direct calculation of buckling strength of imperfect structures abstract this paper is concerned with the direct calculation of buckling strength of an imperfect structure the main assumptions are that for the perfect structure there is a symmetric or a linear fundamental path and the corresponding first critical point is a simple symmetry breaking or a simple bifurcation point an extended system of limit points is proposed for which the newly introduced scaling parameters are regular solution and thus standard methods can be used to compute them using the extended system one can directly obtain the exact buckling loads without tracing the postbuckling paths an efficient implementation of newton s method solving the extended system is presented and numerical examples are given

direct computation of lower bound dynamic buckling loads of imperfection sensitive systems the computation of lower bound dynamic buckling loads of imperfection sensitive systems is examined under step load of ine nite duration an extended system based on the energy criterion for establishing the lower bound dynamic buckling loads without solving the highly nonlinear initial value problems is proposed the newly introduced scaling parameters are nonsingular solutions to the extended system thus standard methods can be used to compute them using the extended system one can directly obtain the lower bound dynamic buckling loads without tracing the postbuckling equilibrium paths an efe cient implementation of newton s method for solving the extended system is presented and numerical examples are given

on approximate type systems generated by l relations the aim of this work is to study approximate type systems induced by l relations in the framework of the general theory of m approximate systems introduced in 42 and its generalizations special attention is payed to the structural properties of lattices of such systems and to the study of connections between categories of such systems and the corresponding categories of sets endowed with l relations

maximum loads of imperfect systems corresponding to stable bifurcation abstract a simple and computationally inexpensive approach is presented for obtaining the maximum load factor of an elastic structure considering reduction of load carrying capacity due to inevitable initial imperfections the structure has a stable bifurcation point if no initial imperfection exists an antioptimization problem is formulated for minimizing the maximum loads reduced by the most sensitive imperfection within the convex bounds on the imperfections of nodal locations and nodal loads the maximum loads may be defined by bifurcation points or deformation constraints a problem of simultaneous analysis and design with energy method is formulated to avoid laborious nonlinear path following analysis the stable bifurcation point is located by minimizing the load factor under constraint on the lowest eigenvalue of the stability matrix it is shown in the examples that a minor imperfection that is usually dismissed is very important in evaluating the maximum load of a flexible structure

postbuckling behavior and imperfection sensitivity of elastic structures by the lyapunov schmidt koiter approach beginning with the work of koiter in 1945 valuable insights into the postbuckling behavior of structures have been gained by lyapunov schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point here this methodology is generalized to include nonlinear prebuckling behavior as well as multiple not necessarily coincident buckling modes the expansion of the reduced equilibrium equations is performed about a reference point which need not coincide with any of the bifurcation points and applies no matter whether the modes are coincident closely spaced or well separated from a variety of possible decompositions of the admissible space of displacements two are incorporated into a finite element program theoretical considerations and numerical examples in which asymptotic results are compared to exact results indicate that one of the decompositions has some important advantages over the other examples include a shallow arch and a beam on elastic foundation problem exhibiting symmetry breaking modal interaction

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

iterative substructuring methods for spectral element discretizations of elliptic systems ii mixed methods for linear elasticity and stokes flow iterative substructuring methods are introduced and analyzed for saddle point problems with a penalty term two examples of saddle point problems are considered the mixed formulation of the linear elasticity system and the generalized stokes system in three dimensions these problems are discretized with mixed spectral element methods the resulting stiffness matrices are symmetric and indefinite the interior unknowns of each element are first implicitly eliminated by using exact local solvers the resulting saddle point schur complement is solved with a krylov space method with block preconditioners the velocity block can be approximated by a domain decomposition method e g of wire basket type which is constructed from a local solver for each face of the elements and a coarse solver related to the wire basket of the elements the condition number of the preconditioned operator is independent of the number of spectral elements and is bounded from above by the product of the square of the logarithm of the spectral degree and the inverse of the discrete inf sup constant of the problem

