dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

hamilton jacobi theory in three dimensional minkowski space via cartan geometry a complete solution to the problem of orthogonal separation of variables of the hamilton jacobi equation in three dimensional minkowski space is obtained the solution is based on the underlying ideas of cartan geometry and ultimately developed into a general new algorithm that can be employed in the study of hamiltonian systems defined by natural hamiltonians within the framework of hamilton jacobi theory to demonstrate its effectiveness we investigate from this viewpoint the morosi tondo integrable system derived as a stationary reduction of the seventh order korteweg de vries flow to show explicitly that the system in question is an orthogonally separable hamiltonian system the latter result is a new characterization of the morosi tondo system

group invariant classification of separable hamiltonian systems in the euclidean plane and the o 4 symmetric yang mills theories of yatsun we present a new and effective method of determining separable coordinate systems for natural hamiltonians with two degrees of freedom in flat riemannian space the method is based on intrinsic properties of the associated killing tensors and their invariants under the group of rigid motions e 2 applications to the hamiltonian systems derived by the late v a yatsun from o 4 symmetric yang mills theories are presented in addition an equivalence between separability of two dimensional hamiltonian systems and the existence of pfaffian quasi bi hamiltonian representations is specified

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

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

fuzzy stochastic fem based homogenization framework for materials with polymorphic uncertainties in the microstructure uncertainties in the macroscopic response of heterogeneous materials result from two sources the natural variability in the microstructure s geometry and the lack of sufficient knowledge regarding the microstructure the first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function the second type of uncertainty is denoted epistemic uncertainty this kind of uncertainty cannot be described using probabilistic methods models considering both sources of uncertainties are called polymorphic in the case of polymorphic uncertainties some combination of stochastic methods and fuzzy arithmetic should be used thus in the current work we examine a fuzzy stochastic finite element method based homogenization framework for materials with random inclusion sizes we analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element the stochastic finite element method is used to obtain the material response in the case of random inclusion radii due to unavoidable noise in experimental data an insufficient number of samples and limited accuracy of the fitting procedure the radii distribution density cannot be obtained exactly thus it is described in terms of fuzzy location and scale parameters the influence of fuzzy input on the homogenized stress measures is analyzed

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

elastoacoustic model with uncertain mechanical properties for ultrasonic wave velocity prediction application to cortical bone evaluation the axial transmission technique can measure the longitudinal wave velocity of an immersed solid an elementary model of the technique is developed with a set of source and receivers placed in a semi infinite fluid coupled at a plane interface with a semi infinite solid the acoustic fluid is homogeneous the solid is homogeneous isotropic and linearly elastic the work is focused on the prediction of the measured velocity apparent velocity when the solid is considered to have random material properties the probability density functions of the random variables modeling each mechanical parameter of the solid are derived following the maximum entropy principle specific attention is paid to the modeling of poisson s ratio so that the second order moments of the velocities remain finite the stochastic solver is based on a monte carlo numerical simulation and uses an exact semianalytic expression of the acoustic response derived with the cagniard de hoop method results are presented for a solid with the material properties of cortical bone the estimated mean values and confidence regions of the apparent velocity are presented for various dispersion levels of the random parameters a sensibility analysis with respect to the source and receivers locations is presented

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

adaptive reduced basis generation for reduced order modeling for the solution of stochastic nondestructive evaluation problems a novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented more specifically the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty in order to achieve this computational efficiency the approach combines reduced basis reduced order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters provided the distributions are known the reduced order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique to then create the reduced order basis the capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty the first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing then the second case study considered characterization of an unknown temperature dependent thermal conductivity of a solid from simulated thermal testing overall the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

a random matrix approach for quantifying model form uncertainties in turbulence modeling abstract with the ever increasing use of reynolds averaged navier stokes rans simulations in mission critical applications the quantification of model form uncertainty in rans models has attracted attention in the turbulence modeling community recently a physics based nonparametric approach for quantifying model form uncertainty in rans simulations has been proposed where reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits xiao et al 2016 however a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints in this work we propose a random matrix approach for quantifying model form uncertainties in rans simulations with the realizability of the reynolds stress guaranteed which is achieved by construction from the cholesky factorization of the normalized reynolds stress tensor furthermore the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints we demonstrate that the proposed approach is able to ensure the realizability of the reynolds stress albeit in a different manner from the physics based approach monte carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent gaussian random fields to the reynolds stress random field with the marginal distributions and correlation structures as specified numerical simulations on a typical flow with separation have shown physically reasonable results which verify the proposed approach therefore the proposed method is a promising alternative to the physics based approach for model form uncertainty quantification of rans simulations the method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

