random matrix eigenvalue problems in structural dynamics natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems considering that the system parameters are known only probabilistically we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems current methods to deal with such problems are dominated by mean centred perturbation based methods here two new approaches are proposed the first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is best in some sense the second approach is based on an asymptotic approximation of multidimensional integrals a closed form expression is derived for a general rth order moment of the eigenvalues two approaches are presented to obtain the probability density functions of the eigenvalues the first is based on the maximum entropy method and the second is based on a chi square distribution both approaches result in simple closed form expressions which can be easily calculated the proposed methods are applied to two problems and the analytical results are compared with monte carlo simulations it is expected that the small randomness assumption usually employed in mean centred perturbation based methods can be relaxed considerably using these methods

asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems we consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients with these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space and the approximation space for the underlying spatial function space sparse polynomial approximation is employed to obtain the first one while classical finite elements are employed to obtain the latter an error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem a series of detailed numerical experiments supports the conclusions of this analysis

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

probability distributions of natural frequencies of uncertain dynamic systems of random eigenvalues commonly encountered in dynamic systems the method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for a fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the expansion coefficients the new decomposition does not require sample points yet it generates a convergent sequence of lower variate estimates of theprobabilitydistributionsofeigensolutions numericalresults includingfrequencydistributionsofapiezoelectric transducer indicate that the decomposition method developed provides accurate convergent and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues

uncertainty analysis of complex structural systems in this paper methods for uncertainty propagation for complex structural systems are discussed only a limited class of methods is available for this purpose it is shown that only numerical procedures allow the uncertainty assessment and reliability estimation of such systems in particular the perturbation procedure the karhunen loeve reduction scheme the polynomial chaos expansion direct and advanced monte carlo simulation mcs as well as random matrix theory meet these requirements it is shown that advanced mcs procedures proved to be the most versatile approach copyright 2009 john wiley sons ltd

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

loudness of low frequency pure tones lateralized by interaural time differences directional loudness is that phenomenon by which the loudness of a sound may vary according to the localization of its source this phenomenon has been mainly observed for high frequency sounds for sources located in the horizontal plane because of the acoustic shadow of the head the left and right ear pressures are modified depending on the source azimuth and the global loudness resulting from a summation process may vary accordingly directional loudness has also been reported to occur at 400 hz where shadowing effects are usually rather small it might therefore be suspected that directional loudness effects could be influenced by other parameters involved in the localization process this study assessed the loudness of low frequency pure tones 200 and 400 hz lateralized with headphones by applying an interaural time difference itd but no interaural level difference it showed small but significant variations of loudness with respect to itd at a low loudness level 40 phon itd values associated with virtual azimuths of 60 and 90 led to a small but significant increase in loudness up to 1 25 db however there was no such effect at a moderate loudness level 70 phon

probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation we present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data for which uncertainties are taken into account by using the nonparametric probabilistic approach such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters consequently the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible two formulations are proposed the first one is the mean square method for which a usual differentiable objective function and an unusual non differentiable objective function are proposed the second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics

computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

probabilistic eigenvalue buckling analysis solved through the ratio of polynomial response surface an efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state is proposed under the assumption that no imperfections are present and that the elastic parameters are uncertain and modeled as random variables the approach allows a deeper investigation of structures which are not sensitive to imperfections the procedure relies on a response surface method adopting simple ratio of polynomials without cross terms as performance function such a relationship approximates analytically the dependence between the buckling load and the basic variables furnishing a limit state equation which is very close to the exact one when a proper experimental design is adopted in this way a monte carlo simulation applied to the response surface leads to a good approximation with low computational effort several numerical examples show the accuracy and effectiveness of the method varying structural complexity correlation between basic variables and their distribution

