Pub Date : 2026-01-09DOI: 10.1016/j.finel.2026.104512
D. Duhamel
Periodic media are widely studied for their industrial applications and unique ability to block wave propagation within certain frequency bands. For one-dimensional periodic systems, the Wave Finite Element (WFE) method efficiently computes dispersion relations and dynamic responses. Research on two-dimensional periodic structures extended this approach to periodicity along two directions, using finite element and reduction techniques such as Craig–Bampton and Bloch mode projection, including cases with damping and anisotropy. Beyond planar geometries, curved and helical periodic structures have been modeled with WFE and semi-analytical finite element methods to capture complex cyclic or screw symmetries for computing dispersion relations. Cylindrical configurations have also been explored, from simple vibration studies to wave propagation in layered or ribbed cylinders and metamaterial shells mainly for dispersion analysis or studies of infinite structures. As real structures are bounded, the present work focuses on finite elastic cylinders with double periodicity, using WFE to compute their dynamic response. Based on finite element matrices of a substructure, circumferential wavenumbers are imposed to obtain axial modes and responses as linear combination of modes. Numerical examples illustrate the method’s effectiveness for modeling finite, doubly periodic cylindrical systems such as homogeneous structures, structures with holes and finally structures with resonators. The low computing time of the present approach allows the consideration of structures with a large number of substructures.
{"title":"Computation of the elastodynamic response of finite doubly periodic cylinders by the wave finite element method","authors":"D. Duhamel","doi":"10.1016/j.finel.2026.104512","DOIUrl":"10.1016/j.finel.2026.104512","url":null,"abstract":"<div><div>Periodic media are widely studied for their industrial applications and unique ability to block wave propagation within certain frequency bands. For one-dimensional periodic systems, the Wave Finite Element (WFE) method efficiently computes dispersion relations and dynamic responses. Research on two-dimensional periodic structures extended this approach to periodicity along two directions, using finite element and reduction techniques such as Craig–Bampton and Bloch mode projection, including cases with damping and anisotropy. Beyond planar geometries, curved and helical periodic structures have been modeled with WFE and semi-analytical finite element methods to capture complex cyclic or screw symmetries for computing dispersion relations. Cylindrical configurations have also been explored, from simple vibration studies to wave propagation in layered or ribbed cylinders and metamaterial shells mainly for dispersion analysis or studies of infinite structures. As real structures are bounded, the present work focuses on finite elastic cylinders with double periodicity, using WFE to compute their dynamic response. Based on finite element matrices of a substructure, circumferential wavenumbers are imposed to obtain axial modes and responses as linear combination of modes. Numerical examples illustrate the method’s effectiveness for modeling finite, doubly periodic cylindrical systems such as homogeneous structures, structures with holes and finally structures with resonators. The low computing time of the present approach allows the consideration of structures with a large number of substructures.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"255 ","pages":"Article 104512"},"PeriodicalIF":3.5,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145929284","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}
Pub Date : 2026-01-08DOI: 10.1016/j.finel.2025.104500
Rafael Abreu , Cristian Mejia , Deane Roehl
Formulating robust integration algorithms for elastoplastic models is crucial for efficient and accurate numerical simulations of materials such as concrete, rocks, and soil. While traditional elastoplastic models typically employ single yield surfaces, more intricate behaviors can be captured using multiple yield surfaces. On the other hand, implementing these models within a finite element framework requires sophisticated numerical methods, particularly implicit integration schemes based on the backward Euler method, to ensure accuracy and stability. In this context, this paper introduces a novel Newton-Raphson-based implicit integration algorithm for multisurface plasticity models, accommodating both plane stress and three-dimensional stress conditions. The proposed algorithm employs well-established smooth complementary functions to handle multisurface plasticity without needing a scheme to identify active surfaces. In addition, this algorithm addresses plane stress plasticity by modifying calculations based on plane strain conditions. The study includes an assessment of the computational efficiency of three complementary functions, considering various finite element problems. The robustness of the algorithm is demonstrated through a series of numerical experiments, highlighting its potential for challenging engineering applications.
