Neural operators generally demonstrate strong predictive performance on in-distribution (ID) problems. However, a critical limitation of existing methods is their significant performance degradation when encountering out-of-distribution (OOD) data. To address this issue, this work introduces continual learning into physics-informed neural operators, with particular emphasis on neural operators built upon the Transolver architecture, and proposes a simple yet effective replay-based continual learning strategy. The proposed method is fully physics-informed and does not require labeled data, relying solely on input fields together with physical constraints for training. When new OOD data become available, a small number of past data are incorporated through a distillation-based constraint to preserve previously acquired knowledge and alleviate catastrophic forgetting. Meanwhile, a transfer learning LoRA is employed to enable rapid adaptation to the new data. The proposed framework is systematically validated on three representative physical problems, including the Darcy flow problem in fluid mechanics, a two-dimensional hyperelastic brain tumor problem in biomechanics, and a three-dimensional linear elastic Triply Periodic Minimal Surfaces problem in solid mechanics. The results demonstrate that the proposed method effectively mitigates catastrophic forgetting on previously learned data while maintaining fast adaptability to new data. Compared with conventional joint training strategies, the proposed method significantly improves training efficiency while reducing additional memory usage and computational cost.

Physics-informed neural networks (PINNs) in energy form, also known as the deep energy method (DEM), offer advantages over strong form of PINNs such as lower-order derivatives and fewer hyperparameters, yet dedicated and user-friendly software for energy form of PINNs remains scarce. To address this gap, we present LM-DEM (Large Language Model-assisted Deep Energy Method), an open-source, Streamlit-based platform for solving vari ational partial differential equations (PDEs) in computational mechanics. LM-DEM integrates large language models (LLMs) for geometry modeling: users can generate Gmsh-compatible geometries directly from natural language descriptions or images, significantly reducing the burden of traditional geometry preprocessing. The solution process is driven by the deep energy method, while finite element solutions can be obtained in parallel. The framework supports built-in problems including Poisson, screened Poisson, linear elasticity, and hyperelas ticity in two and three dimensions, as well as user-defined energy functionals analogous to the UMAT interface in Abaqus. The source code is available at https://github.com/yizheng-wang/LMDEM, and a web-based version is accessible at https://ai4m.llmdem.com. LM-DEM aims to lower the barrier for practitioners and beginners to adopt energy form of PINNs for variational PDE problems.

Numerical analyses of cracked bodies are common tasks in the fracture mechanics evaluation of structures. In finite element calculations, mesh refinement is often required at the crack tip, with specialized meshing strategies and configurations. This technical note reveals a discretization-induced pseudo-constraint phenomenon in crack-tip finite element analysis, in which inappropriate local refinement and mesh transition strategies generate non-physical constraint effect. This pseudo-constraint is often manifested as deviations of characteristic parameters used to quantify constraint effects from their theoretical values. The dependence of the numerical pseudo-constraint on mesh refinement strategies, as well as practical approaches for reducing it, is also investigated. Four common mesh refinement strategies are systematically considered here: circular, rectangular, free, and submodel meshing. It is shown that pseudo-constraints are commonly present in these mesh variations. Moreover, simply refining the crack-tip mesh does not eliminate these pseudo-constraints. In particular, in the free refinement and submodeling strategies, the pseudo-constraint effects are even more significant. Therefore, one should be vigilant in numerical fracture mechanics calculation of constraint, especially for those materials and structures whose fracture behavior is constraint-sensitive.

