Spectral Stochastic Finite Element Method Research Articles

Dual-primal domain decomposition method for uncertainty quantification

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.

Solution of Large Stochastic Finite Element Problems—Application to ECT-NDT

This paper describes an efficient bloc iterative solver for the spectral stochastic finite element method (SSFEM). The SSFEM was widely used to quantify the effect of input data uncertainties on the outputs of finite element models. The bloc iterative solver allows reducing computational cost of the SSFEM. The method is applied on an industrial nondestructive testing (NDT) problem. The numerical performances of the method are compared with those of the nonintrusive spectral projection (NISP).

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

Método del elemento finito estocástico en geotecnia. Enfoque espectral

This paper presents the mathematical tools in which the formulation of Spectral Stochastic Finite Element Method is based. The usefulness of this method to model the spatial variability of heterogeneous materials, and in particular of soils, is illustrated by a practical example in which the propagation of the uncertainty on the deformation modulus to the computed displacement field is assessed. The influence of the correlation length on the distribution of uncertainty is set forth. Finally, the advantages of the method in geotechnical engineering are evaluated and some conclusions are presented.

Structural Reliability Assessment by a Modified Spectral Stochastic Meshless Local Petrov-Galerkin Method

This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.

Extended stochastic FEM for diffusion problems with uncertain material interfaces

This paper is concerned with the prediction of heat transfer in composite materials with uncertain inclusion geometry. To numerically solve the governing equation, which is defined on a random domain, an approach based on the combination of the Extended finite element method (X-FEM) and the spectral stochastic finite element method is studied. Two challenges of the extended stochastic finite element method (X-SFEM) are choosing an enrichment function and numerical integration over the probability domain. An enrichment function, which is based on knowledge of the interface location, captures the C 0-continuous solution in the spatial and probability domains without a conforming mesh. Standard enrichment functions and enrichment functions tailored to X-SFEM are analyzed and compared, and the basic elements of a successful enrichment function are identified. We introduce a partition approach for accurate integration over the probability domain. The X-FEM solution is studied as a function of the parameters describing the inclusion geometry and the different enrichment functions. The efficiency and accuracy of a spectral polynomial chaos expansion and a finite element approximation in the probability domain are compared. Numerical examples of a two-dimensional heat conduction problem with a random inclusion show the spectral PC approximation with a suitable choice of enrichment function is as accurate and more efficient than the finite element approach. Though focused on heat transfer in composite materials, the techniques and observations in this paper are also applicable to other types of problems with uncertain geometry.

Spectral stochastic finite element method for solving 3D stochastic eddy current problems

In this paper, a spectral stochastic finite element method (SSFEM) is described and applied to quantify uncertainties in an industrial eddy current non destructive testing (NDT) problem. A new iterative solver based on the fixed-point method is proposed to solve the large scale linear system. The efficiency of the suggested iterative solver to tackle the SSFEM computational complexity is demonstrated. The SSFEM results and the computation times obtained with 2 and 4 random variables are compared to those given by the anisotropic adaptive sparse grid non intrusive spectral projection (NISP). On this application, the SSFEM dedicated solver is competitive with the adaptive sparse grids strategy.

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

SUMMARYThe repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non‐unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal‐based reduced‐order model of the system. Copyright © 2012 John Wiley & Sons, Ltd.

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.

Quasi-weak and weak formulation of stochastic finite elements on static and dynamic problems—a unifying framework

To model uncertainty of spatial and/or temporal variations widely present in synthetic and natural media, a variety of displacement-based stochastic finite element methods (SFEMs) have been formulated using the standard displacement-based finite elements. In this paper, by distinguishing a quasi-weak form from a weak form in both real and random space, a unifying framework of variational formulation is presented covering both the displacement-based SFEMs and the recently proposed Green-function-based (GFB) SFEM. The study shows that Monte Carlo, perturbation, and weighted integral SFEMs correspond to the quasi-weak form, while the weak form results in spectral SFEM, pseudo-spectral SFEM, and GFB-SFEM. Within the unifying framework, dynamic problems are further addressed especially to demonstrate the unique feature of GFB-SFEM on problems with inputs characterized as random fields or random processes.

Evaluating the response statistics of an uncertain bridge–vehicle system

This paper addresses the statistical prediction of dynamic response of a bridge structure with inherent uncertainty under stochastic moving loads. The uncertainty in bridge structure, which is assumed with Gaussian properties, is modeled by the Spectral Stochastic Finite Element Method. The uncertain moving forces with Gaussian distribution are represented by the Karhunen–Loève expansion. The Polynomial chaos expansion is adopted to represent the uncertain response such that the proposed algorithm is not limited to system with small uncertainty. The bridge is modeled as a simply supported Bernoulli–Euler beam and the bridge–vehicle interaction forces are modeled as Gaussian random processes related to the power spectral density of the surface roughness defined in the ISO standard. The mathematical model of the bridge–vehicle system is formulated and solved with the Newmark-β method. The response statistics of the bridge are compared with those from Monte Carlo simulation. Different levels of uncertainty in both the system parameters and the moving forces are studied to investigate the accuracy and robustness of the proposed method in the stochastic analysis of the bridge–vehicle interaction problem.

