International Journal for Numerical Methods in Engineering最新文献

{"title":"Quantum Time-Marching Algorithms for Solving Linear Transport Problems Including Boundary Conditions","authors":"Sergio Bengoechea,&nbsp;Paul Over,&nbsp;Thomas Rung","doi":"10.1002/nme.70326","DOIUrl":"https://doi.org/10.1002/nme.70326","url":null,"abstract":"<p>This article presents the first complete application of a quantum time-marching algorithm for simulating multidimensional linear transport phenomena with arbitrary boundaries, whereby the success probabilities are problem intrinsic. The method adapts the linear combination of unitaries algorithm to block encode the diffusive dynamics, while arbitrary boundary conditions are enforced by the method of images only at the cost of one additional qubit per spatial dimension. As an alternative to the nonperiodic reflection, the direct encoding of Neumann conditions by the unitary decomposition of the discrete time-marching operator is proposed. All presented algorithms indicate optimal success probabilities while maintaining linear time complexity, thereby securing the practical applicability of the quantum algorithm on fault-tolerant quantum computers. The proposed time-marching method is demonstrated through state-vector simulations of the heat equation in combination with Neumann, Dirichlet, and mixed boundary conditions, showing excellent agreement with classical finite differences.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 8","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70326","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683473","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":"Multiple Time-Scale Homogenization of Coupled Corrosion-Fatigue in Structural Concrete","authors":"Manikandan Gopakumar,&nbsp;Abedulgader Baktheer,&nbsp;Ghandi Kenjo,&nbsp;Fadi Aldakheel","doi":"10.1002/nme.70324","DOIUrl":"https://doi.org/10.1002/nme.70324","url":null,"abstract":"<p>Reinforced concrete structures exposed to chloride-rich environments and cyclic mechanical loading experience simultaneous corrosion of steel reinforcement and fatigue-induced concrete cracking, leading to complex, nonlinear degradation that cannot be accurately captured by conventional sequential analyses. This work presents a <i>fully coupled multiphysics</i> phase-field framework based on a multiple time-scale homogenization strategy, which models the co-evolution of corrosion, chloride transport, fatigue damage, and fracture in both concrete and steel, explicitly capturing the mutual interactions between chemical and mechanical degradation across distinct temporal scales. Unlike traditional approaches, the model resolves feedback mechanisms in which corrosion accelerates fatigue by weakening the steel-concrete interface and inducing microcracks, while cyclic loading enhances chloride ingress and promotes corrosion progression, effects that are difficult to observe experimentally. Numerical studies, including two-dimensional simulations of representative rebar configurations and a three-dimensional beam structure, demonstrate how the homogenized treatment of fast fatigue cycles and slow corrosion processes enables efficient and consistent prediction of degradation, and how the timing, rate, and sequence of cyclic loading relative to corrosion govern crack initiation, corrosion kinetics, and fatigue lifetime. Results show that conventional corrosion-followed-by-fatigue approaches systematically underestimate service life, whereas the proposed multiple time-scale, fully coupled framework provides accurate, physics-based predictions of degradation. This highlights the critical importance of modeling corrosion and fatigue as mutually interacting processes within a unified time-scale homogenization framework and offers new insights into the spatio-temporal interplay between cracking, transport, and corrosion in structural concrete.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70324","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683290","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":"Multiple Time-Scale Homogenization of Coupled Corrosion-Fatigue in Structural Concrete","authors":"Manikandan Gopakumar,&nbsp;Abedulgader Baktheer,&nbsp;Ghandi Kenjo,&nbsp;Fadi Aldakheel","doi":"10.1002/nme.70324","DOIUrl":"10.1002/nme.70324","url":null,"abstract":"<p>Reinforced concrete structures exposed to chloride-rich environments and cyclic mechanical loading experience simultaneous corrosion of steel reinforcement and fatigue-induced concrete cracking, leading to complex, nonlinear degradation that cannot be accurately captured by conventional sequential analyses. This work presents a <i>fully coupled multiphysics</i> phase-field framework based on a multiple time-scale homogenization strategy, which models the co-evolution of corrosion, chloride transport, fatigue damage, and fracture in both concrete and steel, explicitly capturing the mutual interactions between chemical and mechanical degradation across distinct temporal scales. Unlike traditional approaches, the model resolves feedback mechanisms in which corrosion accelerates fatigue by weakening the steel-concrete interface and inducing microcracks, while cyclic loading enhances chloride ingress and promotes corrosion progression, effects that are difficult to observe experimentally. Numerical studies, including two-dimensional simulations of representative rebar configurations and