Finite Elements in Analysis and Design最新文献_第3页

Finite Operator Learning: Bridging neural operators and numerical methods for efficient parametric solution and optimization of PDEs 有限算子学习：桥接神经算子和数值方法的有效参数解和优化的偏微分方程

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-27 DOI: 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.

我们介绍了一种结合神经算子、物理信息机器学习和标准数值方法来求解偏微分方程的方法。该方法统一了上述方法，可以无数据地参数化求解偏微分方程，并提供准确的灵敏度。这些功能支持基于梯度的优化，而不需要典型的灵敏度分析成本，不像伴随方法那样直接随响应函数的数量进行扩展。我们的有限算子学习（FOL）方法最初采用前馈神经网络直接将离散设计空间映射到离散解空间，并且可以与现有的物理信息神经算子技术相结合来恢复连续解域，同时在制定损失项时避免了自动微分的需要。离散化的控制方程，以及设计和解空间，可以从任何成熟的数值技术中推导出来。在这项工作中，我们采用有限元法（FEM）来近似场及其空间导数。由于采用有限元形式，构造满足了Dirichlet边界条件，Neumann边界条件通过弱形式自然地包含在有限元残差中。随后，我们进行Sobolev训练以最小化多目标损失函数，该损失函数包括能量泛函的离散弱形式、边界条件违反以及残差相对于设计变量的平稳性。我们的研究重点是热方程和力学平衡问题。首先，我们主要解决了非均质材料的性能分布，其中基于傅立叶的参数化被用来显著减少设计变量的数量。其次，我们探讨了这些偏微分方程中源项的变化。第三，研究了不同边界条件下的解。在基于梯度优化的背景下，我们研究了微观结构的传热特性的调整。我们的技术也简化为一个有效的无矩阵PDE求解器，可以与标准的可用求解器竞争。通过在复杂的三维几何结构上求解非线性热力学偏微分方程来证明这一点。

{"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}

引用次数: 0

Numerical properties of nodal mass lumping methods for arbitrary-order finite elements 任意阶有限元节点质量集总法的数值性质

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-25 DOI: 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.

在非线性固体动力学中，使用拉格朗日基和其他基的高阶有限元比传统的一阶有限元具有明显的优势，但对使用节点质量集总的显式方法提出了额外的挑战。行和节点质量集总显示出显式方法具有变分一致的数学基础。这一过程中臭名昭著的问题是由不适当的基选择引起的，这些基不适合（次优）节点质量集总，通过任何方法，而不是集总方案。节点积分集总，即通过使用满足Kronecker-Delta性质的基的Gauss-Lobatto正交，也被视为具有固定精度的行和集总（非对角线项固有地和为零）的特定情况。然而，更一般的行和形式在理论上对其他适当的基是合理的，并且允许任意精确的正交规则，这些规则对于畸变很重要，包括高阶形状函数允许的理想曲率。不精确的节点正交集总有时会产生不稳定性。在其他情况下，它捕获较低的模式足以解决精度，但仍然不足以计算最大模式，从而显著减少稳定时间增量大小。高斯-洛巴托节点正交对一致质量矩阵的明显不精确的过度计算及其与行和节点质量集总的等效性也揭示了其他有趣的数值性质。

{"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}

引用次数: 0

Physics-informed meta-learning for elasticity problems with geometric parameterization 几何参数化弹性问题的物理元学习

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-24 DOI: 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.

基于深度能量方法的模型不可知元学习（mml - dem）是一种新的几何参数化元学习框架。单个元训练神经网络可以有效地解决复杂多变拓扑板的各种二维线性弹性问题，包括含有多个混合圆孔和椭圆孔的板。概念研究进一步证明了该框架推广到非均匀边界条件和更复杂的l形孔几何形状的潜力。通过利用最小势能的变分原理，该模型避免了标准物理信息神经网络（pinn）中二阶导数的不稳定梯度。此外，采用几何感知自适应采样方法精确捕获几何不连续周围的高应力区域。在元训练期间，模型学习了适用于各种任务的广泛物理理解。结果表明，这种方法可以非常迅速地适应新的和看不见的几何形状，比从头开始训练特定模型的速度快69倍。与传统的PINN方法相比，mml - dem框架具有更高的精度和稳定性，同时也展示了对训练数据以外任务的强大泛化能力，有效地处理拓扑、边界条件和几何复杂性的变化。这项工作强调了元学习的潜力，将物理信息模拟转化为快速工程设计和分析的实用高效工具。

