Asymptotic homogenization with finite elements for an orthotropic radially microperiodic sphere

This paper proposes a semi-analytical methodology that combines the asymptotic homogenization method (AHM) with the finite elements method (FEM) to solve boundary-value problems with rapidly oscillating coefficients. This approach is motivated by the convergence difficulties observed when this type of problem is addressed directly via FEM, whereas the AHM has shown to be efficacious for obtaining good generic approximations of the exact solution. Illustratively, this AHM-FEM methodology is developed for the mechanical equilibrium problem of a radially microperiodic orthotropic sphere under hydrostatic pressure, which allows its validation by comparing with the AHM analytical solution. Specifically, the effective coefficients and the homogenized and local problems are calculated via AHM, and then their analytical and FEM solutions are obtained. Finally, to validate the semianalytical methodology, the generic solutions are applied in an example and, from the obtained results, a comparison is made between the analytical AHM solution and the semi-analytical AHM-FEM solution.