a three-dimensional beam structure, demonstrate how the homogenized treatment of fast fatigue cycles and slow corrosion processes enables efficient and consistent prediction of degradation, and how the timing, rate, and sequence of cyclic loading relative to corrosion govern crack initiation, corrosion kinetics, and fatigue lifetime. Results show that conventional corrosion-followed-by-fatigue approaches systematically underestimate service life, whereas the proposed multiple time-scale, fully coupled framework provides accurate, physics-based predictions of degradation. This highlights the critical importance of modeling corrosion and fatigue as mutually interacting processes within a unified time-scale homogenization framework and offers new insights into the spatio-temporal interplay between cracking, transport, and corrosion in structural concrete.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70324","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683346","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":"An Explicit, Compact, and Rotation-Agnostic Formulation for Equivalent Nodal Loads for Point-Loaded Beams","authors":"Alexander R. Schock,&nbsp;Mark Reckzin,&nbsp;Robert G. Langlois","doi":"10.1002/nme.70312","DOIUrl":"10.1002/nme.70312","url":null,"abstract":"<p>In time-marching dynamical simulations, treatment of contact forces in deformable bodies represented by finite element meshes requires a compromise between simulation fidelity and computational costs. External forces directly evaluated at the mesh nodes offer better computational performance at the cost of modelling fidelity. Alternatively, externally applied span-wise member loads can be distributed to the mesh nodes through the element's shape functions. The shape functions enable the development of nodal forces and torques that produce consistent deformations to a load applied along the span of the element. For deformable bodies, represented using finite element methods, which undergo large gross translational and angular motions in time-marching simulations, constant reorientation of the shape function matrices is required for each evaluation of force distribution. This can be computationally expensive for larger and stiffer systems. In this work, a compact and orientation-agnostic ‘equivalent’ formulation for expressing nodal loads is presented. The formulation is compared against the consistent formulation for nodal forces of an arbitrarily oriented Euler-Bernoulli beam. Isolated benchmarking tests and sample application tests indicate nearly identical force distributions in time-marching simulations for the consistent and equivalent formulations. Evaluated computational performance metrics reveal that the equivalent formulation results in run-time performance gains. However, the significance of the gains is proportional to the fraction of computational workload associated with the force distribution.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70312","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683252","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":"An Explicit, Compact, and Rotation-Agnostic Formulation for Equivalent Nodal Loads for Point-Loaded Beams","authors":"Alexander R. Schock,&nbsp;Mark Reckzin,&nbsp;Robert G. Langlois","doi":"10.1002/nme.70312","DOIUrl":"https://doi.org/10.1002/nme.70312","url":null,"abstract":"<p>In time-marching dynamical simulations, treatment of contact forces in deformable bodies represented by finite element meshes requires a compromise between simulation fidelity and computational costs. External forces directly evaluated at the mesh nodes offer better computational performance at the cost of modelling fidelity. Alternatively, externally applied span-wise member loads can be distributed to the mesh nodes through the element's shape functions. The shape functions enable the development of nodal forces and torques that produce consistent deformations to a load applied along the span of the element. For deformable bodies, represented using finite element methods, which undergo large gross translational and angular motions in time-marching simulations, constant reorientation of the shape function matrices is required for each evaluation of force distribution. This can be computationally expensive for larger and stiffer systems. In this work, a compact and orientation-agnostic ‘equivalent’ formulation for expressing nodal loads is presented. The formulation is compared against the consistent formulation for nodal forces of an arbitrarily oriented Euler-Bernoulli beam. Isolated benchmarking tests and sample application tests indicate nearly identical force distributions in time-marching simulations for the consistent and equivalent formulations. Evaluated computational performance metrics reveal that the equivalent formulation results in run-time performance gains. However, the significance of the gains is proportional to the fraction of computational workload associated with the force distribution.