random initial imperfections of structures abstract this paper is a theoretical study on random initial imperfections of structures the explicit form of probability density function of the load bearing capacity critical load of structures is derived for random initial imperfections based on a decomposition of the space of imperfection vectors into two orthogonal subspaces the subspace that asymptotically affects the load bearing capacity and the other that does not tight bounds on the range of load bearing capacity are presented for various types of simple critical points by means of the asymptotic theory of statistics we show the inefficiency of a conventional random method that approximates the minimum loadhearing capacity by the minimum load for a number of random initial imperfections the theoretical and empirical probability distribution functions for simple truss structures are compared to show the validity and effectiveness of the present method

on random imperfections for structures of regular polygonal symmetry a theoretical study is made on the effect of random initial imperfections on the load bearing capacities of structures of regular polygonal symmetry the group theoretic method is employed to implement the symmetry in the formulation in view of the concept of critical initial imperfection the explicit form of probability density function of the load bearing capacity of structures is derived for random initial imperfections in particular tight bounds on the range of load bearing capacity are presented for various types of simple and double critical points the theoretical and empirical probability distribution functions are compared for simple truss structures to show the validity and effectiveness of the present method

on stability and the worst imperfection shape in solids with nearly simultaneous eigenmodes abstract of interest here is the influence of imperfections on the stability of elastic systems discrete or continuous with nearly simultaneous modes the majority of such structures show an extreme sensitivity of their first instability load to the shape of these small amplitude but unavoidable imperfections the determination of the worst such possible shape is thus a very important issue from the design standpoint following a general analysis of the stability of arbitrary elastic systems with nearly simultaneous bifurcation eigenmodes in the presence of imperfections conditions are given for the determination of the worst imperfection shape which minimizes the first local load maximum for the case of coincident eigenmodes the answer to the worst shape problem is considerably simplified for it is determined from the equilibrium branch of the perfect structure on which the load drops more rapidly

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

twin characteristic parameter solution for dynamic buckling of columns under elastic compression wave abstract two critical conditions for the dynamic buckling of columns are derived on the basis of the consideration of energy transformation and conservation at the instant when the buckling occurs the first critical condition is that the amount of released compressive deformation energy must be equal to the sum of buckling deformation energy and buckling kinetic energy in the instant course of the dynamic buckling the second is that the rate of energy transformation meets the conservation law in the instant course the governing equations the boundary conditions and the continuity conditions derived by use of the first condition are the same as those obtained by use of hamilton s theorem these equations and conditions are insufficient for the determination of the critical load parameter and the exponent of transverse inertia term involved in the problem a supplementary restraint equation at compression wave front is derived by use of the second condition two characteristic equations for the two parameters are derived by use of the solution of the governing equations and the above mentioned restraint conditions the two characteristic parameters and the corresponding buckling modes are calculated accurately from the solution of the characteristic equations the simple formulas for the relation of the critical force with the buckling time are given the theoretical results predicted by use of the formula are in reasonable agreement with the existent experiment results

a perturbation method for the determination of the buckling strength of imperfection sensitive structures a perturbation method for determining the buckling strength of imperfection sensitive structures is presented the method is based on an extended system which was first used by seydel in computation of limit points of nonlinear systems the effects of various possible sources of structural imperfections on perfect structures with simple buckling loads are investigated by the method the derivations and results of the present exposition are equally applicable to discrete and continuous systems in the finite element implementation of the new formulation the perturbation expansions are determined by solving numerically several linear problems with a single positive definite sub stiffness matrix the method is illustrated with its application to the analyses of imperfection sensitivity of a simply supported column on a linear elastic foundation subjected to axial compression

