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title: | | A free and open source computational tool for solving | (nuclear-related) differential equations in the cloud author: German (a.k.a. Jeremy) Theler email: [email protected] institute: | | International Nuclear Atlantic Conference | Round Table 4 ENFIR ”Best Estimate codes and methods | for Reactor Physics, Thermal Fluid Dynamics and Fuel Behavior” aspectratio: 169 lang: en-US theme: default innertheme: rectangles fonttheme: professionalfonts outertheme: number colorlinks: true sansfont: Carlito monofont: DejaVuSansMono header-includes: \include{syntax.tex} ...

Why $\rightarrow$ How $\rightarrow$ What

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\centering https://www.youtube.com/watch?v=u4ZoJKF_VuA

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IAEA 2D/3D PWR benchmark (1976)

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Vertical PHWR (Atucha)

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Steady-state core-level neutronic calculations

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Transient emergency boron injection---CFD

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Transient emergency boron injection---core-level neutronics

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Transient emergency boron injection---cell-level neutronics

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A little bit of history---College (2004--2008) (v1)

• Solve $\dot{\mathbf{x}} = F(\mathbf{x},t)$

  a. program an ad-hoc numerical method b. use a standard numerical library in C or Python, or c. use a high-level system such as Octave, Maxima, etc.

. . .

• Point reactor equations solved "graphically" with non-free software

  $$ \begin{cases} \dot{\phi}(t) = \displaystyle \frac{\rho(t) - \beta}{\Lambda} \cdot \phi(t) + \sum_{i=1}^{N} \lambda_i \cdot c_i \ \dot{c}_i(t) = \displaystyle \frac{\beta_i}{\Lambda} \cdot \phi(t) - \lambda_i \cdot c_i \end{cases} $$

. . .

• I would rather write the equations in ASCII like

  phi_dot = (rho-Beta)/Lambda * phi + sum(lambda[i], c[i], i, 1, N)
  c_dot[i] = beta[i]/Lambda * phi - lambda[i]*c[i]
  

A little bit of history---Nuclear industry (2008--2014)

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\centering Modern/Advanced $\neq$ Fortran 90

Spatial discretizations

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wasora & milonga (v2)

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Wasora's an advanced suite for reactor analysis

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• Coupled unstructured fine-mesh neutronics and thermal-hydraulics methodology using open software (2018)
• Open software one-step coupled neutronics and CFD thermalhydraulics calculation (2016)
• https://github.com/seamplex/milonga-2015-workshop
• Reactivity coefficient estimation by fuel temperature for Atucha II Nuclear Power Plant from neutron flux measurements (2014)
• On the design basis of a new core-level neutronic code written from scratch (2014)
• Neutron diffusion on unstructured grids: comparison between finite volumes and finite elements (2013)
• Geometric optimization of nuclear reactor cores (2013)
• Unstructured grids and the multigroup neutron diffusion equation (2013)

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FeenoX (v3)

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Software Requirement Specifications

\centering https://seamplex.com/feenox/doc/srs.html

• Industrial-level: open source for V&V
• Extensible: free (as in freedom)
• Cloud-first: programatically-defined
• Arbitrarily scalable for large problems
• Flexible
• Web & mobile-based GUIs
• QA
  • Reproducibility & traceability
  • Automated testing
  • Bug reporting & tracking
• V & V

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Software Design Specifications

\centering https://seamplex.com/feenox/doc/sds.html

• GPLv3+: it is about freedom not price
• No-GUI script-friendly GNU/Linux binary {width=90%}\
• Scalability based on UNIX
  • PETSc/SLEPc (MPI)
  • Gmsh (Metis)
• Flexibility shown in "what"
• Web GUI: https://www.caeplex.com
• Regression testing make check
• https://github.com/seamplex/feenox
• V & V: TODO! (help appreciated)!

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Point kinetics

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$$ \begin{cases} \dot{\phi}(t) = \displaystyle \frac{\rho(t) - \Beta}{\Lambda} \cdot \phi(t) + \sum_{i=1}^{N} \lambda_i \cdot c_i \\ \dot{c}_i(t) = \displaystyle \frac{\beta_i}{\Lambda} \cdot \phi(t) - \lambda_i \cdot c_i \end{cases} $$

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$t$ [s]	$\rho(t)$ [pcm]
0	0
5	0
10	10
30	10
35	0
100	0

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\noindent for $0 < t < 100$ starting from steady-steate conditions at full power. :::

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$ feenox reactivity-from-table.fee > flux.dat
$

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Inverse kinetics

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$ feenox inverse-dae.fee flux.dat > inverse-dae.dat
$ feenox inverse-integral.fee flux.dat > inverse-integral.dat

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2D IAEA PWR Benchmark

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$ gmsh -2 iaea-2dpwr-quarter.geo
$ [...]
$ gmsh -2 iaea-2dpwr-eighth.geo
$ [...]
$ feenox iaea-2dpwr.fee quarter
keff =  1.02986
$ feenox iaea-2dpwr.fee eighth
keff =  1.02975
$

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\centering https://www.seamplex.com/feenox/examples/#iaea-2d-pwr-benchmark

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2D IAEA PWR Benchmark

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2D IAEA PWR Benchmark

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The $S_2$ Stanford Bunny

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• One-group neutron transport
• The Stanford Bunny as the geometry
• $S_2$ method in 3D (8 angular directions)
• Finite elements for spatial discretization

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Cube-sphere

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Conclusions & TODO

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• 21st century: cloud-first approach
• Stick to free and open source software
  • free $\neq$ open source
• More formulations (milonga already had them)
  • FEM transport
  • FVM diffusion & transport
• Massive parallelization
  • MPI through PETSc/SLEPc
  • Metis through Gmsh
• Documentation
  • https://www.seamplex.com/feenox/doc
  • https://www.seamplex.com/feenox/examples
  • Come up with a nice logo
• Build a community!
  • https://github.com/seamplex/feenox/discussions

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