a dimension wise analysis method for the structural acoustic system with interval parameters abstract the interval structural acoustic analysis is mainly accomplished by interval and subinterval perturbation methods potential limitations for these intrusive methods include overestimation or interval translation effect for the former and prohibitive computational cost for the latter in this paper a dimension wise analysis method is thus proposed to overcome these potential limitations in this method a sectional curve of the system response surface along each input dimensionality is firstly extracted the minimal and maximal points of which are identified based on its legendre polynomial approximation and two input vectors i e the minimal and maximal input vectors are dimension wisely assembled by the minimal and maximal points of all sectional curves finally the lower and upper bounds of system response are computed by deterministic finite element analysis at the two input vectors two numerical examples are studied to demonstrate the effectiveness of the proposed method and show that compared to the interval and subinterval perturbation method a better accuracy is achieved without much compromise on efficiency by the proposed method especially for nonlinear problems with large interval parameters

hybrid uncertainty analysis of functionally graded plates via multiple imprecise random field modelling of uncertain material properties abstract this paper presents the mixed probabilistic non probabilistic uncertainty problem regarding the hybrid uncertainty propagation on physical model of functionally graded plates through dissimilarly uncertain characterisation of material properties in the presence of the multiple imprecise random field modelling the spectral stochastic finite element framework is extended to perform mixed uncertainty analysis of functionally graded material whereby the spatial variation of elastic modulus of the plate is uniquely characterised as the multiple imprecise random field modelling more than a purely homogeneous random field this characterisation has originated in heeding the construction of physical models and the significant influences of separately uncertain characteristics propagated on the physical model through different material ingredients are examined to make physical models more reliable and realistic mechanics and mathematics intrusive formulations are then established to explicitly produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation bounded surfaces of statistical moments and bounding distribution functions are vividly depicted for demonstration the feasibility and effectiveness of the presented work are illustrated via test cases of numerical examples

a relaxation procedure for domain decomposition methods using finite elements we present the convergence analysis of a new domain decomposition technique for finite element approximations this technique was introduced in 11 and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent no global preconditioner is needed in the practice but simply single domain finite element solvers are required an optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated applications are given to both elliptic equations and incompressible stokes equations

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

solution of the helmholtz eigenvalue problem via the boundary element method the numerical solution of the helmholtz eigenvalue problem is considered the application of the boundary element method reduces it to that of a non linear eigenvalue problem through a polynomial approximation with respect to the wavenumber the non linear eigenvalue problem is reduced to a standard generalized eigenvalue problem the method is applied to the test problems of a three dimensional sphere with an axisymmetric boundary condition and a two dimensional square

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

epistemic uncertainty propagation in energy flows between structural vibrating systems abstract a dimension wise method for predicting fuzzy energy flows between structural vibrating systems coupled by joints with epistemic uncertainties is established based on its legendre polynomial approximation at 0 both the minimum and maximum point vectors of the energy flow of interest are calculated dimension by dimension within the space spanned by the interval parameters determined by fuzzy those at 0 and the resulted interval bounds are used to assemble the concerned fuzzy energy flows besides the proposed method vertex method as well as two current methods is also applied comparisons among results by different methods are accomplished by two numerical examples and the accuracy of all methods is simultaneously verified by monte carlo simulation

fuzzy interval finite element statistical energy analysis for mid frequency analysis of built up systems with mixed fuzzy and interval parameters abstract this paper introduces mixed fuzzy and interval parametric uncertainties into the fe components of the hybrid finite element statistical energy analysis fe sea model for mid frequency analysis of built up systems thus an uncertain ensemble combining non parametric with mixed fuzzy and interval parametric uncertainties comes into being a fuzzy interval finite element statistical energy analysis fife sea framework is proposed to obtain the uncertain responses of built up systems which are described as intervals with fuzzy bounds termed as fuzzy bounded intervals fbis in this paper based on the level cut technique a first order fuzzy interval perturbation fe sea ffipfe sea and a second order fuzzy interval perturbation fe sea method sfipfe sea are developed to handle the mixed parametric uncertainties efficiently ffipfe sea approximates the response functions by the first order taylor series while sfipfe sea improves the accuracy by considering the second order items of taylor series in which all the mixed second order items are neglected to further improve the accuracy a chebyshev fuzzy interval method cfim is proposed in which the chebyshev polynomials is used to approximate the response functions the fbis are eventually reconstructed by assembling the extrema solutions at all cut levels numerical results on two built up systems verify the effectiveness of the proposed methods

perturbative analysis of the method of particular solutions for improved inclusion of high lying dirichlet eigenvalues the dirichlet eigenvalue or drum problem in a domain omega subset mathbb r 2 becomes numerically challenging at high eigenvalue frequency e in this regime the method of particular solutions mps gives spectral accuracy for many domain shapes it requires a number of degrees of freedom scaling as sqrt e the number of wavelengths on the boundary in contrast to direct discretization for which this scaling is e our main result is an inclusion bound on eigenvalues that is a factor o sqrt e tighter than the classical bound of moler payne and that is optimal in that it reflects the true slopes of curves appearing in the mps we also present an mps variant that cures a normalization problem in the original method while evaluating basis functions only on the boundary this method is efficient at high frequencies where we show that in practice our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

