Computational Homogenization calculation in macroscopic and microscopic structurures. The microscopic BVPs are solved by FFT method. The macroscopic BVPs are solved by FEM method (we use PyFEM framework based on the book Nonlinear Finite Element Analysis of Solids and Structures of Rene' de Borst et. al.)

This is my part of code from the work "A surrogate model for computational homogenization of elastostatics at finite strain using high‐dimensional model representation‐based neural network" Authors: Vien Minh Nguyen‐Thanh, Lu Trong Khiem Nguyen, Timon Rabczuk, Xiaoying Zhuang,

Developement: ntvminh286@gmail.com (institute email: minh.nguyen@ikm.uni-hannover.de)

DOI: https://doi.org/10.1002/nme.6493

Abstract: We propose a surrogate model for two‐scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high‐dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two‐scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT‐based collocation method is applied in tandem with the Newton‐Raphson iteration and the conjugate‐gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann‐Schwinger form without resorting to the reference medium. In this manner, the fixed‐point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT‐based method for homogenization of elasticity at finite strain.