Robert A. Canfield

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.

Polynomial Chaos Expansion with Latin Hypercube Sampling for Estimating Response Variability

A computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented. To deal with the random nature of input parameters of structural models, several efficient probabilistic methods are investigated. Specifically, the polynomial chaos expansion with Latin hypercube sampling is used to represent the response of an uncertain system. Latin hypercube sampling is employed for evaluating the generalized Fourier coefficients of the polynomial chaos expansion. Because the key challenge in uncertainty analysis is to find the most significant components that drive response variability, analysis of variance is employed to find the significant parameters of the approximation model. Several analytical examples and a large finite element model of a joined-wing are used to verify the effectiveness of this procedure.

Comparison of evidence theory and Bayesian theory for uncertainty modeling

Abstract This paper compares Evidence Theory (ET) and Bayesian Theory (BT) for uncertainty modeling and decision under uncertainty, when the evidence about uncertainty is imprecise. The basic concepts of ET and BT are introduced and the ways these theories model uncertainties, propagate them through systems and assess the safety of these systems are presented. ET and BT approaches are demonstrated and compared on challenge problems involving an algebraic function whose input variables are uncertain. The evidence about the input variables consists of intervals provided by experts. It is recommended that a decision-maker compute both the Bayesian probabilities of the outcomes of alternative actions and their plausibility and belief measures when evidence about uncertainty is imprecise, because this helps assess the importance of imprecision and the value of additional information. Finally, the paper presents and demonstrates a method for testing approaches for decision under uncertainty in terms of their effectiveness in making decisions.

Estimation of structural reliability for Gaussian random fields

This research develops a stochastic analysis procedure for Gaussian random fields in structural reliability estimation using an orthogonal transform and a stochastic expansion with Latin Hypercube sampling. The efficiency of the current simulation procedure is achieved by combination of the Karhunen – Loeve transform with stochastic analysis of polynomial chaos expansion. The Karhunen – Loeve transform enables generation of random fields within the framework of Latin Hypercube sampling and dimensionality reduction of the random variables. The polynomial chaos expansion can reduce computational effort of uncertainty quantification in highly nonlinear engineering design applications. In order to show the applicability of the method, the material properties of a cantilever plate and a supercavitating torpedo are treated as random fields.

Finite element model tuning using automated structural optimization system software

A method of adjusting analytical finite element models to measured data is presented. The algorithm uses a mathematical optimization strategy to minimize deviations between measured and analytical modal frequencies and partial mode shapes. A mode tracking algorithm is used to identify and account for mode switching during the optimization process. The algorithm was successfully implemented using the Automated Structural Optimization System Software. Experimental results are presented for tuning a lightly damped 6-m flexible frame structure. The results demonstrate excellent agreement between the tuned model and measured data and illustrate the importance of off-nominal validation before accepting a model for simulation purposes. CCURATE prediction and simulation of the dynamic behav- ior of large flexible space structures require analytical models that agree with measured data. Unfortunately, uncertainty in a finite element model (FEM) implies less than perfect correlation between analytical and measured data. When the disagreement is deemed unacceptable, it is necessary for the design engineer to make ad- justments to the FEM. For large problems the number of potential parameters to adjust, such as elemental areas, elastic moduli, inertia moments, etc., quickly becomes overwhelming. Thus a systematic method is required to ensure the adjustments produce the desired results. To this end, a method is introduced that poses a numerical optimization problem, namely, given a set of measured eigenvalues and partial eigenvectors, determine the values of selected physical parameters of the model that minimize the weighted deviations from the analytical eigenvalues and eigenvectors. Numerous techniques and goals of model tuning, also referred to as model refinement or model identification, have appeared in the literature. The common attribute of these techniques is that they at- tempt to minimize the required modification to the analytical mass and stiffness matrices, assuming the FEM is a reasonable approxi- mation to the physical structure. Sensitivity based approaches have been presented in the literature that adjust the matrices by estab- lishing an objective function based on the difference between the experimental and analytical model data.15 An advantage of those methods is that the updated models are consistent with the FEM formulation, and thus the connectivity is preserved. An alternative method employing an optimal matrix update has been developed by Berman and Nogy6 and Baruch.7 In this method, a perturbation mass, damping, or stiffness matrix is determined that, when added to the analytical matrices, produces the measured result. The advantage of this method is a tuned analytic model that exactly reproduces the experimental data. Its shortcoming is that it does not guarantee the closeness to unmeasured modes not used in the tuning process. This is a result of potentially unrealistic changes in the stiffness matrix, such as the introduction of load paths that physically do not exist. To overcome some of the shortcomings of Baruchs method of stiff- ness matrix adjustments, Kabe8 and Kammer9 introduced objective functions that ensure stiffness terms are corrected in a manner such that the connectivity of the analytical model is preserved. Similar

Robust design of mechanical systems via stochastic expansion

This paper discusses stochastic optimisation using Polynomial Chaos Expansion (PCE) with Latin Hypercube Sampling (LHS). PCE provides a means to quantify the uncertainty of highly non-linear structural models involving inherent randomness of geometric, material, and loading properties. PCE, specifically the non-intrusive formulation, is used to construct surrogates of stochastic responses for optimisation procedures. A standard optimisation algorithm is then used. In particular, the material properties of structural systems are assumed to be uncertain and treated as uncorrelated random variables. Implementation of the method is demonstrated for a three-bar structure and a complex engineering structure of an uninhabited joined-wing aircraft.

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.