Augmented Lagrangian Preconditioner for Large-Scale Hydrodynamic Stability Analysis

A parallel implementation of the steady Navier–Stokes and eigenvalue solvers, developed in the FreeFEM language, suitably interfaced with the PETSc/SLEPc libraries.

The code available in this repository can reproduce the results from the following paper.

@article{moulin2019al,
    Author = {Moulin, Johann and Jolivet, Pierre and Marquet, Olivier},
    Title = {{Augmented Lagrangian Preconditioner for Large-Scale Hydrodynamic Stability Analysis}},
    Year = {2019},
    Volume = {351},
    Pages = {718--743},
    Journal = {Computer Methods in Applied Mechanics and Engineering},
    Publisher = {Elsevier},
    Url = {https://github.com/prj-/moulin2019al}
}

Getting started

Dependencies

Make sure you have access to a recent FreeFEM installation (version 4.3 or above), compiled with the PETSc and PETSc-complex (with SLEPc) plugins. More details about the PETSc and SLEPc options used in the solvers may be found in their respective manual.

Usage example

A small mesh is provided in the Mesh folder. One should be able to launch the following command, which solves the steady Navier–Stokes equations for a Reynolds number of 50 (-Re 50) on the same geometrical configuration as in the paper.

$ mpirun -np 4 FreeFem++-mpi Nonlinear-solver.edp -Re 50 -v 0

The option -v 0 is here to minimize the output generated by FreeFEM, see this tutorial for more information. Then, for performing the actual linear stability analysis.

$ mpirun -np 4 FreeFem++-mpi Eigensolver.edp -v 0

The results are stored in the State/EV folder.

Customization

Here are the main command line parameters.

• Nonlinear-solver.edp
  1. -Newton_tol (default to 1.0e-6), stopping criterion of the Newton method
  2. -mesh (default to FlatPlate3D.mesh), must be stored in the Mesh folder ; if changed, do not forget to set the appropriate Dirichlet boundary conditions in the FreeFEM variational formulations vJ and vRes
  3. -Re (default to 100), Reynolds number
  4. -gamma (default to 0.3), augmentation parameter
• Eigensolver.edp
  1. -shift_real (default to 1.0e-6), real part of the shift
  2. -shift_imag (default to 0.6), imaginary part of the shift
  3. -nev (default to 5), desired number of computed eigenmodes
  4. -recycle (default to 0), number of recycled vectors from the Arnoldi basis between each GMRES cycle

Acknowledgements

• European Research Council (ERC), project AEROFLEX, grant number 638307
• HPC resources of TGCC@CEA under the allocation A0030607519 made by GENCI