International Journal for Numerical Methods in Engineering最新文献

{"title":"A Generalized Peridynamic Model Based on Seth–Hill Bond-Strain Measures for Mixed-Mode Fracture","authors":"NingTao Wang,&nbsp;Hao Yu,&nbsp;HanWei Huang,&nbsp;WenLong Xu,&nbsp;YinBo Zhu,&nbsp;HengAn Wu","doi":"10.1002/nme.70316","DOIUrl":"https://doi.org/10.1002/nme.70316","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, a generalized peridynamic model incorporating Seth–Hill bond-strain measures is proposed to capture mixed-mode fracture behaviors. We begin with the reformulation of the model introduced by Tupek and Radovitzky within the ordinary state-based peridynamic (OSB-PD) framework, where we demonstrate that the three-dimensional shape tensor state satisfies an integral identity equivalent to the fourth-order symmetric identity tensor. Based on this identity, the shape tensor state tailored for two-dimensional problems is constructed, enabling the derivation of the corresponding scalar force state based on Seth–Hill bond-strain measures for linear elastic materials. This generalized model avoids unphysical material interpenetration and enables the decomposition of the scalar force state over the classic model. Moreover, a nonlocal work-conjugate stress tensor is developed for the first time by employing the reformulated scalar force state based on the principle of work conjugacy and the integral identity of the shape tensor state. Finally, the maximum principal stress and Drucker–Prager failure criteria are incorporated into the generalized OSB-PD framework to enable the simulation of mixed-mode brittle fracture. The accuracy and robustness of the proposed model are validated through several benchmark cases, demonstrating accurate stress evaluation and failure prediction. Notably, the model successfully captures complex crack coalescence patterns in rock subjected to uniaxial compression, underscoring its effectiveness in depicting mixed-mode fracture processes.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615238","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":"Immersed Boundary Shell Element Based on Higher Order Shear Deformation Theories for Laminated Composite Shells","authors":"Sichen Liu,&nbsp;Hui Ouyang,&nbsp;Ashok V. Kumar","doi":"10.1002/nme.70321","DOIUrl":"10.1002/nme.70321","url":null,"abstract":"<div>\u0000 \u0000 <p>A mesh that conforms to the geometry is needed for the traditional Finite Element Method (FEM), which is often difficult to generate for complex geometry. It is easier to generate a uniform Cartesian background mesh in which the geometry is embedded. The immersed boundary finite element method (IBFEM) uses a background mesh to approximate the solution while using equations to accurately represent the geometry for finite element analysis. For shell-like structures, a surface representing the mid-surface of the shell is utilized, which passes through the 3D elements of the background mesh. In this article, Higher-Order Shear Deformation Theories (HSDT) are modified for use with immersed boundary shell elements to model laminated composite shells. The rotations and slopes are expressed as derivatives of the displacement field approximated over 3D elements of a background mesh. This requires the displacement field to be tangent-continuous. Therefore, quadratic or higher-order B-spline shape functions that ensure a tangent-continuous displacement field are utilized to formulate the element. Only three degrees of freedom per node are needed for the immersed boundary shell elements. The shell element presented here is formulated assuming that the strains are infinitesimal. For validating the elements and the modified higher order shear deformation theories, several commonly used laminated composite benchmark problems are used to assess its accuracy. The results obtained using this approach are compared with analytical solutions or with solutions from 3D FEA and isogeometric analysis (IGA) for validation.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579882","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":"An Adaptive Numerical Framework for the Electro-Chemical Transport in Direct Internal Reforming Solid Oxide Fuel Cells (DIR-SOFC)","authors":"Albert Costa-Solé,&nbsp;Abel Gargallo-Peiró,&nbsp;Adrià Quintanas-Corominas,&nbsp;Antonio G. Sabato,&nbsp;Marc Torrell,&nbsp;Albert Tarancón,&nbsp;Daniel Mira","doi":"10.1002/nme.70310","DOIUrl":"10.1002/nme.70310","url":null,"abstract":"<div>\u0000 \u0000 <p>This work presents a discretization strategy based on the Continuous Galerkin (CG) formulation with a new Adaptative Mesh Refinement (AMR) strategy to simulate Direct Steam Internal Reforming Solid Oxide Fuel Cells (DIR-SOFC). The governing equations used in the proposed methodology are based on Ohm's law for the electrochemical potentials at each conductive phase and the Dusty Gas Model (DGM) for species diffusion, assuming uniform and constant temperature and pressure. The reactions induced in the different regions of the domain (electrolyte, anode, and cathode) result in sharp variations of the solution on the different interfaces that need to be properly captured. Thus, to enhance the simulation accuracy and efficiency, a new AMR strategy is proposed, refining the mesh according to the error estimators of the potentials. Additionally, a novel iterative approach to accurately approximate the concentration polarization in the anode by employing the Bulter–Volmer equations is also presented for the Direct Internal Reforming (DIR). Finally, we present several examples to validate and assess the capabilities of this formulation and validate the numerical results with reference examples. Furthermore, a parametric analysis is conducted to investigate the impact of temperature and cell current on potentials, chemical composition, and fuel cell efficiency.