Elastoplastic analysis is computationally demanding because its nonlinear, path-dependent constitutive behavior requires incremental loading and repeated iterative solutions. To address this challenge, we propose Plasolver, a physics-informed neural operator framework that combines the efficiency of operator learning with the accuracy and robustness of classical numerical solvers. Plasolver consists of a physics-informed pretraining stage and an optional warm-start stage. During pretraining, the neural operator is trained solely by minimizing the incremental potential energy of elastoplasticity formulated by Simo, without requiring any labeled solution data. It operates directly on unstructured point clouds by encoding spatial coordinates, loading histories, and material properties as unified point-wise prompts. This formulation provides dual invariance to spatial and loading-path discretizations, enabling consistent predictions across different spatial resolutions and different numbers of increments representing the same loading trajectory. The pretrained Plasolver achieves relative errors on the order of 1% while providing approximately two orders of magnitude acceleration over conventional finite element simulations. In the warm-start stage, the pretrained prediction is supplied as the initial solution to a classical iterative solver, preserving its numerical accuracy, robustness, and convergence properties while substantially accelerating convergence. Numerical results show that Plasolver reduces the required number of iterations by approximately 50% compared with conventional zero-initialized solvers and converges to solutions at any prescribed tolerance. Plasolver thus provides an efficient, accurate, and discretization-invariant computational framework for nonlinear, path-dependent elastoplastic problems.

This study addresses the challenge of predicting interfacial shear stress distribution in adhesively bonded scarf-repaired composites subjected to far-field in-plane loading. A quasi-3D semi-analytical reduced-order model is developed by combining orthotropic complex potentials for the in-plane dominant field with a third-order through-thickness enrichment for leading-order interface shear effects. The bonded scarf region is discretized into concentric annular subdomains to represent radial stiffness gradation, while interface equilibrium and displacement compatibility are enforced in a weighted least-squares sense. Direct comparisons with 3D finite element simulations show that the proposed model predicts the dominant interfacial shear-transfer pattern with good accuracy for thin laminates, small scarf angles, and thin adhesive layers, and the relative error of peak interfacial shear stress remains below 6.1%. The framework is intended for membrane/shear-dominated load transfer rather than general transverse loading, bending-dominated response, or full local 3D field reconstruction.

Three-dimensional fatigue lifetime prediction based on the phase-field model is still a big challenge for typical large-scale structures. In this paper, a microstructure based phase-field model is presented for interlayer cracking with cross-scale structure characteristics under thermo-mechanical loading. The presented phase-field fatigue model can be employed to model the thermal fatigue failure of a 3D multilayer materials system, which can reflect the real 3D stress state and large-scale fatigue issues in the electronics engineering scenarios. By introducing a random microstructure field, a relationship is established between the degradation behavior of material parameters and the evolution of the material microstructure field. This study shows that the elastoplastic performance of sintered nano-silver is crucial to crack network formation. The simulation reveals the differences in mud cracking failure modes between different loading conditions and sintering processes of sintered Ag. Samples with better plasticity and higher densification strain exhibit a more evident ductile failure mode, while samples with higher brittleness demonstrate a brittle failure mode without localized debonding phenomena. Overall, this computational framework considers the micrometer-scale microstructure evolution and effectively simulates different failure patterns of nano-silver die-attachment in centimeter-scale power modules, achieving a good balance between accuracy and computational cost.

Physics-Informed Neural Networks (PINNs) have recently emerged as powerful tools for solving partial differential equations (PDEs), with the Deep Energy Method (DEM) proving especially effective in fracture mechanics due to its energy-based formulation. Despite these advances, existing DEM approaches require dense collocation near cracks, face stability challenges, and typically treat discrete and continuous fracture models separately. To overcome these limitations, we introduce the Extended Deep Energy Method (XDEM), a unified deep learning framework that incorporates both displacement discontinuities and crack-tip asymptotics in the discrete setting, while flexibly coupling displacement and phase fields in the continuous setting. This integration enables accurate fracture predictions using uniformly distributed, relatively sparse collocation points. Validation across benchmark problems including stress intensity factor evaluation, straight and kinked crack growth, and complex crack initiation demonstrates that XDEM consistently outperforms standard DEM in accuracy and efficiency. By bridging discrete and phase-field models within a single framework, XDEM establishes a robust foundation for applying AI to fracture mechanics and opens new avenues for predictive modeling in engineering and materials science.

