefactor multiplying the exponentially leading arrhenius factor moreover an estimate for the deviations at finite noise strengths is provided and a supersymmetry type property of the time averaged escape rate is verified for piecewise parabolic potentials the rate expression can be evaluated in closed analytical form while in more general cases as exemplified by a cubic potential an action integral remains to be minimized numerically our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies

beyond the fokker planck equation pathwise control of noisy bistable systems we introduce a new method allowing one to describe slowly time dependent langevin equations through the behaviour of individual paths this approach yields considerably more information than the computation of the probability density in particular scaling laws can be obtained easily the main idea is to show that for sufficiently small noise intensity and slow time dependence the vast majority of paths remain in small space time sets typically in the neighbourhood of potential wells the size of these sets often has a power law dependence on the small parameters with universal exponents the overall probability of exceptional paths is exponentially small with an exponent also showing power law behaviour the results cover time spans up to the maximal kramers time of the system we apply our method to three phenomena characteristic for bistable systems stochastic resonance dynamical hysteresis and bifurcation delay where it yields precise bounds on transition probabilities and the distribution of hysteresis areas and first exit times we also discuss the effect of coloured noise

boundary minimum principles for the unilateral contact problems abstract the aim of the present paper is the derivation of boundary variational principles for the unilateral contact problem using the inequality constrained principles of minimum potential and complementary energy and the equivalent variational inequality formulations we derive first saddle point formulations for the problems using appropriate langrangian functions an elimination technique allows the formulation of two minimum principles on the boundary with respect to the unknown normal displacements and reactions of the contact region respectively it is also shown that these two minimum problems are equivalent to multivalued boundary integral equations involving symmetric operators the theory is illustrated by numerical examples solved both by the fem and the bem

a new explicit predictor multicorrector high order accurate method for linear elastodynamics abstract a new explicit predictor multicorrector high order accurate method is suggested for linear elastodynamics the method is derived from the implicit high order accurate method based on the time continuous galerkin method proposed earlier in our papers the basic unknowns for the method are displacements and velocities accelerations are not calculated the explicit method uses a predictor multicorrector technique with one or two passes in order to reach the fourth order of accuracy and has controllable numerical dissipation for the suppression of spurious high frequency oscillations in contrast to recently suggested explicit high order accurate methods based on the time discontinuous galerkin method the new method is more accurate has a higher order of accuracy and has better algorithmic properties e g a higher stability limit at the same computational efforts presented numerical examples show the performance of the new method the method appears to be competitive for medium and long term analysis when accuracy of numerical solutions arises an issue due to error accumulation

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

mixed fem of higher order for time dependent contact problems abstract in this paper mixed finite element methods of higher order for time dependent contact problems are discussed the mixed methods are based on resolving the contact conditions by the introduction of lagrange multipliers dynamic signorini problems with and without friction are considered involving thermomechanical and rolling contact rothe s method is used to provide a suitable time and space discretization to discretize in time a stabilized newmark method is applied as an adequate time stepping scheme the space discretization relies on finite elements of higher order in each time step the resulting problems are solved by uzawa s method or alternatively by methods of quadratic programming via a suitable formulation in terms of the lagrange multipliers numerical results are presented towards an application in production engineering the results illustrate the performance of the presented techniques for a variety of problem formulations

a self optimizing single step algorithm for the numerical integration of parabolic equations abstract an unconditionally stable single step time marching algorithm for parabolic equations is presented in this algorithm time stepping parameters are chosen according to an error condition such that large time step accuracy is achieved and the solution is also free from oscillations it is different from conventional algorithms such as the crank nicolson scheme in that it uses a matrix of time stepping parameters rather than a single scalar value these parameters can be approximated inexpensively this algorithm is easily programmed and comparisons are made with other methods for different cases of a heat conduction problem it is found that the algorithm gives a more accurate solution for all degrees of freedom for the cases considered it is therefore advantageous over the conventional variably weighted euler method which has a constraint that the same value must be used for all degrees of freedom and that values have to be changed by trial and error for different time step sizes and mesh refinements in order to achieve best results

