two parametric solution to graded yang baxter equation and two parametric uuvgl 1 mod 1 algebra as a hopf algebra the authors discuss a two parametric solution of the graded yang baxter equation ybe and perform the yang baxterization to obtain the solution to the quantum ybe in the formalism developed previously they present the two parametric quantized superalgebra uuvgl 1 mod 1 and prove that this algebra is a hopf algebra with the hopf operations explicitly provided

a modified hybrid uncertain analysis method for dynamic response field of the lsoaac with random and interval parameters abstract for the prediction of dynamic response field of the luffing system of an automobile crane lsoaac with random and interval parameters a hybrid uncertain model is introduced in the hybrid uncertain model the parameters with certain probability distribution are modeled as random variables whereas the parameters with lower and upper bounds are modeled as interval variables instead of given precise values based on the hybrid uncertain model the hybrid uncertain dynamic response equilibrium equation in which different random and interval parameters are simultaneously included in input and output terms is constructed then a modified hybrid uncertain analysis method mhuam is proposed in the mhuam based on random interval perturbation method the first order taylor series expansion and the first order neumann series the dynamic response expression of the lsoaac is developed moreover the mathematical characteristics of extrema of bounds of dynamic response are determined by random interval moment method and monotonic analysis technique compared with the hybrid monte carlo method hmcm and interval perturbation method ipm numerical results show the feasibility and efficiency of the mhuam for solving the hybrid lsoaac problems the effects of different uncertain models and parameters on the lsoaac response field are also investigated deeply and numerical results indicate that the impact made by the randomness in the thrust of the luffing cylinder f is larger than that made by the gravity of the weight in suspension q in addition the impact made by the uncertainty in the displacement between the lower end of the lifting arm and the luffing cylinder a is larger than that made by the length of the lifting arm l

unified response probability distribution analysis of two hybrid uncertain acoustic fields abstract for the response analysis of uncertain acoustic fields with limited information two hybrid uncertain models have been developed one is the hybrid probability and interval model in which the probability and interval variables exist simultaneously the other one is the hybrid interval probability model in which some distribution parameters of probability variables are expressed as interval variables for the unified response probability distribution analysis of acoustic fields under two hybrid uncertain models an inverse mapping hybrid perturbation method imhpm is proposed in imhpm the responses of two hybrid uncertain acoustic fields are converted to invertible functions of probability variables based on the taylor series expansion according to the inverse mapping relationships between responses and probability variables the formal expressions of the response probability distributions are yielded on the basis of the change of variable technique the variational ranges of the response probability distributions are estimated by the interval perturbation technique the effectiveness and efficiency of the proposed method for the unified response probability distribution analysis of two hybrid uncertain acoustic fields are investigated by a numerical example

hybrid uncertain analysis for structural acoustic problem with random and interval parameters abstract in this paper hybrid perturbation monte carlo method hpmcm and hybrid perturbation vertex method hpvm are developed for the hybrid uncertain structural acoustic problem with random and interval variables the random variables are used to treat the uncertain parameters with sufficient information whereas the interval variables are used to treat the uncertain parameters with limited information in hpmcm the intervals of expectation and variance of the frequency response vector are calculated by a combination of random interval matrix perturbation method random interval moment method and monte carlo method in hpvm a monotonicity analysis is conducted for the frequency response vector based on the random interval matrix perturbation method and then the intervals of expectation and variance of the frequency response vector are calculated by employing the random interval moment method and the vertex method numerical results on a shell structural acoustic problem verify the effectiveness and high efficiency of hpvm compared with hpmcm

interval static displacement analysis for structures with interval parameters in this paper a new method to solve the uncertain static displacement problem of structures with interval parameters is presented it is difficult to obtain all possible solutions with sharp bounds even if an optimum scheme is adopted when there are many uncertain parameters with the interval mathematics the interval finite element equation is developed based on the perturbation and the interval extension the upper and lower bounds of the static displacements are obtained in which the sharp bounds are guaranteed by the interval calculation operators two numerical examples a box cantilever beam and an automobile frame are given to illustrate the validity of the present method copyright 2001 john wiley sons ltd

dynamic response analysis of structure under time variant interval process model abstract due to the aggressiveness of the environmental factor the variation of the dynamic load the degeneration of the material property and the wear of the machine surface parameters related with the structure are distinctly time variant typical model for time variant uncertainties is the random process model which is constructed on the basis of a large number of samples in this work we propose a time variant interval process model which can be effectively used to deal with time variant uncertainties with limit information and then two methods are presented for the dynamic response analysis of the structure under the time variant interval process model the first one is the direct monte carlo method dmcm whose computational burden is relative high the second one is the monte carlo method based on the chebyshev polynomial expansion mcm cpe whose computational efficiency is high in mcm cpe the dynamic response of the structure is approximated by the chebyshev polynomials which can be efficiently calculated and then the variational range of the dynamic response is estimated according to the samples yielded by the monte carlo method to solve the dependency phenomenon of the interval operation the affine arithmetic is integrated into the chebyshev polynomial expansion the computational effectiveness and efficiency of mcm cpe is verified by two numerical examples including a spring mass damper system and a shell structure

