Vadim N. Smelyanskiy

Observable and hidden singular features of large fluctuations in nonequilibrium systems

We study local features, and provide a topological insight into the global structure of the probability density distribution and of the pattern of the optimal paths for large rare fluctuations away from a stable state. In contrast to extremal paths in quantum mechanics, the optimal paths do not encounter caustics. We show how this occurs, and what, instead of caustics, are the experimentally observable singularities of the pattern. We reveal the possibility for a caustic and a switching line to start at a saddle point, and discuss the consequences.

A Bayesian estimation of a stochastic predator-prey model of economic fluctuations

In this paper, we develop a Bayesian framework for the empirical estimation of the parameters of one of the best known nonlinear models of the business cycle: The Marx-inspired model of a growth cycle introduced by R. M. Goodwin. The model predicts a series of closed cycles representing the dynamics of labors share and the employment rate in the capitalist economy. The Bayesian framework is used to empirically estimate a modified Goodwin model. The original model is extended in two ways. First, we allow for exogenous periodic variations of the otherwise steady growth rates of the labor force and productivity per worker. Second, we allow for stochastic variations of those parameters. The resultant modified Goodwin model is a stochastic predator-prey model with periodic forcing. The model is then estimated using a newly developed Bayesian estimation method on data sets representing growth cycles in France and Italy during the years 1960-2005. Results show that inference of the parameters of the stochastic Goodwin model can be achieved. The comparison of the dynamics of the Goodwin model with the inferred values of parameters demonstrates quantitative agreement with the growth cycle empirical data.

Data management and decision support for the in-flight SRM

A novel Bayesian framework for the in-flight SRM Failure Decision and Prognostic (FD&P) is introduced and discussed. It is based on a combination of low-dimensional performance models (LPDMs) and a dynamical inference of the parameters of nonlinear flow of combustion products. To verify the method we introduce a high-fidelity model of the overpressure fault based on a system of stochastic partial differential equations (SPDEs). To analyze the deviations of the system parameters fro m the stable burn-back conditions of the SRM we derived a LPDM of the SRM obtained by integrating the SPDEs over the length of the combustion camera. We consider a few fault scen arios, including nozzle failure with neutral and progressive thrust curve, and nozzle bl ocking with time varying fault parameters to model misses or false alarms. Prognostic is accomplished by building the distribution of the predicted values of the fault p arameters as a function of the measurement time. We discuss how the novel Bayesian framework can be extended to encompass the propellant cracking and the case breach faults of t he SRM.

Development of an on-board failure diagnostics and prognostics system for Solid Rocket Booster

We develop a case breach model for the on-board fault diagnostics and prognostics system for sub-scale solid-rocket boosters (SRBs). The model development was motivated by recent ground firing tests, in which a deviation of measured time-traces from the predicted time-series was observed. A modified model takes into account the nozzle ablation, including the effect of roughness of the nozzle surface, the geometry of the fault, and erosion and burning of the walls of the hole in the metal case. The derived low-dimensional performance model (LDPM) of the fault can reproduce the observed time-series data very well. To verify the performance of the LDPM we build a FLUENT model of the case breach fault and demonstrate a good agreement between theoretical predictions based on the analytical solution of the model equations and the results of the FLUENT simulations. We then incorporate the derived LDPM into an inferential Bayesian framework and verify performance of the Bayesian algorithm for the diagnostics and prognostics of the case breach fault. It is shown that the obtained LDPM allows one to track parameters of the SRB during the flight in real time, to diagnose case breach fault, and to predict its values in the future. The application of the method to fault diagnostics and prognostics (FD&P) of other SRB faults modes is discussed.

Physics based model for online fault detection in autonomous cryogenic loading system

We report the progress in the development of the chilldown model for a rapid cryogenic loading system developed at NASA-Kennedy Space Center. The nontrivial characteristic feature of the analyzed chilldown regime is its active control by dump valves. The two-phase flow model of the chilldown is approximated as one-dimensional homogeneous fluid flow with no slip condition for the interphase velocity. The model is built using commercial SINDA/FLUINT software. The results of numerical predictions are in good agreement with the experimental time traces. The obtained results pave the way to the application of the SINDA/FLUINT model as a verification tool for the design and algorithm development required for autonomous loading operation.

