# A Cartesian parametrization for the numerical analysis of material instability

## Abstract

We examine four parametrizations of the unit sphere in the context of material stability analysis by means of the singularity of the acoustic tensor. We then propose a Cartesian parametrization for vectors that lie a cube of side length two and use these vectors in lieu of unit normals to test for the loss of the ellipticity condition. This parametrization is then used to construct a tensor akin to the acoustic tensor. It is shown that both of these tensors become singular at the same time and in the same planes in the presence of a material instability. Furthermore, the performance of the Cartesian parametrization is compared against the other parametrizations, with the results of these comparisons showing that in general, the Cartesian parametrization is more robust and more numerically efficient than the others.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Clemson Univ., Clemson, SC (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

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

- OSTI Identifier:
- 1262242

- Report Number(s):
- SAND-2015-9106J

Journal ID: ISSN 0029-5981; 643723

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal for Numerical Methods in Engineering

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

- Publisher:
- Wiley

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE

### Citation Formats

```
Mota, Alejandro, Chen, Qiushi, Foulk, III, James W., Ostien, Jakob T., and Lai, Zhengshou.
```*A Cartesian parametrization for the numerical analysis of material instability*. United States: N. p., 2016.
Web. doi:10.1002/nme.5228.

```
Mota, Alejandro, Chen, Qiushi, Foulk, III, James W., Ostien, Jakob T., & Lai, Zhengshou.
```*A Cartesian parametrization for the numerical analysis of material instability*. United States. doi:10.1002/nme.5228.

```
Mota, Alejandro, Chen, Qiushi, Foulk, III, James W., Ostien, Jakob T., and Lai, Zhengshou. Thu .
"A Cartesian parametrization for the numerical analysis of material instability". United States. doi:10.1002/nme.5228. https://www.osti.gov/servlets/purl/1262242.
```

```
@article{osti_1262242,
```

title = {A Cartesian parametrization for the numerical analysis of material instability},

author = {Mota, Alejandro and Chen, Qiushi and Foulk, III, James W. and Ostien, Jakob T. and Lai, Zhengshou},

abstractNote = {We examine four parametrizations of the unit sphere in the context of material stability analysis by means of the singularity of the acoustic tensor. We then propose a Cartesian parametrization for vectors that lie a cube of side length two and use these vectors in lieu of unit normals to test for the loss of the ellipticity condition. This parametrization is then used to construct a tensor akin to the acoustic tensor. It is shown that both of these tensors become singular at the same time and in the same planes in the presence of a material instability. Furthermore, the performance of the Cartesian parametrization is compared against the other parametrizations, with the results of these comparisons showing that in general, the Cartesian parametrization is more robust and more numerically efficient than the others.},

doi = {10.1002/nme.5228},

journal = {International Journal for Numerical Methods in Engineering},

number = ,

volume = ,

place = {United States},

year = {Thu Feb 25 00:00:00 EST 2016},

month = {Thu Feb 25 00:00:00 EST 2016}

}

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