stochastic finite element modelling in linear transient heat transfer a stochastic heat transfer variational principle is suggested for transient and steady state heat problems the non statistical formulation is based on the combination of the second order perturbation technique and second moment analysis the principle allows incorporation of system uncertainties into the conventional finite element equations which are solved for the first two probabilistic moments of the nodal random temperature field computational aspects of the problem are discussed numerical algorithms have been worked out and easily adapted to an existing finite element code the formalism presented may be directly employed for a wide class of field problems

buckling of sandwich plates with random material properties using improved plate model an improved higher order zigzag plate theory in a random environment is proposed and it is implemented in a stochastic finite element framework to study the buckling characteristic of sandwich plates with random material properties the theory satisfies transverse shear stress continuity at all of the layer interfaces and the transverse shear stress free condition at the top and bottom surfaces it also includes the effect of transverse deformability to the core the through thickness variation of in plane displacements is assumed to be cubic whereas transverse displacement varies quadratically within the core and it remains constant over the face sheets note that sandwich and composite structures are characterized by inherent uncertainties in their material properties the effect of these uncertainties on the buckling characteristic of sandwich plates is studied by modeling the macromechanical material properties as basic random variables a stochastic finite element method consisting of an efficient c finite element in conjunction with a mean centered first order perturbation approach is developed and the model is employed to evaluate the second order statistics of the buckling loads the published results are used to validate the deterministic part of the proposed approach and its stochastic component is validated with an independent monte carlo simulation

on the growth of nonuniform lattices in pinched negatively curved manifolds we study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices these objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

on semi analytical probabilistic finite element method for homogenization of the periodic fiber reinforced composites the main aim of this paper is a development of the semi analytical probabilistic version of the finite element method fem related to the homogenization problem this approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method it originates from the generalized stochastic perturbation based fem where taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions the hybrid computational implementation of the system maple with homogenization oriented fem code mcceff is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber reinforced composites although numerical illustration deals with a homogenization of a composite with material properties defined as gaussian random variables other composite parameters as well as other probabilistic distributions may be taken into account the methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements finite differences or volumes as well as for meshless numerical strategies copyright 2011 john wiley sons ltd

extension of the certain generalized stresses method for the stochastic analysis of homogeneous and laminated shells abstract the paper presents an extension of the certain generalized stresses method cgsm for the static finite element analysis of homogeneous and laminated shells with variability the basic assumption is that the generalized stresses do not depend on input parameters perturbation the cgsm is a non intrusive method that requires only one finite element analysis with some load cases to calculate the variability of mechanical quantities of interest the uncertain input parameters are material and physical properties uniform random parameters as well as random fields are considered the displacements statistical results mean value standard deviation and distribution are obtained by monte carlo simulations using a semi analytical formula two examples are treated the scordelis lo shell roof and an automotive windscreen the method provides results of good quality and is very economical from a computational time point of view

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

stochastic design sensitivity in structural dynamics a combination of the adjoint variable approach and the second order perturbation method is used in the finite element context an alternative form of the constraint functional that holds for all times is introduced to consider the time response of dynamic sensitivity

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

reduction of dimensions in random elastic soil medium abstract the paper deals with an application of the plane strain analysis in a stochastic three dimensional soil medium in a framework of random elasticity theory the geostatical state of stresses and the problem of a unit force acting in a statistically homogeneous half space are considered only the modulus of elasticity is considered to be random and is modelled as a three dimensional 3 d homogeneous random field as the result of imposed constrains due to the plane strain assumption the additional body and surface forces are induced in order to determine them additional equations must be introduced the equations in a form of constrain relations are proposed in this paper these equations are also valid for a case of uniformly distributed external loading first the two dimensional 2 d problem and its reduction to the uni axial strain state for the gravity forces and uniform unlimited surface loading is considered then it is generalised into a 2 d schematization of the 3 d state next the problem of a unit force acting in a statistically homogeneous half space is considered for a 3 d state of stress and strain the resulting stresses are compared with those for a 2 d state these stresses for the multidimensional state of strain and stress are presented as a sum of two components the first one reflects plane strain state stresses and is given in a form of a 3 d random field this term allows for incorporating a spatial 3 d soil variability into a two dimensional analysis the second component can be treated as a correction term and it represents the longitudinal influence of a 3 d analysis some numerical results are presented in this paper the proposed method can be regarded as a framework for further research aiming at application to a variety of geotechnical problems for which the plane strain state is assumed