frobenius morphisms and representations of algebras by introducing frobenius morphisms f on algebras a and their modules over the algebraic closure f q of the finite field f q of q elements we establish a relation between the representation theory of a over f q and that of the f fixed point algebra a f over f q more precisely we prove that the category mod a f of finite dimensional a f modules is equivalent to the subcategory of finite dimensional f stable a modules and when a is finite dimensional we establish a bijection between the isoclasses of indecomposable a f modules and the f orbits of the isoclasses of indecomposable a modules applying the theory to representations of quivers with automorphisms we show that representations of a modulated quiver or a species over f q can be interpreted as f stable representations of the corresponding quiver over f q we further prove that every finite dimensional hereditary algebra over f q is morita equivalent to some a f where a is the path algebra of a quiver q over f q and f is induced from a certain automorphism of q a close relation between the auslander reiten theories for a and a f is established in particular we prove that the auslander reiten modulated quiver of a f is obtained by folding the auslander reiten quiver of a finally by taking frobenius fixed points we are able to count the number of indecomposable representations of a modulated quiver over f q with a given dimension vector and to generalize kac s theorem for all modulated quivers and their associated kac moody algebras defined by symmetrizable generalized cartan matrices

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

approximate vibration reanalysis of structures thecombinedapproximationsmethodisdevelopedforvibrationreanalysisofstructures inthesolution process the terms of local series approximations are used as basis vectors in global reduced basis approximations the solution steps are straightforward and the method can be readily used with a general e nite element system the computational effort needed by the method is usually much smaller than the effort needed for complete vibration analysis the efe ciency of the calculations and the accuracy of the results can be controlled by the amount of information considered efe cient solutions are obtained by low order approximations and accurate results can be achievedbyconsideringhigher orderterms numericalexamplesshowthataccurateresultsareachievedefe ciently for very large changes in the design nomenclature b matrix dee ned by eq 21 c matrix dee ned by eq 16 e modulus of elasticity i moment of inertia i identity matrix k stiffness matrix kr reduced stiffness matrix l length m mass matrix mr reduced mass matrix r right hand side vector r displacement vector rb matrix of basis vectors s number of basis vectors u upper triangular matrix y vector of coefe cients coefe cient dee ned by eq 35 angle between vectors 1k change in the stiffness matrix 1m change in the mass matrix kronecker delta eigenvalue circular frequency

efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition a new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented the eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis an efficient method for estimating the coefficients with respect to this basis is proposed the method uses a galerkin based approach by orthogonalizing the residual in the eigenvalue eigenvector equation to the subspace spanned by the basis functions used for approximation in this way the stochastic problem is framed as a system of deterministic non linear algebraic equations this system of equations is solved using a newton raphson algorithm although the proposed approach is not based on statistical sampling the efficiency of the proposed method can be significantly enhanced by initializing the non linear iterative process with a small statistical sample synthesized through a monte carlo sampling scheme the proposed method offers a number of advantages over existing methods based on statistical sampling first it provides an approximation to the complete probabilistic description of the eigensolution second it reduces the computational overhead associated with solving the statistical eigenvalue problem finally it circumvents the dependence of the statistical solution on the quality of the underlying random number generator copyright 2007 john wiley sons ltd

a galerkin finite element technique for stochastic field problems a new finite element method is developed to analyse the structures with more than one parameter behaving in a stochastic manner the galerkin weighted residual method is used as a generalization this paper treats the random eigenvalue problems arising when the material property values of the structural systems are having stochastic fluctuations i e the measurement errors can be accounted for when probabilistic modelling is used the free vibration problem of a stochastic beam whose young s modulus and mass density are distributed stochastically is considered using galerkin s weighted residual procedure a stochastic finite element method is developed and implemented to arrive at a random algebraic eigenvalue problem the stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations numerical examples are given it is demonstrated that through this formulation the finite element discretization need not be dependent on the characteristics of stochastic fields describing the fluctuations in the material property values further calculation of covariances between stiffness elements mass elements etc is carried out in a relatively simple manner and is more general in nature