design spaces measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design illustration to structural dynamics for the first time for time discretized operators we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design but also can distinguish in general the quality of computational algorithms for time dependent problems the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts the underlying concepts can be extended to other disciplines as well for further developments in time discretized operators and or for evaluating existing methods from the established measures for computational algorithms the conclusion that the most effective in the sense of convergence namely the stability and accuracy and complexity namely the algorithmic formulation and algorithmic structure computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn with this conclusion and also with the notion of providing new avenues leading to improved algorithms by design as an illustration a novel computational algorithm which departs from the traditional paradigm in the sense of lms methods with which we are mostly familiar with and widely used in commercial software is particularly designed into the perspective design space representation of comparable algorithms and is termed here as the forward displacement non linearly explicit l stable fdel algorithm which is unconditionally consistent and does not require non linear iterations within each time step from the established measures for comparable algorithms simply for illustration purposes the resulting design of the fdel formulation is then compared with the commonly advocated explicit central difference method and the implicit newmark average acceleration method alternately the same conclusion holds true against controllable numerically dissipative algorithms which pertain to the class of linear multi step lms methods for assessing both linear and non linear dynamic cases the conclusions that the proposed new design of the fdel algorithm which is a direct consequence of the present notion of design spaces and measures is the most effective algorithm to date to our knowledge in comparison to the class of second order accurate algorithms pertaining to lms methods for routine and general non linear dynamic situations is finally drawn through rigorous numerical experiments

averaging principle for perturbed random evolution equations and corresponding dirichlet problems we continue the study of exit problems for perturbed random evolution equations corresponding to the weakly coupled elliptic pde systems in the present paper we consider the cases where the corresponding random evolutions stay in a given domain for ever with probability one but do not hinder the exit of the perturbed process we treat such problems by methods based on the averaging principle in such a way we also study the asymptotic behavior of the solutions of the corresponding perturbed dirichlet problems

an ordered line integral method for computing the quasi potential in the case of variable anisotropic diffusion abstract nongradient stochastic differential equations sdes with position dependent and anisotropic diffusion are often used in biological modeling the quasi potential is a crucial function in the large deviation theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths unfortunately the quasi potential can rarely be found analytically it is defined as the solution to a certain action minimization problem in this work the recently introduced ordered line integral method olim is extended for computing the quasi potential for 2d sdes with anisotropic and position dependent diffusion scaled by a small parameter on a regular rectangular mesh the presented solver employs the dynamical programming principle at each step a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule the solver is tested on two examples where analytic formulas for the quasi potential are available the dependence of the computational error on the mesh size the update factor k a key parameter of olims as well as the degree and the orientation of anisotropy is established the effect of anisotropy on the quasi potential and the maximum likelihood paths is demonstrated on the maier stein model the proposed solver is applied to find the quasi potential and the maximum likelihood transition paths in a model of the genetic switch in lambda phage between the lysogenic state where the phage reproduces inside the infected cell without killing it and the lytic state where the phage destroys the infected cell

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

testing transition state theory on kac zwanzig model a variant of the kac zwanzig model is used to test the prediction of transition state theory tst and variational transition state theory vtst the model describes the evolution of a distinguished particle moving in a double well external potential and coupled to n free particles through linear springs while the kac zwanzig model is deterministic under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two state markov chain whose transition rate constants can be computed exactly in suitable limit here these transition rate constants are compared with the predictions of tst and vtst it is shown that the application of tst with a naive albeit natural choice of dividing surface leads to the wrong prediction of the transition rate constants this is due to crossings of the dividing surface that do not correspond to actual transition events however optimizing over the dividing surface within vtst allows one to eliminate completely these spurious crossings and therefore derive the correct transition rate constants for the model the reasons why vtst is successful in this model are discussed which allows one to speculate on the reliability of vtst in more complicated systems

