elastoplastic damage modelling including the gradient of damage formulation and computational aspects a framework for continuum elastoplastic damage modelling which employs irreversible thermodynamics and internal state variables is investigated the damage part of the modelling involves the gradient of damage quantity which together with the equations of motion are issued from a new formulation of the principle of virtual power recently proposed it is shown how the plastic part of the modelling takes its standard format next we consider in detail the variational formulation and subsequent numerical implementation of the elastoplastic damage models the development of an algorithm consistent with the present formulation is given where for the plastic part it leaves the standard return mapping algorithms unchanged application is made to different classical rate independent and rate dependent plastic models with the presence of a damage mechanism the numerical implementation in the context of the finite element method is discussed in detail and a set of representative applications and numerical examples is given

conservation laws for some systems of nonlinear pdes via the symmetry adjoint symmetry pair method abstract this paper applies the symmetry adjoint symmetry pair method to find conservation laws for nonlinear partial differential equations as examples conservation laws are obtained for a combined potential system with nonlinear telegraph and nonlinear diffusion convection equations as special cases as well as for an extended nonlinear wave equation system in addition for both examples conservation laws are obtained for and nonlocally related equivalent systems the paper concludes with a posed open problem

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

a porous thermoelastic problem in this paper a porous thermoelastic problem is numerically considered the variational formulation is written as a coupled system of two hyperbolic equations for the displacement and the porosity fields and a parabolic equation for the temperature field an existence and uniqueness result as well as an energy decay property are recalled then fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward euler scheme to discretize the first order time derivatives a priori error estimates are proved from which the linear convergence is deduced under some additional regularity conditions finally some one and two dimensional numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution

a mixture of thermoelastic solids with two temperatures abstract in this work we study from the numerical point of view a problem involving one dimensional thermoelastic mixtures with two different temperatures that is when each component of the mixture has its own temperature the mechanical problem consists of two hyperbolic equations coupled with two parabolic equations the variational problem is derived in terms of product variables an existence and uniqueness result and an energy decay property are stated then fully discrete approximations are introduced using the finite element method and the backward euler scheme a discrete stability property is proved and a priori error estimates are obtained from which the linear convergence is deduced finally some numerical simulations are described to show the accuracy of the approximation and the behavior of the solution

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

a finite element method to model progressive fracturing abstract in this paper a theoretical basis is established for a finite element iterative algorithm for load displacement analysis of elasticity problems in which the elastic body fractures under load location of fractures is determined by testing a coulomb friction criterion along inter element edges convergence of the algorithm to a physically meaningful solution is proved and is illustrated by results for a cantilever type problem

a dynamic viscoelastic contact problem with normal compliance a dynamic contact problem between a viscoelastic body and a deformable obstacle is numerically considered in this work the contact is modeled by using the well known normal compliance contact condition the variational formulation of this problem is written in terms of the velocity field and it leads to a parabolic nonlinear variational equation an existence and uniqueness result is stated fully discrete approximations are then introduced by using the finite element method to approximate the spatial variable and a hybrid combination of the implicit and explicit euler schemes to discretize the time derivatives an a priori error analysis is recalled then an a posteriori error analysis is provided extending some results already obtained in the study of the heat equation other parabolic equations and the quasistatic case upper and lower bounds are proved finally some two dimensional numerical simulations are presented to demonstrate the accuracy and the behavior of the error estimators

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

differential inclusions and target problems where and how solutions associated to a differential inclusion can or cannot enter a given target is studied for this purpose partitions of the target boundary are associated with the dynamic of the system the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets these kernels determine points such that there exist respectively all solutions starting at these points remain in a given set of constraints the sets that are reached in finite time by viable solutions to the system are also studied finally some applications to control systems with one target are provided and the concept of semipermeable barriers will be generalized

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

invariants for the time dependent harmonic oscillator i two equivalent families of linear invariants and two equivalent families of quadratic invariants are obtained for a harmonic oscillator with variable mass or with variable frequency explicit formulae are obtained for the linear and quadratic invariants in the case of a damped oscillator and the hamiltonian is seen as a special quadratic invariant

a frictionless contact problem for elastic viscoplastic materials with normal compliance numerical analysis and computational experiments in this paper we consider a frictionless contact problem between an elastic viscoplastic body and an obstacle the process is assumed to be quasistatic and the contact is modeled with normal compliance we present a variational formulation of the problem and prove the existence and uniqueness of the weak solution using strongly monotone operators arguments and banach s fixed point theorem we also study the numerical approach to the problem using spatially semi discrete and fully discrete finite elements schemes with implicit and explicit discretization in time we show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions finally we present some numerical results involving examples in one two and three dimensions

a simple and efficient bem implementation of quasistatic linear visco elasticity abstract a simple yet efficient procedure to solve quasistatic problems of special linear visco elastic solids at small strains with equal rheological response in all tensorial components utilizing boundary element method bem is introduced this procedure is based on the implicit discretisation in time the so called rothe method combined with a simple algebraic transformation of variables leading to a numerically stable procedure proved explicitly by discrete energy estimates which can be easily implemented in a bem code to solve initial boundary value visco elastic problems by using the kelvin elastostatic fundamental solution only it is worth mentioning that no inverse laplace transform is required here the formulation is straightforward for both 2d and 3d problems involving unilateral frictionless contact although the focus is to the simplest kelvin voigt rheology a generalization to maxwell boltzmann jeffreys and burgers rheologies is proposed discussed and implemented in the bem code too a few 2d and 3d initial boundary value problems one of them with unilateral frictionless contact are solved numerically

