Homogenization | Tej's Research Page

Asymptotic Homogenization

Displacement of a RVC with circular hole due to unit strain application of {1,0,0}, {0,1,0}, and {0,0,1} during homogenization

Asymptotic Homogenization (AH) is a technique to find effective properties. These properties may be elasticity tensor, thermal conductivity tensor, fluid permeability, etc. Since my research involves structural optimization, I compute effective elasticity tensor.

In AH, a periodic boundary condition is imposed on all the boundaries of the representative unit cell or RVC, and a uniform body force due to a unit strain is imposed on the entire RVC to get the resulting displacement field. This displacement for unit strains {1,0,0}, {0,1,0}, and {0,0,1} for a RVC with circular hole are shown in the above figure. Since the domain is in two-dimension, there were three unit strains applied. In case of a three-dimensional domain, the number of unit strains application is six: three normal and three tangential. An example of displacements due to these strains are shown in the following figures.

Asymptotic Homogenization 3D (normal strain xx)

Asymptotic Homogenization 3D (normal strain yy)

Asymptotic Homogenization 3D (normal strain zz)

Displacement of a 3D RVC with through holes due to unit normal strain application in x,y, and z-axis during homogenization

Asymptotic Homogenization 3D (shear strain yz)

Asymptotic Homogenization 3D (shear strain xz)

Asymptotic Homogenization 3D (shear strain xy)

Displacement of the same RVC due to unit tangential strain application in yz, xz, and xy-axis during homogenization

These displacements are the measure of the heterogeneity of the RVC i.e. if the entire RVC were homogeneous there would not be any displacements. Using these displacements as the measure of heterogeneity, the effective elasticity tensor is computed as a volumetric average. More details can be found in the references.

Andreassen, E., & Andreasen, C. S. (2014). How to determine composite material properties using numerical homogenization. Computational Materials Science, 83, 488–495. https://doi.org/10.1016/j.commatsci.2013.09.006

Hassani, B., & Hinton, E. (1998). A review of homogenization and topology optimization I - Homogenization theory for media with periodic structure. Computers and Structures, 69(6), 707–717. https://doi.org/10.1016/S0045-7949(98)00131-X. (Part II & III as well)