On Two‐Scale Modelling of Heat and Mass Transfer

Abstract

Modelling of macroscopic behaviour of materials, consisting of several layers or components, whose microscopic (at least stochastic) analysis is available, as well as (more general) simulation of non-local phenomena, complicated coupled processes, etc., requires both deeper understanding of physical principles and development of mathematical theories and software algorithms. Starting from the (relatively simple) example of phase transformation in substitutional alloys, this paper sketches the general formulation of a nonlinear system of partial differential equations of evolution for the heat and mass transfer (useful in mechanical and civil engineering, etc.), corresponding to conservation principles of thermodynamics, both at the micro-and at the macroscopic level, and suggests an algorithm for scale-bridging, based on the robust finite element techniques. Some existence and convergence questions, namely those based on the construction of sequences of Rothe and on the mathematical theory of two-scale convergence, are discussed together with references to useful generalizations, required by new technologies.

Reliability of Steel Members Designed in Accordance with the Code Design Concepts

“…1 show lower or higher effect of the epistemic uncertainty, namely for the last four quantities in Table 1 above all where there is not any possibility to find the histograms obtained by measurements. The results are dependent, too, on the definition and mathematical description of the structure limit states, and they can differ, e.g., when taking into consideration time dependent effects [13][14][15], when modeling macro and microscale behaviour of materials [16][17][18], when solving stability problems [19][20][21], and when describing system imperfections for nonlinear solutions [22]. The results are dependent, too, on the definition and mathematical description of the structure limit states, and they can differ, e.g., when taking into consideration time dependent effects [13][14][15], when modeling macro and microscale behaviour of materials [16][17][18], when solving stability problems [19][20][21], and when describing system imperfections for nonlinear solutions [22].…”