Ha-Rok Bae

An approximation approach for uncertainty quantification using evidence theory

Abstract Over the last two decades, uncertainty quantification (UQ) in engineering systems has been performed by the popular framework of probability theory. However, many scientific and engineering communities realize that there are limitations in using only one framework for quantifying the uncertainty experienced in engineering applications. Recently evidence theory, also called Dempster–Shafer theory, was proposed to handle limited and imprecise data situations as an alternative to the classical probability theory. Adaptation of this theory for large-scale engineering structures is a challenge due to implicit nature of simulations and excessive computational costs. In this work, an approximation approach is developed to improve the practical utility of evidence theory in UQ analysis. The techniques are demonstrated on composite material structures and airframe wing aeroelastic design problem.

Uncertainty Quantification of Structural Response Using Evidence Theory

Over the past decade, classical probabilistic analysis has been a popular approach among the uncertainty quantification methods. As the complexity and performance requirements of a structural system are increased, the quantification of uncertainty becomes more complicated, and various forms of uncertainties should be taken into consideration. Because of the need to characterize the distribution of probability, classical probability theory may not be suitable for a large complex system such as an aircraft, in that our information is never complete because of lack of knowledge and statistical data. Evidence theory, also known as Dempster-Shafer theory, is proposed to handle the epistemic uncertainty that stems from lack of knowledge about a structural system. Evidence theory provides us with a useful tool for aleatory (random) and epistemic (subjective) uncertainties. An intermediate complexity wing example is used to evaluate the relevance of evidence theory to an uncertainty quantification problem for the preliminary design of airframe structures. Also, methods for efficient calculations in large-scale problems are discussed.

Gradient projection for reliability-based design optimization using evidence theory

Uncertainty quantification and risk assessment in the optimal design of structural systems has always been a critical consideration for engineers. When new technologies are developed or implemented and budgets are limited for full-scale testing, the result is insufficient datasets for construction of probability distributions. Making assumptions about these probability distributions can potentially introduce more uncertainty to the system than it quantifies. Evidence theory represents a method to handle epistemic uncertainty that represents a lack of knowledge or information in the numerical optimization process. Therefore, it is a natural tool to use for uncertainty quantification and risk assessment especially in the optimization design cycle for future aerospace structures where new technologies are being applied. For evidence theory to be recognized as a useful tool, it must be efficiently applied in a robust design optimization scheme. This article demonstrates a new method for projecting the reliability gradient, based on the measures of belief and plausibility, without gathering any excess information other than what is required to determine these measures. This represents a huge saving in computational time over other methods available in the current literature. The technique developed in this article is demonstrated with three optimization examples.

Using random set theory to calculate reliability bounds for a wing structure

This paper shows how uncertainty can be propagated through a computer model when the input data consists of uncertainty affected by imprecision and randomness. The proper orthogonal decomposition (POD) is used to determine the dominant air pressure distributions on a wing, and the uncertainty in the POD modal amplitude variables is propagated through a finite element (FEM) structural model to calculate the reliability bounds for the wing. It is shown that: (i) when working with imprecise input, the traditional probabilistic approach is inadequate because it would force the designer to add unavailable information; (ii) the proposed mathematical model of uncertainty is capable of capturing the incomplete knowledge upon which important conceptual and preliminary decisions must be made; (iii) this approach indicates if and where resources should be invested to acquire more information; (iv) the proposed techniques can be easily implemented using existing numerical codes; (v) a limited number of model runs is necessary to calculate bounds on the output distributions.

Successive matrix inversion method for reanalysis of engineering structural systems

Over the past several decades, numerous structural analysis techniques have been developed to represent physical systems behavior more realistically, and the structural models, therefore, have become larger and more complex. Even though modern computer power has increased significantly, cost of computational analysis has been a major restrictive factor in a structural system design involving multiple disciplines and repetitive simulations. A new reanalysis technique is developed. The successive matrix inversion (SMI) method is most suitable for reanalysis of structures. The SMI method reproduces exact solutions for any localized modification of the initial system. Several numerical examples are given to demonstrate the efficiency of this method.

