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Abstract We present a machine learning framework capable of consistently inferring mathematical expressions of hyperelastic energy functionals for incompressible materials from sparse experimental data and physical laws. To achieve this goal, we propose a polyconvex neural additive model (PNAM) that enables us to express the hyperelastic model in a learnable feature space while enforcing polyconvexity. An upshot of this feature space obtained via the PNAM is that (1) it is spanned by a set of univariate basis functions that can be re‐parametrized with a more complex mathematical form, and (2) the resultant elasticity model is guaranteed to fulfill the polyconvexity, which ensures that the acoustic tensor remains elliptic for any deformation. To further improve the interpretability, we use genetic programming to convert each univariate basis into a compact mathematical expression. The resultant multi‐variable mathematical models obtained from this proposed framework are not only more interpretable but are also proven to fulfill physical laws. By controlling the compactness of the learned symbolic form, the machine learning‐generated mathematical model also requires fewer arithmetic operations than its deep neural network counterparts during deployment. This latter attribute is crucial for scaling large‐scale simulations where the constitutive responses of every integration point must be updated within each incremental time step. We compare our proposed model discovery framework against other state‐of‐the‐art alternatives to assess the robustness and efficiency of the training algorithms and examine the trade‐off between interpretability, accuracy, and precision of the learned symbolic hyperelastic models obtained from different approaches. Our numerical results suggest that our approach extrapolates well outside the training data regime due to the precise incorporation of physics‐based knowledge. 

Abstract Shock waves in geological materials are characterized by a sudden release of rapidly expanding gas, liquid, and solid particles. These shock waves may occur due to explosive volcanic eruptions or be artificially triggered. In fact, underground explosions have often been used as an engineering solution for large‐scale excavation, stimulating oil and gas recovery, creating cavities for underground waste storage, and even extinguishing gas field fires. As such, hydrocodes capable of simulating the rapid and significant deformation under extreme conditions can be a valuable tool for ensuring the safety of the explosions. Nevertheless, as most of the hydrocodes are often formulated in an Eulerian grid, this setting makes it non‐trivial to track the deformation configuration of the materials without a level set. The objective of this paper is to propose the use of the material point method equipped with appropriate equation of state (EOS) models as a hydrocode suitable to simulate underground explosions of transverse isotropic geomaterials. To capture the anisotropic effect of the common layered soil deposits, we introduce a new MPM hydrocode where an anisotropic version of the Mie‐Gruneisen EOS is coupled with a frictional Drucker‐Prager plasticity model to replicate the high‐strain‐rate constitutive responses of soil. By leveraging the Lagrangian nature of material points to capture the historical dependence and the Eulerian calculation of internal force, the resultant model is capable of simulating the rapid evolution of geometry of the soil as well as the high‐strain‐rate soil mechanics of anisotropic materials. 

Abstract We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as ‐HMX in the geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored free energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored free energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g., strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom‐line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite‐deformation regime. We compare the training efficiency of the neural networks under different Sobolev constraints and assess the accuracy and robustness of the models against MD benchmarks for ‐HMX. 

Abstract For some polycrystalline materials such as austenitic stainless steel, magnesium, TATB, and HMX, twinning is a crucial deformation mechanism when the dislocation slip alone is not enough to accommodate the applied strain. To predict this coupling effect between crystal plasticity and deformation twinning, we introduce a mathematical model and the corresponding monolithic and operator splitting solvers that couple the crystal plasticity material model with a phase field twining model such that the twinning nucleation and propagation can be captured via an implicit function. While a phase field order parameter is introduced to quantify the twinning induced shear strain and corresponding crystal reorientation, the evolution of the order parameter is driven by the resolved shear stress on the twinning system. To avoid introducing an additional set of slip systems for dislocation slip within the twinning region, we introduce a Lie algebra averaging technique to determine the Schmid tensor throughout the twinning transformation. Three different numerical schemes are proposed to solve the coupled problem, including a monolithic scheme, an alternating minimization scheme, and an operator splitting scheme. Three numerical examples are utilized to demonstrate the capability of the proposed model, as well as the accuracy and computational cost of the solvers. 