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

a frictionless contact problem for elastic viscoplastic materials with normal compliance and damage abstract we study a quasistatic frictionless contact problem with normal compliance and damage for elastic viscoplastic bodies the mechanical damage of the material caused by excessive stress or strain is described by a damage function whose evolution is modelled by an inclusion of parabolic type we provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model we then introduce and study a fully discrete scheme for the numerical solutions of the problem an optimal order error estimate is derived for the approximate solutions under suitable solution regularity numerical examples are presented to show the performance of the method

analysis for the strain gradient theory of porous thermoelasticity abstract in this paper we analyse a model involving a strain gradient thermoelastic rod with voids existence and uniqueness as well as an energy decay property are proved by means of the semigroup arguments the variational formulation is derived and then a fully discrete approximation is introduced by using the finite element method to approximate the spatial variable and the implicit euler scheme to discretize the time derivatives a stability result and a priori error estimates are obtained from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions finally some numerical simulations are presented to demonstrate the accuracy of the algorithm and the behaviour of the solution

connection between classical and quantum harmonic oscillators for a time dependent harmonic oscillator we show a connection between the integrals of classical equations of motion and the lewis riesenfeld lr invariant and express the lr invariant in terms of the integrals of classical equations of motion as applications we find the lr invariant for the guth pi hamiltonian and the asymptotic form of the lr invariant for a general time dependent harmonic oscillator

analysis of dynamic nonlinear thermoviscoelastic beam problems abstract in this paper we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles thermal effects are also taken into account and the contact is modeled using the classical signorini condition the existence of solutions is proved by considering approximate problems from a penalization procedure proving some a priori estimates and passing to the limit an exponential decay property is also obtained then fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit euler scheme for the discretization of the time derivatives a stability property is obtained and a priori error estimates are proved from which under some additional regularity conditions the linear convergence of the algorithm is deduced fully discrete approximations of the signorini problem are then introduced proceeding in a similar way for which a stability property is obtained finally some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution

minimal time control of the two tanks gradostat model under a cascade input constraint we study the minimum time control problem of a series of two interconnected chemostats under the input constraint u2 u1 where ui are the respective dilution rates in the tanks this constraint brings controllability issues in the study of the optimal strategies we encounter this difficulty by splitting the state domain into two sub domains one with no lack of controllability of the target and its complementary is such that any optimal control fulfills u1 u2 we explicit the complete optimal synthesis that depends on the position of the target with respect to semi permeable curve that passes through a steady state singular point

contact symmetries of the harmonic oscillator the symmetry group of contact transformations for the one dimensional harmonic oscillator is determined its generators contain two arbitrary functions i e it is an infinite parameter lie group

adjoint symmetries for time dependent second order equations the authors extend part of their previous work on autonomous second order systems sarlet et al 1987 to time dependent differential equations the main subject of the paper concerns the notion of adjoint symmetries they are introduced as a particular type of 1 form whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields it is shown that all interesting properties of adjoint symmetries known from the autonomous theory have their counterparts in the present framework of particular interest is a result establishing that lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries a number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way

some remarks on the lie group of point transformations for the harmonic oscillator some aspects of the symmetry group sl 3 r of the simple harmonic oscillator are revisited a realisation of the general element of the group in newtonian spacetime is obtained with the eight parameters included the group is shown to be sl 3 r in fact without recourse to the lie algebra finally in this way the group law of binary composition of the parameters is calculated easily in the group manifold

on the regularity of semipermeable surfaces in control theory with application to the optimal exit time problem part i in control theory a semipermeable surface is an in general nonsmooth oriented surface that on one hand contains solutions the so called barrier solutions of the controlled system and on the other hand may be crossed by the solutions of this system in only one direction without making any assumption on the regularity of the boundary of the semipermeable surface we show that the barrier solutions contained in this semipermeable surface satisfy the pontryagin principle that this surface is a lipschitz manifold and that it is locally the graph of a semiconcave function applying these results to the optimal exit time function from a given open set yields without any controllability assumption at the boundary of the open set that this function is semiconcave on an open dense subset of its domain

numerical solving of frictionless contact problems in perfectly plastic bodies abstract we consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids under an analogous variational formulation for each one we introduce a discrete approximation via the finite element method and in due course prove convergence also we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences

theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage this paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background the main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results we propose a formulation in terms of the helmholtz free energy incorporating the gradient of the damage field a dissipation potential and the postulate of maximum dissipation as a result the driving force conjugated to damage evolution is identified as the quasi nonlocal energy release rate which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate on the numerical side besides balance of linear momentum the algorithmic consistency condition must be solved in weak form thereby the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming in the examples we compare the behavior in local damage with the performance of the gradient formulation

