A probabilistic approach for representation of interval uncertainty

Multiple sources of errors and uncertainty arise in mechanics computational models and contribute to the uncertainty in the final model prediction. This paper develops a systematic error quantification methodology for computational models. Some types of errors are deterministic, and some are stochastic. Appropriate procedures are developed to either correct the model prediction for deterministic errors or to account for the stochastic errors through sampling. First, input error, discretization error in finite element analysis (FEA), surrogate model error, and output measurement error are considered. Next, uncertainty quantification error, which arises due to the use of sampling-based methods, is also investigated. Model form error is estimated based on the comparison of corrected model prediction against physical observations and after accounting for solution approximation errors, uncertainty quantification errors, and experimental errors (input and output). Both local and global sensitivity measures are investigated to estimate and rank the contribution of each source of error to the uncertainty in the final result. Two numerical examples are used to demonstrate the proposed methodology by considering mechanical stress analysis and fatigue crack growth analysis.

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Error and uncertainty quantification and sensitivity analysis in mechanics computational models

In this paper we investigate the performance of probability estimation methods for reliability analysis. The probability estimation methods typically construct the probability density function (PDF) of a system response using estimated statistical moments, and then perform reliability analysis based on the approximate PDF. In recent years, a number of probability estimation methods have been proposed, such as the Pearson system, saddlepoint approximation, Maximum Entropy Principle (MEP), and Johnson system. However, no general guideline to suggest a most appropriate probability estimation method has yet been proposed. In this study, we carry out a comparative study of the four probability estimation methods so as to derive the general guidelines. Several comparison metrics are proposed to quantify the accuracy in the PDF approximation, cumulative density function (CDF) approximation and tail probability estimations (or reliability analysis). This comparative study gives an insightful guidance for selecting the most appropriate probability estimation method for reliability analysis. The four probability estimation methods are extensively tested with one mathematical and two engineering examples, each of which considers eight different combinations of the system response characteristics in terms of response boundness, skewness, and kurtosis.

A comparative study of probability estimation methods for reliability analysis

This paper proposes formulations and algorithms for reliability-based design optimization (RBDO) of both single and multidisciplinary systems under both aleatory uncertainty (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information). The proposed formulations specifically deal with epistemic uncertainty arising from interval data. When the only information available for an input variable is in the form of interval data, it is likely that the distribution type for the input variable is not known or cannot be specified accurately. This paper uses a four-parameter flexible Johnson family of distributions to represent the uncertainty described by interval data. An efficient approach is proposed to decouple the design analysis from the uncertainty analysis. The proposed methodology for multidisciplinary system optimization does not require any coupled system level analysis. The proposed methods are illustrated through several example problems.

Reliability-based design optimization of multidisciplinary system under aleatory and epistemic uncertainty

A new uncertainty propagation method for problems with parameterized probability-boxes

This paper provides an introduction to the Johnson translation system of probability distributions, and it describes methods for using the Johnson system to model input processes in simulation experiments. The fitting methods based on available data are incorporated into the public-domain software package FITTR1. To handle situations in which little or no data is available, we present a visual interactive method for subjective distribution fitting that has been implemented in the public-domain software package VISIFIT. We present several examples illustrating the use of FITTR1 and VISIFIT for simulation input modeling.

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Input modeling with the Johnson system of distributions

This paper provides an introduction to the Johnson translation system of probability distributions, and it describes methods for using the Johnson system to model input processes in simulation experiments. For situations in which little or no sample information is available, we have developed a visual interactive method to estimate bounded Johnson distributions subjectively; and we have implemented this technique in VESIFIT, a public-domain software package. For fitting all types of Johnson distributions based on sample data, we have implemented several new statistical-estimation methods as well as some standard techniques in FITTR1, another public-domain software package. We present several examples illustrating the use of VISIFIT and FITTR1 for simulation input modeling.

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Modeling Input Processes With Johnson Distributions

The Johnson translation family of distributions provides a variety of distributional shapes for the modelling of empirical data that are readily used in simulation models. We compare a number of methods for estimating the parameters of these distributions, including moment matching (MM), least squares (ordinary- (OLS), weighted- (WLS) and diagonally weighted- (DWLS) least squares), and maximum likelihood (MLE). A sampling study is made to determine the properties of the fitted parameters and estimates based on the fitted parameters, such as the quantiles of the distribution. We restrict attention to the case that the analyst knows the correct distribution when fitting the parameters.

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Sekhar Venkatraman

Papers

Input modeling with the Johnson system of distributions

Modeling Input Processes With Johnson Distributions

Comparison of methods for fitting data using Johnson translation distributions