We propose a Pretrained Finite Element Method (PFEM), a physics-driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics-informed pretraining stage and an optional warm-start stage. In the pretraining stage, a neural operator based on the Transolver architecture is trained solely from governing partial differential equations, without relying on labeled solution data. The model operates directly on unstructured point clouds, jointly encoding geometric information, material properties, and boundary conditions, and produces physically consistent initial solutions with extremely high computational efficiency. PDE constraints are enforced through explicit finite element-based differentiation, avoiding the overhead associated with automatic differentiation. In the warm-start stage, the pretrained prediction is used as an initial guess for conventional FEM solvers, preserving their accuracy, convergence guarantees, and extrapolation capability while substantially reducing the number of iterations required to reach a prescribed tolerance. PFEM is validated on a broad range of benchmark problems, including linear elasticity and nonlinear hyperelasticity with complex geometries, heterogeneous materials, and arbitrary boundary conditions. Numerical results demonstrate strong generalization in the pretraining stage with relative errors on the order of 1%, and speedups of up to one order of magnitude in the warm-start stage compared to FEM with zero initial guesses.

The failure of large-scale composite structures is often governed by their thin interfacial regions, which presents a significant challenge in predicting damage progression and fracture patterns. In this study, we present a multistage progressive damage framework that integrates an energy-regularized internal constraint to control element-level stiffness degradation. This framework ensures physical consistency during failure by enforcing a monotonic increase of dissipated energy, which reflects the irreversible nature of the damage process. The proposed framework is particularly effective in simulating interfacial failure in composite structures, as demonstrated by its application to a 3D scarf-repaired composite. The model accurately reproduces the characteristic zigzag fracture morphology and resolves the failure sequence from interfacial debonding to fiber rupture. Furthermore, a multiaxial failure envelope derived from the simulations provides a quantitative understanding of shear-coupled strength degradation under combined stress states. The energy-regularized formulation is not only numerically stable but also exhibits weak sensitivity to mesh refinement, highlighting its robustness and practical applicability for large-scale structural simulations.

Predicting the strength of 3D scarf-repaired composite structures remains challenging because repair geometry, adhesive properties, and laminate lay-up jointly produce strongly nonlinear strength responses. This study develops an explicit strength-prediction framework that integrates dimensional analysis and symbolic regression. Within this framework, classical laminate theory and dimensional consistency are used to transform the original high-dimensional design problem into a compact set of mechanically meaningful dimensionless variables. A finite-element progressive damage database is constructed over the resulting design domain, and symbolic regression is applied to derive a compact analytical expression for the strength recovery ratio. The resulting strength-prediction formula provides accurate predictions within the sampled parameter ranges and retains an explicit form suitable for engineering design, parameter influence analysis, and rapid repeated evaluation. Additional comparisons with a DA-only power-law model and ANN baseline models show that the proposed SR model attains prediction accuracy close to that of ANN models while preserving a closed-form and interpretable expression. The proposed framework is developed for monotonic uniaxial tensile loading under the modeling assumptions and parameter ranges considered in this study.

Traditional shakedown analysis frameworks typically rely on complex mathematical programming or mechanics-based numerical formulations, requiring substantial domain-specific expertise that limits broader engineering applications. To overcome these challenges, this study proposes a novel shakedown deep neural network (SDNN) that introduces energy-based physics-informed neural networks (PINNs) into shakedown analysis for the first time. Grounded in the upper bound theorem, SDNN constructs an unsupervised energy functional defined as the ratio of plastic dissipation to external work. This functional serves as the sole training loss, effectively eliminating the manual tuning of multiple penalty weights commonly required in physics-informed learning. Beyond determining shakedown limits, SDNN allows for the direct identification of failure mechanisms from predicted cyclic strain increment fields, bypassing the need for costly incremental simulations. To reduce the computational burden of training, this study proposes a strain-guided adaptive importance sampling (SAIS) strategy that dynamically refines sampling distributions. This approach achieves an order-of-magnitude reduction in the required sample size while concurrently enhancing the computational accuracy. The robustness and broad applicability of the proposed method are demonstrated through extensive benchmarks, including complex cases involving combined thermo-mechanical loading and multiple load vertices.

