Hybrid numerical- analytical modeling

The present lecture summarizes some of tiie most recent joint research results from tiie cooperation between the Federal University of Rio de Janeiro, Brasil, and tiie University of Miami, USA, on tiie fransient analysis of both fluid flow and heat transfer within microchannels. This collaborative link is a natural extension of a long term cooperation between the two groups, in the context of fimdamental work on transient forced convection, aimed at tiie development of hybrid numerical-analytical techniques and tiie experimental validation of proposed models md methodologies [1- 9]. The motivation of this new phase of tiie cooperation was thus to extend the previously developed hybrid tools to handle both transient flow and transient convection problems in microchannels within the slip flow regime. 

Liu, H. and Brinson, L.C. (2006) A hybrid numerical-analytical method for modeling the viscoelastic properties of polymer nanocomposites. Journal of Applied Mechanics, 73 (5), 758-768. 

An attempt to exploit positive aspects of the two classes of methods above is made with the so-called hybrid methods. These combine numerical input/output from analytical models with statistical and probabilistic data to define exposure and vulnerability distribution. This allows to reduce the analytical bmden while grotmding results in a geographical context. Applicabihty and reliability of such approaches are constrained by the ability to define certain mechanical and structural characteristics in numerical terms and by the need to define a common approach for treating the various sources of xmcertainty... 

A fire model is a physical or mathematical representation of burning or other processes associated with fires. Mathematical models range from relatively simple formula that can be solved analytically to extensive hybrid sets of differential and algebraic equations that must be solved numerically on a computer. Software to accomplish this is referred to as a computer fire model. 

Regarding handling of model responses, process inversion (calculation of u°p with the help of the model) can be performed implicitly with the help of numerical procedures (the model provides process responses y as functions of inputs u and initial states x), or can be performed explicitly, by developing empirical and/or hybrid neural models off-line (the model provides inputs u as functions of process responses y and initial states x) [ 196, 203-206]. In the first case, model responses are more robust, although model inversion is much faster in the second case. Besides, if the process model can be fairly described by linear or bilinear models, then analytical results can be provided for the optimization problem [40,193,207,208], which makes the real-time implementation of predictive controllers much easier. 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

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