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

analysis of a multidimensional thermoviscoelastic contact problem under the green lindsay theory abstract in this paper we investigate the existence the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation the viscoelastic body is assumed to behave according to kelvin voigt constitutive law with added thermal effects under the green lindsay theory we prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the faedo galerkin method an energy decay property is also shown under the assumption of radial symmetry then a numerical approximation based on the finite element method is proposed a stability result is proved from which the decay of the discrete energy is deduced a priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions finally some numerical experiments are described to support our results

minimal time sequential batch reactors with bounded and impulse controls for one or more species we consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource with the objective being to reach a given level of the resource we allow controls to be bounded measurable functions of time plus possible impulses for the one species case we show that the immediate one impulse strategy filling the whole reactor with one single impulse at the initial time is optimal when the growth function is monotonic for nonmonotonic growth functions with one maximum we show that a particular singular arc strategy precisely defined in section 3 is optimal these results extend and improve former ones obtained for the class of measurable controls only for the two species case with monotonic growth functions we give conditions under which the immediate one impulse strategy is optimal we also give optimality conditions for the singular arc strategy at a level that depends on the initial condition to be optimal the possibility for the immediate one impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations

analysis of a contact problem involving thermoelastic mixtures abstract in this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle the classical normal compliance condition is used for modeling the contact the variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations an existence and uniqueness result is proved using the faedo galerkin method then fully discrete approximations are introduced by using the finite element method and the implicit euler scheme a discrete stability property and a priori error estimates are proved from which the decay of the discrete energy and the linear convergence of the algorithm are deduced finally some numerical simulations are presented to show the convergence and the behavior of the solution

on the symmetries and invariants of the harmonic oscillator a general formulation of noether s theorem is applied to the equation of a harmonic oscillator the definition of symmetry includes the usual lie invariance as a special case and unlike standard formulations generates the full set of invariants i e gives closure under functional composition the analysis for a time dependent oscillator casts doubt on the importance of a known class of invariants the existence of a lagrangian function is shown to be inessential to the analysis

a normal compliance contact problem in viscoelasticity an a posteriori error analysis and computational experiments in this work the numerical approximation of a viscoelastic contact problem is studied the classical kelvin voigt constitutive law is employed and contact is assumed with a deformable obstacle and modelled using the normal compliance condition the variational formulation leads to a nonlinear parabolic variational equation an existence and uniqueness result is recalled then a fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and the implicit euler scheme to discretize time derivatives a priori error estimates recently proved for this problem are recalled then an a posteriori error analysis is provided extending some preliminary results obtained in the study of the heat equation and other parabolic equations upper and lower error bounds are proved finally some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates

pseudo symmetries noether s theorem and the adjoint equation pseudo symmetries were introduced by sarlet and cantrijn 1984 for time dependent non conservative systems they are reconsidered in the context of general autonomous second order systems relying on the new approach to such systems which was presented by sarlet et al they further introduce the notion of adjoint symmetries of a second order system as being associated to invariant 1 forms and show how they may be related to first integrals or to lagrangians under appropriate circumstances their results enable them to clarify a rather unusual account of noether s theorem which was recently given by gordon 1986

numerical analysis of a frictionless viscoelastic contact problem with normal damped response abstract we consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation we model the material s behavior with a nonlinear kelvin voigt constitutive law the process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition we present the variational formulation of the problem including the existence of a unique weak solution to the model we then study the numerical approach to the problem using a fully discrete finite elements scheme with an explicit discretization in time we state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions finally we present some numerical results involving examples in one and two dimensions

evolutionary variational inequalities arising in viscoelastic contact problems we consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials under the smallness assumption of a certain coefficient we prove an existence and uniqueness result using banach s fixed point theorem we then study two numerical approximation schemes of the problem a semidiscrete scheme and a fully discrete scheme for both schemes we show the existence of a unique solution and derive error estimates finally all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem with the finite element method used to discretize the spatial domain

numerical analysis of a viscoelastic mixture problem abstract in this paper we study from the numerical point of view a mixture problem involving a viscoelastic material and an elastic one the mechanical problem is written as a linear system of two coupled hyperbolic partial differential equations an existence and uniqueness result and an energy decay property are recalled then fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward euler scheme to discretize the time derivatives a priori error estimates are proved from which under suitable regularity conditions the linear convergence of the algorithm is derived finally some numerical simulations are presented to demonstrate the accuracy of the approximations and the behaviour of the solutions

numerical analysis and simulations of a dynamic frictionless contact problem with damage the process of dynamic frictionless contact between a viscoelastic body and a reactive foundation which includes material damage is modelled numerically analyzed and simulated contact is modelled with the normal compliance condition the damage of the material resulting from tension or compression is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material the evolution of the damage is governed by a parabolic differential inclusion a fully discrete scheme for the problem is proposed and error estimates on the numerical solutions derived the scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented