Thermomechanical Simulation of a RJH Plate

website: https://github.com/latug0/mfem-mgis-examples/tree/master/ex8

This example models one ring of a RJH fuel assembly. The fuel is U3Si2. The cladding and the stiffeners are ALFENI.

Two problems are strongly coupled at each time step by an IterativeCouplingScheme:

• a heat transfer problem. The power density in the fuel increases linearly during a ramp, then stays constant.

• a mechanical problem at finite strain. It accounts for the thermal expansion, the swelling and the irradiation creep of the fuel, and for the plasticity of ALFENI.

The behaviours are written with MFront. The file MFront/README.md describes them.

Build the mesh

The default mesh mesh/assemblage_hexa.msh is generated by the gmsh script mesh/assemblage_hexa.py. It is copied next to the executable at build time. The script requires the gmsh Python module:

cd mesh
python3 assemblage_hexa.py --output_file assemblage_hexa.msh

The coarser mesh mesh/assemblage_hexa_coarse.msh is generated with:

python3 assemblage_hexa.py --densHaut 5 --densFuelLength 8 --densFuelThick 3 \
  --densCladConn 3 --densStifThick 3 --densStifConn 3 \
  --output_file assemblage_hexa_coarse.msh

Option	Description	Default
--densHaut	Number of elements along the height	10
--densFuelLength	Number of elements along the fuel length	15
--densFuelThick	Number of elements across the fuel thickness	5
--densCladConn	Number of elements across the cladding thickness	5
--densStifThick	Number of elements across the stiffener thickness	5
--densStifConn	Number of elements in the stiffener outside the cladding	5
--output_file	Output mesh file	assemblage_hexa.msh

Run the simulation

This command runs the default simulation on 2 processes:

mpirun -n 2 ./rjh_plate

The results are exported to Paraview in the Results directory:

paraview Results/Mechanics/Mechanics.pvd

The figures show the results at t = 2e6 s. They are computed with mpirun -n 4 ./rjh_plate -et 2e6 -ns 20. The radial displacement is amplified 100 times. The cladding bulges between the stiffeners.

At the end of the simulation, the swelling is compared to its exact value. The end of the power ramp is a time step boundary. The power density is thus linear over each time step, and the swelling model integrates it exactly. The run fails if they differ.

Available options

Command line	Description	Default
--mesh or -m	Mesh file	assemblage_hexa.msh
--libraryU3SI2 or -lU	Material library of the U3Si2 fuel	src/libU3SI2-generic.so
--libraryALFENI or -lA	Material library of the ALFENI cladding and stiffeners	src/libALFENI-generic.so
--linearsolver-thermal or -lsTh	Linear solver of the heat transfer problem	HypreGMRES
--preconditioner-thermal or -pcTh	Preconditioner of the linear solver of the heat transfer problem	HypreBoomerAMG
--linearsolver-mechanics or -lsMc	Linear solver of the mechanical problem	MUMPSSolver
--preconditioner-mechanics or -pcMc	Preconditioner of the linear solver of the mechanical problem	HypreBoomerAMG
--order or -o	Finite element order	1
--refinement or -r	Number of uniform refinements of the mesh	0
--post-processing or -pp, --no-post-processing or -no-pp	Export or not the results to Paraview	export
--verbosity-level or -v	Verbosity level of the linear solvers	0
--debug or -d, --no-debug or -no-d	Print or not the minimum, maximum and mean values of the temperature, of the norm of the displacement, of the swelling and of the power density at the end of the simulation	print
--reference-file or -rf	Reference values of these statistics, one line per field, in the printed format	no comparison
--end-time or -et	End time of the simulation	1e5
--nbsteps or -ns	Number of time steps	1
--t-ramp or -tr	Duration of the power ramp. Its end must be a time step boundary. 0 disables the ramp.	1e5
--h-conv or -hc	Thermal convection coefficient	5e4
--water-pressure or -wp	Coolant pressure on the cladding and the stiffeners	1e6

The iterative solvers are CGSolver, GMRESSolver, BiCGSTABSolver, MINRESSolver, SLISolver, HyprePCG, HypreGMRES and HypreFGMRES. The direct solver is MUMPSSolver. It ignores the preconditioner. The preconditioners are HypreBoomerAMG, HypreDiagScale, HypreEuclid, HypreILU and HypreParaSails.

Iterative solvers are much slower than MUMPS for the mechanics of this problem. With the default mesh, the simulation takes 8 s on 2 processes with MUMPS and 165 s on 4 processes with HyprePCG.

Tests

ctest runs three tests:

• rjh_plate runs the simulation on 2 processes. It compares the statistics of the temperature and of the displacement to the reference values. The full test mode uses the default mesh and assemblage_hexa-statistics.ref. The restricted test mode uses the coarser mesh and assemblage_hexa_coarse-statistics.ref.

• u3si2_swelling compares the swelling model to its exact value under a power ramp.

• robin_test compares the Robin boundary condition to the exact solution of a bar.