A guide to the design of the virtual element methods for second- and fourth-order partial differential equations
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 h...
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AIMS Press
2023-11-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mine.2023100?viewType=HTML |
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| author | Yu Leng Lampros Svolos Dibyendu Adak Ismael Boureima Gianmarco Manzini Hashem Mourad Jeeyeon Plohr |
| author_facet | Yu Leng Lampros Svolos Dibyendu Adak Ismael Boureima Gianmarco Manzini Hashem Mourad Jeeyeon Plohr |
| author_sort | Yu Leng |
| collection | DOAJ |
| description | 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 <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>,<xref ref-type="bibr" rid="b3">3</xref>]</sup>, 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. |
| first_indexed | 2024-03-08T17:24:15Z |
| format | Article |
| id | doaj.art-fd38e71e028c4c0995c278ceb748bffa |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-03-08T17:24:15Z |
| publishDate | 2023-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-fd38e71e028c4c0995c278ceb748bffa2024-01-03T02:49:45ZengAIMS PressMathematics in Engineering2640-35012023-11-015612210.3934/mine.2023100A guide to the design of the virtual element methods for second- and fourth-order partial differential equationsYu Leng 0Lampros Svolos1 Dibyendu Adak2Ismael Boureima 3Gianmarco Manzini 4Hashem Mourad5Jeeyeon Plohr61. T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA1. T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA2. T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA1. T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA2. T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA1. T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA3. XCP-5, Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USAWe 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 <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>,<xref ref-type="bibr" rid="b3">3</xref>]</sup>, 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.https://www.aimspress.com/article/doi/10.3934/mine.2023100?viewType=HTMLfracture mechanicshigh-order phase field modelsvirtual element methodarbitrary-order approximations |
| spellingShingle | Yu Leng Lampros Svolos Dibyendu Adak Ismael Boureima Gianmarco Manzini Hashem Mourad Jeeyeon Plohr A guide to the design of the virtual element methods for second- and fourth-order partial differential equations Mathematics in Engineering fracture mechanics high-order phase field models virtual element method arbitrary-order approximations |
| title | A guide to the design of the virtual element methods for second- and fourth-order partial differential equations |
| title_full | A guide to the design of the virtual element methods for second- and fourth-order partial differential equations |
| title_fullStr | A guide to the design of the virtual element methods for second- and fourth-order partial differential equations |
| title_full_unstemmed | A guide to the design of the virtual element methods for second- and fourth-order partial differential equations |
| title_short | A guide to the design of the virtual element methods for second- and fourth-order partial differential equations |
| title_sort | guide to the design of the virtual element methods for second and fourth order partial differential equations |
| topic | fracture mechanics high-order phase field models virtual element method arbitrary-order approximations |
| url | https://www.aimspress.com/article/doi/10.3934/mine.2023100?viewType=HTML |
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