Doubly Spectral Stochastic Finite-Element Method for Linear Structural Dynamics

Uncertainties in complex dynamic systems play an important role in the prediction of a dynamic response in the mid- and high-frequency ranges. For distributed parameter systems, parametric uncertainties can be represented by random fields leading to stochastic partial differential equations. Over the past two decades, the spectral stochastic finite-element method has been developed to discretize the random fields and solve such problems. On the other hand, for deterministic distributed parameter linear dynamic systems, the spectral finite-element method has been developed to efficiently solve the problem in the frequency domain. In spite of the fact that both approaches use spectral decomposition (one for the random fields and the other for the dynamic displacement fields), very little overlap between them has been reported in literature. In this paper, these two spectral techniques are unified with the aim that the unified approach would outperform any of the spectral methods considered on their own. An ...

3-D Stochastic Spectral Finite-Element Method in Static Electromagnetism Using Vector Potential Formulation

Stochastic spectral finite-element method can be used to take into account some random aspects in the input data (material characteristic, source terms) involved in static electromagnetism problems. Similarly to the deterministic case, two potential formulations can be used in the stochastic case. The vector potential formulation applied to static problems is developed and compared to the scalar potential one, previously developed.

Helicopter aeroelastic analysis with spatially uncertain rotor blade properties

This paper investigates the effects of spatially uncertain material properties on the aeroelastic response predictions (e.g., rotating frequencies, vibratory loads, etc.) of composite helicopter rotor. Initially, the spatial uncertainty is modeled as discrete random variables along the blade span and uncertainty analysis is performed with direct Monte Carlo simulations (MCS). Uncertainty effects on the rotating frequencies vary with the higher-order modes in a non-linear way. Each modal frequency is found to be more sensitive to the uncertainty at certain sections of the rotor blade than uncertainty at other sections. Uncertainty effects on the vibratory hub load predictions are studied in the next stage. To reduce the computational expense of stochastic aeroelastic analysis, a high-dimensional model representation (HDMR) method is developed to approximate the aeroelastic response as functions of blade stiffness properties which are modeled as random fields. Karhunen–Loève expansion and a lower-order expansion are used to represent the input and outputs, respectively, in the HDMR formulation which is similar to the spectral stochastic finite element method. The proposed method involves the approximation of the system response with lower-dimensional HDMR, the response surface generation of HDMR component functions, and Monte Carlo simulation. The proposed approach decouples the computationally expensive aeroelastic simulations and uncertainty analysis. MCS, performed with computationally less expensive HDMR models, shows that spatial uncertainty has considerable influence on the vibratory hub load predictions.

Dynamic analysis of bridge with non-Gaussian uncertainties under a moving vehicle

An analysis method on the bridge–vehicle interaction problem with uncertainties is proposed. The bridge is modeled as a simply supported Euler–Bernoulli beam with non-Gaussian material parameters with a vehicle moving on top modeled by a deterministic four degrees-of-freedom mass–spring system. The non-Gaussian uncertainty in bridge is modeled by the Spectral Stochastic Finite Element Method (Ghanem and Spanos (1991) [17]), and the mathematical model of the coupled bridge–vehicle system, with the road surface roughness assumed as a Gaussian random process, will be solved by the Newmark- β method. The proposed model is verified by the Monte Carlo Simulation with numerical examples. Different levels of uncertainties in both the excitation and system parameters are investigated. Criteria on the selection of both the order of Polynomial Chaos and the threshold for truncation in the Karhunen–Loève expansion are provided. Results show that the proposed algorithm is promising for the dynamic analysis of the bridge–vehicle interaction problem even with a high level of system and excitation uncertainties.

Solution of Dual Stochastic Static Formulations Using Double Orthogonal Polynomials

The solution of stochastic partial differential equations (PDEs) using the spectral stochastic finite-element method (SSFEM) can lead to a very large linear system of equations. If the random input data are independent, it can be shown that the initial linear system can be split into smaller independent linear systems by using double orthogonal polynomials. In this paper, we propose the use of this approach in the case of dual potential formulations in electrokinetics. The method is applied to an electrokinetic problem taking into account the uncertainties on contact resistances.

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.

High dimensional model representation for stochastic finite element analysis

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.

A reduced-order stochastic finite element approach for design optimization under uncertainty

This paper presents a computational framework for structural design optimization under uncertainty. The stochastic static response of linear elastic structures is predicted by a spectral stochastic finite element method (SSFEM) based on a polynomial chaos expansion (PCE). Traditional SSFEM approaches lead to large-scale numerical problems and computational costs rendering these methods impractical for design optimization purposes. To mitigate the computational burden, a Galerkin-based multi-point reduced-order model (ROM) is integrated into SSFEM. The ROM basis is spanned by displacements and derivatives of displacements with respect to design and random variables evaluated at adaptively selected calibration points in the design space. The design sensitivity equations are derived and a formal analysis of the computational costs in terms of floating-point operations is presented. The efficiency of the ROM based approach versus traditional SSFEM is studied with the shape optimization of a shell structure. For this example, the results show that the proposed reduced-order modeling scheme can lower the total turnaround time by more than a factor of 30.

A FETI‐preconditioned conjugate gradient method for large‐scale stochastic finite element problems

AbstractIn the spectral stochastic finite element method for analyzing an uncertain system, the uncertainty is represented by a set of random variables, and a quantity of interest such as the system response is considered as a function of these random variables. Consequently, the underlying Galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled. The solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and/or the level of uncertainty is increased. This paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual–primal finite element tearing and interconnecting method equipped with a Krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right‐hand sides. Preliminary performance results on a Linux Cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable. Copyright © 2009 John Wiley & Sons, Ltd.