Computers & Mathematics with Applications最新文献

Numerical study of magnesium dendrite microstructure under convection: Change of dendrite symmetry 对流条件下镁枝晶微观结构的数值研究：枝晶对称性的变化

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-06 DOI: 10.1016/j.camwa.2024.10.038

Ang Zhang , Minghang Yang , Lang Qin , Jing Cheng , Yuchen Tang , Jinglian Du , Wenbo Yu , Zhihua Dong , Feng Liu , Bin Jiang , Fusheng Pan

Besides diffusion and capillary, convection which is unavoidable under terrestrial condition has remarkable effects on the microstructure evolution during solidification. In this study, a phase-field lattice-Boltzmann model, accelerated by state-of-the-art parallel-adaptive mesh refinement algorithm, is solved to investigate the morphological evolution of the Mg-Gd dendrite under convection. The lengths of the dendrite primary arms are quantified to analyze the asymmetric dendrite patterns under convection. The effects of the multiple factors including the orientation angle, the flow intensity, and the undercooling are elucidated, and the relation between the length ratios and the three independent factors is established through multiple regression analysis. The upstream-downstream arm length difference and the included angle between the primary arms are characterized to illustrate the effect of convection on the evolution of the Mg-Gd dendrite. The 3D morphological selection, together with algorithm performance tests, is further discussed to elucidate the change of morphological symmetry under different growth conditions and to demonstrate the robustness of the numerical scheme. Deep understanding of the synergy between convection-induced solute transport and undercooling-driven growth, which largely determines the morphological selection, can assist guidance for the prediction and control of the magnesium alloy microstructures.

除了扩散和毛细作用外，在陆地条件下不可避免的对流对凝固过程中的微观结构演变也有显著影响。本研究采用最先进的并行自适应网格细化算法，对相场晶格-玻尔兹曼模型进行加速求解，以研究镁钆枝晶在对流作用下的形态演变。通过量化枝晶主臂的长度来分析对流作用下的不对称枝晶形态。阐明了取向角、流动强度和过冷度等多重因素的影响，并通过多元回归分析确定了长度比与三个独立因素之间的关系。通过对流对镁钆树枝晶演化的影响，描述了主臂之间的上下游臂长差和包含角。进一步讨论了三维形态选择以及算法性能测试，以阐明不同生长条件下形态对称性的变化，并证明数值方案的鲁棒性。对对流诱导的溶质传输和欠冷驱动的生长之间的协同作用（这在很大程度上决定了形态选择）的深入理解有助于指导镁合金微结构的预测和控制。

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引用次数: 0

An implementation of hp-FEM for the fractional Laplacian 分数拉普拉斯函数的 hp-FEM 实现

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-06 DOI: 10.1016/j.camwa.2024.10.005

Björn Bahr, Markus Faustmann, Jens Markus Melenk

We consider the discretization of the 1d-integral Dirichlet fractional Laplacian by hp-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of hp-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is

O (N^{5 / 2})

, where N is the problem size. Numerical examples illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for hp-finite element spaces based on shape regular meshes.

我们考虑用 hp 有限元对 1d-integral Dirichlet 分数拉普拉奇进行离散化。我们提出了正交方案来设置刚度矩阵和载荷向量，以保持 hp-FEM 在几何网格上的指数收敛性。这些方案基于高斯-雅可比规则和高斯-列根德规则。我们证明，取略微超过多项式阶数的正交点就足以保持根指数收敛性。建立系统的代数运算总数为 O(N5/2)，其中 N 为问题大小。数值示例说明了这一分析。我们还将分析扩展到基于形状规则网格的高维 hp 有限元空间的分数拉普拉斯。

引用次数: 0

Topology optimization design of labyrinth seal-type devices considering subsonic compressible turbulent flow conditions 考虑亚音速可压缩湍流条件的迷宫式密封装置拓扑优化设计

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-06 DOI: 10.1016/j.camwa.2024.10.029

Luís F.N. Sá , Felipe Silva Maffei , Lucas N.B.S. Ribeiro , Julio Romano Meneghini , Emílio Carlos Nelli Silva

In this work, a topology optimization model for designing devices that operate with multiple relative velocities considering turbulent compressible flows is proposed. The model consists of the Favre-averaged Navier-stokes equations in an axisymmetric domain coupled with a continuous boundary propagation model. The propagation is used to impose different solid behaviors based on which wall it is connected to, for example, solid material in contact with a rotating shaft will have a rotational velocity, while material encrusted in the support will have zero absolute velocity. The implementation is composed of a segregated solver with steps for the FANS equations, the

k - ϵ

turbulent equations, and the propagation model. The sensitivity is obtained with automatic differentiation of the adjoint method and an internal point optimizer is used to update the design variable. A study case of a labyrinth seal is defined to illustrate the methodology by using three different objective functions, maximization of radial velocity, static pressure change rate, and vorticity. The results are designs for small-scale labyrinth seals in real operation conditions.

