A guide to the finite and virtual element methods for elasticity

Berbatov, K.; Lazarov, B. S.; Jivkov, A. P.

doi:10.1016/j.apnum.2021.07.010

Title: A guide to the finite and virtual element methods for elasticity

Abstract

We present a systematic description and comparison of the 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

Authors:

; Lazarov, B. S.;

Publication Date:: 2021-11-01

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE; Engineering and Physical Sciences Research Council (EPSRC)

OSTI Identifier:: 1810424

Alternate Identifier(s):: OSTI ID: 1843569

Report Number(s):: LLNL-JRNL-821129
Journal ID: ISSN 0168-9274; S0168927421002014; PII: S0168927421002014

Grant/Contract Number:: AC52-07NA27344; EP/N026136/1

Resource Type:: Published Article

Journal Name:: Applied Numerical Mathematics

Additional Journal Information:: Journal Name: Applied Numerical Mathematics Journal Volume: 169 Journal Issue: C; Journal ID: ISSN 0168-9274

Publisher:: Elsevier

Country of Publication:: Netherlands

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; linear elasticity; mixed FEM; mixed VFM; implementation; assembly

Citation Formats


                    Berbatov, K., Lazarov, B. S., and Jivkov, A. P. A guide to the finite and virtual element methods for elasticity.  Netherlands: N. p., 2021. 
Web.  doi:10.1016/j.apnum.2021.07.010.

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                    Berbatov, K., Lazarov, B. S., & Jivkov, A. P. A guide to the finite and virtual element methods for elasticity.  Netherlands.  https://doi.org/10.1016/j.apnum.2021.07.010

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                    Berbatov, K., Lazarov, B. S., and Jivkov, A. P. Mon .  
"A guide to the finite and virtual element methods for elasticity".  Netherlands.  https://doi.org/10.1016/j.apnum.2021.07.010.

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@article{osti_1810424,

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

  author       = {Berbatov, K. and Lazarov, B. S. and Jivkov, A. P.},

  abstractNote = {We present a systematic description and comparison of the 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},

  doi          = {10.1016/j.apnum.2021.07.010},

  journal      = {Applied Numerical Mathematics},

  number       = C,

  volume       = 169,

  place        = {Netherlands},

  year         = {2021},

  month        = {11}

}

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Title: A guide to the finite and virtual element methods for elasticity

Abstract

Citation Formats

A stress/displacement Virtual Element method for plane elasticity problems journal, October 2017

PEERS: A new mixed finite element for plane elasticity journal, December 1984

A posteriori error estimates for the virtual element method journal, May 2017

Some basic formulations of the virtual element method (VEM) for finite deformations journal, May 2017

Variational methods for the solution of problems of equilibrium and vibrations journal, January 1943

A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems journal, June 2020

Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach journal, October 2020

Reduced symmetry elements in linear elasticity journal, January 2009

On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes journal, December 2014

A mixed formulation for the finite element solution of contact problems journal, February 1992

A Virtual Element Method for elastic and inelastic problems on polytope meshes journal, October 2015

Gaussian quadrature formulas for triangles journal, January 1973

Mixed stabilized finite element methods in nonlinear solid mechanics journal, August 2010

A computational framework for brittle crack-propagation based on efficient virtual element method journal, July 2019

The virtual element method in 50 lines of MATLAB journal, December 2016

Mixed finite element methods for elliptic problems journal, September 1990

Topology optimization of incompressible media using mixed finite elements journal, July 2007

A Mixed Finite Element Method for Elasticity in Three Dimensions journal, December 2005

The Hitchhiker's Guide to the Virtual Element Method journal, May 2014

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Finite elements for symmetric tensors in three dimensions journal, September 2008

On Locking and Robustness in the Finite Element Method journal, October 1992

A Virtual Element Method for 2D linear elastic fracture analysis journal, October 2018

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case journal, July 2016

Mixed finite element methods for linear elasticity with weakly imposed symmetry journal, October 2007

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Basic Principles of Virtual Element Methods journal, November 2012

Mixed virtual element methods for general second order elliptic problems on polygonal meshes journal, May 2016

Virtual Elements for Linear Elasticity Problems journal, January 2013

Topology optimization using polytopes journal, August 2015

Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem journal, June 2017

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A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle journal, October 2018

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An engineering perspective to the virtual element method and its interplay with the standard finite element method journal, June 2019

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A stress/displacement Virtual Element method for plane elasticity problems
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PEERS: A new mixed finite element for plane elasticity
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A posteriori error estimates for the virtual element method
journal, May 2017

Some basic formulations of the virtual element method (VEM) for finite deformations
journal, May 2017

Variational methods for the solution of problems of equilibrium and vibrations
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A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems
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Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach
journal, October 2020

Reduced symmetry elements in linear elasticity
journal, January 2009

On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes
journal, December 2014

A mixed formulation for the finite element solution of contact problems
journal, February 1992

A Virtual Element Method for elastic and inelastic problems on polytope meshes
journal, October 2015

Gaussian quadrature formulas for triangles
journal, January 1973

Mixed stabilized finite element methods in nonlinear solid mechanics
journal, August 2010

A computational framework for brittle crack-propagation based on efficient virtual element method
journal, July 2019

The virtual element method in 50 lines of MATLAB
journal, December 2016

Mixed finite element methods for elliptic problems
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Topology optimization of incompressible media using mixed finite elements
journal, July 2007

A Mixed Finite Element Method for Elasticity in Three Dimensions
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The Hitchhiker's Guide to the Virtual Element Method
journal, May 2014

Mixed finite elements for elasticity
journal, September 2002

Finite elements for symmetric tensors in three dimensions
journal, September 2008

On Locking and Robustness in the Finite Element Method
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A Virtual Element Method for 2D linear elastic fracture analysis
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Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case
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Mixed finite element methods for linear elasticity with weakly imposed symmetry
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Topology optimization with mixed finite elements on regular grids
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Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem
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Basic Principles of Virtual Element Methods
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Mixed virtual element methods for general second order elliptic problems on polygonal meshes
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Virtual Elements for Linear Elasticity Problems
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Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem
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The Arnold–Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification
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A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle
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Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity
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