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

directional loudness in an anechoic sound field head related transfer functions and binaural summation the effect of sound incidence angle on loudness was investigated using real sound sources positioned in an anechoic chamber eight normal hearing listeners produced loudness matches between a frontal reference location and seven sources placed at other directions both in the horizontal and median planes matches were obtained via a two interval adaptive forced choice 2afc procedure for three center frequencies 0 4 1 and 5khz and two overall levels 45 and 65db spl the results showed that loudness is not constant over sound incidence angles with directional sensitivity varying over a range of up to 10db exhibiting considerable frequency dependence but only minor effects of overall level the pattern of results varied substantially between subjects but was largely accounted for by variations in individual head related transfer functions modeling of binaural loudness based on the at ear signals favored a sound power summation model according to which the maximum binaural gain is only 3db ove

spectral power iterations for the random eigenvalue problem two computationally efficient algorithms are developed for solving the stochastic eigenvalue problem an algorithm based on the power iteration technique is proposed for the calculation of the dominant eigenpairs this algorithm is then extended to find other subdominant random eigenpairs the uncertainty in the operator is represented by a polynomial chaos expansion and a similar representation is considered for the random eigenvalues and eigenvectors the algorithms are distinguished due to their speed in converging to the true random eigenpairs and their ability to estimate a prescribed number of subdominant eigenpairs the algorithms are demonstrated on two examples with close agreement observed with the exact solution and a solution synthesized through monte carlo sampling

parametric instability of stochastic columns columns which have stochastically distributed young s modulus and mass density and are subjected to deterministic periodic axial loadings are considered the general case of a column supported on a winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered material property fluctuations are modeled as independent one dimensional univariate homogeneous real random fields in space in addition to autocorrelation functions or their equivalent power spectral density functions the input random fields are characterized by scale of fluctuations or variance functions for their second order properties the foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables the system equations of boundary frequencies are obtained using bolotin s method for deterministic systems stochastic fem is used to obtain the discrete system with random as well as periodic coefficients statistical properties of boundary frequencies are derived in terms of input parameter statistics a complete covariance structure is obtained the equations developed are illustrated using a numerical example employing a practical correlation structure

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

stochastic interval analysis of natural frequency and mode shape of structures with uncertainties abstract in this paper natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method the bounds of these probabilistic characteristics are then determined by interval arithmetic two examples are given first to illustrate the feasibility of the presented method and the results are verified by monte carlo simulations the presented approach is also applicable to solve pure random and pure interval problems this capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes

a comprehensive overview of a non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics abstract in structural dynamics a predictive model is constructed by developing a mathematical mechanical model of a designed system in order to predict the response of the real system which is the manufactured system realized from the designed system the mathematical mechanical modelling process of the designed system introduces two fundamental types of uncertainties the data uncertainties and the model uncertainties uncertainties have to be taken into account for improving the predictability of the model model uncertainties cannot be modelled by using the usual parametric probabilistic approach recently a general non parametric probabilistic approach of model uncertainties for dynamical systems has been proposed using the random matrix theory this paper gives a comprehensive overview of this approach in developing its foundations in simple terms and in illustrating all the concepts and the tools introduced in the general theory by using a simple example this paper deals with 1 notions of designed systems real systems mean models as predictive models errors and uncertainties 2 the definition of a simple example in linear elastodynamics 3 a comprehensive overview of the non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics 4 a summary of the random matrix ensembles which are necessary for the non parametric modelling of random uncertainties 5 the estimation of the dispersion parameters of the non parametric probabilistic model using experimental data 6 the method to solve the stochastic equation of the dynamical system with non parametric probabilistic model of random uncertainties 7 a numerical simulation and the validation for the simple example

