International Journal for Numerical Methods in Engineering最新文献

{"title":"A Parallel-Computing Based Topology Optimization Method of Bi-Modulus Material Structures With Large Property Difference","authors":"Jianhua Rong,&nbsp;Chenchen Cao,&nbsp;Chenyu Rong,&nbsp;Shuxiang Chen,&nbsp;Junjie Feng,&nbsp;Lei Zhao,&nbsp;Quan Zhou,&nbsp;Jinhu Cai,&nbsp;Zhijun Zhao","doi":"10.1002/nme.70389","DOIUrl":"https://doi.org/10.1002/nme.70389","url":null,"abstract":"<div>\u0000 \u0000 <p>Engineering materials frequently exhibit considerable disparities between their tensile and compressive moduli. However, there are some difficulties in existing topology optimization methods for material structures with large disparities between their tensile and compressive moduli. To address this issue, a parallel-computing based topology optimization method for bi-modulus material structures with large property differences and deformation control is proposed. The main challenges are twofold: (1) the large discrepancy between tensile and compressive moduli usually leads to convergence difficulties within finite element analysis (FEA); (2) the iterative finite element solution procedure imposes a heavy computational burden during the topology optimization process. For the first challenge, new four-node membrane element formulas with rotational DOFs, which can generate a diagonally dominant stiffness matrix, and an FEA modified model algorithm are proposed, and the form of a smooth elastic modulus matrix related to principal strain states is introduced. These measures ensure the diagonal dominance of the element stiffness matrix, and importantly relieve the FEA difficulty of bi-modulus material structures with large differences in tensile and compressive moduli, thereby overcoming convergence issues and improving the accuracy of iterative calculations for displacements and strains. To improve computational efficiency, an algorithm based on the Gershgorin circle theorem is proposed to estimate the largest eigenvalue of a two-level preconditioner matrix integrating diagonal and Chebyshev preconditioners. Subsequently, a novel diagonal-Chebyshev bi-level preconditioned conjugate gradient iterative method based on Graphics Processing Units is developed, which significantly reduces the computation time of bi-modulus material structural optimization. Finally, two numerical examples are presented. Results demonstrate that the proposed method can generate reasonable optimal topologies for the aforementioned bi-modulus material structures, further verifying the feasibility and superior computational efficiency of the proposed optimization methodology.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 15","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148752586","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":"Thermo-Mechanical Topology Optimization: An Immersed FEM Level-Set-Based Approach","authors":"Farzad Tatar,&nbsp;Silvia Monchetti,&nbsp;Elisabetta Manconi,&nbsp;Roberto Brighenti","doi":"10.1002/nme.70398","DOIUrl":"https://doi.org/10.1002/nme.70398","url":null,"abstract":"<p>This paper proposes a multi-physics framework for topology optimization using an immersed level set-finite element model. The work extends the capabilities of a recently developed immersed level-set method to thermo-mechanical problems, including coupling and material-dependent properties. The objective functions are mechanical compliance and heat potential transport capacity. The sensitivity analysis allows capturing both nonlinearities in one velocity extension field. The level set evolution is parametrized using radial basis functions. Separate competing thermal and mechanical objective functions (regardless of their interconnection behavior) are considered using a linear combination. Numerical results confirm the method's effectiveness in handling conflicting performance goals, providing smooth topologies on domains with arbitrary shapes and boundary conditions. The results show that the proposed immersed level set method is an efficient and versatile tool for multi-physics topology optimization, highlighting its potential for advanced engineering design applications.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 15","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70398","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148752587","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 Iterative Solvers With Geometry-Aware Neural Preconditioners for Parametric PDEs","authors":"Youngkyu Lee,&nbsp;Francesc Levrero Florencio,&nbsp;Jay Pathak,&nbsp;George Em Karniadakis","doi":"10.1002/nme.70393","DOIUrl":"https://doi.org/10.1002/nme.70393","url":null,"abstract":"<div>\u0000 \u0000 <p>The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced <i>hybrid</i> solvers by combining the classical solvers with neural operators for a specific geometry, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 15","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148617041","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":"Inductive Differential Constraint Method: Symbolic Regression-Based Partial Differential Equation Regularization for Fatigue Crack Growth Prediction","authors":"Rekisei Ozawa,&nbsp;Yoshitaka Wada","doi":"10.1002/nme.70396","DOIUrl":"https://doi.org/10.1002/nme.70396","url":null,"abstract":"&lt;p&gt;This study introduces the inductive differential constraint method (IDCM), a data-informed structural regularization framework that enhances neural network predictions under data-scarce