Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

Many applications are based on the use of materials with heterogeneous microstructure. Prominent examples are fiber-reinforced composites, multi-phase steels or soft tissue to name only a few. The modeling of structures composed of such materials is suitably carried out at different scales. At the micro scale, the detailed microstructure is taken into account, whereas the modeling at the macro scale serves to include sophisticated structural geometries with complex boundary conditions. The procedure is crucially based on an intelligent bridging between the scales. One of the methods derived for this purpose is the meanwhile well established FE2 method which, however, leads to a very high computational effort. Unfortunately, this impedes the use of the FE2 method and similar methodologies for practically relevant problems as they occur e.g. in production or medical technology. The goal of the present paper is to significantly improve computational efficiency by using model reduction. The suggested procedure is very generally applicable. It holds for large deformations as well as for all relevant types of inelasticity. An important merit of the work is the computation of the consistent tangent operator based on the reduced stiffness matrix of the microstructure. In this way a very fast (in most cases quadratic) convergence within the Newton iteration at macro level is achieved.