本研究提出了一种拓扑优化模型，用于设计在考虑湍流可压缩流的情况下以多种相对速度运行的设备。该模型由轴对称域中的法夫尔平均纳维-斯托克斯方程和连续边界传播模型组成。例如，与旋转轴接触的固体材料将具有旋转速度，而包覆在支架上的材料将具有零绝对速度。该方法由一个分离式求解器组成，其步骤包括 FANS 方程、k-ϵ 湍流方程和传播模型。灵敏度通过邻接法自动微分获得，内部点优化器用于更新设计变量。通过使用三种不同的目标函数（径向速度最大化、静压变化率最大化和涡度最大化）定义了一个迷宫密封研究案例，以说明该方法。结果是在实际运行条件下对小型迷宫密封进行的设计。

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引用次数: 0

An implicit GNN solver for Poisson-like problems 泊松类问题的隐式 GNN 求解器

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-05 DOI: 10.1016/j.camwa.2024.10.036

Matthieu Nastorg , Michele-Alessandro Bucci , Thibault Faney , Jean-Marc Gratien , Guillaume Charpiat , Marc Schoenauer

This paper presents Ψ-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems on general unstructured meshes with mixed boundary conditions. By leveraging the Implicit Layer Theory, Ψ-GNN models an “infinitely” deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical pre-requisite for physical applications, and is able to adapt to any initially provided solution. Ψ-GNN is trained using a physics-informed loss, and the training process is stable by design. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, Ψ-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

本文提出了一种新颖的图神经网络（GNN）方法--Ψ-GNN，用于求解具有混合边界条件的一般非结构网格上无处不在的泊松 PDE 问题。通过利用隐含层理论，Ψ-GNN 建立了一个 "无限 "深的网络模型，从而避免了根据经验调整所需的消息传递层数以实现求解。它的原始架构明确考虑了边界条件（物理应用的一个重要前提条件），并能适应任何最初提供的解决方案。Ψ-GNN使用物理信息损失进行训练，训练过程在设计上是稳定的。此外，Ψ-GNN 方法的一致性已在理论上得到证明，其灵活性和泛化效率也已在实验中得到证实：同一学习模型可以准确处理各种尺寸的非结构网格以及不同的边界条件。据我们所知，Ψ-GNN 是第一个基于物理信息的 GNN 方法，它可以处理各种非结构域、边界条件和初始解，同时还能提供收敛保证。

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引用次数: 0

Modular parametric PGD enabling online solution of partial differential equations 实现偏微分方程在线求解的模块参数化 PGD

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-05 DOI: 10.1016/j.camwa.2024.10.037

Angelo Pasquale , Mohammad-Javad Kazemzadeh-Parsi , Daniele Di Lorenzo , Victor Champaney , Amine Ammar , Francisco Chinesta

In the present work, a new methodology is proposed for building surrogate parametric models of engineering systems based on modular assembly of pre-solved modules. Each module is a generic parametric solution considering parametric geometry, material and boundary conditions. By assembling these modules and satisfying continuity constraints at the interfaces, a parametric surrogate model of the full problem can be obtained. In the present paper, the PGD technique in connection with NURBS geometry representation is used to create a parametric model for each module. In this technique, the NURBS objects allow to map the governing boundary value problem from a parametric non-regular domain into a regular reference domain and the PGD is used to create a reduced model in the reference domain. In the assembly stage, an optimization problem is solved to satisfy the continuity constraints at the interfaces. The proposed procedure is based on the offline–online paradigm: the offline stage consists of creating multiple pre-solved modules which can be afterwards assembled in almost real-time during the online stage, enabling quick evaluations of the full system response. To show the potential of the proposed approach some numerical examples in heat conduction and structural plates under bending are presented.