Understanding constitutive relationships is essential for the modeling, design, and practical application of advanced metals and alloys under extreme conditions. Currently, most constitutive models are phenomenological. While crystal plasticity finite element (CPFE) modeling can incorporate physical equations at the microscale, its high computational cost makes it difficult to apply in engineering problems. There is an urgent need to develop a multiscale neural network constitutive surrogate model to improve efficiency and address the issue of historical dependence. To address the limitations above, a multiscale autoregressive physics-guided neural network (MRPGNN) model is developed to obtain the evolution of the slip system rotation, resolved shear stress, back stress, isotropic hardening term, and state variables describing dislocation hardening through physics-based propagation via update equations, considering finite deformations. It is developed using initial state and macroscopic loading history as input, evolution of all physical quantities on the slip system with loading history serve as the outputs. The proposed MRPGNN model achieves two orders of magnitude higher efficiency than CPFE modeling while also effectively addressing cumulative errors. A film cooling hole (FCH) structural component is used to validate the robustness of the MRPGNN model and assess its potential for practical applications. Research shows that the MRPGNN model demonstrates high prediction accuracy and robustness. The MRPGNN model has the potential to be integrated with advanced technologies in the future, allowing for a deeper exploration of multiscale constitutive relationships by incorporating more complex deformation mechanisms (such as climb and diffusional creep and so on) through modifications to the update equations in the deep learning framework.

GB local stress is central to the initiation and evolution of long-term creep damage in polycrystalline superalloys. Owing to the high-dimensional nonlinear relationships between the GB stress response and multiple crystallographic, microstructural, and micromechanical characteristics, it remains challenging to identify the key characteristics governing GB stress and to elucidate their mechanisms of influence. Dislocation-climb-affected crystal-plasticity finite-element simulations of minimal grain clusters are combined with an integrated causation-guided machine-learning framework, in which mechanics-informed descriptors are analyzed by causation entropy (CE) to identify governing mechanisms and then distilled into a reduced-order regression form for interpretable prediction of GB normal stress. Unlike conventional black-box surrogate models, the proposed framework combines physically meaningful descriptors, CE-based mechanism identification, and an explicit compact regression form, enabling the dominant physical variables and their effects on GB normal stress to be directly interpreted. Among 18 physically motivated characteristics, the GB inclination angle, the slip transmission, the climb-related Schmid-type indicator, and the elastic-modulus mismatch are found to be dominant, revealing the coupled roles of interfacial geometry, crystallographic compatibility, creep stress relaxation, and micromechanical contrast. The identified characteristics hierarchy and functional representation remain effective under multiaxial loading and can be extended to tricrystal systems through physically interpretable nonlocal augmentation when a purely local description becomes insufficient, demonstrating strong physical consistency and robust generalizability across physical conditions.

Homogenization is a fundamental tool for studying multiscale physical phenomena. Traditional numerical homogenization methods, heavily reliant on finite element analysis, demand significant computational resources, especially for complex geometries, materials, and high-resolution problems. To address these challenges, we propose PreFine-Homo, a novel numerical homogenization framework comprising two phases: pretraining and fine-tuning. In the pretraining phase, a Fourier Neural Operator (FNO) is trained on large datasets to learn the mapping from input geometries and material properties to displacement fields. In the fine-tuning phase, the pretrained predictions serve as initial solutions for iterative algorithms, drastically reducing the number of iterations needed for convergence. The pretraining phase of PreFine-Homo delivers homogenization results up to 1000 times faster than conventional methods, while the fine-tuning phase further enhances accuracy. Moreover, the fine-tuning phase grants PreFine-Homo improved generalization capabilities, enabling continuous learning and improvement as data availability increases. We validate PreFine-Homo by predicting the effective elastic tensor for 3D periodic materials, specifically Triply Periodic Minimal Surfaces (TPMS). The results demonstrate that PreFine-Homo achieves high precision, exceptional efficiency, robust learning capabilities, and strong extrapolation ability, establishing it as a powerful tool for multiscale homogenization tasks. The source code is publicly available at: https://github.com/yizheng-wang/HomoGenius.

