The paper presents the cellular automaton (CA) model for tracking the development of dendritic structure in non-equilibrium solidification conditions of binary alloy. Thermal, diffusion and surface phenomena have been included in the mathematical description of solidification. The methodology for calculating growth velocity of the liquid-solid interface based on solute balance, considering the distribution of the alloy component in the neighborhood of moving interface has been proposed. The influence of solidification front curvature on the equilibrium temperature was determined by applying the Gibbs Thomson approach. Solute and heat transfer equations were solved using the finite difference method assuming periodic boundary conditions and Newton cooling boundary condition at the edges of the system. The solutal field in the calculation domain was obtained separately for solid and liquid phase. Numerical simulations were carried out for the Al-4 wt.% Cu alloy at two cooling rates 15 K/s and 50 K/s. Microstructure images generated on the basis of calculations were compared with actual structures of castings. It was found that the results of the calculations are agreement in qualitative terms with the results of experimental research. The developed model can reproduce many morphological features of the dendritic structure and in particular: generating dendritic front and primary arms, creating, extension and coarsening of secondary branches, interface inhibition, branch fusion, considering the coupled motion and growth interaction of crystals.

A numerical model of binary alloy crystallization, based on the cellular automaton technique, is presented. The model allows to follow the

crystallization front movement and to generate the images of evolution of the dendritic structures during the solidification of a binary

alloy. The mathematic description of the model takes into account the proceeding thermal, diffusive, and surface phenomena. There are

presented the results of numerical simulations concerning the multi-dendritic growth of solid phase along with the accompanying changes

in the alloying element concentration field during the solidification of Al + 5% wt. Mg alloy. The model structure of the solidified casting

was achieved and compared with the actual structure of a die casting. The dendrite interaction was studied with respect to its influence on

the generation and growth of the primary and secondary dendrite arms and on the evolution of solute segregation both in the liquid and in

the solid state during the crystallization of the examined alloy. The morphology of a single, free-growing dendritic crystal was also

modelled. The performed investigations and analyses allowed to state e.g. that the developed numerical model correctly describes the

actual evolution of the dendritic structure under the non-equilibrium conditions and provides for obtaining the qualitatively correct results

of simulation of the crystallization process.

Modern geological maps are 3D, or even 4D models representing the complex structure of the Earth. They interpret a variety of geological information and represent it in graphical form.

It is presented in detail how the selection of a structural model describing the behaviour of a steel hall transverse frame when subject to fire exposure in a more or less complex way may affect the fire resistance evaluation for such a frame. In the examples compiled in this paper the same typical one-aisle and single-story steel hall is subjected to simulated fire action, each time following the same fire development scenario.Aresultant fire resistance is identified individually in each case, using various computational models, on an appropriate static equilibrium path obtained numerically. The resulting estimates vary, not only in the quantitative sense, but also in terms of their qualitative interpretation. It is shown that the greater the simplification of the model used, the more overstated the estimated fire resistance is in relation to its real value. Such an overestimation seems to be dangerous to the user, as it gives him an illusory but formally unjustified sense of the guaranteed safety level.

The paper deals with recent developments in the fields of applied mathematics and operational research triggered by the needs of effective support of environmental policy-making processes that require interdisciplinary science-based advice. Mathematical models developed for this purpose demand new modeling paradigms for an adequate integration of pertinent knowledge. and creation of knowledge needed for rational decision-making. The article first summarizes the model-based support for problem solving from the point of view of actual decision-makers. Next, il discusses the model representation of the knowledge pertinent lo a given decision-making problem, and the recently developed modeling technology supporting the whole process of modeling complex problems. The last parł deals with novel methods and tools for integrated management of risks related to natural catastrophes. The presented methodology is illustrated by its application lo actual environmental policy-making support.