Spectral Stochastic Finite Element Method Research Articles

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.

Current Calculation in Electrokinetics Using a Spectral Stochastic Finite Element Method

Electrical engineering often deals with global quantities as current, magnetic flux, difference of potentials, power or torque. Previous works have shown that spectral stochastic finite element method makes it possible to spread input data uncertainties (material characteristics) to the local output which are now random. The next step consists in deriving the random global quantities from the random distribution of the fields. It has been implemented in the extended stochastic version of a 3D FEM software. The approach has been successfully validated on a 3D electrical current density problem by comparing with Monte Carlo simulation method. Finally, this method has been applied to an industrial device.

Solution of randomly excited stochastic differential equations with stochastic operator using spectral stochastic finite element method (SSFEM)

This paper considers the solution of the stochastic differential equations (SDEs) with random operator and/or random excitation using the spectral SFEM. The random system parameters (involved in the operator) and the random excitations are modeled as second order stochastic processes defined only by their means and covariance functions. All random fields dealt with in this paper are continuous and do not have known explicit forms dependent on the spatial dimension. This fact makes the usage of the finite element (FE) analysis be difficult. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used to represent these processes to overcome this difficulty. Then, a spectral approximation for the stochastic response (solution) of the SDE is obtained based on the implementation of the concept of generalized inverse defined by the Neumann expansion. This leads to an explicit expression for the solution process as a multivariate polynomial functional of a set of uncorrelated random variables that enables us to compute the statistical moments of the solution vector. To check the validity of this method, two applications are introduced which are, randomly loaded simply supported reinforced concrete beam and reinforced concrete cantilever beam with random bending rigidity. Finally, a more general application, randomly loaded simply supported reinforced concrete beam with random bending rigidity, is presented to illustrate the method.

Inversion of Robin coefficient by a spectral stochastic finite element approach

This paper investigates a variational approach to the nonlinear stochastic inverse problem of probabilistically calibrating the Robin coefficient from boundary measurements for the steady-state heat conduction. The problem is formulated into an optimization problem, and mathematical properties relevant to its numerical computations are investigated. The spectral stochastic finite element method using polynomial chaos is utilized for the discretization of the optimization problem, and its convergence is analyzed. The nonlinear conjugate gradient method is derived for the optimization system. Numerical results for several two-dimensional problems are presented to illustrate the accuracy and efficiency of the stochastic finite element method.

3-D Spectral Stochastic Finite Element Method in Electromagnetism

Spectral stochastic finite element method (SSFEM), used in mechanics to take into account random aspects of input data, has been implemented, as an extended version, in the 3-D finite element method (FEM) software CARMEL dedicated to electromagnetic field computation. As a test case, this approach has been applied to a 3-D electrostatic problem and successfully validated by comparing with the Monte Carlo simulation method involving usual "deterministic" CARMEL resolutions

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed 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 modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.

Efficient solution of stochastic systems: Application to the embankment dam problem

The embankment dam problem of the benchmark study is treated using the newly developed Stochastic Model Reduction for Polynomial Chaos Representations method. The elastic and shear moduli of the material, in the present problem, are modeled as two stochastic processes that are explicit functions of the same process possessing a relatively low correlation length. The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process. In such a setting, the spectral stochastic finite element method (SSFEM) for the specified spatial discretization is computationally prohibitive. The approach adopted in this paper enables the stochastic characterization of a fine mesh problem based on the high dimensional polynomial chaos solution of a coarse mesh analysis. After relatively reducing the dimensionality of the problem through a Karhunen–Loeve representation of the stochastic variables, the SSFEM solution consisting of a high dimensional polynomial in Gaussian independent variables is obtained for the coarse mesh problem. Then the attained solution is used to define a new basis for solving the fine mesh problem. The paper presents some new algorithms for the estimation of chaos coefficients in the presence of complex non-Gaussian dependencies. A numerical convergence study is presented together with a discussion of the results.

Application of the spectral stochastic finite element method for performance prediction of composite structures

This paper demonstrates an application of the spectral stochastic finite element method (SSFEM) for predicting the performance of a composite structure with variable material constitutive properties. The SSFEM can provide an efficient means of calculating statistical moments (e.g. mean and variance) and probability distributions associated with the probabilistic behaviour of a structure, and should therefore be suitable as an alternative to Monte Carlo Simulation for assessing structural reliability. A summary of the SSFEM theory is given, and it is shown how the method can be used to calculate the variability of strain and stress in a composite structure with variable material properties. An example is presented to demonstrate application of the method, and solutions are compared with results from other methods of analysis.

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.

Uncertainty propagation in finite deformations––A spectral stochastic Lagrangian approach

This work proposes a method for quantifying uncertainty propagation in finite deformation problems using the spectral stochastic finite element method (SSFEM). A spectral expansion of the current configuration of a deforming body is proposed to compute the stochastic deformation gradient which is in turn used to compute the stochastic analogs of the various quantities which appear in large deformation analysis such as strain and stress measures and consistent moduli. A total Lagrangian approach to the stochastic large deformation problem is presented. Model problems in large deformation elasto-plasticity are considered highlighting the features of the methodology developed. Rigorous comparisons with Monte-Carlo solutions are presented. It is shown that the proposed approach can estimate the probability density function and response statistics of the field variables with satisfactory accuracy.

