# Continuously differentiable PIC shape functions for triangular meshes

## Abstract

In this study, a new class of continuously-differentiable shape functions is developed and applied to two-dimensional electrostatic PIC simulation on an unstructured simplex (triangle) mesh. It is shown that troublesome aliasing instabilities are avoided for cold plasma simulation in which the Debye length is as small as 0.01 cell sizes. These new shape functions satisfy all requirements for PIC particle shape. They are non-negative, have compact support, and partition unity. They are given explicitly by cubic expressions in the usual triangle logical (areal) coordinates. The shape functions are not finite elements because their structure depends on the topology of the mesh, in particular, the number of triangles neighboring each mesh vertex. Nevertheless, they may be useful as approximations to solution of other problems in which continuity of derivatives is required or desired.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1429644

- Report Number(s):
- SAND-2017-10597J

Journal ID: ISSN 0021-9991; 657436

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 362; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Particle-in-cell; Unstructured mesh; Triangle mesh; Implicit; Continuous derivative

### Citation Formats

```
Barnes, D. C.
```*Continuously differentiable PIC shape functions for triangular meshes*. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.02.002.

```
Barnes, D. C.
```*Continuously differentiable PIC shape functions for triangular meshes*. United States. doi:10.1016/j.jcp.2018.02.002.

```
Barnes, D. C. Wed .
"Continuously differentiable PIC shape functions for triangular meshes". United States.
doi:10.1016/j.jcp.2018.02.002.
```

```
@article{osti_1429644,
```

title = {Continuously differentiable PIC shape functions for triangular meshes},

author = {Barnes, D. C.},

abstractNote = {In this study, a new class of continuously-differentiable shape functions is developed and applied to two-dimensional electrostatic PIC simulation on an unstructured simplex (triangle) mesh. It is shown that troublesome aliasing instabilities are avoided for cold plasma simulation in which the Debye length is as small as 0.01 cell sizes. These new shape functions satisfy all requirements for PIC particle shape. They are non-negative, have compact support, and partition unity. They are given explicitly by cubic expressions in the usual triangle logical (areal) coordinates. The shape functions are not finite elements because their structure depends on the topology of the mesh, in particular, the number of triangles neighboring each mesh vertex. Nevertheless, they may be useful as approximations to solution of other problems in which continuity of derivatives is required or desired.},

doi = {10.1016/j.jcp.2018.02.002},

journal = {Journal of Computational Physics},

number = ,

volume = 362,

place = {United States},

year = {Wed Mar 21 00:00:00 EDT 2018},

month = {Wed Mar 21 00:00:00 EDT 2018}

}