topology optimization of continuum structures under uncertainty a polynomial chaos approach abstract a computational method for topology optimization in the presence of uncertainty is proposed the method combines the spectral stochastic approach for the representation and propagation of uncertainties with an existing deterministic topology optimization technique the idea in spectral stochastic formulations is to add an extra dimension a random dimension to the problem where the stochastic variability of the input parameters and outputs of interest can be modeled the resulting compact representations for the response quantities allow for efficient and accurate calculation of sensitivities of response statistics with respect to the design variables which are then fed into a gradient based optimizer that searches for the optimum design from a variety of frameworks for uncertainty informed optimization robust topology optimization is chosen to demonstrate the applicability of the method examples from continuum topology optimization under uncertainty in material properties are presented it is also shown that results obtained from the proposed method are in excellent agreement with those obtained from a monte carlo based optimization algorithm

an input normal form homotopy for the l sup 2 optimal model order reduction problem in control system analysis and design finding a reduced order model optimal in the l sup 2 sense to a given system model is a fundamental problem the problem is very difficult without the global convergence of homotopy methods and a homotopy based approach has been proposed the issues are the number of degrees of freedom the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm a homotopy algorithm based on the input normal form characterization of the reduced order model is developed here and is compared with the homotopy algorithms based on hyland and bernstein s optimal projection equations the main conclusions are that the input normal form algorithm can be very efficient but can also be very ill conditioned or even fail

buckling of stochastically heterogeneous beams using a functional perturbation method abstract the buckling load and its probabilistic nature average and variance of bernoulli beams with stochastic material bending stiffness properties is derived analytically by a new functional perturbation method fpm a buckling shape function is assumed based on the homogeneous solution and additional terms to account for the morphology effects the buckling load in the transcendental equation is treated as a functional of the bending modulus stiffness or compliance field applying a functional perturbation to the above equation the buckling load is found analytically to any desired degree of accuracy as a function of material morphology the fpm is executed using both stiffness and compliance statistical data the impact of each of the two data sources on the solution accuracy is examined showing that compliance based solutions are accurate for small correlation lengths statically indeterminate problems can be treated with no additional effort an example of a simply supported beam is solved in detail comparison with previous studies where stochastic finite element and monte carlo simulation were used showed the relative accuracy and insight capabilities of the method the clamped free case is also studied to demonstrate that symmetry conditions used for homogeneous beams to find the buckling load on the basis of a simply supported case are not valid for heterogeneous beams

some exact solutions for the bending of beams with spatially stochastic stiffness to the best of the knowledge of the authors there are no exact solutions available for the bending of beams with spatially stochastic stiffness investigators are therefore utilizing various approximate techniques in the present study a new method is developed to obtain exact solutions for first and second moments of displacements for statically determinate beams that have spatially random stiffness the method is based on the full probabilistic characterization of the random stiffness so that the solutions are valid for any value of the coefficient of variation of the stiffness the deterministic governing equations and boundary conditions derived for both first and second moments allow apparently for the first time in the literature the exact solutions for the mean and covariance functions of the displacements to be determined two governing equations are uncoupled from each other and can be solved separately several exact solutions for the mean and covariance functions are obtained to illustrate the application of the method it is hoped that the exact solutions determined will serve as benchmark solutions to enable the researchers to check the accuracy of various approximate analytical and numerical techniques on the test solutions presented in this study

response variability of stochastic frame structures using evolutionary field theory the integral form for the variance of the response of stochastic statically indeterminate structural systems involving the so called variability response function vrf and the spectral density function of the stochastic field modelling the uncertain system properties is established for the first time in this paper using evolutionary spectra theory the vrf is a function depending on deterministic parameters related to the geometry boundary conditions mean material properties and loading of the structural system no approximations are involved in the derivation of the integral form however a conjecture has to be made that is validated using monte carlo simulations the uncertain system property considered is the inverse of the elastic modulus flexibility closed form expressions can be derived in principle for the vrf of any statically determinate or indeterminate frame system using a flexibility based formulation alternatively a fast monte carlo simulation approach is provided to numerically evaluate the vrf it is shown in closed form and numerically that the vrf for statically indeterminate structures is a function of the standard deviation ff of the stochastic field modeling the inverse of the elastic modulus although the vrf depends on ff it appears to be independent of the functional form of the spectral density function of the stochastic field modeling the uncertain system properties for statically determinate structures the vrf is independent of ff the integral form can be used to compute the variance of the system response as well as its upper bound with minimal computational effort it also provides an excellent insight into the mechanisms controlling the response variability the upper bounds for the response variance are spectral and probability distribution free requiring knowledge of only the variance of the inverse of the elastic modulus the proposed bounds are realizable in the sense that it is possible to determine the probabilistic characteristics of the stochastic field that produces them several numerical examples are provided demonstrating the capabilities of the methodology