rough fuzzy approximations on two universes of discourse in rough set theory the lower and upper approximation operators can be constructed via a variety of approaches various fuzzy generalizations of rough approximation operators have been made over the years this paper presents a framework for the study of rough fuzzy sets on two universes of discourse by means of a binary relation between two universes of discourse a covering and three relations are induced to a single universe of discourse based on the induced notions four pairs of rough fuzzy approximation operators are proposed these models guarantee that the approximating sets and the approximated sets are on the same universes of discourse furthermore the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized and some interesting properties are investigated finally the connections of these approximation operators are made and conditions under which some of these approximation operators are equivalent are obtained

singular perturbations for sensitivity analysis in symmetric bifurcation buckling a direct procedure for the evaluation of imperfection sensitivity in bifurcation problems is presented the problems arise in the context of the general theory of elastic stability koiter s theory for discrete structural systems in which the total potential energy is employed together with a stability criterion based on energy derivatives the imperfection sensitivity of critical states such as bifurcations and trifurcations is usually represented as a plot of the critical load versus the amplitude of the imperfection considered however such plots have a singularity at the point with 0 so that a regular perturbation expansion of the solution is not possible in this work we describe a direct procedure to obtain the sensitivity of the critical load eigenvalue of the bifurcation problem and the sensitivity of the critical direction eigenvector of the bifurcation problem using singular perturbation analysis the perturbation expansions are constructed as a power series in terms of the imperfection amplitude in which the exponents and the coe cients are the unknowns of the problem the solution of the exponents is obtained by means of trial and error using a least degenerate criterion or by geometrical considerations to compute the coe cients a detailed formulation is presented which employs the conditions of equilibrium and stability at the critical state and their contracted forms the formulation is applied to symmetric bifurcations and the coe cients are solved up to third order terms in the expansion the algorithm is illustrated by means of a simple example a beam on an elastic foundation under axial load for which the coe cients are computed and the imperfection sensitivity is plotted copyright 2001 john wiley sons ltd

optimization of imperfection sensitive symmetric systems for specified maximum load factor a method is presented for optimum design of a symmetric structure which reaches an unstable bifurcation point as the load factor is increased reduction of the maximum load level due to the antisymmetric imperfection is considered and a straightforward algorithm is proposed for calculating the magnitude of reduction of the load factor corresponding to the most critical mode of antisymmetric imperfection the sensitivity coefficients of the bifurcation load factor are calculated by using the interpolation method developed by the authors and an approximate formulation is presented for sensitivity analysis of the maximum load factor it is shown in the examples that the errors in the sensitivity coefficients do not lead to any significant difference in the optimum design

post critical analysis of structures with a nonlinear pre buckling state in the presence of imperfections in the context of the geometrically nonlinear analysis of structures by continuation methods a technique for the determination of the imperfection sensitivity is carried out the technique is based on galerkin s reduction of equilibrium equations in the subspace of extrapolation branches emanating from the bifurcation points the pre and post critical behaviour in the neighbourhood of bifurcation points however is well approximated provided that extrapolations have a large convergence domain to this end an efficient asymptotic representation of the primary and secondary equilibrium paths and procedures that overcome the ill conditioning of the systems defined in the critical points is used here

the use of pointwise interpolation in domain decomposition methods with nonnested meshes in this paper the author develops a new technique and a corresponding theory for schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations the theory provides an optimal convergence of an additive schwarz algorithm that is constructed with a nonnested coarse space and a not necessarily shape regular subdomain partitioning the theory is also applicable to the graph partitioning algorithms recently developed by cai and saad overlapping domain decomposition algorithms for general sparse matrices preprint 93 027 univ of minnesota and farhat and lesoine internat j numer methods engrg 36 1993 pp 745 764 for problems defined on unstructured meshes

stability boundaries of two parameter non linear elastic structures abstract the load bearing capacity of structures can be influenced by variations in parameters such as initial geometric defects multi parameter loadings material specifications and temperature this paper aims to introduce a new formulation to trace the stability boundaries of two parameter elastic structures the proposed procedure can find a set of critical points both limit and bifurcation ones via a modified newton s method in the authors formulation the residual force is set to zero and a critically constraint is satisfied simultaneously numerical examples presented in this paper demonstrate the efficiency of the suggested method

