A phase field model based on an orthogonality-based energy split and incorporating the effect of in-situ stress is proposed to simulate hydraulic fracturing in transversely isotropic rocks under complex stress conditions. The driving force is reconstructed through volumetric–deviatoric decomposition in a transformed strain space, thereby rigorously satisfying the energy orthogonality condition required for tension–compression asymmetry in anisotropic materials. Meanwhile, the influence of in-situ stress field is incorporated into the energy functional. Within the framework of Biot poroelasticity and phase field fracture theory, the proposed PFM is formulated and validated through a series of numerical examples and experimental comparisons. Then the fracture propagation mechanisms under bedding anisotropy, perforation angle, and in-situ stress difference are then systematically discussed. The anisotropy of the critical energy release rate exerts the strongest control on the fracture trajectory, followed by elastic modulus anisotropy, whereas permeability anisotropy primarily governs pore-pressure redistribution and peak pressure response despite its weaker influence on the final fracture path.

Generating high-quality meshes for arbitrary geometries remains a fundamental bottleneck in computational engineering, often demanding heuristic tuning and semi-manual workflows. In this paper, we introduce Dmsh, a first fully automated reinforcement learning pipeline that unifies geometric decomposition and quadrilateral mesh generation within a single learning-based framework. Dmsh decomposes the problem through three coordinated agents handling topology simplification, geometric regularization, and mesh generation. The meshing process is formulated as a Markov Decision Process and solved using a parametric Soft Actor-Critic architecture with decoupled critics, enabling efficient exploration of a hybrid discrete-continuous action space. A curriculum learning strategy ensures scalability from simple domains to highly complex geometries, suppressing seed variance. By design, the recursive decomposition enables parallel meshing of subregions, yielding globally conforming all-quadrilateral meshes without post hoc correction. Across a wide range of benchmarks, Dmsh consistently outperforms existing methods in automation, robustness, and mesh quality, establishing a new paradigm for learning-based mesh generation.

The performance of aero-engine gas path components exhibits high correlation and strong coupling. Performance degradation and faults have a significant impact on the safety and reliability of aircraft. Previous research primarily focused on establishing direct mapping relationships between measurable parameters and fault categories or performance, while neglecting the propagation and evolution mechanisms of multi-factor coupling in fault information. This paper proposes a propagation and evolution graph method, utilizing the gas path structure and fault information propagation laws as physical constraints, to construct fault propagation subgraphs and whole life cycle (WLC) fault evolution graphs, thereby enabling multi-indicator deep fault diagnosis. Firstly, based on parameter correlation analysis, a three-layer multi-factor coupling representation of fault sources, performance parameters, and measurement parameters is determined, and a symbolic directed graph method is used to construct single-condition point fault propagation subgraphs, revealing the information propagation laws of different faults. Then, a multi-dimensional fusion distance is established to achieve a WLC large-scale fusion graph for tracking the fault evolution process. Thereby, deep fault diagnosis of multiple indicators such as fault state, fault mode, and fault components is realized. Finally, various comparative experiments are designed, fully validating that the proposed method improves the accuracy and interpretability of multi-indicator fault diagnosis, providing an important basis for subsequent research on graph theory.

Metasurfaces composed of mechanical resonators distributed on the free surface of a semi-infinite elastic medium provide a promising platform for manipulating surface wave propagation. However, the design of metasurfaces with targeted surface wave functionalities remains challenging because of the intricate frequency-dependent coupling between the resonators and the substrate, as well as the resulting high-dimensional design space with strong interactions. In this work, an optimization method is proposed for determining resonator positions to achieve desired functionalities, such as Rayleigh wave focusing and isolation. A multiple scattering formulation is first developed to characterize the dynamic interactions between Rayleigh waves and mechanical resonators. On this basis, a gradient-descent optimization framework is established, in which the gradient of the objective function is derived analytically. Using the proposed method, Rayleigh wave focusing at prescribed target locations and wave isolation within the target regions are achieved. Furthermore, the framework is extended to multi-objective optimization, enabling symmetric and antisymmetric focusing under left and right source excitations. Frequency-dependent wave focusing and isolation are also achieved by combining resonators with different resonance frequencies. The proposed analytical and optimization framework paves the way for the design of advanced devices for surface wave localization, transmission, and isolation, with potential applications in vibration mitigation, sensing, and energy harvesting.

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.

A conventional constant-volume compressed air energy storage system (CAES) is integrated with a constant-pressure configuration to propose an underground CAES system employing a staged multi-level compression strategy. A comprehensive thermodynamic model is developed encompassing the surface compression units, heat exchangers, expansion turbines, and the subterranean storage cavern. Moreover, a fully coupled simulation framework is implemented in COMSOL Multiphysics to capture the combined thermodynamic response of both surface equipment and the underground cavern throughout charge-discharge cycles. Results indicate that the proposed constant-pressure CAES exhibits distinctive compression and expansion behavior compared with traditional CAES, particularly in terms of compressor power, air thermal evolution, and turbine performance. Parametric studies further reveal that operational parameters, such as charging rate, heat exchanger efficiency, and hydrostatic pressure, substantially affect energy consumption, discharge capability, and overall round-trip efficiency. These insights elucidate the thermodynamic characteristics and key performance drivers of the system, offering guidance for its parameters optimization.

This study proposes a novel deep learning-based framework for probabilistic analysis of slopes in spatially variable soils under saturated seepage conditions. Two types of deep neural networks are involved, including physics-informed neural networks (PINNs) for simulating saturated seepage fields and convolutional neural networks (CNNs) for building a surrogate of the deterministic model for slope stability responses. The soil spatial variability is captured by random fields, with specific realizations generated using the fast Fourier transform-moving averaging technique. The governing partial differential equation for saturated seepage within slopes is solved numerically using PINNs. A deterministic response model of slope stability is then developed using the discretized limit analysis method by incorporating the computed seepage fields from PINNs. A small number of random field realizations are generated and fed into the developed deterministic model to compute their corresponding responses, which together constitute the input-output pairs of the training set. On this basis, the CNN model is trained and subsequently acts as a surrogate model that can rapidly predict slope stability outcomes for new random field realizations. Based on the trained CNN model, an efficient probabilistic slope stability analysis is performed through Monte Carlo simulations. Validation examples are used to verify the developed PINN-based deterministic model and the CNN-based surrogate model. The numerical results reveal the effects of several factors on the reliability of slopes in spatially variable soils while considering saturated seepage. The proposed deep learning-based framework can serve as a robust alternative for probabilistic analysis of geotechnical structures in spatially variable soils.

Fracturing phenomena driven by temperature variations present substantial challenges across geotechnical engineering applications. Accurate computational representation of such behavior demands robust numerical architectures that can reliably capture discontinuous crack evolution within materials characterized by pronounced directional dependencies. This paper introduces a novel approach that integrates dual-horizon non-ordinary state-based peridynamics (DH-NOSBPD) with a variational damage model for simulating thermomechanical fracture in anisotropic media. The proposed framework overcomes the limitations of conventional peridynamic methods in representing anisotropic thermal and mechanical coupling while eliminating numerical instabilities inherent in bond-breaking criteria. A staggered coupling strategy is employed to synchronize thermal and mechanical field updates, incorporating anisotropic constitutive relationships for both heat conduction and stress-strain behavior. The variational damage model introduces a history-dependent scalar damage field derived from strain energy density, thereby circumventing the spurious energy release and mesh dependence associated with abrupt bond deletion. This approach yields physically consistent crack evolution. Numerical examples validate the framework's accuracy in anisotropic heat transfer, mechanical deformation, and complex fracture patterns under combined thermo-mechanical loading. The model demonstrates superior stability and predictive capability compared to conventional bond-breaking approaches.

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.

In this paper, we develop, analyze, and demonstrate a robust and efficient numerical method by utilizing the recently developed methodology of subdivision-based isogeometric analysis (IGA) for solving the Cahn–Hilliard–Darcy (or Cahn–Hilliard–Hele-Shaw) system confined on complex curved manifolds. The subdivision-based IGA enables the accurate representation for complex curved manifolds of flexible geometry and arbitrary topology, while bringing its essential properties of hierarchical refinement and high smoothness. Incorporating the scalar auxiliary variables (SAV) approach and the zero energy contribution (ZEC) idea together, the original model is reformulated with two auxiliary variables so that a decoupled and linearized scheme can be easily constructed and theoretically guaranteed for the unconditional energy stability. Numerical examples are provided to verify the robustness, stability and accuracy of the spatiotemporal fully discrete scheme, including the convergence test, the phase-separation phenomena, and the Saffman–Taylor fingering instability with different viscosity ratios on various complex curved manifolds driven by rotational forces.

This study proposes a Cosserat non-ordinary state-based peridynamic model (CNOSBPD) for simulating the elastic-brittle fracture process in rock-like materials. By introducing independent microrotation degrees of freedom into material points, an explicit expression for the rotational-related stiffness coefficients is derived based on the principle of strain energy density equivalence. Furthermore, a generalized mixed-mode fracture criterion for the Cosserat model is established to simultaneously determine tensile and shear failures. The model is validated through numerical examples, including uniaxial loading of a plate with a hole, four-point bending of a double-notched beam, and uniaxial compression of pre-cracked rock specimens. The results demonstrate that the proposed CNOSBPD model can reasonably reproduce the characteristics of crack initiation, propagation, and instability evolution, showing good agreement with experimental observations in terms of crack deflection, coalescence and evolving crack patterns, and load-carrying capacity degradation. Compared with the non-ordinary state-based peridynamic model (NOSBPD) that neglects microrotation, the proposed model enhances the representation of shear effects during crack initiation and propagation, leading to predictions of fracture morphology and evolutionary processes that are more consistent with experimental observations.

Maxwell stress refers to the mechanical stress exerted on a dielectric material due to the presence of electric fields. It plays a significant role in the interaction between a dielectric material and the surrounding free space under finite deformation. Previous research on finite deformation of flexoelectricity mainly adopted a modified form of Maxwell stress, potentially not able to correctly capture some physical phenomena, such as the compression of a dielectric droplet in an electric field. In this work, we propose a consistent and complete variational principle for flexoelectricity, in which the Maxwell stress emerges naturally from the derivation, without introducing additional assumptions. An Isogeometric analysis-based numerical framework is developed accordingly and verified by both linear and nonlinear benchmark cases compared with experimental results. The present framework successfully captures and quantifies the behaviors of conductive liquids and soft dielectric solids subjected to an external electric field. Finally, a novel scenario is investigated in which a flexoelectric beam immersed in free space is analyzed, showing the interesting distribution of Maxwell stress-induced tractions at opposing boundaries. The test demonstrates that a higher dielectric constant can effectively enhance the material's stiffness in response to the external electric loading.

Inertial effect plays a significant role in the fracture process of structures under dynamic loading. The inertial force at the crack tip often influences the crack pattern. To effectively describe dynamic fracture problem, a phase field model that accounts for phase field inertia is developed in this paper. First, a phase field inertial parameter expression determined by the material parameters is given by analogizing the displacement field and the phase field, finding the similarity of some parameters in the two field variables, and establishing a phase field model that includes the inertial energy of the phase field in the total energy functional. Secondly, the mechanical equilibrium equations and phase field evolution equations under dynamic conditions are obtained based on the Lagrangian kinetic equations. Then, the model is implemented using the finite element discretization strategy and staggered approach. Finally, the presented model is validated through several numerical examples. The impact of phase field inertia on crack patterns and energy characteristics is analyzed in detail and compared with experiment and reference numerical results. It is demonstrated that the phase field inertial effect affects the distribution of strain energy, fracture dissipated energy and kinetic energy in the system, naturally capturing the decrease in terminal crack velocity and preventing undesirable crack patterns. This further improves the phase field theoretical framework for dynamic fracture.

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.

Assembly interfaces are inherent in practical rotor systems and significantly influence rotor dynamics. Tuning dynamic behavior through interface property design represents a promising yet underexplored strategy. This study focuses on interface geometry and aims to regulate rotor dynamics by proactively designing the interface shape. The shape is modeled using sequentially connected control points, with adjustments made by varying their positions. Influence of interface shape on rotor dynamics is determined through a parametric correlation analysis. An adaptive multi-objective optimization approach that combines Kriging and multi-objective genetic algorithm is employed to identify the optimal interface shape within a finite element framework. Rotor dynamics experiments are conducted to validate the simulation results. The proposed design methodology is applied to a bolted thin-shell rotor system under both thermal-structural coupled and isothermal conditions. Results demonstrate that the designed interface shape increases critical speed and reduces unbalanced vibration.

In this study, cyclic air injection–withdrawal model tests were conducted using a large-scale physical model system to investigate the long-term deformation characteristics of the rock mass. Based on the experimental observations, a spatiotemporal evolution equation of deformation was established, and an equivalent deformation modulus approach was proposed to characterize the long-term deformation effect. By embedding this evolution law into a multi-field coupled framework, a numerical solution method considering long-term deformation of rock mass was developed. The results indicate that the rock mass exhibits pronounced spatiotemporal evolution characteristics under cyclic internal pressure. Temporally, deformation shows an exponential-type growth with asymptotic stabilization, accompanied by significant accumulation of residual deformation. Spatially, the influence of cyclic pressure attenuates rapidly with radial distance and becomes negligible beyond approximately three times the cavern radius. The proposed model effectively captures the evolution of structural mechanical responses, including cavern displacement and stresses in the steel liner and reinforcement. Thermodynamic effects lead to variations of approximately 13