parallel solutions of very large sparse lyapunov equations by balanced bbd decompositions in this paper a method for the parallel solution of large sparse matrix equations is presented the method is based on the balanced bordered block diagonal bbd decomposition which is applied in conjunction with the successive over relaxation sor iterative method to solve large lyapunov equations in the kronecker sum representation a variety of experimental results will be presented including solutions of lyapunov equations with matrices as large as 1993 spl times 1993

efficient bounds for the monte carlo neumann solution of stochastic systems the numerical solution of uncertainty propagation problems in mechanics can be computationally demanding in this article a well known property of the neumann series is explored in order to derive lower and upper bounds for expected value and autocorrelation of stochastic system responses uncertainties in system parameters are represented as parameterized stochastic processes monte carlo simulation is employed to obtain samples of system response from which lower and upper bounds of expected value and autocorrelation are computed the proposed methodology is applied to two example problems involving beam bending and axial thermo elasticity it is shown that accurate and efficient bounds can be obtained for a proper choice of operator norm with as few as one or two terms in the neumann expansion the monte carlo neumann bounding scheme proposed herein is shown to be an efficient alternative for the solution of uncertainty propagation problems in mechanics

stochastic finite element method for random temperature in concrete structures this paper proposes numerical methods for random temperature field in mass concrete structures to research various kinds of random variability including both random environmental temperature and random material properties first presented based on local average method of random field are the formulas of random variational principle as well as the corresponding formulas of stochastic fem then discussed is the stochastic spectral response finite element method given the complex frequency functions as random functions of material parameters to resolve the effect of random environmental temperature finally the taylor expansion series is referred to simplify the non stationary process and suggest the corresponding formulas for the effect of random hydration heat of concrete the paper demonstrates that these proposed methods can help to study conveniently the random variability of temperature which could appear as both random processes and random fields in mass concrete structures numerical examples calculated by the methods theoretical and monte carlo simulation are also provided

generalized variability response functions for two dimensional elasticity problems the ability to determine probabilistic characteristics of response quantities in structural mechanics e g displacements stresses as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters the concept of the variability response function vrf has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response the key property of the vrf in its classical sense is its independence from the marginal probability distribution function pdf and the spectral density function sdf of the uncertain system parameters it depends only on the deterministic structural configuration and boundary conditions proofs have been provided for the existence of vrfs for linear and some nonlinear statically determinate beams for statically indeterminate structures the monte carlo based generalized variability response function gvrf methodology has been proposed recently as a generalization of the vrf concept to indeterminate linear and some nonlinear beams the methodology computes gvrfs which are analogous to vrfs for statically determinate structures and evaluates their dependence or lack thereof on the pdf and sdf of the random field thereby providing an estimate of the accuracy of the gvrf in this paper the gvrf methodology is extended to problems involving two dimensional linear continua whose stochasticity is characterized by statistically homogeneous random fields after detailing the gvrf methodology for two dimensional random fields two numerical examples are provided gvrfs are computed for the displacement response and for the effective compliance of linear plane stress systems

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

a comparison of homotopies for alternative formulations of the l 2 optimal model order reduction problem abstract a fundamental problem in control systems theory is finding a reduced order model that is optimal in the l 2 sense to a given full order system model the numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous a number of homotopy based approaches have been developed the primary numerical issues are the number of degrees of freedom in the homotopy parameter vector the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm this paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms the results show that the numerical well conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent

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

frequency weighted model reduction via optimal projection the problem of order reduction with frequency weighting is considered necessary conditions characterizing the reduced order model which minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting are given the problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input when the weighting matrix is singular at infinity the reduced order model has a direct transmission term even if the original system is strictly proper this additional degree of freedom can be used to obtain better approximation over a prespecified frequency range

