A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rationalB-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method.
Nguyen-Xuan, H., Hoang, T., & Nguyen, V. P. (2014). An isogeometric analysis for elliptic homogenization problems. Computers and Mathematics with Applications, 67(9), 1722-1741. https://doi.org/10.1016/j.camwa.2014.01.001