Extended multiscale finite element method based on polyhedral coarse grid elements for heterogeneous materials and structures

In this paper, a three-dimensional multiscale approach based on the theoretical framework of the multiscale finite element method is proposed to solve the behaviors of heterogeneous materials and structures containing irregular polyhedral inclusions with arbitrary numbers of vertices and faces. The heterogeneous structures are meshed with arbitrary polyhedral elements with coarse-scale sizes, according to the geometric information of irregular inclusions. The mechanical information about the polyhedral inclusions at the fine-scale are constructed through multiscale base functions for the macroscopic deformations of the structures. In these constructions, an extended linear boundary condition is proposed to construct the multiscale base functions for the irregular polyhedral inclusions. The equivalent stiffness matrix of the polyhedral coarse-grid elements can be derived with the multiscale base functions, and the mechanical response of the structures can be calculated at the macroscale. This work further concentrates on the applications of the proposed method to simulate the smart materials and structures that composed of polyhedral heterogeneous inclusions, such as bio-inspired smart materials with inspirations from nastic movements of plants. The multiscale base functions for both fluid and solid phases of the actuators are constructed for such smart materials with multiphysics characteristics. Numerical examples about reinforced concrete structures, heterogeneous cantilever beam and smart wing structures are analyzed by the proposed multiscale method, and the results are compared with those calculated by the standard finite element method. A good correlation between the two methods is observed, providing the evidence that the proposed method can accurately predict the behaviors of the heterogeneous materials with polyhedral inclusions.