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

an adaptive spectral galerkin stochastic finite element method using variability response functions a methodology is proposed in this paper to construct an adaptive sparse polynomial chaos pc expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields the proposed methodology utilizes the concept of variability response function vrf in order to compute an a priori low cost estimate of the spatial distribution of the second order error of the response as a function of the number of terms used in the truncated karhunen lo eve kl expansion this way the influence of the response variance to the spectral content correlation structure of the random input is taken into account through a spatial variation of the truncated kl terms the criterion for selecting the number of kl terms at different parts of the structure is the uniformity of the spatial distribution of the second order error this way a significantly reduced number of pc coefficients with respect to classical pc expansion is required in order to reach a uniformly distributed target second order error this results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method ssfem copyright c 2014 john wiley sons ltd

new formulation of fem for deterministic and stochastic beams through generalization of fuchs approach this paper proposes an alternative way of constructing the global stiffness matrix of the finite element method for bending beams it also applies the new formulation to first and second moment analysis of stochastic beams which involve spatially uncertain bending stiffness originating from fuchs idea of decoupling the shear and bending components in the bending beam the element level stiffness matrix is diagonalized the generalized stress strain strain displacement and equilibrium relationships are assembled respectively and then are combined to form the global stiffness matrix the advantage of the new formulation is that the bending stiffness explicitly appears in the global stiffness matrix the mean vector and covariance matrix of the displacement of the beam are then obtained in terms of probabilistic characteristics of the uncertain bending stiffness this is in contrast to the conventional finite element method in stochastic setting which is based on the perturbation technique the example is given to illustrate the efficacy of the new formulation and its application to bending of stochastic beams

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a note on balanced controller reduction the standard linear quadratic gaussian lqg optimal controller is not minimal in general even if the plant is minimal hence the order of the optimal controller can be less than that of the plant this note shows that the original balancing method of moore 1 can be used to generate these reduced order optimal controllers but the newer balancing methods recently proposed for controller reduction might not generate the reduced order optimal controllers even when they exist

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness abstract the paper is to develop stochastic isogeometric analysis for the free vibration of functionally graded plates with spatially varying random material properties the isogeometric analysis method is employed in the new stochastic analysis scheme which is called stochastic isogeometric analysis siga the elastic modulus and mass density are modeled as homogeneous gaussian random fields along the plane of structure the governing equation of stochastic isogeometric analysis for free vibration of functionally graded plates is derived in conjunction with perturbation expansions to predict the first and second moments of eigenvalues in order to verify proposed method the brute force monte carlo simulation is employed the mean variance and cov coefficient of variation predicted by siga and those predicted by monte carlo simulation show good agreement the numerical examples demonstrate that the randomness of material properties affects significantly the structural responses of the functionally graded plates the correlation between elastic modulus and mass density is also observed to have significant effect on the response cov of eigenvalue

a parameterization of minimal plants we present the input normal riccati parameterization a generalization of kabamba s input normal form which allows the continuous parameterization of the set of minimal linear systems of a given order that have distinct singular values the input normal riccati form has no requirement that the underlying system be stable we also present formulas for the use of the input normal riccati parameterization for the synthesis of systems via gradient methods

a new balanced canonical form for stable multivariable systems a new balanced canonical form is presented for stable multivariable linear systems overlapping continuous block balanced canonical forms were introduced by hanson ober for the stable single input single output siso case as a generalisation of the balanced canonical form introduced by ober 1987 for the siso case in the search for a generalization of these results to the multivariable case a new multivariable balanced canonical form was discovered which is of interest in its own right and is presented in this paper the new canonical form has a number of nice properties the integer invariants that appear in the canonical form are the multiplicities of the hankel singular values and a number of new invariants which are in one to one objective correspondence with the kronecker indexes of subsystems truncation of the state vector leads to stable minimal models in canonical form in the siso case the canonical form coincides with ober s balanced canonical form the reachability matrix of a system in canonical form with identical singular values is positive upper triangular

