Abstract
The extended multiscale finite element method (EMsFEM) has been more and more widely used in the multiscale analysis of the mechanical properties of heterogeneous materials and structures, since it has no assumptions of scale separation and periodicity compared to the asymptotic homogenization method and the representative volume element (RVE) method. To further facilitate the promotion and application of the EMsFEM, this paper discourses the basic ideas, principles, and implementations of this method in detail, and releases the corresponding MATLAB code for free use by researchers. On the work of this paper, scholars can carry out further research on different multiscale computation problems. In the EMsFEM, numerical basis functions play a key role as a bridge, establishing a connection between the microscopic heterogeneous properties of materials and their macroscopic equivalent mechanical quantities. In this paper, four approaches are introduced to construct the numerical shape functions and their implementation processes are also presented. Finally, some typical numerical examples are given for verifying the effectiveness of the EMsFEM. The MATLAB code can be download from the website https://github.com/hliu-whu/EMsFEM.
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