A Spectral Formulation of Stochastic Finite Elements

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.

Chapter 3 presents the fundamentals of the Stochastic Finite Element Method in the framework of the stochastic formulation of the virtual work principle. The resulting stochastic partial differential equations are solved with either non-intrusive Monte Carlo simulation methods, or intrusive approaches such as the versatile spectral stochastic finite element method. Additional approximate methodologies such as the Neumann and Taylor series expansion methods are also presented together with some exact analytic solutions that are available for statically determinate stochastic structures. The concept of the variability response function is then developed and generalized for general stochastic finite element systems.

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.

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

Collocation-based stochastic finite element analysis for random field problems

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.

Fabrication tolerance often causes spatial variations in material properties in electromagnetic systems. In this paper, these spatial variations are modeled using Karhunen–Loéve expansion, which is truncated for a finite dimensional representation. Uncertainty quantification is performed by incorporating this model into an intrusive polynomial chaos expansion based spectral stochastic finite element method. Using a numerical example, it has been shown that the stochastic response obtained by the proposed approach is accurate and computationally efficient. The impact of correlation length of the covariance function used in the spectral expansion is analyzed.

Karhunen-Loeve(KL) expansion of auto-covariance of input Gaussian random process is carried out using standard normal variables as the basis of KL expansion. Input random processes for many practicle problems have non-Gaussian random process description, so an algorithm is developed to find the basis random variables of KL expansion for non-Gaussian processes. The basis random variables of KL expansion are obtained via nonlinear transformation of marginal cumulative distribution function using standard deviation. Results are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential. In all the cases, it is found that the proposed algorithm matches very well with the known solutions and can be applied to solve non-Gaussian process using Spectral Stochastic Finite Element Method.

For the safety of civil structures, it is required to predict the possibility of ground surfce 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 spectral stochastic finite element method. In this paper, we report the current state of the simulator. We calculated the ground surface deformation caused by lateral sliding of numerical model of earthqauke surface fault and examined numerical results qualitatively and quantitatively in the meaning of character and probability by comparing with the result of model experiments for lateral sliding of a sand box. The numerical result is consistent with the result of the model experiment in the meaning of character of configuration of Riedel shears and amount of the base slip at which the Riedel shears appear on the surface. But the experiment result couldn't be reproduced in the meaning of probabilistic character like a standrad deviation.

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.

Spectral stochastic finite element analysis of periodic random thermal creep stress in concrete

Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.

The lifespan of a reinforced concrete (RC) structure can be greatly influenced by the spatial variability of its material characteristics which, in particular, explains the observed or measured reduction of the tensile strength at first crack when the volume under tension increases. This paper discusses the ability of accounting for the spatial variability of the tensile strength of concrete in RC structures using a stochastic finite element (SFE) method based on random field simulations. In this work, the generation of random fields on the concrete tensile strength aims at computing the force corresponding to the first crack occurrence, and the reduced tensile strength of the structure. The method can be applied in particular to large‐sized structures, which show a pronounced size effect, for different types of loading. The method consists of, first, estimating the mean of the random field, using the analytical approach of the weakest link and localization method (WL ). Then, the discretized random field is defined on a particular 2D or 3D grid, and it is finally projected on the finite element mesh of the studied structure. The study of the parameters that influence the prediction of the cumulative density functions (CDFs) of the rupture force or the tensile strength is highlighted using experimental series of concrete beams having different volumes and subjected to 4‐point bending loading. Moreover, the SFE method is applied to a RC tie‐beam under tensile loading, characterized by a weak stress gradient, which complicates the prediction of crack positions.

Wave propagation in thin pretwisted composite strips with an embedded delamination