Efficient Successive Reanalysis Technique for Engineering Structures

Engineering structural design requires many simulations of the structural system after modifications are made to the initial structure. The computational cost of these repetitive simulations can be prohibitive in many engineering applications, such as optimization, reliability analysis, and so forth. In this work, the successive matrix inversion method is improved to include the capability to update not only the inverse of the modified stiffness matrix, but also the modified system response vector. This is done by introducing an influence vector storage matrix and a vector-updating operator that makes it possible to perform sequential reanalyses. The improved successive matrix inversion method has a wider applicable range of modification than the original method. The applicability of the proposed successive matrix inversion method is demonstrated with two popular engineering problems: sensitivity analysis and reliability analysis of an aircraft wing structure.

Reliability-Based Design Optimization under Imprecise Uncertainty

Many reliability-based design optimization (RBDO) techniques have been developed to obtain a reliable design accurately and efficiently in the presence of uncertainties. Probability theory has primarily been used for uncertainty analysis in RBDO. However, in a situation in which information of uncertain variables is imprecise and incomplete, probabilistic techniques are not appropriate to describe the propagation of uncertainty. In this work, evidence theory, also called Dempster-Shafer theory, is employed to handle imprecise uncertain variables as an alternative to classical probability theory. Since the uncertainty quantification method of evidence theory is not compatible with prevailing probabilistic RBDO techniques, we propose an efficient optimization strategy by defining a plausibility function in a trustregion. The procedure for handling imprecise uncertain variables is presented on a practical large-scale structural RBDO problem.

Engineering Design Exploration Using Locally Optimized Covariance Kriging

Surrogate models are used in many engineering applications where actual function evaluations are computationally expensive. Kriging is a flexible surrogate model best suited for interpolating nonlinear system responses with a limited number of training points. It is commonly used to alleviate the high computational cost associated with design exploration techniques: for example, uncertainty quantification and multidisciplinary design optimization. However, when the underlying function shows varying degrees of nonlinear behavior within a design domain of interest, kriging, with a stationary covariance structure, can result in low-quality predictions and an overly conservative expected mean squared error. This effect is often amplified by data collected adaptively and unevenly during iterative design explorations. In this paper, the locally optimized covariance kriging method is proposed to capture the nonstationarity of the underlying function behavior. In locally optimized covariance kriging, the nonstati...

Sequential Subspace Reliability Method with Univariate Revolving Integration

A new computational framework called the sequential subspace reliability method (SSRM) is presented. This method decomposes the multidimensional random space into multiple two-dimensional subspaces. In this manner, SSRM is able to approximate bivariate interaction effects. When the reliability estimate contribution is calculated subspace by subspace, the final assessment is updated in a progressive manner. The iterative history of sequential reliability assessment can be used to understand the complexity and convergence behavior of the limit state function of interest. In a decision-making situation, the flexibility of the proposed SSRM to provide iterative updates on reliability estimation becomes especially valuable in dealing with large-scale and complex problems under the constraints of limited time and resources. To calculate the individual subspace contributions, a novel univariate revolving integration (URI) method is proposed. The URI method takes advantage of the axisymmetric nature of a joint pr...

Locally-Optimized Covariance Kriging for Engineering Design Exploration

To alleviate computational challenges in uncertainty quantification and multidisciplinary design optimization, Kriging has gained its popularity due to its high accuracy and flexibility interpolating non-linear system responses with collected data. One of the benefits of using Kriging is the availability of expected mean square error along with a response prediction at any location of interest. However, a stationary covariance structure, as is the case with the typical Kriging methodology, used with data collected adaptively from an optimal data acquisition strategy will result in lower quality predictions across the entire sample space. In this paper, a Locally Optimized Covariance Kriging (LOC-Kriging) method is proposed to address the difficulties of building a Kriging model with unevenly distributed adaptive samples. In the proposed method, the global non-stationary covariance is approximated by constructing and aggregating multiple local stationary covariance structures. An optimization problem is formulated to find a minimum number of LOC windows and a membership weighting function is used to combine the LOC-Krigings across the entire domain. This paper will demonstrate that LOC-Kriging improves efficiency and provides more reliable predictions and estimated error bounds than a stationary covariance Kriging, especially with adaptively collected data.