The ubiquity of uncertainty in computational estimates of reality and the necessity for its quantification…
" has been recently recognized by the National Academies and the National Research Council. In this regard, a large portion of the engineering mechanics/dynamics community has focused on multi-scale/multi-physics problems with stochastic media properties, random excitations and uncertain initial/boundary conditions. Two main challenges associated with uncertainty treatment relate to the (A) modeling, and the (B) propagation of the uncertainties.
(A) relates to the development of methodologies for the interpretation/analysis of measured available data, as well as for the estimation of pertinent stochastic models, i.e. quantification of the underlying stochastic process/field statistics. First, measured data most often exhibit a time/space-varying behavior; thus, statistical estimators based on joint time/space-frequency analysis tools (e.g. wavelets) need to be developed. Second, multi-scale computational analyses necessitate the development and application of multi-scale statistical descriptors capable of capturing complex uncertainty relationships across scales. Third, most often there are limited, incomplete and/or missing data; thus, techniques are required not only for estimating stochastic process/field statistics subject to vastly sparse/limited/incomplete data, but also for quantifying the uncertainty of the estimates as well.
(B) relates to the development of methodologies for determining response/reliability statistics of complex systems, i.e. development of analytical and/or numerical methodologies for solving high-dimensional nonlinear stochastic (partial / fractional) differential equations efficiently. First, Monte Carlo Simulation (MCS) has been the most versatile approach for addressing this challenge. However, it can be computationally prohibitive for relatively large-scale complex systems, or when the quantity of interest has a small probability of occurrence (e.g. failure probability); thus, there is a need for developing efficient analytical/numerical solution methodologies. Second, the ever-increasing available computational capabilities, as well as advanced experimental setups have contributed to a highly sophisticated modeling of engineering systems and related excitations. As a result, the form of the governing dynamics equations has become highly complex from a mathematics perspective (e.g. generalized/fractional calculus). Clearly, the solution of such equations has become a much more challenging task than it used to be a decade ago.
Recent and ongoing work to address challenge (A) relates to the development of a versatile framework for spectral analysis and stochastic process/field statistics quantification subject to vastly sparse/limited/incomplete data based on the emerging concept of compressive sensing, and on joint time-frequency analysis tools such as wavelets. Preliminary work indicates satisfactory accuracy for up to 80% missing data.
Recent and ongoing work to address challenge (B) relates to the development of a general and robust computational framework, based on functional/path integrals theory, for stochastic analysis of complex engineering mechanics/dynamics systems. Preliminary work indicates that functional integrals exhibit significant accuracy, and efficiency (in terms of computational cost) and can be, potentially, a promising alternative to computationally demanding MCS.