在本研究中，我们提出了一种基于预先解决的模块组装的工程系统代用参数模型的新方法。每个模块都是考虑到几何参数、材料和边界条件的通用参数解决方案。通过组装这些模块并满足接口处的连续性约束，可以获得完整问题的参数代用模型。本文采用 PGD 技术和 NURBS 几何表示法为每个模块创建参数模型。在该技术中，NURBS 对象允许将支配边界值问题从参数非规则域映射到规则参考域，而 PGD 则用于在参考域中创建简化模型。在装配阶段，需要解决一个优化问题，以满足界面上的连续性约束。所提议的程序基于离线-在线范例：离线阶段包括创建多个预先解决的模块，这些模块随后可以在在线阶段几乎实时地组装起来，从而实现对整个系统响应的快速评估。为了展示所提方法的潜力，我们介绍了一些热传导和弯曲结构板的数值示例。

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引用次数: 0

Investigating the impact of vessel geometry on cerebral aneurysm formation using multi-phase blood flow models 利用多相血流模型研究血管几何形状对脑动脉瘤形成的影响

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-05 DOI: 10.1016/j.camwa.2024.10.039

Dimitrios S. Lampropoulos, Maria Hadjinicolaou

Cerebral aneurysms represent a life-threatening condition associated with considerable morbidity and mortality rates. The formation of cerebral aneurysms is influenced by various factors, including vessel geometry, blood flow characteristics, and hemodynamic forces. In this study, we investigate the impact of vessel geometry on the formation of cerebral aneurysms utilizing computational fluid dynamics (CFD) simulations for multi-phase blood flow models.

More precisely, we employ the Finite Volume Method to numerically solve the Navier-Stokes equations for simulating blood flow. To accurately capture the intricate nature of blood behavior, we utilize a multiphase blood flow model, as blood consists of red blood cells, white blood cells, and platelets suspended in the blood plasma.

Our results demonstrate that the local curvature of the vessel has a pronounced effect on the blood flow patterns and hemodynamic forces within the vessel. Specifically, our simulations indicate that an increase in vessel curvature can lead to the formation of regions of high stress and flow stagnation, both of which are known to be associated with an increased risk of aneurysm formation.

The current study provides significant insights into the impact of vessel geometry on the formation of cerebral aneurysms. The obtained results may aid in designing treatment and preventive strategies for cerebral aneurysms, while also contributing to the existing body of knowledge on the subject. Additionally, the approach developed in this study can be applied to investigate various other vascular pathologies, including arterial stenosis and atherosclerosis.

脑动脉瘤是一种威胁生命的疾病，发病率和死亡率都很高。脑动脉瘤的形成受多种因素影响，包括血管几何形状、血流特征和血液动力。在本研究中，我们利用计算流体动力学（CFD）模拟多相血流模型，研究血管几何形状对脑动脉瘤形成的影响。更确切地说，我们采用有限体积法数值求解纳维-斯托克斯方程，模拟血流。为了准确捕捉血液行为的复杂性质，我们采用了多相血流模型，因为血液由悬浮在血浆中的红细胞、白细胞和血小板组成。我们的结果表明，血管的局部曲率对血管内的血流模式和血液动力有明显影响。具体来说，我们的模拟结果表明，血管曲率的增加会导致高应力区和血流停滞区的形成，而这两种情况都与动脉瘤形成风险的增加有关。目前的研究对血管几何形状对脑动脉瘤形成的影响提供了重要的见解，所获得的结果可能有助于设计脑动脉瘤的治疗和预防策略，同时也对现有的相关知识体系有所贡献。此外，本研究开发的方法还可用于研究其他各种血管病变，包括动脉狭窄和动脉粥样硬化。

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引用次数: 0

Developing PDE-constrained optimal control of multicomponent contamination flows in porous media 开发多孔介质中多成分污染流的 PDE 约束优化控制

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-04 DOI: 10.1016/j.camwa.2024.10.033