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

static and dynamic local and global bifurcations in nonlinear autonomous structural systems discrete damped or undamped gradient systems described by nonlinear autonomous ordinary differential equations ode of motion are examined in detail emphasis is given to the relationship between static with possibly existing dynamic bifurcations criteria for dynamic bifurcations and stability of precritical critical and postcritical states associated with the nature of jacobian eigenvalues are also presented cases of discrepancies between local and global dynamic analysis regarding the stability of equilibriums are reported for the first time finally using a simple energy criterion exact dynamic buckling loads for vanishing but nonzero damping are readily obtained

statistics of normally distributed initial imperfections abstract this paper offers a theoretical study on the probabilistic nature of critical loads buckling loads of structures subject to normally distributed initial imperfections explicit form of probability density function of critical loads are derived for various types of critical points double bifurcation points of structures with regular polygonal symmetry are dealt with by means of the group theoretic bifurcation theory the distribution of minimum values of the critical loads is investigated to present a statistical design index the theoretical and empirical probability density functions for simple structures are compared to show the validity and effectiveness of this method the method is quite efficient when it is directly applicable otherwise the explicit forms at least can greatly supplement the inefficiency of the conventional random method

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

towards the theory of m approximate systems fundamentals and examples the concept of an m approximate system is introduced basic properties of the category of m approximate systems and in a natural way defined morphisms between them are studied it is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of m approximate systems

energy based dynamic buckling estimates for autonomous dissipative systems the dynamic buckling global response of a nonlinear 3 degree of freedom dissipative model under a step loading of infinite duration is thoroughly discussed geometrically imperfect models with symmetric or antisymmetric imperfections losing their static stability through a limit point and an asymmetric bifurcation point respectively are considered emphasis is given to the combined effect of nonlinearities geometric and or material and damping exact approximate and lower upper bound estimates based on energy criteria for establishing the dynamic buckling response of such autonomous models without solving the highly nonlinear initial value problem are assessed the reliability and efficiency of the proposed readily obtained estimates is illustrated via numerical simulation the accuracy of which is checked using energy balance considerations certain interesting byproducts associated with a postlimit point bifurcation and breakdown of the symmetry of deformation are also revealed

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

the fast adaptive composite grid fac method for elliptic equation the fast adaptive composite grid fag method is a systematic process for solving differential boundary value problems fac uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution it can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation this paper develops both the theoretical and practical aspects of fac as it applies to elliptic problems 1 introduction the need for local resolution in physical models occurs frequently in practice special local features of the forcing function operator coefficients boundary and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution it is important that the discretization and solution processes account for this locally that is that the local phenomena do not precipitate a dramatic increase in the overall computation unfortunately this objective of efficiently adapting to local features is often in conflict with the solution process equation solvers can degrade or even fail to apply in the presence of varying discretization scales data structures that account for irregular grids can be cumbersome the computer architecture may not be able to effectively account for grid irregularity e g vectorizability may be inhibited etc in fact even the discretization process itself may find difficulty with this objective for finite differences it is problematic to develop accurate difference formulae for irregular grids for finite elements this objective is reflected in the substantial overhead costs needed to automate the discretization the fast adaptive composite grid method fac 11 is a discretization and solution method designed to achieve efficient local resolution by constructing the

asymptotic post buckling fem analysis using corotational formulation finite element asymptotic post buckling analysis being based on fourth order expansions of the strain energy requires that nonlinear structural modeling be accurate to same order at least with respect to the rigid motions of the elements a corotational description is proposed here as a general tool to satisfy this requirement of objectivity by referring each element to a local frame which moves rotates with the element so filtering its rigid motion in this description nonlinearity of the problem derives essentially from the change of reference from the global fixed frame to the local one the strain energy being governed by their relative rotations in finite kinematics this noticeably complicates the algebra for obtaining exact expressions of its variations quite simple basic expressions for the first four corotational derivatives of the strain energy are provided allowing the set up of a fourth order accurate asymptotic analysis starting from standard finite elements based on linear or simplified nonlinear local modelings the formulation is implemented for the analysis of 3d beam assemblages and several numerical results are presented and discussed showing the effectiveness and robustness of the proposed approach in reproducing the nonlinear equilibrium path in both cases of monomodal and coupled multimodal buckling

