convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction diffusion problems the numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems the quasi optimal order error estimates are proved in the e weighted h 1 norm valid uniformly up to a logarithmic factor in the singular perturbation parameter by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

galerkin projected residual method applied to diffusion reaction problems a stabilized finite element method is presented for scalar and linear second order boundary value problems the method is obtained by adding to the galerkin formulation multiple projections of the residual of the differential equation at element level these multiple projections allow the generation of appropriate number of free stabilization parameters in the element matrix depending on the local space of approximation and on the differential operator the free parameters can be determined imposing some convergence and or stability criteria or by postulating the element matrix with the desired stability properties the element matrix of most stabilized methods such as gls and ggls methods can be obtained using this new method with appropriate choices of the stabilization parameters we applied this formulation to diffusion reaction problems optimal rates of convergency are numerically observed for regular solutions

the hp finite element method for problems in mechanics with boundary layers abstract we consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics such as plate and shell problems we present guidelines for the effective resolution of such layers in the context of existing commercial p and hp finite element fe version codes we show that if high order spectral elements are available then just two elements are sufficient to approximate these layers at a near exponential rate independently of the problem parameters thickness or reynolds number we present hp mesh design principles for situations where both corner singularities and boundary layers are present

quasioptimal uniformly convergent finite element methods for the elliptic boundary layer problem we consider the bilinear finite element method for the singularly perturbed elliptic boundary value problem in two dimensions we construct our method on a shishkin mesh through the boundary layer correction technique and elaborate barrier functions the method is shown to be convergent uniformly in the perturbation parameter of order n 2 ln2 n in the l2 norm numerical results confirm our theoretical analysis

amenable groups and stabilizers of measures on the boundary of a hadamard manifold let x be a symmetric space of noncompact type or more generally a hadamard manifold i e a complete simply connected riemannian manifold of nonpositive sectional curvature we consider a finite positive borel measure on the ideal boundary x see sect 2 for the definitions let g be the isometry group of x and g the subgroup which stabilizes the measure in the case of a symmetric space we obtain the following result

a finite difference method on layer adapted meshes for an elliptic reaction diffusion system in two dimensions an elliptic system of m 2 singularly perturbed linear reaction diffusion equations coupled through their zero order terms is considered on the unit square this system does not in general satisfy a maximum principle it is solved numerically using a standard difference scheme on tensor product bakhvalov and shishkin meshes an error analysis for these numerical methods shows that one obtains nodal o n 2 convergence on the bakhvalov mesh and o n 2 ln 2 n convergence on the shishkin mesh where n mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter the analysis is much simpler than previous analyses of similar problems even in the case of a single reaction diffusion equation as it does not require the construction of an elaborate decomposition of the solution numerical results are presented to confirm our theoretical error estimates

comparison of dynamic response of structures with uncertain but bounded parameters using non probabilistic interval analysis method and probabilistic approach abstract now by combining the finite element analysis and interval mathematics a new non probabilistic set theoretical models that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures in which bounds on the magnitude of uncertain parameters are only required not necessarily knowing the probabilistic distribution densities instead of conventional optimization studies where the minimum possible response is sought here an uncertainty modeling is developed as an anti optimization problem of finding the least favorable response and the most favorable response under the constraints within the set theoretical description in this study we presented the non probabilistic interval analysis method for the dynamical response of structures with uncertain but bounded parameters under the condition of the interval vector which contains the uncertain but bounded parameters determined from probabilistic statistical information or stochastic sample test through comparing between the non probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples we can see that the region of the dynamical response of structures with uncertain but bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach in other words the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method this kind of results is coincident with the meaning of the probabilistic theory and interval mathematics