Khan Enaet Hossain, Dong Liang, Hongmei Zhu

This paper develops a robust and efficient PDE-constrained optimal control model for multicomponent pollutions in porous media, which takes into account nonlinear multi-component contamination flows of groundwater. The objective of the pollution optimal control is to identify the optimal injection rates from the top part boundary of domain, which can minimize the least squares error between the concentrations that are simulated and the allowable observed concentrations at observation sites, combining with the affection of costs associated with reducing emissions at injection locations. To discrete the constrained governing system of nonlinear multi-component flows, the splitting improved upwind finite difference scheme is developed for multicomponent PDEs system involving nonlinear chemical reactions of multicomponent pollutants and the pollutant injected rates on the upper part boundary. We employ the differential evolution (DE) optimization algorithm to solve the optimization. We numerically demonstrate the effectiveness of our model by analyzing the flow simulation on a simple geometric aquifer and identifying the optimal injection rates by minimizing the concentration derivation and the abatement costs. We also investigate the simulation of the contamination flow in a more realistic-shaped aquifer, which further validates our model's robustness and efficacy. The developed PDE-constrained control model and algorithm can be applied to applications of groundwater pollution control.

本文针对多孔介质中的多组分污染建立了一个稳健高效的 PDE 约束优化控制模型，该模型考虑了地下水的非线性多组分污染流。污染优化控制的目标是确定域顶边界的最佳注入率，使模拟浓度与观测点允许观测浓度之间的最小二乘误差最小，同时考虑减少注入点排放的相关成本。为了离散非线性多组分流动的约束系统，我们针对多组分 PDEs 系统开发了分裂改进的上风有限差分方案，该方案涉及多组分污染物的非线性化学反应和上部边界的污染物注入率。我们采用微分演化（DE）优化算法来解决优化问题。我们通过分析简单几何含水层上的流动模拟，并通过最小化浓度推导和减排成本确定最佳注入率，从数值上证明了我们模型的有效性。我们还研究了形状更逼真的含水层中的污染流模拟，进一步验证了模型的稳健性和有效性。所开发的 PDE 受限控制模型和算法可应用于地下水污染控制领域。

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引用次数: 0

A gas-surface interaction algorithm for discrete velocity methods in predicting rarefied and multi-scale flows: For Maxwell boundary model 用于预测稀薄和多尺度流动的离散速度法的气体-表面相互作用算法：用于麦克斯韦边界模型

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-04 DOI: 10.1016/j.camwa.2024.10.034

Jianfeng Chen , Sha Liu , Yong Wang , Congshan Zhuo , Yanguang Yang , Chengwen Zhong

The discrete velocity method (DVM) for rarefied flows and the unified methods (based on the DVM framework) for flows in all regimes, from continuum one to free molecular one, have worked well as precise flow solvers over the past decades and have been successfully extended to other important physical fields. Both DVM and unified methods endeavor to model the gas-gas interaction physically. However, for the gas-surface interaction (GSI) at the wall boundary, they have only use the full accommodation boundary up to now, which can be viewed as a rough Maxwell boundary with a fixed accommodation coefficient (AC) at unity, deviating from the real value. For example, the AC for metal materials typically falls in the range of 0.8 to 0.9. To overcome this bottleneck and extend the DVM and unified methods to more physical boundary conditions, an algorithm for Maxwell boundary with an adjustable AC is established into the DVM framework. The Maxwell boundary model splits the distribution of the bounce-back molecules into specular ones and Maxwellian (normal) ones. Since the bounce-back molecules after the spectral reflection does not math with the discrete velocity space (DVS), both macroscopic conservation (from numerical quadrature) and microscopic consistency in the DVS are hard to achieve in the DVM framework. In this work, this problem is addressed by employing a combination of interpolation methods for mismatch points in DVS and an efficient numerical error correction method for micro-macro consistency. On the other hand, the current Maxwell boundary for DVM takes the generality into consideration, accommodating both the recently developed efficient unstructured velocity space and the traditional Cartesian velocity space. Moreover, the proposed algorithm allows for calculations of both monatomic gases and diatomic gases with internal degrees in DVS. Finally, by being integrated with the unified gas-kinetic scheme within the DVM framework, the performance of the present GSI algorithm is validated through a series of benchmark numerical tests across a wide range of Knudsen numbers.