the exit problem at weak noise the two variable quasipotential and the kramers problem a new approach to the weak noise analysis of exit problems removes an intrinsic contradiction of an existing method it applies to both the mean time and the location of the exits novel results mainly concern the exits from an entire domain of attraction

postintegration methods for numerical integration of non linear dynamic equations of motion the paper presents a family of methods with postintegration for numerical integration of non linear dynamic equations of motion of discrete systems both explicit and implicit algorithms of these methods are conditionally stable but can give solutions of accuracy of a few orders higher than the known methods with preintegration corresponding to them a general analysis of stability and accuracy of single step methods is carried out the postintegration methods have been tested on a single degree of freedom non linear system

harnack principle for weakly coupled elliptic systems abstract we prove the harnack inequality for the weakly coupled elliptic system s where su big begin align l 1 cdots l n end align big u qu and u big begin align u 1 cdot cdot cdot u n end align big l k k 1 n are second order elliptic operators with holder continuous coefficients and q is a matrix valued function with singular entries in the case that q is irreducible a full harnack principle is proved

disease extinction in the presence of random vaccination we investigate disease extinction in an epidemic model described by a birth death process we show that in the absence of vaccination the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point with an unusual critical exponent even a comparatively weak poisson distributed random vaccination leads to an exponential increase in the extinction rate with the exponent that strongly depends on the vaccination parameters

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

consistent use of finite elements in time and the performance of various recurrence schemes for the heat diffusion equation the use of the finite element concept in the time domain has resulted in a reinterpretation of many previously recorded recurrence formulae and the discovery of new ones an important new feature common to all such schemes is the fact that a weighted forcing term is used this precludes the application of unrealistic forcing terms resulting in a direct finite difference derivation and smooths oscillation due to sudden changes of boundary condition this problem is discussed in the context of various two and three level time schemes

diffusion across characteristic boundaries with critical points we consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with i an asymptotically stable equilibrium point or limit cycle and ii an equilibrium point surrounded by closed trajectories the mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points uniformly valid asymptotic solutions are constructed for each of the problems for the asymptotically stable equilibrium point the method of matched asymptotic expansions with the integral condition of matkowsky and schuss is employed a method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories the influence on the solutions of the critical points on the boundary is exhibited and explained an application to the physical pendulum is given finally our r

noise induced escape in an excitable system we consider the stochastic dynamics of escape in an excitable system the fitzhugh nagumo fhn neuronal model for different classes of excitability we discuss first the threshold structure of the fhn model as an example of a system without a saddle state we then develop a nonlinear nonlocal stability approach based on the theory of large fluctuations including a finite noise correction to describe noise induced escape in the excitable regime we show that the threshold structure is revealed via patterns of most probable optimal fluctuational paths the approach allows us to estimate the escape rate and the exit location distribution we compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli unlike the commonly used local analysis of the stable state our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the langevin equation

variable order variable step algorithms for second order systems part 2 the codes we develop a new class of multistage algorithms for second order systems of ordinary differential equations ode s based on methods by zienkiewicz et al 13 our new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques in a companion paper we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

a novel non linearly explicit second order accurate l stable methodology for finite deformation hypoelastic hypoelasto plastic structural dynamics problems a novel non linearly explicit second order accurate l stable computational methodology for integrating the non linear equations of motion without non linear iterations during each time step and the underlying implementation procedure is described emphasis is placed on illustrative non linear structural dynamics problems employing both total updated lagrangian formulations to handle finite deformation hypoelasticity hypoelasto plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non linear structural dynamics applications

superelement method for transient dynamic analysis of structural systems the method represents a generalization of existing static and dynamic superelement substructure algorithms natural frequencies and modal shapes of a fixed boundary substructure are used to obtain equations of motion of a superelement an implicit time stepping algorithm is presented for the numerical solution of a resulting superelement system of equations of motion the method is believed to be effective for large scale problems of structural analysis