a phase based interior penalty discontinuous galerkin method for the helmholtz equation with spatially varying wavenumber this paper is concerned with an interior penalty discontinuous galerkin ipdg method based on a flexible type of non polynomial local approximation space for the helmholtz equation with varying wavenumber the local approximation space consists of multiple polynomial modulated phase functions which can be chosen according to the phase information of the solution we obtain some approximation properties for this space and a priori l2l2 error estimates for the h convergence of the ipdg method using duality argument we also provide ample numerical examples to show that building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces

stochastic finite element analysis of structures in the presence of multiple imprecise random field parameters the research work presents the study on non deterministic problems in the presence of the multiple imprecise random field uncertainties by extending the spectral stochastic finite element framework the standard random field is expanded to characterize the behaviour of physical model with imprecise randomness more appropriately for real engineering problems based on available sources of uncertainty the young s modulus and body force of structures are considered as imprecise random fields with bounded statistical moments mathematical expressions and solution procedure are developed to produce the uncertain but bounded statistical characteristics of responses namely interval mean value interval standard deviation and bounding distribution functions the bounding probability density and cumulative distribution of the imprecise random response are then effectively visualized by means of polynomial chaos expansion the feasibility and effectiveness of the presented method are illustrated by three numerical examples

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

tolerance analysis of antenna arrays through interval arithmetic an analytical method based on interval analysis ia is proposed to predict the impact of the manufacturing tolerances of the excitation amplitudes on the radiated array pattern by expressing the array factor according to the rules of the interval arithmetic the radiation features of the linear array are described in terms of intervals whose bounds are analytically determined as functions of the nominal value and the tolerances of the array amplitudes a set of representative numerical experiments dealing with different radiated beams and linear array sizes is reported and discussed to point out the features and potentials of the proposed approach

fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters summary this paper proposes a fuzzy interval perturbation method fipm and a modified fuzzy interval perturbation method mfipm for the hybrid uncertain temperature field prediction involving both interval and fuzzy parameters in material properties and boundary conditions interval variables are used to quantify the non probabilistic uncertainty with limited information whereas fuzzy variables are used to represent the uncertainty associated with the expert opinions the level cut method is introduced to decompose the fuzzy parameters into interval variables fipm approximates the interval matrix inverse by the first order neumann series while mfipm improves the accuracy by considering higher order terms of the neumann series the membership functions of the interval temperature field are eventually derived using the fuzzy decomposition theorem three numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed methods for solving heat conduction problems with hybrid uncertain parameters pure interval parameters and pure fuzzy parameters respectively copyright 2015 john wiley sons ltd

consolidating boundary methods for finding the eigenstates of billiards the plane wave decomposition method a widely used means of numerically finding eigenstates of the helmholtz equation in billiard systems is described as a variant of the mathematically well established boundary integral method bim a new unified framework encompassing the two methods is discussed furthermore a third numerical method which we call the gauge freedom method is derived from the bim equations this opens the way to further improvements in eigenstate search techniques

interval arithmetic for pattern tolerance analysis of parabolic reflectors an innovative analytic strategy for the estimation of the average pattern behavior of reflector antennas when surface deformations are present on the parabolic disc is proposed since measuring the surface deviations from a reference ideal shape is a complex task the root mean square rms surface deformations are modeled as intervals and their impact on the power pattern as well as the sensitivity of the beam features is efficiently predicted by exploiting the rules of the interval arithmetic throughout interval analysis ia the result is the definition of closed form relationships between the bounds upper and lower values of the radiated power pattern and the rms interval surface deformations uniformly and non uniformly distributed perturbations of the ideal surface are analyzed to assess the effectiveness of proposed approach comparisons with state of the art solutions are also carried out by varying the deformation thickness as well as the reflector geometry and illumination

boundary quasi orthogonality and sharp inclusion bounds for large dirichlet eigenvalues we study eigenfunctions phi j and eigenvalues e j of the dirichlet laplacian on a bounded domain omega subset mathbb r n with piecewise smooth boundary we bound the distance between an arbitrary parameter e 0 and the spectrum e j in terms of the boundary l 2 norm of a normalized trial solution u of the helmholtz equation delta e u 0 we also bound the l 2 norm of the error of this trial solution from an eigenfunction both of these results are sharp up to constants hold for all e greater than a small constant and improve upon the best known bounds of moler payne by a factor of the wavenumber sqrt e one application is to the solution of eigenvalue problems at high frequency via for example the method of particular solutions in the case of planar strictly star shaped domains we give an inclusion bound where the constant is also sharp we give explicit constants in the theorems and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy the proof makes use of a new quasi orthogonality property of the boundary normal derivatives of the eigenmodes theorem 1 3 of interest in its own right namely the operator norm of the sum of rank 1 operators partial n phi j langle partial n phi j cdot rangle over all e j in a spectral window of width sqrt e a sum with about e n 1 2 terms is at most a constant factor independent of e larger than the operator norm of any one individual term

