# A Newton solution for the Superhomogenization method: The PJFNK-SPH

## Abstract

This work presents two novel topics regarding the Superhomogenization method: 1) the formalism for the implementation of the method with the linear Boltzmann Transport Equation, and 2) a Newton algorithm for the solution of the nonlinear problem that arises from the method. These new ideas have been implemented in a continuous finite element discretization in the MAMMOTH reactor physics application. The traditional solution strategy for this nonlinear problem uses a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE’s Preconditioned Jacobian-Free Newton Krylov method to allow for a direct solution. The PJFNK-SPH can converge problems that were either intractable or very difficult to converge with the traditional iterative approach, including geometries with reflectors and vacuum boundary conditions. This is partly due to the underlying Scalable Nonlinear Equations Solvers in PETSc, which are integral to MOOSE and offer Newton damping, line search and trust region methods. The PJFNK-SPH has been implemented and tested for various discretizations of the transport equation included in the Rattlesnake transport solver. Speedups of five times for diffusion and ten to fifteen times for transport were obtained when compared to the traditional Picard approach. The three test problems cover a wide range of applications includingmore »

- Authors:

- Idaho National Lab. (INL), Idaho Falls, ID (United States)
- Polytechnique Montréal, QC (Canada)

- Publication Date:

- Research Org.:
- Idaho National Lab. (INL), Idaho Falls, ID (United States)

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE)

- OSTI Identifier:
- 1469340

- Alternate Identifier(s):
- OSTI ID: 1476738

- Report Number(s):
- INL/JOU-17-41953-Rev000

Journal ID: ISSN 0306-4549; PII: S0306454917303079

- Grant/Contract Number:
- AC07-05ID14517

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Annals of Nuclear Energy (Oxford)

- Additional Journal Information:
- Journal Volume: 111; Journal ID: ISSN 0306-4549

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 22 GENERAL STUDIES OF NUCLEAR REACTORS; Superhomogenization; equivalence; homogenization; Rattlesnake; PJFNK; MOOSE; MAMMOTH

### Citation Formats

```
Ortensi, Javier, Wang, Yaqi, Laurier, Alexandre, Schunert, Sebastian, Hébert, Alain, and DeHart, Mark.
```*A Newton solution for the Superhomogenization method: The PJFNK-SPH*. United States: N. p., 2017.
Web. doi:10.1016/j.anucene.2017.09.027.

```
Ortensi, Javier, Wang, Yaqi, Laurier, Alexandre, Schunert, Sebastian, Hébert, Alain, & DeHart, Mark.
```*A Newton solution for the Superhomogenization method: The PJFNK-SPH*. United States. doi:10.1016/j.anucene.2017.09.027.

```
Ortensi, Javier, Wang, Yaqi, Laurier, Alexandre, Schunert, Sebastian, Hébert, Alain, and DeHart, Mark. Tue .
"A Newton solution for the Superhomogenization method: The PJFNK-SPH". United States. doi:10.1016/j.anucene.2017.09.027. https://www.osti.gov/servlets/purl/1469340.
```

```
@article{osti_1469340,
```

title = {A Newton solution for the Superhomogenization method: The PJFNK-SPH},

author = {Ortensi, Javier and Wang, Yaqi and Laurier, Alexandre and Schunert, Sebastian and Hébert, Alain and DeHart, Mark},

abstractNote = {This work presents two novel topics regarding the Superhomogenization method: 1) the formalism for the implementation of the method with the linear Boltzmann Transport Equation, and 2) a Newton algorithm for the solution of the nonlinear problem that arises from the method. These new ideas have been implemented in a continuous finite element discretization in the MAMMOTH reactor physics application. The traditional solution strategy for this nonlinear problem uses a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE’s Preconditioned Jacobian-Free Newton Krylov method to allow for a direct solution. The PJFNK-SPH can converge problems that were either intractable or very difficult to converge with the traditional iterative approach, including geometries with reflectors and vacuum boundary conditions. This is partly due to the underlying Scalable Nonlinear Equations Solvers in PETSc, which are integral to MOOSE and offer Newton damping, line search and trust region methods. The PJFNK-SPH has been implemented and tested for various discretizations of the transport equation included in the Rattlesnake transport solver. Speedups of five times for diffusion and ten to fifteen times for transport were obtained when compared to the traditional Picard approach. The three test problems cover a wide range of applications including a standard Pressurized Water Reactor lattice with control rods, a Transient Reactor Test facility control rod supercell and a prototype fast-thermal reactor. The reference solutions and initial cross sections were obtained from the Serpent 2 Monte Carlo code. The SPH-corrected cross sections yield eigenvalues that are near exact, relative to reference solutions, for reflected geometries, even with reflector regions. In geometries with vacuum boundary conditions the accuracy is problem dependent and solutions can be within a few to a few hundred pcm. The root-mean-square error in the power distribution is below 0.8% of the reference Monte Carlo. There is little benefit from SPH-corrected transport in typical scoping calculations, but for more detailed analyses it can yield superior convergence of the solution in some of the test problems. This PJFNK-SPH approach is currently being used in the modeling of the Transient Test Reactor at Idaho National Laboratory, where full reactor core SPH-corrected cross sections are employed to reduce the homogenization errors in transient or multi-physics calculations. This base implementation of the PJFNK-SPH provides an extremely robust solver and a springboard to further improve the Superhomogenization method in order to better preserve neutron currents, one of the primary deficiencies of the method.},

doi = {10.1016/j.anucene.2017.09.027},

journal = {Annals of Nuclear Energy (Oxford)},

issn = {0306-4549},

number = ,

volume = 111,

place = {United States},

year = {2017},

month = {10}

}