stochastic boundary elements for two dimensional potential flow in homogeneous domains abstract a stochastic boundary element formulation is presented for the analysis of two dimensional steady state potential flow through homogeneous domains the operator of the governing differential equation is assumed to be random and is described by a set of correlated random variables the perturbation method in conjunction with the boundary element method is employed to derive the systems of equations for the unknown boundary variables and their respective first and second order derivatives with respect to the random variables these quantities are then used to calculate the desired response statistics a general procedure is developed which is next applied for the specific cases of random geometric configuration and random material parameter the random geometric configuration is modeled using a finite set of correlated random variables the random material parameter is modeled as a homogeneous random field which allows the use of deterministic fundamental solutions and integral representations for homogeneous domains the random field is first discretized into a set of correlated random variables and then the general procedure is applied a transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations the results for the boundary variables are used to calculate the response statistics of internal potentials these calculations require the modeling of the interior of the domain under consideration several models for representing the interior of the domain are presented for both random configuration and random material parameter and their influence on the response statistics is analysed distributed sources are considered in the present study using the particular integral approach a number of numerical examples are presented to demonstrate the validity of the present formulations the results obtained from the present analyses are compared with those obtained from monte carlo simulations with 5000 samples and a good agreement of results is observed

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

computation of heat conduction in materials with random variable thermophysical properties a method for the direct computation of mean values and variances of the temperature in conduction heated objects with random variable thermophysical properties is presented this method is based on a taylor expansion of the finite element formulation of the heat conduction equation and offers a powerful alternative to the computationally expensive monte carlo method both steady state and transient problems are considered some example problems are solved and the results discussed the simulations indicate that the variability of the thermophysical properties may cause a considerable variability of the temperature within the heated object this may have important consequences on the design of heating operations in food process engineering

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

long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos abstract polynomial chaos pc is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems in the case of a structural system under dynamic loading the pc method is combined with a time integration scheme the pc method with galerkin projection converts the stochastic pde to a set of simultaneous deterministic pde which can be solved using any standard time integration scheme however the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response the probability density functions of the responses are not of constant type as the time progresses however in the case of generalized pc gpc these are approximated using initially considered gpc thus loses its efficiency time dependent general polynomial chaos tdgpc can be considered to overcome these difficulties where polynomials are generated on the fly to match with the pdf of the responses the present study considered an investigation of structural mechanics responses with both gaussian and non gaussian random system properties under dynamic loading using tdgpc an adaptive updation scheme based on the rms value of the coefficients of pc expansion is investigated for a proper updation of pc expansion the curse of dimensionality of pc expansion is addressed by considering only the dominant components of the responses evaluated using kl expansion further starting with lower order pc the order of pc expansion is increased only after the first updation thus reducing computational complexities

an element free galerkin method for probabilistic mechanics and reliability a stochastic element free galerkin method was developed for reliability analysis of linear elastic structures with spatially varying random material properties a random field representing material properties was discretized into a set of random variables with statistical properties obtained from the statistical properties of random field in conjunction with meshless formulations the first order reliability method was employed to predict the full probabilistic characteristics of a structural response unlike the stochastic finite element method the stochastic mesh free method does not require a structured mesh instead only a scattered set of nodal points is required in the domain of interest as well there is no need for fixed connectivities between nodes numerical examples show good agreement between the results of the developed method and monte carlo simulation furthermore the stochastic meshless method provides convergent solutions of the probability of failure since mesh generation of complex structures can be far more time consuming and costly effort than solution of a discrete set of equations the developed meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems

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

piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system abstract this paper proposes numerical developments based on polynomial chaos pc expansions to process stochastic eigenvalue problems efficiently these developments are applied to the problem of linear stability calculations for a simplified brake system the stability of a finite element model of a brake is investigated when its friction coefficient or the contact stiffness are modeled as random parameters getting rid of the statistical point of view of the pc method but keeping the principle of a polynomial decomposition of eigenvalues and eigenvectors the stochastic space is decomposed into several elements to realize a low degree piecewise polynomial approximation of these quantities an approach relying on continuation principles is compared to the classical dichotomy method to build the partition moreover a criterion for testing accuracy of the decomposition over each cell of the partition without requiring evaluation of exact eigenmodes is proposed and implemented several random distributions are tested including a uniform like law for description of friction coefficient variation results are compared to monte carlo simulations so as to determine the method accuracy and efficiency some general rules relative to the influence of the friction coefficient or the contact stiffness are also inferred from these calculations

stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion this paper presents the theory and application of the generalized polynomial chaos expansion for the stochastic free vibration of orthotropic plates specifically the stochastic analysis of orthotropic plates under the uncertainties in elasticity moduli is investigated the uncertain moduli eigen frequencies and eigen modes of the plates are represented by truncated polynomial chaos expansions with arbitrary random basis the expansions are substituted in the governing differential equations to calculating the polynomial chaos coefficients of the eigen frequencies and the eigen modes distribution functions of the uncertain moduli are derived from experimental data where the pearson model is used to identify the type of density functions this realization then is employed to construct random orthogonal basis for each uncertain parameter because of available experimental modal analysis data this paper provides a useful practical example on the efficacy of polynomial chaos where the statistical moments and the probability distributions of modal responses are compared with experimental results

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

stochastic boundary elements for two dimensional potential flow in non homogeneous media abstract a stochastic boundary element formulation is presented for the treatment of two dimensional problems of steady state potential flow in non homogeneous media that involve a random operator in the governing differential equation the randomness is introduced through the material parameter of the domain which is described as a non homogeneous random field the random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem this leads to differential equations for the unknown potential and its first and second order derivatives respectively evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field an approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order these solutions are introduced into the differential equations and upon equating similar order terms a sequence of poisson s equations is obtained for each order a transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables the resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first and second order derivatives which are then used to compute the desired response statistics quadratic conforming boundary elements are used in the boundary integration and four node quadrilateral cells are used in the domain integration strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object the singular domain integrals are calculated analytically direct solution techniques are used to calculate the response variables and their derivatives respectively a number of example problems are presented and the results are compared with those obtained from monte carlo simulations a good agreement of the results is observed

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

a generalized dimension reduction method for multidimensional integration in stochastic mechanics summary a new generalized multivariate dimension reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads material properties and geometry the method involves an additive decomposition of an n dimensional response function into at most s dimensional functions where s n an approximation of response moments by moments of input random variables and a moment based quadrature rule for numerical integration a new theorem is presented which provides a convenient means to represent the taylor series up to a specific dimension without involving any partial derivatives a complete proof of the theorem is given using two lemmas also proved in this paper the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method eight numerical examples involving elementary mathematical functions and solid mechanics problems illustrate the proposed method results indicate that the multivariate dimension reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods such as first and second order taylor expansion methods statistically equivalent solutions quasi monte carlo simulation and the fully symmetric interpolatory rule while the accuracy of the dimension reduction method is comparable to that of the fourth order neumann expansion method a comparison of cpu time suggests that the former is computationally far more efficient than the latter copyright 2004 john wiley sons ltd

a small perturbation automatic differentiation method for determining uncertainty in computational electromagnetics a new methodology for approximating uncertainties in computational electromagnetics cem solutions is discussed the proposed method involves the use of a taylor series expansion to account for small perturbations uncertainty in the input parameters and subsequent use of an automatic differentiation technique throughout the numerical calculation chain this approach is denoted as small perturbation automatic differentiation spad method and provides certain benefits over other cem uncertainty methods available in the literature the method is illustrated in a couple of examples involving a moment method solution of the electric field integral equation

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

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation a numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods the radius of convergence of eigenvalue perturbation methods based on taylor series is known to decrease when eigenvalues are getting closer to each other this phenomenon knwon as veering in structural dynamics is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective when this degeneracy referred to as exceptional point ep is close to the real axis the veering becomes stronger the main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity to do so two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter this yields a new robust eigenvalue reconstruction scheme which is compared to taylor and puiseux series in all cases theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model this system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter computationally efficient the proposed algorithm could also be relevant for actual numerical models of large size arising from other applications involving parametric eigenvalue problems e g waveguides rotating machinery or instability problems such as squeal or flutter

