A Meshless Approach in Solution of Multiscale Solidification Modeling
This paper describes an overview of a new meshless solution procedure for calculation of one-domain coupled macroscopic heat, mass, momentum and species transfer problems as well as phase-field concepts of grain evolution. The solution procedure is defined on the macro  as well as on the micro levels  by a set of nodes which can be non-uniformly distributed. The domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis functions (RBF) collocation on a related sub-set of nodes. The time-stepping is performed in an explicit way. All governing equations are solved in their strong form, i.e. no integrations are performed. The polygonisation is not present and the formulation of the method is practically independent of the problem dimension. The solution can be easily and efficiently adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients. The concept and the results of the multiscale solidification modeling with the new approach are compared with the classical mesh-based  approach. The method turns out to be extremely simple to code and accurate, inclusion of the complicated physics can easily be looked over. The coding in 2D or 3D is almost identical.
A. Roósz, V. Mertinger, P. Barkóczy and Cs. Hoó
B. Šarler et al., "A Meshless Approach in Solution of Multiscale Solidification Modeling", Materials Science Forum, Vol. 649, pp. 211-216, 2010