A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-01-01 DOI:10.3934/mine.2023100

Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, Jeeyeon Plohr

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引用次数: 0

Abstract

We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by ^[1,2,3], we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $ C^0 $ and $ C^1 $ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

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二阶和四阶偏微分方程虚元法的设计指南

我们讨论了弹性材料断裂扩展相场建模中出现的两个偏微分方程数值逼近的两种一致性虚元方法的设计和实现细节。两个偏微分方程是:(i)描述动量平衡的线性双曲方程和(ii)模拟材料损伤的四阶椭圆方程。受[<xref - ref-type="bibr" rid="b1">1</xref>，<xref -type="bibr" rid="b2">2</xref>，<xref -type="bibr" rid="b3">3</xref>]的启发，我们开发了一种新的符合VEM，用于两个方程的离散化，该模型易于实现，即利用单个投影算子可以实现不同的项。对于二阶和四阶偏微分方程，我们分别使用C^0和C^1虚元。对于这两个方程，我们回顾了虚元近似的公式，并讨论了有关实现的细节。</ </abstract>

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.