A guide to the finite and virtual element methods for elasticity

Kiprian Berbatov; Boyan Lazarov; Andrey Jivkov

doi:10.1016/j.apnum.2021.07.010

A guide to the finite and virtual element methods for elasticity

Kiprian Berbatov, Boyan Lazarov, Andrey Jivkov

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Abstract

We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.

Original language	English
Pages (from-to)	351-395
Number of pages	45
Journal	Applied Numerical Mathematics
Volume	169
Early online date	19 Jul 2021
DOIs	https://doi.org/10.1016/j.apnum.2021.07.010
Publication status	Published - Nov 2021

Keywords

Linear elasticity
(Mixed) FEM
(Mixed) VEM
Implementation
Assembly

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Advanced materials

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10.1016/j.apnum.2021.07.010

APNUM2021Final published version, 1.9 MBLicence: CC BY

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title = "A guide to the finite and virtual element methods for elasticity",

abstract = "We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.",

keywords = "Linear elasticity, (Mixed) FEM, (Mixed) VEM, Implementation, Assembly",

author = "Kiprian Berbatov and Boyan Lazarov and Andrey Jivkov",

note = "Funding Information: The authors acknowledge gratefully the support of EPSRC via grant EP/N026136/1 “Geometric Mechanics of Solids”. The work of the second author was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 . Funding Information: This document is the results of the research project funded by the Engineering and Physical Sciences Research Council. Publisher Copyright: {\textcopyright} 2021 The Authors",

year = "2021",

month = nov,

doi = "10.1016/j.apnum.2021.07.010",

language = "English",

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journal = "Applied Numerical Mathematics",

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AU - Berbatov, Kiprian

AU - Lazarov, Boyan

AU - Jivkov, Andrey

N1 - Funding Information: The authors acknowledge gratefully the support of EPSRC via grant EP/N026136/1 “Geometric Mechanics of Solids”. The work of the second author was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 . Funding Information: This document is the results of the research project funded by the Engineering and Physical Sciences Research Council. Publisher Copyright: © 2021 The Authors

PY - 2021/11

Y1 - 2021/11

N2 - We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.

AB - We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.

KW - Linear elasticity

KW - (Mixed) FEM

KW - (Mixed) VEM

KW - Implementation

KW - Assembly

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U2 - 10.1016/j.apnum.2021.07.010

DO - 10.1016/j.apnum.2021.07.010

M3 - Article

SN - 0168-9274

VL - 169

SP - 351

EP - 395

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -

A guide to the finite and virtual element methods for elasticity

Abstract

Keywords

Research Beacons, Institutes and Platforms

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