design of low order linear phase iir filters via orthogonal projection this paper presents an indirect linear phase iir filter design technique based on a reduction of linear phase fir filters the desired filter is obtained by minimizing the l sub 2 norm of the difference between the original fir filter and the lower order iir filter we first establish a relationship between the hankel singular values of the discarded part of the fir filter and the l sub 2 norm of the corresponding filter approximation error based on model truncation this result motivates us to propose a simple finite search method that will achieve better approximation results than commonly used truncation methods such as the balanced truncation bt and the impulse response gramian irg methods we then develop an iterative algorithm for finding an optimal iir filter based on a matrix projection of the original fir filter the convergence of the proposed algorithm is established filters designed using the proposed algorithm are compared with those obtained by other techniques with respect to the amplitude response and group delay characteristics in the passband numerical examples show that the proposed algorithm offers the best performance

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

an efficient 3d stochastic finite element method for failure probability analysis of high temperature components abstract the loadings on high temperature components are generally complex and the discreteness of the material strength is usually great therefore the two dimensional 2d failure probability analysis model and the deterministic finite element method dfem cannot be applied to evaluate the failure probability of asymmetrical three dimensional 3d components to overcome the drawbacks of the 2d model and the dfem an efficient 3d stochastic finite element method sfem is proposed in this paper with this method the failure probability of components subjected to complex loadings can be estimated by using the statistical analysis of the von mises stresses of element nodes meanwhile ansys and matlab were employed to carry out 3d parametric modeling solving and statistical analysis the proposed method is efficient as is verified for two cases and it can also be easily applied in practical engineering

response variability of laminate composite plates due to spatially random material parameter young s modulus is at the center of attention in the stochastic finite element analysis since the parameter plays an important role in determining structural behavior however the other material parameter of poisson s ratio is another independent material parameter that governs the behavior of structural systems accordingly the independent estimation of the influence of this parameter on the uncertain response of a system is of importance from the perspective of stochastic analysis to this end we propose a formulation to determine the response variability in laminated composite plates due to the spatial randomness of poisson s ratio to filter out the independent contribution of random poisson s ratio a decomposition of the constitutive matrix into several sub matrices by using the taylor s expansion is needed which makes the random poisson s ratio simple enough to be included in the formulation to validate the adequacy of the proposed formulation several examples are chosen and the results are compared with those given by monte carlo analysis by means of the formulation suggested here it is expected that an extension of the formulation to include the effect of correlations between random poisson s ratio and other structural and or geometrical parameters will be achieved with ease resulting in a more practical estimation of the response variability of laminated composite plates

stochastic beam equations under random dynamic loads the fourth order partial differential equation representing beams under random loading is considered a general solution for this equation is obtained using the eigenfunction and variation of parameters techniques also the average and the variance of the beam deflection shear and bending moment are obtained the load is divided into a deterministic function and a randomly perturbed function representing the expected error in the deterministic load the general closed form solution is obtained in stochastic integral terms some important statistical moments of the solution process are computed and illustrated

topology optimization of continuum structures with stress constraints and uncertainties in loading summary topology optimization using stress constraints and considering uncertainties is a serious challenge since a reliability problem has to be solved for each stress constraint for each element in the mesh in this paper an alternative way of solving this problem is employed where uncertainty quantification is performed through the first order perturbation approach with proper validation by monte carlo simulation uncertainties are considered in the loading magnitude and direction the minimum volume problem subjected to local stress constraints is formulated as a robust problem where the stress constraints are written as a weighted average between their expected value and standard deviation the augmented lagrangian method is used for handling the large set of local stress constraints whereas a gradient based algorithm is used for handling the bounding constraints it is shown that even in the presence of small uncertainties in loading direction different topologies are obtained when compared to a deterministic approach the effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied where the main observed result is loss of symmetry in optimal topologies this article is protected by copyright all rights reserved

flexibility based upper bounds on the response variability of simple beams spectral and probability distribution free upper bounds on the response variability of both statically determinate and indeterminate beams are established in the present paper based on exact closed form analytic expressions derived for the variance of the response displacement a conjecture has to be made in the case of statically indeterminate beams in order to establish these bounds the conjecture is supported through an argument postulating the existence of an integral form for the variance of the response displacement and through a brute force optimization procedure providing numerical validation such bounds require knowledge of only the variance of the stochastic field modeling the inverse of the elastic modulus and are realizable in the sense that it is possible to fully determine the probabilistic characteristics of the stochastic field modeling the inverse of the elastic modulus that produces them furthermore it is possible to fully determine also the corresponding stochastic field modeling the elastic modulus that produces these bounds these spectral and probability distribution free bounds can also be computed numerically using a so called fast monte carlo simulation procedure that does not require a closed form analytic expression for the response displacement making this approach much more general numerical examples are provided involving a statically determinate and a statically indeterminate beam

