Isogeometric analysis method for heterogeneous periodic microstructures

A method of isogeometric analysis (IGA) based on NURBS basis functions is applied to homogenization problems for periodic heterogeneous media and composite plates with in-plane periodicity. Since the treatment of the combination of different materials in IGA models is not trivial especially for periodicity constraints and has not been reported in the literature, the first priority is to clearly specify points at issue in the numerical modeling, or equivalently mesh generation, for IG homogenization analysis (IGHA). The most awkward, but important issue is how to generate patches for NURBS representation of the geometry of a rectangular parallelepiped unit cell to realize appropriate deformations in consideration of the convex-hull property of IGA. The issue arises from the introduction of multiple control points located at angular points in the heterogeneous unit cell, which must satisfy multiple point constraint (MPC) conditions associated with periodic boundary conditions (PBCs). Although some countermeasures may be conceivable, we suggest the use of multiple patches along with double MPC that impose PBCs and the continuity conditions between different patches simultaneously. Several numerical examples of numerical material and plate tests are presented to demonstrate the validity of the proposed method of IG unit cell modeling for IGHA.