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

a bayesian collocation method for static analysis of structures with unknown but bounded uncertainties abstract this paper proposes a bayesian collocation method bcm for static analysis of structures with unknown but bounded uncertainties or interval uncertainties unlike the existing collocation methods which construct surrogate model using preselected samples at fixed points the bcm chooses samples sequentially and adaptively based on bayesian inference theorem the bcm constructs a crude gaussian process surrogate model of structural response with selected samples then two acquisition functions named upper expected improvement ei up function and lower expected improvement ei lo function are defined simultaneously to guide the selection of next samples the samples will be collocated to the place where extrema of structural response are most likely to be found the procedure will repeat until the convergence criterion is met the extrema of the selected samples can be regarded as structural response bounds fewer samples are needed compared with existing collocation methods two numerical examples are used to demonstrate the accuracy and efficiency of the proposed method the results are compared with those obtained by existing methods the application of the method is performed by an engineering example the results show the validity of the proposed method

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

sequential optimization and mixed uncertainty analysis method for reliability based optimization a reliability based optimization method under both aleatory and epistemic uncertainties is studied the mixed uncertainties are analyzed by combined probability and evidence theory if the mixed uncertainty analysis is directly embedded in reliability based optimization to quantify the uncertain features of each search point it would be computationally prohibitive to address this problem a sequential optimization and mixed uncertainty analysis method is proposed to decompose the reliability based optimization problem into separate deterministic optimization and mixed uncertainty analysis subproblems which are solved sequentially and alternately until convergence is achieved the research focus is how to transform the reliability based optimization problem into its quasi equivalent deterministic formulation according to the information obtained in the uncertainty analysis it is proposed to first decompose the total reliability target into each focal element of the epistemic uncertainties so as to sim

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

fuzzy boundary element method for the analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches the uncertainties present may be associated with parameters that are vague imprecise or linguistic for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical system a fuzzy boundary element method for the analysis of imprecisely defined systems is developed starting from the basic concepts of fuzzy sets and fuzzy arithmetic the various steps of the boundary element method are redefined using fuzzy concepts the resulting fuzzy equations are solved using a fuzzified version of gaussian elimination procedure coupled with truncation the truncation method limits the growth of interval ranges of response parameters to obtain realistic and accurate solutions the procedure is illustrated by considering the analysis of a potential flow problem the methodology is applicable to uncertain systems that are described in linguistic terms as well as those that are described by incomplete information the approach represents a unique methodology that enables the analysis and design of many engineering systems more realistically

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

on the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators the standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed they are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds further the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues a new method is presented which combines multigrid techniques with the lanczos process the resulting scheme requires o mn arithmetic operations and o n storage requirement where n is the number of unknowns and m the number of needed eigenvalues the discretization of the considered differential operators is realized by means of p finite elements and is applicable on general geometries numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods at least on current workstations the ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a pc shows the potential of this method practical applications are found e g in the numerical simulation of quantum billiards

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

three dimensional free vibration analysis of thick cylindrical shells resting on two parameter elastic supports three dimensional free vibration of a circular cylindrical shell in contact with an elastic medium will be studied the response of the elastic medium is formulated by the winkler pasternak model the layerwise theory in conjunction with a three dimensional form of hamilton s principle is used to obtain the transversely discretized equations of motion and the related boundary conditions the differential quadrature method is employed to discretize the resulting equations in the axial direction and for the solution procedure to validate the formulation the results are compared with the available exact solutions and also with ansys finite elements solutions by examining the results for thick circular shells under various boundary conditions and on different elastic foundations the influence of such parameters and in particular those due to elastic foundations are studied

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

robust and efficient solution of the drum problem via nystrom approximation of the fredholm determinant the drum problem finding the eigenvalues and eigenfunctions of the laplacian with dirichlet boundary condition has many applications yet remains challenging for general domains when high accuracy or high frequency is needed boundary integral equations are appealing for large scale problems yet certain difficulties have limited their use we introduce the following two ideas to remedy this 1 we solve the resulting nonlinear eigenvalue problem using boyd s method for analytic root finding applied to the fredholm determinant and we show that this is many times faster than the usual iterative minimization of a singular value 2 we fix the problem of spurious exterior resonances via a combined field representation this also provides the first robust boundary integral eigenvalue method for non simply connected domains we implement the new method in two dimensions using spectrally accurate nystrom product quadrature we prove exponential convergence of the determinant at roots for domains with

