efficient modification scheme of stress strain tensor for wrinkled membranes a new modification scheme of the stress strain tensor for wrinkled membranes is presented on the basis of the tension field theory the scheme is applicable to finite element analysis of partly wrinkled membranes with arbitrary shapes derivation of the modification scheme rests on an introduction of the so called wrinkle strain and a simplification of the virtual work equation of wrinkled membranes because all of the modifications required to account for wrinkling are totally confined within the stress strain relations of membranes the scheme can be easily implemented with existing finite element codes furthermore the modified stress strain tensor automatically leads to the consistent tangent stiffness matrix where changes in both the wrinkling direction and the amount of wrinkliness are taken into account three numerical examples are treated to show the accuracy and effectiveness of the proposed modification scheme

order and stepsize control in extrapolation methods the paper presents a new theory for joint order and stepsize control in extrapolation methods this theory defines a locally optimal order that can be determined along any trajectory to be computed in addition shannon s information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non stiff integration by the new device moreover a significant increase in reliability robustness and portability of the extrapolation codes is achieved

global aspects of turbulence induced by heteroclinic cycles in competitive diffusion lotka volterra equation abstract this paper presents detailed numerical results of the competitive diffusion lotka volterra equation may leonard type first we derive the global phase diagrams of attractors in the parameter space including the system size where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis but the transition borders become complex when multi basin structures appear the complex aspects of the transition borders are studied in the case when the system size decreases next we show the statistical aspects of the turbulence with special attention to the onset of the supercritical hopf bifurcation several characteristic quantities such as correlation length correlation time lyapunov spectra and lyapunov dimension are investigated in detail near the onset of turbulence our data show the critical scaling law near the onset only in the restricted parameter domain however even when the critical indices are not determined accurately it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations these scaling relations are discussed in comparison with the result derived by the phase reduction method lastly we make a conjecture about the stability of an ecosystem based on the bifurcation diagram the ecosystem obeying the lotka volterra equation in the case of may leonard type is stabilized more as the system size increases

some new results concerning index 3 differential algebraic equations on introduit la notion de tractabilite d indice 3 on donne un critere matriciel pour tester cette tractabilite pour l equation differentielle algebrique f x t x t t 0

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

classical nonlinear response of a chaotic system i collective resonances we develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems in contrast to behavior in integrable or almost integrable systems the nonlinear classical response in chaotic systems vanishes at long times the exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices we calculate the linear and second order response in one of the simplest chaotic systems free classical motion on a compact surface of constant negative curvature the response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories we demonstrate the relevance of the model for the interpretation of spectroscopic experiments

an algorithm for finite element analysis of partly wrinkled membranes a generalization and numerical implementation is presented of the stein hedgepeth 1961 continuum theory for the analysis of partly wrinkled membranes such as the tensioned membrane surfaces which are found in spacecraft structural components the approach has its basis in experimental observations showing that when wrinkles develop in a membrane parallel to the x direction the associated overall contraction in the y direction exceeds that predicted by the poisson s ratio effect

new numerical method for two dimensional partially wrinkled membranes based on the introduction of a nonnegative adjustable parameter into poisson s ratio a new numerical method is proposed to solve numerically the partially wrinkled membrane problems with arbitrary complex shapes three fundamental issues in this new method are addressed first a nonnegative adjustable parameter is introduced into the constitutive law automatically representing unwrinkled taut and wrinkled states of a membrane second a parametric variational principle is developed for the new membrane model third by the variational principle the original membrane problem is converted to a nonlinear mathematical programming problem the variational principle and the mathematical programming formulation lay a foundation for efficient numerical analysis of wrinkled membrane structures using finite element discretization three typical examples are given to show the effectiveness of this method

analysis of descriptor systems using numerical algorithms in this paper we analyze numerical methods for the solution of the large scale dynamical system e dot y t ay t g t y t 0 y 0 where e and a are matrices possibly singular systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables special cases of such systems are algebraic differential systems we discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure detect nonsolvable systems resolve initial value consistency problems and are applicable to stiff descriptor systems we also present an algorithm for the control of the local truncation error on only the state variables

