# HYDROCODE SENSITIVITIES BY MEANS OF AUTOMATIC DIFFERENTIATION

## Abstract

The purpose of this project has been to provide sensitivities of results from an Eulerian hydrodynamics computer code (hydrocode) for use in design-optimization and uncertainty analyses. We began by applying an equation-based sensitivity technique used successfully in the early eighties that was applied to reactor-safety thermal-hydraulics problems, which is called Differential Sensitivity Theory (DST). The methodology is as follows: the system of partial differential equations (the forward or physical PDEs) is assembled, and differentiated with respect to the model parameters of interest; the adjoint equations are then determined using the inner-product rules of Hilbert spaces; and finally, the resulting adjoint PDEs are solved using straightforward numerical operators. The forward-variable solutions when needed for the adjoint solutions are provided by the original computer code that solves the physical (or forward) problem. In the present hydrocode application, acceptable results were obtained for one-material, one-dimensional problems. The DST results were then improved by means of ''compatible'' finite difference operators. We have seen, however, that DST techniques do not produce accurate values for sensitivities to all of the parameters of interest and for problems with discontinuities such as a multi-material problem. To obtain accurate sensitivities for arbitrary numerical resolution a more code-based approach wasmore »

- Authors:

- Publication Date:

- Research Org.:
- Los Alamos National Lab., NM (US)

- Sponsoring Org.:
- US Department of Energy (US)

- OSTI Identifier:
- 774349

- Report Number(s):
- LA-UR-01-514

TRN: US0102612

- DOE Contract Number:
- W-7405-ENG-36

- Resource Type:
- Conference

- Resource Relation:
- Conference: Conference title not supplied, Conference location not supplied, Conference dates not supplied; Other Information: PBD: 1 Jan 2001

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ACCURACY; COMPUTER CODES; HILBERT SPACE; HYDRODYNAMICS; PARTIAL DIFFERENTIAL EQUATIONS; SENSITIVITY; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
R. HENNINGER, A. CARLE, and P. MAUDLIN.
```*HYDROCODE SENSITIVITIES BY MEANS OF AUTOMATIC DIFFERENTIATION*. United States: N. p., 2001.
Web.

```
R. HENNINGER, A. CARLE, & P. MAUDLIN.
```*HYDROCODE SENSITIVITIES BY MEANS OF AUTOMATIC DIFFERENTIATION*. United States.

```
R. HENNINGER, A. CARLE, and P. MAUDLIN. Mon .
"HYDROCODE SENSITIVITIES BY MEANS OF AUTOMATIC DIFFERENTIATION". United States. https://www.osti.gov/servlets/purl/774349.
```

```
@article{osti_774349,
```

title = {HYDROCODE SENSITIVITIES BY MEANS OF AUTOMATIC DIFFERENTIATION},

author = {R. HENNINGER and A. CARLE and P. MAUDLIN},

abstractNote = {The purpose of this project has been to provide sensitivities of results from an Eulerian hydrodynamics computer code (hydrocode) for use in design-optimization and uncertainty analyses. We began by applying an equation-based sensitivity technique used successfully in the early eighties that was applied to reactor-safety thermal-hydraulics problems, which is called Differential Sensitivity Theory (DST). The methodology is as follows: the system of partial differential equations (the forward or physical PDEs) is assembled, and differentiated with respect to the model parameters of interest; the adjoint equations are then determined using the inner-product rules of Hilbert spaces; and finally, the resulting adjoint PDEs are solved using straightforward numerical operators. The forward-variable solutions when needed for the adjoint solutions are provided by the original computer code that solves the physical (or forward) problem. In the present hydrocode application, acceptable results were obtained for one-material, one-dimensional problems. The DST results were then improved by means of ''compatible'' finite difference operators. We have seen, however, that DST techniques do not produce accurate values for sensitivities to all of the parameters of interest and for problems with discontinuities such as a multi-material problem. To obtain accurate sensitivities for arbitrary numerical resolution a more code-based approach was then tried. We attempted to apply automatic differentiation (AD) in the forward mode using Automatic Differentiation of Fortran (ADIFOR, version 2.0) and the Tangent-linear and Adjoint Model Compiler (TAMC) in the forward and adjoint modes. We were successful for one-dimensional problems in both modes but failed to obtain accurate sensitivities in the adjoint mode for two-dimensional problem. Here we present the successful results for two-dimensional problems in both the forward and adjoint modes using ADIFOR, version 3.0. In what follows, we describe AD methods in the context of their use for a hydrocode. We then examine setup time, results, accuracy, and computer run times for three test problems obtained by ADIFOR. Finally, we outline our plans for future work.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2001},

month = {1}

}