[Paper Review] A learning-based multiscale method and its application to inelastic impact problems

Burigede Liu, Nikola Kovachki+5 more|arXiv (Cornell University)|Feb 14, 2021

Composite Material Mechanics65 references59 citations

TL;DR

This paper proposes a learning-based multiscale method that trains a deep neural network to approximate the solution operator of a fine-scale crystal plasticity model, enabling high-fidelity simulation of inelastic impact problems at a fraction of the cost of concurrent multiscale methods. The approach eliminates the need for a priori identification of state variables and achieves fidelity comparable to first-principles multiscale modeling with only ~10× the cost of empirical models.

ABSTRACT

The macroscopic properties of materials that we observe and exploit in engineering application result from complex interactions between physics at multiple length and time scales: electronic, atomistic, defects, domains etc. Multiscale modeling seeks to understand these interactions by exploiting the inherent hierarchy where the behavior at a coarser scale regulates and averages the behavior at a finer scale. This requires the repeated solution of computationally expensive finer-scale models, and often a priori knowledge of those aspects of the finer-scale behavior that affect the coarser scale (order parameters, state variables, descriptors, etc.). We address this challenge in a two-scale setting where we learn the fine-scale behavior from off-line calculations and then use the learnt behavior directly in coarse scale calculations. The approach draws from recent successes of deep neural networks, in combination with ideas from model reduction. The approach builds on the recent success of deep neural networks by combining their approximation power in high dimensions with ideas from model reduction. It results in a neural network approximation that has high fidelity, is computationally inexpensive, is independent of the need for a priori knowledge, and can be used directly in the coarse scale calculations. We demonstrate the approach on problems involving the impact of magnesium, a promising light-weight structural and protective material.

Motivation & Objective

• To address the high computational cost and reliance on a priori knowledge in concurrent multiscale modeling of inelastic materials.
• To develop a surrogate model that captures fine-scale behavior without requiring explicit identification of order parameters or state variables.
• To enable accurate macroscopic simulations using data-driven approximation of the fine-scale solution operator.
• To demonstrate the method's effectiveness on polycrystalline magnesium impact problems with varying impact velocities and plate thicknesses.

Proposed method

• The method trains a deep neural network to learn the solution operator mapping macroscopic strain history to fine-scale stress and internal variable responses.
• It leverages model reduction techniques to ensure the learned map is independent of spatial discretization and resolution.
• The fine-scale behavior is generated via offline simulations of crystal plasticity at the microscale using FFT-based solvers.
• The trained neural network surrogate is directly embedded into the coarse-scale finite element simulation, replacing the need for repeated fine-scale solves.
• The framework uses a two-scale setting where the coarse scale is solved with the learned fine-scale response, enabling efficient and accurate macroscopic dynamics.
• The approach avoids the need for handcrafted descriptors by learning the mapping directly from simulation data.

Experimental results

Research questions

• RQ1 Can a data-driven surrogate model replace repeated fine-scale simulations in multiscale inelastic impact problems without requiring a priori knowledge of state variables?
• RQ2 How does the fidelity of the proposed learning-based method compare to concurrent multiscale and empirical models in simulating polycrystalline magnesium impact?
• RQ3 Can the neural network surrogate capture complex wave propagation and deformation modes such as elastic waves and bending in thin plates?
• RQ4 What is the computational cost of the proposed method relative to empirical models and concurrent multiscale simulations?
• RQ5 Can the method be extended to other multiscale phenomena such as phase transitions or stress-assisted diffusion?

Key findings

• The proposed method achieves simulation fidelity comparable to concurrent multiscale modeling while incurring only about ten times the computational cost of an empirical model.
• The method successfully captures complex physical phenomena such as radially propagating elastic waves, plastic deformation, and plate bending under varying impact velocities and thicknesses.
• For a single plate impact simulation, the online computational cost was approximately 10 seconds on a CPU, compared to orders of magnitude higher cost for concurrent multiscale methods.
• The one-time offline training cost of the neural network was comparable to a single concurrent multiscale simulation, making the method scalable for repeated use.
• The framework demonstrated robustness across parametric variations, including impact velocity and plate thickness, with accurate prediction of von Mises stress, deviatoric strain, and volumetric strain evolution.
• The method enables direct use of the surrogate in coarse-scale simulations without requiring explicit identification of internal variables or order parameters.