Cracow University of Technology
Institute for Computational Civil Engineering
The lecture will focus on development of the Multiscale FEM (MsFEM) [1,2], the method for numerical homogenization that enables discretization by coarse meshes not complying with material heterogeneities. The necessary, key information from the micro scale is incorporated into the macro level by special shape functions, constructed on the y. We use the hp-adaptive FEM  on both levels, in the context of both the displacement and mixed formulations . Application of such homogenization methods was motivated by their convenience for the parallel computing and absence of requirements for either the scale separation or the structure periodicity.
We have modified the MsFEM to be able to use higher order approximation and obtained fast p as well as h convergences of the results. The method requires solution of local boundary value problems defined in small subdomains that typically are assumed to be either oversampled coarse elements or the elements themselves but with additional boundary value problems defined on their edges and faces. We increased the efficiency of the method by constructing global basis functions in patches of elements that coincide with supports of the global shape functions. These subdomains reduce to one or at most two finite elements for the mixed formulation.
We have examined the proposed approaches for selected solid mechanics problems, including elastic and visco-elastic deformations, sandwich structures, FGM materials and short fiber reinforced composites. All the results confirm efficiency of the higher order approximation for the displacement and mixed versions of the MsFEM.
1. Hou T. and Wu X., A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134, pp.169-189, 1997.
2. Cecot W. and Oleksy M., High order FEM for multigrid homogenization, Computers and Mathematics with Applications, 70 (7) pp. 1391-1400, 2015.
3. Demkowicz L., Computing with hp-adaptive finite elements. Volume 1: One and two dimensional elliptic and maxwell problems, Chapman & Hall CRC, USA (Boca Raton), 2007.
4. Arbogast T., Mixed Multiscale Methods for Heterogeneous Elliptic Problems, Vol. 83 of Lecture Notes in Computational Science and Engineering. Numerical Analysis of Multiscale Problems, Springer-Verlag, Berlin Heidelberg, 2012, Ch. 8.
Hosted by: Professor Tarek Zohdi, 6117 Etcheverry Hall, 510- 642-9172, email@example.com