{"title":"Implicit numerical integration of multisurface plasticity for both plane stress and three-dimensional stress conditions","authors":"Rafael Abreu , Cristian Mejia , Deane Roehl","doi":"10.1016/j.finel.2025.104500","DOIUrl":"10.1016/j.finel.2025.104500","url":null,"abstract":"<div><div>Formulating robust integration algorithms for elastoplastic models is crucial for efficient and accurate numerical simulations of materials such as concrete, rocks, and soil. While traditional elastoplastic models typically employ single yield surfaces, more intricate behaviors can be captured using multiple yield surfaces. On the other hand, implementing these models within a finite element framework requires sophisticated numerical methods, particularly implicit integration schemes based on the backward Euler method, to ensure accuracy and stability. In this context, this paper introduces a novel Newton-Raphson-based implicit integration algorithm for multisurface plasticity models, accommodating both plane stress and three-dimensional stress conditions. The proposed algorithm employs well-established smooth complementary functions to handle multisurface plasticity without needing a scheme to identify active surfaces. In addition, this algorithm addresses plane stress plasticity by modifying calculations based on plane strain conditions. The study includes an assessment of the computational efficiency of three complementary functions, considering various finite element problems. The robustness of the algorithm is demonstrated through a series of numerical experiments, highlighting its potential for challenging engineering applications.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"255 ","pages":"Article 104500"},"PeriodicalIF":3.5,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145929283","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}
Pub Date : 2026-01-06DOI: 10.1016/j.finel.2025.104502
Loc V. Tran , Thang N. Dao , Vuong Nguyen Van Do
The shear deformation plate theory is applicable to moderate and thick plates by accounting for transverse shear effects. In an effort to optimize the number of unknowns, this study proposes a new form for shear-locking free Reissner-Mindlin plate theory that employs only a single variable–the bending deflection. Consequently, the governing equation is expressed as a fourth-order partial differential equation, retaining the same form as the classical Kirchhoff–Love theory, while fully considering shear deformation. Based on that, analytical solutions for transverse displacement are derived for rectangular plates with arbitrary slenderness ratios. Additionally, the weak form of the plate problem is derived and includes the second- and third-order derivatives. To address these higher-order continuity requirements, a conforming Galerkin method based on an isogeometric analysis (IGA) is adopted. In particular, the basis functions-based IGA with order p ≥ 3, naturally satisfy the C2-continuity requirement mandated by the proposed model. Moreover, these basis functions facilitate a straightforward enforcement of natural boundary conditions, such as prescribed slopes and curvatures, that are inherent in the present plate formulation. Numerical examples demonstrate that the proposed model, despite adopting a single unknown, provides highly accurate results for thin and thick plates and achieves high convergence rates for all quantities of interest.
