Reducing the microstructure design space of 2nd order homogenization techniques using discrete Fourier Transforms

This paper presents a spectral method of reducing the size of the design space and the computational expense of 2nd order homogenization of elastic properties. A deconvolution is employed to separate the structural and property information in 2nd order homogenization for computational efficiency. This 2nd order homogenization method is then validated against the 1st order method and finite element analysis. It is demonstrated that the space may be further reduced by selecting only the dominant frequencies of the microstructure function in Fourier space. This allows for a significant reduction of the number of terms needed for the representation of the microstructure compared to the terms needed using the full 2nd order homogenization method. This new framework is then used to design and optimize a microstructure for a selected set of three material properties on the edge of the 1st order properties closure (the theoretical envelope of all possible property combinations for a given combination of materials and homogenization method) resulting in only 0.1–5.8% difference from the desired properties.

► A spectral framework for 2nd order elastic homogenization is developed.
► This efficient spectral space is further compacted by using only dominant terms.
► The capability of this method is evaluated using FEA.
► The reduced space is used for a test design problem.