{"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}

引用次数: 0

A novel data compression method for GPU accelerated large-scale isogeometric topology optimization with order-ascending strategy 一种新的GPU数据压缩方法加速了大规模等高几何拓扑优化

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-18 DOI: 10.1016/j.finel.2025.104503

Tao Nie, Jianli Liu, Wanpeng Zhao, Tao Zhang, Ruichen Zhang, Jinpeng Han, Zhaohui Xia

This paper aims to address the common challenges of storage overhead and computational inefficiencies that arise in isogeometric topology optimization (ITO) when dealing with large-scale problems. To tackle these issues, the paper proposes a novel framework that combines a highly efficient data storage strategy with Graphics Processing Unit (GPU) accelerated optimization. By utilizing control point pairs and removing redundant matrix storage, the Isogeometric Compressed Sparse Row (IGA-CSR) technique effectively reduces storage requirements. Furthermore, the paper presents an order-ascending optimization strategy to avoid intensive calculations caused by large degrees of freedom in the early stage. What's more, the introduction of Graphics Processing Unit further improves the optimization process. Combining these methods, an efficient optimization framework is proposed, which allows efficient optimization even for problems that involve tens of millions of degrees of freedom via single NVIDIA GeForce RTX 3090 GPU with 24 GB. Validation through two 3D benchmark examples reveals that the IGA-CSR method shows the best performance comparing with existing methods in memory consumption. At the same time, it enhances computational efficiency about 65.4 % comparing with conventional second-order isogeometric topology optimization via GPU acceleration.

本文旨在解决在处理大规模问题时等几何拓扑优化（ITO）中出现的存储开销和计算效率低下的常见挑战。为了解决这些问题，本文提出了一个将高效数据存储策略与图形处理单元（GPU）加速优化相结合的新框架。等几何压缩稀疏行（IGA-CSR）技术通过利用控制点对和去除冗余矩阵存储，有效地降低了存储需求。在此基础上，提出了一种递进优化策略，避免了前期由于自由度过大而导致的计算量过大。此外，图形处理单元的引入进一步改善了优化过程。结合这些方法，提出了一个高效的优化框架，即使在涉及数千万个自由度的问题上，也可以通过单个NVIDIA GeForce RTX 3090 24 GB GPU进行高效优化。通过两个三维基准算例的验证表明，IGA-CSR方法在内存消耗方面比现有方法表现出最好的性能。同时，与传统的二阶等几何拓扑优化算法相比，该算法的计算效率提高了65.4%。

{"title":"A novel data compression method for GPU accelerated large-scale isogeometric topology optimization with order-ascending strategy","authors":"Tao Nie, Jianli Liu, Wanpeng Zhao, Tao Zhang, Ruichen Zhang, Jinpeng Han, Zhaohui Xia","doi":"10.1016/j.finel.2025.104503","DOIUrl":"10.1016/j.finel.2025.104503","url":null,"abstract":"<div><div>This paper aims to address the common challenges of storage overhead and computational inefficiencies that arise in isogeometric topology optimization (ITO) when dealing with large-scale problems. To tackle these issues, the paper proposes a novel framework that combines a highly efficient data storage strategy with Graphics Processing Unit (GPU) accelerated optimization. By utilizing control point pairs and removing redundant matrix storage, the Isogeometric Compressed Sparse Row (IGA-CSR) technique effectively reduces storage requirements. Furthermore, the paper presents an order-ascending optimization strategy to avoid intensive calculations caused by large degrees of freedom in the early stage. What's more, the introduction of Graphics Processing Unit further improves the optimization process. Combining these methods, an efficient optimization framework is proposed, which allows efficient optimization even for problems that involve tens of millions of degrees of freedom via single NVIDIA GeForce RTX 3090 GPU with 24 GB. Validation through two 3D benchmark examples reveals that the IGA-CSR method shows the best performance comparing with existing methods in memory consumption. At the same time, it enhances computational efficiency about 65.4 % comparing with conventional second-order isogeometric topology optimization via GPU acceleration.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104503"},"PeriodicalIF":3.5,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145784760","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}