analysis of discrete vibratory systems with parameter uncertainties part i eigensolution abstract a new analytical method is proposed for the estimation of eigensolutions for undamped and proportionally damped discrete vibratory systems when the system parameters are uncertain or random variables given several simplifying assumptions a direct product technique is used to estimate the first and second moments of eigensolutions in terms of the moments of system parameter matrices the proposed methodology is reasonably accurate and computationally fast when compared with existing methods such as the first perturbation method and the monte carlo simulation application of the new method is demonstrated through several single and multi degree of freedom systems it is seen that the standard deviation predictions match well with the results yielded by the monte carlo simulation unlike the first order perturbation method especially when moderately large random fluctuations are considered the impulse response issue is addressed in the companion paper part ii

statistics of complex eigenvalues in friction induced vibration abstract self excited vibrations appear in many mechanical systems with sliding contacts there are several mechanisms whereby friction can cause the self excited vibration to become unstable of these mechanisms mode coupling is thought to be responsible for generating annoying high frequency noise and vibration in brakes conventionally in order to identify whether a system is stable or not complex eigenvalue analysis is performed however what has recently received much attention of researchers is the variability and uncertainty of input variables in the stability analysis of self excited vibrations for this purpose a second order perturbation method is extended and employed in the current study the moments of the output distribution along with its joint moment generating function are used for quantifying the statistics of the complex eigenvalues moreover the eigen derivatives required for the perturbation method are presented in a way that they can deal with the asymmetry of the stiffness matrix and non proportional damping since the eigen derivatives of such systems are complex valued numbers it is mathematically more informative and convenient to derive the statistics of the eigenvalues in a complex form without decomposing them into two real valued real and imaginary parts then the variance and pseudo variance of the complex eigenvalues are used for determining the statistics of the real and imaginary parts the reliability and robustness of the system in terms of stability can also be quantified by the approximated output distribution

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions a new generalized probabilistic approach of uncertainties is proposed for computational model in structural linear dynamics and can be extended without difficulty to computational linear vibroacoustics and computational non linear structural dynamics this method allows the prior probability model of each type of uncertainties model parameter uncertainties and modeling errors to be separately constructed and identified the modeling errors are not taken into account with the usual output prediction error method but with the nonparametric probabilistic approach of modeling errors recently introduced and based on the use of the random matrix theory the theory an identification procedure and a numerical validation are presented then a chaos decomposition with random coefficients is proposed to represent the prior probabilistic model of random responses the random germ is related to the prior probability model of model parameter uncertainties the random coefficients arc related to the prior probability model of modeling errors and then depends on the random matrices introduced by the nonparametric probabilistic approach of modeling errors a validation is presented finally a future perspective is introduced when experimental data are available the prior probability model of the random coefficients can be improved in constructing a posterior probability model using the bayesian approach

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

higher order eigenpair perturbations a matrix method is presented for the exact calculation of large eigenpair perturbations arising from design variable changes this method of forward analysis is presented that takes into account all order of the perturbation expansion

canonical basis for type a4 i monomial elements the global crystal basis or canonical basis plays an important role in the theory of the quantum groups and their representations the tight monomials are the simplest elements in the canonical basis based on lusztig and xi s work the regions of tight monomials in quantized enveloping algebra of type a4 are determined in this paper 2003 elsevier science usa all rights reserved

an approximate solution scheme for the algebraic random eigenvalue problem a reduced basis formulation is presented for the efficient solution of large scale algebraic random eigenvalue problems this formulation aims to improve the accuracy of the first order perturbation method and also allow the efficient computation of higher order statistical moments of the eigenparameters in the present method the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for ritz analysis of the governing random eigenvalue problem this leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest since only two basis vectors are used to represent each eigenvector explicit expressions for the random eigenvalues and eigenvectors can readily be derived this enables the statistics of the random eigenparameters and the forced response to be efficiently computed numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system it is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

an efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions summary this paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on gaussian process model gpm the proposed gpm based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions because of the use of gpm the closed form expressions of mean and variance can be derived by decomposing high dimensional integrals into one dimensional integrals this paper details on how to efficiently compute the one dimensional integrals when the parameters are either uniformly or normally distributed the one dimensional integrals can be analytically evaluated while when parameters do not follow normal or uniform distributions this paper adopts the effective gaussian quadrature technique for the fast computation of the one dimensional integrals as a result the developed gpm method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions the proposed gpm method is applied to a collection of examples and its accuracy and efficiency is compared with monte carlo simulation which is used as benchmark solution results show that the proposed gpm method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency copyright 2016 john wiley sons ltd

an improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem a technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem the improved method very simply replaces the usual b norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by bp norms where bp is the perturbed b matrix with increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions numerical studies are presented to support the analysis

interaural correlation and the binaural summation of loudness the effect of interaural correlation on the loudness for noisebands was measured using a loudness matching task in naive listeners the task involved a sequence of loudness comparisons for which the intensity of one stimulus in a given comparison was varied using a one up one down adaptive rule the task provided an estimate of the level difference in decibels for which two stimulus conditions have equal loudness giving measures of loudness difference in equivalent decibel units dbequiv concurrent adaptive tracks measured loudness differences between 1 0 and 1 and between these binaural stimuli and the monaural case for various noisebands for all noisebands monaural stimuli required approximately 6db higher levels than 1 for equal loudness for most noisebands 1 and 1 were almost equal in loudness with 1 being slightly louder in the majority of measurements while 0 was about 2dbequiv louder than 1 or 1 however noisebands with significant high frequency energy showed smaller differences for 3745 4245hz 0 was only about 0 85dbequiv louder than 1 and for 100 5000hz it was non significantly louder perhaps 0 7dbequiv

polynomial chaos representation of spatio temporal random fields from experimental measurements two numerical techniques are proposed to construct a polynomial chaos pc representation of an arbitrary second order random vector in the first approach a pc representation is constructed by matching a target joint probability density function pdf based on sequential conditioning a sequence of conditional probability relations in conjunction with the rosenblatt transformation in the second approach the pc representation is obtained by having recourse to the rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target spearman s rank correlation coefficient srcc matrix both techniques are applied to model an experimental spatio temporal data set exhibiting strong non stationary and non gaussian features the data consists of a set of oceanographic temperature records obtained from a shallow water acoustics transmission experiment 1 the measurement data observed over a finite denumerable subset of the indexing set of the random process is treated as a collection of observed samples of a second order random vector that can be treated as a finite dimensional approximation of the original random field a set of properly ordered conditional pdfs that uniquely characterizes the target joint pdf in the first approach and a set of target marginal pdfs and a target srcc matrix in the second approach are estimated from available experimental data digital realizations sampled from the constructed pc representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy the relative advantages and disadvantages of the two proposed techniques are also highlighted

a method for solving stochastic eigenvalue problems ii we present a novel approach for calculating the stochastic eigenvalues of a linear operator this is an extension of our earlier work on the subject and introduces a significant simplification that reduces the amount of numerical work required when evaluating the polynomial chaos expansions improves the accuracy and enables higher harmonics to be studied examples based on several types of problem involving differential equations and matrices have been analysed and detailed numerical results obtained

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

dynamic analysis using a reduced basis of exact modes and ritz vectors the problem of obtaining an approximate solution for the response of a structure under dynamic forces is addressed the structure is modeled using mass and stiffness matrices and has n degrees of freedom an dimensional approximation is sought in a subspace defined by an arbitrary m orthogonal array a residual vector which is smaller in magnitude the more accurate the approximation is derived the residual contains two terms one is the error involved representing the force vector in the subspace the other is smaller the better the array leads to m approximate solutions of the exact eigenproblem numerical experiments are performed using arrays that are combinations of exact eigenvectors and of ritz vectors in one case when 11 different 10 dimensional bases are used to approximate the solution of a 60 degree of freedom problem it is found that a root mean square measure of relative error is smallest for the basis formed using the 6 lowest exact eigenvectors plus 4 ritz vectors generated from the spatial distribution of the loading