{"title":"Isogeometric implementation of a single-variable shear deformable plate theory","authors":"Loc V. Tran , Thang N. Dao , Vuong Nguyen Van Do","doi":"10.1016/j.finel.2025.104502","DOIUrl":"10.1016/j.finel.2025.104502","url":null,"abstract":"<div><div>The shear deformation plate theory is applicable to moderate and thick plates by accounting for transverse shear effects. In an effort to optimize the number of unknowns, this study proposes a new form for shear-locking free Reissner-Mindlin plate theory that employs only a single variable–the bending deflection. Consequently, the governing equation is expressed as a fourth-order partial differential equation, retaining the same form as the classical Kirchhoff–Love theory, while fully considering shear deformation. Based on that, analytical solutions for transverse displacement are derived for rectangular plates with arbitrary slenderness ratios. Additionally, the weak form of the plate problem is derived and includes the second- and third-order derivatives. To address these higher-order continuity requirements, a conforming Galerkin method based on an isogeometric analysis (IGA) is adopted. In particular, the basis functions-based IGA with order <em>p</em> ≥ 3, naturally satisfy the <em>C</em><sup>2</sup>-continuity requirement mandated by the proposed model. Moreover, these basis functions facilitate a straightforward enforcement of natural boundary conditions, such as prescribed slopes and curvatures, that are inherent in the present plate formulation. Numerical examples demonstrate that the proposed model, despite adopting a single unknown, provides highly accurate results for thin and thick plates and achieves high convergence rates for all quantities of interest.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"255 ","pages":"Article 104502"},"PeriodicalIF":3.5,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145904202","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}
Pub Date : 2026-01-06DOI: 10.1016/j.finel.2025.104509
Christian Toderascu , Badadjida Wintiba , Karim Ehab Moustafa Kamel , Thierry J. Massart , Tine Tysmans
Textile Reinforced Cement (TRC) composites provide slender concrete material solutions. Using 3D textile reinforcements significantly improves the bending performance in the post-cracking stage compared to 2D textiles. Yet, no computational model takes into account explicitly the woven connections in 3D TRC. This contribution develops a novel strategy for generating the complex geometry of a mesoscale through-thickness Representative Volume Element (RVE) of 3D TRC shells, thanks to advanced geometric algorithms such as Rotation Minimising Frames (RMF), subsequently meshed and used in finite element (FE) analysis. The RVE with realistic reinforcement geometry enables the numerical evaluation of different fine scale processes contributing to the composite material performance, in particular the effect of the woven reinforcement architecture. RVE simulations under two types of bending loading conditions, using computational homogenisation procedures, illustrate that the proposed approach enables the investigation of the average macroscopic bending properties of 3D TRC based on the fine scale morphology of their reinforcement, together with an assessment of local strain fields.
{"title":"Modelling of 3D woven textile reinforced cement composites behaviour accounting for through-thickness reinforcement","authors":"Christian Toderascu , Badadjida Wintiba , Karim Ehab Moustafa Kamel , Thierry J. Massart , Tine Tysmans","doi":"10.1016/j.finel.2025.104509","DOIUrl":"10.1016/j.finel.2025.104509","url":null,"abstract":"<div><div>Textile Reinforced Cement (TRC) composites provide slender concrete material solutions. Using 3D textile reinforcements significantly improves the bending performance in the post-cracking stage compared to 2D textiles. Yet, no computational model takes into account explicitly the woven connections in 3D TRC. This contribution develops a novel strategy for generating the complex geometry of a mesoscale through-thickness Representative Volume Element (RVE) of 3D TRC shells, thanks to advanced geometric algorithms such as Rotation Minimising Frames (RMF), subsequently meshed and used in finite element (FE) analysis. The RVE with realistic reinforcement geometry enables the numerical evaluation of different fine scale processes contributing to the composite material performance, in particular the effect of the woven reinforcement architecture. RVE simulations under two types of bending loading conditions, using computational homogenisation procedures, illustrate that the proposed approach enables the investigation of the average macroscopic bending properties of 3D TRC based on the fine scale morphology of their reinforcement, together with an assessment of local strain fields.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"255 ","pages":"Article 104509"},"PeriodicalIF":3.5,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145903371","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}
Pub Date : 2026-01-02DOI: 10.1016/j.finel.2025.104505
Laura Rinaldi, Alvise Sommariva, Marco Vianello
We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.
