approximate optimal projection for reduced order models summary a critical component of a reduced order model is the projection that maps the original high fidelity system on to the reduced order basis in this manuscript we develop a projection for linear systems that is optimal in an operator independent norm we derive an expression for this projection as a galerkin projection plus another component that is interpreted as the effect of the scales that live outside the reduced order basis we note that the exact form of this projection does not lead to a viable computational method because it involves the inverse of the original high fidelity operator we approximate this inverse by an inexpensive preconditioner and create a practical method whose costs are of the same order as the galerkin method we test the performance of this method on heat conduction and advection diffusion problems while using the incomplete lu preconditioner as an approximate to the inverse of the original operator and conclude that it provides more accurate results than the galerkin projection copyright 2015 john wiley sons ltd

estimation of time to stationarity in geometrically nonlinear structural responses abstract the determination of the time to stationarity is defined here through the convergence to its stationary limit of the nonstationary variance this function is estimated through monte carlo simulations and a two step autoregressive based modeling approach is employed to minimize the effects of randomness introduced by the limited number of simulations that can be carried out the methodology is applied extensively to nonlinear single degree of freedom models but is also demonstrated on 8 mode reduced order models of clamped clamped straight and curved beams in the latter case it is found that the time to stationarity is strongly dependent on the excitation level i e varying from one case to another by a factor at least larger than 4 thereby emphasizing the interest in estimating it beforehand it is finally shown that the single degree of freedom results tabulated here may be used for these reduced order models to obtain a first estimate of the time to stationarity

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

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

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

proper orthogonal decomposition pod of a class of vibroimpact oscillations abstract this study employs the method of proper orthogonal decomposition pod also known as the karhunen loeve k l method to extract dominant coherent structures modes that approximate the system behavior from time series data these mode shapes can be used in a galerkin reconstruction process to obtain lower dimensional models for the structural systems under consideration the mode shapes and the energies obtained by this process can also be used to predict certain kinds of periodic or non periodic non linear motions the k l method has been applied successfully to fluid dynamical thermal processes and signal processing however only a handful of works exist in the area of vibration analysis and structural mechanics an extensive k l analysis is performed numerically for two vibroimpacting systems a beam and a rotor lower dimensional models are created and used to study non linear energy transmission from low to higher k l modes extensive reconstructions are also performed to prove the efficacy of the k l method to provide accurate low dimensional dynamical models in addition experimental investigations are presented for the case of the overhung impacting rotor and qualitative comparisons with the theoretical analysis are presented

uncertainty based experimental validation of nonlinear reduced order models abstract this paper presents the first results of a combined experimental computational investigation focused on the validation of reduced order models of geometrically nonlinear structures in the presence of uncertainty the validation approach considered here is based on the premise that the model is valid if the experimental results can be considered as random sample responses of the stochastic system of which the reduced order model is the mean for the situation considered here the power spectra of the experiments should lie within the 2nd and 98th percentiles of the response forming the uncertainty band of the stochastic model nominally clamped clamped beams are considered to demonstrate the entire process the construction of two mean reduced order models and their stochastic counterparts are first performed then the validation effort is carried out by comparing experimentally obtained power spectra and their corresponding computational uncertainty bands this process leads for both reduced order models to a very good representation of the important upper envelope 98th percentile of the experimental data but a less good fit of the lower envelope 2nd percentile

a nonparametric probabilistic approach for quantifying uncertainties in low dimensional and high dimensional nonlinear models summary a nonparametric probabilistic approach for modeling uncertainties in projection based nonlinear reduced order models is presented when experimental data are available this approach can also quantify uncertainties in the associated high dimensional models the main underlying idea is twofold first to substitute the deterministic reduced order basis rob with a stochastic counterpart second to construct the probability measure of the stochastic reduced order basis srob on a subset of a compact stiefel manifold in order to preserve some important properties of a rob the stochastic modeling is performed so that the probability distribution of the constructed srob depends on a small number of hyperparameters these are determined by solving a reduced order statistical inverse problem the mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations its potential is demonstrated through several example problems from computational structural dynamics copyright 2016 john wiley sons ltd

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

nonparametric probabilistic approach of uncertainties for elliptic boundary value problem the paper is devoted to elliptic boundary value problems with uncertainties such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media model uncertainties are induced by the mathematical physical process which allows the boundary value problem to be constructed from the design system if experiments are not available the bayesian approach cannot be used to take into account model uncertainties recently a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non linear dynamical systems nevertheless the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem for instance for a linear elastostatic problem relative to an elastic bounded domain we then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems the theory and its validation are presented

