Approximations by Piecewise Smooth Functions

Physical laws often cannot be described in closed form, or their representation in explicit form is very expensive. Frequently, the only information available is a set of measurement data. In these cases the laws are often represented as piecewise smooth interpolating functions (linear, splines,. .. ) or simple tables. 

The so-called characteristic lines can be defined by using the variables given in Fig. 6.33  

The characteristic line k is given as function of tythresh(2 (0) 0 which itself depends on the state variables x t) and time. 

Sometimes damping elements are described by such characteristic lines. In this case the damping force fn depends on the relative velocity Vreh This yields 

If i thresh crosses One of the threshold values Si this leads to a discontinuity. The treatment with standard switching algorithms involves the computation of all n/b + l switching functions 

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D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math., 42[5] 577-685,1989. 

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