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

global pointwise error estimates for uniformly convergent finite element methods for the elliptic boundary layer problem this paper continues our discussion for the anisotropic model problem e2 2u x2 2 y2 a x y u f x y in 1 there we constructed a bilinear finite element method on a shishkin type mesh the method was shown to be convergent independent of the small parameter in the order of n 2ln2n in the l2 norm where n2 is the total number of mesh points in this paper the method is shown to be convergent independent of in the order of n 2ln3 n in the l norm in the whole computational domain which explains the uniform convergence phenomena we found in the numerical results in 1 another numerical experiment is presented here which confirms our theoretical analysis published by elsevier science ltd

a posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes abstract based on conservative flux reconstruction we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes without the assumption that the meshes are shape regular our estimates give the upper bounds on the error only containing the alignment measure factor and therefore could provide actual numerical bounds if the alignment measure was approximated very well two resulting error estimators are presented to be equivalent to each other up to a data oscillation one of which can be directly constructed without solving local neumann problems and provide computable error bound numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure

combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters abstract this paper is devoted to a fundamental problem of accounting for parameter uncertainties in random vibrations of structures in contrast to the overwhelming majority of random vibration studies where perfect knowledge is assumed for the parameters of the excitation this crucial conjecture is dispensed with the probabilistic characteristics of the excitation are assumed to be given as depending on some parameters which are not known in advance we postulate that some imprecise knowledge is available namely that these parameters belong to a bounded convex set in the case where this convex set is represented by an ellipsoid closed form solutions are given for the upper and lower bounds of the mean square displacement of the structures for the first time in the literature the system uncertainty in the random vibrations is dealt with as an anti optimization problem of finding the least favorable values of the mean square response the approach developed here opens a new avenue for tackling parameter uncertainty which is often encountered in various branches of engineering

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

uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems convection diffusion type in this paper we consider the standard bilinear finite element method fem and the corresponding streamline diffusion fem for the singularly perturbed elliptic boundary value problem e 2u x2 2u y2 b x y u a x y u f x y in the two space dimensions by using the asymptotic expansion method of vishik and lyusternik 36 and the technique we used in 21 22 we prove that the standard bilinear fem on a shishkin type mesh achieves first order uniform convergence rate globally in l2 norm for both the ordinary exponential boundary layer case and the parabolic boundary layer case extensive numerical results are carried out for both cases the results show that our methods perform much better than either the classical standard or streamline diffusion fem

unified uncertainty analysis under probabilistic evidence fuzzy and interval uncertainties abstract the uncertainty analysis of structures generally involves uncertain parameters of different types in order to derive predictions regarding uncertain structural responses it is crucial to represent the uncertainty appropriately according to the underlying information available this paper presents a unified framework for uncertainty analysis under probabilistic evidence fuzzy and interval uncertainties by which the quantities with sufficient data sparse data and subjective information can be simultaneously considered a taylor expansion based unified uncertainty analysis t uua method is first proposed for small uncertainty problems by temporarily neglecting the evidence fuzzy and interval variables the probability evidence interval fuzzy model is degraded into a random problem in which the expectations and variances of responses can be obtained as functions in terms of evidence interval and fuzzy uncertainties then through dealing with the evidence variables the previous expectations and variances are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments bpas the fuzziness is then discretized by using cut technique and thus the expectations and variances are further expressed as functions of only intervals afterwards by reconsidering the interval uncertainties the bounds of the expectations and variances are computed via combining taylor expansion with interval arithmetic in addition a dimensional reduction dr efficient global optimization ego based unified uncertainty analysis dr ego uua method is also presented to solve the large uncertainty problems the framework of dr ego uua is similar as t uua however in dr ego uua the second moments of responses are computed by dr integrations and their upper and lower bounds are calculated by the ego finally three numerical examples are investigated to demonstrate the effectiveness of the proposed methods

detecting unknown lateral forces on a bar by vibration measurement abstract in this paper a method is presented for using standard vibration measurements for detecting and characterizing spatially uncertain lateral step forces on a damped hinged hinged bar the analysis leads to the specification of the sensor positions and the sampling frequency an algorithm for signal processing is proposed and tested by computer simulation uncertainty in the excitation force profile is described by employing the non probabilistic set theoretical technique called convex modelling three different convex models of force profile uncertainty are discussed

a non uniform haar wavelet method for numerically solving two dimensional convection dominated equations and two dimensional near singular elliptic equations abstract a non uniform haar wavelet based collocation method has been developed in this paper for two dimensional convection dominated equations and two dimensional near singular elliptic partial differential equations in which traditional haar wavelet method produces oscillatory solutions or low accurate solutions the main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations this process is done by discretizing space variables with non uniform haar wavelets to confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods finite element method finite difference method and polynomial collocation method numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes

curvature of leaves in amenable foliations 1 statement of results if m is a smooth manifold a basic problem is to examine the relationship between the fundamental group ir1 m and the possible riemannian geometries on m a fundamental result in this direction due to s t yau 22 and d gromoll and j wolf 11 asserts that if m is compact with non positive sectional curvature and i1x m is solvable then m is flat and in particular r1r m is a finite extension of an abelian group this result has been generalized from the compact case to the finite volume case and in fact to certain infinite volume manifolds as well by s s chen and p eberlein 3 the point of this paper is to prove an analogue of these results for foliations

eigenvalue bounds of structures with uncertain but bounded parameters many analysis and design problems in engineering and science involve uncertainty to varying degrees this paper is concerned with the structural vibration problem involving uncertain material or geometric parameters specified as bounds on these parameters this produces interval stiffness and mass matrices and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics however tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem often the stiffness and mass matrices can be formed as a non negative decomposition in the uncertain structural parameters in this case the eigenvalue bounds may be obtained from the parameter vertex solutions even more efficiently using interval extension from interval mathematics the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs the parameter vertex solution algorithm is compared with deif s solution the eigenvalue inclusion principle and the interval perturbation method in numerical examples

classification of negatively pinched manifolds with amenable fundamental groups we give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups namely they are precisely the mobius band and the products of r with the total spaces of flat vector bundles over closed infranilmanifolds

uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems i reaction diffusion type abstract we consider the bilinear finite element method on a shishkin mesh for the singularly perturbed elliptic boundary value problem 2 2 u x 2 2 u y 2 a x y u f x y in two space dimensions by using a very sophisticated asymptotic expansion of han et al 1 and the technique we used in 2 we prove that our method achieves almost second order uniform convergence rate in l2 norm numerical results confirm our theoretical analysis

amenable isometry groups of hadamard spaces one of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology since the universal covering space of a complete space of nonpositive curvature is contractible this amounts to relations between the geometry of the space or its universal covering space and the algebraic structure of its fundamental group a paradigm is the well known result of avez av that the fundamental group of a compact riemannian manifold m of nonpositive sectional curvature has exponential growth if and only if m is not flat gromov observed that the proof of avez yields a somewhat stronger statement the fundamental group of m is amenable if and only if m is flat see g1 p 93 zimmer generalized this to the case where m is complete and of finite volume zi further generalizations were obtained by anderson an and burger and schroeder bs the objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are alexandrov spaces of nonpositive curvature for the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature it is convenient to study the isometric action of the fundamental group on the universal covering space in many situations it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature this is the case in most of our arguments below let x be a hadamard space that is a complete simply connected geodesic space of nonpositive curvature in the sense of alexandrov a flat in x is a

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

stability of elastic bars on uncertain foundations using a convex model a method is presented for evaluating the buckling load of weightless prismatic rigid bars resting on uncertain foundations which are modeled using linear extensional elastic springs the uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model an ellipsoidal bound is used which defines the uncertainty of the foundation in terms of a size parameter and the deviations of the elastic spring constants from their nominal values the size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses the semiaxes of the ellipsoid are the deviations of the foundation spring constants from their nominal values and they determine the shape of the ellipsoid a first order analysis shows that the reduction in the buckling load when uncertainty in the foundation s spring stiffnesses is present is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid for the same uncertainty in the spring elements different reductions in the buckling load result for beams with multimode buckling

convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction diffusion problems the bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems in each case the quasi optimal order error estimates are proved in the e weighted h 1 norm uniformly in singular perturbation parameter e up to a logarithmic factor by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

robust coupling of dpg and bem for a singularly perturbed transmission problem abstract we consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the laplacian in the exterior connected through standard transmission conditions we establish a dpg scheme coupled with galerkin boundary elements for its discretization and prove its robustness for the field variables in so called balanced norms our coupling scheme is the one from fuhrer et al 2016 adapted to the singularly perturbed case by using the scheme from heuer and karkulik 2015 essential feature of our method is that optimal test functions have to be computed only locally we report on various numerical experiments in two dimensions