过去几十年来，用于稀薄流动的离散速度法（DVM）和用于从连续流动到自由分子流动等各种流动状态的统一方法（基于 DVM 框架）作为精确的流动求解器发挥了很好的作用，并成功地扩展到其他重要的物理领域。DVM 和统一方法都致力于为气体与气体之间的相互作用建立物理模型。然而，对于壁面边界的气-面相互作用（GSI），迄今为止它们都只使用了全容纳边界，这可以看作是一个粗糙的麦克斯韦边界，其容纳系数（AC）固定为 1，偏离真实值。例如，金属材料的 AC 通常在 0.8 至 0.9 之间。为了克服这一瓶颈，并将 DVM 和统一方法扩展到更多的物理边界条件，我们在 DVM 框架中建立了可调节 AC 的麦克斯韦边界算法。麦克斯韦边界模型将反弹分子的分布分为镜面分子和麦克斯韦（正常）分子。由于光谱反射后的反弹分子与离散速度空间（DVS）不存在数学关系，因此在 DVM 框架中很难实现 DVS 的宏观守恒（来自数值正交）和微观一致性。在这项工作中，通过结合使用 DVS 中错配点的插值方法和高效的数值纠错方法来实现微观-宏观一致性，解决了这一问题。另一方面，目前用于 DVM 的麦克斯韦边界考虑到了通用性，既能容纳最近开发的高效非结构化速度空间，也能容纳传统的笛卡尔速度空间。此外，所提出的算法允许在 DVS 中计算单原子气体和具有内部度数的双原子气体。最后，通过与 DVM 框架内的统一气体动力学方案的集成，本 GSI 算法的性能通过一系列宽范围克努森数的基准数值测试得到了验证。

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引用次数: 0

Mathematical and numerical analysis of reduced order interface conditions and augmented finite elements for mixed dimensional problems 混合维度问题的减阶界面条件和增强有限元的数学与数值分析

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.camwa.2024.10.028

Muriel Boulakia , Céline Grandmont , Fabien Lespagnol , Paolo Zunino

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments.

在本文中，我们对基于虚构域方法结合降阶界面耦合条件的方法的数学特性很感兴趣，这些方法最近已被引入模拟三维-一维流体-结构或结构-结构耦合问题。为了深入了解这些方法的近似特性，我们通过考虑二维域中的泊松问题和小夹杂物上的非均质 Dirichlet 边界条件，对它们进行了简化研究。利用虚构域方法，结合与 Dirichlet 边界约束相关的拉格朗日乘数在傅里叶有限维空间上的投影，可以得到近似的简化问题，从而得到具有缺陷界面条件的泊松问题。在分析了简化问题解的存在性之后，我们证明了当孔的大小趋近于零时，它向原始完整问题的收敛性，收敛速度取决于有限维空间的模数。我们的估计结果特别强调了一个事实，即要获得拉格朗日乘数的良好收敛性，需要考虑比第一个傅里叶模式（恒定模式）更多的模式。在处理实际耦合问题（如流体-结构问题）时，这是一个关键问题。接下来，我们分析了在计算网格不适合小包容界面的情况下，使用有限元法对简化问题进行数值离散化的方法。与这类问题的标准情况一样，由于求解缺乏规律性，求解的收敛性并不理想。此外，当网格大小和包含体大小处于同一数量级时，收敛会表现出众所周知的锁定效应。这种锁定效应在增加模式数时更加明显，对拉格朗日乘法器收敛速度的影响也更大。为了解决这些问题，我们提出并分析了一种稳定方法和一种丰富方法，即在不改变拉格朗日乘法器近似空间的情况下增加额外的基函数。最后，我们通过数值实验说明了数值策略的特性。

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引用次数: 0

A nonconforming extended virtual element method for Stokes interface problems 斯托克斯界面问题的不符扩展虚拟元素法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.camwa.2024.10.027

Yuxiang Huang , Feng Wang , Jinru Chen

In this paper, we propose a nonconforming extended virtual element method, which combines the extended finite element method with the nonconforming virtual element method, for solving Stokes interface problems with the unfitted-interface mesh. By introducing some stabilization terms and penalty terms, as well as some special terms defined on non-cut edges of interface elements in the discrete bilinear form, we prove the discrete inf-sup condition and obtain optimal error estimates. It is shown that all results are not only independent of the mesh size and the viscosity coefficient, but also the interface position. Numerical experiments are performed to verify theoretical results.

本文提出了一种非拟合扩展虚元法，它将扩展有限元法与非拟合虚元法相结合，用于求解非拟合界面网格的斯托克斯界面问题。通过引入一些稳定项和惩罚项，以及离散双线性形式中定义在界面元素非切边上的一些特殊项，我们证明了离散 inf-sup 条件，并获得了最优误差估计值。结果表明，所有结果不仅与网格大小和粘度系数无关，而且与界面位置无关。我们进行了数值实验来验证理论结果。

引用次数: 0