引用次数: 0

Partition of Unity-based four-node tetrahedral element for nonlinear structural analysis of nearly incompressible hyperelastic materials 近乎不可压缩超弹性材料非线性结构分析中基于单位的四节点四面体单元划分

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-17 DOI: 10.1016/j.finel.2025.104464

Taejung Lim , Minh-Chien Trinh , Hyungmin Jun

This study introduces a refined four-node tetrahedral finite element employing the Partition of Unity method for nonlinear static and modal analysis of nearly incompressible hyperelastic materials. The proposed Partition of Unity-based element effectively reduces volumetric locking and improves solution accuracy without increasing the number of nodes. The Partition of Unity method enriches the displacement field by incorporating additional polynomial basis functions, enabling higher-order displacement approximation, and effectively alleviating volumetric locking. Mooney–Rivlin and Neo-Hookean material models are integrated with the penalty method, ensuring robust handling of nearly incompressible behavior. Large deformations are addressed using a total Lagrangian formulation. In addition, a displacement-based direct iterative nonlinear modal analysis procedure is employed to derive nonlinear natural frequencies and corresponding mode shapes. In nonlinear static analysis, the proposed element is validated through various numerical cases including blocks under compression, cylinders under large deformation, mesh distortion sensitivity analysis, and tires under compression. The present element effectively alleviates the volumetric locking phenomenon and provides excellent performance even when using a coarse mesh. Nonlinear modal analysis has been performed on cases such as free vibration of distorted plates, truncated cylindrical shells, and hyperelastic soft robots. The proposed elements effectively capture nonlinear natural frequencies and mode shapes even with distorted and coarse meshes.

本文介绍了一种改进的四节点四面体有限元，该有限元采用统一分割法对几乎不可压缩的超弹性材料进行了非线性静力和模态分析。该方法在不增加节点数量的情况下，有效地减少了体积锁定，提高了求解精度。统一分割法通过加入额外的多项式基函数丰富了位移场，实现了高阶位移逼近，有效缓解了体积锁定。Mooney-Rivlin和Neo-Hookean材料模型与惩罚方法相结合，确保了几乎不可压缩行为的稳健处理。使用全拉格朗日公式来处理大变形。此外，采用基于位移的直接迭代非线性模态分析方法推导了非线性固有频率和相应的模态振型。在非线性静力分析中，通过压缩砌块、大变形圆柱体、网格畸变敏感性分析和轮胎压缩等多种数值案例对所提出的单元进行了验证。该元件有效地缓解了体积锁紧现象，即使在使用粗网格时也能提供出色的性能。本文对变形板、截短圆柱壳和超弹性软体机器人的自由振动进行了非线性模态分析。提出的单元即使在扭曲和粗糙的网格中也能有效地捕获非线性固有频率和模态振型。

{"title":"Partition of Unity-based four-node tetrahedral element for nonlinear structural analysis of nearly incompressible hyperelastic materials","authors":"Taejung Lim , Minh-Chien Trinh , Hyungmin Jun","doi":"10.1016/j.finel.2025.104464","DOIUrl":"10.1016/j.finel.2025.104464","url":null,"abstract":"<div><div>This study introduces a refined four-node tetrahedral finite element employing the Partition of Unity method for nonlinear static and modal analysis of nearly incompressible hyperelastic materials. The proposed Partition of Unity-based element effectively reduces volumetric locking and improves solution accuracy without increasing the number of nodes. The Partition of Unity method enriches the displacement field by incorporating additional polynomial basis functions, enabling higher-order displacement approximation, and effectively alleviating volumetric locking. Mooney–Rivlin and Neo-Hookean material models are integrated with the penalty method, ensuring robust handling of nearly incompressible behavior. Large deformations are addressed using a total Lagrangian formulation. In addition, a displacement-based direct iterative nonlinear modal analysis procedure is employed to derive nonlinear natural frequencies and corresponding mode shapes. In nonlinear static analysis, the proposed element is validated through various numerical cases including blocks under compression, cylinders under large deformation, mesh distortion sensitivity analysis, and tires under compression. The present element effectively alleviates the volumetric locking phenomenon and provides excellent performance even when using a coarse mesh. Nonlinear modal analysis has been performed on cases such as free vibration of distorted plates, truncated cylindrical shells, and hyperelastic soft robots. The proposed elements effectively capture nonlinear natural frequencies and mode shapes even with distorted and coarse meshes.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104464"},"PeriodicalIF":3.5,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145784761","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}

引用次数: 0

An axisymmetric finite-volume method for thermal stress problems in heterogeneous materials 非均质材料热应力问题的轴对称有限体积法

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-16 DOI: 10.1016/j.finel.2025.104498

Chenqi Li , Lingkuan Xuan , Jingfeng Gong , Hongyu Guo , Le Gu

To reduce the computational cost of axisymmetric problems and to extend the applicability of the cell-vertex finite volume method (CV-FVM), this paper develops an axisymmetric cell-vertex finite volume method (ACV-FVM) for transient thermal stress analysis in heterogeneous materials with axisymmetric structures. The reduced two-dimensional domain is discretized using 3-node triangular ring elements and 4-node quadrilateral ring elements. A numerical solver based on the ACV-FVM is implemented in C++ and applied to solve thermo-mechanical coupling problems involving homogeneous materials, multilayered materials, functionally graded materials, and materials with temperature-dependent properties. The numerical results show good agreement with analytical solutions and other numerical results. The findings indicate that, compared to nodal-based output schemes, element-center-based output significantly suppresses spurious stress oscillations in multilayered materials. The proposed method has been successfully applied to the thermal stress analysis of a cylinder liner with thermal barrier coatings. Results reveal that temperature-dependent material properties lead to an approximate 1.5 % increase in temperature and a 3.4 % increase in thermal stress at the same location, highlighting the necessity of considering temperature-dependent thermo-mechanical behavior in such analyses.

为了降低轴对称问题的计算成本，扩大胞-顶点有限体积法（CV-FVM）的适用性，本文提出了一种用于轴对称非均质材料瞬态热应力分析的胞-顶点有限体积法（ACV-FVM）。采用3节点三角形环单元和4节点四边形环单元对二维域进行离散化。采用c++语言实现了基于ACV-FVM的数值求解器，并将其应用于均质材料、多层材料、功能梯度材料和温度相关材料的热-力耦合问题的求解。数值结果与解析解和其他数值结果吻合较好。研究结果表明，与基于节点的输出方案相比，基于单元中心的输出方案显著抑制了多层材料中的虚假应力振荡。该方法已成功地应用于热障涂层气缸套的热应力分析。结果表明，在同一位置，温度相关的材料性能导致温度升高约1.5%，热应力增加3.4%，突出了在此类分析中考虑温度相关的热力学行为的必要性。

{"title":"An axisymmetric finite-volume method for thermal stress problems in heterogeneous materials","authors":"Chenqi Li , Lingkuan Xuan , Jingfeng Gong , Hongyu Guo , Le Gu","doi":"10.1016/j.finel.2025.104498","DOIUrl":"10.1016/j.finel.2025.104498","url":null,"abstract":"<div><div>To reduce the computational cost of axisymmetric problems and to extend the applicability of the cell-vertex finite volume method (CV-FVM), this paper develops an axisymmetric cell-vertex finite volume method (ACV-FVM) for transient thermal stress analysis in heterogeneous materials with axisymmetric structures. The reduced two-dimensional domain is discretized using 3-node triangular ring elements and 4-node quadrilateral ring elements. A numerical solver based on the ACV-FVM is implemented in C++ and applied to solve thermo-mechanical coupling problems involving homogeneous materials, multilayered materials, functionally graded materials, and materials with temperature-dependent properties. The numerical results show good agreement with analytical solutions and other numerical results. The findings indicate that, compared to nodal-based output schemes, element-center-based output significantly suppresses spurious stress oscillations in multilayered materials. The proposed method has been successfully applied to the thermal stress analysis of a cylinder liner with thermal barrier coatings. Results reveal that temperature-dependent material properties lead to an approximate 1.5 % increase in temperature and a 3.4 % increase in thermal stress at the same location, highlighting the necessity of considering temperature-dependent thermo-mechanical behavior in such analyses.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104498"},"PeriodicalIF":3.5,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145784762","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}

引用次数: 0

The Gradient Complete Stabilization Method (GCSM) for scalar diffusive–convective–reactive problems 标量扩散-对流-反应问题的梯度完全稳定方法

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-14 DOI: 10.1016/j.finel.2025.104501

E.G. Dutra do Carmo , E.F. Fontes Jr. , M.F.F. Santos , W.J. Mansur

Purely convective and convective–diffusive problems with dominant convection, presenting high gradients in directions misaligned with the convective field, are typically stabilized using nonlinear methods, even when the underlying problem is linear. This not only leads to an increase in computational cost but also degrades the accuracy of the gradient of the approximate solution. Therefore, it is desirable to obtain a method that completely stabilizes the approximated solution gradient while ensuring optimal approximation rates for it. In this sense, the Gradient Complete Stabilization Method (GCSM) is proposed in this paper. A rigorous mathematical analysis of the method is performed by elaborating the variational formulation of the diffusive–convective–reactive problem. A robust set of theorems is defined and proved, including the Fundamental Identity Theorem, which plays a central role in enabling gradient stabilization with optimal convergence rates. Several numerical experiments are conducted, comparing accuracy from GCSM against a classic discontinuity capture method, the Consistent Approximate Upwind (CAU). The results demonstrate a marked improvement in performance achieved by the proposed method, especially in the final example, which involves both internal and external boundary layers. In this case, the GCSM delivers solutions that are nearly oscillation-free.

具有优势对流的纯对流和对流扩散问题，在与对流场不对齐的方向上呈现高梯度，通常使用非线性方法来稳定，即使潜在问题是线性的。这不仅会导致计算成本的增加，而且会降低近似解的梯度精度。因此，我们希望找到一种既能使近似解梯度完全稳定，又能保证最优近似速率的方法。在这个意义上，本文提出了梯度完全稳定方法（GCSM）。通过阐述扩散-对流-反应问题的变分公式，对该方法进行了严格的数学分析。定义并证明了一组鲁棒定理，其中包括基本恒等定理，它在实现具有最优收敛速率的梯度镇定中起着核心作用。通过几个数值实验，比较了GCSM与经典的不连续捕获方法一致近似迎风（CAU）的精度。结果表明，该方法的性能有了显著的提高，特别是在最后的例子中，同时涉及到内部和外部边界层。在这种情况下，GCSM提供了几乎无振荡的解决方案。

{"title":"The Gradient Complete Stabilization Method (GCSM) for scalar diffusive–convective–reactive problems","authors":"E.G. Dutra do Carmo , E.F. Fontes Jr. , M.F.F. Santos , W.J. Mansur","doi":"10.1016/j.finel.2025.104501","DOIUrl":"10.1016/j.finel.2025.104501","url":null,"abstract":"<div><div>Purely convective and convective–diffusive problems with dominant convection, presenting high gradients in directions misaligned with the convective field, are typically stabilized using nonlinear methods, even when the underlying problem is linear. This not only leads to an increase in computational cost but also degrades the accuracy of the gradient of the approximate solution. Therefore, it is desirable to obtain a method that completely stabilizes the approximated solution gradient while ensuring optimal approximation rates for it. In this sense, the Gradient Complete Stabilization Method (GCSM) is proposed in this paper. A rigorous mathematical analysis of the method is performed by elaborating the variational formulation of the diffusive–convective–reactive problem. A robust set of theorems is defined and proved, including the Fundamental Identity Theorem, which plays a central role in enabling gradient stabilization with optimal convergence rates. Several numerical experiments are conducted, comparing accuracy from GCSM against a classic discontinuity capture method, the Consistent Approximate Upwind (CAU). The results demonstrate a marked improvement in performance achieved by the proposed method, especially in the final example, which involves both internal and external boundary layers. In this case, the GCSM delivers solutions that are nearly oscillation-free.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104501"},"PeriodicalIF":3.5,"publicationDate":"2025-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753512","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}

引用次数: 0

Regularizing the linearly extrapolated BDF2 scheme for incompressible flows with time relaxation 带时间松弛的不可压缩流线性外推BDF2格式的正则化

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-10 DOI: 10.1016/j.finel.2025.104491

Sean Breckling , Jorge Reyes , Sidney Shields , Clifford Watkins

This paper presents a highly-efficient finite element scheme for the time relaxation model (TRM). The efficiency is achieved through the second-order BDF2 time-stepping scheme with linear extrapolation (BDF2LE). The accuracy of the scheme is also greatly enhanced through the use of the divergence-free Scott-Vogeulis finite elements, and van Cittert approximate deconvolution. A complete finite element analysis is provided, which includes rigorous proofs for the stability, well-possessedness, and convergence of both velocity and pressure solutions. We also demonstrate that the inclusion of the linear time relaxation term preserves the long-time stability of the unregularized BDF2LE scheme. Finally, numerical experiments are presented that demonstrate the added stability and accuracy that time relaxation can provide.

本文提出了一种求解时间松弛模型（TRM）的高效有限元方案。通过线性外推的二阶BDF2时间步进方案（BDF2LE）实现了效率。通过使用无散度的Scott-Vogeulis有限元和van Cittert近似反卷积，该方案的精度也大大提高。给出了完整的有限元分析，包括速度解和压力解的稳定性、完备性和收敛性的严格证明。我们还证明了线性时间松弛项的加入保留了非正则BDF2LE格式的长期稳定性。最后，给出了数值实验，证明了时间松弛可以提供额外的稳定性和准确性。

引用次数: 0

Integral constitutive equations based temporal finite element modeling for the static viscoelastic problem 基于积分本构方程的静态粘弹性问题时间有限元建模

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-05 DOI: 10.1016/j.finel.2025.104490

Fengling Chen, Yiqian He, Haitian Yang

A stepwise spatial–temporal finite element algorithm is developed to provide a general numerical tool for solving static viscoelastic problems with integral constitutive equations. The displacement, strain and stress are formulated by the hybrid basis functions based Temporal Finite Element Method (TFEM), and are incorporated into the constitutive relations. The framework is established based on the virtual work principle and the weighted residual technique, and is convenient to cooperate with kinds of numerical schemes for boundary value problems such as FEM and SBFEM. Two criteria are proposed to numerically evaluate error propagation during the step-marching process, which can be used to determine appropriate time-step sizes for prescribed temporal shape functions and spatial FE meshes. Compared with the TFEM algorithm based on differential viscoelastic constitutive equations, the present approach overcomes the order-restriction limitation by employing integral constitutive equations with Prony-series based relaxation moduli. Numerical examples demonstrate the capability and accuracy of the proposed method in handling viscoelastic problems involving material heterogeneity, stress singularity, various relaxation moduli, and different loading forms. The obtained results with various configurations of temporal shape functions and step sizes, exhibit good agreement with analytical solutions and ABAQUS simulations.

提出了一种分步时空有限元算法，为求解具有积分本构方程的静态粘弹性问题提供了一种通用的数值工具。位移、应变和应力由基于混合基函数的时间有限元法（TFEM）表示，并纳入本构关系。该框架是基于虚功原理和加权残差技术建立的，可方便地与有限元法、单轴有限元法等边值问题的多种数值格式配合使用。提出了两种步进过程误差传播数值评价准则，可用于确定指定时间形状函数和空间有限元网格的适当时间步长。与基于粘弹性微分本构方程的TFEM算法相比，该方法采用基于prony级数的松弛模量的积分本构方程，克服了阶数限制。数值算例验证了该方法处理材料非均质性、应力奇异性、不同松弛模量和不同加载形式等粘弹性问题的能力和准确性。所得结果与解析解和ABAQUS仿真结果吻合较好。

{"title":"Integral constitutive equations based temporal finite element modeling for the static viscoelastic problem","authors":"Fengling Chen, Yiqian He, Haitian Yang","doi":"10.1016/j.finel.2025.104490","DOIUrl":"10.1016/j.finel.2025.104490","url":null,"abstract":"<div><div>A stepwise spatial–temporal finite element algorithm is developed to provide a general numerical tool for solving static viscoelastic problems with integral constitutive equations. The displacement, strain and stress are formulated by the hybrid basis functions based Temporal Finite Element Method (TFEM), and are incorporated into the constitutive relations. The framework is established based on the virtual work principle and the weighted residual technique, and is convenient to cooperate with kinds of numerical schemes for boundary value problems such as FEM and SBFEM. Two criteria are proposed to numerically evaluate error propagation during the step-marching process, which can be used to determine appropriate time-step sizes for prescribed temporal shape functions and spatial FE meshes. Compared with the TFEM algorithm based on differential viscoelastic constitutive equations, the present approach overcomes the order-restriction limitation by employing integral constitutive equations with Prony-series based relaxation moduli. Numerical examples demonstrate the capability and accuracy of the proposed method in handling viscoelastic problems involving material heterogeneity, stress singularity, various relaxation moduli, and different loading forms. The obtained results with various configurations of temporal shape functions and step sizes, exhibit good agreement with analytical solutions and ABAQUS simulations.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104490"},"PeriodicalIF":3.5,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685751","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}

引用次数: 0

Unsymmetric Serendipity finite elements: Performance analysis 非对称偶然性有限元：性能分析

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-12-03 DOI: 10.1016/j.finel.2025.104487

S. Eisenträger , E. Woschke , E.T. Ooi

This paper presents a comparative analysis of the conventional finite element method (FEM) and the unsymmetric finite element method (UFEM) for Serendipity elements (

p \leq 3

), focusing on two factors: (i) achievable accuracy and (ii) computational costs. The UFEM, based on a Petrov–Galerkin formulation, uses metric shape functions as trial functions and parametric shape functions as test functions. This unique approach enhances the resistance against mesh distortion, as it ensures polynomial completeness of the Ansatz space of unsymmetric finite elements. Hence, higher accuracy can be achieved in complex geometries. However, the unsymmetric nature of UFEM leads to increased computational costs as a result of the added complexity of solving the resulting system of equations. This study provides a quantitative evaluation of the computational burden associated with achieving specific error thresholds for both methods. By analyzing a range of benchmark problems, we identify scenarios in which each method performs optimally, offering practical insights for selecting the appropriate approach based on accuracy demands and computational constraints. Our findings suggest that, while UFEM can produce superior accuracy, its computational efficiency depends on application-specific requirements and available resources.

本文对Serendipity单元（p≤3）的传统有限元法（FEM）和非对称有限元法（UFEM）进行了比较分析，重点关注两个因素：(i)可实现的精度和（ii）计算成本。UFEM基于Petrov-Galerkin公式，使用度量形状函数作为试验函数，参数形状函数作为测试函数。这种独特的方法增强了对网格变形的抵抗，因为它保证了非对称有限元的Ansatz空间的多项式完备性。因此，在复杂的几何形状中可以达到更高的精度。然而，UFEM的非对称性质导致计算成本的增加，这是由于求解所得到的方程组的复杂性增加的结果。本研究提供了与实现两种方法的特定误差阈值相关的计算负担的定量评估。通过分析一系列基准问题，我们确定了每种方法执行最佳的场景，为基于精度要求和计算约束选择合适的方法提供了实用的见解。我们的研究结果表明，虽然UFEM可以产生更高的精度，但其计算效率取决于特定应用的要求和可用资源。

{"title":"Unsymmetric Serendipity finite elements: Performance analysis","authors":"S. Eisenträger , E. Woschke , E.T. Ooi","doi":"10.1016/j.finel.2025.104487","DOIUrl":"10.1016/j.finel.2025.104487","url":null,"abstract":"<div><div>This paper presents a comparative analysis of the conventional finite element method (FEM) and the unsymmetric finite element method (UFEM) for Serendipity elements (<span><math><mrow><mi>p</mi><mo>≤</mo><mn>3</mn></mrow></math></span>), focusing on two factors: (i) achievable accuracy and (ii) computational costs. The UFEM, based on a Petrov–Galerkin formulation, uses <em>metric</em> shape functions as <em>trial</em> functions and <em>parametric</em> shape functions as <em>test</em> functions. This unique approach enhances the resistance against mesh distortion, as it ensures polynomial completeness of the Ansatz space of unsymmetric finite elements. Hence, higher accuracy can be achieved in complex geometries. However, the unsymmetric nature of UFEM leads to increased computational costs as a result of the added complexity of solving the resulting system of equations. This study provides a quantitative evaluation of the computational burden associated with achieving specific error thresholds for both methods. By analyzing a range of benchmark problems, we identify scenarios in which each method performs optimally, offering practical insights for selecting the appropriate approach based on accuracy demands and computational constraints. Our findings suggest that, while UFEM can produce superior accuracy, its computational efficiency depends on application-specific requirements and available resources.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"254 ","pages":"Article 104487"},"PeriodicalIF":3.5,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145659097","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}

引用次数: 0