Global and combined global-local response sensitivity analyses of uncertain structures based on model distance measures

Abstract

The work reported in this thesis is in the area of global response sensitivity analysis (GRSA) of engineering systems in which the uncertainties associated with external actions and/or system parameters need to be explicitly modelled using a suitable mathematical framework. The uncertainties in loads and system parameters propagate through the relevant input-output relationship and manifest as uncertainties in the response variables of interest. The problem of GRSA consists of decomposing a measure of uncertainty in a chosen response variable into a set of constituents each of which quantifies the influence of a specified input or load variable or their group wise interactions. The quantification of relative importance of different input variables in a model enables the ranking of input variables according to their relative importance. This facilitates effective model reductions and also helps in planning experiments while gathering empirical data for arriving at the uncertainty models. The thesis pursues two intertwining themes, within the context of GRSA of structural systems: a) development of notions of global response sensitivity indices based on the concept of measures of distance between two mathematical models (viz., a fiducial model in which all the sources of uncertainties are included in characterising the response, and a set of altered models, in which uncertainties in one or more of the input variables are deliberately suppressed), and b) embedding these developments within the context of alternative uncertainty modelling frameworks, namely, probabilistic, non-probabilistic (including intervals, convex functions, and fuzzy variables), combined probabilistic and non-probabilistic modelling (within the context of a given problem), and polymorphic modelling frameworks. The studies involving probabilistic models allow for uncertain variables to be modelled as a set of random variables (which could, in general, be dependent and non-Gaussian distributed) and/or as a set of random processes evolving space and/or time. The model distance measures considered include norm, Hellinger distance, Kullback-Leibler divergence, and Kantorovich distance. The study demonstrates the equivalence of Sobol’s indices and norm based indices for the special case when the uncertain variables are modelled as a set of independent random variables. The studies involving non-probabilistic uncertain input variables employ Hausdorff distance measures to characterize the global response sensitivity indices. Newer definitions of model distance measures are introduced when dealing with polymorphic uncertain variables. This part of the study allows for different forms of polymorphic uncertain variable modelling, namely, fuzzy random variables, probability-box variables, fuzzy probability based random variables, and fuzzy probability based fuzzy random variables. The study also involves development of appropriate computational strategies (which combine Monte Carlo simulations and numerical optimization schemes) for evaluating the various global response sensitivity indices. The illustrations are drawn mainly from problems of structural mechanics involving static/dynamic and linear/nonlinear behaviours.