Numerical integration in the virtual element method with the scaled boundary cubature scheme

Chin, Eric B.; Dassi, Franco; Manzini, Gianmarco; Sukumar, N.

doi:10.1002/nme.7549

Numerical integration in the virtual element method with the scaled boundary cubature scheme

Journal Article · Thu Jul 04 04:00:00 EDT 2024 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.7549· OSTI ID:2391042

Chin, Eric B. ^[1]; ^[2]; ^[3]; ^[4]

Lawrence Livermore National Laboratory Livermore California USA
Dipartimento di Matematica e Applicazioni Università di Milano‐Bicocca Milan Italy
Theoretical Division Los Alamos National Laboratory Los Alamos New Mexico USA
Department of Civil and Environmental Engineering University of California Davis California USA

Abstract

The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

View Accepted Manuscript (Publisher)

Sponsoring Organization:: USDOE

Grant/Contract Number:: 89233218CNA000001; AC52-07NA27344

OSTI ID:: 2391042

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering; ISSN 0029-5981

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

References (29)

Virtual element methods for parabolic problems on polygonal meshes: Vem For Parabolic Problems Vacca, Giuseppe; Beirão da Veiga, Lourenco Numerical Methods for Partial Differential Equations, Vol. 31, Issue 6 https://doi.org/10.1002/num.21982	journal	July 2015
Virtual Element Methods in Engineering Sciences Wriggers, Peter; Aldakheel, Fadi; Hudobivnik, Blaž https://doi.org/10.1007/978-3-031-39255-9	book	October 2023
Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics Wachspress, Eugene https://doi.org/10.1007/978-3-319-21614-0	book	October 2015
Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains Antonietti, Paola F.; Cangiani, Andrea; Collis, Joe Lecture Notes in Computational Science and Engineering, p. 281-310 https://doi.org/10.1007/978-3-319-41640-3_9	book	October 2016
Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra Chin, Eric B.; Lasserre, Jean B.; Sukumar, N. Computational Mechanics, Vol. 56, Issue 6 https://doi.org/10.1007/s00466-015-1213-7	journal	October 2015
Product Gauss cubature over polygons based on Green’s integration formula Sommariva, A.; Vianello, M. BIT Numerical Mathematics, Vol. 47, Issue 2 https://doi.org/10.1007/s10543-007-0131-2	journal	May 2007
Bricks for the mixed high-order virtual element method: Projectors and differential operators Dassi, F.; Vacca, G. Applied Numerical Mathematics, Vol. 155 https://doi.org/10.1016/j.apnum.2019.03.014	journal	September 2020
Gauss–Green cubature and moment computation over arbitrary geometries Sommariva, Alvise; Vianello, Marco Journal of Computational and Applied Mathematics, Vol. 231, Issue 2 https://doi.org/10.1016/j.cam.2009.05.014	journal	September 2009
High-order Virtual Element Method on polyhedral meshes Beirão da Veiga, L.; Dassi, F.; Russo, A. Computers & Mathematics with Applications, Vol. 74, Issue 5 https://doi.org/10.1016/j.camwa.2017.03.021	journal	September 2017
A C1 Virtual Element Method on polyhedral meshes Beirão da Veiga, L.; Dassi, F.; Russo, A. Computers & Mathematics with Applications, Vol. 79, Issue 7 https://doi.org/10.1016/j.camwa.2019.06.019	journal	April 2020
Bend 3d mixed virtual element method for Darcy problems Dassi, Franco; Fumagalli, Alessio; Scotti, Anna Computers & Mathematics with Applications, Vol. 119 https://doi.org/10.1016/j.camwa.2022.05.023	journal	August 2022
Virtual Element Methods for plate bending problems Brezzi, Franco; Marini, L. Donatella Computer Methods in Applied Mechanics and Engineering, Vol. 253 https://doi.org/10.1016/j.cma.2012.09.012	journal	January 2013
A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle Artioli, E.; de Miranda, S.; Lovadina, C. Computer Methods in Applied Mechanics and Engineering, Vol. 340 https://doi.org/10.1016/j.cma.2018.06.020	journal	October 2018
Extended virtual element method for the Laplace problem with singularities and discontinuities Benvenuti, E.; Chiozzi, A.; Manzini, G. Computer Methods in Applied Mechanics and Engineering, Vol. 356 https://doi.org/10.1016/j.cma.2019.07.028	journal	November 2019
A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems Dassi, F.; Lovadina, C.; Visinoni, M. Computer Methods in Applied Mechanics and Engineering, Vol. 364 https://doi.org/10.1016/j.cma.2020.112910	journal	June 2020
Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries Chin, Eric B.; Sukumar, N. Computer Methods in Applied Mechanics and Engineering, Vol. 380 https://doi.org/10.1016/j.cma.2021.113796	journal	July 2021
The mixed virtual element method on curved edges in two dimensions Dassi, Franco; Fumagalli, Alessio; Losapio, Davide Computer Methods in Applied Mechanics and Engineering, Vol. 386 https://doi.org/10.1016/j.cma.2021.114098	journal	December 2021
Extended virtual element method for two-dimensional linear elastic fracture Benvenuti, E.; Chiozzi, A.; Manzini, G. Computer Methods in Applied Mechanics and Engineering, Vol. 390 https://doi.org/10.1016/j.cma.2021.114352	journal	February 2022
Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates Arnold, D. N.; Brezzi, F. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 19, Issue 1 https://doi.org/10.1051/m2an/1985190100071	journal	January 1985
Divergence free virtual elements for the stokes problem on polygonal meshes da Veiga, Lourenco Beirão; Lovadina, Carlo; Vacca, Giuseppe ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 2 https://doi.org/10.1051/m2an/2016032	journal	February 2017
The Virtual Element Method with curved edges Beirão da Veiga, L.; Russo, A.; Vacca, G. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 53, Issue 2 https://doi.org/10.1051/m2an/2018052	journal	March 2019
VORO++ : A three-dimensional Voronoi cell library in C++ Rycroft, Chris H. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 19, Issue 4 https://doi.org/10.1063/1.3215722	journal	December 2009
Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$ Wang, Ming; Xu, Jinchao Mathematics of Computation, Vol. 82, Issue 281 https://doi.org/10.1090/S0025-5718-2012-02611-1	journal	June 2012
A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes Antonietti, P. F.; da Veiga, L. Beira͂o; Mora, D. SIAM Journal on Numerical Analysis, Vol. 52, Issue 1 https://doi.org/10.1137/13091141X	journal	January 2014
Basic Principles of Virtual Element Methods BeirÃO Da Veiga, L.; Brezzi, F.; Cangiani, A. Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 01 https://doi.org/10.1142/S0218202512500492	journal	November 2012
The Hitchhiker's Guide to the Virtual Element Method Beirão da Veiga, L.; Brezzi, F.; Marini, L. D. Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 08 https://doi.org/10.1142/S021820251440003X	journal	May 2014
The Stokes complex for Virtual Elements in three dimensions Beirão da Veiga, L.; Dassi, F.; Vacca, G. Mathematical Models and Methods in Applied Sciences, Vol. 30, Issue 03 https://doi.org/10.1142/S0218202520500128	journal	March 2020
Hybridization of the virtual element method for linear elasticity problems Dassi, Franco; Lovadina, Carlo; Visinoni, Michele Mathematical Models and Methods in Applied Sciences, Vol. 31, Issue 14 https://doi.org/10.1142/S0218202521500676	journal	December 2021
Vem++, a c++ library to handle and play with the Virtual Element Method Dassi, Franco https://doi.org/10.21203/rs.3.rs-4432259/v1	preprint	June 2024

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