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

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

nonlinear free vibration analysis of composite plates with material uncertainties a monte carlo simulation approach abstract in this paper the effects of dispersion in material properties on free vibration response of composite plates with geometric nonlinearity in von karman sense are investigated the higher order shear deformation theory is employed for the study reported here an efficient c0 finite element formulation is developed for the analysis using monte carlo simulation the second order statistics i e mean and standard deviation of the nonlinear free vibration response of the composite laminates are obtained for different thickness and amplitude ratios the input statistics mean and variance of the material property are assumed to be known as a priori the computed results demonstrate the influence of the variations in the material properties and amplitudes on the nonlinear free vibration response of the composite plates the results are compared with those available in the literature

stochastic boundary elements in elastostatics abstract a stochastic boundary element formulation for the treatment of boundary value problems in two dimensional elastostatics that involve a random operator is presented a general perturbation procedure is formulated for the set of correlated random variables governing the response of the solid this procedure is then specialized for the cases of a random geometry and b random material properties the problems involving a random configuration are analyzed using the random variable model and those with a random material property are analyzed using the random field model the random field is first discretized into a set of correlated random variables which are then transformed into an uncorrelated set to simplify the analysis the derivatives of the boundary element matrices appearing in the systems of equations from the perturbation of the random variables are derived analytically the direct solution methods are used to obtain the response variables and their first and second order derivatives respectively quadratic conforming boundary elements are employed in the boundary element discretization and the strongly singular terms of the boundary element matrices and their first and second order derivatives are obtained using the conditions associated with rigid body motions of the solid the present formulation has been evaluated for a number of example problems through comparisons with the solutions obtained by monte carlo simulation a good agreement of the results is observed

probabilistic entropy in homogenization of the periodic fiber reinforced composites with random elastic parameters summary the purpose of this elaboration is to develop an efficient method for a determination of the probabilistic entropy loss in the homogenization process of the periodic composites with random material characteristics a definition of this entropy convenient for the gaussian continuous distribution is adopted and implemented into the computer algebra system maple homogenization of the fiber reinforced composite with randomly defined young s moduli of the constituents is carried out in the fem and homogenization oriented code based on the four noded plane strain elements probabilistic procedure has triple character and is alternatively based on the monte carlo simulation on the generalized stochastic perturbation based analysis and on the recently developed semi analytical determination of homogenized tensor using the response function method because of this triple usage of the probabilistic methods it is possible to make a detailed comparison of all those techniques especially in view of the entropy variation during the homogenization this procedure may be linked with other homogenization techniques also for various constitutive models and or for the upper and lower bounds on the effective tensor components copyright 2012 john wiley sons ltd

supersensitivity due to uncertain boundary conditions summary we study the viscous burgers equation subject to perturbations on the boundary conditions two kinds of perturbations are considered deterministic and random for deterministic perturbations we show that small perturbations can result in o 1 changes in the location of the transition layer for random perturbations we solve the stochastic burgers equation using different approaches first we employ the jacobi polynomial chaos which is a subset of the generalized polynomial chaos for stochastic modeling converged numerical results are reported up to seven significant digits and we observe similar stochastic supersensitivity for the mean location of the transition layer subsequently we employ up to fourth order perturbation expansions we show that even with small random inputs the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output two types of distributions are considered uniform distribution and a truncated gaussian distribution with no tails various solution statistics including the spatial evolution of probability density function at steady state are studied copyright 2004 john wiley sons ltd

stochastic galerkin methods for elliptic interface problems with random input in this work we consider random elliptic interface problems namely the media in elliptic equations have both randomness and interfaces a galerkin method using bi orthogonal polynomials is used to convert the random problem into an uncoupled system of deterministic interface problems a principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in random spaces with which the total degree of freedom can be significantly reduced then immersed finite element methods are introduced to solve the resulting system convergence results are given both theoretically and numerically

on the symmetric scalar curvature problem on sn abstract we discuss some existence results dealing with the scalar curvature problem on s n in the presence of various symmetries

