# Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics

## Abstract

We analyze several new and existing approaches for limiting tensor quantities in the context of deviatoric stress remapping in an ALE numerical simulation of elastic flow. Remapping and limiting of the tensor component-by-component is shown to violate radial symmetry of derived variables such as elastic energy or force. Therefore, we have extended the symmetry-preserving Vector Image Polygon algorithm, originally designed for limiting vector variables. This limiter constrains the vector (in our case a vector of independent tensor components) within the convex hull formed by the vectors from surrounding cells – an equivalent of the discrete maximum principle in scalar variables. We compare this method with a limiter designed specifically for deviatoric stress limiting which aims to constrain the J _{2} invariant that is proportional to the specific elastic energy and scale the tensor accordingly. We also propose a method which involves remapping and limiting the J _{2} invariant independently using known scalar techniques. The deviatoric stress tensor is then scaled to match this remapped invariant, which guarantees conservation in terms of elastic energy.

- Authors:

- Czech Technical Univ. in Prague, Praha (Czech Republic)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1338785

- Report Number(s):
- LA-UR-17-20068

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Klima, Matej, Kucharik, MIlan, Shashkov, Mikhail Jurievich, and Velechovsky, Jan.
```*Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics*. United States: N. p., 2017.
Web. doi:10.2172/1338785.

```
Klima, Matej, Kucharik, MIlan, Shashkov, Mikhail Jurievich, & Velechovsky, Jan.
```*Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics*. United States. doi:10.2172/1338785.

```
Klima, Matej, Kucharik, MIlan, Shashkov, Mikhail Jurievich, and Velechovsky, Jan. Fri .
"Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics". United States.
doi:10.2172/1338785. https://www.osti.gov/servlets/purl/1338785.
```

```
@article{osti_1338785,
```

title = {Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics},

author = {Klima, Matej and Kucharik, MIlan and Shashkov, Mikhail Jurievich and Velechovsky, Jan},

abstractNote = {We analyze several new and existing approaches for limiting tensor quantities in the context of deviatoric stress remapping in an ALE numerical simulation of elastic flow. Remapping and limiting of the tensor component-by-component is shown to violate radial symmetry of derived variables such as elastic energy or force. Therefore, we have extended the symmetry-preserving Vector Image Polygon algorithm, originally designed for limiting vector variables. This limiter constrains the vector (in our case a vector of independent tensor components) within the convex hull formed by the vectors from surrounding cells – an equivalent of the discrete maximum principle in scalar variables. We compare this method with a limiter designed specifically for deviatoric stress limiting which aims to constrain the J2 invariant that is proportional to the specific elastic energy and scale the tensor accordingly. We also propose a method which involves remapping and limiting the J2 invariant independently using known scalar techniques. The deviatoric stress tensor is then scaled to match this remapped invariant, which guarantees conservation in terms of elastic energy.},

doi = {10.2172/1338785},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri Jan 06 00:00:00 EST 2017},

month = {Fri Jan 06 00:00:00 EST 2017}

}