invariant and attractor sets for fuzzy dynamical systems abstract in this work we establish conditions for the existence and stability of invariant sets for dynamical systems defined on metric space of fuzzy subsets of r n we show that the results presented in this work generalize some conditions of existence and stability of equilibrium points and periodicity of fuzzy differential equations solutions

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

on the stability of fuzzy dynamical systems abstract in this work we study the asymptotic behavior of fuzzy solutions obtained by using zadeh s extension at the deterministic solutions of initial value problems we obtain some result regarding the existence of fuzzy equilibrium points that generalize the already known results as show earlier the membership function of the fuzzy equilibrium points may be obtained in a relatively simple way also we study the projection of the fuzzy solution on the appropriate subspaces of the phase space several examples are given to illustrate the obtained results

a survey of parametrized variational principles and applications to computational mechanics abstract this survey paper describes recent developments in the area of parametrized variational principles pvps and selected applications to finite element computational mechanics a pvp is a variational principle containing free parameters that have no effect on the euler lagrange equations the theory of single field pvps based on gauge functions also known as null lagrangians is a subset of the inverse problem of variational calculus that has limited value on the other hand multifield pvps are more interesting from theoretical and practical standpoints following a tutorial introduction the paper describes the recent construction of multifield pvps in several areas of elasticity and electromagnetics it then discusses three applications to finite element computational mechanics the derivation of high performance finite elements the development of element level error indicators and the construction of finite element templates the paper concludes with an overview of open research areas

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

selective mass scaling for explicit finite element analyses due to their inherent lack of convergence problems explicit finite element techniques are widely used for analysing non linear mechanical processes in many such processes the energy content in the

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

fuzzy differential equations for universal oscillators abstract this study considers universal oscillator fuzzy differential equations the differential inclusion method is applied to this problem by considering the universal oscillator fuzzy differential equations as universal oscillator uncertain dynamical systems three types of damped oscillators are considered comprising underdamped critically damped and overdamped oscillators the existence and uniqueness of the solutions to each case are discussed and proofs are given under some certain conditions for universal oscillator uncertain dynamical systems with an undamped forcing function the existence and uniqueness of the solutions and big solutions are also verified while the scopes of the trajectories to the solutions are obtained based on the inclusion relationship between solutions and big solutions we also prove the existence and uniqueness of solutions to universal oscillator uncertain dynamical systems with a damped forcing function by using other more direct conditions

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics summary the spatial discretisation of a continuum by the fem introduces dispersion errors to the numerical solution of stress wave propagation errors of phase and group velocities and the scatter of wave propagation are induced when these propagating phenomena are modelled by the fem the speed of a single harmonic wave depends on its frequency parasitic effects do not exist in an ideal unbounded continuum with higher order finite elements there are optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefronts in principle the dispersion errors and the spurious oscillations cannot be removed from these solutions but can be suppressed to a certain extent for reliable numerical solutions by the fem the dispersion errors should be analysed and determined the dispersion properties of a plane square biquadratic serendipity finite element are examined and compared with a bilinear one dispersion analysis is carried out for the consistent and lumped mass matrices a dispersive improved lumped mass matrix for biquadratic serendipity elements is proposed the paper closes with the recommendation of a choice of permissible dimensionless wavelengths for bilinear and biquadratic serendipity finite element meshes copyright 2013 john wiley sons ltd

dispersion corrected explicit integration of the wave equation a simple dispersion correction is introduced into the centered difference explicit time integration scheme for wave propagation problems the extra term introduces a change of the wave speed depending on local wave characteristics a rigorous theoretical analysis is provided for the scalar wave equation on a regular mesh and a notable improvement in the ability to conserve wave fronts is demonstrated by two numerical examples the correction is reinterpreted to apply to a typical finite element formulation in terms of mass and stiffness matrices resulting in a general dispersion correction of the explicit centered difference time integration algorithm for acoustic and elastic wave propagation problems

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

some properties of solutions for a class of semi linear uncertain dynamical systems in this paper some properties of solutions are obtained for semi linear fuzzy differential equations understood as differential inclusions i e semi linear uncertain dynamical systems the definitions of the solution to the semi linear uncertain dynamical system and its corresponding big solution are given the existence of the two kinds of solutions is established and the inclusion relation between them is also obtained

direct and sparse construction of consistent inverse mass matrices general variational formulation and application to selective mass scaling summary classical explicit finite element formulations rely on lumped mass matrices a diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector recently non diagonal mass matrices for explicit finite element analysis fea have received attention due to the selective mass scaling sms technique sms allows larger time step sizes without substantial loss of accuracy however an expensive solution for accelerations is required at each time step in the present study this problem is solved by directly constructing the inverse mass matrix first a consistent and sparse inverse mass matrix is built from the modified hamiltons principle with independent displacement and momentum variables usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix rmm secondly a variational mass scaling technique is applied to the rmm it is based on the penalized hamiltons principle with an additional velocity variable and a free parameter using element wise bases for velocity and a local elimination yields variationally scaled rmm thirdly examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed copyright 2014 john wiley sons ltd

novel higher order mass matrices for isogeometric structural vibration analysis abstract a set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using nurbs the proposed method for the construction of higher order mass matrices contains two essential steps firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed this reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix the desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices a generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices a series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis

variationally consistent inertia templates for b spline and nurbs based fem inertia scaling and customization abstract in this contribution variationally consistent inertia templates for b spline and nurbs based finite elements are proposed as a unified concept for two different purposes customization of the template allows construction of masses and reciprocal masses with desired properties like higher order accuracy or improved dispersion behavior inertia scaling allows substantial speed up for explicit dynamics by increased critical time steps the derivation of the template is based on a three field parametrized functional as in previous works of the authors group but with modified primary variables namely displacement velocity and mass specific linear momentum the latter allows for mass preservation for non constant density throughout the domain and is therefore an enhancement to the formulation proposed in tkachuk and bischoff 2015 with the focus on b spline and nurbs based finite elements the proposed template provides alternatives to the row sum lumped mass matrix which is only 2nd order accurate independent of the polynomial order earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances furthermore higher order reciprocal masses can be constructed from the template they are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems for non linear problems only small overhead is expected but this paper focuses on linear problems only and mainly undistorted meshes tuning the method towards inertia scaling a reduction of the maximum eigenfrequency by 25 40 is obtained in the examples herein whereas the accuracy is higher than for lumped or consistent mass

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

periodic orbits for fuzzy flows abstract periodic solutions are present in many of the various mathematical models that describe physical chemical or biological phenomena in this work we investigate the existence of periodic solutions for fuzzy initial value problem we will show that fuzzy solutions can present periodic points and we will develop tools of qualitative analysis for such solutions some models are presented to illustrate the results obtained in this paper

lumped mass finite element implementation of continuum theories with micro inertia summary continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro structured materials such terms often called micro inertia are convenient and straightforward extensions of classical continuum theories furthermore the critical time step is usually increased via the inclusion of micro inertia however the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro inertia terms in this paper we will develop a solution algorithm based on neumann expansions by which this disadvantage is avoided altogether the micro inertia terms are translated into modifications of the residual force vector so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained the numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail numerical examples are used to illustrate the various aspects of the algorithm copyright 2013 john wiley sons ltd