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

an invariant subspace based approach to the random eigenvalue problem of systems with clustered spectrum the repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix which makes capturing the behavior of these eigenpairs very difficult similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean in addition the methods to solve the random eigenvalue problem often differ in characterizing the problem which leads to different interpretations of the solution thus the solutions obtained from different methods become mathematically incomparable these two issues the difficulty of solving and the non unique characterization are addressed here a different approach is used where instead of tracking a few individual eigenpairs the corresponding invariant subspace is tracked the spectral stochastic finite element method is used for analysis where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors however the main concept of tracking the invariant subspace remains mostly independent of any such representation the approach is successfully implemented in response prediction of a system with repeated natural frequencies it is found that tracking only an invariant subspace could be sufficient to build a modal based reduced order model of the system copyright c 2012 john wiley sons ltd

extended kirsch combined method for eigenvalue reanalysis based on the kirsch combined method and the rayleigh quotient a new eigenvalue reanalysis method for modified structures is developed using this method the accuracy of the eigenvalues reanalysis of large modifications of structural parameters is improved as an application of the method a numerical example is given to illustrate the accuracy by comparing the second order perturbation method with the proposed method if the large modifications of structural parameters are made the calculated results show that the proposed method is effective

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

sampling of closely spaced ordered set of uniformly distributed random variables summary when the individual pdfs of closely spaced random variables such as natural frequencies of a structure overlap generation of sample sets by assuming the frequencies to be independent random variables can lead to incorrect sets of frequencies in the sense that the frequencies do not remain as ordered sets rejection of such disordered sample sets results in individual density functions that are significantly different from the distributions initially assumed for sampling each random variable one way to overcome this constraint in the simulation of an ordered set of random variables is to consider them in an implicit manner using a joint pdf in this paper we present a formulation for a joint density function that is developed using fundamental probability approaches the formulation ensures that the sampled random variables always remain as ordered sets and maintain the individual density functions for each variable application of the proposed formulation is illustrated for cases with not just two closely spaced variables but also for a case with multiple closely spaced variables such that the pdfs of more than two random variables overlap with each other an expression is presented to determine the exact number of terms needed in the formulation however it is also illustrated that only two terms are sufficient in most applications even when the exact number of terms needed is very high copyright 2011 john wiley sons ltd

tight monomials and the monomial basis property abstract we generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized cartan matrices and a monomial basis property of those associated with symmetric classical cartan matrices to their respective symmetrizable case we then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word as an application we develop an algorithm to compute all tight monomials in the rank 2 dynkin case

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

reanalysis of linear and nonlinear structures using iterated shanks transformation reanalysis of structures is a common task in two important fields of structural engineering namely optimization and reliability it implies the solution of the kinematic equations for each new set of data in order to simplify this task several approximate methods have been developed in this paper a simple method for reanalysis applicable to either linear or nonlinear structures is proposed it is based on the improvement of the neumann resolvent series which diverges beyond a certain radius by means of the iterated shanks transformation the method is compared with two relevant proposals intended to improve the convergence of the neumann series namely the pade series and the scaled approximation methods the accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

approximate methods for stochastic eigenvalue problems we consider the discretization and solution of eigenvalue problems of elliptic operators with random coefficients for solving the resulting systems of equations we present a new and efficient spectral inverse iteration based on the stochastic galerkin approach with respect to a polynomial chaos basis the curse of dimensionality inherent in normalization over parameter spaces is avoided by a solution of a non linear system of equations defining the galerkin coefficients for reference we also present an algorithm for adaptive stochastic collocation functionality of the algorithms is demonstrated by applying them on four examples of a given model problem convergence of the galerkin based method is analyzed and the results are tested against the collocated reference solutions and theoretical predictions