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

rough approximations on a complete completely distributive lattice with applications to generalized rough sets the generalizations of rough sets considered with respect to similarity relation covers and fuzzy relations are main research topics of rough set theory however these generalizations have shown less connection among each other and have not been brought into a unified framework which has limited the in depth research and application of rough set theory in this paper the complete completely distributive ccd lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed these definitions result from the concept of cover introduced on a ccd lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers when t similarity relation is considered the existing fuzzy rough sets are the special cases of our proposed approximations on a ccd lattice thus these generalizations of rough sets are brought into a unified framework and a wider mathematical foundation for rough set theory is established

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

non overlapping domain decomposition methods for adaptive hp approximations of the stokes problem with discontinuous pressure fields abstract this paper proposes a framework for non overlapping domain decomposition methods applied to viscous incompressible flows governed by stokes equations modelled by the hp version of the finite element method preconditioning methods developed earlier for elliptic problems are extended to this case suitable condition number bounds are obtained for both the hierarchical hp preconditioners of the type proposed in 9 and neumann neumann preconditioners proposed in 10 systematic procedures for constructing compatible hp approximations for incompressible flow are also described

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

finite element asymptotic analysis of slender elastic structures a simple approach an asymptotic method directly derived from koiter s theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here the order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features theoretical definitions eigenproblem numerical techniques suitable fem implementation which make such an approach competitive with respect to the classic step by step methods for plane frames and 3d pin jointed trusses the performances of the algorithm numerical accuracy and computational cost are compared with those of riks are length method

design sensitivity analysis and optimization for nonlinear buckling of finite dimensional elastic conservative structures the purpose of this review paper is to summarize the existing methods of design sensitivity analysis and optimization of elastic conservative finite dimensional systems with respect to nonlinear buckling behavior difficulties related to geometrical nonlinear singular behaviors are discussed in detail characteristics of optimized structures are demonstrated in reference to snapthrough behavior hill top branching and degenerate critical points a new optimization result of a flexible truss that fully utilizes the snapthrough behavior is also presented

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

an asymptotic lower bound dynamic buckling estimate for imperfect systems under impact loading a regular perturbation procedure is presented for the determination of lower bound dynamic buckling loads of discrete dissipative or nondissipative imperfect structural systems under impact loading the procedure is based on the energy criterion for establishing the lower bound dynamic buckling estimate of autonomous systems without solving a set of differential equations of motion numerically presented by kounadis and his associates attention is focused on the effect of imperfections on the lower bound estimate as an expansion in terms of fractional powers of imperfection amplitude the lower bound dynamic load is constructed by using an inverse expansion of the aforementioned procedure application of the procedure is demonstrated by two examples

chaotic behavior resulting in transient and steady state instabilities of pressure loaded shallow spherical shells abstract in this paper the axisymmetric dynamic behavior and snap through buckling of thin elastic shallow spherical shells under harmonic excitation is investigated based on marguerre kinematical assumptions the governing partial differential equations of motion for a pre loaded cap are presented in the form of a compatibility equation and a transverse motion equation the continuous model is reduced to a finite degree of freedom system using the galerkin method and a fourier bessel approach results show that pre loaded shells may exhibit co existing stable equilibrium states and that with the application of sufficiently large dynamic loads the structure escapes from the well corresponding to pre buckling configurations to another this escape load may be much lower than the corresponding quasi static buckling load indeed complex resonances can occur until the system snaps through often signalling the loss of stability as parameters are slowly varied steady state instabilities may occur these can include jumps to resonance subharmonic period doubling bifurcations cascades to chaos etc moreover a sudden pulse of excitation may lead to a transient failure of the system in this paper we examine how spherical caps under harmonic loading may be assessed in an engineering context with a view to design against steady state instabilities as well as the various modes of transient failure steady state and transient stability boundaries are presented in which special attention is devoted to the determination of the critical load conditions from this theoretical analysis dynamic buckling criteria can be properly established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading

postbuckling analysis using a general purpose code a new capability for solving postbuckling problems in shell structures is described the matrix theory to adapt newton s method to nonlinear finite element shell analysis is outlined first the matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular the matrix theory suggests a change of variables as part of the usual iterative procedure in newton s method the change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures numerical results from a new option that has been programmed in an existing general purpose code are presented the analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples

nonlinear dynamic buckling of discrete structural systems under impact loading abstract nonlinear dynamic buckling of discrete dissipative or nondissipative nonlinearly elastic structural systems geometrically perfect or imperfect under impact loading is thoroughly examined applying the law of impulse momentum one can determine analytically the initial conditions valid immediately after impact thereafter the response of the system is governed by a set of autonomous highly nonlinear ordinary differential equations using an efficient and qualitative analysis exact and lower upper bound dynamic buckling estimates very useful for structural design purposes are established without integrating the aforementioned nonlinear initial value problem the analysis is supplemented by a variety of numerical results of a two degrees of freedom dissipative model from these results it is clearly shown how the point attractor response of the system is changed after a sudden jump to dynamic buckling occurring through a global dynamic bifurcation

a multiscale darcy brinkman model for fluid flow in fractured porous media the aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures the darcy s law is assumed in the porous domain while the stokes brinkman equations are considered in the fractures we address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain and describe the fracture as if it were an interface between porous regions we derive the corresponding interface model governing the fluid flow in the fracture and in the porous media and establish the well posedness of the coupled problem further we introduce a finite element scheme for the approximation of the coupled problem and discuss solution strategies we conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non reduced one

quasi optimal schwarz methods for the conforming spectral element discretization the spectral element method is used to discretize self adjoint elliptic equations in three dimensional domains the domain is decomposed into hexahedral elements and in each of the elements the discretization space is the set of polynomials of degree n in each variable a conforming galerkin formulation is used the corresponding integrals are computed approximately with gauss lobatto legendre gll quadrature rules of order n and a lagrange interpolation basis associated with the gll nodes is used fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method the conforming finite element space on the gll mesh consisting of piecewise q1 or p1 functions produces a stiffness matrix kh that is known to be spectrally equivalent to the spectral element stiffness matrix kn kh is replaced by a preconditioner tilde k h which is well adapted to parallel computer architectures the preconditioned operator is then tilde k h 1 k n techniques for nonregular meshes are developed which make it possible to estimate the condition number of tilde k h 1 k n where tilde k h is a standard finite element preconditioner of kh based on the gll mesh two finite element based preconditioners the wirebasket method of smith and the overlapping schwarz algorithm for the spectral element method are given as examples of the use of these tools numerical experiments performed by pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions

dynamic stability criteria of nonlinear elastic damped undamped systems under step loading the nonlinear dynamic stability of simple nonlinearly elastic systems with or without viscous damping under step loading of infinite or finite duration is discussed the analysis refers to discrete structural systems with one or two degrees of freedom which under the action of the same loading applied statically lose their stability by snapping considering the stability of motion in the large in the sense of lagrange conditions for a bounded or an unbounded motion and thereby dynamic buckling criteria are properly established for one degree of freedom systems it is found that regardless of the length of time of loading application as well as regardless of whether or not damping is included the motion ceases to be bounded when the phase point velocity vanishes using this criterion exact or approximate lower bound dynamic buckling loads for damped or undamped systems under step loading of infinite or finite duration can be obtained without solving the strongly nonlinear differential equation of motion