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

response bounds for complex systems with a localised and uncertain nonlinearity predicting the vibration response of complex nonlinear structures is a significant challenge the response may involve many modes of the structure nonlinearity precludes the use of efficient techniques developed for linear systems and there is often uncertainty associated with the nonlinear law even to the extent that its functional form is not always known this paper builds on a recently developed method for handling this class of problem in a novel way the method exploits the fact that nonlinearities are often spatially localised and seeks the best and worst case system response with respect to a chosen metric by regarding the internal nonlinear force as an independent excitation to the underlying linear system constraints are used to capture what is thought to be known about the nonlinearity without needing to specify a particular law this paper focuses on the case of systems with a single point nonlinearity but with arbitrarily complex underlying linear dynamics driven by a sinusoidal force excitation semi analytic upper and lower bounds are proposed for root mean square response metrics subject to constraints which specify that the nonlinearity should be a combination of a passive b displacement limited and or c force saturating the concept of equivalent linear bounds is also introduced for cases where the response metric is thought to be dominated by the same frequency as the input the bounds corresponding to a passive and displacement limited nonlinearity are compared with monte carlo experimental and numerical results from an impacting beam test rig the bounds corresponding to a passive and force saturating nonlinearity are compared with numerical results for a friction damped beam the global upper and lower bounds are satisfied for all input frequencies but are generally found to be rather conservative the equivalent linear bounds show remarkably good agreement for predicting the range of root mean square velocity responses finally the principle of maximum entropy is used to estimate the response distributions which was found to give surprisingly good agreement with experimental and numerical data

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

post buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation the paper presents a complete experimental validation of an advanced computational methodology adapted to the nonlinear post buckling analysis of geometrically nonlinear structures in presence of uncertainty a mean nonlinear reduced order computational model is first obtained using an adapted projection basis the stochastic nonlinear computational model is then constructed as a function of a scalar dispersion parameter which has to be identified with respect to the nonlinear static experimental response of a very thin cylindrical shell submitted to a static shear load the identified stochastic computational model is finally used for predicting the nonlinear dynamical post buckling behavior of the structure submitted to a stochastic ground motion

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 combined modal finite element analysis technique for the dynamic response of a non linear beam to harmonic excitation abstract in this paper a method is proposed for modelling large deflection beam response involving multiple vibration modes significant savings in computational time can be obtained compared with the direct integration non linear finite element method the deflections from a number of static non linear finite element test cases are transformed into modal co ordinates using the modes of the underlying linear system regression analysis is then used to find the unknown coupled non linear modal stiffness coefficients the inclusion of finite element derived modal masses and an arbitrary damping model completes the governing non linear equations of motion the response of the beam to excitation of an arbitrary nature may then be found using time domain numerical integration of the reduced set of equations the work presented here extends upon the work of previous researchers to include non linearly coupled multi modal response the particular benefits of this approach are that no linearization is imposed and that almost any commercial finite element package may be employed without modification the proposed method is applied to the case of a homogeneous isotropic beam fully simply supported and fully clamped boundary conditions are considered for the free vibration case results are compared to those of previous researchers for the case of steady state harmonic excitation results are compared with the direct integration non linear finite element method using abaqus in all cases excellent agreement is obtained

development of optimally disordered critical random excitation the problem of determining the critical power spectral density psd function of a partially specified stationary gaussian load process which maximizes the response of a linear system has been considered the partial specification of the load is given only in terms of its total average energy the critical input psd turns out to be highly narrow banded which fails to capture the erratic nature of the excitation consequently the trade off curve between the maximum linear system response and the disorder in the input process quantified in terms of its entropy rate has been generated the pareto optimization theory is used to tackle the conflicting objectives of simultaneous maximization of the system response and the input entropy rate consequently the non linear multi objective optimization has been carried out using a multi criteria genetic algorithm scheme an illustrative example of determining the critical input of an axially vibrating rod excited by a partially specified stationary gaussian load process has been considered

reliability based critical earthquake load models part 2 nonlinear structures the problem of determining critical stochastic earthquake excitation models for simple nonlinear systems under single point or multi point nonstationary seismic inputs is considered the earthquake acceleration components are obtained by multiplying known deterministic enveloping functions with zero mean gaussian stationary random processes with unknown auto power spectral density functions for single point excitations and power spectral density matrix for multi point excitations the definition of critical earthquake input is based on the notion of a performance function the system is considered to have failed if the maximum response over a given time interval exceeds specified limits the critical excitations are defined as those that minimize the hasofer lind reliability index associated with this performance function the computation of this index in turn is based on the use of response surface to model the limit surface near the check point here the quantity to be optimally determined is taken to be the unknown input power spectral density function for single point excitations or the input power spectral density matrix for multi point excitations the excitations are taken to satisfy constraints on total average energy zero crossing rate lower bounds on entropy rate and other positivity and bounding requirements that are of mathematical nature the resulting constrained nonlinear optimization problems are solved using the sequential quadratic programming method illustrative examples for computing random critical excitations for singly supported and multiply supported oscillators that have cubic force displacement relations are provided