{"title":"Effective numerical integration on complex shaped elements by discrete signed measures","authors":"Laura Rinaldi, Alvise Sommariva, Marco Vianello","doi":"10.1016/j.finel.2025.104505","DOIUrl":"10.1016/j.finel.2025.104505","url":null,"abstract":"<div><div>We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104505"},"PeriodicalIF":3.5,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884254","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}
Pub Date : 2025-12-31DOI: 10.1016/j.finel.2025.104508
{"title":"Editorial for the Special Issue on Digital Twins in Analysis and Design","authors":"","doi":"10.1016/j.finel.2025.104508","DOIUrl":"10.1016/j.finel.2025.104508","url":null,"abstract":"","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104508"},"PeriodicalIF":3.5,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145894406","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}
The goal of this paper is to develop an enriched beam model using the asymptotic expansion method, that takes into account localized bearing effects by incorporating kinematics that capture the influence of external forces and support conditions. A stiffness matrix is derived for this model and a force-based method is introduced to include additional degrees of freedom for the implementation of support conditions or for the use of stiffeners. The aim is to demonstrate that the proposed enriched beam model, which considers the effects of external force distribution and localized effects specifically those resulting from supports, can be as effective as a volumetric or shell model of the same problem. The formulation is tested on two thin-walled box-girder benchmarks. For an unstiffened girder, the enriched element reproduces shell-model deflections with a maximum error of 0.83%. When five diagonal struts are inserted, global displacements and strut axial forces remain within 7% and 6% of the reference shell solution, respectively. This approach offers significant computational efficiency while effectively handling complex configurations, including struts and stiffeners.
{"title":"Localized bearing effects using an enriched beam model","authors":"Laila Limni , Youssef Derrazi , Mohammed-Khalil Ferradi","doi":"10.1016/j.finel.2025.104507","DOIUrl":"10.1016/j.finel.2025.104507","url":null,"abstract":"<div><div>The goal of this paper is to develop an enriched beam model using the asymptotic expansion method, that takes into account localized bearing effects by incorporating kinematics that capture the influence of external forces and support conditions. A stiffness matrix is derived for this model and a force-based method is introduced to include additional degrees of freedom for the implementation of support conditions or for the use of stiffeners. The aim is to demonstrate that the proposed enriched beam model, which considers the effects of external force distribution and localized effects specifically those resulting from supports, can be as effective as a volumetric or shell model of the same problem. The formulation is tested on two thin-walled box-girder benchmarks. For an unstiffened girder, the enriched element reproduces shell-model deflections with a maximum error of 0.83%. When five diagonal struts are inserted, global displacements and strut axial forces remain within 7% and 6% of the reference shell solution, respectively. This approach offers significant computational efficiency while effectively handling complex configurations, including struts and stiffeners.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104507"},"PeriodicalIF":3.5,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884253","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}
Pub Date : 2025-12-27DOI: 10.1016/j.finel.2025.104506
Shahed Rezaei , Reza Najian Asl , Kianoosh Taghikhani , Ahmad Moeineddin , Michael Kaliske , Markus Apel
We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach unifies aforementioned methods and we can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities. These capabilities enable gradient-based optimization without the typical sensitivity analysis costs, unlike adjoint methods that scale directly with the number of response functions. Our Finite Operator Learning (FOL) approach originally employs feed-forward neural networks to directly map the discrete design space to the discrete solution space, and can alternatively be combined with existing physics-informed neural operator techniques to recover continuous solution fields, while avoiding the need for automatic differentiation when formulating the loss terms. The discretized governing equations, as well as the design and solution spaces, can be derived from any well-established numerical techniques. In this work, we employ the Finite Element Method (FEM) to approximate fields and their spatial derivatives. Thanks to the finite-element formulation, Dirichlet boundary conditions are satisfied by construction, and Neumann boundary conditions are naturally included in the FE residual through the weak form. Subsequently, we conduct Sobolev training to minimize a multi-objective loss function, which includes the discretized weak form of the energy functional, boundary conditions violations, and the stationarity of the residuals with respect to the design variables. Our study focuses on the heat equation and the mechanical equilibrium problem. First, we primarily address the property distribution in heterogeneous materials, where Fourier-based parameterization is employed to significantly reduce the number of design variables. Second, we explore changes in the source term in such PDEs. Third, we investigate the solution under different boundary conditions. In the context of gradient-based optimization, we examine the tuning of the microstructure’s heat transfer characteristics. Our technique also simplifies to an efficient matrix-free PDE solver that can compete with standard available solvers. This is demonstrated by solving a nonlinear thermal and mechanical PDE on a complex 3D geometry.