basis random variables in finite element analysis galerkin s method is applied to random operator equations appropriate hilbert spaces are defined for random functions and solutions are projected into these spaces allowing the first and second moment properties of the solution to be calculated an equivalent energy based approach similar to the rayleigh ritz method is developed from which a stochastic finite element technique is derived several one and two dimensional example problems are solved and the results discussed

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

higher order weighted integral stochastic finite element method and simplified first order application it is well known that expansion based stochastic methods are approximate schemes as they are based on a first or at most second order series expansion on the basic variable e g displacement therefore expansion based stochastic analysis schemes are bound to show small response variability when compared with monte carlo simulation mcs results and application of these schemes is limited to stochastic problems with relatively small variability in order to overcome these general drawbacks of the expansion methods we suggest a higher order stochastic field function that can be employed in the expansion based stochastic analysis scheme of the weighted integral method we then propose a new weighted integral formulation using the higher order stochastic field function the new formulation is not only applicable to stochastic problems with a high degree of uncertainty but also can reproduce the phenomenon of accelerated increase in the response variability when the coefficient of variation of the stochastic field increases as observed in the mcs in order to show the validity of the proposed formulation we provide two numerical examples and the results are discussed in detail

stochastic finite element analysis of shells with combined random material and geometric properties abstract in this work an extended stochastic formulation of the triangular composite facet shell element tric is presented for the case of combined uncertain material young s modulus poisson s ratio and geometric thickness properties these properties are assumed to be described by uncorrelated two dimensional homogeneous stochastic fields the stochastic finite element analysis of shell structures is performed using the spectral representation method for the description of the random fields in conjunction with monte carlo simulation mcs for the computation of the response variability useful conclusions regarding the influence of each one of the structural parameters on the response variability are derived from the numerical tests examined

an exact solution for the static and dynamic analysis of fe discretized uncertain structures the probabilistic analysis of fe discretized uncertain linear structures in the static and dynamic domain is the goal of this paper in particular a procedure able to give the exact relationship between the response and the random variables representing the structural uncertainties is presented under the assumption that a point discretization method is used for the representation of the uncertain random field this procedure is based on the properties of the structural deformation modes and in particular on the number of principal deformation modes of the fe type used for the structural discretization the use of these relationships leads to the exact probabilistic characterization of the response once that the probabilistic description of the uncertainty is fixed moreover they give the approximate relationship when only some of the random variables chosen by a sensitivity procedure are included in the analysis this allows to obtain an optimum level of accuracy through a reduced computational effort

analysis of mean and mean square response of general linear stochastic finite element systems a general finite element based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields a flexibility based formulation is followed that does not involve any approximations closed formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper these integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function mrf and the variability response function for the mean square response vrf1 the mrf and the vrf1 have many similarities with the classical variability response function vrf used to estimate the variance of the response displacement the existence of the mrf and the vrf1 depends on a conjecture that is validated numerically using a brute force monte carlo simulation approach a so called finite element method based fast monte carlo simulation procedure fem fmcs is introduced for the accurate and efficient numerical evaluation of the mrf and the vrf1 this methodology is established for the analysis of stochastic beam frame structures as well as for more general stochastic finite element systems i e plane stress and shell problems numerical examples are provided including a statically indeterminate beam a plane stress problem and a shell structure the mrf and vrf1 are used to perform sensitivity parametric analyses with respect to various probabilistic characteristics involved in the problem i e correlation distance standard deviation and to establish realizable upper bounds on the mean value of the response displacement throughout this paper the vrf is computed in parallel to the mrf and the vrf1 it should be noted that although the concept of the vrf has been introduced in an earlier work this is the first time that a flexibility based approach has been used to estimate the vrf for two dimensional problems