a novel interval finite element method based on the improved interval analysis abstract static analysis of linear elastic structures with uncertain parameters subjected to deterministic loads is addressed the uncertain structural properties are modeled as interval variables with assigned lower bound and upper bound a novel interval finite element method is formulated in the framework of the improved interval analysis via extra unitary interval recently proposed to limit the conservatism affecting the classical interval analysis the key idea of the novel method is to associate an extra unitary interval to each uncertain parameter in order to keep physical properties linked to the finite elements in both the assembly and solution phases this allows one to reduce overestimation and perform standard assembly of the interval element matrices the lower bound and upper bound of interval displacements and stresses are evaluated by applying two different strategies both based on the so called interval rational series expansion for deriving the approximate explicit inverse of the interval global stiffness matrix numerical examples concerning 2d and 3d structures with uncertain young s modulus are presented to demonstrate the accuracy and efficiency of the proposed procedure

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

interval and subinterval perturbation finite element boundary element method for low frequency uncertain analysis of structural acoustic systems abstract uncertainties exist widely in the structural acoustic systems due to the physical imperfections model inaccuracies and system complexities to deal with the uncertainties the interval perturbation and subinterval perturbation techniques for the first time are extended and integrated into the hybrid finite element boundary element method ipfem bem in this work firstly the structural part of structural acoustic systems is modeled using finite element method fem and acoustic cavity is modeled by boundary element method bem then the interval perturbation technique is introduced to establish the interval perturbation equations of the system for large uncertainty levels the interval of parameters is divided into smaller sub intervals to improve the accuracy of proposed ipfem bem sub interval perturbation finite element boundary element method sipfem bem the proposed ipfem bem and sipfem bem are verified by two numerical examples the results obtained by ipfem bem sipfem bem and monte carlo method are compared it is found that ipfem bem and sipfem bem is reliable for the frequency response analysis of structural acoustic systems with small or large uncertain levels

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

a polynomial expansion approach for response analysis of periodical composite structural acoustic problems with multi scale mixed aleatory and epistemic uncertainties abstract the response analysis of periodical composite structural acoustic problem with multi scale mixed aleatory and epistemic uncertainties is investigated based on homogenization method in this paper the aleatory uncertainties are presented by bounded random variables whereas the epistemic uncertainties are presented by interval variables and evidence variables when dealing with the combination of bounded random variables interval variables and evidence variables enormous computation is needed to estimate the output probability bounds of the sound pressure response of the periodical composite structural acoustic system to reduce the involved computational cost but without losing accuracy by transforming all of the bounded random variables and interval variables into evidence variables appropriately an evidence theory based polynomial expansion method epem is developed in which the gegenbauer series expansion is employed to approximate the variation range of the response with respect to evidence variables by using epem the probability bounds of the response can be obtained efficiently a numerical example is used to validate the proposed method and two engineering examples are given to demonstrate its efficiency

interval and subinterval homogenization based method for determining the effective elastic properties of periodic microstructure with interval parameters abstract the main issue of this paper is to present an interval homogenization based method ihm for the estimation of the effective elastic tensor for microscopic material properties with interval uncertainty based on the finite element method fem for homogenization analysis the interval effective elastic tensor is expanded by the first or second order taylor series for obtaining its upper and lower bounds in order to guarantee the accuracy of ihm the subinterval technique is introduced into the ihm for estimating the effective elastic tensor numerical examples of a unit rectangular voids cell and a unit particle filled polymers cell are implemented through comparing the numerical results with the results obtained by the monte carlo simulation the effectiveness and the efficiency of the ihm for homogenization analysis with interval uncertainty are verified the influence of the microscopic interval uncertainty in material properties on a homogenized macroscopic elastic property of an inhomogeneous material has also been investigated

on the flexural vibration of rectangular plates approached by using simple polynomials in the rayleigh ritz method abstract sets of simple polynomials are proposed for use as admissible functions in the rayleigh ritz method for the study of the flexural vibration of rectangular plates the plates may be subject to a number of different complicating factors including the presence of in plane forces concentrated masses point and line spring supports and special orthotropy of the plate material the sets of polynomials are shown to be equivalent to the orthogonally generated polynomials used very successfully in earlier studies but are more easily generated and the subsequent evaluation of the integrals in the resulting eigenvalue problem are more simply performed numerical results are presented for several problems illustrating the applicability of the method and its accuracy

reliability based design optimization with dependent interval variables summary reliability based design optimization rbdo ensures design requirements are satisfied at required reliability levels in the presence of uncertainty the recent rbdo methodologies can accommodate two types of uncertainty aleatory type represented by random variables and epistemic type represented by interval variables however only independent interval variables can be handled the objective of this research is to introduce dependent interval variables into rbdo the dependency of interval variables comes from equality and inequality constraints with the intervals the reliability is also an interval and its lower bound is considered to preserve conservativeness an efficient optimization method is developed to solve rbdo problems the method is an extension of the existing framework of sequential optimization and reliability analysis with the inclusion of inequality constraints of interval variables the new method is efficient because it decouples the optimization process from the process of reliability and interval analyses the new method is applied to the rbdo of a slider mechanism with joint clearances which are known as dependent interval variables copyright 2012 john wiley sons ltd

