Common simulation methods for heat conduction from the perspective of Cellular Automata

摘要

In the field of engineering the modeling with Cellular Automata has achieved a higher popularity over the last decades. The reason for this lies in its rather simple architecture which allows one to investigate systems of high complexity. For a tribological problem, dealing with the growth and destruction of characteristic surface structures in automotive brake systems, which determine friction and wear of the system, such a model could have been introduced successfully. In tribology in general and for the brake system in particular heat generation and heat conduction play an important role either. So the authors were confronted with the task to integrate the respective partial differential equation into the model. Different common simulation methods - Finite Elements, Finite Differences and a new sophisticated Finite Volume model -have been tested with identical meshes. Therefore in this paper a general investigation of these three methods is carried out. With the help of a unit cube the matrices governing the equations of motion (conductivity matrix, capacity matrix) are analyzed with respect to the question which nodes interact with each other. This information can be interpreted as a type of neighborhood which crucially characterizes the respective method. In this context the methods are compared with each other and the correlation towards the results is pointed out. It can be seen that for this problem Finite Elements generate results clearly worse than Finite Differences and Finite Volumes. Reasons for that can be found in the neighborhood within the Finite Element model, the consequences are exposed. Finally the paper dedicates to the question of the correlation between Cellular Automata and these methods within the frame of system theory.