A surrogate model for computational homogenization of elastostatics at finite strain using the HDMR-based neural network approximator

TL;DR

A surrogate model for two-scale computational homogenization of elastostatics at finite strains by using a neural network architecture that mimics a high-dimensional model representation to solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium.

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 macro-energy 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 values 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 the 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. A standard finite element method is employed to solve the equilibrium equation at the macroscale. As for mircoscopic 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 and thus avoid the fixed-point iteration that might require quite strict numerical stability condition in the nonlinear regime.

Figures (18)

• Figure 1: A continuum body of heterogeneous materials. The heterogeneities in the body around the material point $\overline{\mathbf{X}}$ can be "averaged out" to understand the mechanical response of the body at $\overline{\mathbf{X}}$ as if it is a homogeneous body by looking at a representative volume element enclosing the material point $\overline{\mathbf{X}}$. In this way, we could realize the entire heterogeneous body as the homogeneous one by looking at infinite number of RVEs surrounding all material points.
• Figure 2: Surrogate model for computational homogenization by means of approximator of macro-energy density.
• Figure 3: Architecture of the HDMR-based neural network function. A function $f$ of multidimensional variable $(x_1,\ldots, x_D)$ is approximated by a summation of $L$ component functions. One component is a neural network with two hidden layers which uses in order the linear activation functions and $\tanh$ activation functions. It should be interpreted that $\overline{F}_{ij}^{k} = \overline{F}_{ij}$ for all the component functions.
• Figure 4: Mathematical toy problem. (left) Problem setting of the mathematical heterogeneous bar subject to a traction. (right) With $\mu(0) = 3/2$, the energy function $\psi(0,\epsilon)= (1+\epsilon)^{3/2} - 3/2\epsilon - 1$ is plotted against the $\epsilon$-coordinate.
• Figure 5: Comparison between homogenization solution and full-field solution. (left) The $100$ macroscopic strain data $\overline{\epsilon}$ is uniformly distributed in the range $[0, 2]$, which can be seen by its histogram plot. (right) The homogenized solution is obtained by using the neural network macro-energy density with $1000$ sampling data (red dots) and by the two-scale FE-FFT method (black dashed line). The full-field solution is obtained by using standard FEM with a high number of elements.