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Title: Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities

Abstract

The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes. In the VEM, the basis functions are implicit (virtual) — they are not known nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form at the element-level into two parts: a consistent term that reproduces a given polynomial space and a correction term that ensures stability. In this study, we consider a low-order extended virtual element method (X-VEM) that is in the spirit of the extended finite element method for crack problems. Herein, we focus on the two-dimensional Laplace crack problem. In the X-VEM, we enrich the standard virtual element space with additional discontinuous functions through the framework of partition-of-unity. The nodal basis functions in the VEM are chosen as the partition-of-unity functions, and we study means to stabilize the standard and enriched sub-matrices that constitute the element stiffness matrix. Numerical experiments are performed on the problem of a cracked membrane under mode III loading to affirm the accuracy, and to establish the optimal convergence in energy of the method.

Authors:
 [1];  [2];  [2];  [3]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Univ. di Ferrara, Ferrara (Italy)
  3. Univ. of California, Davis, CA (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1434461
Report Number(s):
LA-UR-18-23443
DOE Contract Number:  
AC52-06NA25396
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
mathematics; enrichment; partition of unity; VEM; polygonal mesh; crack

Citation Formats

Manzini, Gianmarco, Benvenuti, Elena, Chiozzi, Andrea, and Sukumar, N. Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities. United States: N. p., 2018. Web. doi:10.2172/1434461.
Manzini, Gianmarco, Benvenuti, Elena, Chiozzi, Andrea, & Sukumar, N. Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities. United States. doi:10.2172/1434461.
Manzini, Gianmarco, Benvenuti, Elena, Chiozzi, Andrea, and Sukumar, N. Fri . "Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities". United States. doi:10.2172/1434461. https://www.osti.gov/servlets/purl/1434461.
@article{osti_1434461,
title = {Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities},
author = {Manzini, Gianmarco and Benvenuti, Elena and Chiozzi, Andrea and Sukumar, N.},
abstractNote = {The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes. In the VEM, the basis functions are implicit (virtual) — they are not known nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form at the element-level into two parts: a consistent term that reproduces a given polynomial space and a correction term that ensures stability. In this study, we consider a low-order extended virtual element method (X-VEM) that is in the spirit of the extended finite element method for crack problems. Herein, we focus on the two-dimensional Laplace crack problem. In the X-VEM, we enrich the standard virtual element space with additional discontinuous functions through the framework of partition-of-unity. The nodal basis functions in the VEM are chosen as the partition-of-unity functions, and we study means to stabilize the standard and enriched sub-matrices that constitute the element stiffness matrix. Numerical experiments are performed on the problem of a cracked membrane under mode III loading to affirm the accuracy, and to establish the optimal convergence in energy of the method.},
doi = {10.2172/1434461},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri Apr 20 00:00:00 EDT 2018},
month = {Fri Apr 20 00:00:00 EDT 2018}
}

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