Mathematical and Critical Physics Analysis of Engineering Problems: Old-New Way of Doing Things

In a modern world, importance of computer modeling for solving complex engineering problems cannot be overstated. However, in a number of critical engineering problems computational models cannot provide unique answer and so further physical and analytical insight is required to guide computer simulations. Such an insight becomes even more valuable when off-nominal regimes of operation have to be considered. To deal with complexity of the physical process at the interface of multiple engineering systems a new discipline is emerging - operational physics of critical missions. This discipline combines an old-good physics based approach to modeling engineering problems with modern advanced technologies for analyzing continuous and discrete volving multiple modes of operation in uncertain environments, unknown state variables, heterogeneous software and hardware components. In this paper the new approach is illustrated using as an example analysis of the critical physics phenomena that lead to Challenger accident including physics of cryogenic explosion and propagation of detonation waves, internal ballistics of SRMs in the presence of the case breach fault, and monitoring of the structural integrity of the spacecraft.

IVHM System for a case breach fault in Large Segmented SRMs

An analysis of the case breach fault in a large segmented SRM is presented in the context of development of the IVHM system. The internal ballistic of the SRM is simulated using a 1D model that takes into account grain geometry, propellant regression rate including erosive burning and surface friction, nozzle ablation, and case breach fault dynamics. The model is integrated in quasisteady approximation by solving a boundary value problem for the spatial distribution of the flow variables in the combustion chamber at each time step of steady burning. The model can simulate very accurately nominal and off-nominal regime of internal ballistic of segmented SRM and can be applied to an analysis various fault modes of large SRM. The model calculations are verified by comparison with the results of simulations of axi-symmetric high-fidelity model (developed by the third party). The model is used to simulate case breach fault at a given location along the motor axis. The fault diagnostic and prognostic (FD&P) system is developed in two steps. First, the diagnostics of the case breach fault is performed using stationary solution for the nozzle stagnation pressure, which is known to hold surprisingly well in large SRMs. The later approximation is further improved by introducing an effective design curve that relates the total burning area to the propellant burn distance. The prognostics of the case breach fault dynamics and internal ballistics of SRM in off-nominal regime is achieved using scaling equations developed earlier for malfunction study of RSRM ballistic failure. The results of these predictions are compared with the results of integration forward in time 1D model of internal ballistics of SRM in off-nominal regime for the given parameters of the case breach fault.

Bayesian inferential framework for diagnosis of non-stationary systems

A Bayesian framework for parameter inference in non-stationary, nonlinear, stochastic, dynamical systems is introduced. It is applied to decode time variation of control parameters from time-series data modelling physiological signals. In this context a system of FitzHugh-Nagumo (FHN) oscillators is considered, for which synthetically generated signals are mixed via a measurement matrix. For each oscillator only one of the dynamical variables is assumed to be measured, while another variable remains hidden (unobservable). The control parameter for each FHN oscillator is varying in time. It is shown that the proposed approach allows one: (i) to reconstruct both unmeasured (hidden) variables of the FHN oscillators and the model parameters, (ii) to detect stepwise changes of control parameters for each oscillator, and (iii) to follow a continuous evolution of the control parameters in the quasi-adiabatic limit.

Some Novel Features of Nonequilibrium Systems

In this chapter, we explore two novel features of nonequilibrium systems—fluctuation-induced transport and the formation and significance of nonequilibrium singularities. These phenomena are excellent examples of some of the interesting things that can happen in fluctuating nonequilibrium systems. They also serve as reminders that much of our intuition about noise formed from an understanding of equilibrium systems can leave us unprepared for the variety and complexity of nonequilibrium phenomena.