Creep void evolution is a key damage process in high-temperature alloys, because void growth, shape change, and coalescence promote damage localization and thereby accelerate the transition to tertiary creep and eventual rupture. Existing studies based on single-void unit cells have clarified the effects of stress state, crystallographic orientation, and void geometry on creep void evolution, but have paid little attention to how void behavior is changed by the underlying creep deformation mechanism itself, especially dislocation climb. Here, this issue is addressed using a finite-strain crystal-plasticity finite-element framework with crystallographic slip and dislocation climb. A three-dimensional periodic single-crystal unit cell containing a pre-existing intragranular spherical void is analyzed under multiaxial loading, and a climb-enabled constitutive description calibrated for the high-temperature IN617 alloy is systematically compared with a matched slip-only reference over prescribed stress-triaxiality and Lode-parameter combinations and crystallographic orientations. Within the adopted periodic single-void unit-cell and constitutive framework, the results show that admitting climb produces a distinct creep-void response rather than merely accelerating the overall deformation. In particular, climb strengthens the net tendency toward void growth, causes the coalescence criterion to be reached over a wider stress-state range within the early Eeq≤1 observation window, weakens collapse at low triaxiality, and makes stress triaxiality a more dominant control of the global cell-averaged creep response while reducing that response’s sensitivity to the Lode parameter and crystallographic orientation. These trends are interpreted in terms of two model-internal features of climb: an admissible non-zero plastic volumetric contribution and an explicit hydrostatic-stress contribution to the microscopic driving force. Overall, the study reveals a distinct micromechanical pathway for creep void evolution when climb becomes active and provides a basis for future analyses based on single-void unit cells and creep-damage models that account for mechanism-dependent void evolution.

Numerical methods for contact mechanics are of great importance in engineering applications, enabling the prediction and analysis of complex surface interactions under various conditions. In this work, we propose an energy-based physics-informed neural network (PINN) framework for solving frictionless contact problems under large deformation. Inspired by microscopic Lennard-Jones potential, a surface contact energy is used to describe the contact phenomena. To ensure the robustness of the proposed PINN framework, relaxation, gradual loading and output scaling techniques are introduced. In the numerical examples, the well-known Hertz contact benchmark problem is conducted, demonstrating the effectiveness and robustness of the proposed PINN framework. Moreover, challenging contact problems with the consideration of geometrical and material nonlinearities are tested. It has been shown that the proposed PINN framework provides a reliable and powerful tool for nonlinear contact mechanics. More importantly, the proposed PINN framework exhibits competitive computational efficiency to the commercial FEM software when dealing with those complex contact problems. The codes used in this manuscript are available at https://github.com/JinshuaiBai/energy_PINN_Contact.

Shakedown analysis provides a powerful tool for evaluating the load-carrying capacity or safety boundary against plastic failure of structures subjected to variable repeated loads, and is widely used in the design of engineering structures. In this paper, the shakedown theorem is extended to solve the quasi-periodic dynamic shakedown problem by considering the inertia effect. A numerical formulation and algorithm based on the linear matching method (LMM) framework is proposed for dynamic shakedown analysis of three-dimensional structures under quasi-periodic dynamic loads. The corresponding principle and numerical iteration process are presented. The virtual elastic stress response of structure is calculated through the time-history analysis and the dynamic load vertices are related to the frequency of the quasi-periodic dynamic loads. The numerical procedure has been implemented into ABAQUS by the usage of user subroutines, where only a series of linear elastic analyses with varying moduli in time and space are essentially conducted. Three numerical examples show the accuracy and effectiveness of the proposed method for dynamic shakedown analysis of engineering structures. The obtained results indicate that not only the amplitude but also the frequency of the quasi-periodic load is the significant factor affecting the shakedown domain of the structure, and when the frequency of the applied load is closer to the natural frequency of the structure, the shakedown domain becomes narrower.

