Stochastic Newmark scheme

In this paper, we are concerned with numerical resolution of the dynamical equation of linear oscillators. An answer is given to the following question: can one use widely implemented Newmark's algorithm when the acceleration is a white noise instead of a regular deterministic one? After showing drawbacks of such a practice, a modification of the usual Newmark schemes is introduced, to be used for the numerical resolution of such a stochastic dynamical equation. The modification is only in the stochastic part, so that already implemented softwares can be used, the input only having to be carefully adapted. After that, mathematical analysis of this new scheme is fulfilled. First and second moments, as well as the power spectral density resulting from the use of this scheme are compared with the corresponding quantities resulting from the theoretical process, and also from usual Newmark scheme and Euler schemes. Almost sure convergence of this new scheme is proved.     

PVU and wave-particle splitting schemes for Euler equations of gas dynamics

A new way of flux-splitting, termed as the wave-particle splitting is presented for developing upwind methods for solving Euler equations of gas dynamics. Based on this splitting, two new upwind methods termed as Acoustic Flux Vector Splitting (AFVS) and Acoustic Flux Difference Splitting (AFDS) methods are developed. A new Boltzmann scheme, which closely resembles the wave-particle splitting, is developed using the kinetic theory of gases. This method, termed as Peculiar Velocity based Upwind (PVU) method, uses the concept of peculiar velocity for upwinding. A special feature of all these methods is that the unidirectional and multidirectional parts of the flux vector are treated separately. Extensive computations done using these schemes demonstrate the soundness of the ideas.     

On the efficacy of stochastic collocation,stochastic Galerkin,and stochastic reduced order models for solving stochastic problems

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method. Rather, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.     

Analysis of one-dimensional stochastic finite elements using neural networks

This paper explores the applicability of neural networks for analyzing the uncertainty spread of structural responses under the presence of one-dimensional random fields. Specifically, the neural network is intended to be a partial surrogate of the structural model needed in a Monte Carlo simulation, due to its associative memory properties. The network is trained with some pairs of input and output data obtained by some Monte Carlo simulations and then used in substitution of the finite element solver. In order to minimize the size of the networks, and hence the number of training pairs, the Karhunen–Loéve decomposition is applied as an optimal feature extraction tool. The Monte Carlo samples for training and validation are also generated using this decomposition. The Nyström technique is employed for the numerical solution of the Fredholm integral equation. The radial basis function (RBF) network was selected as the neural device for learning the input/output relationship due to its high accuracy and fast training speed. The analysis shows that this approach constitutes a promising method for stochastic finite element analysis inasmuch as the error with respect to the Monte Carlo simulation is negligible.     

Data assimilation and uncertainty analysis of environmental assessment problems—an application of Stochastic Transfer Function and Generalised Likelihood Uncertainty Estimation techniques

Stochastic Transfer Function (STF) and Generalised Likelihood Uncertainty Estimation (GLUE) techniques are outlined and applied to an environmental problem concerned with marine dose assessment. The goal of both methods in this application is the estimation and prediction of the environmental variables, together with their associated probability distributions. In particular, they are used to estimate the amount of radionuclides transferred to marine biota from a given source: the British Nuclear Fuel Ltd (BNFL) repository plant in Sellafield, UK. The complexity of the processes involved, together with the large dispersion and scarcity of observations regarding radionuclide concentrations in the marine environment, require efficient data assimilation techniques. In this regard, the basic STF methods search for identifiable, linear model structures that capture the maximum amount of information contained in the data with a minimal parameterisation. They can be extended for on-line use, based on recursively updated Bayesian estimation and, although applicable to only constant or time-variable parameter (non-stationary) linear systems in the form used in this paper, they have the potential for application to non-linear systems using recently developed State Dependent Parameter (SDP) non-linear STF models. The GLUE based-methods, on the other hand, formulate the problem of estimation using a more general Bayesian approach, usually without prior statistical identification of the model structure. As a result, they are applicable to almost any linear or non-linear stochastic model, although they are much less efficient both computationally and in their use of the information contained in the observations. As expected in this particular environmental application, it is shown that the STF methods give much narrower confidence limits for the estimates due to their more efficient use of the information contained in the data. Exploiting Monte Carlo Simulation (MCS) analysis, the GLUE technique is used to estimate how the errors involved in the STF model structure and observations influence the model outputs and errors. The discussion on updating information originating from different locations using GLUE procedure is also given. A final aim of the paper is to use the results obtained in this particular example to explore the differences between the GLUE and STF approaches.     