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579885","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":"A New Shape Sensing Approach for Multilayer Composites Beams With Variable Cross-Section Under Nonlinear Deformations","authors":"Tao Jiang,&nbsp;Xiao-fan Sun,&nbsp;Jie-zhong Huang,&nbsp;Dong-sheng Li,&nbsp;Chun-xu Qu,&nbsp;Lulu Chen","doi":"10.1002/nme.70323","DOIUrl":"10.1002/nme.70323","url":null,"abstract":"<div>\u0000 \u0000 <p>Composite materials are widely used in engineering structures such as aircraft, wind turbine blades, and bridges because of their light weight, high strength, and fatigue resistance. Owing to their use in complex working environments and heavy load conditions, these multilayer composite variable-section structures are very prone to geometric nonlinear deformations, which can affect their working performance and even lead to damage. Shape sensing is an effective approach for monitoring the deformation of these composite structures in practical applications. However, existing shape sensing methods are not capable of simultaneously addressing geometric nonlinearity, variable cross-section effects, and multilayer composite coupling, which are essential for accurate reconstruction of practical large-scale structures. Therefore, this paper proposed a new method—the rotation angle approximation (RAA) algorithm for shape sensing applications. This method can calculate the axial angle by constructing a least-squares error functional between the theoretical and actual curvatures, thereby reconstructing the deformed shape of the multilayer composite variable-section beam. The accuracy and feasibility of the method are validated through numerical simulations. Specifically, a high-fidelity 16-MW wind turbine blade model was built. The shear web, a representative multilayer composite beam with variable cross-section, was selected to further demonstrate the algorithm's capability in reconstructing large deformations of complex real-world structures. The results show that the RAA can accurately predict the deformed shapes of multilayer composite variable-section beams.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579735","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":"Robust Inverse Material Design With Physical Guarantees Using the Voigt-Reuss Net","authors":"Sanath Keshav,&nbsp;Felix Fritzen","doi":"10.1002/nme.70296","DOIUrl":"https://doi.org/10.1002/nme.70296","url":null,"abstract":"&lt;p&gt;We apply the Voigt-Reuss net, a spectrally normalized neural surrogate introduced in [38], for forward and inverse mechanical homogenization with a key guarantee that all predicted effective stiffness tensors satisfy Voigt-Reuss bounds in the Löwner sense during training, inference, and gradient-driven optimization. The approach operates in a bounded spectral space by reparametrizing each effective tensor relative to its Voigt and Reuss bounds, ensuring that all outputs reside within a unit-cube domain and are mapped back via a deterministic inverse transform to physically admissible tensors. For 3D elasticity, a fully connected Voigt-Reuss net is trained on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∼&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;.&lt;/mo&gt;\u0000 &lt;mn&gt;18&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ sim 1.18 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; million high-fidelity homogenization labels of stochastic biphasic microstructures. The model ingests 236 image-derived morphological descriptors and phase parameters that encode bulk and shear moduli, enabling a single surrogate to represent material combinations spanning 4 orders of magnitude. Due to the rotational invariance of the descriptor set, the surrogate recovers the isotropic projection of the effective stiffness (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;.&lt;/mo&gt;\u0000 &lt;mn&gt;998&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {R}^2ge 0.998 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for isotropy-related components). However, anisotropy-revealing entries remain unlearnable from the available features. At the tensor level, the relative Frobenius error exhibits a median of approximately 1.8% (mean approximately 3.6%) and approaches the irreducible isotropic-projection floor, far outperforming all alternative surrogates considered. In 2D plane-strain elasticity, spectral normalization is integrated with a differentiable microstructure renderer and a convolutional regressor, yielding a surrogate that maps generator parameters to effective stiffness tensors for highly anisotropic and high-contrast composites. Voigt-Reuss net is compared against vanilla and Cholesky regressors trained with identical architectures, data, and training procedures. The unconstrained surrogates frequently violate Voigt/Reuss bounds and even positive definiteness, whereas Voigt-Reuss net produces no violations by design and also enables robust inverse design. Utilizing the surrogate with b","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70296","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579819","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":"Application