Computational Procedure for Finite Element Analysis of Functionally Graded Metamaterials

The extremely fine meshes needed to run problems with details on two scales have promoted several computational homogenization techniques.However, these techniques typically require periodicity of the deformations and the separation of scales.In functionally graded metamaterials, two scales become relevant, but since every metamaterial cell may differ slightly from the neighboring ones, neither the separation of scales nor periodicity conditions may be assumed to hold.Nevertheless, the fact that every metamaterial cell differs only slightly and that we are typically concerned with the global behavior may be considered to speed up the computational procedure.This paper introduces a novel method specifically designed for metamaterials to address these challenges by structuring the metamaterial in macroelements, the stiffness matrix of elements being reduced prior to assembling in the global matrix.The advantage is taken from the fact that macroelement matrices differ only in a few terms.This strategy significantly improves computational efficiency, enabling the handling of very large meshed structures in metamaterials with improved performance compared to conventional FEA techniques.