regimes by enforcing invariant differential structures extracted from simulation data. Unlike physics-informed neural networks, which rely on known governing equations and dense collocation sampling, IDCM constructs data-informed structural regularization terms from discovered symbolic relations without assuming them to constitute universal physical laws. The key novelty lies in transforming case-dependent symbolic expressions into parameter-free differential constraints. The proposed approach employs symbolic regression to identify explicit functional relationships from limited data. Systematic differentiation then eliminates case-specific parameters, yielding a parameter-free ordinary differential equation, which is subsequently extended to a partial differential equation. The resulting invariant constraint is incorporated into the training loss and evaluated at independently generated collocation points over the continuous input-variable space, promoting consistency with the discovered response structure without geometric discretization. Under fixed training settings, application to a symmetric two-crack configuration under fixed-amplitude loading demonstrates that IDCM improves interpolation robustness within the considered geometric family, reducing validation loss from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mo&gt;.&lt;/mo&gt;\u0000 &lt;mn&gt;40&lt;/mn&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ 3.40times 1{0}^{-2} $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;.&lt;/mo&gt;\u0000 &lt;mn&gt;89&lt;/mn&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ 1.89times 1{0}^{-3} $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; compared to conventional data-driven neural networks. Validation in this study focuses on unseen intermediate initial vertical distances within the simulated range of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 ","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 15","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70396","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616159","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":"A Novel Uncertainty-Integrated Design of Force Reconstruction and Control for Compliant Space Docking","authors":"Chen Yang,&nbsp;Mingchang Chen,&nbsp;Bowen Zhan,&nbsp;Wanze Lu,&nbsp;Haibo Zhang","doi":"10.1002/nme.70395","DOIUrl":"https://doi.org/10.1002/nme.70395","url":null,"abstract":"<div>\u0000 \u0000 <p>In the critical tasks of spacecraft assembly and docking, the implementation of force reconstruction and control technology greatly improves the success rate of the task. Thus, a groundbreaking uncertainty-integrated design method for force reconstruction and control for compliant operation tasks of space docking is proposed in this study. To bolster the accuracy of the contact force in the space cabin docking process, force reconstruction based on singular value decomposition (SVD) is introduced into the control loop to achieve more refined control effects. To realize the compliant operation throughout the space-docking process, an impedance control method is incorporated into the integrated design to accomplish the motion/force control. Based on the interval uncertainty analysis method, the entire study of inverse dynamics and motion/force control in the space docking system is analyzed with the interval dimension-wise analysis (IDWA), so as to complete the uncertainty-integrated design of force reconstruction and control. Furthermore, a novel interval control criterion is established to facilitate a judicious switching between the simulated environmental contact force and the accurately identified contact force in the uncertainty-integrated design of bounded force reconstruction and control. Numerical examples of cabin docking are shown to demonstrate the effectiveness of the proposed method. The designed uncertainty-integrated method can provide theoretical support for the development of compliant space docking tasks in the actual spacecraft engineering to improve and optimize the loading environment in the assembly process.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 15","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616161","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":"Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning","authors":"Kianoosh Taghikhani,&nbsp;Yusuke Yamazaki,&nbsp;Jerry Paul Varghese,&nbsp;Markus Apel,&nbsp;Reza Najian Asl,&nbsp;Shahed Rezaei","doi":"10.1002/nme.70380","DOIUrl":"https://doi.org/10.1002/nme.70380","url":null,"abstract":"<p>We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics-informed neural operator based on conditional neural fields or Fourier neural operators is trained to approximate the nonlinear parametric solution of the governing equations. This establishes a continuous mapping between the parameter and solution spaces, which can then be evaluated for a given parameter at any spatial resolution. Second, since the neural approximation may not be exact, it is subsequently refined using a Newton-based correction initialized by the neural output. To evaluate the effectiveness of this hybrid approach, we compare three solution strategies: (i) the standard Newton-Raphson solver used in NFEM, which is robust and accurate but computationally demanding; (ii) physics-informed neural operators, which provide rapid inference but may lose accuracy outside the training distribution and resolution; and (iii) the neural-initialized Newton (NiN) strategy, which combines the efficiency of neural operators with the robustness of NFEM. The results demonstrate that the proposed hybrid approach reduces computational cost while preserving accuracy. This effect becomes even more pronounced when correcting solutions at significantly higher resolutions, effectively enabling a reduced-order modelling strategy during the training phase without loss of accuracy after inference. The conducted studies are intentionally designed to cover a broad spectrum of problems, including variations in material properties, boundary conditions, complex geometries, and nonlinear multi-physics scenarios, thereby demonstrating the robustness and general applicability of the approach.