{"title":"Finite Operator Learning: Bridging neural operators and numerical methods for efficient parametric solution and optimization of PDEs","authors":"Shahed Rezaei , Reza Najian Asl , Kianoosh Taghikhani , Ahmad Moeineddin , Michael Kaliske , Markus Apel","doi":"10.1016/j.finel.2025.104506","DOIUrl":"10.1016/j.finel.2025.104506","url":null,"abstract":"<div><div>We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach unifies aforementioned methods and we can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities. These capabilities enable gradient-based optimization without the typical sensitivity analysis costs, unlike adjoint methods that scale directly with the number of response functions. Our Finite Operator Learning (FOL) approach originally employs feed-forward neural networks to directly map the discrete design space to the discrete solution space, and can alternatively be combined with existing physics-informed neural operator techniques to recover continuous solution fields, while avoiding the need for automatic differentiation when formulating the loss terms. The discretized governing equations, as well as the design and solution spaces, can be derived from any well-established numerical techniques. In this work, we employ the Finite Element Method (FEM) to approximate fields and their spatial derivatives. Thanks to the finite-element formulation, Dirichlet boundary conditions are satisfied by construction, and Neumann boundary conditions are naturally included in the FE residual through the weak form. Subsequently, we conduct Sobolev training to minimize a multi-objective loss function, which includes the discretized weak form of the energy functional, boundary conditions violations, and the stationarity of the residuals with respect to the design variables. Our study focuses on the heat equation and the mechanical equilibrium problem. First, we primarily address the property distribution in heterogeneous materials, where Fourier-based parameterization is employed to significantly reduce the number of design variables. Second, we explore changes in the source term in such PDEs. Third, we investigate the solution under different boundary conditions. In the context of gradient-based optimization, we examine the tuning of the microstructure’s heat transfer characteristics. Our technique also simplifies to an efficient matrix-free PDE solver that can compete with standard available solvers. This is demonstrated by solving a nonlinear thermal and mechanical PDE on a complex 3D geometry.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104506"},"PeriodicalIF":3.5,"publicationDate":"2025-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840875","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}
Pub Date : 2025-12-25DOI: 10.1016/j.finel.2025.104504
Kent T. Danielson , William M. Furr
Higher-order finite elements using Lagrange and other bases provide distinct benefits over traditional first-order ones in nonlinear solid dynamics but pose additional challenges for explicit methods using nodal mass lumping. Row-summation nodal mass lumping is shown to have a variationally consistent mathematical foundation for explicit methods. Notorious problems with this procedure are caused by improper selection of bases that are not well-suited (suboptimal) for nodal mass lumping, by any method, and not by the lumping scheme. Nodal integration lumping, i.e., via Gauss-Lobatto quadrature using bases that satisfy the Kronecker-Delta property, is also seen to be just a specific case of row-summation lumping (off-diagonal terms innately sum to zero) with a fixed precision. The more general row-sum form, however, is theoretically sound for other appropriate bases and permits arbitrary precision quadrature rules that is shown can be important with distortion, including desirable curvature permitted by higher-order shape functions. Imprecise nodal quadrature lumping can sometimes produce instabilities. In other cases, it captures lower modes sufficiently for solution accuracy but still inadequately computes the largest mode to thus reduce the stable time increment size noticeably. The distinct imprecise over-calculation of the consistent mass matrix by Gauss-Lobatto nodal quadrature and its equivalency to row-summation nodal mass-lumping also reveals additional interesting numerical properties.