domain decomposition using spectral expansions of steklov poincare operators ii a matrix formulation we describe a matrix formulation of the iterative domain decomposition method in natarajan siam j sci comput 16 1995 pp 470 495 from one point of view this method can be regarded as a preconditioning technique for the interface schur complement operator obtained from a decomposition into nonoverlapping subdomains from another point of view it can be viewed as a method of the schwarz type for overlapping subdomains but with an overlap between the physical space in one subdomain and the leading components of the eigenspace induced by the steklov poincare operator in the complementary domain

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

scalable domain decomposition solvers for stochastic pdes in high performance computing abstract stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non linear problems with billions of unknowns for stochastic modeling it is therefore essential to design robust parallel and scalable algorithms that can efficiently utilize high performance computing to tackle such large scale systems domain decomposition based iterative solvers can handle such systems although these algorithms exhibit excellent scalabilities significant algorithmic and implementational challenges exist to extend them to solve extreme scale stochastic systems using emerging computing platforms intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in house implementation sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi level iterative solvers parallel sparse matrix vector operations are used to reduce the floating point operations and memory requirements numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non gaussian stochastic process scalability of the solvers with respect to the number of random variables is also investigated

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

addressing the curse of dimensionality in ssfem using the dependence of eigenvalues in kl expansion on domain size this work is aimed at reducing the dimensionality in the spectral stochastic finite element method ssfem thus the computational cost through a domain decomposition dd method this reduction hinges on some new mathematical results on domain size dependence of the karhunen loeve kl expansion it has been reported in the literature that for few covariance kernels a lower domain size leads to a faster convergence of the kl eigenvalues this observation leads to an early truncation of the kl expansion and this reduction in stochastic dimensionality brings down the total computational cost in this work first we mathematically show the generalization of this faster convergence that is for any arbitrary covariance kernel this is achieved via developing a bound on eigenvalues as a function of the domain size then we prove that for a chosen number of terms in the kl expansion with any kernel for a one dimensional process the approximation error in the trace norm reduces with the domain size based on this domain size dependence we propose an algorithm for solving a stochastic elliptic equation in a dd framework the computational cost gain is demonstrated by a numerical study and is observed that the serial implementation of the proposed algorithm is about an order of magnitude faster compared to the existing method the cost saving increases with the stochastic dimensionality in the global domain the structure of this algorithm provides scope for parallelization which would help in efficiently solving large scale problems the sharpness of the proposed eigenvalue bounds is also tested for gaussian and exponential kernels the generalization opens avenues for developing further dd based ssfem solvers c 2016 elsevier b v all rights reserved

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

a domain decomposition approach for uncertainty analysis this paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations pdes the system is split into local components using domain decomposition our domain decomposed uncertainty quantification dduq approach performs uncertainty analysis independently on each local component in an offline phase and then assembles global uncertainty analysis results using precomputed local information in an online phase at the heart of the dduq approach is importance sampling which weights the precomputed local pde solutions appropriately so as to satisfy the domain decomposition coupling conditions to avoid global pde solves in the online phase a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions

schwarz preconditioners for stochastic elliptic pdes abstract increasingly the spectral stochastic finite element method ssfem has become a popular computational tool for uncertainty quantification in numerous practical engineering problems for large scale problems however the computational cost associated with solving the arising linear system in the ssfem still poses a significant challenge the development of efficient and robust preconditioned iterative solvers for the ssfem linear system is thus of paramount importance for uncertainty quantification of large scale industrially relevant problems in the context of high performance computing the preconditioner must scale to a large number of processors therefore in this paper a two level additive schwarz preconditioner is described for the iterative solution of the ssfem linear system the proposed preconditioner can be viewed as a generalization of the mean based block diagonal preconditioner commonly used in the literature for the numerical illustrations two dimensional steady state diffusion and elasticity problems with spatially varying random coefficients are considered the performance of the algorithm is investigated with respect to the geometric parameters strength of randomness dimension and order of the stochastic expansion

