Generalization of the multi-scale finite element method to plane elasticity problems

In this paper, according to the governing differential equations of problem, the theory to construct the shape functions in the multi-scale finite element method is established for plane elasticity problems. An approach is then suggested to numerically solve the shape functions via the corresponding homogeneous governing equations on an element level. The linear, quadratic and cubic shape functions are finally obtained by prescribing the appropriate boundary conditions. Typical numerical experiments are conducted, including bending of a homogeneous beam, bending of a beam with voids, as well as bending of a beam with a random material distribution and with an oscillatory material property. The current work shows that the multi-scale finite element method has a prominent advantage in solution efficiency even for classic problems, and therefore can be implemented on a considerably coarse mesh for problems with complex microstructures, as well as for large scale problems to effectively save the solution cost.