Inverse design of anisotropic microstructures using physics-augmented neural networks

TL;DR

This work addresses inverse design of anisotropic microstructures in finite-strain hyperelastic composites by learning a physics-informed forward surrogate based on partially input convex neural networks (pICNNs) that enforce polyconvexity via invariants of the right Cauchy-Green tensor and structure tensors. The model jointly learns the anisotropy class and preferred directions and then solves inverse design problems to identify microstructural design parameters that yield target mechanical responses, using CMA-ES and FE integration for multiscale inversion. It introduces stress- and energy-normalization strategies to guarantee a zero-stress reference state and stable tangent moduli, enabling accurate parameter recovery across isotropic, transversely isotropic, and orthotropic cases, including RVEs with single inclusions and fiber reinforcements. The approach integrates with a decoupled FE workflow to invert fiber orientations and optimize macroscopic stress fields, demonstrating robustness to changes in anisotropy direction and offering a path toward topology optimization and functionally graded materials in anisotropic contexts.

Abstract

Composite materials often exhibit mechanical anisotropy owing to the material properties or geometrical configurations of the microstructure. This makes their inverse design a two-fold problem. First, we must learn the type and orientation of anisotropy and then find the optimal design parameters to achieve the desired mechanical response. In our work, we solve this challenge by first training a forward surrogate model based on the macroscopic stress-strain data obtained via computational homogenization for a given multiscale material. To this end, we use partially Input Convex Neural Networks (pICNNs) to obtain a polyconvex representation of the strain energy in terms of the invariants of the Cauchy-Green deformation tensor. The network architecture and the strain energy function are modified to incorporate, by construction, physics and mechanistic assumptions into the framework. While training the neural network, we find the type of anisotropy, if any, along with the preferred directions. Once the model is trained, we solve the inverse problem using an evolution strategy to obtain the design parameters that give a desired mechanical response. We test the framework against synthetic macroscale and also homogenized data. For cases where polyconvexity might be violated during the homogenization process, we present viable alternate formulations. The trained model is also integrated into a finite element framework to invert design parameters that result in a desired macroscopic response. We show that the invariant-based model is able to solve the inverse problem for a stress-strain dataset with a different preferred direction than the one it was trained on and is able to not only learn the polyconvex potentials of hyperelastic materials but also recover the correct parameters for the inverse design problem.

Figures (24)

• Figure 1: Implementation schematic for partially input convex neural network employed. This network is convex in $x_0$ which would be the invariants in our framework whereas it can take an arbitrary form with respect to $y_0$ which represent the design variables in our framework. Here, $\Theta_c$ represents a positive, monotonically nondecreasing, and convex activation function whereas $\Theta_a$ represents an arbitrary activation function. The red color indicates the weights constrained to be positive to ensure convexity and monotonicity with respect to inputs $x_0$.
• Figure 2: Physics-augmented neural network model for the solution of the forward problem.
• Figure 3: RVE of size 1$\times$1$\times$1 with a single spherical inclusion in the center.
• Figure 4: Local stress fields for different values of $\phi$ and $\frac{\mu_1}{\mu_2}$ under uniaxial loading of the single inclusion RVE. Shown is the component $S_{11}$ of the second Piola-Kirchhoff stress tensor for a macroscopi stretching of $\lambda_1 = 1.45$ in the $x_1$ direction applied via periodic boundary conditions.
• Figure 5: RVE with fibers oriented along $\bm{n} = (0,0,1)$.