A mesh-in-element method for the theory of porous media

While direct homogenisation approaches such as the FE${\hspace{0.17em}}^2$ method are subject to the assumption of scale separation, the mesh-in-element (MIEL) approach is based on an approach with strong scale coupling, which is based on a discretization with finite elements. In this contribution we propose a two-scale MIEL scheme in the framework of the theory of porous media (TPM). This work is a further development of the MIEL method which is based on the works of the authors A. Ibrahimbegovic, R.L. Taylor, D. Markovic, H.G. Matthies, R. Niekamp (in alphabetical order); where we find the physical and mathematical as well as the software coupling implementation aspects of the multi-scale modeling of heterogeneous structures with inelastic constitutive behaviour, see for example, [Eng Comput, 2005;22(5-6):664-683.] and [Eng Comput, 2009;26(1/2):6-28.]. Within the scope of this contribution, the necessary theoretical foundations of TPM are provided and the special features of the algorithmic implementation in the context of the MIEL method are worked out. Their fusion is investigated in representative numerical examples to evaluate the characteristics of this approach and to determine its range of application.