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70380","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616545","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":"A Matrix-Free, Scalable, and Robust Implicit Material Point Method","authors":"Sam J. V. Sutcliffe,&nbsp;William M. Coombs,&nbsp;Wangcheng Zhang,&nbsp;Ravindra Duddu,&nbsp;Giuliano Pretti,&nbsp;Charles E. Augarde","doi":"10.1002/nme.70384","DOIUrl":"https://doi.org/10.1002/nme.70384","url":null,"abstract":"<p>An implicit material point method is presented using a new dynamic relaxation solver, allowing for a simple and efficient quasi-static formulation that may be readily implemented in existing explicit-dynamic codes. The use of dynamic relaxation and aggregation is shown to produce an extremely robust material point method scheme, with good convergence characteristics across a range of challenging problems. A new simple fictitious mass matrix formulation based on estimated wave-speed is derived and presented, allowing for the trivial restructuring of explicit dynamic codes to implicit to solve quasi-static problems. Finally, a range of large deformation quasi-static problems, both elastic and elasto-plastic, are used to demonstrate the convergence characteristics of the method.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70384","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148616045","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 Energy Method for Large Deformation Analysis of Isotropic and Inhomogeneous Hyperelastic Ellipsoidal Pressurized Structures","authors":"Nasser Firouzi,&nbsp;Shaofan Li,&nbsp;Timon Rabczuk","doi":"10.1002/nme.70366","DOIUrl":"https://doi.org/10.1002/nme.70366","url":null,"abstract":"&lt;p&gt;The accurate prediction of displacement and stress fields in pressure vessels is essential for the safe and reliable design of these structures, particularly when dealing with nonlinear behavior such as that of hyperelastic functionally graded materials (FGMs). Conventional finite element method (FEM) approaches, while robust, often require extensive meshing and may encounter challenges in graded or strongly nonlinear domains. In this work, we present a Deep Energy Method (DEM) framework, based on physics-informed neural networks, to analyze the axisymmetric deformation of ellipsoidal vessels subjected to internal and external pressure loading. The formulation employs automatic differentiation and a hyperelastic constitutive model to evaluate the strain-energy functional, with pressure boundary conditions incorporated through natural work terms. Both homogeneous and exponentially graded materials are studied, enabling direct assessment of how material gradation influences the structural response. Three neural network architectures with varying depth and width are examined to evaluate the effect of model complexity on accuracy and convergence. The DEM solution is obtained by minimizing the total potential energy, comprising the internal strain energy and pressure work, to train a neural network representation of the displacement field. Convergence analyses confirm stable optimization for all cases, with graded vessels showing fast and smooth loss reduction due to the stabilizing effect of material gradation. DEM predictions of displacement agree closely with FEM benchmarks in both expansion-type (internal pressure) and contraction-type (external pressure) scenarios, accurately capturing global deformation modes as well as local responses near the inner and outer boundaries. To further ensure numerical robustness, a Monte Carlo sampling strategy is also employed for a selected test case, yielding displacement and von Mises stress fields fully consistent with structured-grid DEM and FEM solutions. von Mises stress fields derived from DEM match FEM references for both homogeneous and graded vessels. While stress concentrations appear near loaded boundaries in homogeneous materials, gradation leads to smoother stress transitions, and DEM successfully reproduces these patterns. Qualitative comparisons and magnitude-wise agreement across all network architectures demonstrate the method's capability to reflect nonlinear constitutive behavior and spatial material variation. The findings highlight several advantages of DEM: it provides smooth mesh-free solutions, naturally accommodates material gradation, and embeds the variational energy principle directly within the learning process. The method shows strong potential for extension to more complex geometries and nonlinear multiphysics problems, although sensitivity to optimization parameters and training cost remain important considerations. It also offers a powerful and reliable alternative to FEM for","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70366","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615652","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":"Numerical