Spectral Stochastic Finite Element Method Research Articles

A semi-intrusive stochastic perturbation method for lift prediction and global sensitivity analysis

AbstractSensitivity analysis plays an important role in finding an optimal design of a structure under uncertainty. Quantifying relative importance of random parameters, which leads to a rank ordering, helps in developing a systematic and efficient way to reach the optimal design. In this work, lift prediction and sensitivity analysis of a potential flow around a submerged body is considered. Such flow is often used in the initial design stage of structures. The flow computation is carried out using a vortex-panel method. A few parameters of the submerged body and flow are considered as random variables. To improve the accuracy in lift prediction in a computationally efficient way, a new semi-intrusive stochastic perturbation method is proposed. Accordingly, a perturbation is applied at the linear system solving level involving the inuence coefficient matrix, as opposed to using perturbation in the lift quantity itself. This proposed method, which is partially analogous to the intrusive or Galerkin projection methods in spectral stochastic finite element methods, is found to be more accurate than using perturbation directly on the lift and faster than a direct simulation. The proposed semi-intrusive stochastic perturbation method is found to yield faster estimates of the Sobol’ indices, which are used for global sensitivity analysis. From global sensitivity analysis, the flow parameters are found to be more important than the parameters of the submerged body.

Spectral stochastic finite element method in vibroacoustic analysis of fiber-reinforced composites

The attractive high strength and low weight of fiber-reinforced composites (FRC) make them ideal in many applications as defense and automotive industry. Such materials, however, possess some level of uncertainties which have to be taking in account for a reliable performance and quality control. In this paper, non-sampling based spectral stochastic finite element method is applied to vibroacoustic analysis of FRC plates with uncertain elastic and damping parameters. The impact of parameter uncertainties on the acoustical transmission loss is investigated. The results denote various impacts of uncertainties in different frequency ranges. Further, the calculated random transmission loss with few realizations on collocation points show a reasonable accuracy compared to results obtained from large number of realizations by sampling-based stochastic Monte Carlo simulations, and consequently, very efficient in terms of computational time.

Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method

ABSTRACTThis article aims to present a combination of stochastic finite element and spectral finite element methods as a new numerical tool for uncertainty quantification. One of the well-established numerical methods for reliability analysis of engineering systems is the stochastic finite element method. In this article, a commonly used version of the stochastic finite element method is combined with the spectral finite element method. Furthermore, the spectral finite element method is a numerical method employing special orthogonal polynomials (e.g., Lobatto) and quadrature schemes (e.g., Gauss-Lobatto-Legendre), leading to suitable accuracy, and much less domain discretization with excellent convergence as well. The proposed method of this article is a hybrid method utilizing efficiencies of both methods for analysis of stochastically linear elastostatic problems. Moreover, a spectral finite element method is proposed for numerical solution of a Fredholm integral equation followed by the present method, to provide further efficiencies to accelerate stochastic computations. Numerical examples indicate the efficiency and accuracy of the proposed method.

Graph theoretical methods for efficient stochastic finite element analysis of structures

Stochastic finite element method (StFEM) is a robust tool for uncertainty quantification of engineering systems having random properties. Nevertheless, the matrices involved in this method are very large compared to their deterministic counterparts. Thus, the computational aspects of StFEM are of great importance to be optimized. In this paper, an efficient StFEM is developed for analysis of structures. For this purpose, a method based on graph concepts is presented and extended to StFEM and recently developed stochastic spectral finite element method (StSFEM) procedures. Here, mathematical remedies are incorporated to enhance the analysis performance. Firstly, a graph theoretical method is presented for swift numerical solution of Fredholm integral equation arising from Karhunen–Loève expansion, which greatly reduces the existing computational cost, and can even be applied to the domain without symmetry. Secondly, a preconditioner is applied to decompose the matrices to Kronecker products of sub-matrices, and then graph product rules are utilized to solve the governing linear equation of cyclically symmetric models without inversing the final matrix, while only a small matrix is inversed instead. The proposed method provides significant improvement in the stochastic structural analysis. Illustrative examples demonstrate the efficiency and accuracy of the present method as a swift analysis.

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–Loève (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.

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

SummaryThis paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

The determination of limit-cycles plays an important role in characterizing complex dynamical systems, such as unsteady fluid flows. In practice, dynamical systems are described by models equations involving parameters which are seldom exactly known, leading to parametric uncertainties. These parameters can be suitably modeled as random variables, so if the system possesses almost surely a stable time periodic solution, limit-cycles become stochastic, too. This paper introduces a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method with polynomial chaos (PC) expansions. The method is designed to overcome the limitation of PC expansions related to convergence breakdown for long term integration. First, a stochastic time scaling of the model equations is determined to control the phase-drift of the stochastic trajectories and allowing for accurate low order PC expansions. Second, using the rescaled governing equations, we aim at ...