Stochastic inverse heat conduction using a spectral approach

AbstractAn adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L2 optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L2 optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.

６．地震の力学問題 三次元不確定波動場のスペクトル確率有限要素解析の並列計算

Consideration of uncertainty is essential in simulation of the earthquake motion. since material properties of the ground has many uncertain factors. Spectral Stochastic Finite Element Method (SSFEM) is a powerful computation tool for three dimensional wave propagation problems in the uncertain media. Although SSFEM is more efficient than Monte Carlo simulation, its computation involves hugematrices and requires long computation time. This paper presents an algorithm suitable for parallel computation of SSFEM with MPI. Numerical examples show that the presented scheme can reduce computation timewithout losing accuracy.

６．地震の力学問題 スペクトル確率有限要素法の地表地震断層問題への適用

For the safety of civil structures, it is required to predict the possibility of ground surface deformation caused by a surface earthquake fault. In order to provide quantitative estimate of hazards due to surface earthquake fault, the authors have been developing a numerical simulator based on the NonLinear Spectral Stochastic Finite Element Method. In this paper, we report the current state of the simulator. We calculated the ground surface deformation caused by Nojima fault and Chulungpu fault and examined numerical results qualitatively and quantitatively in the meaning of character and probability by comparing with the real data based on trench investigation. The numerical result is consistent with the real data based on trench investigation in the meaning of configuration of shear bands and amount of the base slip at which the shear bands appear on the surface.

Comparison of finite element reliability methods

The spectral stochastic finite element method (SSFEM) aims at constructing a probabilistic representation of the response of a mechanical system, whose material properties are random fields. The response quantities, e.g. the nodal displacements, are represented by a polynomial series expansion in terms of standard normal random variables. This expansion is usually post-processed to obtain the second-order statistical moments of the response quantities. However, in the literature, the SSFEM has also been suggested as a method for reliability analysis. No careful examination of this potential has been made yet. In this paper, the SSFEM is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis. This approach is compared with the direct coupling of a FORM reliability code and a finite element code. The two procedures are applied to the reliability analysis of the settlement of a foundation lying on a randomly heterogeneous soil layer. The results are used to make a comprehensive comparison of the two methods in terms of their relative accuracies and efficiencies.

Polynomial Chaos Decomposition for the Simulation of Non-Gaussian Nonstationary Stochastic Processes

A method is developed for representing and synthesizing random processes that have been specified by their two-point correlation function and their nonstationary marginal probability density functions. The target process is represented as a polynomial transformation of an appropriate Gaussian process. The target correlation structure is decomposed according to the Karhunen-Loeve expansion of the underlying Gaussian process. A sequence of polynomial transformations in this process is then used to match the one-point marginal probability density functions. The method results in a representation of a stochastic process that is particularly well suited for implementation with the spectral stochastic finite element method as well as for general purpose simulation of realizations of these processes.

Adaptive data refinement in the spectral stochastic finite element method

AbstractOne version of the stochastic finite element method involves representing the solution with respect to a basis in the space of random variables and evaluating the co‐ordinates of the solution with respect to this basis by relying on Hilbert space projections. The approach results in an explicit dependence of the solution on certain statistics of the data. The error in evaluating these statistics, which is directly related to the amount of available data, can be propagated into errors in computing probabilistic measures of the solution. This provides the possibility of controlling the approximation error, due to limitations in the data, in probabilistic statements regarding the performance of the system under consideration. In addition to this error associated with data resolution, is added the more traditional error, associated with mesh resolution. This latter also contributes to polluting the estimated probabilities associated with the problem. The present paper will develop the above concepts and indicate how they can be coupled in order to yield a more meaningful and useful measure of approximation error in a given problem. Copyright © 2002 John Wiley & Sons, Ltd.

スペクトル確率有限要素法を用いた横ずれ断層運動に伴う地表地盤の変状とその確率分布に関する研究

スペクトル確率有限要素法を用いた横ずれ断層運動に伴う地表地盤の変状とその確率分布に関する研究

A Stochastic Projection Method for Fluid Flow: I. Basic Formulation

We describe the construction and implementation of a stochastic Navier–Stokes solver. The solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation. The uncertainty quantification scheme is adapted from the spectral stochastic finite element method (SSFEM), which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension. In the SSFEM formalism, the stochastic dependence is represented in terms of the polynomial chaos system, and the coefficients in the corresponding spectral representation are obtained using a Galerkin approach. It is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection–diffusion equations for the various uncertainty fields, and in a decoupled system of pressure projection steps. This leads to a very efficient stochastic solver, whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel.

The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables

A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.

Higher-Order Sensitivity of Heat Conduction Problems to Random Data Using the Spectral Stochastic Finite Element Method

The spectral formulation of the stochastic finite element method is applied to the problem of heat conduction in a random medium. Specifically, the conductivity of the medium, as well as its heat capacity are treated as uncorrelated random processes with spatial random fluctuations. This paper introduces the basic concepts of the spectral stochastic finite element method using a simple one-dimensional heat conduction examples. The implementation of the method is demonstrated for both Gaussian and log-normal material properties. Moreover, the case of the material properties being modeled as random variables is presented as a simple digression of the formulation for the stochastic process case. Both Gaussian and log-normal models for the material properties are treated.