Zero-crossing problem

The zero-crossing problem is function root-finding during the integration of a differential system. In other words, during the integration of ODE/DAE systems, there may be a need to calculate at which value of the independent variable t a certain function of the dependent variables y is zeroed. 

If the function to be zeroed has a single root and changes sign during the integration of the system, an object from the BzzFunctionRoot class can be used. The root-finding classes can be combined with classes for differential systems based on multivalue algorithms (see Vol. 4 - Buzzi-Ferraris and Manenti, in press). 

Consider the following ordinary differential equation (ODE) system  

Please note that the differential system is iteratively integrated until the variable y j 0.7. At each integration step, the mesh point and the corresponding value of are both collected. As the value of 0.7, the integration is stopped and the object z from the BzzFunctionRoot class is initialized on the left with the values collected in the last but one mesh point, and on the right with the values collected in the last mesh point (where the function yR [ 1 ] -. 7 changes sign). 

Within the function BzzFunctionRootRobertsonVariant to be zeroed, the object z uses the object oFindRoot from the class, which has integrated the differential system. oFindRoot allows the vector y to be calculated in each point within the two mesh points tO and tNextMesh of the last integration step. 

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This section illustrates a set of case studies in which root-finding plays an important role in chemical engineering including the calculation of the volume of a nonideal gas, bubble point, and zero-crossing. However, these scenarios also crop up in several other areas. For instance, the calculation of the volume of a nonideal gas is a typical problem in fiuid dynamics, whereas the zero-crossing problem is very common in all disciplines involving differential and differential-algebraic systems as convolutions models, such as the optimal control for electrical and electronic purposes. 

These problems can be avoided by the use of RFl-free or zero-crossing SCR circuitry. A characteristic very much like time proportioning is used, but the power is always turned on and off at the instant when the power line voltage is zero, as shown in Fig. 4.34. 

The problem of deriving the pdf of zero-crossing wave periods is very difficult. Rice derived a pdf for the zero-crossing interval. He treated the interval where I t) [Eq. (7.1)] crosses the 0 line upward at t = 0 and crosses the 0 line downward between r and t - - dr. 

The applications of the periodic wave theory for zero-crossing wave properties raise problems. There have been very few studies about the physical properties of zero-crossing waves, of wave period-wavelength/celerity, of wave height/period — water particle velocities, etc. 

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