reduction of multivariate mixtures and its applications abstract we consider fast deterministic algorithms to identify the best linearly independent terms in multivariate mixtures and use them to compute up to a user selected accuracy an equivalent representation with fewer terms importantly the multivariate mixtures do not have to be a separated representation of a function one algorithm employs a pivoted cholesky decomposition of the gram matrix constructed from the terms of the mixture to select what we call skeleton terms and the other uses orthogonalization for the same purpose both algorithms require o r 2 n p d r n operations where n is the initial number of terms in the multivariate mixture r is the number of selected linearly independent terms and p d is the cost of computing the inner product between two terms of a mixture in d variables for general gaussian mixtures p d d 3 since we need to diagonalize a d d matrix whereas for separated representations p d d there is no need for diagonalization due to conditioning issues the resulting accuracy is limited to about one half of the available significant digits for both algorithms we also describe an alternative algorithm that is capable of achieving higher accuracy but is only applicable in low dimensions or to multivariate mixtures in separated form we describe a number of initial applications of these algorithms to solve partial differential and integral equations and to address several problems in data science for data science applications in high dimensions we consider the kernel density estimation kde approach for constructing a probability density function pdf of a cloud of points a far field kernel summation method and the construction of equivalent sources for non oscillatory kernels used in both computational physics and data science and finally show how to use the new algorithm to produce seeds for subdividing a cloud of points into groups

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

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

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

reliability analysis of structures with interval uncertainties under stationary stochastic excitations reliability analysis of linear discretized structural systems with interval uncertainties subjected to stationary gaussian random excitation is addressed under the assumption of independent up crossings of a specified threshold an efficient procedure for the evaluation of the bounds of the interval reliability function of the generic response process is presented the first step of the proposed approach is to derive approximate expressions of the interval mean value and spectral moments of the response along with the associated bounds to this aim the improved interval analysis via extra unitaryinterval is applied in conjunction with a novel series expansion of the inverse of an interval matrix with modifications called interval rational series expansion irse then the lower bound and upper bounds of the interval reliability function are readily evaluated by properly combining the bounds of the interval mean value and spectral moments of the response two numerical examples are provided to demonstrate the accuracy of the proposed procedure and its usefulness in view of decision making in engineering practice

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

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

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

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

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

inverse interval matrix the inverse interval matrix is defined as the narrowest interval matrix containing the inverses of all the matrices from a given interval matrix both theoretical and practical results concerning computation of the inverse interval matrix are presented in particular explicit formulas are given for the inverse of an interval matrix with radius of rank one

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

the encounter of interval and probabilistic approaches to structural reliability at the design point abstract the consideration of uncertainties in structural analysis has traditionally been made in the field of structural reliability within the framework of classical probability theory in many cases however there is no sufficient probabilistic information to substantiate such an approach in recent years several non probabilistic methods have been proposed as alternatives one of them is the interval analysis oriented to estimate the range of variation of output structural variables given the ranges of the input ones the method however requires the use of special arithmetics which renders its application unpractical for structural analysis based on finite elements or other numerical techniques in this paper a monte carlo method that can be applied to probabilistic as well as to interval approaches to reliability analysis is proposed it is based on the optimal properties of the classical design point of probabilistic structural reliability in particular it is shown that the order statistics of the output variable is concealed in a plot defined by the design point vector this facilitates the selection of the relevant samples for either interval or probability based analysis the examples show that the desired interval of the response or the failure probability can be accurately estimated on such a basis the paper also discusses the use of monte carlo methods for both reliability and interval analysis from the point of view of the proposed representation

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

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

a dynamic evolution scheme for structures with interval uncertainties by using bidirectional sequential kriging method abstract this paper proposes a bidirectional sequential kriging bsk method for nonlinear interval uncertainty quantification of dynamic systems different from existing surrogate based methods which build surrogate model at preselected samples the proposed method constructs surrogate models sequentially small amount of initial samples are generated by latin hypercube sampling lhs and a crude kriging surrogate model ksm is constructed then two index functions named lower confidence bound lcb and upper confidence bound ucb respectively are defined for sample collocation the index functions guide the search for the optima of structural response after iterations a sophisticated surrogate model will be obtained and the interval bounds of dynamic response can be calculated quickly with some auxiliary algorithms such as genetic algorithm ga numerical examples and engineering application are studied to verify the effectiveness of bsk the results show that compared with existing methods the proposed method can achieve higher accuracy with good efficiency in nonlinear interval analysis

