Jowita Rychlewska

ON THE MODELING OF THE HYPERBOLIC HEAT TRANSFER PROBLEMS IN PERIODIC LATTICE-TYPE CONDUCTORS

The aim of this contribution is to propose, compare, and apply two kinds of simplified mathematical models for the analysis of hyperbolic problems describing heat transfer in dense periodic lattice-type conductors of an arbitrary form. The considerations are based on the Cattaneo-type constitutive heat transfer law. We begin with the formulation of a discrete model represented by a system of ordinary differential equations that have a finite difference form with respect to the spatial coordinates. By using some smoothness operations we derive continuum models from the aforementioned finite difference formulation. The general results are illustrated and compared in the example of a temperature wave propagating in a certain special lattice-type periodic conductor.

On the Mathematical Modelling of Functionally Graded Composites with a Determistinic Microstructure

The object of analysis is continuum modelling of compositeswith a deterministic space-varying (nonperiodic) material microstructure. The aim of the paper is to propose a new modelling method which makes it possible to describe response of a composite by means of PDEs with smooth (averaged) coefficients. In contrast to the known asymptotic homogenization theory the proposed approach leads to model equations describing phenomena depending on the microstructure size. The presented paper summarizes new results obtained in Poland during the last decade. Instead of limit passage with the microstructure size to zero the idea of the method is strictly related to the concept of tolerance relation and tolerance parameter as a certain upper bound for negligibles. That is why the proposed approach is referred to as tolerance modelling method.