A stochastic model for localized disturbances and its applications

A stochastic model for local disturbances, particularly for a temporal harmonic with random modulations in amplitude and/or phase, is proposed in this paper. Results for the second moment responses of a linear single-degree-of-freedom system to this type of stochastic loading demonstrate a significant change in response characteristics due to a small uncertainty. A local phenomenon may last much longer and resonance may be smeared to a broad range. Integrated with wavelet transform, the proposed approach may be used to model a random process with non-stationary frequency content. Especially, it can be effectively used for Monte Carlo simulation to generate large size of samples that have similar characteristics in time and frequency domains as a pre-selected mother sample has. The technique has a great potential for the case where uncertainty study is warranted but the available samples are limited.     

A comparison of stochastic dominance and stochastic DEA for vendor evaluation

Vendor selection involves decisions balancing a number of conflicting criteria. Data envelopment analysis (DEA) is a mathematical programming approach capable of identifying non-dominated solutions, as well as assessing relative efficiency of dominated solutions. A simple multi-attribute utility function can be applied to a small set of alternatives, providing a tool to assess relative value, but is subject to error if estimated measures are not precise. This paper compares stochastic DEA with a multiple-criteria model in a vendor selection model involving multiple criteria, reporting simulation experiments varying the degree of uncertainty involved in model parameters.     

Hybrid Subset Simulation method for reliability estimation of dynamical systems subject to stochastic excitation

A hybrid Subset Simulation approach is proposed for reliability estimation for general dynamical systems subject to stochastic excitation. This new stochastic simulation approach combines the advantages of the two previously proposed Subset Simulation methods, Subset Simulation with Markov Chain Monte Carlo (MCMC) algorithm and Subset Simulation with splitting. The new method employs the MCMC algorithm before reaching an intermediate failure level and splitting after reaching the level to exploit the causality of dynamical systems. The statistical properties of the failure probability estimators are derived. Two examples are presented to demonstrate the effectiveness of the new approach and to compare with the previous two Subset Simulation methods. The results show that the new method is robust to the choice of proposal distribution for the MCMC algorithm and to the intermediate failure events selected for Subset Simulation.     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.     

Comparative study of numerical explicit schemes for impact problems

Explicit numerical schemes are used to integrate in time finite element discretization methods. Unfortunately, these numerical approaches can induce high-frequency numerical oscillations into the solution. To eliminate or to reduce these oscillations, numerical dissipation can be introduced. The paper deals with the comparison of three different explicit schemes: the central-difference scheme which is a non-dissipative method, the Hulbert-Chung dissipative explicit scheme and the Tchamwa-Wielgosz dissipative scheme. Particular attention is paid to the study of these algorithms' behavior in problems involving high-velocity impacts like Taylor anvil impact and bullet-target interactions. It is shown that Tchamwa-Wielgosz scheme is efficient in filtering the high-frequency oscillations and is more dissipative than Hulbert-Chung explicit scheme. Although its convergence rate is only first order, the loss of accuracy remains limited to acceptable values.     

Monte Carlo simulation of gas flow through the KATRIN DPS2-F differential pumping system

The Karlsruhe Tritium Neutrino experiment (KATRIN) is a large vacuum system and aims to measure the electron neutrino mass from the β decay of tritium with unprecedented sensitivity. To achieve this purpose, the tritium gas flow has to be significantly reduced along the beamline by means of a modular differential pumping system. This paper studies systematically the vacuum performance of one of the differential pumping systems (known as DPS2-F) based on turbomolecular pumps. The flow reduction rates in the complete system are described by a matrix equation as a function of the capture factor of the turbomolecular pumps employed. The results show that a total reduction factor greater than 10⁵ can be attained, which is one of the prerequisites to achieve XHV conditions in the spectrometers used in the downstream end of the experiment.     

