finite approximations of frobenius perron operators a solution of ulam s conjecture to multi dimensional transformations abstract we prove that ulam s piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise c2 expanding transformation or a jablonski transformation s 0 1 n rn 0 1 n this solves an extension of ulam s conjecture to multi dimensions and generalizes the convergence result given by t y li for one dimensional transformations

γ convergence of multiscale periodic functionals depending on the time derivative and the curl operator abstract we study the convergence of a family of multiscale periodic quadratic integral functionals defined in a product space whose densities depend on the time derivative and on the curl of solenoidal fields through the multiscale convergence in time space and the multiscale young measures in time space associated with relevant sequences of pairs an explicit representation of the limit density is given by means of an infinite dimensional minimization problem

assumed strain stabilization procedure for the 9 node lagrange shell element an assumed strain strain interpolation method is used to construct a stabilization matrix for the 9 node shell element the stabilization procedure can be justified based on the hellinger reissner variational method it involves a projection vector which is orthogonal to both linear and quadratic fields in the local co ordinate system of each quadrature point all terms in the development involve 2 2 quadrature in the 9 node element example problems show good accuracy and an almost optimal rate of convergence

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

a finite element method for the frobenius perron operator equation we present a finite element framework to numerically solve the absolutely continuous invariant measure associated with a non singular transformation both convergence analysis and error estimates of the method are given for a class of mappings in r n

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

multi scale young measures we introduce multi scale young measures to deal with problems where multi scale phenomena are relevant we prove some interesting representation results that allow the use of these families of measures in practice and illustrate its applicability by treating from this perspective multi scale convergence and homogenization of multiple integrals

γ convergence of quadratic functionals with oscillating linear perturbations abstract we explore the convergence of quadratic functionals perturbed by linear terms with oscillatory coefficients in particular we are interested in understanding the interaction between the oscillations of the quadratic part and the linear term we also look at this issue from the perspective of the associated equations of optimality as in homogenization

higher order shell elements based on a cosserat model and their use in the topology design of structures for the purpose of carrying out topology optimization of shell structures we show that it is advantageous to use shell elements based on a classical shell theory such as the cosserat theory rather than elements based on the degenerated solid approach since the shell thickness appears explicitly in the formulation thereby greatly simplifying the sensitivity analysis one of the well known shell elements based on the cosserat shell theory is the four node element presented by simo et al comput methods appl mech engrg 72 1989 267 73 1989 53 although one could use this element the use of lower order elements often results in instabilities such as the checkerboard instability in the resulting topologies this provides the motivation for developing higher order shell elements based on a classical shell theory in this work we present the formulation and implementation details for six node and seven node triangular and nine node quadrilateral shell elements which are based on the variational formulation of simo et al and show that good accuracy is obtained even with coarse meshes in fairly demanding problems we also present the topology optimization formulation and some examples of optimal topologies that are obtained using this formulation

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

on γ convergence in divergence free fields through young measures the explicit characterization of the limit energy density of sequences of general non periodic functionals defined on divergence free fields is achieved by means of the div young measure associated with relevant sequences of functions through a minimization problem

a quadratic assumed natural strain curved triangular shell element abstract in this paper a six node triangular c 0 curved shell element is developed by the assumed natural or covariant strain ans method all the sampled natural strains are optimal with respect to the derivative of a prescribed cubic field in a subparametric element numerical examinations reveal that the element shows no sign of locking passes all the patch tests and yields satisfactory accuracy

reiterated ergodic algebras and applications we redefine the homogenization algebras without requiring the separability assumption we show that this enables one to treat more complicated homogenization problems than those solved by the previous theory in particular we exhibit an example of algebra which contrary to the algebra of almost periodic functions induces no homogenization algebra we prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized reynolds type equations and to some nonlinear hyperbolic equations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

refined consistent formulation of a curved triangular finite rotation shell element the present paper considers a finite rotation formulation for curved shell elements with rotations about the element sides as nodal degrees of freedom attention is mainly on the derivation of a consistent finite rotation formulation significant simplifications of the governing equations are presented these simplifications lead to more efficient finite element implementations numerical examples demonstrate the differences between the present consistent and previous approximate formulations