of Group-Theoretical Approaches in Structural Natural Frequency Analyses","authors":"Shiyao Sun,&nbsp;Kapil Khandelwal","doi":"10.1002/nme.70308","DOIUrl":"https://doi.org/10.1002/nme.70308","url":null,"abstract":"<p>Group theory has profoundly advanced physics and chemistry in systems with symmetries. Yet its use in structural engineering applications has not yet been fully explored beyond the aesthetics of symmetric designs. This work addresses two significant gaps that have limited the broader adoption of group-theoretic methods in structural vibration analysis and clarifies their implications for structural design when multiple eigenvalues arise. First, a problem-independent approach is presented with detailed derivations for constructing group representations for symmetric structures directly from the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$$ G $$</annotation>\u0000 </semantics></math>-invariance for structural vibration analysis. This method applies effectively to both dihedral groups and the higher-order Platonic groups, including tetrahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>d</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {T}_d $$</annotation>\u0000 </semantics></math>), octahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>O</mi>\u0000 <mi>h</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {O}_h $$</annotation>\u0000 </semantics></math>), and icosahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>h</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {I}_h $$</annotation>\u0000 </semantics></math>) symmetries. The method used in this work scales well with structural complexity and enables both explicit and canonical block diagonalizations. Second, this work provides a comprehensive guide to applying finite-point-group representations in structural vibration analysis and proves that the dimensions of irreducible representations determine eigenfrequency multiplicities. Although no optimization is performed in this work, this theoretical result has direct implications for structural optimization, resolving longstanding misconceptions about the coalescence of eigenvalues by showing that symmetry is the origin of repeated eigenfrequencies. The theoretical developments are validated on truss structures with dihedral and higher-order symmetries, accurately predicting their eigenfrequency distributions.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70308","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615321","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 Scalable Reduced-Order Model for the Steady Navier–Stokes Equations","authors":"Seung Whan Chung,&nbsp;Siu Wun Cheung,&nbsp;Youngsoo Choi,&nbsp;Pratanu Roy,&nbsp;Thomas Roy,&nbsp;Tiras Y. Lin,&nbsp;Du T. Nguyen,&nbsp;Christopher Hahn,&nbsp;Eric B. Duoss,&nbsp;Sarah E. Baker","doi":"10.1002/nme.70314","DOIUrl":"10.1002/nme.70314","url":null,"abstract":"<div>\u0000 \u0000 <p>Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, the component reduced order modeling method has been developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has only been demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier–Stokes flow, with application to general nonlinear physics in view. The large-scale, global domain is decomposed into a combination of small-scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected from each small-scale unit component. Velocity bases are augmented with a pressure supremizer to satisfy the inf–sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: A third-order tensor projection operator and the empirical quadrature procedure. The proposed method is demonstrated on the flow over arrays of five different unit objects, achieving a 23-fold speedup with less than 4% relative error in domains up to 256 times larger than the unit components. Furthermore, a numerical experiment with the pressure supremizer strongly indicates the need for a supremizer for stable pressure prediction. A comparison between the tensorial approach and the empirical quadrature procedure revealed a slight advantage of the empirical quadrature procedure. The framework is compared with an alternating Schwarz-based reduced-order approach, demonstrating improved efficiency and robustness for the DG-based global solver while retaining flexibility for sub-scale iterative solvers. The method is further extended to a coupled advection–diffusion and Navier–Stokes system, illustrating its applicability to multi-physics problems and its potential for more general, inter-coupled nonlinear systems.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579883","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":"Deep Collocation Method: A Numerical Algorithm Combining Reversible Neural Network and Spectral Differentiation for Solving the Incompressible Navier–Stokes Equations","authors":"Ruilin Chen,&nbsp;Guihui Ma,&nbsp;Ming Fang","doi":"10.1002/nme.70317","DOIUrl":"10.1002/nme.70317","url":null,"abstract":"<div>\u0000 \u0000 <p>Neural networks have the potential in approximating the solutions of the Navier–Stokes equations. To address the high computational complexity of high-order derivatives based on automatic differentiation and the high memory requirements of deep neural network training, a novel Deep Collocation Method (DCM) that combines reversible neural network (RevNet) and spectral differentiation is