design optimization for dynamic response of vibration mechanical system with uncertain parameters using convex model the concept of uncertainty plays an important role in the design of practical mechanical system the most common methods for solving uncertainty problems are to model the parameters as a random vector a natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution however the drawback of this approach is that the probability distribution is difficult to obtain in this paper we use the non probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions using the convex model theory a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived because the uncertain parameters can be selected as the optimization parameters the present method can provide more information about the optimization results than those obtained by the deterministic optimization the present method is implemented for a torsional vibration system the numerical results show that the method is effective

corrigendum corrigendum to uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems i reaction diffusion type comput math appl 35 3 1998 57 70

mesh refinement strategies for solving singularly perturbed reaction diffusion problems abstract we consider the numerical approximation of a singularly perturbed reaction diffusion problem over a square two different approaches are compared namely adaptive isotropic mesh refinement and anisotropic mesh refinement thus we compare the h refinement and the shishkin mesh approaches numerically with pltmg software 1 it is shown how isotropic elements lead to over refinement and how anisotropic mesh refinement is much more efficient in thin boundary layers

lardy s regularization of a singularly perturbed elliptic pde in this paper we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives viewing it as an ill posed problem lardy s regularization method is applied to approximate the solution convergence of the regularized solution to the original is proved numerical examples have been included for illustrating the method

fixed points and amenability in non positive curvature consider a proper cocompact cat 0 space x we give a complete algebraic characterisation of amenable groups of isometries of x for amenable discrete subgroups an even narrower description is derived implying mathbf q linearity in the torsion free case we establish levi decompositions for stabilisers of points at infinity of x generalising the case of linear algebraic groups to text is x a geometric counterpart of this sheds light on the refined bordification of x a la karpelevich and leads to a converse to the adams ballmann theorem it is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the euclidean factor this fact is needed for the study of cat 0 lattices various fixed point results are derived as illustrations

high order finite element methods for singularly perturbed elliptic and parabolic problems we develop a framework for applying high order finite element methods to singularly perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects such as excess diffusion and nonphysical oscillations to boundary and interior layers this approach is more suited for use with adaptive mesh refinement and order variation techniques than other problem dependent methods quadrature rules developed for two point convection diffusion and reaction diffusion problems are used with finite element software to solve examples involving ordinary and partial differential equations numerical artifacts are confined to layers for all combinations of meshes orders and singular perturbation parameters that were tested radau or lobatto quadrature used with the finite element method to solve respectively convection and reaction diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

a robust numerical method for a control problem involving singularly perturbed equations we consider an unconstrained linear quadratic optimal control problem governed by a singularly perturbed convection reaction diffusion equation we discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by duran and lombardi in du an and lombardi 2005 2006 we prove convergence of this scheme in addition when the state equation is a singularly perturbed reaction diffusion equation we derive quasi optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes we present several numerical experiments when the state equation is both a reaction diffusion and a convection reaction diffusion equation these numerical experiments reveal a competitive performance of the proposed solution technique

convex models for optimizing diagnosis of uncertain slender obstacles on surfaces abstract blockage of a flowing compressible fluid has in the past been assessed by measuring shifts in acoustic eigenfrequencies this method has proven successful in many situations but measurements in complex geometries can present ambiguities partial blockage of the flow over a surface can be assessed by measuring pressure oscillations detection of small obstacles requires careful measurement design in this paper a technique for establishing the optimal deployment of pressure sensors is presented the primary uncertainty with respect to which the design is optimized is the spatial uncertainty in the shape and location of the blockage simulations are presented for subconic flow

a global uniformly convergent finite element method for a quasi linear singularly perturbed elliptic problem abstract in this paper we construct a bilinear finite element method based on a special piecewise uniform mesh for solving a quasi linear singularly perturbed elliptic problem in two space dimensions a quasi optimal global uniform convergence rate o n 2x ln2 nx n 2y ln2 ny was obtained which is independent of the perturbation parameter here nx and ny are the number of elements in the x and y directions respectively