localized behavior in the lyapunov vectors for quasi one dimensional many hard disk systems we introduce a definition of a localization width whose logarithm is given by the entropy of the distribution of particle component amplitudes in the lyapunov vector different types of localization widths are observed for example a minimum localization width where the components of only two particles are dominant we can distinguish a delocalization associated with a random distribution of particle contributions a delocalization associated with a uniform distribution and a delocalization associated with a wavelike structure in the lyapunov vector using the localization width we show that in quasi one dimensional systems of many hard disks there are two kinds of dependence of the localization width on the lyapunov exponent index for the larger exponents one is exponential and the other is linear differences due to these kinds of localizations also appear in the shapes of the localized peaks of the lyapunov vectors the lyapunov spectra and the angle between the spatial and momentum parts of the lyapunov vectors we show that the krylov relation for the largest lyapunov exponent ensuremath lambda ensuremath sim ensuremath ensuremath rho mathrm ln ensuremath rho as a function of the density ensuremath rho is satisfied apart from a factor in the same density region as the linear dependence of the localization widths is observed it is also shown that there are asymmetries in the spatial and momentum parts of the lyapunov vectors as well as in their x and y components

lyapunov instability of rough hard disk fluids the dynamical instability of rough hard disk fluids in two dimensions is characterized through the lyapunov spectrum and the kolmogorov sinai entropy h ks for a wide range of densities and moments of inertia i for small i the spectrum separates into translation dominated and rotation dominated parts with increasing i the rotation dominated part is gradually filled in at the expense of translation until such a separation becomes meaningless at any density the rate of phase space mixing given by h ks becomes less and less effective the more the rotation affects the dynamics however the degree of dynamical chaos measured by the maximum lyapunov exponent is only enhanced by the rotational degrees of freedom for high density gases but is diminished for lower densities surprisingly no traces of lyapunov modes were found in the spectrum for larger moments of inertia the spatial localization of the perturbation vector associated with the maximum exponent however persists for any i

the simulation problem for high order linear differential systems in this paper we consider linear constant coefficient ordinary differential equations of arbitrary order of the form r ddt w m ddt f with r and m given but otherwise arbitrary polynomial matrices to these equations we associate a behavior b defined as the set of all vector valued distributions w f that solve the equations themselves we think of f as a given distribution while w is to be found in such a way that the equations defining the behavior are satisfied alongside r ddt w m ddt f we also consider initial conditions obtained by imposing the value at time t 0 of a linear combination of the variables w and their derivatives yielding s ddt w 0 ta with s a polynomial matrix t a fixed real matrix and a a real vector the three main issues we address are i solvability of the behavior meaning the problem of finding conditions that assure that corresponding to a or any given vector distribution f a trajectory w exists such that w f belong to b this question is formalized and answered in section 4 ii the index of the behavior which means investigating how the smoothness of the given distribution f is related to smoothness of the solution w section 5 is dedicated to this issue iii compatibility of initial conditions in which case the initial conditions s ddt w 0 ta are considered alongside the behavior b we first check that s ddt w 0 is well defined if it is we provide conditions under which there exists a unique distribution w such that w f belong to b and such that s ddt w 0 ta are also satisfied for a or any given a this problem is addressed in section 6 we show how our results generalize to behaviors defined by equations of arbitrary order classical properties of behaviors defined by first order systems such as eddtw aw f

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

a simple computer implementation of membrane wrinkle behaviour via a projection technique a simple computer implementation of membrane wrinkle behaviour is presented within the classical elastic plane stress constitutive model in the present method a projection technique is utilized for modelling of the wrinkle mechanisms in which the total strains in wrinkled membranes are decomposed into elastic and zero strain energy parts and a projection matrix that extracts the elastic parts from the total strains is derived the resulting modified elasticity matrix that represents the stress strain relations in wrinkled membranes is thus obtained as product of the classical elasticity matrix and the projection matrix the modified elasticity matrix is straightforward to implement within the context of the finite element method numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method copyright 2007 john wiley sons ltd

analysis of 2 d bimodular materials and wrinkled membranes based on the parametric variational principle and co rotational approach summary in the numerical analysis of 2 d bimodular materials strain discontinuity is problematic and the traditional iterative algorithm is frequently unstable this paper develops a stable algorithm for the large displacement and small strain analyses of 2 d bimodular materials and structures geometrically nonlinear formulations are based on the co rotational approach using the parametric variational principle pvp a unified constitutive equation is created to resolve the problem induced by strain discontinuity in the local coordinate system because the traditional stress iteration is not required the local linear stiffness matrix does not need to be updated when computing the global stiffness matrix and the nodal internal force vector the nonlinear problem is ultimately transformed into a complementarity problem that is simply solved by combing the newton raphson scheme and the mature quadratic programming algorithm numerical examples demonstrate that the pvp algorithm presents better convergence behavior than the traditional iterative algorithm by incorporating the concept of material modification the new algorithm is also be successfully extended to the wrinkling analysis of thin membranes copyright 2014 john wiley sons ltd