uncertainty propagation in sea for structural acoustic coupled systems with non deterministic parameters abstract considering limited available information on uncertainties in structural acoustic coupled systems two methods namely the vertex method and the legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the statistical energy analysis sea approach for the vertex method an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded which is well suited for an interval input vector in arbitrary dimension instead of calculating the extremum of the response of interest a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the legendre orthogonal polynomial approximation following the theoretical analysis of the accuracy and efficiency of the proposed methods their validation is performed by one numerical example and two applications

further studies on the vibration of plates with curved edges including complicating effects abstract the ritz method is used to obtain an eigenvalue equation for the free vibration of a class of thin flat plates which involve curved boundaries defined by polynomial expressions the class of plates is such that each plate may be discretized into four 90 sectorial elements allowing it to have up to four sections of outer boundary and up to four sections of inner boundary each described by polynomials in the absence of symmetry or where it is not utilized the elements are joined together through the use of very stiff translational and rotational springs which enforce the required continuity conditions a number of complicating effects have been included such as the presence of internal point or line supports concentrated masses and stepped thickness geometry natural frequency parameters are given for several plates for which comparison results exist and the accuracy of the approach demonstrated results are also given for several plates of varying complexity which have not previously been treated in the open literature

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

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

application of a three dimensional shell theory to the free vibration of shells arbitrarily deep in one direction abstract a three dimensional shell theory is presented which is applicable to doubly curved thick open shells which are arbitrarily deep have a large side length to radius of curvature ratio in one principal direction but are shallow in the other direction the strain displacement equations for the proposed deep shallow shell theory are expressed in cartesian co ordinates and the limits of applicability of these equations are discussed these equations are then used in a ritz variational formulation with algebraic polynomials as trial functions to solve for the natural frequencies of a number of doubly curved shell problems a novel approach is also proposed in which penalty functions are introduced to enforce continuity of displacements at two opposite ends of a shell of rectangular platform increasing the range of problems which can be treated to include closed shells such as cylinders barrels cooling tower type structures toroids rings etc a sub class of shells of revolution

a dual layer dimension wise fuzzy finite element method for structural analysis with epistemic uncertainties abstract epistemic uncertainty propagation is gaining momentum but remains a great challenge in the numerical analysis of a mechanical system a dual layer dimension wise analysis procedure for the fuzzy finite element method is proposed the core algorithm layer is to identify the minimal and maximal min max points of each slice of the response surface at the zero cut the data processing layer aims to filtrate the min max points of each response slice at each nonzero cut subsequently the min max input vectors at any alpha cut are dimension wisely assembled and the response interval at alpha cut is calculated by deterministic finite element analysis at the min max input vectors finally the fuzzy description of the system response is recomposed two examples are investigated to verify both the accuracy and efficiency of the proposed method

assessment of uncertainty on structural dynamic responses with the short transformation method abstract the predictive capability of finite element models is limited by their deterministic nature typically not all model parameters are exactly known while even small deviations may have significant effects on the predicted response parameter uncertainty should therefore be taken into account e g with fuzzy arithmetic the absence of fuzzy solvers led to interval arithmetic as a numerical alternative the transformation method tm presented by m hanss replaces interval arithmetic with a set of deterministic computations for each interval all parameter extrema are combined in every possible way in a design of experiments terminology the tm is a so called full factorial design the tm is applicable if the output is monotonic in the inputs unlike interval arithmetic it does not overestimate the response uncertainty as only parameter combinations are evaluated that actually occur in this paper the tm has been applied to visualise uncertain frequency response functions frfs obtained with modal superposition this yields accurate results when validated against monte carlo data but the computation time is rather high the short transformation method stm is proposed as an attractive alternative to the original tm a full set of deterministic computations combining all interval extrema is only performed at the lowest interval for higher levels a smaller set is evaluated this allows reconstructing the fuzzy frf from a much lower number of deterministic computations with only a small reduction in the accuracy of frfs both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters

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

an interval finite element approach for the calculation of envelope frequency response functions this paper focusses on the application of the interval finite element method in dynamic analyses it describes a methodology for calculating frequency response function envelopes from a finite element model containing imprecise parameters defined as interval uncertainties the resulting envelope functions give a conservative approximation of the possible range of the frequency response function taking into account that the uncertain parameters in the model can adopt any value in their presumed uncertainty intervals the methodology is based on the modal superposition principle it consists of an interval aritmethic algorithm which processes the results of a preliminary global optimization performed on the modal parameters the algorithm is constructed such that it optimally combines the advantages of both the anti optimization and the interval arithmetic strategy for general numerical interval calculations in the first stage of the development the modal parameter ranges of each individual mode are independently combined in the modal response contributions this yields the modal rectangle mr method in order to remedy the high conservatism inherent to the mr method the exact eigenfrequency ranges are added to the analysis this results in the modal rectangle method with eigenfrequency interval correction mre a second improvement consists of adding extra delimiters to the mre modal parameter range approximation this is achieved by performing an extra optimization on the modal response contributions at discrete frequencies the method is referred to as the locally optimized modal rectangle method with eigenfrequency interval correction omre finally a numerical example illustrates the different algorithms copyright 2004 john wiley sons ltd