an interpolation scheme for the approximation of dynamical systems abstract the demand of many industries for highly accurate predictions of the structural response of dynamical systems leads to a rapidly growing complexity of the established numerical models due to the thereby arising large computational efforts the use of model reduction techniques becomes an essential task model reduction procedures in the context of deterministic structural analysis are already well developed and widely used such as guyan reduction and component mode synthesis techniques however the assumption of deterministically known structural properties is in most cases not realistic hence the rational propagation of uncertainties becomes an indispensable task in the robust prediction of the structural response the associated multiple simulation runs of complex models bring the topic of model reduction to the foreground of technical and scientific interest in this work a novel method which approximates eigenfrequencies and mode shape vectors needed for the evaluation of frequency response functions will be presented this meta model of structural matrices is established by employing interpolation between the matrices evaluated at some supporting points the technique has no difficulties with closely spaced eigenfrequencies and mode switching and it can be employed over the entire frequency range on the full model but also on the substructures and it is hence applicable together with deterministic model reduction techniques

canonical basis for typeb2 abstract a quantized enveloping algebra has a remarkable basis called the canonical basis or global crystal basis for type a 2 the canonical basis is described explicitly the purpose of this paper is to find the canonical basis for type b 2 the main result is theorem 2 2

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

a hybrid approach for the time domain analysis of linear stochastic structures abstract a new hybrid approach for the time domain analysis of linear stochastic structures with uncorrelated or correlated random variables is proposed this new hybrid approach combines the modal approach the second order perturbation technique and the number theoretical method ntm in the hybrid approach an approximate stochastic model of the finite element fe model is developed in the sense of first order accuracy which provides second order estimations of the mean and covariance matrices of structural responses compared with the fe model the proposed model is more convenient in terms of the computation effort it requires by employing the ntm to evaluate the statistical moments of solutions of the approximate model the secular terms contained in the results of the perturbation stochastic finite element method are eliminated two numerical examples are presented to demonstrate the accuracy and efficiency of the method proposed

eigensolution reanalysis of modified structures using epsilon algorithm based on the neumann series expansion and epsilon algorithm a new eigensolution reanalysis method is developed in the solution process the basis vectors can be obtained using the matrix perturbation or the neumann series expansion to construct the vector sequence and then using the epsilon algorithm table to obtain the approximate eigenvectors the approximate eigenvalues are computed from the rayleigh quotients the solution steps are straightforward and it is easy to implement with the general finite element analysis system two numerical examples a 40 storey frame and a chassis structure are given to demonstrate the application of the present method by comparing with the exact solutions and the kirsch method solutions it is shown that the excellent results are obtained for very large changes in the design and that the accuracy of the epsilon algorithm is higher than that of the kirsch method and the computation time is less than that of the kirsch method

modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties this paper presents two efficient modal approaches as an alternative to direct formulations for the time harmonic dynamic analysis of structures with random material and shape parameters in both approaches the structural eigenproblem is solved and complemented by a sensitivity analysis to the random parameters the modal perturbation stochastic finite element method sfem then condenses the response sensitivities to assess the response variability the mixed perturbation monte carlo sfem assesses the response statistics by sampling the structural eigenmodes according to the perturbation estimation of their probability distribution functions pdfs geometric uncertainties are handled through an appropriate shape parameterisation and a shape design sensitivity analysis two numerical applications examine both approaches in terms of accuracy variability level and computational requirements the applications involve a plate bending problem with random young modulus or edge length and a plate with a random flatness default particular observations related to the influence of the parameter pdfs in simulation based methods are also provided

a solution of the random eigenvalue problem by crossing theory abstract a method is developed for finding probabilistic descriptors of the eigenvalues of stochastic real valued symmetric matrices the method is based on the mean rate at which the characteristic polynomial of a stochastic matrix crosses level zero the proposed approach can also be applied when additional information is available on the value of the first or the first few eigenvalues several examples are presented in the paper to demonstrate the application and usefulness of the method

