Coupling finite element method with meshless finite difference method in thermomechanical problems

This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain. The coupled approach is applied in solving thermomechanical initial-boundary value problem where the heat transport in the domain is non-stationary. In this method, the domain is divided into two subdomains for FEM and MFDM, respectively. Contrary to other coupling techniques, the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level. In the weak form of thermomechanical initial-boundary value problem (variational principle), the appropriate additional coupling integrals are defined a-priori. Subsequently, the FEM and the MFDM approximations, which may differ from each other, are provided to the formulation. It is assumed that there exists a very thin layer of material between the subdomains, which is not spatially discretized. The width of this layer may be considered the coupling parameter and it is the same for both, thermal and mechanical parts. Similar approach is applied to essential boundary conditions (e.g. prescribed temperature and displacements). Consequently, the consistent formulation of the mixed problem for the coupled FEM-MFDM method is derived. The analysis is illustrated with two- and three-dimensional examples of mechanical and thermomechanical problems.