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

exact bounds for the static response set of structures with uncertain but bounded parameters abstract the numerical estimation of the static displacement bounds of structures with uncertain but bounded parameters is considered in this paper by representing each uncertain but bounded parameter as an interval number or vector a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations in which the coefficient matrix and the right hand side term are respectively the interval matrix and the interval vector in this study we present two new simple mathematical proofs of the vertex solution theorem using cramer s rule for solving linear interval equations different from the other proof methods to find the upper and lower bounds on the set of solutions the first takes advantage of optimization theory while the second is based on interval extension by means of a typical example considered first by hansen it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the oettli prager criterion examples of a three stepped beam and a 10 bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method

fuzzy meshless local petrov galerkin method for analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical mechanical or structural system the uncertainties present may be associated with parameters that are vague imprecise or linguistic a fuzzy meshless local petrov galerkin approach is developed in this work for the analysis of imprecisely defined systems starting from the basic concepts of fuzzy sets fuzzy arithmetic and fuzzy calculus differentiation and integration the various steps of the meshless local petrov galerkin approach involved in the derivation of the equations are redefined using fuzzy concepts the resulting fuzzy equations are derived using a fuzzy version of gaussian elimination procedure coupled with truncation a one dimensional heat transfer problem involving conduction convection and radiation with vaguely defined thermal pro

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

asymptotic rate of quantum ergodicity in chaotic euclidean billiards the quantum unique ergodicity que conjecture of rudnick and sarnak is that every eigenfunction n of the laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit eigenvalue en that is strong scars are absent we study numerically the rate of equidistribution for a uniformly hyperbolic sinai type planar euclidean billiard with dirichlet boundary condition the drum problem at unprecedented high e and statistical accuracy via the matrix elements n m of a piecewise constant test function a by collecting 30 000 diagonal elements up to level n 7 105 we find that their variance decays with eigenvalue as a power 0 48 0 01 close to the semiclassical estimate of feingold and peres this contrasts with the results of existing studies which have been limited to en a factor 102 smaller we find strong evidence for que in this system we also compare off diagonal variance as a function of distance from the diagonal against feingold peres or spectral measure at the highest accuracy 0 7 thus far in any chaotic system we outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions 2006 wiley periodicals inc

reliability analysis for multidisciplinary systems with random and interval variables tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems most of the approaches are only capable of dealing with random variables modeled by probability distributions both random and interval variables however may exist in multidisciplinary systems their propagation through coupled subsystems make s reliability analysis computationally expensive in this work a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems the framework is an extension of an existing unified uncertainty analysis framework for single disciplinary problems the new framework involves probabilistic analysis and interval analysis both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially the first order reliability method is used for probabilistic analysis three supporting algorithms are developed the effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application

three dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels abstract the free vibration problem of a homogeneous isotropic thick cylindrical shell or panel subjected to a certain type of simply supported edge boundary conditions is considered for this problem the governing equations of three dimensional linear elasticity are employed and solved by using a new iterative approach which in practice leads to the prediction of the exact frequencies of vibration in the particular case of a flat plate or a complete shell excellent agreement is achieved between results based on the present approach and corresponding results based on other exact analyses available in the literature for certain cylindrical panel geometries comparisons are also made between corresponding numerical results based on the present analysis and certain two dimensional shell theories

test of a numerical approach to the quantization of billiards a method for computing large numbers of eigenvalues of self adjoint elliptic operators j comput phys 184 321 2003 is tested in numerical studies of the spectral properties of quantum billiards to this extent we study a time reversal invariant quantum billiard of threefold symmetry that undergoes a transformation in its symmetry properties from c 3v to c3 thereby a transition from gaussian orthogonal to gaussian unitary ensemble statistics is observed verifying earlier experimental indications and theoretical predictions at the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes

interval analysis of structures with uncertain but bounded axial stiffness abstract a method to evaluate the static response of structures with interval axial stiffnesses is proposed the procedure overcomes the drawbacks arising in traditional interval analysis due to dependency phenomenon the inversion of the interval stiffness matrix is pursued by means of a novel approximate relationship based on an interval valued sherman morrison woodbury formula numerical results on truss and frame structures evidence the accuracy and the effectiveness of the proposed approach to evaluate bounds of displacements and internal forces

reliability analysis of stiffened aircraft panels using adjusting mean value method with regard to real world engineering structures an adjusting mean value called the self adjusting mean value smv is proposed for improving the efficiency and robustness of a maximum performance

