hho-ns.hpp

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// Implementation of the HHO scheme, with full gradient, for the Navier-Stokes equations
//
// Authors: Daniele Di Pietro (daniele.di-pietro@umontpellier.fr),
//          Jerome Droniou (jerome.droniou@umontpellier.fr)
//
 
/*
 *
 *  This implementation of HHO was developped following the principles described in 
 * Appendix B of the book
 *
 * The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications. 
 *  D. A. Di Pietro and J. Droniou. Modeling, Simulation and Applications, vol. 19. 
 *  Springer International Publishing, 2020, xxxi + 525p. doi: 10.1007/978-3-030-37203-3. 
 *  url: https://hal.archives-ouvertes.fr/hal-02151813.
 *
 * If you use this code or part of it for a scientific publication, please cite the book
 * above as a reference for the implementation.
 *
 */
 
#ifndef HHOSTOKES_HPP
#define HHOSTOKES_HPP
 
#include <iostream>
 
#include <boost/math/constants/constants.hpp>
 
#include <Eigen/Sparse>
#include <unsupported/Eigen/SparseExtra>
 
#include <mesh.hpp>
// #include <mesh_builder.hpp>
#include <local_static_condensation.hpp>
 
#include <vhhospace.hpp>
#include <BoundaryConditions/BoundaryConditions.hpp>
#include <integralweight.hpp>
 
#include "ns-solutions.hpp"
 
namespace HArDCore2D
{
 
  //------------------------------------------------------------------------------
  // Navier-Stokes model
  //------------------------------------------------------------------------------
 
 
  struct NavierStokes
  {
    typedef Eigen::SparseMatrix<double> SystemMatrixType;
    
    typedef std::function<VectorRd(const VectorRd &)> MomentumForcingTermType;
    typedef std::function<double(const VectorRd &)> CompressibilityForcingTermType;
    typedef std::function<VectorRd(const VectorRd &)> VelocityType;
    typedef std::function<MatrixRd(const VectorRd &)> VelocityGradientType;
    typedef std::function<double(const VectorRd &)> PressureType;
    typedef std::function<VectorRd(const VectorRd &)> PressureGradientType;
    typedef IntegralWeight ViscosityType;
 
    NavierStokes(
                 const VHHOSpace & vhho_space,      
                 const GlobalDOFSpace & p_space,    
                 const BoundaryConditions & BC,     
                 bool use_threads,                  
                 bool stokes = false,               
                 std::ostream & output = std::cout  
                 );
 
    void assembleLinearisedSystem(
                                  const NavierStokesSolutions::IExactSolution * isolution, 
                                  const Eigen::VectorXd & uph_old 
                                  );
                    
    inline size_t nloc_sc_u() const
    {
      return m_nloc_sc_u;
    }
 
    inline size_t nloc_sc_p() const
    {
      return m_nloc_sc_p;
    }
 
    inline size_t numSCDOFs_u() const
    {
      return m_vhhospace.mesh().n_cells() * m_nloc_sc_u;
    }
 
    inline size_t numSCDOFs_p() const
    {
      return m_vhhospace.mesh().n_cells() * m_nloc_sc_p;
    }
 
    inline size_t numSCDOFs() const
    {
      return numSCDOFs_u() + numSCDOFs_p();
    }
 
    inline size_t numDirDOFs() const
    {
      return m_BC.n_dir_edges()*m_vhhospace.numLocalDofsEdge();
    }
    
    inline size_t dimVelocity() const
    {
      return m_vhhospace.dimension();
    }
 
    inline size_t dimPressure() const
    {
      return m_pspace.dimension();
    }
    
    inline size_t numNonSCDOFs() const
    {
      return dimVelocity() + dimPressure() - numSCDOFs() + 1;
    }
 
    inline size_t sizeSystem() const
    {
      return numNonSCDOFs() - numDirDOFs();
    }
 
    inline const VHHOSpace & vhhospace() const
    {
      return m_vhhospace;
    }
    
    inline const GlobalDOFSpace & pspace() const
    {
      return m_pspace;
    }
 
    inline const Mesh & mesh() const
    {
      return m_vhhospace.mesh();
    }
 
    inline const SystemMatrixType & systemMatrix() const {
      return m_A;
    }
 
    inline SystemMatrixType & systemMatrix() {
      return m_A;
    }
 
    inline const Eigen::VectorXd & systemVector() const {
      return m_b;
    }
 
    inline Eigen::VectorXd & systemVector() {
      return m_b;
    }
 
    inline const double & stabilizationParameter() const {
      return m_stab_par;
    }
 
    inline double & stabilizationParameter() {
      return m_stab_par;
    }
    
    inline const SystemMatrixType & scMatrix() const {
      return m_sc_A;
    }
 
    inline Eigen::VectorXd & scVector() {
      return m_sc_b;
    }
    
    inline bool & isStokes() { return m_stokes; }
 
    Eigen::VectorXd interpolate(
                                const VelocityType & u, 
                                const PressureType & p, 
                                const int doe_cell = -1, 
                                const int doe_face = -1 
                                ) const;    
 
    std::vector<double> computeEnergyNorms(
                                           const std::vector<Eigen::VectorXd> & list_dofs   
                                           ) const;
 
  private:
    std::pair<Eigen::MatrixXd, Eigen::VectorXd>
    _compute_local_contribution(
                                size_t iT, 
                                const NavierStokesSolutions::IExactSolution * isolution, 
                                const Eigen::VectorXd & uph_old 
                                );
 
    LocalStaticCondensation _compute_static_condensation(const size_t & iT) const;
 
    void _assemble_local_contribution(
                                      size_t iT,                                               
                                      const std::pair<Eigen::MatrixXd, Eigen::VectorXd> & lsT, 
                                      std::list<Eigen::Triplet<double> > & A1,                 
                                      Eigen::VectorXd & b1,                                    
                                      std::list<Eigen::Triplet<double> > & A2,                 
                                      Eigen::VectorXd & b2                                     
                                      );
    
    const VHHOSpace & m_vhhospace;
    const GlobalDOFSpace & m_pspace; 
 
    const BoundaryConditions & m_BC;
    bool m_use_threads;
    bool m_stokes;
    std::ostream & m_output;
    const size_t m_nloc_sc_u; // Number of statically condensed velocity DOFs in each cell
    const size_t m_nloc_sc_p; // Number of statically condensed pressure DOFs in each cell
    SystemMatrixType m_A;     // Matrix and RHS of statically condensed system
    Eigen::VectorXd m_b;
    SystemMatrixType m_sc_A;  // Static condensation operator and RHS (to recover statically condensed DOFs)
    Eigen::VectorXd m_sc_b;
    double m_stab_par;
  };
 
} // namespace HArDCore2D
 
#endif