[Paper Review] Machine-learning convex and texture-dependent macroscopic yield from crystal plasticity simulations

Jan N. Fuhg, Lloyd van Wees+4 more|arXiv (Cornell University)|Jan 28, 2022

Machine Learning in Materials Science4 citations

TL;DR

This paper proposes a machine learning framework using input convex neural networks (ICNNs) to learn convex, texture-dependent macroscopic yield surfaces from crystal plasticity finite element simulations. By ensuring convexity and enabling differentiable yield function evaluation via subgradients, the method allows direct integration into finite element time-integration schemes, enabling efficient multiscale modeling of polycrystalline materials with complex textures.

ABSTRACT

The influence of the microstructure of a polycrystalline material on its macroscopic deformation response is still one of the major problems in materials engineering. For materials characterized by elastic-plastic deformation responses, predictive computational models to characterize crystal-plasticity (CP) have been developed. However, due to their large demand of computational resources, CP simulations cannot be straightforwardly implemented in hierarchical computational models such as FE$^{2}$. This bottleneck intensifies the need for the development of macroscopic simulation tools that can be directly informed by microstructural quantities. Using a 3D Finite-Element solver for CP, we generate a macroscopic yield function database based on general loading conditions and crystallographic texture. We furthermore assume an independence of the yield function to hydrostatic pressure of the yield function. Leveraging the advancement in statistical modeling we describe and apply a machine learning framework for predicting macroscopic yield as a function of crystallographic texture. The convexity of the data-driven yield function is guaranteed by using partially input convex neural networks as the predictive tool. Furthermore, in order to allow for the predicted yield function to be directly incorporated in time-integration schemes, as needed for the Finite Element method, the yield surfaces are interpreted as the boundaries of signed distance function level sets.

Motivation & Objective

• To develop a data-driven, convex yield function that captures texture-dependent anisotropic plasticity in polycrystalline materials.
• To overcome the computational bottleneck of crystal plasticity simulations in hierarchical models like FE2 by replacing them with fast, learned yield surfaces.
• To ensure the predicted yield function is convex and differentiable at non-smooth points (e.g., vertices) for direct use in finite element time-integration schemes.
• To incorporate microstructural texture as the primary input, enabling predictive modeling without relying on phenomenological assumptions.

Proposed method

• Generate a high-fidelity database of macroscopic yield surfaces using 3D finite element crystal plasticity simulations under general loading conditions.
• Assume hydrostatic pressure independence of the yield function to reduce dimensionality and improve generalization.
• Train input convex neural networks (ICNNs) on the simulated yield surface data to ensure the predicted yield function is convex by design.
• Represent the yield surface as the zero level set of a signed distance function to enable robust evaluation and gradient computation.
• Use automatic differentiation with subgradient techniques to compute derivatives at non-differentiable points (e.g., vertices), enabling use in Newton-Raphson solvers.
• Validate the method on synthetic and simulated data, demonstrating accurate fitting of non-smooth, convex functions including vertices and facets.

Experimental results

Research questions

• RQ1 Can a data-driven yield function be learned from crystal plasticity simulations that accurately captures texture-dependent anisotropy?
• RQ2 How can convexity of the learned yield function be guaranteed without relying on restrictive phenomenological assumptions?
• RQ3 Can the predicted yield function be directly used in time-integration schemes of the finite element method, even at non-differentiable points like vertices?
• RQ4 To what extent can ICNNs approximate non-smooth, convex functions such as those with sharp vertices or facets?
• RQ5 Can the framework be extended to complex textures and embedded into multiscale finite element codes for structural analysis?

Key findings

• The ICNN-based framework successfully learns convex yield surfaces from crystal plasticity simulations, with the model achieving near-perfect fit to the ground truth function, including exact vertex values (e.g., 1.99999 at x=0 for |x|-2).
• The method enables accurate subgradient computation at non-differentiable points, as demonstrated by successful convergence of a Newton-Raphson loop starting from a vertex position.
• The predicted yield function is globally convex and differentiable in the subgradient sense, allowing direct integration into standard finite element time-integration schemes.
• The framework avoids the need for manual smoothing or hand-coded subdifferential formulations, which are common in classical approaches.
• The approach is generalizable and can be extended to complex textures and embedded into structural finite element codes for multiscale modeling.
• The method significantly reduces computational cost compared to full crystal plasticity simulations while preserving physical fidelity and convexity.