A POD-TANN approach for the multiscale modeling of materials and macroelement derivation in geomechanics

TL;DR

The results indicate that the POD‐TANN approach not only offers accuracy in reproducing the studied constitutive responses, but also reduces computational costs, making it a practical tool for the multiscale modeling of heterogeneous inelastic geomechanical systems.

Abstract

This paper introduces a novel approach that combines Proper Orthogonal Decomposition (POD) with Thermodynamics-based Artificial Neural Networks (TANN) to capture the macroscopic behavior of complex inelastic systems and derive macroelements in geomechanics. The methodology leverages POD to extract macroscopic Internal State Variables from microscopic state information, thereby enriching the macroscopic state description used to train an energy potential network within the TANN framework. The thermodynamic consistency provided by TANN, combined with the hierarchical nature of POD, allows to reproduce complex, non-linear inelastic material behaviors as well as macroscopic geomechanical systems responses. The approach is validated through applications of increasing complexity, demonstrating its capability to reproduce high-fidelity simulation data. The applications proposed include the homogenization of continuous inelastic representative unit cells and the derivation of a macroelement for a geotechnical system involving a monopile in a clay layer subjected to horizontal loading. Eventually, the projection operators directly obtained via POD, are exploit to easily reconstruct the microscopic fields. The results indicate that the POD-TANN approach not only offers accuracy in reproducing the studied constitutive responses, but also reduces computational costs, making it a practical tool for the multiscale modeling of heterogeneous inelastic geomechanical systems.

Figures (17)

• Figure 1: Sketch of the energy network of the POD-aided TANN used for the discovery of the macroscopic Helmholtz free energy density function of the homogenized micro-structured medium.
• Figure 2: a) Computational model of the RUC, representation of the field of maximum principal microscopic plastic strains. b) Normalized reconstruction error $(\Psi - \Psi)/\Psi_{max}$ of the macroscopic energy based on considered number of POD modes. $\tilde{\mu}_{\Psi}$ and $\tilde{\sigma}_{\Psi}$ are the mean and standard deviation of $err_{\Psi}$ over all the data samples in the training set; $\Psi_{max}$ is the maximum energy value in the training set. c) Loss curves the macroscopic stress and the regularization term on the rate of dissipation, considering 13, 17 and 25 POD modes.
• Figure 3: Prediction in inference mode of the TANN's energy network trained considering 25 POD modes of the RUC with the ellipsoidal inclusion. The RUC is subjected to an unseen random 3D strain path. Stress components are reported as a functions of the conjugate strain components.
• Figure 4: a) Prediction in inference mode of the TANN's energy network trained considering 25 POD modes. On the right there is the microscopic elastic strain field corresponding to a point in the first reloading branch of the path. b) Reconstructed elastic strain field considering 25, 50, 75 and 100 POD modes.
• Figure 5: a) Computational model of the RUC, representation of the field of maximum principal microscopic plastic strain. b) Normalized reconstruction error $(\Psi - \Psi)/\Psi_{max}$ of the macroscopic energy based on considered number of POD modes. $\tilde{\mu}_{\Psi}$ and $\tilde{\sigma}_{\Psi}$ are the mean and standard deviation of $err_{\Psi}$ over all the data samples in the training set; $\Psi_{max}$ is the maximum energy value in the training set. c) Loss curves of the macroscopic stress and the regularization term for imposition of the second Law, considering 15, 20 and 25 POD modes.