</p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"127 7","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.70312","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147683258","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":"An Optimized Algorithm for the Mitigation of Numerical Oscillations in Engineering Simulations: A Case Study in Grain Aeration Modeling","authors":"Daniel Rigoni,&nbsp;Marcio Augusto Villela Pinto,&nbsp;Jotair Elio Kwiatkowski Jr.","doi":"10.1002/nme.70304","DOIUrl":"10.1002/nme.70304","url":null,"abstract":"<div>\u0000 \u0000 <p>Spurious oscillations are a recurring challenge in numerical simulations of advection-dominated transport, often degrading stability and predictive accuracy. Artificial viscosity is commonly employed to mitigate these effects, but its coefficient is usually tuned empirically, limiting reproducibility and scalability. This study introduces a predictive framework in which the viscosity coefficient is derived analytically from discretization parameters through a closed-form law obtained via offline optimization guided by a smoothness metric. The methodology is demonstrated for grain aeration, a coupled heat-moisture transport problem of high practical relevance. The mathematical model was solved using finite differences with the Leith scheme, known for enhanced robustness under realistic aeration conditions. Verification based on apparent order-of-convergence analysis of the discretization error confirmed that the stabilized formulation recovered second-order accuracy, while the unstabilized model exhibited order degradation. Validation against experimental data showed accuracy comparable to manual calibration but with greater stability. Smoothness analysis revealed oscillations only in the energy balance, with mass-balance equations remaining naturally smooth. Once trained, the predictive law added negligible computational cost (3.5 s per run vs. 162.1 s for mesh refinement - a well-known technique for reducing oscillations in numerical solutions). The approach eliminates empirical tuning, ensures convergence under experimental conditions, and achieves substantial computational savings for coupled transport problems.</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":"147579884","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 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":"147615320","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":"Efficient Energy-Stable Discontinuous Galerkin Scheme for the Non-Isothermal Cahn–Hilliard–Navier–Stokes Two-Phase Fluid Flow System","authors":"Guang-An Zou,&nbsp;Meiting Wang,&nbsp;Kejia Pan,&nbsp;Yin Yang,&nbsp;Xiaofeng Yang","doi":"10.1002/nme.70319","DOIUrl":"10.1002/nme.70319","url":null,"abstract":"<p>In this article, we propose a novel numerical framework for the non-isothermal Cahn–Hilliard–Navier–Stokes two-phase flow system, which couples the incompressible Navier–Stokes equations, the Cahn–Hilliard phase-field equation, and the heat transport equation to capture temperature-dependent two-phase flow dynamics. The proposed scheme achieves three major advances: (i) unconditional energy stability through a combined scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) approach, (ii) linearity and full decoupling of all variables while using a second-order temporal discretization, and (iii) efficient implementation via discontinuous Galerkin (DG) spatial discretization together with a second-order projection method for the Navier–Stokes equations. We rigorously prove the unconditional energy stability of the scheme and present key details of its decoupled implementation. Extensive 2D and 3D simulations, including droplet deformation, bubble coalescence, and interfacial instabilities in stratified binary fluids, are presented to demonstrate the accuracy, efficiency, and robustness of the proposed numerical method, thereby confirming its effectiveness for energy-stable simulation of non-isothermal two-phase incompressible flows.</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.70319","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579737","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":"Robust Inverse Material Design With Physical Guarantees Using the Voigt-Reuss Net","authors":"Sanath Keshav,&nbsp;Felix Fritzen","doi":"10.1002/nme.70296","DOIUrl":"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":"147579880","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 Model Reduction of Multi-Scale Electrochemical Ion Transport","authors":"Vinh Tu,&nbsp;Fredrik Larsson,&nbsp;Kenneth Runesson,&nbsp;Ralf Jänicke","doi":"10.1002/nme.70313","DOIUrl":"10.1002/nme.70313","url":null,"abstract":"<p>In this paper, we develop a Numerical Model Reduction (NMR) framework for multi-scale modeling of electro-chemically coupled ion transport. Upon introducing the governing equations and employing Variationally Consistent Homogenization, a two-scale model, consisting of a macro-scale and a sub-scale part, is obtained. Instead of solving for the computationally expensive FE² simulation, where the macro-scale and sub-scale problems are solved in a nested fashion, we exploit NMR by training a surrogate model that replaces the sub-scale finite element simulations. The surrogate model is trained by performing Proper Orthogonal Decomposition on snapshots of the primary fields. Each macro-scale quadrature point is no longer occupied by a Representative Volume Element simulation; instead, it is replaced by a surrogate model that consists of a system of Ordinary Differential Equations. In this way, a computationally efficient solution scheme for solving two-scale problems is obtained.</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.70313","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147579881","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}