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

a further look at newmark houbolt etc time stepping formulae this paper looks at some well known methods for the numerical integration of the vibration equation from the point of view of various groupings of the independent equations involved new forms are given of the general three parameter method introduced by zienkiewicz 1 some new one step methods are introduced these are of great practical importance because of the ease with which the size of time step can be changed

stochastic perturbations to conservative dynamical systems on the plane i convergence of invariant distributions we consider a nonlinear system on the plane given by an oscil lator with homoclinic orbits the above system is subjected to a perturbation composed of a deterministic part and a random white noise part assum ing the existence of a finite invariant measure to the perturbed system we deal with the convergence of the measures to a limit measure as the pertur bation parameter tends to zero the limit measure is constructed in terms of the action function of the unperturbed oscillator and the strong local l2 convergence of the associated densities is proved

computational algorithms for transient analysis the burden of weight and consequences towards formalizing discrete numerically assigned dna algorithmic markers wp family abstract different from the way we have been looking in the past at developments encompassing modal type and a wide class of step by step time stepping approaches for transient and dynamic field problems and significantly different from the way these have been developed and described in standard text books over the years this paper outlines the theoretical ideas and basis towards providing a generalized methodology and formulations leading to integral and a wide class of integration operators for the resulting solution of these transient equation systems it is herein postulated that integral operators and a wide class of integration operators pertain to and emanate from the same family with the burden which is being carried by a virtual field or weighted function field specifically introduced for the time discretization is strictly enacted in a mathematically consistent manner so as to first permit obtaining the adjoint operator of the original semi discretized equation system subsequently the selection or burden carried by the virtual or weighted function fields originally introduced to facilitate the time discretization process determines the formal development and outcome of exact integral operators approximate integral operators and a wide class of integration operators thus the present developments not only serve as a prelude towards the formal developments for exact integral operators but also demonstrate that the resulting approximate integral operators and a wide class of integration operators and known methods are simply subsets of the generalizations of a standardized w p family and emanate from the principles presented herein the burden of weight and the resulting consequences towards subsequently enabling a formal basis for the selection of discrete numerically assigned dna algorimmic markers to particularly identitfy a wide class of algorithms is described for transient field problems and dynamic field problems respectively

a finite element finite strain formulation for modeling colliding blocks of gent materials the present paper is devoted to the analysis of the contact impact problems with coulomb friction and large deformation between two hyperelastic bodies of gent model the total lagrangian formulation is adopted to describe the geometrically non linear behavior for the finite element implementation the explicit expression of the incremental law of gent model is derived a first order algorithm is applied for the numerical integration of the time discretized equation of motion efficiency and accuracy of the resulting method is illustrated on a two dimensional static contact problem and a three dimensional dynamic contact problem as compared with ansys simulations

switching path distribution in multidimensional systems we explore the distribution of paths followed in fluctuation induced switching between coexisting stable states we introduce a quantitative characteristic of the path distribution in phase space that does not require a priori knowledge of system dynamics the theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance the experimental and theoretical results on the shape and position of the distribution are in excellent agreement with no adjustable parameters in addition the experiment provides the first demonstration of the lack of time reversal symmetry in switching of systems far from thermal equilibrium the results open the possibility of efficient control of the switching probability based on the measured narrow path distribution

dynamic analysis of two dimensional frictional contact by linear complementarity problem formulation two dimensional dynamic contact is formulated as a linear complementarity problem using contact compatibility conditions and the coulomb friction law for the time integration of dynamic response displacements are approximated as second order polynomials and a new scheme of considering velocity discontinuities is presented the efficiency of the method is shown with three numerical examples

