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Conventional inverse problems for cohesive zones often utilize homogenized responses of the effective media to identify a fixed set of material parameters prescribed a priori. However, the mixed-mode loading conditions of composites or natural materials may exhibit interfacial relations that are difficult to anticipate. This article presents a model-discovery framework for directly identifying cohesive zone models inferred from displacement fields across the interface, without fixing on a specific form of equations. We develop a differentiable version of the Material Point Method (MPM) with interface elements formulated to capture the traction-separation law at a pre-existing crack or bonded interface. Ensuring the differentiability of the MPM solver enables us to backpropagate the mismatch between simulated and measured (e.g., DIC/DVC) displacement fields through the time integrator and interface physics. Using only kinematics and equilibrium as constraints, numerical experiments suggest that the method may recover (i) a Mode-I traction-separation curve in a double-cantilever-beam test and (ii) a mixed-mode law for a circular interface shear test. These numerical results demonstrate that displacement-only experiments, combined with a differentiable solver, offer a promising pathway for identifying rich and potentially nonparametric cohesive laws. 

Cook’s membrane is one of the most popular boundary value problems used to benchmark the performance of finite element models. Despite its popularity, the analytical solution to this boundary value problem remains unknown. As such, Richardson’s extrapolation, which provides a highly accurate displacement at the tip, is often used in verification exercises for finite element software used for analyses and designs. This paper introduces machine learning algorithms, particularly (1) the family of neural additive models and their subsequent symbolic approximations, (2) Kolmogorov-Arnold networks, (3) physics-informed neural networks as well as (4) the classical finite element method, (5) physics-informed polynomials and (6) brute-force symbolic regression algorithm to obtain new analytical solutions that may supplement Richardson’s extrapolation for the verification exercise. We consider two cases: one with a compressible linear-elastic model and the other with an incompressible neo-Hookean model, for which analytical solutions are unknown. Due to the floating-point representation, we did not seek an analytical solution with no error. Instead, we compare the accuracy, complexity, and interpretability of the displacement field solutions obtained from these methods and seek the optimal trade-off. We find that the best analytical solutions for the linear elastic and incompressible neo-Hookean cases are both obtained via the projected neural additive models followed by a post-processing step, with (1) errors in the orders of 10^-7 and 10^-5 respectively, and (2) complexities an order less than the counterparts obtained from Kolmogorov-Arnold networks. The training algorithms and results are open-source to facilitate third-party verification and further efforts to surpass the benchmark performance established in this paper. 

Inferring accurate and precise material models necessary for high-fidelity predictions has been a central challenge in constitutive modeling. Both traditional regression methods and modern machine-learning approaches require specialized data labels, which often cannot be sufficiently obtained from experiments. This data demand makes many sophisticated models impractical for real-world problems. Improvements in digital image correlation techniques have enabled accurate measurements of displacement data, providing an alternative kinematics-based approach for model identification. However, for materials undergoing plastic deformation, fracture, and damage, the corresponding inverse problem could be inherently challenging due to the dependence on loading history. We overcome this by formulating an inverse problem to discover interpretable plasticity models parameterized by neural networks (NN) from kinematic observations, leveraging a differentiable simulator with a smooth constitutive update that enables backpropagation for the NN training. The ability to use kinematic observations to infer complex material models may pave the way for a massive generation of material models that can be game-changing for emerging applications such as the design of metamaterials, response surface analyses, and the design of experiments. 

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. 

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. 

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. 

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 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. 

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.