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

schwarz preconditioners for the spectral element discretization of the steady stokes and navier stokes equations the q n q n 2 spectral element discretization of the stokes equation gives rise to an ill conditioned indefinite symmetric linear system for the velocity and pressure degrees of freedom we propose a domain decomposition method which involves the solution of a low order global and several local problems related to the vertices edges and interiors of the subdomains the original system is reduced to a symmetric equation for the velocity which can be solved with the conjugate gradient method we prove that the condition number of the iteration operator is bounded from above by c 1 log n 3 beta n where c is a positive constant independent of the degree n and the number of subdomains and beta n is the inf sup condition of the pair q n q n 2 we also consider the stationary navier stokes equations in each newton step a non symmetric indefinite problem is solved using a schwarz preconditioner by using an especially designed low order global space and the solution of local problems analogous to those decribed above for the stokes equation we are able to present a complete theory for the method we prove that the number of iterations of the gmres method at each newton step is bounded from above by c 1 log n 3 beta n the constant c does not depend on the number of subdomains or n and it does not deteriorate as the newton iteration proceeds

mixed hp finite element methods for problems in elasticity and stokes flow we consider the mixed formulation for the elasticity problem and the limiting stokes problem in bbb r d d 2 3 we derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and subject to a maximum loss of o k frac d 1 2 with respect to the polynomial degree k we obtain asymptotic rates of convergence that are optimal up to o k epsilon in the displacement velocity and up to o k frac d 1 2 epsilon in the pressure with epsilon 0 arbitrary both rates being optimal with respect to h several choices of elements are discussed with reference to properties desirable in the context of the hp version

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

a nonconforming multigrid method for the stationary stokes equations an optimal order w cycle multigrid method for solving the stationary stokes equations is developed using p1 nonconforming divergence free finite elements

nonlinear analysis based on homogeneous incremental systems abstract the performance of path following techniques depends on so called constraint equations which are used in the augmented incremental systems it is common practice in the literature that these equations are explicitly defined following the arbitrary geometrical or physical ideas in the present paper their role is discussed in order to establish a reliable method of augmentation it is shown that the explicitly defined constraint equation is not necessary the implicit formulation is proposed which is based on the rank analysis performed on the rectangular matrix of the homogeneous incremental system the proposed method allows to solve problems where the well known continuation schemes fail the theoretical considerations are illustrated by the numerical examples

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

a dual primal feti method for incompressible stokes equations in this paper a dual primal feti method is developed for incompressible stokes equations approximated by mixed finite elements with discontinuous pressures the domain of the problem is decomposed into nonoverlapping subdomains and the continuity of the velocity across the subdomain interface is enforced by introducing lagrange multipliers by a schur complement procedure the solution of an indefinite stokes problem is reduced to solving a symmetric positive definite problem for the dual variables i e the lagrange multipliers this dual problem is solved by the conjugate gradient method with a dirichlet preconditioner in each iteration step both subdomain problems and a coarse level problem are solved by a direct method it is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains numerical experiments demonstrate the scalability of this new method

rough sets based on complete completely distributive lattice in this paper a pair of rough approximation operators on a complete completely distributive ccd lattice based on an ordinary binary relation is defined this kind of rough sets can be seen as a unified framework for the study of rough sets based on ordinary binary relations rough fuzzy sets and interval valued rough fuzzy set moreover depending on classes of binary relations this paper defines several classes of rough sets on ccd lattices and investigates properties of these classes finally two generalized rough set models on two ccd lattices are given at the end of this paper

rough operations on boolean algebras in this paper we introduce two pairs of rough operations on boolean algebras first we define a pair of rough approximations based on a partition of the unity of a boolean algebra we then propose a pair of generalized rough approximations on boolean algebras after defining a basic assignment function between two different boolean algebras finally some discussions on the relationship between rough operations and some uncertainty measures are given to provide a better understanding of both rough operations and uncertainty measures on boolean algebras

