Multilevel Adaptive Algorithm for Multiscale Analysis of Heterogeneous Materials

Abstract

A multilevel adaptive FEM is proposed for multiscale mechanical analysis of heterogeneous materials. The algorithm is developed based on the extended multiscale FEM (EMSFEM), which has been proven to be an efficient method for multiscale mechanical analysis. A residual force-based error estimate is introduced, which is able to detect both global and local errors economically. The concept of microscopic numerical base function (NBF) is introduced, which can be used to refine the local results at the regions with a high-stress gradient level by level. Two efficient methods for updating the microscopic NBFs were developed, i.e., the hierarchical NBFs scheme and the residual force NBFs scheme. A hierarchical linear solver is designed to solve the multilevel equation system in each iteration step. In the algorithm, the calculations at the current iteration step take full advantage of the results of previous iteration steps, which is helpful to save computational resources. The numerical examples demonstrate the stability, flexibility, and convergence of the method.