Monte Carlo estimation of the differential importance measure: application to the protection system of a nuclear reactor

In this paper, we perform a reliability/availability analysis by means of the Monte Carlo simulation method and illustrate a perturbation method, inherited from the particle transport field, which allows to compute first-order, differential sensitivity indexes with little additional computational effort. The proposed method is illustrated first on a simple case study and, successively, on a nuclear safety system, the reactor protection system. In these applications, the sensitivity indexes are used to compute the differential importance measure, recently introduced to respond to the need of the analyst/decision maker to get information about the importance of changes in the stochastic properties of system components.     

A family of numerical schemes for kinematic flows with discontinuous flux

Multiphase flows of suspensions and emulsions are frequently approximated by spatially one-dimensional kinematic models, in which the velocity of each species of the disperse phase is an explicitly given function of the vector of concentrations of all species. The continuity equations for all species then form a system of conservation laws which describes spatial segregation and the creation of areas of different composition. This class of models also includes multi-class traffic flow, where vehicles belong to different classes according to their preferential velocities. Recently, these models were extended to fluxes that depend discontinuously on the spatial coordinate, which appear in clarifier–thickener models, in duct flows with abruptly varying cross-sectional area, and in traffic flow with variable road surface conditions. This paper presents a new family of numerical schemes for such kinematic flows with a discontinuous flux. It is shown how a very simple scheme for the scalar case, which is adapted to the “concentration times velocity” structure of the flux, can be extended to kinematic models with phase velocities that change sign, flows with two or more species (the system case), and discontinuous fluxes. In addition, a MUSCL-type upgrade in combination with a Runge–Kutta-type time discretization can be devised to attain second-order accuracy. It is proved that two particular schemes within the family, which apply to systems of conservation laws, preserve an invariant region of admissible concentration vectors, provided that all velocities have the same sign. Moreover, for the relevant case of a multiplicative flux discontinuity and a constant maximum density, it is proved that one scalar version converges to a BV _t entropy solution of the model. In the latter case, the compactness proof involves a novel uniform but local estimate of the spatial total variation of the approximate solutions. Numerical examples illustrate the performance of all variants within the new family of schemes, including applications to problems of sedimentation, traffic flow, and the settling of oil-in-water emulsions.     

Assessment of three preconditioning schemes for solution of the two‐dimensional Euler equations at low Mach number flows

Three preconditioners proposed by Eriksson, Choi and Merkel, and Turkel are implemented in a 2D upwind Euler flow solver on unstructured meshes. The mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn, and their eigenvalues and eigenvectors are compared with each other. For this purpose, these preconditioning schemes are expressed in a unified formulation. A cell‐centered finite volume Roe's method is used for the discretization of the preconditioned Euler equations. The accuracy and performance of these preconditioning schemes are examined by computing steady low Mach number flows over a NACA0012 airfoil and a two‐element NACA4412–4415 airfoil for different conditions. The study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low Mach number flows. The study indicates that the preconditioning methods implemented provide nearly the same results in accuracy; however, they give different performances in convergence rate. It is demonstrated that although the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality, the condition number of the system of equations plays an important role, and it determines the convergence characteristics of solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Tsallis entropy in dual homogenization of random composites using the stochastic finite element method

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 10th‐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.     

On sampling-based schemes for probability of failure sensitivity analysis

In this paper we discuss the accuracy of probability of failure sensitivity analysis with sampling-based schemes. Three approaches commonly employed in literature are discussed: the Weak sensitivity analysis, the Direct employment of finite difference schemes and the Common Random Variable approach. Theoretical estimates for the bias, the coefficient of variation and the mean square error for each approach are presented. The results hold for a single random variable and the extension to more general situations should be pursued in future works. These results lead to the conclusion that the Common Random Variable approach is superior to the Direct approach from the theoretical point of view. The Weak approach, on the other hand, is equivalent to the Common Random Variable approach with central finite difference formula. The choice between these latter two approaches is then a matter of computational efficiency. The results of this work should contribute to further development of efficient algorithms for the problem under study.     

An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic-load due to wind or sea waves. Practical methods for numerical solution of the probability density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability density equations. The ST method also includes a control for the size of the time-step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping matrix picture. The work suggests a few directions for future research.