Uncertainty Quantification of Aeroelastic Stability

The application of uncertainty analysis for the prediction of aeroelastic stability, using probabilistic and non-probabilistic methodologies, is considered in this chapter. Initially, a background to aeroelasticity and possible instabilities, in particular “flutter,” that can occur in aircraft is given along with the consideration of why Uncertainty Quantification (UQ) is becoming an important issue to the aerospace industry. The Polynomial Chaos Expansion method and the Fuzzy Analysis for UQ are then introduced and a range of different random and quasi-random sampling techniques as well as methods for surrogate modeling are discussed. The implementation of these methods is demonstrated for the prediction of the effects that variations in the structural mass, resembling variations in the fuel load, have on the aeroelastic behavior of the Semi-Span Super-Sonic Transport wind-tunnel model (S4T). A numerical model of the aircraft is investigated using an eigenvalue analysis and a series of linear flutter analyses for a range of subsonic and supersonic speeds. It is shown how the Probability Density Functions (PDF) of the resulting critical flutter speeds can be determined efficiently using both UQ approaches and how the membership functions of the aeroelastic system outputs can be obtained accurately using a Kriging predictor.

Evidence-Based Uncertainty Modeling

Evidence is the essence of any decision making process. However in any situation the evidences that we come across are usually not complete. Absence of complete evidence results in uncertainty, and uncertainty leads to belief. The framework of Dempster-Shafer theory which is based on the notion of belief is overviewed in this chapter. Methods of combining different sources of evidences are surveyed. Relationship of probability theory and possibility theory to evidence theory is exhibited. Extension of the classical Dempster-Shafer Structure to fuzzy setting is discussed. Finally uncertainty measurement in the frame work of Dempster-Shafer structure is dealt with.

Uncertain Static and Dynamic Analysis of Imprecisely Defined Structural Systems

This chapter presents the static and dynamic analysis of structures with uncertain parameters using fuzzy finite element method. Uncertainties presents in the parameters are modelled through convex normalised fuzzy sets. Fuzzy finite element method converts the structures into fuzzy system of linear equations and fuzzy eigenvalue problem for static and dynamic problems respectively. As such method to solve fuzzy system of linear equations, fully fuzzy system of linear equations and fuzzy eigenvalue problems are presented. These methods are applied to various structural problems to find out the fuzzy static and dynamic responses of the structures. Also the chapter analyses the numerical solution of uncertain fractionally damped spring-mass system. Uncertainties considered in the initial condition of the system.

Fuzzy Finite Element Method in Diffusion Problems

Diffusion is an important phenomenon in various fields of science and engineering. It may arise in a variety of problems viz. in heat transfer, fluid flow problem and atomic reactors etc. As such these diffusion equations are being solved throughout the globe by various methods. It has been seen from literature that researchers have investigated these problems when the material properties, geometry (domain) etc. are in crisp (exact) form which is easier to solve. But in real practice the parameters used in the modelled physical problems are not crisp because of the experimental error, mechanical defect, measurement error etc. In that case the problem has to be defined with uncertain parameters and this makes the problem complex. In this chapter related uncertain differential equation of various diffusion problems will be investigated using finite element method, which may be called fuzzy or interval finite element method.

MV-Partitions and MV-Powers

In this chapter, the authors generalize the Boolean partition to semisimple MV-algebras. MV-partitions together with a notion of refinement is tantamount a construction of an MV-power, analogous to Boolean power construction (Mansfield, 1971). Using this new notion we introduce the corresponding theory of MV-powers.

Numerical Solution of Fuzzy Differential Equations and its Applications

Theory of fuzzy differential equations is the important new developments to model various science and engineering problems of uncertain nature because this theory represents a natural way to model dynamical systems under uncertainty. Since, it is too difficult to obtain the exact solution of fuzzy differential equations so one may need reliable and efficient numerical techniques for the solution of fuzzy differential equations. In this chapter we have presented various numerical techniques viz. Euler and improved Euler type methods and Homotopy Perturbation Method (HPM) to solve fuzzy differential equations. Also application problems such as fuzzy continuum reaction diffusion model to analyse the dynamical behaviour of the fire with fuzzy initial condition is investigated. To analyse the fire propagation, the complex fuzzy arithmetic and computation are used to solve hyperbolic reaction diffusion equation. This analysis finds the rate of burning number of trees in bounds where wave variable/ time are defined in terms of fuzzy. Obtained results are compared with the existing solution to show the efficiency of the applied methods.

Uncertainty Modeling Using Expert’s Knowledge as Evidence

In this paper we discuss the uncertainty modeling using evidence theory. In practice, very often availability of data is incomplete in the sense that sufficient amount of data which is required may not be possible to collect. Therefore, uncertainty modeling in that case with this incomplete data set is not possible to carry out using probability theory or Monte Carlo method. Fuzzy set theory or any other imprecision based theory is applicable in this case. With a view to this expert’s knowledge is represented as the input data set. Belief and plausibility are the two bounds (lower and upper) of the uncertainty of this imprecision based system. The fundamental definitions and the mathematical structures of the belief and plausibility fuzzy measures are discussed in this chapter. Uncertainty modeling using this technique is illustrated with a simple example of contaminant transport through groundwater.

A New Approach for Suggesting Takeover Targets Based on Computational Intelligence and Information Retrieval Methods

In recent years researchers in financial management have shown considerable interest in predicting future takeover target companies in merger and acquisition (M&A) scenarios. However, most of these predictions are based upon multiple instances of previous takeovers. Now consider a company that is at the early stage of its acquisition spree and therefore has only limited data of possibly only a single previous takeover. Traditional studies on M&A, based upon statistical records of multiple previous takeovers, may not be suitable for suggesting future takeover targets for this company since the lack of history data strongly limits the applicability of statistical techniques. The challenge then is to extract as much knowledge as possible from the single/limited takeover history in order to guide this company during future takeover selections. Under such an extreme case, the authors present a new algorithmic approach for suggesting future takeover targets for acquiring companies based on solely one previous history of acquisition. The approach is based upon methods originating from information retrieval and computational intelligence. The proposal is exemplified upon a case study using real financial data of companies from the Indian software industry.

Hybrid Set Structures for Soft Computing

A Major problem in achieving an effective computational systems is the presence of inherent uncertainty in the computational problem itself. Among various techniques proposed to address this, the technique of soft computing is of significant interest. Further, Generalized set structures like fuzzy sets, rough sets, multisets etc. have already proven their role in the context of soft computing. The computational techniques based on one of these structures alone will not always yield the best results but a fusion of two or more of them can often give better results. Such structures are regarded as hybrid set structures. This chapter surveys an analysis of various hybrid set structures which are quite useful tools for soft computing and shows how this hybridization can help in improving modeling real situations.

Interval Mathematics as a Potential Weapon against Uncertainty

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.