Shuping Huang

Implementation of Karhunen -Loeve expansion for simulation using a wavelet-Galerkin scheme

Abstract The feasibility of implementing Karhunen–Loeve (K–L) expansion as a practical simulation tool hinges crucially on the ability to compute a large number of K–L terms accurately and cheaply. This study presents a simple wavelet-Galerkin approach to solve the Fredholm integral equation for K–L simulation. The proposed method has significant computational advantages over the conventional Galerkin method. Wavelet bases provide localized compact support, which lead to sparse representations of functions and integral operators. Existing efficient numerical scheme to obtain wavelet coefficients and inverse wavelet transforms can be taken advantage of solving the integral equation. The computational efficiency of the wavelet-Garlekin method is illustrated using two stationary covariance functions (exponential and squared exponential) and one non-stationary covariance function (Wiener–Levy). The ability of the wavelet-Galerkin approach to compute a large number of eigensolutions accurately and cheaply can be exploited to great advantage in implementing the K–L expansion for practical simulation.

Validation and error estimation of computational models

This paper develops a Bayesian methodology for assessing the confidence in model prediction by comparing the model output with experimental data when both are stochastic. The prior distribution of the response is first computed, which is then updated based on experimental observation using Bayesian analysis to compute a validation metric. A model error estimation methodology is then developed to include model form error, discretization error, stochastic analysis error (UQ error), input data error and output measurement error. Sensitivity of the validation metric to various error components and model parameters is discussed. A numerical example is presented to illustrate the proposed methodology.

Simulation of second-order processes using Karhunen–Loeve expansion

Abstract A unified and practical framework is developed for generating second-order stationary and non-stationary, Gaussian and non-Gaussian processes with a specified marginal distribution function and covariance function. It utilizes the Karhunen–Loeve expansion for simulation and an iterative mapping scheme to fit the target marginal distribution function. The proposed method has three main advantages: (a) processes with Gaussian-like marginal distribution can be generated almost directly without iteration, (b) distributions that deviate significantly from the Gaussian case can be handled efficiently and (c) non-stationary processes can be generated within the same unified framework. Four numerical examples are used to demonstrate the validity and convergence characteristics of the proposed algorithm. Based on these examples, it was shown that the proposed algorithm is more robust and general than the commonly used spectral representation method.

Geotechnical probabilistic analysis by collocation-based stochastic response surface method: An Excel add-in implementation

A general probabilistic method called collocation-based stochastic response surface method (CSRSM) was previously developed. It involves the propagation of input uncertainties through a computation model to arrive at a random output vector. It is assumed that the unknown random output can be expanded using a polynomial chaos basis with corresponding unknown coefficients. The unknown coefficients are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as ‘black boxes’. The roots of the Hermite polynomial provide efficient collocation points to evaluate the coefficients in the stochastic response surface. An Excel add-in is developed to produce the basis functions (multi-dimensional Hermite polynomials) without resorting to symbolic algebra practitioners. This is a major practical advantage that would bring realistic probabilistic analyses within reach of the practitioners. Full Excel implementation details are illustrated using a simple slope problem involving six input random variables. A second problem (sum of exponential random variables) is studied to examine CSRSM over a wider range of conditions. It also provides further validation because the solution is available in closed-form. The results show the ease and successful implementation of the proposed Excel-based CSRSM. However, the add-in is unable to handle correlated input parameters thus far. Future development work is needed.

Statistical validation of simulation models

This paper investigates various statistical methodologies for validating simulation models in automotive design. Validation metrics to compare model prediction with experimental observation, when there is uncertainty in both, are developed. Two types of metrics based on Bayesian analysis and principal components analysis are proposed. The validation results are also compared with those obtained from classical hypothesis testing. A fatigue life prediction model for composite materials and a residual stress prediction model for a spot-welded joint are validated, using the proposed methodology.

Uncertainty Quantification Using Multi-Dimensional Hermite Polynomials

The general stochastic problem involves the propagation of input uncertainties through a computation model to arrive at a random output vector. This paper presents the application of the multi-dimensional Hermite polynomials to reduce an unknown random output vector into a significantly simpler unknown vector of numbers. The unknown numbers are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as “black boxes”. A simple laterally loaded pile example involving two input random variables demonstrated that a third- or fourth-order Hermite expansion is adequate to reproduce probabilities of failure between 10 -3 and 10 -4 . A simple and efficient 2-term recurrence method for obtaining Hermite polynomials of any order in the case of two random dimensions is proposed. To our knowledge, this proposal appears to be original.

Human Reliability Analysis for Visual Inspection in Aviation Maintenance by a Bayesian Network Approach

Visual inspection plays an important role in aviation maintenance. Human reliability analysis (HRA) in this field is necessary and can bring benefits to better human error management. Because of the lack of safety data, the present paper aims to introduce the Bayesian network (BN) approach to perform HRA in visual inspection, which permits the utilization of multi-disciplinary sources of objective and subjective information. In this paper, significant influence factors of visual inspection are identified according to the Human Factors Analysis and Classification System–Maintenance Extension. Then a network representing the visual inspection performance model is constructed. Expert opinions, data fusion from accident reports, and related literature are utilized in the step of obtaining parameters. Two canonical models used in probabilistic network model building, the Noisy-OR gates and the Recursive Noisy-OR rule, are applied to generate conditional probabilities from parameters obtained by the absolute probability judgment technique. Through the BN inference, the inspection reliability can be assessed, and some conclusions and recommendations are drawn that could provide theoretical base and data support to make interventions for safety management of visual inspection.

Karhunen-Loeve Expansion for Simulation of Non-Stationary Gaussian Processes Using the Wavelet-Galerkin Approach

A unified procedure based on K-L expansion to simulate stationary and non-stationary, Gaussian and non-Gaussian processes has earlier been proposed by the authors. As an extension to the unified simulation procedure, this paper adopts the wavelet-Galerkin approach to solve Fredholm integral equation to improve the performance of the K-L method. The validity and convergence characteristics of the wavelet-Garlekin method for solving integral equations are illustrated using the non-stationary Wiener process. The wavelet-Galerkin solutions are more accurate and faster to compute than those derived using the conventional Garlerkin method. This ability to compute a very large number of K-L terms rapidly and accurately provides a practical and direct means of overcoming known K-L limitations associated with non-smooth covariance functions and long weakly-correlated processes.