A catalytic converter in an automobile's exhaust system provides an environment for a chemical reaction where unburned hydrocarbons completely combust, in such a way that pollution is reduced. An enormous effort is being made with the purpose of developing appropiate supports and the catalyst itself. In some cases the design of supports geometry is what makes a process more optimal. Our objective is to model catalysis processes which occurs in an automobile's exhaust system. Due to the great difficulty of the problem, the first step is to study the asymptotic behaviour of catalysis supports in a linear elasticity problem. Since the computational domain in catalysis processes are beams with reticulated structure and the finite element method is not suitable for this type of structures, it is necessary to obtain an equivalent mathematical model defined over the domain without holes. This model has to approach as far as possible the global supports behaviour. This is the main objective of homogenization theories.
In this work, a new procedure, called the unfolding method, developed by Cioranescu, Damlamian and Griso (see [1]), is applied to solve this first approach. The catalysis support is a structure made of beams, placed periodically and with inner holes. We introduce a decomposition of the displacements in such a structure and, by proving some convergence results, we obtain three one-dimensional uncoupled limit problems: The first problem defines the longitudinal displacement and the second one gives the transverse bending of the structure, while the third one defines the torsion angle. The general form ot these problems is well-known in the classic theory of beams, the only difference appears in the new homogenized coefficients.

References

1. D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Math. Acad. Sci. Paris 335 (2002), no. 1, 99-104.

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