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

dynamics of structural systems with spatial randomness the objective of this paper is the analysis of the dynamics of structural systems with randomly varying parameters the formulation and analysis are based on the theory of random integral equations the general analysis including error estimation is applied to the problems of harmonic vibration of i a beam with randomly varying density and ii a uniform beam resting on a randomly inhomogeneous winkler foundation for these applications the analytical formulae for the mean and correlation function of the response are illustrated numerically and graphically

an efficient 3d stochastic finite element method real life structural systems are characterized by their inherent or externally induced uncertainties in the design parameters this study proposes a stochastic finite element tool efficient to take account of these uncertainties here uncertain structural parameter is modeled as homogeneous gaussian stochastic field and commonly used two dimensional 2d local averaging technique is extended and generalized for 3d random field this is followed by cholesky decomposition of respective covariance matrix for digital simulation by expanding uncertain stiffness matrix about its reference value the neumann expansion method is introduced blended with direct monte carlo simulation this approach involves decomposition of stiffness matrix only once for the entire simulated structure thus substantial saving of cpu time and also the scope of tackling several stochastic fields simultaneously are the basic advantages of the proposed algorithm accuracy and efficiency of this method with reference to example problem is also studied here and numerical results validate its superiority over direct simulation method or first order perturbation approach

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

ne mcs technique for stochastic structural response sensitivity in structural sensitivity analysis deterministic procedure is insufficient to provide complete information stochastic analysis is a holistic approach for computing response sensitivity considering uncertainties in structural parameters present study focuses an efficient stochastic finite element method for random response sensitivity analysis randomness of design variables is modeled through local averaging technique followed by cholesky decomposition for digital simulation neumann expansion blended with monte carlo simulation ne mcs has been employed in finite element solution for response sensitivity numerical examples are presented to illustrate the ne mcs technique solution employing direct monte carlo simulation and perturbation are also presented to show the efficiency and accuracy of the proposed technique

stress based topology optimization of continuum structures under uncertainties abstract this work addresses the use of the topology optimization approach to the design of continuum structures with failure constraints under the hypothesis of uncertainties in the spatial distribution of young s modulus to this end the first order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to represent the random field the objective is the minimization of the amount of material used in the design subjected to local stress constraints under uncertainties the probability of failure is bounded by the one sided chebychev inequality since the exact probability distribution function of the stress constraints is not known in advance the effective probability of failure of the obtained optimal designs is validated with the use of the monte carlo simulation approach indicating that the probability of failures of the topologies obtained with the stochastic approach is within the bounds provided by the one sided chebychev inequality the optimization problem is solved by means of the augmented lagrangian method in order to address the large number of constraints associated to this kind of formulation it is shown that the correlation length and the number of standard deviations considered in the formulation play an important role in both the obtained topology and effective probability of failure

efficient bounds for the monte carlo neumann solution of stochastic thermo elasticity problems abstract the numerical solution of stochastic thermo elasticity problems can be computationally demanding in this article a well known property of the neumann series is explored in order to derive lower and upper bounds for expected value and second order moment of the stochastic temperature and displacement responses uncertainties in axial stiffness and conductivity are represented as parameterized stochastic processes monte carlo simulation is employed to obtain a few samples of the stochastic temperature and displacement fields from which lower and upper bounds of expected value and second order moment are computed the proposed methodology is applied to two linear one dimensional thermo elastic example problems it is shown that accurate and efficient bounds can be obtained for a proper choice of operator norm with as few as one or two terms in the neumann expansion the monte carlo neumann bounding scheme proposed herein is shown to be an efficient alternative for the solution of stochastic thermo elasticity problems

effect of multiple uncertain material properties on the response variability of in plane and plate structures abstract in the majority of research works on stochastic finite element analysis the main concern is put on the effect of uncertain elastic modulus on the response variability this is mainly due to the importance of this parameter in the stochastic response of structures but also to the difficulties in introducing other material and or geometrical uncertain parameters in the stochastic formulation in this paper a formulation is suggested in order to determine the response variability of in plane and plate structures with multiple uncertain material parameters i e elastic modulus and poisson s ratio to accomplish this use is made of the general equation for n th joint moment and a polynomial expansion on the coefficients of constitutive matrix for computational purposes the infinite series of stochastic sub stiffness matrices is truncated preserving sufficient accuracy to demonstrate the validity of the proposed formulation some numerical examples are provided and the results are compared with those obtained by means of classical monte carlo simulation combined with the local average scheme