killing tensors warped products and the orthogonal separation of the hamilton jacobi equation we study killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the hamilton jacobi equation this work is motivated primarily by the case of spaces of constant curvature where warped products are abundant we first characterize killing tensors which have a natural algebraic decomposition in warped products we then apply this result to show how one can obtain the killing stackel space ks space for separable coordinate systems decomposable in warped products this result in combination with benenti s theory for constructing the ks space of certain special separable coordinates can be used to obtain the ks space for all orthogonal separable coordinates found by kalnins and miller in riemannian spaces of constant curvature next we characterize when a natural hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by benenti can be reduced finally we use this characterization and concircular tensors a special type of torsionless conformal killing tensor to develop a general algorithm to determine when a natural hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature

equivalence problem for the orthogonal webs on the 3 sphere we solve the equivalence problem for the orthogonally separable webs on the 3 sphere under the action of the isometry group this continues a classical project initiated by olevsky in which he solved the corresponding canonical forms problem the solution to the equivalence problem together with the results by olevsky forms a complete solution to the problem of orthogonal separation of variables for the hamilton jacobi equation defined on the 3 sphere via orthogonal separation of variables it is based on invariant properties of characteristic killing tensors in addition to properties of the corresponding algebraic curvature tensor and the associated ricci tensor the result is illustrated by a nontrivial application to a natural hamiltonian defined on the 3 sphere

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

orthogonal separation of variables for the hamilton jacobi and wave equations in three dimensional minkowski space we give a complete solution of the eisenhart integrability conditions in three dimensional minkowski space obtaining 39 orthogonally separable webs and 58 inequivalent metrics in adapted coordinate systems which permit orthogonal separation of variables for the associated hamilton jacobi and wave equations the corresponding transformations from canonical separable coordinates to canonical pseudo cartesian coordinates are listed for each of the 58 cases and characteristic killing tensors are given for each of the 39 webs

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

invariant classification of orthogonally separable hamiltonian systems in euclidean space the problem of the invariant classification of the orthogonal coordinate webs defined in euclidean space is solved within the framework of felix klein s erlangen program the results are applied to the problem of integrability of the calogero moser model

superintegrable systems on 3 dimensional curved spaces eisenhart formalism and separability the eisenhart geometric formalism which transforms an euclidean natural hamiltonian h t v into a geodesic hamiltonian t with one additional degree of freedom is applied to the four families of quadratically superintegrable systems with multiple separabilities in the euclidean plane first the separability and superintegrability of such four geodesic hamiltonians t r r a b c d in a three dimensional curved space are studied and then these four systems are modified with the addition of a potential u r leading to h r t r u r second we study the superintegrability of the four hamiltonians h r h r r where r is a certain position dependent mass that enjoys the same separability as the original system h r all the hamiltonians here studied describe superintegrable systems on non euclidean three dimensional manifolds with a broken spherical symmetry

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

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

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

covariants joint invariants and the problem of equivalence in the invariant theory of killing tensors defined in pseudo riemannian spaces of constant curvature the invariant theory of killing tensors itkt is extended by introducing the new concepts of covariants and joint invariants of product vector spaces of killing tensors defined in pseudo riemannian spaces of constant curvature the covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by nontrivial killing tensors of valence two defined in the euclidean and minkowski planes illustrative examples are provided