brief detecting period doubling bifurcation an approximate monodromy matrix approach a quasi analytical approach is developed for detecting period doubling bifurcation emerging near a hopf bifurcation point the new algorithm employs higher order harmonic balance approximations hbas to compute the monodromy matrix useful for the study of limit cycle bifurcations prediction of the period doubling bifurcation is accomplished very accurately by using this algorithm along with a detailed approximation error analysis without using numerical integration of the dynamical system an example is given to illustrate the results

a state space error estimate for pod deim nonlinear model reduction this paper derives state space error bounds for the solutions of reduced systems constructed using proper orthogonal decomposition pod together with the discrete empirical interpolation method deim recently developed for nonlinear dynamical systems siam j sci comput 32 2010 pp 2737 2764 the resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected pod basis vectors both in galerkin projection of the reduced system and in the deim approximation of the nonlinear term the analysis is particularly relevant to ode systems arising from spatial discretizations of parabolic pdes the derivation clearly identifies where the parabolicity is crucial it also explains how the deim approximation error involving the nonlinear term comes into play

stochastic reduced order models for uncertain geometrically nonlinear dynamical systems a general methodology is presented for the consideration of both parameter and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems the approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model the mean model uncertainty both from parameters and model is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model i e while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model this randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion affects all coefficients of the reduced order model and is computationally easily achieved the reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented these applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems

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

generalized polynomial chaos and random oscillators we present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions which are more efficient than the classical wiener hermite expansions the approach is general but here we present results for linear oscillators only with random forcing or random coefficients in this context we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution we have also performed computational comparisons with monte carlo simulations which show that the new approach can be orders of magnitude faster especially for compact distributions

time domain formulation in computational dynamics for linear viscoelastic media with model uncertainties and stochastic excitation the paper is devoted to the computational time domain formulation of linear viscoelastic systems submitted to a nonstationary stochastic excitation and in the presence of model uncertainties which are modeled in the framework of the probability theory the objective is to introduce and to develop an adapted and complete formulation of such a problem in the context of computational mechanics a reduced order model in the time domain with stochastic excitation is derived from the computational model for the reduced order model the stochastic modeling of both computational model parameter uncertainties and modeling errors is carried out using the nonparametric probabilistic approach and the random matrix theory we present a new formulation of model uncertainties to construct the random operators for viscoelastic media we then obtained a linear stochastic integro differential equation side with random operators and with a stochastic nonhomogeneous part stochastic excitation a time discretization of this side is proposed in a first step the side is transformed to a linear ito stochastic differential equation isde with random operators then the isde is discretized using an extension of the stormer verlet scheme which is a particularly well adapted algorithm for long time good behavior of the numerical solution finally for the stochastic solver and statistical estimations of the random responses we propose to use the monte carlo simulation for gaussian and non gaussian excitations

buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections abstract in this paper the effect of random geometric imperfections on the critical load of isotropic thin walled cylindrical shells under axial compression with rectangular cutouts is presented second moment characteristics of geometric imperfections are estimated by data of available measurements a simulation procedure based on the karhunen loeve expansion is applied for generating realizations of geometric imperfections nonlinear static finite element analyses are carried out for the calculation of the response statistics of the critical load of the cylindrical shells cumulative distribution functions of the critical load as obtained by direct monte carlo simulation are presented furthermore the individual and combined effects of random boundary and geometric imperfections on the limit loads of isotropic thin walled cylindrical shells under axial compression are also treated again second moment characteristics of these imperfections are estimated by data of available measurements of imperfections a simulation procedure also based on the karhunen loeve expansion is applied for generating realizations of both boundary and geometric imperfections a nonlinear static finite element analysis using the general purpose code stags c c rankin f a brogan w a loden h d cabiness stags structural analysis of general shells user manual lmsc p032594 version 3 0 lockheed martin missiles and space co inc palo alto ca usa 1998 is carried out for the calculation of the buckling response of the cylindrical shells finally the cumulative distribution functions of the limit load using direct monte carlo simulation are shown