interval field model and interval finite element analysis uncertain parameters with inherent spatial variability are commonly encountered in engineering modeling of this kind of spatial uncertainty plays a fundamental role in structural uncertainty analysis which provides a necessary basis for subsequent uncertainty propagation through the system an interval field model for quantification of spatial uncertain parameters is proposed by which only the upper and lower bounds of the spatial uncertain parameters rather than their precise probability distributions are required the dependency can be fully considered by the proposed interval field model with the information of dependency an interval k l expansion is presented as a combination of deterministic functions with uncorrelated standard interval variables through which the continuous spatial interval field with dependency can be expressed only by very limited intervals necessary mathematical illustrations are provided for the proposed interval field model and the interval k l expansion the sampling method for the interval field model is given providing a robust numerical analysis basis for subsequent structural uncertainty analysis when the interval field model is applied in finite element analysis of structures with spatial uncertain parameters the non deterministic equilibrium equations with interval factors is then formulated the mcs method and the perturbation method are developed for solution of the derived interval equilibrium equations the feasibility and validity of the proposed interval field model and corresponding interval finite element methods are verified by numerical examples where the upper and lower bounds of the responses such as the displacements and the stresses of structures with spatial uncertain parameters are computed and compared

on solving elliptic stochastic partial differential equations abstract a model elliptic boundary value problem of second order with stochastic coefficients described by the karhunen loeve expansion is addressed this problem is transformed into an equivalent deterministic one the perturbation method and the method of successive approximations is analyzed rigorous error estimates in the framework of sobolev spaces are given

numerical methods for second order shape sensitivity analysis with applications to heat conduction problems in this paper two finite element based schemes for second order shape sensitivity analysis are presented in the first formulation the av dd method the first order shape sensitivity equation is derived and expressed in terms of state and adjoint variables the resultant equation is then directly differentiated to obtain the second order shape sensitivity equation in the second formulation the dd av method the functional of concern is differentiated twice to yield the second order sensitivity equation in which the second order shape derivatives can be eliminated by introducing a proper adjoint equation a thermal fin problem and a thermal insulation layer problem have been studied to validate the proposed schemes it is shown that both methods yield identical results though the dd av method is computationally more efficient

probabilistic nonlinear finite element analysis of composite structures a probabilistic finite element analysis procedure for laminated composite shells is developed a total lagrangian finite element formulation employing a degenerated three dimensional laminated composite shell element with the full green lagrange strains and first order shear deformable kinematics is used the first order second moment technique for probabilistic finite element analysis of random fields is employed and results are presented in the form of mean and variance of the structural response reliability calculations are made by using the first order reliability method combined with sensitivity derivatives from the finite element analysis both ply level and micromechanics level random variables are incorporated the latter by means of the aboudi micromechanics model two sample problems are solved to verify the accuracy of the procedures developed and to quantify the variability of certain material type structure combinations in general the procedure is quite effective in determining the response statistics and reliability for linear and geometric nonlinear behavior of laminated composite shells

stochastic modeling and vibration analysis of rotating beams considering geometric random fields abstract geometric parameters such as the thickness and width of a beam are random for various reasons including manufacturing tolerance and operation wear due to these random parameter properties the vibration characteristics of the structure are also random in this paper we derive equations of motion to conduct stochastic vibration analysis of a rotating beam using the assumed mode method and stochastic spectral method the accuracy of the proposed method is first verified by comparing analysis results to those obtained with monte carlo simulation mcs the efficiency of the proposed method is then compared to that of mcs finally probability densities of various modal and transient response characteristics of rotating beams are obtained with the proposed method

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

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

an improved response function based stochastic meshless method for problems in elasto statics abstract the current study proposes an improved response function irf based stochastic element free galerkin method sefgm for the analysis of problems in elasto statics wherein young s modulus is modelled as a homogeneous random field with symmetric distribution characteristics the proposed sefgm approximates displacement as the sum of a deterministic part and a stochastic part the stochastic part is modelled with the help of an irf which is a function of discretized set of random variables moving least square shape functions are employed to discretize the random field utilizing taylor series expansions of stiffness matrix and force vector and irf approximation of displacement explicit expressions for system responses in terms of random variables are derived stochastic informations of system responses are evaluated by employing monte carlo simulation mcs on the response function which eliminates the need of construction and simulation of system matrices at each set of sample generation 1d and 2d numerical examples in elasto statics are solved using proposed method results are validated with those obtained by direct simulation of system of equations using mcs and also compared with other methods like second order perturbation and ad hoc response function based sefgm normalized computational times required for all the methods are also compared it is found that the proposed method is computationally efficient and can produce accurate results even for higher coefficient of variation of input random fields

