An adaptive relaxation algorithm for multiscale problems and application to nematic elastomers

The relaxation of nonconvex variational problems involving free energy densities W which depend on the deformation gradient is frequently characterized by a hierarchy of structures at different and well-separated length scales. A wide range of these structures can be characterized as the superposition of one-dimensional oscillations on different length scales which are referred to as laminates of finite order. During a finite element simulation, the relaxed energy Wqc needs to be evaluated in each time step in each Gauss point in the triangulation. In this paper, an algorithmic scheme is presented that allows for the efficient computation of an approximation of the relaxed energy based on laminates of finite order in a large number of points. As an application, the relaxed energy for thin sheets of anisotropic nematic elastomers is studied in detail.