{"title":"Numerical properties of nodal mass lumping methods for arbitrary-order finite elements","authors":"Kent T. Danielson , William M. Furr","doi":"10.1016/j.finel.2025.104504","DOIUrl":"10.1016/j.finel.2025.104504","url":null,"abstract":"<div><div>Higher-order finite elements using Lagrange and other bases provide distinct benefits over traditional first-order ones in nonlinear solid dynamics but pose additional challenges for explicit methods using nodal mass lumping. Row-summation nodal mass lumping is shown to have a variationally consistent mathematical foundation for explicit methods. Notorious problems with this procedure are caused by improper selection of bases that are not well-suited (suboptimal) for nodal mass lumping, by any method, and not by the lumping scheme. Nodal integration lumping, i.e., via Gauss-Lobatto quadrature using bases that satisfy the Kronecker-Delta property, is also seen to be just a specific case of row-summation lumping (off-diagonal terms innately sum to zero) with a fixed precision. The more general row-sum form, however, is theoretically sound for other appropriate bases and permits arbitrary precision quadrature rules that is shown can be important with distortion, including desirable curvature permitted by higher-order shape functions. Imprecise nodal quadrature lumping can sometimes produce instabilities. In other cases, it captures lower modes sufficiently for solution accuracy but still inadequately computes the largest mode to thus reduce the stable time increment size noticeably. The distinct imprecise over-calculation of the consistent mass matrix by Gauss-Lobatto nodal quadrature and its equivalency to row-summation nodal mass-lumping also reveals additional interesting numerical properties.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104504"},"PeriodicalIF":3.5,"publicationDate":"2025-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145822959","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}
Pub Date : 2025-12-24DOI: 10.1016/j.finel.2025.104499
Misganaw Abebe , Min-Geun Kim , Bonyong Koo
In this paper, Model-Agnostic Meta-Learning based on Deep Energy Method (MAML-DEM), a novel meta-learning framework is developed for the geometric parameterization. A single meta-trained neural network efficiently solves diverse 2D linear elasticity problems in plates with complex and varying topologies, including those containing multiple mixed circular and elliptical holes. Conceptual studies further demonstrate the framework's potential to generalize to non-uniform boundary conditions and more complex L-shaped hole geometries. By leveraging the variational principle of minimum potential energy, the model avoids the unstable gradients linked to second-order derivatives in standard Physics-Informed Neural Networks (PINNs). Additionally, a geometry-aware adaptive sampling method is employed to capture high-stress areas around geometric discontinuities precisely. During meta-training, the model learns a broad physical understanding applicable across various tasks. Results show that this approach can adapt very quickly to new and unseen geometries, achieving speeds up to 69x faster than training a specific model from scratch. The MAML-DEM framework exhibits superior accuracy and stability over conventional PINN methods, while also demonstrating strong generalization capability to tasks beyond its training data, effectively handling variations in topology, boundary conditions, and geometric complexity. This work highlights the potential of meta-learning to transform physics-informed simulations into practical and efficient tools for rapid engineering design and analysis.
{"title":"Physics-informed meta-learning for elasticity problems with geometric parameterization","authors":"Misganaw Abebe , Min-Geun Kim , Bonyong Koo","doi":"10.1016/j.finel.2025.104499","DOIUrl":"10.1016/j.finel.2025.104499","url":null,"abstract":"<div><div>In this paper, Model-Agnostic Meta-Learning based on Deep Energy Method (MAML-DEM), a novel meta-learning framework is developed for the geometric parameterization. A single meta-trained neural network efficiently solves diverse 2D linear elasticity problems in plates with complex and varying topologies, including those containing multiple mixed circular and elliptical holes. Conceptual studies further demonstrate the framework's potential to generalize to non-uniform boundary conditions and more complex L-shaped hole geometries. By leveraging the variational principle of minimum potential energy, the model avoids the unstable gradients linked to second-order derivatives in standard Physics-Informed Neural Networks (PINNs). Additionally, a geometry-aware adaptive sampling method is employed to capture high-stress areas around geometric discontinuities precisely. During meta-training, the model learns a broad physical understanding applicable across various tasks. Results show that this approach can adapt very quickly to new and unseen geometries, achieving speeds up to 69x faster than training a specific model from scratch. The MAML-DEM framework exhibits superior accuracy and stability over conventional PINN methods, while also demonstrating strong generalization capability to tasks beyond its training data, effectively handling variations in topology, boundary conditions, and geometric complexity. This work highlights the potential of meta-learning to transform physics-informed simulations into practical and efficient tools for rapid engineering design and analysis.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104499"},"PeriodicalIF":3.5,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145822960","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}
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