A multiscale eigenelement method and its application to periodical composite structures

A uniform multiscale method for 2D static and dynamic analyses of heterogeneous materials

“…The multiscale eigenelement method was developed by Xing et al . , which brings the micro characteristics within the macro element to the nodal degrees of freedom on the boundaries of the macro element by static condensation. In this method, all of the boundary nodes are kept as the macro nodes, and the static equation only needs to be solved once on the microscopic structure when calculating the base functions for the periodic structures.…”

“…During the past years, there have been a number of multiscale methods developed within the framework of small deformation elasticity or elasto‐plastic theory for the heterogeneous materials . The most typical methods include the asymptotic homogenization method , the representative volume element method , the heterogeneous multiscale method , the multiscale eigenelement method , etc. The first two methods are based on the homogeneous theory, and usually limited to problems with periodic microstructure.…”

“…Numerical examples also indicate that the uniform extended multiscale finite element method can obtain sufficiently accurate results with less computational cost compared with the standard FEM.During the past years, there have been a number of multiscale methods developed within the framework of small deformation elasticity or elasto-plastic theory for the heterogeneous materials [1]. The most typical methods include the asymptotic homogenization method [2][3][4][5], the representative volume element method [6][7][8], the heterogeneous multiscale method [10][11][12][13][14][15], the multiscale eigenelement method [16,17], etc. The first two methods are based on the homogeneous theory, and usually limited to problems with periodic microstructure.…”

“…For the elliptic boundary value problems, this method can be carried out at the macroscopic level with some undetermined parameters that can be obtained by solving the micro problems in the subdomains, then the micro heterogeneous properties are reflected to the macroscopic level and the original multiscale problems can be solved at the macroscopic level. The multiscale eigenelement method was developed by Xing et al [16,17], which brings the micro characteristics within the macro element to the nodal degrees of freedom on the boundaries of the macro element by static condensation. In this method, all of the boundary nodes are kept as the macro nodes, and the static equation only needs to be solved once on the microscopic structure when calculating the base functions for the periodic structures.…”

“…Thus, this method is efficient and easy to implement. It has been used for the static and dynamic analyses [17].The multiscale dynamic problems of heterogeneous materials have also been studied by many scholars. Cao et al[18] studied the second-order Helmholtz equations with rapidly oscillating coefficients over general convex domains by using the multiscale asymptotic homogenization method.…”