nonparametric stochastic modeling of structures with uncertain boundary conditions coupling between substructures the focus of this investigation is on the formulation and validation of a novel approach for the inclusion of uncertainty in the modeling of the boundary conditions of linear structures and of the coupling between linear substructures this work is particularly relevant to complex structures assembled from simpler substructures as in aerospace applications first a mean structural dynamic model that includes boundary condition coupling flexibility is obtained using classical substructuring concepts the application of the nonparametric stochastic modeling approach to this mean model is next described and thus permits the consideration of both model and parameter uncertainty finally the identification of the two parameters of this stochastic model the mean boundary condition coupling flexibility and the overall level of uncertainty is shown and discussed the methodology is demonstrated on three different structural dynamic models of a rectangular plate and of two different wings

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

harmonic balance analysis of period doubling bifurcations with implications for control of nonlinear dynamics the harmonic balance method is applied to the analysis of period doubling bifurcations in a general class of nonlinear feedback systems compact conditions for the prediction and stability analysis of period doubling bifurcations are obtained specializations of these conditions for systems in which the nonlinear subsystem is static are given the implications of the results to control design are illustrated through the development of a design procedure for solving a representative control problem the objectives of the control design can include delay of a given period doubling bifurcation as well as stability objectives the results are illustrated with a detailed example

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

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

uncertainty quantification in computational linear structural dynamics for viscoelastic composite structures this paper deals with the analysis of a stochastic reduced order computational model in computational linear dynamics for linear viscoelastic composite structures in the presence of uncertainties the computational framework proposed is based on a recent theoretical work that allows for constructing the stochastic reduced order model using the nonparametric probabilistic approach in the frequency domain the generalized damping matrix and the generalized stiffness matrix of the stochastic computational reduced order model are random matrices due to the causality of the dynamical system these two frequency dependent random matrices are statistically dependent and are linked by a compatibility equation induced by the causality of the system involving a hilbert transform the computational aspects related to the nonparametric stochastic modeling of the reduced stiffness matrix and the reduced damping matrix that are frequency dependent random matrices are presented a dedicated numerical approach is developed for obtaining an efficient computation of the cauchy principal value integrals involved in those equations for which an integration over a broad frequency domain is required a computational analysis of the propagation of uncertainties is conducted for a composite viscoelastic structure in the frequency range it is shown that the uncertainties on the damping matrix have a strong influence on the observed statistical dispersion of the stiffness matrix

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

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

acceleration of uncertainty propagation through lagrange multipliers in partitioned stochastic method abstract a partitioned stochastic method psm is proposed for the solution of static structural mechanics problems with uncertainties whose solution vectors are the displacements for each partition and lagrange multipliers along with the partition interfaces the proposed partitioned stochastic method employs three stochastic basis selection steps an arbitrary initial choice of displacement random bases a set of conjugate bases for the lagrange multipliers and finally modification of the displacement bases affected by those of the lagrange multipliers the present psm thus propagates the uncertainties instantly across partitioned substructures resulting in an improved rate of convergence numerical experiments illustrate the proposed psm outperforms a conventional partitioned solution method for structural mechanics problems

buckling analysis of imperfect shells with stochastic non gaussian material and thickness properties in this paper the effect of material and thickness spatial variation on the buckling load of isotropic shells with random initial geometric imperfections is investigated to this purpose a random spatial variability of the elastic modulus as well as of the thickness of the shell is introduced in addition to the random initial geometric deviations of the shell structure from its perfect geometry the main novelty of this paper compared to previous works is that a non gaussian assumption is made for the distribution of the two aforementioned uncertain parameters i e the modulus of elasticity and the shell thickness which are described by two dimensional uni variate 2d 1v homogeneous non gaussian stochastic fields the initial geometric imperfections are described as a 2d 1v gaussian non homogeneous stochastic field with properties derived from corresponding experimental measurements numerical examples are presented focusing on the influence of the non gaussian assumption on the variability of the buckling load which is calculated by means of the monte carlo simulation method it is shown that the choice of the marginal probability distribution for the description of the material and thickness variability is crucial since it affects significantly the statistics of the buckling load of imperfection sensitive shell type structures

uncertain buckling its past present and future an authoritative review on stochastic buckling of structures was written by amazigo some quarter century ago the present review summarizes some of the developments which took place since then a brief overview of the effect of uncertainty in the initial geometric imperfections elastic moduli applied forces and thickness variation is given for the benefit of the thoughtful reader the review is of critical nature