pinching pontrjagin classes and negatively curved vector bundles we prove several finiteness results for the class a b n of n manifolds that have fundamental groups isomorphic to and that can be given complete riemannian metrics of sectional curvatures within a b where a b 0 in particular if m is a closed negatively curved manifold of dimension at least three then only finitely many manifolds in the class a b 1 m n are total spaces of vector bundles over m furthermore given a word hyperbolic group and an integer n there exists a positive e e n such that the tangent bundle of any manifold in the class 1 e 1 n has zero rational pontrjagin classes

convex models and interval analysis method to predict the effect of uncertain but bounded parameters on the buckling of composite structures a new non probabilistic set theoretical models that is interval analysis method is developed in this study to predict the variability or uncertainty in buckling loads of composite structures resulting from the unavoidable scatter in structural parameters by mathematically analyzing and proving the width of the upper and lower bounds on the critical buckling loads calculated by the interval analysis method is sharper than those that are obtained by convex models method moreover the interval analysis method has less computational cost than convex models method comparison between convex models and interval analysis method is performanced through numerical examples for the specific cases considered the results of numerical examples indicate that uncertainties in structural parameters have significant effect on the critical buckling loads of composite structures

comparison of static response of structures using convex models and interval analysis method in this paper by combining the finite element analysis and non probabilistic convex models we present the numerical algorithm of non probabilistic convex models and interval analysis method for the static displacement of structures with uncertain but bounded parameters under the condition of the box or interval vector determined from the ellipsoid of the uncertain but bounded structural parameter vector by comparing the numerical algorithm of non probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non probabilistic convex models is tighter than those produced by the interval analysis method copyright 2003 john wiley sons ltd

probabilistic interval reliability of structural systems the probabilistic reliability approach is the most widely used method for reliability analysis the recent research shows that the reliabilities of structural systems strongly depend on the parameters of the probability model it is possible that the little error in the estimation of the parameters may lead to the remarkable error of the resulting probability in this study we introduce the interval approach into the conventional reliability theory we present a novel approach which allows us to obtain the system failure probability interval from the statistical parameter intervals of the basic variables this approach is a combination of the two techniques namely the classical reliability theory and the interval analysis in the end of this paper we show the feasibility of the proposed approach through two examples of the truss systems

on splitting theorems for cat 0 spaces and compact geodesic spaces of non positive curvature in this paper we show some splitting theorems for cat 0 spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non positive curvature a cat 0 group is said to be rigid if determines its boundary up to homeomorphisms of a cat 0 space on which acts geometrically c croke and b kleiner have constructed a non rigid cat 0 group as an application of the splitting theorems for cat 0 spaces we obtain that if 1 and 2 are rigid cat 0 groups then so is 1 2

a unified method for the response analysis of interval random variable models of acoustic fields with uncertain but bounded parameters summary for the response analysis of engineering systems with uncertain but bounded parameters three uncertain models have been considered according to the available probability distribution information one is the bounded random model in which the uncertain parameters are well defined with sufficient probability distribution information and described as bounded random variables the second one is the interval model in which the uncertain parameters are expressed as interval variables without giving any information of probability distribution the last one is the bounded hybrid uncertain model which includes both bounded random variables and interval variables on the basis theory of gegenbauer polynomial approximation theory a unified interval and random gegenbauer series expansion irgse method is proposed and extended for the response prediction of three uncertain models of acoustic fields with uncertain but bounded parameters in irgse the uncertain but bounded variables with different probability distribution information including interval variables and bounded random variables with different probability density functions pdfs are transformed into the function of unitary variables defined on associated with the corresponding polynomial parameter of gegenbauer series expansion gse the coefficients of gse is calculated by gauss gegenbauer integration method by using irgse the responses of three uncertain acoustic models are approximated uniformly by gse through which the interval and random analysis can be easily implemented by many numerical solvers two numerical examples are applied to investigate the effectiveness of the proposed method

a robust finite element method for a singularly perturbed elliptic problem with two small parameters abstract in this paper we consider a bilinear finite element method fem for a singularly perturbed elliptic problem with two small parameters by using butuzov asymptotic expansion 1 and the technique we developed in 2 4 we prove that our fem on a special piecewise uniform mesh converges independently of small parameters numerical results show that our method perform much better than the classical fem published by elsevier science ltd

