Solving heterogeneous material problems are of importance in many fields. If the heterogeneities are small compared to the scale of the whole problem, a standard finite element analysis often becomes computationally too large. Multi-scale homogenization is a technique that reduces the amount of calculations, but still manages to capture the heterogeneous properties. The domain of the problem is divided into Representative Volume Elements(RVEs), which in turn are discretized through ordinary finite elements. Periodic boundary conditions have to be applied to the RVEs for homogenization to be possible, and a common way to maintain these boundary conditions is by Multi-Point Constraints (MPC). A limitation with MPC is that it does not maintain the periodic boundary conditions in a correct manner when the RVE boundary nodes are non-matching, which they in general are.
In this thesis, Localized Lagrange Multipliers(LLM) are used to maintain the periodic boundary conditions over the RVE, in order to handle RVEs with non-matching grids. Mathematical homogenization theory in 2D and derivations of MPC and the LLM method are given. A computer program solving two-scale computational homogenization problems in 2D using both LLM and MPC has been implemented in MATLAB. Several RVEs with different boundary situations and material compositions are analyzed, and the results from the LLM and MPC analysis are compared.
The results show that LLM is more suitable than MPC to handle the periodic boundary conditions in multi-scale homogenization. LLM deals with all situations that MPC does. In addition, it produces reliable solutions when the boundary nodes are nonmatching.