By my understanding Asymptotic homogenization (AH) is a mathematical method used to approximate the behavior of a heterogeneous material with complex microstructure by representing it as an equivalent homogeneous material with effective material properties. The goal of AH is to capture the macroscopic behavior of the material under external loading, while taking into account the influence of the microstructure on the material's response.

AH is based on the assumption that the material can be represented as a periodic array of unit cells, each of which has a specific microstructure and material properties. By analyzing the behavior of a single unit cell under external loading, it is possible to determine the effective material properties of the entire material.

So to perform AH in Abaqus, you will need to define a unit cell model that represents the microstructure and material properties of the unit cell. This can be done using the finite element method (FEM) or other numerical techniques. You will also need to define a boundary condition that represents the external loading on the unit cell.

After this, you will then need to set up an Asymptotic Homogenization analysis in Abaqus. This can be done using the Asymptotic Homogenization procedure in the Abaqus/Standard interface. In this procedure, you will specify the unit cell model, the boundary condition, and the type of effective material properties you want to compute (e.g., elastic, plastic, or thermal). You will also need to specify the number of terms to include in the expansion of the displacement field and the number of sampling points to use in the integration of the unit cell.

Once the Asymptotic Homogenization analysis is set up, you can then run the analysis in Abaqus to compute the effective material properties of the unit cell. The results of this analysis will include the effective material properties of the unit cell, as well as plots of the displacement field and the strain field within the unit cell.

I have shown that AH is exactly the same as the RVE analysis with periodic boundary conditions (PBCs). See my paper at Article An Introduction to Micromechanics