Application of stochastic finite element approaches to wood-based products

Due to the natural growing process of wood, the mechanical properties of wooden boards are subject to high variability, mainly introduced by knots and the resulting deviation of the wood fibre directions around them. This variability has a great impact on the serviceability limit state performance of wood-based products, such as glued-laminated timber, and thus should be considered within design concepts. Numerous applications of random process models for numerically representing the fluctuation of the mechanical properties along wooden boards can be found in the literature. But, the corresponding mechanical probabilistic investigation, however, is limited almost exclusively to Monte Carlo simulations so far. For this reason, the focus of this work is laid on alternative probabilistic approaches, in particular the perturbation and the spectral stochastic finite element method. Both methods are combined with several discretization methods for the random process, programmed in a consistent environment, and compared to the Monte Carlo simulation, regarding computational effort as well as quality of results. For this purpose, the second-order moments (mean and standard deviation) of the system response of a glued-laminated timber beam with random lamination stiffness are computed. The performance of the different approaches is compared and, in particular, the influence of the variability of the ‘raw’ material on the structural response is shown. Well-known effects, such as the decrease in the variability of effective properties of GLT with increasing number of lamellas, are numerically reproduced and quantified. Moreover, a significant influence of the correlation length, specifying the rate of material property fluctuation within each lamella, on the effective stiffness of the resulting glued-laminated timber beams is demonstrated.

Stochastic finite element approaches for wood-based products: theoretical framework and review of methods

As a result of the natural growing process of wood, each timber lamella shows a high variability in its mechanical properties, particularly strength and stiffness. It is assumed that those properties are governed by a random process and, subsequently, that the effective stiffness of glued laminated timber also varies randomly. Therefore, a probabilistic approach is necessary. Hence, the latest achievements in probabilistic timber engineering are reviewed and compared. Numerous works rely on random process models for the representation of stiffness and/or strength distributions in single timber lamellas. The statistical evaluation of those random process models, however, is limited almost exclusively to Monte Carlo simulation (MCS) so far. Therefore, this work aims at giving an overview of alternative ways to compute the effective stiffness, by reviewing the framework of stochastic finite element methods. Random process models for the representation of the stiffness distribution in single lamellas are discussed, and the two most promising alternatives to the MCS for computation of effective stiffness parameters, the perturbation and the spectral stochastic finite element method, are evaluated in terms of accuracy and efficiency. Finally, this paper shows alternative and more efficient ways of exploring the stochastic nature of wood, delivering a new basis for more reliable design concepts for timber products.

A Study of Probabilistic FEMs for a Slope Reliability Analysis Using the Stress Fields

In this paper, the applicability and the effectiveness of the probabilistic finite element methods (FEMs) such as the perturbation method, and the Spectral Stochastic Finite Element Method (SSFEM) applied to the reliability analysis of the slope stability have been studied. The results were checked by the Monte Carlo simulation and a direct coupling ap-proach combining the deterministic finite elements code and First Order Reliability Method (FORM) algorithm. These methods are presented considering the spatial variation of soil strength parameters and Young modulus. The random field is used to describe the spatial variation. Also, the reliability analysis is conducted using a performance function formulat-ed in terms of the stochastic stress mobilized along the sliding surface. The present study shows that the perturbation method and SSFEM can be considered as practical methods to conduct a second moment analysis of the slope stability taking into account the spatial variability of soil properties since good results are obtained with acceptable estimated rela-tive errors. Finally, the perturbation method is performed to delimit the location of the critical probabilistic sliding surfac-es and to evaluate the effect of the correlation length of soil strength parameters on the safety factor. In addition, the two methods are used to estimate the probability density and the cumulative distribution function of the factor of safety.

An adaptive spectral Galerkin stochastic finite element method using variability response functions

SummaryA methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.