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

stochastic second order perturbation approach to the stress based finite element method abstract the paper is devoted to the application of the second order perturbation second probabilistic moment method to the stress based finite element method fem the approach is introduced for the linear elastic heterogeneous medium up to the second order variational equations of the complementary energy principle are presented together with an additional stochastic finite element discretization based on airy and prandtl stress functions the numerical examples shown in this paper illustrate the probabilistic stress and strain tensors in the cantilever beam under shear loading and torsioned square beam with randomly defined material and geometrical parameters the results obtained in the tests can be applied in probabilistic analyses of the boundary value problems having any closed form mathematical solutions as well as in the stress based stochastic fem analysis of solids and structures

balanced canonical forms for system identification parameterizations of linear state space systems by balanced canonical forms are considered the structure in particular the topological properties of these parameterizations are investigated based on these results an algorithm for identification obtained by optimizing a likelihood type criterion function is proposed and analyzed in the algorithm both the estimation of real valued parameters and of integer parameters defining dynamic specification is incorporated the analysis of the algorithm emphasizes consistency results

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

topology optimization of continuum structures subjected to uncertainties in material properties summary this work addresses the use of the topology optimization approach to the design of robust continuum structures under the hypothesis of uncertainties with known second order statistics to this end the second order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to discretize the random field the objective function is a weighted sum of the expected compliance and its standard deviation the optimization problem is solved using a traditional optimality criteria method it is shown that the correlation length plays an important role in the obtained topology and statistical moments when only the minimization of the standard deviation is considered resulting in more and thinner reinforcements as the correlation length decreases it is also shown that the minimization of the expected value is close to the minimization of the deterministic compliance for small variations of young s modulus this article is protected by copyright all rights reserved

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

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

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

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

introducing loading uncertainty in topology optimization uncertainty is an important consideration in structural design and optimization to produce robust and reliable solutions this paper introduces an efficient and accurate approach to robust structural topology optimization the objective is to minimize expected compliance with uncertainty in loading magnitude and applied direction where uncertainties are assumed normally distributed and statistically independent this new approach is analogous to a multiple load case problem where load cases and weights are derived analytically to accurately and efficiently compute expected compliance and sensitivities illustrative examples using a level set based topology optimization method are then used to demonstrate the proposed approach

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

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

a formulation for stochastic finite element analysis of plate structures with uncertain poisson s ratio up to now the young s modulus is mainly dealt within the analysis of response variability however since the poisson s ratio is the other material constant which influences the behavior of structures the independent evaluation of the effects of this parameter on the response variability is of importance in this paper a formulation to determine the response variability in plate structure due to the randomness of poisson s ratio is given to filter out the independent contributions of randomness in poisson s ratio to the response variability the constitutive matrix has to be decomposed into several sub matrices in order to include the poisson s ratio in the constitutive relation as a non linear parameter a polynomial expansion of poisson s ratio is introduced to demonstrate the validity of the proposed formulation an example is chosen and the results are compared with those obtained by means of monte carlo simulation through the formulation proposed in this paper it becomes possible for the non statistical weighted integral stochastic approach to deal with all the uncertain material parameters in its application

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

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

response variability due to randomness in poisson s ratio for plane strain and plane stress states summary in the ordinary structural materials one of the parameters that can be assumed to have spatial uncertainty is poisson s ratio therefore the independent evaluation of the effects of this parameter on the response variability is of importance the difficulties in obtaining the response variability due to randomness in poisson s ratio lie in the fact that the poisson s ratio enters the stiffness matrix as a non linear parameter in this paper a formulation to determine the response variability in plane strain and plane stress states due to the randomness in the poisson s ratio is given the formulation is accomplished by means of the stochastic decomposition of the constitutive matrix into several sub matrices taking into consideration of the polynomial expansion on the coefficients of constitutive relation to demonstrate the validity of the proposed formulation some example structures are chosen and the results are compared with those obtained by means of monte carlo simulation through the formulation proposed in this study it becomes possible for the weighted integral stochastic finite element analysis to consider all the uncertain material parameters in its application