probability densities for noisy delay bifurcations the delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise a new asymptotic approximation for the time dependent probability density function gives a complete description of the process into the transition region and is easily interpreted in terms of the noisy dynamics it is also used to calculate mean transition times the method is applied to two nonlinear systems with noise a one dimensional canonical model for a steady bifurcation and the noisy fitzhugh nagumo model

a unified set of single step algorithms part 4 backward error analysis applied to the solution of the dynamic vibration equation this paper applies the method of backward error analysis to the numerical solution of the dynamic vibration equation by the single step algorithms proposed in references 1 3 and obtains some general rules to help with the choice of parameters

finite elements in space and time for dynamic contact problems dynamic obstacle and signorini problems are discretized by continuous and discontinuous finite elements in space and time the resulting discrete problems are attributed to a standard implicit time stepping scheme through relaxation of impact phenomena and suitable numerical integration the method can cope with dynamic contact problems which is shown by an analysis of a model problem moreover numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems which are not fully covered by the theory copyright 2008 john wiley sons ltd

constitutive relation error estimator for transient finite element analysis the concept of error in the constitutive relation allows defining a consistent error measurement for dynamic finite element simulations a solution which satisfies exactly the kinematic constraints the initial conditions and the equilibrium equation is built from the solution to the discretised problem this construction utilizes as best as possible the properties of the integration scheme used to obtain a satisfactory effectivity index the quality of the numerical solution is assessed by the way in which the constitutive relations are satisfied results from various tests conducted on a one degree of freedom problem and on a 1d example display the estimator s good behaviour the effectivity index remains at a reasonable value and on the basis of the first test a comparison of the error in the constitutive relation with different estimators drawn from the literature is presented

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

large deviations from the thermodynamic limit in globally coupled maps abstract systems of a large number n of globally coupled maps have become popular as a relatively simple prototype of high dimensional dynamics showing many interesting and typical phenomena like synchronisation cluster formation and multistability and having potential applications in systems like josephson junction arrays or in biophysical models while much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit n there is still need for a sound theory about the asymptotic behaviour of finite n systems as n approaches infinity this article introduces a method by which it is possible to obtain asymptotic estimates for finite size effects that show up as long term deviations from the thermodynamic limit behaviour this method works for systems of globally coupled noisy maps and is based upon the concept of quasipotentials originally developed by freidlin wentzell and others for describing the influence of small random perturbations on the long term behaviour of dynamical systems the problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed all the concepts described in this article are illustrated with a simple example

on integral representations of twisted group rings let be a primitive 2n root of unity for n 2 and z g be the twisted group ring where g 2 1 and where ga z with ga the complex conjugate of the purpose of this paper is to find the non isomorphic indecomposable lattices

a methodology for the formulation of error estimators for time integration in linear solid and structural dynamics in this article we present a novel methodology for the formulation of a posteriori error estimators applicable to time stepping algorithms of the type commonly employed in solid and structural mechanics the estimators constructed with the presented methodology are accurate and can be implemented very efficiently more importantly they provide reliable error estimations even in non smooth problems where many standard estimators fail to capture the order of magnitude of the error the proposed methodology is applied as an illustrative example to construct an error estimator for the newmark method numerical examples of its performance and comparison with existing error estimators are presented these examples verify the good accuracy and robustness predicted by the analysis copyright 2005 john wiley sons ltd

the asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries we consider diffusion random perturbations of a dynamical systemst in a domaing rm which in particular may be invariant under the action ofst continuing the study of k1 k4 we find the asymptotic behavior of the principal eigenvalue of the corresponding generator when the diffusion term tends to zero

generalized midpoint finite element dynamic analysis of elastoplastic systems the application of a four parameter generalized midpoint integration method to small displacement elastoplastic dynamic problems is considered the integration of the governing relations over a time step is formulated as a non linear algebraic problem finite step problem by making use of generalized variables finite element modelling uniqueness and an extremum characterization of the solution of the finite step problem are discussed sufficient conditions for convergence of various iterative algorithms including newton raphson are given the b stability of the proposed procedure is assessed