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a solution of the random eigenvalue problem by a dimensional decomposition method summary this paper presents a dimensional decomposition method for obtaining probabilistic descriptors of realvalued eigenvalues of positive semi definite random matrices the method involves a novel function decomposition allowing lower variate approximations of eigenvalues lower dimensional numerical integration for statistical moments and lagrange interpolation facilitating efficient monte carlo simulation for probability density functions compared with commonly used perturbation and recently developed asymptotic methods no derivatives of eigenvalues are required by the new method developed results of numerical examples from structural dynamics indicate that the decomposition method provides excellent estimates of moments and probability densities of eigenvalues for various cases including closely spaced modes and large statistical variations of input copyright 2006 john wiley sons ltd

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

robust updating of uncertain computational models using experimental modal analysis in this paper a methodology is presented to perform the robust updating of complex uncertain dynamic systems with respect to modal experimental data in the context of structural dynamics because both model uncertainties and parameter uncertainties must be considered in the computational model the uncertain computational model is constructed by using the nonparametric probabilistic approach we present an extension to the probabilistic case of the input error methodology for modal analysis adapted to the deterministic updating problem it is shown that such an extension to the robust updating context induces some conceptual difficulties and is not straightforward the robust updating formulation leads us to solve a mono objective optimization problem in the presence of inequality probabilistic constraints a numerical application is presented to show the efficiency of the proposed method

random eigenvalue problems in structural dynamics experimental investigations an experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function frf under dynamic loading is presented experiments were conducted on one hundred nominally identical realizations of a fixed fixed beam each realisation was obtained by placin g twelve identical masses at random spatial locations generated by a computer along the length of the beam the total random mass is about 2 of the total mass of the beam these experiments contain parametric uncertainty in the mass matrix and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist predictions from monte carlo simulation of deterministic finite elements model are compared with experiments it is co ncluded that the method of estimation of natural frequencies from frfs and the spatial location of the measurements has significant influence upon the first two moments mean and standard deviation of the natural frequency ensemble furthermore whilst the monte carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies the probability density function differ appreciably within the limits of the sample size investigated in this study

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

polynomial chaos representation of a stochastic preconditioner a method is developed in this paper to accelerate the convergence in computing the solution of stochastic algebraic systems of equations the method is based on computing via statistical sampling a polynomial chaos decomposition of a stochastic preconditioner to the system of equations this preconditioner can subsequently be used in conjunction with either chaos representations of the solution or with approaches based on monte carlo sampling in addition to presenting the supporting theory the paper also presents a convergence analysis and an example to demonstrate the significance of the proposed algorithm

rapid reanalysis of the generalized eigenvalue problem of locally modified linear dynamical systems abstract a computationally efficient method for the reanalysis of the damped generalized eigenvalue problems associated with linear dynamical systems with local damping and stiffness modifications is developed no restrictions are imposed on the damping of the system and the modifications in the damping and the stiffness can be independent of each other the application of the developed method to the undamped generalized eigenvalue problem with mass and stiffness modifications is also discussed the proposed method achieves up to three orders of magnitude gains in computational efficiency compared to the conventional solution procedures

iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems a new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non symmetric matrix is proposed the algorithm extends the deterministic inverse power method using a spectral discretization approach the convergence and accuracy of the algorithm is studied for both symmetric and non symmetric matrices the method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors

eigenvalue error analysis of viscously damped structures using a ritz reduction method the efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem an analysis of this technique in the case of a single damper is presented a priori and a posteriori error estimates are generated and tested on numerical examples comparison theorems with modally truncated models and a markov parameter matching reduced order model are derived these theorems corroborate the heuristic that residual flexibility methods improve low frequency approximation of the system the analysis leads to other techniques for eigenvalue approximation approximate closed form solutions are derived that include a refinement to eigenvalue derivative methods for approximation an efficient newton scheme is also developed a numerical example is presented demonstrating the effectiveness of each of these methods