is the helmholtz equation really sign indefinite the usual variational or weak formulations of the helmholtz equation are sign indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared this is often for a good reason since in bounded domains under certain boundary conditions the solution of the helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the laplacian and thus the variational problem cannot be sign definite however even in cases where the solution is unique for all wavenumbers the standard variational formulations of the helmholtz equation are still indefinite when the wavenumber is large this indefiniteness has implications for both the analysis and the practical implementation of finite element methods in this paper we introduce new sign definite also called coercive or elliptic formulations of the helmholtz equation posed in either the interior of a star shaped domain with impedance boundary conditions or the exterior of a star shape

an explicit kernel split panel based nystrom scheme for integral equations on axially symmetric surfaces a high order accurate explicit kernel split panel based fourier nystrom dis cretization scheme is developed for integral equations associated with the helm holtz equation in axially symmetric domains extensive incorporation of ana lytic information about singular integral kernels and on the y computation of nearly singular quadrature rules allow for very high achievable accuracy also in the evaluation of elds close to the boundary of the computational domain

boundary value problems and hardy spaces associated to the helmholtz equation in lipschitz domains abstract in this paper we introduce and discuss in the clifford algebra framework certain hardy like spaces which are well suited for the study of the helmholtz equation u k 2 u 0 in lipschitz domains of r n 1 in particular in the second part of the paper these results are used in connection with the classical boundary value problems for the helmholtz equation in lipschitz domains in arbitrary space dimensions in this setting existence uniqueness and optimal estimates are obtained by inverting the corresponding layer potential operators on l p for sharp ranges of p s also a detailed discussion of the helmholtz eigenvalues of lipschitz domains is presented

two non probabilistic set theoretical models for dynamic response and buckling failure measures of bars with unknown but bounded initial imperfections this paper is concerned with the problem of comparison of two non probabilistic set theoretical models for dynamic response and buckling failure measures of bars with unknown but bounded initial imperfections two kinds of non probabilistic set theoretical models are convex models and interval analysis models in convex models and interval analysis models the uncertain quantities are considered to be unknown except that they belong to given sets in an appropriate vector space in this case all information about the dynamic response and buckling failure measures of bars is provided by the set of dynamic responses and buckling failure measures consistent with the constraints on the uncertain quantities the dynamic response estimate is actually a set in appropriate response space rather than a single vector the set estimate is the smallest calculable set which contains the uncertain dynamic response but it is usually impractical to calculate this set two set estimate methods are developed which can calculate the time varying box or hyperrectangle i e interval vector in the response space that contains the true dynamic response comparison between convex models and interval analysis models in mathematical proofs and numerical calculations shows that under the condition of the outer enclosed ellipsoid from a hyperrectangle or an interval vector the set dynamic response predicted by interval analysis models is smaller than that yielded by convex models under the condition of the outer enclosed hyperrectangle or an interval vector from an ellipsoid the dynamic response set calculated by convex models is smaller than that obtained by interval analysis models

homogenization based interval analysis for structural acoustic problem involving periodical composites and multi scale uncertain but bounded parameters this paper presents a homogenization based interval analysis method for the prediction of coupled structural acoustic systems involving periodical composites and multi scale uncertain but bounded parameters in the structural acoustic system the macro plate structure is assumed to be composed of a periodically uniform microstructure the equivalent macro material properties of the microstructure are computed using the homogenization method by integrating the first order taylor expansion interval analysis method with the homogenization based finite element method a homogenization based interval finite element method hifem is developed to solve a periodical composite structural acoustic system with multi scale uncertain but bounded parameters the corresponding formulations of the hifem are deduced a subinterval technique is also introduced into the hifem for higher accuracy numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented method for a periodical composite structural acoustic system with multi scale uncertain but bounded parameters

nondeterministic possibilistic approaches for structural analysis and optimal design the development of methods to take into account uncertainties in structural analysis and in design optimization is attracting both the scientific and the industrial communities in this domain possibilistic methods in which uncertainties are defined by fuzzy numbers appear as an alternative to the classical probabilistic methods such as the monte carlo simulations or the stochastic finite element method the principal difficulty of possibilistic methods is that they lead to the resolution of systems of equations in which the coefficients are defined by intervals several approaches for the direct solution of such interval linear equations systems are presented and compared the vertex method is taken as reference it is shown that the problems that are solved are mathematically different according to the method used for static linear analysis a cost effective iterative solution of the vertex is proposed it is based on neumann series expansions and solves an optimization subproblem to compute extrema of the structural responses the effectiveness of the method is illustrated by the solution of truss problems classical in the optimization literature extension of the method to inverse problems of design is also considered

sharp norm estimates of layer potentials and operators at high frequency abstract in this paper we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency 2 in the high frequency regime the key problem is the dependence of mapping norms on the parameter for single layer potentials we find that the l 2 l 2 norms decay in the rate of decay depends on the curvature of the norm is 3 4 in piecewise smooth domains and 5 6 if the boundary is positively curved the double layer potential however displays uniform l 2 l 2 bounds independent of curvature by various examples we show that all our estimates on layer potentials are sharp appendix a by galkowski gives bounds l 2 l 2 for the single and double layer operators at high frequency that are sharp modulo log in this case both the single and double layer operator bounds depend upon the curvature of the boundary

diagrammatic representation for interval arithmetic abstract the paper presents a diagrammatic representation of a standard interval space the so called mr diagram and shows how to represent and perform interval arithmetic and derive its various properties using the diagram the representation is an extension and refinement of the is diagram representation devised earlier by the author to represent interval relations first the mr diagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations second diagrammatic constructions for all standard arithmetic operations are presented several examples of the use of these constructions to aid reasoning about various simple though nontrivial properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

an efficient fredholm method for the calculation of highly excited states of billiards a numerically efficient fredholm formulation of the billiard problem is presented the standard solution in the framework of the boundary integral method in terms of a search for roots of a secular determinant is reviewed first we next reformulate the singularity condition in terms of a flow in the space of an auxiliary one parameter family of eigenproblems and argue that the eigenvalues and eigenfunctions are analytic functions within a certain domain based on this analytic behavior we present a numerical algorithm to compute a range of billiard eigenvalues and associated eigenvectors by only two diagonalizations

universal grey system theory for analysis of uncertain structural systems the uncertainty present in many structural systems is commonly modeled using probabilistic fuzzy or interval approaches depending on the nature of uncertainty present the probabilistic approach

the vertex solution theorem and its coupled framework for static analysis of structures with interval parameters summary this work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory the core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations then the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters following discussions about the computational efforts a coupled framework based on vertex method vm is established which allows us to solve many large scale engineering problems with uncertainties using deterministic finite element software compared with the previous works the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy to use interval finite element software numerical examples demonstrate the good accuracy of vm meanwhile the implementation of vm and availability of the coupled framework are demonstrated by engineering example copyright 2017 john wiley sons ltd

modified sub interval perturbation finite element method for 2d acoustic field prediction with large uncertain but bounded parameters abstract based on the sub interval perturbation analysis a modified sub interval perturbation finite element method is proposed to determine the bounds of sound pressure in the 2d acoustic field with large uncertain but bounded parameters in the proposed method the inversion of the invertible sub interval dynamic stiffness matrix is approximated by a modified approximate interval value sherman morrison woodbury formula to overcome the drawbacks arising from the dependency phenomenon of parameters and the unpredictable effect of neglecting the higher order terms in neumann series the modified sub interval perturbation procedures are implemented in a numerical finite element framework numerical results on a 2d acoustic tube and a 2d acoustic cavity of a car with large uncertain but bounded parameters evidence the remarkable accuracy and effectiveness of the proposed method

uncertainty based structural optimization using universal grey number theory several types of uncertainties are encountered in the design of machines and structures the geometric parameters or dimensions are specified in terms of nominal values and tolerances the experime

fast computation of high frequency dirichlet eigenmodes via spectral flow of the interior neumann to dirichlet map we present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the dirichlet laplacian in a smooth star shaped domain in d d 2 conventional boundary based methods require a root search in eigenfrequency k hence take o n3 effort per eigenpair found where n o kd 1 is the number of unknowns required to discretize the boundary our method is o n faster achieved by linearizing with respect to k the spectrum of a weighted interior neumann to dirichlet ntd operator for the helmholtz equation approximations to the square roots kj of all o n eigenvalues lying in k k where o 1 are found with o n3 effort we prove an error estimate with c independent of k we present a higher order variant with eigenvalue error scaling empirically as o 5 and eigenfunction error as o 3 the former improving upon the scaling method of vergini and saraceno for planar domains d 2 with an assumption of absence of spectral concentration we also prove rigorous error bounds that are close to those numerically observed for d 2 we compute robustly the spectrum of the ntd operator via potential theory nystrom discretization and the cayley transform at high frequencies 400 wavelengths across with eigenfrequency relative error 10 10 we show that the method is 103 times faster than standard ones based upon a root search 2014 wiley periodicals inc

eigenvalue bounds for symmetric matrices with entries in one interval in this paper we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval based on some matrix computation theories a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results furthermore the ideas are spread to general symmetric interval matrices finally numerical examples are presented which show the validity of the proposed methods

analysis of mechanical systems using interval computations applied to finite element methods this paper addresses the problem of mechanical systems in which parameters are uncertain and bounded interval calculation is used to find an envelope of transfer functions for mechanical systems modelled with finite elements within this context a new formulation has been developed for finite element problems involving bounded parameters to avoid the problems of overestimation an iterative algorithm is introduced which leads to a conservative solution for linear mechanical problems a method to ensure the convergence of this algorithm is also proposed this new algorithm has been tested on simple mechanical systems and leads to a conservative envelope of the transfer functions