We develop a Newton/Picard hybrid integration algorithm for finitestrain associative plasticity with separable yield functions thatare not required to be $1$-homogeneous. The elastic state is advancedby an exponential update of the elastic right Cauchy--Green tensor$\Ce$, and hardening follows a power-balance rule discretized bybackward Euler. The plastic multiplier is obtained from a one-dimensionalsafeguarded Newton-like corrector with frozen flow direction $\bm{N}$,followed by an outer Picard iteration for $\bm{N}$. Under standardregularity assumptions (convex elastic domain, $sigma_{\textrm{eq}}\in\mathcal{C}^{2}$, monotone hardening, and sufficiently small increments), we establish uniqueness of the plastic multiplier along the exact associated-flow path and, for accepted increments satisfying a contraction bound, local strict monotonicity also along the frozen-direction path. We also prove local contractivity of the Picard update and derive a closed-form algorithmic tangent in the intermediate configuration, written as a rank-one correction to a dressed elastic modulus; the full reference-configuration tangent is obtained by automatic differentiation of the exponential push-forward based on the converged local state. Quadratic convergence of the global Newton--Raphson solver is expected when the local Picard loop is run to convergence; for truncated runs the rate is super-linear, as the tangent approximates the exact linearization of the finite Picard map (see Section~\ref{sec:s5}). A Fortran implementation is included. Iso-error maps for a principal-value Hosford criterion ($n=8$, $1$-homogeneous, convex) for polylactic acid (PLA) and a Ghorbel paraboloidal criterion for polycarbonate (non-$1$-homogeneous)---both $\mathcal{C}^{2}$ in the neighborhood of the yield surface---show accuracy close to that of a fully coupled return-mapping scheme for the tested configurations. In the present implementation, the Newton/Picard method is approximately $5$--$7.5\times$ faster than the chosen Newton baseline with staggered hardening on a 3D framed-beam benchmark, depending on mesh density and the number of Picard updates $N_{n}$. Because the hardening treatments differ, this is an overall algorithmic comparison rather than a controlled isolation of the scalar corrector alone.

Phase-field modeling of brittle fracture removes the need to track cracks explicitly by recasting their evolution as the minimization of an energy functional. In return it requires a discretization dense enough to resolve a localization band whose width is set by a regularization length and whose path is not known in advance. We propose a mesh-free discretization in which a single neural network represents the displacement and phase fields and is trained by minimizing the incremental energy directly. The coordinates enter the network through a multiresolution feature encoding built from C^1 quadratic B-spline grids, so the finest scale the representation can express is set by choice rather than reached through slow training, and the energy is estimated by stratified Monte Carlo integration on points redrawn at every optimizer iteration. This pairing proves critical, since the crack fails to advance both when the integration points are held fixed and when the encoding is too coarse to represent the band, while each ingredient tolerates a wide range of settings once the other is in place. Because the representation is globally C^1, the second- and the fourth-order fracture energy densities run on the identical discretization. Across six problems, from single-edge-notched tension and shear to a thick-walled ring on a single spline patch, the computed load-displacement curves follow staggered finite element references at matched regularization length, with peak loads within about 1

Spatial variability in mechanical properties significantly affects fracture evolution in materials. A strict random phase field model suitable for spatially heterogeneous materials is developed, in which a gradient term of the critical energy release rate is introduced. Compared with the simple integration of traditional finite element models with random fields, which involves merely replacing deterministic mechanical properties in the governing equations, a more accurate coupling approach is adopted by incorporating a coordinate-dependent critical energy release rate into the energy functional, which is reformulated from existing models. The governing equation of the phase field is derived through variational principles. Unlike traditional random phase field models, the proposed model captures the spatial variation in the gradient of the critical energy release rate and effectively characterizes the directional rate of change in fracture toughness. The proposed model is implemented by using COMSOL Multiphysics and MATLAB, and validated through the rock fracture experiment. Quasi-static and dynamic fracture simulations reveal that in the presence of pronounced spatial heterogeneity, incorporating the critical energy release rate gradient can significantly alters fracture behavior in heterogeneous materials, often producing effects not captured by traditional random phase field models. Therefore, the rigorous random phase field model is indispensable for understanding and predicting the fracture behavior of spatially heterogeneous materials such as rocks.