# 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:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. di Ferrara, Ferrara (Italy)
- 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}

}