karhunen loeve decomposition of random fields based on a hierarchical matrix approach summary the simulation of the behavior of structures with uncertain properties is a challenging issue because it requires suitable probabilistic models and adequate numerical tools nowadays it is possible to perform probabilistic investigations of the structural performance which take into account a space variant uncertainty characterization of the structures given a structural solver and the probabilistic models the reliability analysis of the structural response depends on the continuous random fields approximation which is carried out by means of a finite set of random variables the paper analyzes the main aspects of discretization in the case of 2d problems the combination of the well known karhunen loeve series expansion the finite element method and the hierarchical matrices approach is proposed in the paper copyright 2013 john wiley sons ltd

on solving nonlinear transient heat transfer problems with random parameters abstract a variational methodology is proposed for nonlinear transient heat transfer systems with random parameters defined by their first two probabilistic moments the finite element formalism is based on the combination of the second order perturbation and mean based second moment analysis a hierarchical system of first order partial differential equations is obtained and solved for the first two probabilistic moments of the random temperature field numerical aspects of the problem are discussed and illustrative results are given the approach presented may be employed for a wide class of field problems

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

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

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

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

an iterative polynomial chaos approach for solution of structural mechanics problem with gaussian material property abstract a new approach for solution of stochastic structural mechanics problem with random coefficient in the framework of polynomial chaos pc expansion is proposed the basic strategy of the proposed method is to iteratively construct a pc expansion with reduced numbers of unknown coefficients the method is based on orthogonal expansion of stochastic responses and generation of an iterative pc based on the responses of the previous iteration the polynomials are evaluated using gram schmidt orthogonalization process the number of random variables in pc expansion is reduced by considering only the dominant components of the response characteristics which is evaluated using karhunen loeve kl expansion in case of random material field problem kl expansion is used to discretize the random field the usefulness of the proposed method in terms of accuracy and computational efficiency is examined the results of linear static structural mechanics problems are compared with mcs and pc method the proposed method is observed to be computationally more efficient and accurate than pc method

a fourier karhunen loeve discretization scheme for stationary random material properties in sfem in order to overcome the computational difficulties in karhunen loeve k l expansions of stationary random material properties in stochastic finite element method sfem analysis a fourier karhunen loeve f k l discretization scheme is developed in this paper by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical sfem simulations the proposed f k l discretization scheme exhibits a number of advantages over the widely used k l expansion scheme based on fe meshes including better computational efficiency in terms of memory and cpu time convenient a priori error control mechanism better approximation accuracy of random material properties explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes copyright 2007 john wiley sons ltd

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

stochastic evolution of microcracks in continua abstract a multifield description of continua endowed by diffused microcracks recently proposed by the authors mariano and augusti math mech sol 3 1998 183 200 within a deterministic context is extended to cover some of the stochastic aspects of microcrack distribution and evolution the microcrack state of the continuum is described by a tensor field defined as the second order approximation of the microcrack density function m p n t that represents the distribution of the number of microcracks in each direction n in a neighborhood mesodomain of the point p m p n t is considered as a submartingale process in each point p leading to a stochastic field over the continuum body generalized measures of internal actions represent the interactions voids voids and void matrix they perform work in the variation of relevant conjugated fields invariance requirements on the overall power allows to deduce both the usual balance of forces and the balance of generalized internal actions obtaining a model different from the classical internal variable models not only conceptually but also formally the introduction of a damage entropy flux whose divergence is the production of configurational entropy allows to include damage criteria within the context of clausius duhem inequality the basic features of an appropriate finite element discretization are formalized in the case of linear elastic brittle materials the stochastic distribution of microcracks is considered through the stochastic nature of the elements of the stiffness matrix among other assets the multifield approach overcomes the mesh dependence of the numerical results obtained on the basis of the more used internal variable schemes

stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty the problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method a randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered the discretized parametric fields are interpolated within each element with multidimensional lagrange polynomials and integrated into the weak formulation the proposed discretized random field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables the description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain a method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral galerkin framework is studied in detail the treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems the convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme

free vibration of laminated spherical panels with random material properties randomness in the material properties is inherent in all engineering materials composites exhibit a greater scatter compared to conventional materials because of the larger number of parameters associated with its fabrication and manufacturing for accurate prediction of its behavior the composite material properties have been modelled as random variables in the present study higher order shear deformation theory including rotatory inertia effects has been employed in developing the system equations and first order perturbation technique has been adopted for the solution an approach has been presented for obtaining the analytical solution for generalized eigenvalue problem associated with free vibrations mean and variance of the natural frequency have been obtained for cross ply spherical laminates projected in rectangular plan form with different boundary conditions

computational uncertainty analysis in multiresolution materials via stochastic constitutive theory abstract a stochastic constitutive theory is proposed in this work to propagate microstructure uncertainties in computational multiscale continuum models to bulk multiresolution material behavior ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized in detail and this research transmits these uncertainties to coarser material resolutions by introducing a stochastic constitutive theory deduced from volume element simulations to implement the stochastic upscaling process two advanced uncertainty quantification methods are examined statistical copula functions and random process polynomial chaos expansion both methods confront the mathematical difficulty in randomizing constitutive laws by capturing the marked correlation among constitutive parameters seen in complex materials thus the results proffer a more accurate probabilistic estimation of constitutive material behavior the contribution of this work is twofold uncertainty is propagated from heterogeneous material structure to material property via the stochastic constitutive theory and rigorous data driven mathematics are formalized to represent complicated dependence structures in multivariate statistical distributions to the authors knowledge this is the first work in multiresolution mechanics that presents an approach to computationally derive correlation functions from numerical experiments as opposed for instance to assuming one a priori the method put forth in this research though quite general is applied to a mathematical example and plastic high strength steel alloy for demonstration results include stochastic constitutive curve confidence intervals for the material stress strain response and qualitative comparisons of the two stochastic methods detailed herein

tsallis entropy in dual homogenization of random composites using the stochastic finite element method summary this work concerns an application of the tsallis entropy to homogenization problem of the fiber reinforced and also of the particle filled composites with random material and geometrical characteristics calculation of the effective material parameters is done with two alternative homogenization methods the first is based upon the deformation energy of the representative volume element rve subjected to the few specific deformations while the second uses explicitly the so called homogenization functions determined under periodic boundary conditions imposed on this rve probabilistic homogenization is made with the use of three concurrent non deterministic methods namely monte carlo simulation iterative generalized stochastic perturbation technique as well as the semi analytical approach the last two approaches are based on the least squares method with polynomial basis of the statistically optimized order this basis serves for further differentiation in the tenth order stochastic perturbation technique while semi analytical method uses it in probabilistic integrals these three approaches are implemented all as the extensions of the traditional finite element method fem with contrastively different mesh sizes and they serve in computations of tsallis entropies of the homogenized tensor components as the functions of input coefficient of variation

explicit solution to the stochastic system of linear algebraic equations α1a1 α2a2 αmam x b abstract this paper presents a novel solution strategy to the stochastic system of linear algebraic equations 1 a 1 2 a 2 m a m x b arising from stochastic finite element modelling in computational mechanics in which i i 1 m denote random variables a i i 1 m real symmetric deterministic matrices b a deterministic random vector and x the unknown random vector to be solved the system is first decoupled by simultaneously diagonalizing all the matrices a i via a similarity transformation and then it is trivial to invert the sum of the diagonalized stochastic matrices to obtain the explicit solution of the stochastic equation system unless all the matrices a i share exactly the same eigen structure the joint diagonalization can only be approximately achieved hence the solution is approximate and corresponds to a particular average eigen structure of the matrix family specifically the classical jacobi algorithm for the computation of eigenvalues of a single matrix is modified to accommodate multiple matrices and the resulting jacobi like joint diagonalization algorithm preserves the fundamental properties of the original version including its convergence and an explicit solution for the optimal givens rotation angle three numerical examples are provided to illustrate the performance of the method