parameterized reduced order models from a single mesh using hyper dual numbers abstract in order to assess the predicted performance of a manufactured system analysts must consider random variations both geometric and material in the development of a model instead of a single deterministic model of an idealized geometry with idealized material properties the incorporation of random geometric variations however potentially could necessitate the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed which would require an impractical number of man hours to complete this research advances a recent approach to uncertainty quantification by developing parameterized reduced order models these parameterizations are based upon taylor series expansions of the system s matrices about the ideal geometry and a component mode synthesis representation for each linear substructure is used to form an efficient basis with which to study the system the numerical derivatives required for the taylor series expansions are obtained via hyper dual numbers and are compared to parameterized models constructed with finite difference formulations the advantage of using hyper dual numbers is two fold accuracy of the derivatives to machine precision and the need to only generate a single mesh of the system of interest the theory is applied to a stepped beam system in order to demonstrate proof of concept the results demonstrate that the hyper dual number multivariate parameterization of geometric variations which largely are neglected in the literature are accurate for both sensitivity and optimization studies as model and mesh generation can constitute the greatest expense of time in analyzing a system the foundation to create a parameterized reduced order model based off of a single mesh is expected to reduce dramatically the necessary time to analyze multiple realizations of a component s possible geometry

nonlinear reduced order models for thermoelastodynamic response of isotropic and functionally graded panels the focus of this investigation is on the development and validation of thermoelastic reduced order models for the geometrically nonlinear response and temperature of heated structures the reduced order modeling approach is based on a modal type expansion of both displacements and temperatures in the undeformed unheated configuration a set of coupled nonlinear differential equations governing the time varying generalized coordinates of the response and temperature expansion are derived from finite thermoelasticity using a galerkin approach furthermore the selection of the basis functions to be used in these reduced order models is discussed and the numerical evaluation of the model coefficients is addressed this approach is first validated on an isotropic beam subjected to both thermal effects and external loads the thermal effects are large enough to induce a significant buckling of the panel while the time varying loads lead to snap throughs ranging in frequency from infrequent to continuous validation to a functionally graded material panel in similar conditions is then performed in both cases the reduced order modeling predicted temperatures and responses are found to very closely match their full finite element counterparts

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

adapted polynomial chaos expansion for failure detection abstract in this paper we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions the performance of the two methods is demonstrated by a predator prey model and a chemical reaction problem

computational aspects for constructing realizations of polynomial chaos in high dimension this paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension the method proposed consists of 1 constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and 2 performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the gram schmidt orthogonalization algorithm which is not stable in high dimension a brief review of polynomial chaos expansion with arbitrary measure is given the statistically independent realizations of multivariate monomials are introduced the centered statistically independent realizations of orthonormal multivariate polynomials are developed finally a quantification of the errors induced by the usual methods is given

a computational procedure to estimate the stochastic dynamic response of large non linear fe models abstract an algorithm for the computation of the stochastic non stationary non linear response of large fe models is presented in stochastic analysis the response is described by the mean and covariance function and possibly by the probability distribution of the non linear response to estimate the covariance matrix the method of equivalent statistical linearization is applied for linearizing all non linear elements the large fully populated and symmetric covariance matrix of dimension 2 n is described by the so called karhunen loeve expansion representation which allows one to employ a feasible description in this context an efficient procedure is suggested to determine the minimal number of karhunen loeve vectors necessary to assure a sufficiently accurate representation this method in fact allows one to employ any available deterministic integration scheme to compute the karhunen loeve vectors to increase the computational efficiency modal coordinates are used to represent the linear sub system special attention is given to the effects of mode truncation which are generally not negligible for large fe systems moreover the suggested approach has no limitation w r t the size of the model in terms of degrees of freedom dofs or the number of non linear elements the feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25 50 and 100 thousand dofs containing hysteretic damping elements

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

remarks on the efficiency of pod for model reduction in non linear dynamics of continuous elastic systems the first objective of this paper is to analyse the efficiency of the reduced models constructed using the proper orthogonal decomposition pod basis and the lin basis in non linear dynamics for continuous elastic systems the pod basis is the hilbertian basis constructed with the pod method while the lin basis is the hilbertian basis derived from the generalized continuous eigenvalue problem associated with the underlying conservative part of the continuous elastic system and usually called the eigenmodes of vibration the efficiency of the pod basis or the lin basis is related to the rate of convergence in the frequency domain of the solution constructed with the reduced model with respect to its dimension a basis will be more efficient than another if the reduced order solution of the galerkin projection converges to the solution of the dynamical system more rapidly than the reduced order solution of the other

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques