International Journal for Numerical Methods in Engineering最新文献

{"title":"A Quasicontinuum Method With Optimized Local Maximum-Entropy Interpolation and Heaviside Enrichment for Heterogeneous Lattices","authors":"Benjamin Werner,&nbsp;Ondřej Rokoš,&nbsp;Jan Zeman","doi":"10.1002/nme.70390","DOIUrl":"https://doi.org/10.1002/nme.70390","url":null,"abstract":"<p>Lattice systems are indispensable for modeling and analyzing physical phenomena in materials with discrete or heterogeneous micro- or meso-structures. However, the computational requirements for practical engineering applications of lattice systems remain high. The quasicontinuum (QC) method addresses this by reducing the system of equations using a finite element mesh interpolation, rather than considering all nodes of a fully resolved lattice. Nevertheless, interfaces between separate phases in heterogeneous materials, such as concrete, require fine meshes throughout the domain, diminishing the effectiveness of QC. Enrichment strategies originally introduced in the extended finite element method (XFEM) can also account for material interfaces in discrete systems using nonconforming meshes, thereby resolving this issue. In a previous study, we applied Heaviside enrichment to investigate concrete mesostructures with the extended QC method, achieving a tenfold reduction in the number of unknowns while maintaining similar accuracy compared to discretizations with fully resolved interfaces. In the present study, we employ the meshless local maximum entropy (LME) interpolation, which transitions seamlessly from widespread meshfree to linear basis functions. Additionally, we combine LME interpolation with Heaviside enrichment and systematically investigate the role of the locality parameter and its optimization in heterogeneous lattices. This combination of optimized LME basis functions with Heaviside enrichment leads to an order-of-magnitude improvement in displacement accuracy while using the same number of degrees of freedom (DOF) compared to QC with linear interpolation. Moreover, we identify optimized distributions of the LME locality parameter and propose simple, non-optimized rules that deliver comparable accuracy at a fraction of the computational cost. Results from three numerical examples show that the optimal locality-parameter fields are nonuniform near interfaces and can be approximated by simple pattern-based rules that retain much of the benefit of full optimization.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70390","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148534013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Flow Feature Aligned Structured Mesh Generation via Sweeping Cross-Field Generated Multiblocks","authors":"Yibin Wang,&nbsp;Siqiang Deng,&nbsp;Ning Qin,&nbsp;Pengcheng Cui,&nbsp;Huan Li,&nbsp;Bing Chen","doi":"10.1002/nme.70388","DOIUrl":"https://doi.org/10.1002/nme.70388","url":null,"abstract":"<div>\u0000 \u0000 <p>For accurately resolving flow features such as shocks and vortices, it is desirable for the mesh edges to be aligned or orthogonal to these features, which minimizes interpolation errors and allows the Rankine-Hugoniot condition across discontinuities to be satisfied. For this, we develop a three-dimensional structured mesh generation method that aligns hexahedral grids with flow features to improve resolution accuracy and efficiency. For this purpose, a novel sweeping strategy is introduced. We first generate two-dimensional cross-sectional meshes using the cross-field method, and then extrude them into volumetric blocks. Flow features extracted from an initial solution are embedded as geometric constraints. A direction-aware mechanism can handle two alternatives. If the feature's sweep direction aligns with the geometry, the feature curve is embedded directly into the cross-sectional block. On the other hand, if they differ, a separate mesh domain is created for the feature and coupled via an overlapping grid strategy. Numerical tests on transonic flows show that the method captures λ-shocks, tip vortices and wakes with far fewer cells than uniform refinement, while achieving comparable or better resolution. The resulting anisotropic grids suppress numerical dissipation along critical flow structures, offering a practical balance between computational cost and solution fidelity.