A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials (1807.09829v4)

Abstract: In this paper, a new data-driven multiscale material modeling method, which we refer to as deep material network, is developed based on mechanistic homogenization theory of representative volume element (RVE) and advanced machine learning techniques. We propose to use a collection of connected mechanistic building blocks with analytical homogenization solutions which avoids the loss of essential physics in generic neural networks, and this concept is demonstrated for 2-dimensional RVE problems and network depth up to 7. Based on linear elastic RVE data from offline direct numerical simulations, the material network can be effectively trained using stochastic gradient descent with backpropagation algorithm, enhanced by model compression methods. Importantly, the trained network is valid for any local material laws without the need for additional calibration or micromechanics assumption. Its extrapolations to unknown material and loading spaces for a wide range of problems are validated through numerical experiments, including linear elasticity with high contrast of phase properties, nonlinear history-dependent plasticity and finite-strain hyperelasticity under large deformations. By discovering a proper topological representation of RVE with fewer degrees of freedom, this intelligent material model is believed to open new possibilities of high-fidelity efficient concurrent simulations for a large-scale heterogeneous structure. It also provides a mechanistic understanding of structure-property relations across material length scales and enables the development of parameterized microstructural database for material design and manufacturing.

• The paper introduces a novel deep material network that predicts nonlinear behavior with training errors typically below 1%.
• It employs a hierarchical two-layer architecture integrating machine learning with classical homogenization to capture microstructural details.
• The method demonstrates robust extrapolation across diverse loading conditions, offering scalable and interpretable insights for material design.

Overview of "A Deep Material Network for Multiscale Topology Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials"

The paper presents an innovative machine learning approach for multiscale material modeling, termed the "deep material network." This approach seeks to address the computational challenges faced when modeling heterogeneous materials by effectively capturing their complex behaviors at different scales. The deep material network is based on a simplified hierarchical architecture with two-layer mechanistic building blocks, enabling the capture of intricate microstructural details without sacrificing computational efficiency.

The methodology integrates advanced machine learning techniques with classical multiscale modeling concepts, leveraging the analytical mechanics-based homogenization theory. It is particularly noteworthy for its ability to predict nonlinear material behavior, including scenarios involving large deformations, plasticity, and varying material laws, without further calibration or making additional micromechanical assumptions.

Strong Numerical Results and Methodological Rigor

• The deep material network was tested on several types of representative volume elements (RVEs), achieving average relative training errors typically below 1% with sufficient network depth ($N\geq5$).
• The method demonstrates exceptional predictive capabilities for unseen material and loading conditions, maintaining extrapolation errors within a low range even for cases with high material contrast.
• The paper showcases the application in nonlinear plasticity and finite-strain hyperelasticity, areas where traditional reduced-order models often struggle.

Theoretical and Practical Implications

The introduction of the deep material network has significant implications for both theoretical advancements and practical applications in computational material science:

• Theoretical Understanding: By using mechanistic deep learning, the network not only reduces computation costs but also provides insights into the structure-property relations inherent in heterogeneous materials. The approach bridges the gap between the purely data-driven methods and physics-informed modeling, allowing for enhanced interpretability of the material behavior in complex multiscale settings.
• Computational Efficiency: One of the standout features is the reduced computational time, which scales linearly with the number of degrees of freedom in the system. This efficiency gain is critical for concurrent multiscale simulations and potentially transformative for application in large-scale industrial problems involving complex morphologies and mechanical laws.
• Material Design and Manufacturing: The network's ability to parameterize the microstructural data implicitly suggests new methodologies for material design. Its adaptability and efficiency make it a promising candidate for designing materials with tailored properties, informed by both computational models and empirical data.

Future Directions

The authors mention several promising extensions and applications, including the adaptation of the network to three-dimensional (3D) problems, multiphysics phenomena, and the modeling of materials with even more complex mechanical behaviors like viscoelasticity. There is also potential in utilizing the network for real-time simulation in industrial settings, particularly in fields where traditional high-fidelity models fail due to computational limits.

Overall, the deep material network offers a scalable and efficient framework for advancing material modeling strategies, with potential broad-spectrum applications ranging from academic research to practical industrial deployment. The approach effectively leverages the synergy of machine learning and physics, setting a benchmark for future developments in multiscale modeling techniques.