proposed. The memory-efficient RevNet is employed to approximate the solutions, while Chebyshev differentiation matrices are derived to rapidly and accurately compute the spatiotemporal derivatives of the approximate solutions, making the computational complexity of high-order derivatives comparable to that of the forward pass, and significantly reducing memory requirements. A unified spectral discretization scheme is applied in both temporal and spatial dimensions, enabling seamless integration of the neural network and the spatiotemporal Chebyshev differentiation matrices. It effectively incorporates global spatiotemporal information into the residual computation of the governing equations at each collocation point, enabling the neural network to actively account for nonlocal effects of the flow field. Validation results demonstrate that DCM performs well various benchmark flow problems and enables efficient training of deep models even on memory-constrained devices.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147585160","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":"Consistent PINNs for Higher-Order Elliptic PDEs","authors":"Shiv Mishra,&nbsp;Arbaz Khan","doi":"10.1002/nme.70320","DOIUrl":"10.1002/nme.70320","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we develop a consistent physics-informed neural networks (CPINNs) framework for higher-order elliptic PDEs, with emphasis on the biharmonic equation under nonhomogeneous Dirichlet boundary conditions. Unlike standard PINNs, whose loss functions are often inconsistent with the natural stability norms associated with higher-order differential operators and may therefore result in unstable or nonconvergent approximations, the proposed CPINNs formulation is grounded in a norm-equivalent discrete representation of the underlying variational problem. This structure-preserving design guarantees convergence in the natural stability norm <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>(</mo>\u0000 <mi>Ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {H}^2left(Omega right) $$</annotation>\u0000 </semantics></math>. Specifically, (i) we introduce a consistent loss functional <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℒ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>∗</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {mathcal{L}}^{ast } $$</annotation>\u0000 </semantics></math>, (ii) consistency proofs and norm equivalence between the continuous and discrete formulations, (iii) a priori error analysis, and (iv) convergence rates under Besov regularity assumptions using optimal recovery theory. Numerical experiments validate the theoretical predictions and demonstrate the superior performance of the CPINNs formulation over traditional loss designs.</p>\u0000 </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147585283","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":"Application of Group-Theoretical Approaches in Structural Natural Frequency Analyses","authors":"Shiyao Sun,&nbsp;Kapil Khandelwal","doi":"10.1002/nme.70308","DOIUrl":"https://doi.org/10.1002/nme.70308","url":null,"abstract":"<p>Group theory has profoundly advanced physics and chemistry in systems with symmetries. Yet its use in structural engineering applications has not yet been fully explored beyond the aesthetics of symmetric designs. This work addresses two significant gaps that have limited the broader adoption of group-theoretic methods in structural vibration analysis and clarifies their implications for structural design when multiple eigenvalues arise. First, a problem-independent approach is presented with detailed derivations for constructing group representations for symmetric structures directly from the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$$ G $$</annotation>\u0000 </semantics></math>-invariance for structural vibration analysis. This method applies effectively to both dihedral groups and the higher-order Platonic groups, including tetrahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>d</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {T}_d $$</annotation>\u0000 </semantics></math>), octahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>O</mi>\u0000 <mi>h</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {O}_h $$</annotation>\u0000 </semantics></math>), and icosahedral (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>h</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {I}_h $$</annotation>\u0000 </semantics></math>) symmetries. The method used in this work scales well with structural complexity and enables both explicit and canonical block diagonalizations. Second, this work provides a comprehensive guide to applying finite-point-group representations in structural vibration analysis and proves that the dimensions of irreducible representations determine eigenfrequency multiplicities. Although no optimization is performed in this work, this theoretical result has direct implications for structural optimization, resolving longstanding misconceptions about the coalescence of eigenvalues by showing that symmetry is the origin of repeated eigenfrequencies. The theoretical developments are validated on truss structures with dihedral and higher-order symmetries, accurately predicting their eigenfrequency distributions.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70308","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147615237","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}