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Surface and Boundary Corrections in Peridynamics Using Optimized Nodal Influence Weights","authors":"Arman Shojaei,&nbsp;Greta Ongaro,&nbsp;Alexander Hermann,&nbsp;Stewart A. Silling,&nbsp;Patrizia Trovalusci,&nbsp;Christian J. Cyron","doi":"10.1002/nme.70385","DOIUrl":"https://doi.org/10.1002/nme.70385","url":null,"abstract":"<p>Peridynamics (PD), similarly to some other nonlocal continuum theories, exhibits truncated interaction horizons near free surfaces, cracks, and voids in bounded domains. This loss of neighbors causes artificial surface effects and inconsistencies in the derivative and energy operators that persist under discretization refinement, limiting PD accuracy and robustness. First, the analysis and numerical isolation of the effects of horizon truncation on these operators are carried out, showing how it induces surface softening in bond-based and stiffening/softening in state-based PD formulations. Then, a purely PD-based surface correction is proposed to restore full-horizon behavior while preserving the original horizon radius and bond topology. Each node is assigned a scalar influence weight, and every bond is scaled by the average of the endpoint weights to maintain symmetry. The nodal influence weights are computed in a preprocessing optimization step enforcing agreement between truncated- and full-horizon derivative and energy operators. Unlike existing approaches, the proposed method does not rely on ghost particles, variable horizons, or reference solutions from classical continuum mechanics (CCM). Benchmarks in one, two, and three dimensions, including a dynamic brittle fracture test, show that the optimized nodal influence weights reduce boundary-induced surface effects, improve energy consistency, and accelerate convergence of PD solutions. Analytical evaluation of the calibration targets over full-horizons also improves interior volume integration accuracy. The proposed scheme is simple to implement in existing PD codes, incurs only a modest preprocessing cost, performs robustly for flat and curved boundaries on both regular and irregular point sets, produces reliable crack paths, and applies to both bond-based and ordinary state-based PD formulations. An open-source Python implementation, <span>perifit</span>, is provided.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70385","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Acceleration Scheme for Nonlocal Gradient Damage Methods","authors":"Shqipron Shala,&nbsp;Haim Waisman","doi":"10.1002/nme.70386","DOIUrl":"https://doi.org/10.1002/nme.70386","url":null,"abstract":"<div>\u0000 \u0000 <p>There is a growing interest for using nonlocal damage mechanics type methods for modeling damage in solid mechanics, due to their practical implementation and ability to incorporate complex failure criteria. However, it is well known that the gradient damage methodology is computationally demanding, as it significantly increases the number of degrees of freedom in a system and requires the solution of a coupled system. In this study, we introduce a new acceleration method for localizing the gradient damage theory. Our approach adaptively modifies the number of degrees of freedom in the system, without altering the solution scheme, by introducing a criterion that adaptively switches between local and nonlocal formulations. The criteria is based on an integral approach and is applicable to both isotropic and anisotropic damage formulations, highlighting its versatility. The method is implemented in the finite element software ABAQUS by invoking several user modules. The performance of the acceleration scheme is validated through various numerical examples employing isotropic and anisotropic damage models. The results show that the scheme significantly reduces the computational cost of nonlocal gradient continuum damage analysis. Furthermore, it is shown that the acceleration does not affect the quality of the results.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Extension of Mesh-Distortion-Insensitive Petrov-Galerkin Finite Element Formulations to Linear Elastodynamics","authors":"Felix Zähringer,&nbsp;Peter Betsch","doi":"10.1002/nme.70377","DOIUrl":"https://doi.org/10.1002/nme.70377","url":null,"abstract":"<p>Numerous mesh-distortion-insensitive Petrov-Galerkin finite element formulations have been developed for the simulation of elastostatic problems. This contribution presents initial results on how these formulations can be extended to linear elastodynamics. In this context, three different approaches are examined, building upon an established 8-node elastostatic formulation. The first two approaches demonstrate that preexisting strategies reach their limits when applied to such unsymmetric formulations. In contrast, the novel third approach enables stable simulations. Moreover, the resulting formulation appears to be less sensitive to mesh distortion than a standard 8-node Bubnov-Galerkin formulation.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70377","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Electro-Chemo-Mechanical Multi-Component Multi-Phase-Field Modelling Framework","authors":"Jan Hauck,&nbsp;Hamzeh Beiranvand,&nbsp;Stephan Wulfinghoff","doi":"10.1002/nme.70378","DOIUrl":"https://doi.org/10.1002/nme.70378","url":null,"abstract":"<p>In this work we present the framework for a fully coupled electro-chemo-mechanical multi-component multi-phase continuum model. It is based on the variational framework for generalized standard materials and couples finite mechanical deformations and stresses, chemical volume changes, Maxwell–Stefan type multi-component diffusion, chemical reactions, electric fields, electric currents and phase transformations of multiple phases. The presented framework fulfills the balance laws for mass, charge, particles, momentum and is consistent with the second law of thermodynamics. The model is implemented and tested with the finite elements method. Examples including mechanically induced diffusion, reaction driven phase transformation, chemical and phase dependent stress development and ionic diffusion driven electric field formation show some of the coupled multi-physical effects that the framework is capable to represent.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70378","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148533712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hybrid Equilibrium Element Formulation for Large Displacement Nonlinear Elasticity","authors":"Francesco Parrinello,&nbsp;Simona Lo Franco,&nbsp;Guido Borino","doi":"10.1002/nme.70383","DOIUrl":"https://doi.org/10.1002/nme.70383","url":null,"abstract":"<p>The present paper proposes an equilibrium-based finite element formulation for the analysis of large displacement and large strain nonlinear elasticity problems. The proposed formulation is based on the Hybrid Equilibrium Element (HEE) and is developed in an updated Lagrangian framework. The solution is expressed in terms of a Cauchy stress field that strongly satisfies the domain and the boundary equilibrium equations in the deformed configuration, at the end of each loading step. The Cauchy stress field is independently modeled for each finite element, and the inter-element and boundary equilibrium conditions are imposed by the Lagrangian multiplier methods, for which the Lagrangian variable coincides with the element side displacement. The proposed formulation is developed in the theoretical framework of the <i>Virtual Stress Principle</i>, and the complementary weak form of the governing equations is derived in an <i>incremental</i> updated Lagrangian form. The proposed nonlinear HEE formulation is developed with two <i>hyperelastic</i> constitutive models, the Hencky model and the Neo-Hookean model. The proposed HEE formulations for large displacement and large deformation nonlinear elasticity have been implemented in a finite element code, using a triangular nine-node HEE with the quadratic stress field, and the results of three numerical simulations are presented, showing encouraging results for both compressible and quasi-incompressible materials.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70383","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148532929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vision Transformers for Modeling of Nonlinear Heterogeneous Materials","authors":"Yijing Zhou,&nbsp;Shabnam J. Semnani","doi":"10.1002/nme.70369","DOIUrl":"https://doi.org/10.1002/nme.70369","url":null,"abstract":"<p>Multiscale simulations of nonlinear heterogeneous materials and composites are challenging due to the prohibitive computational costs of high-fidelity simulations. Recently, machine learning (ML) based approaches have emerged as promising alternatives to traditional multiscale methods. However, existing ML surrogate constitutive models struggle in capturing long-range dependencies and generalization across microstructures. The recent advancements in attention-based Transformer architectures open the door to a more powerful class of surrogate models. Attention mechanism has demonstrated remarkable capabilities in natural language processing and computer vision. In this work, we introduce a surrogate (meta) model, namely ViT-Transformer, using a Vision Transformer (ViT) encoder and a Transformer-based decoder, which are both driven by the self-attention mechanism. The ViT encoder extracts microstructural features from material images, while the decoder is a masked Transformer encoder that combines the latent geometrical features with the macroscopic strain input sequence to predict the corresponding stress response. To enhance training, we propose a random extract training algorithm that improves robustness to sequences of variable lengths. We design and construct a compact yet varied dataset containing different numbers of fibers and fiber volume fractions via data augmentation, and validate the surrogate model using various composite material images and loading scenarios. Several numerical examples are provided to show the effectiveness and accuracy of the ViT-Transformer model and the training algorithm.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70369","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148532930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deep Learning-Based Phase-Field Modelling of Brittle Fracture in Anisotropic Media","authors":"Nojus Plungė,&nbsp;Peter Brommer,&nbsp;Rachel S. Edwards,&nbsp;Emmanouil G. Kakouris","doi":"10.1002/nme.70381","DOIUrl":"https://doi.org/10.1002/nme.70381","url":null,"abstract":"<p>This work presents a variational physics-informed deep learning framework for phase-field modelling of brittle crack propagation in anisotropic media. Previous variational energy-based neural network (NN) approaches have focused on second-order, isotropic phase-field fracture formulations and have relied on fully connected NNs. In contrast, the present work introduces, for the first time within a variational deep learning setting, a family of higher-order anisotropic phase-field models through a generalised crack density functional. The fracture problem is solved by minimising the governing energy functional using the deep energy method (DEM). The trial space is enriched with higher-order B-spline basis functions to represent higher-order gradients accurately and stably, thereby eliminating the need for conventional automatic differentiation. A ReZero-based residual NN architecture is adopted, for the first time in fracture mechanics. It stabilises the optimisation of the non-convex energy functional and improves convergence under strong coupling between displacement and phase fields. Random Fourier feature (RFF) enrichment is also introduced, enhancing the representation of steep displacement gradients and narrow damage zones near crack tips. The methodology is assessed for isotropic behaviour and cubic and orthotropic anisotropies. Numerical examples demonstrate direction-dependent crack growth in anisotropic cases, highlighting the capability of the method to accurately capture this behaviour.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 13","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70381","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148466691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalised Phase Field Modelling of Fracture in Gyroid Structures: Numerical and Experimental Insights Using Additively Manufactured Polylactic Acid (PLA) Specimens","authors":"Deison Preve,&nbsp;Jasen Zenzerović,&nbsp;Emanuele Avoledo,&nbsp;Manish Kumar,&nbsp;Enrico Salvati","doi":"10.1002/nme.70379","DOIUrl":"https://doi.org/10.1002/nme.70379","url":null,"abstract":"<p>This study presents a three-dimensional cohesive phase field framework for simulating quasi-static fracture in Gyroid lattice structures. To the authors' knowledge, this is one of the first experimentally validated applications of a diffuse cohesive phase field model to describe damage and fracture evolution in additively manufactured polylactic acid (PLA) lattice specimens. Due to the softening-like mechanical behaviour exhibited by the 3D-printed PLA specimens, the present work employs a high-order cohesive generalised phase field model capable of capturing material softening once the maximum load-carrying capacity is reached and energy dissipation becomes dominant. This modelling choice is essential to reproduce the gradual post-peak degradation observed experimentally, which would not be adequately described by a purely brittle AT1/AT2-type formulation. After validation in a two-dimensional case, the approach is extended to the analysis of 3D-printed Gyroid specimens subjected to tensile and compressive loading. Different Gyroid configurations, including unit cells and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>×</mo>\u0000 <mn>2</mn>\u0000 <mo>×</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$$ 2times 2times 2 $$</annotation>\u0000 </semantics></math> clusters, were tested under tensile and compressive loading regimes using an in situ tensile/compression testing stage, with deformations and failure modes directly monitored through in situ imaging. Each experiment was recorded from multiple angles during tensile and compressive loading of the additively manufactured Gyroid specimens and compared frame by frame with simulation outputs, revealing good agreement in the prediction of crack initiation and propagation paths. The comparison between experimental and numerical results confirms the model's ability to capture the main tensile failure mechanisms and the softening-driven fracture evolution in architected PLA lattices, while the compressive response is interpreted with caution in the late post-peak regime due to mechanisms such as buckling, contact and densification. This versatile modelling framework lays the foundation for future studies incorporating additional physical phenomena—such as surface coatings, residual stresses, or micron-scale effects—and for extension to other triply periodic minimal surface (TPMS) topologies beyond the Gyroid.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 13","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70379","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148466694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}