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

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel abstract the paper presents a theoretical study of the interaction between a nonlinear model of a moving vehicle velocity v and a plate elastically mounted in a tunnel an efficient approach to the solution of the problem of vehicle slab track tunnel soil interaction is developed on the basis of a coupling of the finite element and integral transform methods fem and itm according to this approach the tunnel which may have an arbitrary shape and a portion of the surrounding soil is modelled by finite elements while the soil half space is described by the itm the corresponding solution is found using the solutions for the uniform half space and for the continuum with a cylindrical cavity to exploit the invariance of the structure in longitudinal direction x for this direction additionally to the time frequency transform t a space wavenumber transform x k x is used the case of a half space is analyzed using a fourier transform also in the second horizontal direction y k y and an analytical solution for the z direction on the basis of exponential functions in the case of the infinite continuum with the cavity a fourier series for the circumferential direction and a series of cylindrical functions for the radial direction are used for the solution regarding the cross sectional coordinates y z the tunnel structure that may have an arbitrary shape and a portion of the surrounding soil will be modelled by finite elements in a x k x t transformed domain in order to observe the boundary conditions at the surface of the half space as well as at the surface of the cavity the superposition of the two solutions has to be performed after an inverse fourier transform ift in the k x y z domain the solution for the complete system floating slab track tunnel half space is obtained in the k x y z domain once the system transfer function h with vk x for this complete coupled system is found the displacements of the plate can be calculated in time domain by the ift of the product p h i e the convolution of the loading p t with the impulse response function h t which completely represents the behavior of the coupled system in the context of the coupling of systems in a relative movement the problems of differential algebraic equations dae have to be observed

quasi bi hamiltonian systems and separability two quasi bi hamiltonian systems with three and four degrees of freedom are presented these systems are shown to be separable in terms of nijenhuis coordinates moreover the most general pfaffian quasi bi hamiltonian system with an arbitrary number of degrees of freedom is constructed in terms of nijenhuis coordinates and its separability is proved

physical consequences of a nonparametric uncertainty model in structural dynamics abstract one of the main claims of the nonparametric model of random uncertainty introduced by soize 2000 3 is its ability to account for model uncertainty the present paper investigates this claim by examining the statistics of natural frequencies total energy and underlying dispersion equation yielded by the nonparametric approach for two simple systems a thin plate in bending and a one dimensional finite periodic mass spring chain results for the plate show that the average modal density and the underlying dispersion equation of the structure are gradually and systematically altered with increasing uncertainty the findings for the mass spring chain corroborate the findings for the plate and show that the remote coupling of nonadjacent degrees of freedom induced by the approach suppresses the phenomenon of mode localization this remote coupling also leads to an instantaneous response of all points in the chain when one mass is excited in the light of these results it is argued that the nonparametric approach can deal with a certain type of model uncertainty in this case the presence of unknown terms of higher or lower order in the governing differential equation but that certain expectations about the system such as the average modal density may conflict with these results

cost reduction of stochastic galerkin method by adaptive identification of significant polynomial chaos bases for elliptic equations abstract one widely used and computationally efficient method for uncertainty quantification using spectral stochastic finite element is the stochastic galerkin method here the solution is represented in polynomial chaos expansion and the residual of the discretized governing equation is projected on the polynomial chaos bases this results in a system of deterministic algebraic equations with the polynomials chaos coefficients as unknown however one impediment for its large scale applications is the curse of dimensionality that is the exponential growth of the number of polynomial chaos bases with the stochastic dimensionality and degree of expansion here for a stochastic elliptic problem an adaptive selection of polynomial chaos bases is proposed accordingly during the first few iterations in the preconditioned conjugate gradient method for solving the system of linear algebraic equations the chaos bases with maximal contribution in an appropriately defined metric to the solution are first identified subsequently only these bases are retained for further iterations until convergence is achieved using numerical studies a three times cost saving over the existing method is observed furthermore for enhancing the computational cost gain the stochastic galerkin method is reformulated as a generalized sylvester equation this step allowed efficient usage of the sparsity of moments of product of polynomial chaos bases through numerical studies on problems with large stochastic dimensionality an additional cost saving of up to one order of magnitude twenty times is observed this amounts to sixty times speedup over the existing method when adaptive selection and generalized sylvester equation formulation are used together the proposed methodology can be easily incorporated in an existing standard stochastic galerkin method solver for elliptic problems