Shakedown analysis of functionally graded (FG) structures under coupled cyclic mechanical-thermal loading is critical to ensuring structural integrity. However, prior research has primarily focused on specific geometries, simplified loading conditions, or idealized assumptions, leaving a need for a general numerical approach. This paper develops a general numerical method based on the stress compensation method (SCM) for the accurate shakedown assessment of FG structures, addressing the combined challenges of complex geometries, multi-axial stress states, temperature-dependent material properties, and arbitrary cyclic loading. The numerical method is validated using the analytical solution of FG Bree plates. Additionally, the study investigates FG plates containing volumetric defects, quantifying the distinct sensitivities of alternating plasticity-controlled and ratchetingcontrolled shakedown boundaries to defect size. Furthermore, for thick-walled FG cylindrical pressure vessels, with radially nonlinear variations in material properties and temperature-dependent yield strength, the shakedown limits under constant internal pressure and cyclic thermal loading conditions are evaluated. An optimization strategy based on shakedown analysis, is proposed to optimize the material gradient distribution and enhance load-bearing capacity. Finally, the results substantiate the accuracy and efficiency of the developed method, offering a reliable and versatile tool for design optimization of FG structures and safety evaluation of pressure vessels in engineering applications.

Crystal plasticity finite element (CPFE) modeling has emerged as a leading mesoscopic modeling approach by integrating fully resolved microstructures and physics-based microscale deformation mechanisms into the constitutive modeling of crystal materials. However, this approach demands substantial computational resources, which limits its application for complex polycrystalline microstructures subjected to extreme loadings such as creep. In this study, a physics-informed intelligent constitutive model is proposed to accelerate the simulation of the creep response and life of polycrystalline microstructures and applied to nickel alloy Inconel 617. The model is trained with physical constraints and creep data generated by CPFE simulations that explicitly consider the interaction between dislocation glide and climb, grain boundary sliding and opening, and various grain orientations. To address the computational challenges and data redundancy issues associated with polycrystalline representative volume elements, two dimensionality reduction methods, namely principal component analysis and homogenized fabric tensor condensation, are proposed and studied. An autoregressive physics-informed neural network model is then developed using initial state and loading conditions as input, while creep time, evolution of creep strain, and grain orientation are the outputs. The model is trained using CPFE modeling data to predict the high-temperature creep behavior of Inconel 617. The latter demonstrates better predictive performance and delivers six orders of higher efficiency compared to direct numerical simulation using CPFE. The developed model is further used to study the rapid construction of intelligent constitutive and texture description, which shows improved efficiency and accuracy in predicting the creep behavior. The effect of texture description is further studied by using the proposed model. The fabric tensor is demonstrated to be an effective microstructural indicator.

Shakedown theory has been widely used to evaluate the load-bearing capacity of elastic-plastic structures under cyclic load. The stress compensation method (SCM) shows good performance in the shakedown analysis of homogeneous materials in previous work. In this paper, the numerical procedure of the SCM is extended for shakedown analysis of metal matrix composites (MMCs) made of heterogeneous materials. To validate the applicability and accuracy of the extended SCM, the calculated results of the method are compared with those available in the literature and the results of step-by-step analyses. Based on the computational homogenization method, the effective strength of MMCs is determined through shakedown analysis on representative volume elements (RVEs). Two kinds of RVEs individually from periodic particle-reinforced metal matrix composite and continuous fiber-reinforced aluminum matrix composite are investigated. The shakedown boundaries under various combinations of thermal and mechanical loads are given. Both the ratchetting mechanism and reverse plasticity mechanism to determine the different regimes of the shakedown boundary are revealed. Additionally, we illustrate that both stress approach and strain approach are feasible for the shakedown analysis of MMCs. The boundary conditions for constructing the self-equilibrium stress fields for stress-controlled and strain-controlled shakedown problems are formulated. The corresponding shakedown boundaries are determined and compared. A discrepancy exists in the shakedown boundaries obtained under the two loading approaches. The discrepancy is strongly correlated with the macroscopic residual strain and macroscopic residual stress.