Anisotropy and Band-Gap Formation in Finite Element Wave Propagation: Analysis and Mitigation Using Spline-Based Elements","authors":"Wiktor Waszkowiak,&nbsp;Arkadiusz Żak,&nbsp;Łukasz Doliński,&nbsp;Paweł Kowalski","doi":"10.1002/nme.70397","DOIUrl":"https://doi.org/10.1002/nme.70397","url":null,"abstract":"<p>Numerical anisotropy and dispersion are inherent consequences of spatial discretization in finite element modeling of wave propagation. In two-dimensional problems, these effects manifest not only as direction-dependent phase and group velocities but also as angularly dependent numerical band gaps that may entirely suppress wave propagation. Such phenomena are often overlooked when dispersion characteristics are examined only along a limited set of symmetry directions. This paper presents a systematic analysis of numerical anisotropy in two-dimensional wave propagation models, focusing on the spectral consequences of discretization-induced discontinuities. Dispersion relations are examined using Bloch-type formulations combined with mode assignment, extended Brillouin zone representations, and angular mapping, enabling a consistent interpretation of multimodal and directional effects. A spline-based finite element formulation (spFEM) is proposed and extended from 1-D to 2-D problems. The method employs piecewise polynomial shape functions with enforced higher-order global continuity, increasing the information carried by each node and reducing inter-element derivative discontinuities. The performance of spFEM is assessed against classical FEM and a time-domain spectral finite element method (TD-SFEM) using Reissner–Mindlin plate models with comparable numbers of degrees of freedom. The results show that classical FEM and TD-SFEM formulations exhibit pronounced numerical anisotropy and direction-dependent band gaps, severely limiting their usable frequency ranges. In contrast, the proposed spFEM formulation significantly suppresses these artefacts and provides a substantially wider usable frequency range at comparable computational cost. The study demonstrates that analyzing numerical behavior beyond nominal applicability limits is essential for understanding error mechanisms and for the systematic design of improved finite element formulations.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70397","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615651","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":"A Hierarchical Deep Symbolic AI Method for Closed-Form, Interpretable Constitutive Modeling","authors":"Haozhe Yu,&nbsp;Mohammad Madadi,&nbsp;Pu Zhang,&nbsp;Lin Cheng","doi":"10.1002/nme.70387","DOIUrl":"https://doi.org/10.1002/nme.70387","url":null,"abstract":"<p>Recent advances in advanced manufacturing have enabled the development of multifunctional materials such as soft composites, whose complex nonlinear behaviors demand accurate and interpretable constitutive modeling. Traditional phenomenological formulations rely heavily on assumed functional forms and expert intuition, requiring extensive trial and error and offering limited generality. In contrast, data-driven approaches based on neural networks provide strong predictive capability but often require large datasets and lack interpretability, stability, and physical consistency. To address these challenges, we propose a hierarchical deep symbolic artificial intelligence (HDSAI) method for discovering constitutive laws directly from data while providing closed-form analytical expressions. The method constructs constitutive relations hierarchically in a layer-by-layer manner using symbolic mathematical operators as building blocks. Each layer combines and transforms the outputs from the previous one, progressively enriching functional complexity while maintaining analytical transparency. A layer-wise evolutionary and selection algorithm is introduced to optimize the symbolic operators within and across layers, efficiently balancing model accuracy and parsimony. The HDSAI method offers three key advantages: (1) deeper symbolic composition while avoiding uncontrolled symbolic-tree explosion; (2) flexible control of model complexity via progressive symbolic evolution; and (3) enhanced stability and generalization compared with existing symbolic regression and neural-network-based models. Benchmark studies against state-of-the-art symbolic regression including GPTIPS, PySR, and LoHM methods demonstrate that the proposed HDSAI method not only achieves higher numerical accuracy and stability but also successfully rediscovers the exact ground-truth constitutive expressions that existing methods fail to capture. This capability underscores the effectiveness of the hierarchical architecture in navigating complex search spaces and preserving physically interpretable functional forms. Systematic investigations of key parameters, including population size, layer depth, data availability, and noisy level of the data, further reveal that the hierarchical design enhances convergence robustness and expressiveness, enabling HDSAI to efficiently identify compact, physically meaningful constitutive laws with superior interpretability and generalization.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 14","pages":""},"PeriodicalIF":3.3,"publicationDate":"2026-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70387","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148615487","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}