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

a stochastic computational multiscale approach application to mems resonators abstract the aim of this work is to develop a stochastic multiscale model for polycrystalline materials which accounts for the uncertainties in the micro structure at the finest scale we model the micro structure using a random voronoi tessellation each grain being assigned a random orientation then we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso scale a random field of the meso scale elasticity tensor can thus be generated based on the information obtained from the sve simulations finally using a stochastic finite element method these meso scale uncertainties are propagated to the coarser scale as an illustration we study the resonance frequencies of mems micro beams made of poly silicon materials and we show that the stochastic multiscale approach predicts results in agreement with a monte carlo analysis applied directly on the fine finite element model i e with an explicit discretization of the grains

on the quantification of damping model uncertainty abstract the characterization of uncertainty in the damping forces in a vibrating structure is of significant interest in structural dynamics the most common approach is to use a viscous damping matrix with uncertain coefficients the viscous damping is not the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model there are many functional forms which can be used as damping models in this paper two approaches are proposed to quantify such model form uncertainty associated with the use of the viscous damping model not only the model parameters the first approach is based on an ensemble of equivalent damping functions and the second approach is based on the random matrix theory the results obtained from the two methods are compared using numerical examples it was observed that the random matrix theory can be used to quantify damping model uncertainty in some cases

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

equivalence problem for the orthogonal separable webs in 3 dimensional hyperbolic space the equivalence problem for the orthogonal separable coordinate webs determined by killing two tensors defined in hyperbolic 3 space is solved within the framework of the invariant theory of killing tensors this work completes and extends the results presented in the classical 1950 paper by olevsky math sb 27 379 426 1950

modeling and quantification of model form uncertainties in eigenvalue computations using a stochastic reduced model a feasible nonparametric probabilistic approach for modeling and quantifying model form uncertainties associated with a computational model designed for the solution of a generalized eigenvalue problem is presented it is based on the construction of a stochastic projection based reduced order model sprom associated with a high dimensional model hdm using three innovative ideas the substitution of the deterministic reduced order basis rob with a stochastic counterpart srob featuring a reduced number of hyperparameters the construction of this srob on a subset of a compact stiefel manifold in order to guarantee the linear independence of its column vectors and the satisfaction of any constraints of interest and the formulation and solution of a reduced order inverse statistical problem to determine the hyperparameters so that the mean value and statistical fluctuations of the eigenvalues predicted using the sprom match target values obtained from available data consequently the proposed approach for modeling model form uncertainties can be interpreted as an effective approach for extracting from data fundamental information and or knowledge that are not captured by a deterministic computational model and incorporating them in this model its potential for quantifying model form uncertainties in generalized eigencomputations is demonstrated for a natural vibration analysis of a small scale replica of an x 56 type aircraft made of a composite material for which ground ground vibration test data are available

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

surface ground vibration due to a moving train in a tunnel two dimensional model abstract vibration of a surface of a two dimensional 2d elastic layer generated by a point load moving uniformly along a beam which is located inside the layer is investigated theoretically it is supposed that the layer possesses a small viscosity is fixed along the bottom and has a traction free surface the beam is described by the euler bernoulli model and located parallel both to the surface and the bottom of the layer the surface vibration is analysed under three types of the load namely constant harmonically varying and a stationary random load for the deterministic loads the vector displacement of an observation point on the layer surface is analyzed along with the amplitude spectrum of vibration in this point for the random load main attention is paid to the variance of vibration at the observation point qualitative features of obtained results are discussed via kinematic analyses of the wave propagation in the structure

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized

reduced chaos decomposition with random coefficients of vector valued random variables and random fields we develop a stochastic functional representation that is adapted to problems involving various forms of epistemic uncertainties including modeling error and data paucity the new representation builds on the polynomial chaos decomposition and eventually yields a karhunen loeve expansion with random multiplicative coefficients in this expansion one set of uncertainty is captured in the usual manner as uncorrelated scalar random variables another component of the uncertainty statistically independent from the first is captured by constructing the usually deterministic functions in the kl expansion as random functions we think of the first set of uncertainties as associated with a coarse scale model and of the second set as associated with subscale fluctuations not captured in the coarse scale description

invariant classification of the rotationally symmetric r separable webs for the laplace equation in euclidean space an invariant characterization of the rotationally symmetric r separable webs for the laplace equation in euclidean space is given in terms of invariants and covariants of a real binary quartic canonically associated with the characteristic conformal killing tensor which defines the webs

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

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

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

reduced models for the medium frequency dynamics of stochastic systems in this paper a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium frequency range in this frequency range both traditional modal analysis and statistical energy analysis sea procedures well suited for low and high frequency vibration analysis respectively lead to computational and conceptual difficulties the uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical the methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods in particular the karhunen loeve and polynomial chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest