Computers & Mathematics with Applications最新文献_第7页

A dual reduced-order modeling method for nonlinear thermoelastic analysis of 3D honeycomb structure 三维蜂窝结构非线性热弹性分析的双降阶建模方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-01 DOI: 10.1016/j.camwa.2025.11.010

Zhen Yin , Ke Liang

The in-plane stiffness of honeycomb structures is several orders of magnitude lower than the material itself, making them an ideal sandwich structure for flexible deformation mechanisms of high-speed aircrafts in a high-temperature environment. Finite element method is widely used to simulate the in-and-out-of-plane large deformation of honeycomb structures; however the high computational cost is still a major bottleneck for fast analysis and optimization design. The existing reduced-order methods are mainly applicable to buckling problems, rather than the in-plane and out-of-plane large deformation case of honeycomb structures. In this work, a geometrically nonlinear reduced-order method considering both the thermal expansion and temperature-dependent material properties is proposed for in-and-out-of-plane large deformation analysis of honeycomb structures subjected to a high temperature field. The dual reduced-order models with only one degree of freedom are constructed for the temperature rise and mechanical loading phases, respectively, based on the third-order equilibrium equations and perturbation method. The constructional efficiency of the reduced system is largely improved by zeroing the fourth-order strain energy variation using the two-field Hellinger-Reissner variational principle and three-dimensional (3D) solid element. The nonlinear predictor solved by the reduced-order model can be corrected when its numerical accuracy is not satisfactory in path-following analysis. Various numerical examples demonstrate that the proposed method has a superior path-following capability for 3D finite element analysis of honeycomb structures.

蜂窝结构的面内刚度比材料本身低几个数量级，是高温环境下高速飞机柔性变形机构的理想夹层结构。有限元法被广泛应用于蜂窝结构面内外大变形的模拟；然而，高计算成本仍然是快速分析和优化设计的主要瓶颈。现有的降阶方法主要适用于屈曲问题，而不适用于蜂窝结构的面内面外大变形情况。在这项工作中，提出了一种考虑热膨胀和温度相关材料特性的几何非线性降阶方法，用于高温场下蜂窝结构的面内外大变形分析。基于三阶平衡方程和微扰法，分别建立了温升阶段和机械加载阶段的单自由度双降阶模型。利用双场Hellinger-Reissner变分原理和三维实体单元将四阶应变能变化归零，大大提高了简化系统的构造效率。用降阶模型求解的非线性预测器在路径跟踪分析中，当其数值精度不能令人满意时，可以进行修正。数值算例表明，该方法对蜂窝结构的三维有限元分析具有良好的路径跟踪能力。

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引用次数: 0

Three-dimensional narrow volume reconstruction method with unconditional stability based on a phase-field lagrange multiplier approach 基于相场拉格朗日乘子法的无条件稳定三维窄体重构方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-25 DOI: 10.1016/j.camwa.2025.11.009

Renjun Gao, Xiangjie Kong, Dongting Cai, Boyi Fu, Junxiang Yang

Reconstruction of an object from points cloud is essential in prosthetics, medical imaging, computer vision, etc. We present an effective algorithm for an Allen–Cahn-type model of reconstruction, employing the Lagrange multiplier approach. Utilizing scattered data points from an object, we reconstruct a narrow shell by solving the governing equation enhanced with an edge detection function derived from the unsigned distance function. The specifically designed edge detection function ensures the energy stability. By reformulating the governing equation through the Lagrange multiplier technique and implementing a Crank–Nicolson time discretization, we can update the solutions in a stable and decoupled manner. The spatial operations are approximated using the finite difference method, and we analytically demonstrate the unconditional stability of the fully discrete scheme. Comprehensive numerical experiments, including reconstructions of complex 3D volumes such as characters from Star Wars, validate the algorithm’s accuracy, stability, and effectiveness. Additionally, we analyze how specific parameter selections influence the level of detail and refinement in the reconstructed volumes. To facilitate the interested readers to understand our algorithm, we share the computational codes and data in https://github.com/cfdyang521/C-3PO/tree/main.

从点云中重建物体在假肢、医学成像、计算机视觉等领域是必不可少的。本文采用拉格朗日乘子方法，提出了一种有效的allen - cahn型重构模型算法。利用物体上分散的数据点，我们通过求解控制方程，利用无符号距离函数衍生的边缘检测函数来重建窄壳。特别设计的边缘检测功能保证了能量的稳定性。通过拉格朗日乘子技术重新表述控制方程并实现Crank-Nicolson时间离散化，我们可以以稳定和解耦的方式更新解。利用有限差分法对空间运算进行近似，并解析地证明了完全离散格式的无条件稳定性。全面的数值实验，包括重建复杂的三维体，如《星球大战》中的人物，验证了算法的准确性、稳定性和有效性。此外，我们分析了具体的参数选择如何影响重建体的细节和细化水平。为了方便有兴趣的读者理解我们的算法，我们在https://github.com/cfdyang521/C-3PO/tree/main中分享了计算代码和数据。

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引用次数: 0

A second order accurate, and structure-preserving numerical scheme for the thermodynamical consistent Keller–Segel–Navier–Stokes model 热力学一致Keller-Segel-Navier-Stokes模型的二阶精确且保持结构的数值格式

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-24 DOI: 10.1016/j.camwa.2025.11.005

Rui Wang , Cheng Wang , Yuzhe Qin , Zhengru Zhang

In this work, we propose a Keller–Segel–Navier–Stokes (KSNS) model for describing chemotactic phenomena, formulated within the framework of the Energetic Variational Approach (EnVarA). A second-order accurate numerical scheme is developed that rigorously preserves three fundamental properties in discrete sense: the positivity of cell density, mass conservation of cell density, and total energy dissipation. The Keller–Segel subsystem is reformulated as a coupling between an

H^{- 1}

gradient flow with non-constant mobility and an

L^{2}

gradient flow, enabling the effective treatment of the nonlinear and singular logarithmic energy potential via a modified Crank-Nicolson scheme. Artificial regularization terms are introduced to enforce positivity preservation. For the fluid dynamics component, we adopt a second-order semi-implicit time discretization. The marker-and-cell (MAC) finite difference approximation is used as the spatial discretization, which ensures a discretely divergence-free velocity field. The proposed numerical method guarantees unique solvability, mass conservation, and total energy stability. Furthermore, through detailed asymptotic expansions and rigorous error analysis, we establish optimal convergence rates. A series of numerical experiments are presented to validate the effectiveness and robustness of both the physical model and the numerical scheme.

在这项工作中，我们提出了一个Keller-Segel-Navier-Stokes （KSNS）模型来描述化学趋化现象，该模型是在能量变分方法（EnVarA）的框架内制定的。建立了一种二阶精确的数值格式，该格式严格地保留了三个基本性质：细胞密度的正性、细胞密度的质量守恒和总能量耗散。Keller-Segel子系统被重新表述为具有非恒定迁移率的H - 1梯度流和L2梯度流之间的耦合，从而能够通过改进的Crank-Nicolson格式有效地处理非线性和奇异对数能量势。引入人工正则化项来加强正性保持。对于流体力学分量，采用二阶半隐式时间离散。采用标记单元有限差分近似作为空间离散化，保证了离散的无发散速度场。所提出的数值方法保证了独特的可解性、质量守恒性和总能量稳定性。此外，通过详细的渐近展开和严格的误差分析，我们建立了最优收敛速率。通过一系列数值实验验证了物理模型和数值方案的有效性和鲁棒性。

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引用次数: 0

Pre-classification based stochastic reduced-order model for time-dependent complex system 基于预分类的时变复杂系统随机降阶模型

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-22 DOI: 10.1016/j.camwa.2025.11.006

Meixin Xiong , Liuhong Chen , Yulan Ning , Ju Ming , Zhiwen Zhang

We propose a novel stochastic reduced-order model (SROM) for complex systems by combining statistical analysis tools. Based on the generalizability of distance in the centroidal Voronoi tessellation (CVT) method and the minimization of projection error in proper orthogonal decomposition (POD), we define a time-dependent generalized CVT clustering. Each generalized centroid corresponds to a set of cluster-based POD (CPOD) basis functions. Then, using the clustering results as the training dataset, the classification mechanism of the system input can be obtained by applying the naive Bayesian method. For a given input sample, the predicted label obtained by the classifier is used to determine a set of CPOD basis functions for model reduction. Rigorous error analysis is shown, and a discussion of the Navier-Stokes equation with random parameters is given to provide a context for the application of this SROM. Numerical experiments verify that the accuracy of our SROM is improved compared with the standard POD method.

结合统计分析工具，提出了一种复杂系统的随机降阶模型。基于质心Voronoi镶嵌法（CVT）中距离的可泛化性和适当正交分解（POD）中投影误差的最小化，定义了一种时变广义CVT聚类。每个广义质心对应一组基于聚类的POD （CPOD）基函数。然后，将聚类结果作为训练数据集，应用朴素贝叶斯方法得到系统输入的分类机制。对于给定的输入样本，使用分类器获得的预测标签来确定一组CPOD基函数，用于模型约简。给出了严格的误差分析，并讨论了带有随机参数的Navier-Stokes方程，为该存储器的应用提供了一个背景。数值实验表明，与标准的POD方法相比，该方法的精度得到了提高。

引用次数: 0

Prediction-correction method for nonlinear partial differential equations based on sparse grid interpolation techniques 基于稀疏网格插值技术的非线性偏微分方程预测校正方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-20 DOI: 10.1016/j.camwa.2025.11.007

Chenglong Xu , Yining Qiu , Keyan Wang , Bihao Su

This article proposes a prediction-correction method for solving nonlinear partial differential equations (PDEs) in complex multidimensional domains. Our approach employs the Feynman-Kac formula, which establishes a connection between stochastic differential equations and PDEs, along with multidimensional function interpolation and corresponding integration methods based on sparse grid nodes. First, we derive the relationship between solutions of discrete-time PDEs using the Feynman-Kac formula. Next, we establish a discretization scheme through integration over sparse grid points and then construct the expression of each time-layer solution via interpolation on these points. Prediction-correction techniques are applied to address the integration of nonlinear unknown functions. While deriving a prediction-correction numerical method, this paper also presents a rigorous theoretical error analysis of the method. The effectiveness and convergence of the proposed method are validated through two types of examples. Compared with traditional numerical methods, our method enables efficient handling of more complex and multidimensional regions.

提出了一种求解复杂多维域非线性偏微分方程的预测-修正方法。我们的方法采用费曼-卡茨公式，该公式建立了随机微分方程与偏微分方程之间的联系，以及基于稀疏网格节点的多维函数插值和相应的积分方法。首先，我们利用费曼-卡茨公式推导了离散时间偏微分方程解之间的关系。其次，通过对稀疏网格点的积分建立离散化方案，然后在这些点上通过插值构造每个时间层解的表达式。应用预测校正技术解决非线性未知函数的积分问题。本文在推导预测校正数值方法的同时，对该方法进行了严格的理论误差分析。通过两类算例验证了该方法的有效性和收敛性。与传统的数值方法相比，该方法能够有效地处理更复杂和多维的区域。

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引用次数: 0

Convergence and numerical simulations of the continuous data assimilation for Biot's poroelasticity system Biot孔隙弹性系统连续数据同化的收敛性与数值模拟

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-20 DOI: 10.1016/j.camwa.2025.11.008

Seungmin Lee, Sanghyun Lee

Real-world applications often suffer from incomplete or uncertain data, which can undermine the accuracy of physical models. In our work, we tackle this challenge by applying continuous data assimilation (CDA) to Biot's poroelasticity system, a model that captures how fluid-saturated porous materials, such as soils or biological tissues, deform under stress. We utilize a nudging term in the conservation of mass equation to integrate sparse observational data directly into the pressure field. We then use an iterative fixed-stress method to decouple the pressure and displacement problem, making the process both stable and efficient. Our approach employs finite element methods along with an interpolation strategy that adapts to different data resolutions. Several numerical tests illustrate that the proposed method reliably drives the model's predictions towards the true state, even in complex scenarios involving heterogeneous permeability or nonlinear permeability, and including a three dimensional setup. This study is the first to rigorously apply CDA to Biot's poroelasticity system, offering a promising avenue for more accurate predictive modeling in fields ranging from geomechanics to biomechanics.

现实世界的应用经常受到不完整或不确定数据的影响，这可能会破坏物理模型的准确性。在我们的工作中，我们通过将连续数据同化（CDA）应用于Biot的孔隙弹性系统来解决这一挑战，该系统是一个捕获流体饱和多孔材料（如土壤或生物组织）在应力下如何变形的模型。我们利用质量守恒方程中的轻推项将稀疏观测数据直接积分到压力场中。然后，我们使用迭代固定应力方法来解耦压力和位移问题，使过程既稳定又高效。我们的方法采用有限元方法以及适应不同数据分辨率的插值策略。几个数值试验表明，即使在涉及非均质渗透率或非线性渗透率的复杂场景中，包括三维设置，所提出的方法也能可靠地将模型的预测推向真实状态。该研究首次将CDA严格应用于Biot的孔隙弹性系统，为从地质力学到生物力学等领域更准确的预测建模提供了一条有希望的途径。

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引用次数: 0

Meshless and FEM approximations for mixed polytopal elements 混合多面体单元的无网格和有限元逼近

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-19 DOI: 10.1016/j.camwa.2025.11.003

Witold Cecot , Marta Oleksy , Mateusz Dryzek

This paper presents a comparison of various approaches to approximating the unknown functions within the mixed ultra-weak formulation, using Voronoi polygon discretization and stabilization via the Discontinuous Petrov–Galerkin (DPG) methodology. The primary objective is to identify the basis functions that offer the best approximability. We consider meshless functions, which have not previously been applied to fully mixed formulations (stress–displacement in mechanics), and compare several of their variants with polynomial approximations. Numerical tests are conducted for the Laplace equation on both square and L-shaped domains, for regular as well as singular solutions. Our study reveals that, with one exception, the polynomial approximation—particularly the orthogonal Legendre type—exhibits faster convergence, most likely due to its superior interpolation properties in the two-field formulation. The findings of this study can be extended to other elliptic problems, including those in solid mechanics, where mixed formulations and polygonal finite elements offer significant advantages.

本文介绍了利用Voronoi多边形离散化和不连续Petrov-Galerkin （DPG）方法稳定混合超弱公式中未知函数的各种近似方法的比较。主要目标是确定提供最佳逼近性的基函数。我们考虑无网格函数，它以前没有被应用于完全混合的公式（力学中的应力-位移），并将它们的几种变体与多项式近似进行比较。对拉普拉斯方程在正方形和l形域、正则解和奇异解上进行了数值试验。我们的研究表明，除了一个例外，多项式近似-特别是正交Legendre型-表现出更快的收敛，很可能是由于其在双场公式中的优越插值特性。本研究的发现可以推广到其他椭圆问题，包括固体力学中的混合公式和多边形有限元提供显著优势的问题。

引用次数: 0

A weaker adversarial neural networks method for linear second order elliptic PDEs 线性二阶椭圆偏微分方程的弱对抗神经网络方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-13 DOI: 10.1016/j.camwa.2025.11.004

Zhizhong Kong , Rui Sheng , Jerry Zhijian Yang , Juntao You , Cheng Yuan

In this paper, we propose the Weaker Adversarial Network (WerAN), a novel extension of the Weak Adversarial Network (WAN) framework designed for partial differential equations with singularities. By applying integration by parts twice in the derivation of weak form, our approach relaxes the constraints on the solution space in WAN. This adjustment transitions the implementation-level requirements from

H^{1}

to

L^{2}

and the theoretical requirements from

W^{2, \infty}

to

W^{1, \infty}

. Furthermore, we provide a systematic error analysis of the proposed method. Our theoretical investigations affirm the convergence of the WerAN method and offer insights into selecting network parameters and sample sizes across the domain and its boundary. We also present several numerical examples to demonstrate the effectiveness of WerAN in addressing both smooth and singular problems.

在本文中，我们提出了弱对抗网络（WerAN），这是弱对抗网络（WAN）框架的一个新的扩展，该框架是为具有奇异性的偏微分方程设计的。通过在弱形式的推导中两次应用分部积分，我们的方法放宽了广域网中解空间的约束。这一调整将实现层需求从H1转变为L2，理论需求从W2，∞转变为W1，∞。此外，我们还对所提出的方法进行了系统误差分析。我们的理论研究证实了WerAN方法的收敛性，并为跨域及其边界选择网络参数和样本大小提供了见解。我们还给出了几个数值例子来证明WerAN在解决光滑和奇异问题方面的有效性。

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引用次数: 0

Pinning phenomena in numerical schemes of the Allen–Cahn equation Allen-Cahn方程数值格式中的钉钉现象

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-12 DOI: 10.1016/j.camwa.2025.11.001

Junseok Kim , Zhengang Li , Xinpei Wu , Soobin Kwak

In this study, we investigate the artificial pinning phenomena that emerge in numerical simulations of high-order Allen–Cahn (AC) equations. The AC equation is widely used to describe interface motion during phase separation processes. However, discretized numerical solutions may exhibit nonphysical behavior such as interface immobilization, or “pinning,” particularly when the ratio between the interfacial thickness model parameter (ϵ) and the spatial numerical mesh size (h) is below a critical threshold. The present work systematically analyzes how this ratio,

P = ϵ / h

, influences interface dynamics and determines the critical values at which pinning begins to occur for various nonlinearity degrees (α) of the polynomial potential. A fully explicit finite difference method is used to simulate the AC equation in both two- and three-dimensional settings. A bisection algorithm is introduced to accurately identify the critical pinning threshold

P^{⁎}

as a function of α. Numerical experiments demonstrate that as α increases,

P^{⁎}

decreases, indicating that the solution becomes more sensitive to spatial discretization and requires finer grids to avoid artificial pinning. Additionally, computational results show that higher values of α yield thicker and smoother interface transitions, and the study also investigates the influence of these values on phase morphology under random initial conditions. The numerical results in this study provide quantitative guidance for selecting discretization parameters in phase-field simulations and emphasize the importance of balancing the interface resolution with model complexity. These insights contribute to the development of accurate and robust numerical schemes for simulating interfacial dynamics in complex physical systems.

在本研究中，我们研究了高阶Allen-Cahn （AC）方程数值模拟中出现的人为钉住现象。交流方程被广泛用于描述相分离过程中的界面运动。然而，离散数值解可能表现出非物理行为，如界面固定或“钉住”，特别是当界面厚度模型参数（柱）与空间数值网格尺寸(h)之间的比率低于临界阈值时。本研究系统地分析了这个比值P= λ /h如何影响界面动力学，并确定了多项式电位的各种非线性度（α）开始发生钉住的临界值。采用全显式有限差分法模拟了二维和三维环境下的交流方程。引入了一种对分算法，以准确地识别临界钉住阈值P作为α的函数。数值实验表明，随着α的增加，P - P - P减小，表明溶液对空间离散化更加敏感，需要更细的网格来避免人为钉住。此外，计算结果表明，α值越高，界面转变越厚、越光滑，并研究了随机初始条件下α值对相形态的影响。本研究的数值结果为相场模拟中离散化参数的选择提供了定量指导，并强调了平衡界面分辨率与模型复杂性的重要性。这些见解有助于开发精确和强大的数值方案来模拟复杂物理系统中的界面动力学。

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引用次数: 0

Discontinuous Galerkin finite element method for a fourth order parabolic variational inequality 一类四阶抛物型变分不等式的不连续Galerkin有限元法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-11-11 DOI: 10.1016/j.camwa.2025.11.002

Kamana Porwal, Ritesh Singla

In this article, we propose and analyze a discontinuous Galerkin finite element method for the numerical approximation of a fourth-order parabolic variational inequality. In the fully-discrete discontinuous Galerkin scheme, we implement time discretization using the implicit backward Euler method, while spatial discretization is achieved through the utilization of a piecewise quadratic finite element space. The convergence analysis is conducted under a reasonable regularity assumption on the exact solution u, to be specific we assume

u_{t} \in L^{2} (0, S; L^{2} (Ω))

, and the obstacle constraints are incorporated at the vertices of the triangulation. We derive an optimal order (with respect to the regularity)

a p r i o r i

error estimate in the energy norm. Additionally, we present some numerical experiment results to illustrate the performance of the proposed method.

本文提出并分析了四阶抛物型变分不等式数值逼近的不连续Galerkin有限元法。在完全离散的不连续Galerkin格式中，我们使用隐式后向欧拉方法实现时间离散，而利用分段二次元有限元空间实现空间离散。收敛性分析是在对精确解u的合理正则性假设下进行的，即假设ut∈L2（0，S;L2（Ω）），并在三角剖分的顶点处加入障碍物约束。我们在能量范数中推导出一个最优阶（相对于正则性）先验误差估计。此外，我们还给出了一些数值实验结果来说明所提出方法的性能。

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引用次数: 0