A hybrid method for stochastic response analysis of a vibrating structure

Response analysis of a linear structure with uncertainties in both structural parameters and external excitation is considered here. When such an analysis is carried out using the spectral stochastic finite element method (SSFEM), often the computational cost tends to be prohibitive due to the rapid growth of the number of spectral bases with the number of random variables and the order of expansion. For instance, if the excitation contains a random frequency, or if it is a general random process, then a good approximation of these excitations using polynomial chaos expansion (PCE) involves a large number of terms, which leads to very high cost. To address this issue of high computational cost, a hybrid method is proposed in this work. In this method, first the random eigenvalue problem is solved using the weak formulation of SSFEM, which involves solving a system of deterministic nonlinear algebraic equations to estimate the PCE coefficients of the random eigenvalues and eigenvectors. Then the response is estimated using a Monte Carlo (MC) simulation, where the modal bases are sampled from the PCE of the random eigenvectors estimated in the previous step, followed by a numerical time integration. It is observed through numerical studies that this proposed method successfully reduces the computational burden compared with either a pure SSFEM of a pure MC simulation and more accurate than a perturbation method. The computational gain improves as the problem size in terms of degrees of freedom grows. It also improves as the timespan of interest reduces.

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.

Practical Application of the Stochastic Finite Element Method

The stochastic finite element method is an extension of the FEM that considers the uncertainty of a system that arises through variations in initial conditions, materials or geometry. Systems which display a measurable degree of disorder can be studied efficiently using a probabilistic approach. Different scenarios can be randomly generated with the SFEM to study the behaviour of systems that take into account prior knowledge of the differing variations in properties. This review paper introduces the most commonly used techniques: direct Monte Carlo simulation, the perturbation method and the spectral stochastic finite element method. It then looks at the currently available software for the SFEM and provides examples from the disciplines of materials science, biomechanics and engineering to illustrate different procedures by which the SFEM is practically used. The aim of the paper is to help scientists and engineers quickly assess how they might apply SFEM to their own research and guide them towards key publications.

Spectral Stochastic Finite Element Method for Electromagnetic Problems with Random Geometry

Abstract In electromagnetic problems, the problem geometry may not always be exactly known. One example of such a case is a rotating machine with random-wound windings. While spectral stochastic finite element methods have been used to solve statistical electromagnetic problems such as this, their use has been mainly limited to problems with uncertainties in material parameters only. This paper presents a simple method to solve both static and time-harmonic magnetic field problems with source currents in random positions. By using an indicator function, the geometric uncertainties are effectively reduced to material uncertainties, and the problem can be solved using the established spectral stochastic procedures. The proposed method is used to solve a demonstrative single-conductor problem, and the results are compared to the Monte Carlo method. Based on these simulations, the method appears to yield accurate mean values and variances both for the vector potential and current, converging close to the results obtained by time-consuming Monte Carlo analysis. However, further study may be needed to use the method for more complicated multi-conductor problems and to reduce the sensitivity of the method on the mesh used.

Assessment of the probability of failure for EC nondestructive testing based on intrusive spectral stochastic finite element method

This work is undertaken to study the reliability of eddy current nondestructive testing (ED-NDT) when the defect concerns a change of physical property of the material. So, an intrusive spectral stochastic finite element method (SSFEM) is developed in the case of 2D electromagnetic harmonic equation. The electrical conductivity is considered as random variable and is developed in series of Hermite polynomials. The developed model is validated from measurements on NDT device and is applied to the assessment of the reliability of failure in steam generator tubing of nuclear power plants. The exploitation of the model concerns the impedance calculation of the sensor and the assessment of the reliability of failure. The random defect geometry is also considered and results are given.

Schwarz preconditioners for stochastic elliptic PDEs

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.

A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

The determination of stable limit-cycles plays an important role in quantifying the characteristics of dynamical systems. In practice, exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves, resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method using polynomial chaos (PC). We are able to overcome the difficulties of PC regarding its well-known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the rescaled governing equations, we aim at determini...

Solución en dominios cuadrados de ecuaciones de difusión-convección mediante elementos finitos estocásticos espectrales en conjunto con el método de Petrov Galerkin en contracorriente

In this paper we have developed the numerical solution of two problems of diffusion-convection (DC), using the finite element method of Streamline Upwind Petrov-Galerkin (SUPG). The parameters that define the behavior of the equations are modeled as stochastic fields, specifically, are used: the convective velocity, diffusion and heat capacity as values of random type. Therefore, we have included SUPG method to DC, with dominant convection, with the stochastic spectral finite element method. Each parameter was described by Karhunen-Loève expansion, while the unknown is represented by the polynomial expansion of the chaos. The objectives of the paper are: 1. To study the influence of stochastic fields in solving problems with SUPG DC and 2. Get the solution of each expansion unknown variable. The results show the versatility of the method for solving different physical problems due to the generality of the statistical description of the solution and the richness in the representation of the areas where there is the greater variability in response. The patterns shown in the unknown uncertainty depends on the dynamics of diffusion, convective velocity and the type of solution used.

A domain decomposition method of stochastic PDEs: An iterative solution techniques using a two-level scalable preconditioner

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.