the explicit inverse of the stiffness matrix abstract in two recent papers the author has given the exact analytic expression of the internal forces in linear elastic structures composed of uniform prismatic elements it was shown that the member forces are the ratios of two multilinear homogeneous polynomials in the unimodal stiffnesses of the elements of the structure the order of the polynomials is equal to the number of nodal degrees of freedom of the structure the number of terms of each polynomial is equal to the number of statically determinate stable substructures which can be derived from the original structure the coefficients of the polynomials can be computed by employing the equilibrium equations and by enforcing global compatibility of deformations it was found empirically that the coefficients of the polynomial in the denominator were numerically equal to the square of the determinants of the statics matrices of the respective statically determinate substructures as a consequence the denominator became the sum of the stiffness matrices of the statically determinate substructures this is in fact the binet cauchy form of the determinant of the stiffness matrix of the structure bearing in mind that the inverse of the stiffness matrix can be expressed as the ratio of the adjoint matrix of the stiffness matrix divided by the determinant of the stiffness matrix it became clear that the expressions of the stress resultants stem from an explicit expression of the adjoint the explicit expression of the adjoint of the stiffness matrix lies at the heart of this paper it is shown that the adjoint is a congruent transformation of the n 1 compound of the stiffness matrix where n is the number of degrees of freedom of the structure this cleared the way to use the binct cauchy theorem on the product of compound matrices to obtain an explicit expression for the adjoint and ipso facto for the inverse of the stiffness matrix having now the displacements of the structure the expression of the stress resultants which was obtained independently emerges in a very elegant manner the member forces in a structure can be expressed as the weighted sum of the member forces in all its determinate substructures when subjected to the applied loads the weighting factors are the ratios of the determinants of he stiffness matrices of the substructures to the determinant of the stiffness matrix of the original structure both the explicit inverse of the stiffness matrix and the expression of the internal forces in the structure are at present of a theoretical nature the number of terms involved in the polynomials is simply excessive for common engineering structures however ongoing reseach may yield more applicable expressions to be used for instance in he field of automated design of structures the theory is illustrated with the explicit analysis of a stayed mast

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

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

robust assessment of collapse resistance of structures under uncertain loads based on info gap model abstract the paper proposes a pair of novel mathematical programming based approaches which directly determine the worst collapse load limit in one case and the best limit in the other case of rigid perfectly plastic structures subjected to uncertain but bounded applied loads using an info gap model the methods take advantage of the important properties in which the worst collapse load limit defined for an uncertain static formulation is equivalent to the most favourable solution of the uncertain kinematic limit analysis problem and vice versa for the best collapse load limit the formulation for capturing the worst collapse load limit robust worst case solution takes the form as a mathematical program with equilibrium constraints that is processed using a penalty algorithm whilst that for the best collapse load limit is a standard linear programming problem the efficiency and robustness of the proposed schemes are evidenced from a number of successfully solved examples

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

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

a response surface approach for the static analysis of stochastic structures with geometrical nonlinearities a response surface approach for the finite element analysis of uncertain structures undergoing large displacements is presented this method is based on the use of ad hoc response surface functions built up by ratios of polynomials as opposite to commonly used linear or quadratic polynomials such functions are insensitive to the sampling point positions once the response surface form is defined response statistics can be approximated by analytical relationships or statistical simulation taking full advantage of sensitivity analysis numerical investigations demonstrated that a remarkable accuracy is achieved in the evaluation of both statistical moments and probability density functions of the response

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

a new approach for the stochastic analysis of finite element modelled structures with uncertain parameters a method for evaluating the static response of uncertain finite element fe discretised structures is presented the method is comparable with the perturbation procedures from a computational point of view but it overcomes the drawbacks related to these procedures in fact the present method gives excellent level of accuracy even for high amount of uncertainties moreover it produces the exact solution for statically determinate structures the accuracy is remarkable even in the evaluation of high order statistics and probability density function of the response so that reliability consideration can be drawn

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

a method to derive approximate explicit solutions for structural mechanics problems the availability of explicit solutions i e analytical relationships between the structural response and the design variables allows a more direct and plain treatment of several structural problems this paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout fixed node positions the proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept which allows a non conventional assembly of the global stiffness matrix the exact inversion of that matrix is a trivial task for the case of statically determinate structures structures with few redundancies or few design variables an approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the sherman morrison woodbury formula 2006 elsevier ltd all rights reserved

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

rank revealing qr factorizations and the singular value decomposition t chan has noted that even when the singular value decomposition of a matrix a is known it is still not obvious how to find a rank revealing qr factorization rrqr of a if a has numerical rank deficiency this paper offers a constructive proof of the existence of the rrqr factorization of any matrix a of size m x n with numerical rank r the bounds derived in this paper that guarantee the existence of rrqr are all of order f i in comparison with chan s 0 2 it has been known for some time that if a is only numerically rank one deficient then the column permutation l7 of a that guarantees a small rnn in the qr factorization of a n can be obtained by inspecting the size of the elements of the right singular vector of a corresponding to the smallest singular value of a to some extent our paper generalizes this well known result we consider the interplay between two important matrix decompositions the singular value decomposition and the qr factorization of a matrix a in particular we are interested in the case when a is singular or nearly singular it is well known that for any a e r m x n a real matrix with rn rows and n columns where without loss of generality we assume rn n there are orthogonal matrices u and v such that where c is a diagonal matrix with nonnegative diagonal elements we assume that a 2 a2 2 2 on 2 0 the decomposition 0 1 is the singular value decomposition svd of a and the ai are the singular values of a the columns of v are the right singular vectors of a and the columns of u are the left singular vectors of a mathematically in terms of the singular values received december 1 1990 revised february 8 199 1 199 1 mathematics subject classification primary 65f30 15a23 15a42 15a15