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

asymptotic post buckling analysis of rectangular plates by hc finite elements the present paper extends the finite element perturbation approach already presented for pin jointed and framed structures15 to rectangular thin plates koiter s asymptotic strategy2 is coupled with a high continuity finite element discretization of the plate 22 the consistency of the discrete model is discussed from the kinematical and numerical points of view and several numerical tests are reported it appears that use of the hc elements makes the perturbation algorithm insensitive to the locking phenomenon occurring in the evaluation of the postbuckling behaviour it also allows the use of very fine discretization meshes at low computational cost

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems spectral collocation approximations based on legendre gauss lobatto lgl points for helmholtz equations as well as for the linear elasticity system in rectangular domains are studied the collocati

critical initial imperfection of structures abstract a method is introduced for determining the critical initial imperfection of discretized structures that decreases the load bearing capacity most rapidly the effects of imperfections on simple critical points such as limit points of loads and simple bifurcation points are theoretically investigated based on the idea of the lyapunov schmidt decomposition developed in bifurcation theory imperfection sensitivity varies with the types of points nonetheless critical imperfection pattern is expressed in the same formula regardless of the types among various imperfections the most influential can be found in a quantitative manner the validity and the usability of the proposed method are illustrated through its application to simple example structures

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part ii convergence theory in this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric indefinite linear systems arising from mixed finite element discretizations of elliptic problems the algorithms include schwarz methods and iterative refinement methods on locally refined grids the implementation of schwarz and iterative refinement algorithms have been discussed in part i a discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived

axiomatic approaches to rough approximation operators on complete completely distributive lattices we proposed a pair of rough approximation operators on a complete completely distributive lattice ccd lattice for short in 2015 in this paper we further discuss its properties and study the axiomatic approaches to the rough approximation operators through these axioms fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a ccd lattice we also discuss the axiomatic approaches to generalized rough sets on ccd lattices

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

point attractors and dynamic buckling of autonomous systems under step loading abstract in this study the dynamic response of autonomous mainly dissipative multi d o f systems under step loading is re examined based on the geometrical point of view of the theory of non linear dynamical systems and the rapidly developing theory of attractors the investigation focuses on limit point like systems with snapping as their salient feature it is found that dynamic buckling through a saddle or its neighborhood although leading to a large amplitude motion may be associated with a point attractor response on the pre buckling fixed point depending on the amount of damping considered in close conjunction with the motion channel geometry and the total potential characteristics of all stable and complementary equilibria for such systems only a straightforward fully non linear dynamic analysis can provide valid information on the global dynamic stability since the shape of the total potential hypersurface may become very complicated rendering energy aspects practically not applicable a 2 d o f model simulating an asymmetric suspended roof is comprehensively analyzed to capture the above findings and a parametric investigation is carried out revealing a variety of new dynamic response types and leading to a more accurate insight of the stability of motion in the large

elastic postbuckling analysis via finite element and perturbation techniques part 1 formulation the main equations for the equilibrium stability and critical state analysis of discrete elastic systems are presented following the works of thompson but in such a way that the original set of generalized coordinates and loads are preserved in the total potential energy this introduces differences in the resulting equations in bifurcation analysis but does not introduce any new feature regarding the physics of the problem the new formulation is approximated by means of a standard finite element approach based on interpolation of displacements in which the derivatives of the potential energy are approximated the terms retained are those of moderately large rotation theory the energy analysis is finally related to the more conventional finite element notation in terms of stiffness matrices and it is shown how in such a way it can be included in present day codes part 2 of the paper deals with applications to the analysis of shells of revolution using a semi analytical approximation two cases are presented in detail bifurcation in axisymmetric and in asymmetric modes and the results show good correlation with those of other authors the influence of load and geometric imperfections is evaluated