# Survey on Modelling and Techniques for Friction Estimation in Automotive Brakes
## 2 Techniques for the Estimation of BLCF in Automotive Disc Brakes
### An Extension of the Model-Based Approach--The Cellular Automata Model  
The cellular automata were formerly introduced by [PERSON] in the 1940s but their application as a generic calculation technique was formulated by [PERSON] [53]. Their field of application ranges among problems of biology, transport and physics but their use in tribological systems is relatively recent [54]. As frequently mentioned, the tribological behaviour of the system is determined by the growth and destruction of characteristic hard structures on a mesoscopic scale [17]. Hereto, the cellular automata have been revealed as an effective tool to render the patch life cycle, consisting of the growth and destruction phase or alternatively in the growth and detachment process with the ability to depict the particle flow. The integration of the cellular automata model with a friction law in macroscopic models, such as [PERSON]'s model, allows for tightening the gap between the surface topography dynamics and the macroscopic phenomena. The actual aim of the cellular automata is to obtain a guess at the frictional behaviour at the pad-disc contact with relation to the patches' dynamics. Cellular automata are characterized by the following elements: (i) a regular lattice, whose elements are the cells, represents the spatial discretization of the space; (ii) a finite set of inner variables for each cell; (iii) a finite set of rules and functional relations among the neighbouring cells that leads the cellular development; (iv) a transition function concerning the cell neighbourhood and updating the inner variables at each time increment. The lattice is constituted by the spatial discretization of the pad-disc contact under the form of cells. The patch size in pad-disc contact is approximately some hundred microns, as experimentally assessed. That is why it is necessary to choose a lattice so that the cellular size is sufficiently small compared to the characteristic patch dimensions. For instance, the authors in [54; 55] define each cell as 10 \(\upmu\)m \(\times\) 10 \(\upmu\)m in order to depict the patch growth in detail (Figure 7). In accordance with many authors, each cell contains a set of four inner variables. The first inner variable is the status of a cell that can be \"patch\" or \"not patch\" depending on whether the cell belongs to a patch or not. The second variable is the friction power of a cell caused by the local friction force and the relative velocity of the cell. For each cell, the temperature is also computed as the third inner variable, by considering the mutual interaction among neighbouring cells. Finally, the particle density stands as an index for the wear volume: the more the generated particles agglomerate, the more the density increases. It is worth noting that the four properties are tied to each other by means of rules. The interactions among the cells' inner variables occur on different timescales, therefore every single process has to be simulated against different time steps, as shown in Figure 8. For instance, the phenomena of patch destruction and flow occur on a time scale of \(\upmu\)s; instead, the processes of patch birth and growth occur on a time scale ranging between \(s\) and \(ms\)[54]. This means that in the timescale of \(\upmu\)s the heat flux and the wear particle stream must be computed, whereas these terms can be assumed to have stationary values in the timescale of seconds. This explains why it is necessary to separate each phenomenon and integrate it with a sample time aimed at minimizing the computational burden. It is expected that there exist mutual influences among the abovementioned variables and each interaction withstands a rule constituting the cellular automata algorithm. Several authors introduce a particular set or rules derived from physical laws and experimental assessments, which describe the interactions between the patches, friction coefficient, wear and temperature.