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

an explicit hybrid stabilized 9 node lagrangian shell element abstract this paper presents an explicit hybrid stabilized 9 node lagrangian shell element in the formulation the stress modes pertinent to the stabilization are contravariant in nature and their products with the displacement derived covariant strain are explicit linear functions of the nodal geometric parameters the element is particularly efficient because the leverage matrix or the stabilization vectors can be constructed without resorting to numerical integration or the gram schmidt orthogonalization accuracy of the element is promising and comparable to the state of the art stabilized and hybrid elements

reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media abstract we present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures moreover we study the convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence free fields applying the properties of multiscale young measures associated with sequences of divergence free fields

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

nonlinear thin shell analysis using a curved triangular element abstract this paper describes a simple but robust curved triangular thin shell element the element has twelve degrees of freedom namely the displacement components in the corner nodes and the rotations about the element sides the element is based upon the well known facet element with constant membrane strains and constant stress couples and is augmented to take into account the influence of the changes of curvature on the membrane deformations due to the quadratic character of this influence curved geometries can be modelled in order to permit arbitrarily large rotational increments reference vectors are attached to each element side these reference vectors are updated by means of orthogonal transformations the latter are determined uniquely by the subsequent orientations of the associated element sides the changes of curvature are evaluated using these reference vectors and a corotated reference configuration to describe time independent elastoplasticity a layered shell model is applied while besseling s fraction model is adopted for each layer a number of numerical examples illustrate the performance of the element

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

phenomenological invariant based finite element model for geometrically nonlinear analysis of thin shells abstract a kirchhoff love type curved triangular finite element is proposed for geometrically nonlinear analysis of elastic isotropic shells undergoing small strains but large displacements the finite element formulation is based on the expression of the strain energy in terms of invariants of the strain and curvature change tensors of the shell middle surface the element sides are chosen as three independent directions for determining the strains and curvature changes the emphasis is put on improvement of the bending behavior of the element so that the element is able to undergo finite curvature changes recursive relations are obtained for exactly calculating the coefficients of the first and second order variations of the strain energy of the finite element which are necessary to formulate the equilibrium and stability conditions of the discrete model of a shell a shell finite element with 15 degrees of freedom is developed and tested numerical examples are presented to demonstrate the accuracy and mesh convergence of the finite element solutions

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

hybrid gradient smoothing technique with discrete shear gap method for shell structures abstract in order to enhance the performance of the discrete shear gap technique dsg for shell structures the coupling of hybrid gradient smoothing technique h gst with dsg using triangular elements hs dsg3 is presented to solve the governing partial differential equations of shell structures in the formulation hs dsg3 we firstly employ the node based gradient smoothing technique n gst to obtain the node based smoothed strain field then a scale factor 0 1 is used to reconstruct a new strain field which includes both the strain components from standard dgs3 and the strain components from node based smoothed dsg3 ns dsg3 the hs dsg3 takes advantage of the overly soft ns dsg3 model and the overly stiff dsg3 model and has a relatively appropriate stiffness of the continuous system therefore the degree of the solution accuracy can be improved significantly several typical benchmark numerical examples have been investigated and it is demonstrated that the present hs dsg3 can provide better numerical solutions than the original dsg3 for shell structures

on γ convergence of pairs of dual functionals abstract the paper considers a slightly modified notion of the convergence of convex functionals in uniformly convex banach spaces and establishes that under standard coercitivity and growth conditions the convergence of a sequence of functionals f j to f implies that the corresponding sequence of dual functionals f j converges in an analogous sense to the dual to f functional f

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

the projection method for computing multidimensional absolutely continuous invariant measures we present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewisec2 expanding mappings on a bounded region rn the method is based on the galerkin projection principle for solving an operator equation in a banach space with the help of the modern notion of functions of bounded variation in multidimension we prove the convergence of the algorithm

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

stability in terms of two measures of impulsive integro differential equations via variation of the lyapunov method this paper establishes several criteria on stability in terms of two measures for impulsive integro differential equations with fixed moments of impulse effects by using the generalized variation of parameters technique unified with the lyapunov second method

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by