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

parametric quadratic programming method for dynamic contact problems with friction based on the parametric variational principle a general but effective parametric quadratic programming technique satisfying various contact conditions is established for dynamic analysis of contact problems with friction and damping the discretization with respect to time and space leads to a static linear complementary problem lcp for each time step which is solved by a quadratic optimization algorithm such as the lemke algorithm etc thus the convergence and numerical stability of the solution can be guaranteed the substructure condensation technique is implemented to handle the unknown contact boundary condition so that the computation effect is considerably reduced as an application of the method presented three dynamic contact examples and numerical results are given

a numerical analysis of a class of problems in elastodynamics with friction on utilise les methodes d elements finis pour analyser le comportement dynamique de corps lineairement elastiques ayant une partie de leurs frontieres soumise au frottement et a des tractions normales prescrites on etablit des estimations d erreur on presente un algorithme de resolution des problemes a deux dimensions et on decrit plusieurs essais numeriques

on the representation type of local twisted group rings the purpose of this paper is to examine locally the representation type of twisted group rings it is proved that a twisted group ring over a local field is of finite representation type if and only if g1 3 and g1 g2 or g1 2 where g1 is the ith ramification group of the extension i 1 2

integral representations of a certain twisted group ring soit k un corps global et r un domaine de dedekind avec corps quotient k soit l k une extension de galois de degre 2 de groupe de galois g et s la fermeture integrale de r dans l on etudie l anneau de groupe twiste sog de g sur s est un r ordre dans la k algebre simple centrale a log l algebre de groupe twistee de g sur l on etudie les treillis

hierarchical higher order dissipative methods for transient analysis this work focuses on devising an efficient hierarchy of higher order methods for linear transient analysis equipped with an effective dissipative action on the spurious high modes of the response the proposed strategy stems from the norsett idea and is based on a multi stage algorithm designed to hierarchically improve accuracy while retaining the desired dissipative behaviour computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved as standard time integration methods and in addition all the stages should share the same coefficient matrix this target is achieved by rationally formulating the methods based on the discontinuous collocation approach the resultant procedure is shown to be well suited for adaptive solution strategies in particular it embeds two natural tools to effectively control the error propagation one estimates the local error through the next stage solution which is one order more accurate the other through the solution discontinuity at the beginning of the current time step which is permitted by the present formulation the performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems some typical numerical tests in transient heat conduction and elasto dynamics are presented copyright 2006 john wiley sons ltd

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

a linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics the present paper deals with the theoretical and numerical treatment of dynamic unilateral problems the governing equations are formulated as an equivalent variational inequality expressing d alembert s principle in its inequality form the discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples

gradient particle solutions of fokker planck equations for noisy delay bifurcations a gradient particle method is used to compute probability densities for processes with delay bifurcations that are sensitive to very small noise the method is particularly useful for computing the probability density in the transition region where asymptotic approximations may not be valid for a one dimensional steady bifurcation problem and a two dimensional fitzhugh nagumo model we solve the fokker planck equation fpe and compare the results with direct simulations and asymptotic approximations

the invariant density of a chaotic dynamical system with small noise abstract we consider a chaotic dynamical system perturbed by noise and calculate an approximate invariant density when the noise level is small because of the special structure of the dynamical system the effective support of the invariant density is much smaller than the noiseless attractor this behavior is captured by the asymptotic form of the invariant density which is given explicitly

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

towards a theory of transition paths we construct a statistical theory of reactive trajectories between two pre specified sets a and b i e the portionsof the path of a markov process during which the path makes a transition from a to b this problem is relevant e g in the context of metastability in which case the two sets a and b are metastable sets though the formalism we propose is independent of any such assumptions on a and b we show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so called committor functions which give the probability that the process reaches first b before reaching a either backward or forward in time using these objects we obtain i the distribution of reactive trajectories which gives the proportion of time reactive trajectories spend in sets outside of a and b ii the hitting point distribution of the reactive trajectories on a surface which measures where the reactive trajectories hit the surface when they cross it iii the last hitting point distribution of the reactive trajectories on the surface iv the probability current of reactive trajectories the integral of which on a surface gives the net average flux of reactive trajectories across this surface v the average frequency of reactive trajectories which gives the average number of transitions between a and b per unit of time and vi the traffic distribution of reactive trajectories which gives some information about the regions the reactive trajectories visit regardless of the time they spend in these regions

application of the generalized α method to the time integration of the cable dynamics equations abstract the box method is a popular finite difference scheme for solving the cable dynamics equations in ocean engineering applications in this paper we analyze the shortcomings of the temporal integration defined by the box method and propose a new procedure that retains the box method s spatial discretization but employs the generalized method for temporal discretization the stability and accuracy of the generalized method as applied to the cable dynamics equations are developed this analysis procedure demonstrates the formulation of the semi discrete governing equations using spatial finite differences and the applicability of amplification matrix stability analysis to finite difference problems an example is presented that demonstrates the robustness of the new approach

an analysis of metal forming processes using large deformation elastic plastic formulations a concise survey of the literature related to the large deformation elasto plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformation analysis starting from the principle of virtual work the so called total lagrangian and updated lagrangian formulations are derived based on some fundamental assumptions in linearizing the nonlinear equations by introducing the zaremba jaumann co rotational increment to the cauchy stress tensor the classical prandtl reuss equations are generalized for describing the elastic plastic material behavior to allow a proper consideration of the contact conditions in the incremental analysis a general friction law with an associated isotropic coulomb sliding rule is obtained by the similarity between dry friction and plasticity finite element discretizations and approximations are applied to the resulting formulation of the updated lagrangian approach four example problems are solved to test the formulations developed in this paper the emphasis is made toward the numerical accuracy of the finite element solutions

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

a local space time discontinuous finite element method we present a locally enriched space time finite element method for solving hyperbolic problems with discontinuities the space time formulation is coupled to a standard semidiscrete finite element method the discontinuities are captured by a space time version of the extended finite element method which treats arbitrary moving discontinuities since the discontinuities are local the enriched space time method is only needed around the discontinuities which provides significant computational savings the coupling is implemented through a weak enforcement of the continuity of the flux between the space time and semidiscrete domains in a manner similar to discontinuous galerkin methods the method is illustrated through one dimensional problems it displays remarkable accuracy in capturing moving discontinuities both in their amplitude and speed

on the consideration of unilateral contact and friction in cracks the boundary integral method the scope of the present paper is the study of cracks having a given geometry by taking into account unilateral contact and friction phenomena between the two sides of the crack the arising problem is highly non classical because of the inequality relations describing the unilateral contact and friction interface conditions the indirect biem is extended appropriately in order to treat this problem numerical examples concerning the calculation of stress intensity factors under the unilateral contact and friction interface conditions illustrate the developed method

large deviations results for the exit problem with characteristic boundary abstract we consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region d we show the standard large deviations results for the exit distribution and mean exit time as obtained by wentzell and freidlin under the assumption of nontangential drift b n

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

a combined space time extended finite element method the newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities zienkiewicz proposed an approach using finite element concept in time which allows a new look at the newmark method the idea of this paper is to propose thanks to this approach the use of a time partition of the unity method denoted time extended finite element method tx fem for improved numerical simulations of time discontinuities an enriched basis of shape functions in time is used to capture with a good accuracy the non polynomial part of the solution this formulation allows a suitable form of the time stepping formulae to study stability and energy conservation the case of an enrichment with the heaviside function is developed and can be seen as an alternative approach to time discontinuous galerkin method t dgm stability and accuracy properties of which can be derived from those of the tx fem then space and time x fem stx fem are combined to obtain a unified space time discretization this combined stx fem appears to be a suitable technique for space time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities copyright 2005 john wiley sons ltd

master equation simulations of a model of a thermochemical system master equation approach is used to study the influence of fluctuations on the dynamics of a model thermochemical system for appropriate values of parameters the deterministic description of the system gives the subcritical or supercritical hopf bifurcations for small systems containing 100 000 particles close to the supercritical hopf bifurcation the stochastic trajectories obtained from numerical simulations do not allow to distinguish between damped oscillations around a stable focus and sustained oscillations around a small stable limit cycle this uncertainty disappears if the number of particles in the system is increased up to 1 000 000 close to subcritical hopf bifurcation the stochastic trajectory of the system jumps from the basin of attraction of a stable focus to the basin of attraction of a stable limit cycle in this case the time dependencies of temperature and concentration of reactant in the system are apparently similar to intermittent chaotic oscillations the mean first passage time for the transitions from the stable focus to the stable limit cycle show the characteristic exponential dependence on the number of particles this passage time depends very strongly on the bifurcation parameter reaction heat which determines the distance between the stable focus and an unstable limit cycle

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

truncation error and stability analysis of iterative and non iterative thomas gladwell methods for first order non linear differential equations the consistency and stability of a thomas gladwell family of multistage time stepping schemes for the solution of first order non linear differential equations are examined it is shown that the consistency and stability conditions are less stringent than those derived for second order governing equations second order accuracy is achieved by approximating the solution and its derivative at the same location within the time step useful flexibility is available in the evaluation of the non linear coefficients and is exploited to develop a new non iterative modification of the thomas gladwell method that is second order accurate and unconditionally stable a case study from applied hydrogeology using the non linear richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non iterative formulation

quantifier elimination in applied mechanics problems with cylindrical algebraic decomposition abstract the method of cylindrical algebraic decomposition cad of the k dimensional space constitutes a classical technique for the efficient solution of quantifier elimination qe problems in algorithmic computer aided algebra here we apply this method to some applied mechanics problems under appropriate constraints at first we study the problem of a straight elastic beam under a restriction on the maximum permissible deflection along this beam which can easily be reduced to the construction of a one dimensional cad as well as the problem of a circular isotropic elastic medium where a stress component should not exceed a critical value which requires the construction of a three dimensional cad in both these problems we derive also the required quantifier free formulae qffs not including the fundamental variables but only the parameters involved much more difficult cad qff derivation applications concerning an elliptical elastic medium again with an upper bound for a stress component a special case of failure by yielding in fracture mechanics related to sih s strain energy density factor and a frictionless contact problem for an elastic half plane are also considered and explicitly solved with the help of already available cad produced results although evidently cad is not expected to produce qffs in extremely difficult problems finally additional possible applications of cad cqe to applied mechanics problems are also suggested

a new approach in non linear mechanics the large time increment method a new algorithm family taking into account the entire loading process in a single large time increment is proposed to compute structures with physical non linearities and is tested on some examples in elastoplasticity the method considerably reduces the number of transfers between local and global levels hence the numerical cost of calculation is also diminished

the exit distribution on the stochastic separatrix in kramers exit problem kramers exit problem is concerned with noise activated escape from a potential well in the case when the noise strength varepsilon temperature measured in units of potential barrier height is small this becomes a singular perturbation problem unexpectedly the noise induced escape from the domain of attraction of the stable equilibrium at the bottom of the well does not occur at points on the separatrix where the energy is minimal rather the distribution of exit points on the separatrix in the phase plane is shown to be spread away from the saddle point where the energy is minimal it is shown that actually most of the probability of the exit points on the separatrix is located at a distance o sqrt varepsilon from the saddle point and the probability vanishes altogether at the saddle point in this problem large deviations theory fails to predict the distribution of the exit point for finite noise we construct an asymptotic approximation to the solution of the fokker planck eq