PINN enhanced extended multiscale finite element method for fast mechanical analysis of heterogeneous materials

The extended multiscale finite element method (EMsFEM) shows great efficiency and accuracy for analyzing the mechanical behavior of heterogeneous materials, especially for non-periodic multiscale materials. The conventional EMsFEM requires solving boundary value problems repeatedly on each coarse-scale element to construct the numerical base functions related to the material parameters of fine-scale element, which constitutes the main part of computational resources. This paper presents a physics-informed neural network (PINN) enhanced EMsFEM to further improve the efficiency of multiscale mechanical analysis. Since the boundary value problems are based on the same solution domain and boundary conditions, a PINN is elaborately designed to solve them described by mechanical equations. The input parameters of PINN contain the material parameters of the fine-scale elements inside the coarse-scale element; therefore, the PINN can quickly map the heterogeneous material properties to the displacements inside the coarse-scale element and greatly improve the construction efficiency of the numerical base functions. To enhance the computational accuracy, the domain decomposition technique is applied to characterize the heterogeneity of the elements, and an unbiased construction method is developed to obtain the numerical base functions that simultaneously ensure the computational consistency and normalization condition. In addition, to further improve the computational efficiency, the construction process of numerical base functions is simplified according to the approximately ergodic property of the network for randomly physical fields. Several representative numerical examples are presented to demonstrate the high efficiency and accuracy of the proposed PINN-enhanced EMsFEM. The method is of high universality since the PINN does not need to be retrained as the geometry of the entire domain and loading of the problem change, the network structure is only related to the length ratio of the coarse- and fine-scale elements.