Abstract Supervised machine learning via artificial neural network (ANN) has gained significant popularity for many geomechanics applications that involves multi‐phase flow and poromechanics. For unsaturated poromechanics problems, the multi‐physics nature and the complexity of the hydraulic laws make it difficult to design the optimal setup, architecture, and hyper‐parameters of the deep neural networks. This paper presents a meta‐modeling approach that utilizes deep reinforcement learning (DRL) to automatically discover optimal neural network settings that maximize a pre‐defined performance metric for the machine learning constitutive laws. This meta‐modeling framework is cast as a Markov Decision Process (MDP) with well‐defined states (subsets of states representing the proposed neural network (NN) settings), actions, and rewards. Following the selection rules, the artificial intelligence (AI) agent, represented in DRL via NN, self‐learns from taking a sequence of actions and receiving feedback signals (rewards) within the selection environment. By utilizing the Monte Carlo Tree Search (MCTS) to update the policy/value networks, the AI agent replaces the human modeler to handle the otherwise time‐consuming trial‐and‐error process that leads to the optimized choices of setup from a high‐dimensional parametric space. This approach is applied to generate two key constitutive laws for the unsaturated poromechanics problems: (1) the path‐dependent retention curve with distinctive wetting and drying paths. (2) The flow in the micropores, governed by an anisotropic permeability tensor. Numerical experiments have shown that the resultant ML‐generated material models can be integrated into a finite element (FE) solver to solve initial‐boundary‐value problems as replacements of the hand‐craft constitutive laws. 

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiment, verification, and validation. 

Topology optimization algorithms often employ a smooth density function to implicitly represent geometries in a discretized domain. While this implicit representation offers great flexibility to parametrize the optimized geometry, it also leads to a transition region. Previous approaches, such as the Solid Isotropic Material Penalty (SIMP) method, have been proposed to modify the objective function aiming to converge toward integer density values and eliminate this non-physical transition region. However, the iterative nature of topology optimization renders this process computationally demanding, emphasizing the importance of achieving fast convergence. Accelerating convergence without significantly compromising the final solution can be challenging. In this work, we introduce a machine learning approach that leverages the message-passing Graph Neural Network (GNN) to eliminate the non-physical transition zone for the topology optimization problems. By representing the optimized structures as weighted graphs, we introduce a generalized filtering algorithm based on the topology of the spatial discretization. As such, the resultant algorithm can be applied to two- and three-dimensional space for both Cartesian (structured grid) and non-Cartesian discretizations (e.g. polygon finite element). The numerical experiments indicate that applying this filter throughout the optimization process may avoid excessive iterations and enable a more efficient optimization procedure. 

The yield surface of a material is a criterion at which macroscopic plastic deformation begins. For crystalline solids, plastic deformation occurs through the motion of dislocations, which can be captured by discrete dislocation dynamics (DDD) simulations. In this paper, we predict the yield surfaces and strain-hardening behaviors using DDD simulations and a geometric manifold learning approach. The yield surfaces in the three-dimensional space of plane stress are constructed for single-crystal copper subjected to uniaxial loading along the [100] and [110] directions, respectively. With increasing plastic deformation under loading, the yield surface expands nearly uniformly in all directions, corresponding to isotropic hardening. In contrast, under [110] loading, latent hardening is observed, where the yield surface remains nearly unchanged in the orientations in the vicinity of the loading direction itself but expands in other directions, resulting in an asymmetric shape. This difference in hardening behaviors is attributed to the different dislocation multiplication behaviors on various slip systems under the two loading conditions. 

This paper introduces a neural kernel method to generate machine learning plasticity models for micropolar and micromorphic materials that lack material symmetry and have internal structures. Since these complex materials often require higher-dimensional parametric space to be precisely characterized, we introduce a representation learning step where we first learn a feature vector space isomorphic to a finite-dimensional subspace of the original parametric function space from the augmented labeled data expanded from the narrow band of the yield data. This approach simplifies the data augmentation step and enables us to constitute the high-dimensional yield surface in a feature space spanned by the feature kernels. In the numerical examples, we first verified the implementations with data generated from known models, then tested the capacity of the models to discover feature spaces from meso-scale simulation data generated from representative elementary volume (RVE) of heterogeneous materials with internal structures. The neural kernel plasticity model and other alternative machine learning approaches are compared in a computational homogenization problem for layered geomaterials. The results indicate that the neural kernel feature space may lead to more robust forward predictions against sparse and high-dimensional data. 

This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.