robust estimation and filtering in the presence of bounded noise optimal algorithms for robust estimation and filtering are constructed the noise is considered a deterministic variable belonging to a set described by a norm previous results obtained for complete one to one and approximate information by m milanese and r tempo ibid vol 30 p 730 8 1985 are extended to partial and approximate information this information is considered useful in dealing with dynamic systems not completely identifiable and or with two different sources of noise such as process and measurement noise for different norms characterizing the noise optimal algorithms in a min max sense are shown in particular for hilbert norms a linear optimal algorithm is the well known minimum variance estimator for l sub infinity norm an optimal algorithm computable by linear programming is presented state estimation is formalized in the context of the general theory for stable systems an approximate state estimation is obtained by neglecting higher order powers an upper bound determining the approximation introduced is derived a numerical example illustrates the application of the theory

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

probabilistic analysis of non self adjoint mechanical systems with uncertain parameters a new approach to analyze mechanical systems that have an uncertain distribution of material properties and loadings for the variability of their dynamic response parameters is developed the methodology is demonstrated through solving for the probabilistic moments of eigenvalues of a non self adjoint structural system material properties are modelled using random fields and uncertain loadings are modelled using random variables by treating the random fluctuations of vibratory response to be the stochastic perturbations to mean response second order moments of eigenvalues are obtained to circumvent the practical difficulty of obtaining the exact correlation models for parameter variability second order moments of eigenvalues are evaluated in terms of variance functions both the sensitivity of eigenvalues to the variability in system parameters and the effects on the eigenvalue variability of correlation properties of uncertain parameters are demonstrated

at infinity of finite dimensional cat 0 spaces we show that any filtering family of closed convex subsets of a finite dimensional cat 0 space x has a non empty intersection in the visual bordification x over bar x boolean or partial derivative x using this fact several results known for proper cat 0 spaces may be extended to finite dimensional spaces including the existence of canonical fixed points at infinity for parabolic isometries algebraic and geometric restrictions on amenable group actions and geometric superrigidity for non elementary actions of irreducible uniform lattices in products of locally compact groups

an adaptive characteristic petrov galerkin finite element method for convection dominated linear and nonlinear parabolic problems in one space variable a new adaptive finite element method for convection dominated problems is presented a special feature of the method is that it is based on a petrov galerkin scheme for spatial approximation at a typical time step which employs test functions chosen so that the approximate solution coincides with the exact solution at the nodes of finite element grid this procedure makes possible the derivation of truly local a posteriori error estimates and a very effective solver numerical examples are discussed which illustrate the efficiency and effectively of the method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

topology optimization of structures with interval random parameters abstract the aim of this paper is to present a robust topology optimization methodology for structures with imprecise probability uncertainty in this paper the imprecise probability uncertainties are treated with an interval random model in which the probability variables are used to model the uncertain parameters and some distribution parameters of probability variables are expressed as interval variables instead of a precise value in the presented methodology the deterministic topology optimization techniques and the hybrid stochastic interval perturbation method hsipm are combined to obtain robust topology designs for structures with interval random parameters the exploitation of hsipm transforms the problem of topology optimization with interval random parameters into an augmented deterministic topology optimization problem this provides a computationally cheap alternative to monte carlo based optimization algorithms several numerical examples are presented to demonstrate the effectiveness of the proposed method

detrimental or serendipitous effect of human error on reliability of structures abstract this paper is directed to a single objective to illustrate the possible error assoictated with an interpretation of experimental data in terms of the stochastic dependency of involved random variables in view of evaluating the structural reliability it is shown that in some circumstances the error may prove to be serendipitous more importantly however in some other instances it may turn out to be detrimental it is concluded that the error associated with reliability calculations should become a part of any serious implementation of probabilistic design for structural components or large scale structures

showalter regularization of a singularly perturbed pde in this paper we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives viewing it as an ill posed problem showalter s regularization method is applied to approximate the solution convergence of the regularized solution to the original is proved a numerical example has been included for illustrating the method

global uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems higher order elements abstract in this paper we develop a general higher order finite element method for solving singularly perturbed elliptic linear and quasilinear problems in two space dimensions we prove that a quasioptimal global uniform convergence rate of 0 n x m 1 in m 1 n x n v m 1 in m 1 n v in l 2 norm is obtained for a reaction diffusion model by using the m th order m 2 tensor product element thus answering some open problems posed by roos in h g roos layer adapted grids for singular perturbation problems z angew math mech 78 5 1998 291 309 and h g ross m stynes and l tobiska numerical methods for singularly perturbed differential equations springer verlag berlin 1996 278 here n x and n v are the number of partitions in the x and y directions respectively numerical results are provided supporting our theoretical analysis

lavrentiev regularization of a singularly perturbed elliptic pde this paper considers a reaction diffusion equation with a small parameter multiplied with the derivatives considering this as an ill posed problem lavrentiev regularization method is used to formulate the related well posed problem pointwise a priori estimates of the regularized solutions and of the regularization error are also derived the regularized equations are solved numerically via the central difference scheme it is suggested that any other suitable discretization might also be used to solve the regularized problems also it is found that with a comparatively small number of grid points the numerical solution of the regularized equation comes up as a good approximation to the original solution

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

a deterministic method to predict the effect of unknown but bounded elastic moduli on the buckling of composite structures abstract a new non probabilistic method is developed in this study to predict the variability in buckling loads of composite plates and shells stemming from the unavoidable scatter in elastic moduli the available measurements of elastic moduli are fitted by the four dimensional uncertainty box and appropriate ellipsoid for the specific cases considered the maximum buckling load of the composite plate exceeds by 11 the value of the buckling load based on the calculation using the nominal values of elastic moduli in the case of the cylindrical shell an appropriate exceedance constitutes 9

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2

hybrid uncertain analysis of acoustic field with interval random parameters abstract for the acoustic field prediction without sufficient information an interval random model is introduced to treat with the uncertain parameters in which the probability distribution parameters of random variables are modeled as interval variables instead of given precise values based on the interval random model the hybrid uncertain acoustic dynamic equilibrium equation is constructed and a hybrid uncertain analysis method is proposed in the proposed method the interval random matrix and vector are expanded by the first order taylor series and the sound pressure response vector is obtained by the matrix perturbation technique a monotonic analysis is conducted for the sound pressure response vector and based on it the lower and upper bounds of the sound pressure response vector are obtained by the vertex method according to the response bounds the intervals of expectation and variance of the sound pressure response vector are calculated by the random interval moment method two numerical examples which include a simple problem with explicit expression and a complex engineering application are presented to demonstrate the effectiveness of the present method

interval and random analysis for structure acoustic systems with large uncertain but bounded parameters abstract for the response analysis of the structure acoustic system with uncertain but bounded parameters three bounded uncertain models are introduced one is the bounded random model in which all of the uncertain but bounded parameters are described as bounded random variables with well defined probability distribution the second one is the interval model in which all of the uncertain but bounded parameters are described as interval variables due to the limited information the third one is the bounded hybrid uncertain model in which the interval variables and the bounded random variables exist simultaneously based on the parametric gegenbauer polynomial which is formulated for bounded random model recently the gegenbauer series expansion method gsem is developed for the response prediction of the structure acoustic system under these three bounded uncertain models within gsem the response of these three bounded uncertain models of the structure acoustic system can be approximated by the unified gegenbauer series with different values of polynomial parameter then the interval and random analysis for these three bounded uncertain models of the structure acoustic system are conducted on the basis of gegenbauer series owing to the orthogonal property of gegenbauer polynomial the analytical solution of the expectation and variance of gegenbauer series with respect to the bounded random variables can be readily obtained the bounds of gegenbauer series with respect to the interval variables are determined by the monte carlo simulation the gsem is applied to solve a shell structure acoustic system under these three bounded uncertain models inspired by the convergence behavior of gsem the relative improvement criterion is established to estimate the required retained order of gegenbauer series for large uncertain problems the results on numerical examples show that gsem with the estimated retained order can achieve a prescribed accuracy and good efficiency for structure acoustic systems with large uncertain but bounded parameters