a finite element method for elasto plastic structures and contact problems by parametric quadratic programming in this paper the stiffness matrix of a contact element is introduced by means of a penalty function expression of the contact pressure and frictional force the contact condition and the flow rule are expressed by the same form as in a non associated plastic flow problem a unified pqp parametric quadratic programming model related to contact problems as well as to elasto plastic structures is constructed a series of pqp formulae for contact problems and elastic plastic structures is derived in the text and some numerical examples are illustrated as well

the application of rosenbrock wanner type methods with stepsize control in differential algebraic equations two rosenbrock wanner type methods for the numerical treatment of differential algebraic equations are presented both methods possess a stepsize control and an index 1 monitor the first method dae34 is of order 3 4 and uses a full semi implicit rosenbrock wanner scheme the second method rkf4da is derived from the runge kutta fehlberg 4 5 pair where a semi implicit rosenbrock wanner method is embedded in order to solve the nonlinear equations the performance of both methods is discussed in artificial test problems and in technical applications

rosenbrock type methods adapted to differential algebraic systems we consider the numerical solution of differential algebraic systems of index one given in kronecker canonical form the methods described here are derived from the rosenbrock approach hence they do not require the solution of nonlinear systems of equations but one evaluation of the jacobian and one lu decomposition per step by construction the s stage method coincides with a solver for nonlinear equations of order s 1 if the stepsize is set to zero in this sense the adaptation to differential algebraic equations is performed the special structure of the method leads to simplified order conditions and to an easy implementation some particular methods up to order 4 are given especially an embedded 4 stage method of order 4 3 is derived

smooth approximations to nonlinear complementarity problems it is well known that a nonlinear complementarity problem ncp can be formulated as a system of nonsmooth equations chen and mangasarian comput optim appl 5 1996 pp 97 138 proposed a class of parametric smooth functions by twice integrating a probability density function as a result the nonsmooth equations can be approximated by smooth equations this paper refines the smooth functions proposed by chen and mangasarian and investigates their structural properties the refinement allows us to establish the existence uniqueness and limiting properties of the trajectory defined by the solutions of these smooth equation approximations in addition global error bounds for the ncp with a uniform p function are obtained

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

integral algebraic equations theory of collocation methods i our analysis of collocation solutions for general systems of linear integral algebraic equations iaes is based on the notions of the tractability index and the nu smoothing property of a volterra integral operator these are used to decouple the given iae system into the inherent system of regular second kind volterra integral equations vies and a system of first kind vies this decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions numerical examples illustrate the theoretical results

mixing lyapunov instability and the approach to equilibrium in a hard sphere gas we present maximum lyapunov exponentsl 1 and related kolmogorov sinai entropies hks for a gas of hard spheres at various densities the time scales defined by l 1 and hks are compared with the collision time the decay time of typical autocorrelation functions and the relaxation time of a one particle distribution at low densities the lyapunov time t l 1 l 1 is much smaller than the collision time t c whereas at high densities we findt l t c we discuss consequences for kinetic theory and numerical simulations the mixing properties in phase space are investigated for the two dimensional lorentz gas it is numerically verified that the kolmogorov sinai time t ks 1 hks is the characteristic time for this relaxation process s1063 651x 97 50301 1 mixing in phase space is a necessary condition for the relaxation of a nonequilibrium state towards equilibrium and therefore for statistical mechanics to apply 1 due to the exponential instability characterized by a positive kolmogorov sinai entropy hks a number of initially close phase points are eventually uniformly distributed over the energy surface the characteristic time for this mixing process in phase space is the kolmogorov sinai time t ks 51 h ks 2 3 in order to relate this time to the typical relaxation time of a nonequilibrium state and to decay times of equilibrium autocorrelation functions we performed a series of relaxation experiments on a hard sphere gas such experiments were pioneered by alder and wainwright in 1958 4 and repeated by several authors since then 5 6 consider a system of n identical smooth hard spheres with diameter s and mass m prepared with velocities equal in magnitude but pointing in random directions we monitor the time evolution of this nonequilibrium state by computing the reduced single particle distribution f p t of the magnitude p5upu of the momenta f p t is obtained as an average over 1000 runs with different initial conditions starting from the d distribution at time t50 f p t converges towards the equilibrium maxwell boltzmann distribution f 0 p to describe the relaxation of this nonequilibrium state we use

tension field theory a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations the principal contribution is a partial differential equation describing a geometrical property of tension trajectories this is one of a system of two equations which describes the state of stress independently of the deformation this system is strongly elliptic at any stable solution whereas the deformation is described by a system of parabolic type controllable solutions i e those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone are obtained the general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples existing small strain theories are shown to correspond to a singular limit of the general theory at which the underlying system changes from elliptic to parabolic type

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems