conv_shape.h

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/*
 * Copyright (c) 2011-2015:  G-CSC, Goethe University Frankfurt
 * Author: Andreas Vogel
 * 
 * This file is part of UG4.
 * 
 * UG4 is free software: you can redistribute it and/or modify it under the
 * terms of the GNU Lesser General Public License version 3 (as published by the
 * Free Software Foundation) with the following additional attribution
 * requirements (according to LGPL/GPL v3 §7):
 * 
 * (1) The following notice must be displayed in the Appropriate Legal Notices
 * of covered and combined works: "Based on UG4 (www.ug4.org/license)".
 * 
 * (2) The following notice must be displayed at a prominent place in the
 * terminal output of covered works: "Based on UG4 (www.ug4.org/license)".
 * 
 * (3) The following bibliography is recommended for citation and must be
 * preserved in all covered files:
 * "Reiter, S., Vogel, A., Heppner, I., Rupp, M., and Wittum, G. A massively
 *   parallel geometric multigrid solver on hierarchically distributed grids.
 *   Computing and visualization in science 16, 4 (2013), 151-164"
 * "Vogel, A., Reiter, S., Rupp, M., Nägel, A., and Wittum, G. UG4 -- a novel
 *   flexible software system for simulating pde based models on high performance
 *   computers. Computing and visualization in science 16, 4 (2013), 165-179"
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 * GNU Lesser General Public License for more details.
 */
 
#ifndef __H__UG__LIB_DISC__SPATIAL_DISC__ELEM_DISC__DENSITY_DRIVEN_FLOW__FV1__CONV_SHAPE__
#define __H__UG__LIB_DISC__SPATIAL_DISC__ELEM_DISC__DENSITY_DRIVEN_FLOW__FV1__CONV_SHAPE__
 
#include <boost/mpl/range_c.hpp>
#include <boost/mpl/for_each.hpp>
 
// ug4 headers
#include "conv_shape_interface.h"
#include "lib_disc/spatial_disc/disc_util/fv1_geom.h"
#include "lib_disc/spatial_disc/disc_util/hfv1_geom.h"
 
namespace ug{
 
// No Upwind
 
template <int TDim>
class ConvectionShapesNoUpwind
  : public IConvectionShapes<TDim>
{
  public:
    typedef IConvectionShapes<TDim> base_type;
 
    typedef ConvectionShapesNoUpwind<TDim> this_type;
 
    static const int dim = TDim;
 
  protected:
  //  explicitly forward some function
    using base_type::set_non_zero_deriv_diffusion_flag;
    using base_type::conv_shape;
    using base_type::D_vel;
    using base_type::conv_shape_diffusion;
    using base_type::non_zero_deriv_diffusion;
    using base_type::register_update_func;
 
  public:
    ConvectionShapesNoUpwind()
    {
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
      set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
      boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
      boost::mpl::for_each< boost::mpl::range_c<int, 1, dim+1> >(RegisterRefDimFunc(this));
    }
 
    template <typename TFVGeom>
    bool update(const TFVGeom* geo,
          const MathVector<dim>* Velocity,
          const MathMatrix<dim, dim>* DiffDisp,
                bool computeDeriv);
 
  private:
    
    struct RegisterElemFunc
    {
      this_type* m_pThis;
      RegisterElemFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TElem> void operator() (TElem &)
      {
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1Geometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, HFV1Geometry<TElem, dim> >();
      }
    };
  
    template <typename TElem, typename TFVGeom>
    void register_func_for_elem_fvgeom()
    {
      typedef bool (this_type::*TFunc) (const TFVGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
    }
    
    struct RegisterRefDimFunc
    {
      this_type* m_pThis;
      RegisterRefDimFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TRefDim> void operator() (TRefDim &) {m_pThis->register_func_for_refDim<TRefDim::value> ();}
    };
 
    template <int refDim>
    void register_func_for_refDim()
    {
      typedef DimFV1Geometry<refDim, dim> TGeom;
      typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
    }
};
 
template <int TDim>
template <typename TFVGeom>
bool
ConvectionShapesNoUpwind<TDim>::
update(const TFVGeom* geo,
       const MathVector<dim>* Velocity,
       const MathMatrix<dim, dim>* DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
 
//  \todo: think about: this should be something like scvf.num_sh()
  const size_t numSH = geo->num_sh();
 
//  loop subcontrol volume faces
  for(size_t ip = 0; ip < geo->num_scvf(); ++ip)
  {
  //  get subcontrol volume face
    const typename TFVGeom::SCVF& scvf = geo->scvf(ip);
 
  //  Compute flux
    const number flux = VecDot(scvf.normal(), Velocity[ip]);
 
  //  Write Shapes
    for(size_t sh = 0; sh < scvf.num_sh(); sh++)
      conv_shape(ip, sh) = flux * scvf.shape(sh);
 
  //  this is introduced here, hopefully temporarily: The problem is, that
  //  for hanging nodes the number of shape function is not the number of
  //  corners, but scvf.num_sh() currently returns the number of corners.
  //  this is actually enough to interpolate the function, but still we
  //  should reset the interpolation adding for hanging dofs to zero
    for(size_t sh = scvf.num_sh(); sh < numSH; sh++)
      conv_shape(ip, sh) = 0.0;
 
  //  Write Derivatives if wanted
    if(computeDeriv){
      for (size_t sh = 0; sh < scvf.num_sh(); sh++)
        VecScale(D_vel(ip, sh), scvf.normal(), scvf.shape(sh));
 
      // temporary, see above
      for(size_t sh = scvf.num_sh(); sh < numSH; sh++)
        VecSet(D_vel(ip, sh), 0.0);
    }
 
  //  The shapes do not depend of the diffusion tensor
  }
 
//  we're done
  return true;
}
 
// Full Upwind
 
template <int TDim>
class ConvectionShapesFullUpwind
  : public IConvectionShapes<TDim>
{
  public:
    typedef IConvectionShapes<TDim> base_type;
 
    typedef ConvectionShapesFullUpwind<TDim> this_type;
 
    static const int dim = TDim;
 
  protected:
  //  explicitly forward some function
    using base_type::set_non_zero_deriv_diffusion_flag;
    using base_type::conv_shape;
    using base_type::D_vel;
    using base_type::conv_shape_diffusion;
    using base_type::non_zero_deriv_diffusion;
    using base_type::register_update_func;
 
  public:
    ConvectionShapesFullUpwind()
    {
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
      set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
      boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
      boost::mpl::for_each< boost::mpl::range_c<int, 1, dim+1> >(RegisterRefDimFunc(this));
    }
 
    template <typename TFVGeom>
    bool update(const TFVGeom* geo,
          const MathVector<dim>* Velocity,
          const MathMatrix<dim, dim>* DiffDisp,
                bool computeDeriv);
 
  private:
    
    struct RegisterElemFunc
    {
      this_type* m_pThis;
      RegisterElemFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TElem> void operator() (TElem &)
      {
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1Geometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, HFV1Geometry<TElem, dim> >();
      }
    };
  
    template <typename TElem, typename TFVGeom>
    void register_func_for_elem_fvgeom()
    {
      typedef bool (this_type::*TFunc) (const TFVGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
    }
    
    struct RegisterRefDimFunc
    {
      this_type* m_pThis;
      RegisterRefDimFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TRefDim> void operator() (TRefDim &) {m_pThis->register_func_for_refDim<TRefDim::value> ();}
    };
 
    template <int refDim>
    void register_func_for_refDim()
    {
      typedef DimFV1Geometry<refDim, dim> TGeom;
      typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
    }
};
 
template <int TDim>
template <typename TFVGeom>
bool ConvectionShapesFullUpwind<TDim>::
update(const TFVGeom* geo,
       const MathVector<dim>* Velocity,
       const MathMatrix<dim, dim>* DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
 
//  \todo: think about: this should be something like scvf.num_sh()
  const size_t numSH = geo->num_sh();
 
//  loop subcontrol volume faces
  for(size_t ip = 0; ip < geo->num_scvf(); ++ip)
  {
  //  get subcontrol volume face
    const typename TFVGeom::SCVF& scvf = geo->scvf(ip);
 
  //  Compute flux
    const number flux = VecDot(scvf.normal(), Velocity[ip]);
 
    size_t from = scvf.from();
    size_t to = scvf.to();
 
  //  if e.g. hanging nodes are involved, no upwind can be performed...   
    if((from >= scvf.num_sh()) || (to >= scvf.num_sh())){
    //  No upwind...
    //  Write Shapes
      for(size_t sh = 0; sh < scvf.num_sh(); sh++)
        conv_shape(ip, sh) = flux * scvf.shape(sh);
 
      UG_ASSERT(scvf.num_sh() == numSH, "sh's have to match!");
 
    //  Write Derivatives if wanted
      if(computeDeriv){
        for (size_t sh = 0; sh < scvf.num_sh(); sh++)
          VecScale(D_vel(ip, sh), scvf.normal(), scvf.shape(sh));
      }
      
      continue;
    }
    
  //  Full upwind below...
  //  Choose Upwind corner
    size_t up = (flux >= 0) ? from : to;
 
  //  Write Shapes
    for(size_t sh = 0; sh < scvf.num_sh(); ++sh) conv_shape(ip, sh) = 0.0;
 
    conv_shape(ip, up) = flux;
 
  //  Write Derivatives if wanted
    if(computeDeriv)
    {
      for(size_t sh = 0; sh < numSH; ++sh) VecSet(D_vel(ip, sh), 0.0);
      D_vel(ip, up) = scvf.normal();
    }
 
  //  The shapes do not depend of the diffusion tensor
  }
 
//  we're done
  return true;
}
 
// Weighted Upwind
 
template <int TDim>
class ConvectionShapesWeightedUpwind
  : public IConvectionShapes<TDim>
{
  public:
    typedef IConvectionShapes<TDim> base_type;
 
    typedef ConvectionShapesWeightedUpwind<TDim> this_type;
 
    static const int dim = TDim;
 
  protected:
  //  explicitly forward some function
    using base_type::set_non_zero_deriv_diffusion_flag;
    using base_type::conv_shape;
    using base_type::D_vel;
    using base_type::conv_shape_diffusion;
    using base_type::non_zero_deriv_diffusion;
    using base_type::register_update_func;
 
  public:
    ConvectionShapesWeightedUpwind() : m_weight(0.5)
    {
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
      set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
      boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
      boost::mpl::for_each< boost::mpl::range_c<int, 1, dim+1> >(RegisterRefDimFunc(this));
    }
 
    ConvectionShapesWeightedUpwind(number weight)
    {
      set_weight(weight);
 
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
      set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
      boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
      boost::mpl::for_each< boost::mpl::range_c<int, 1, dim+1> >(RegisterRefDimFunc(this));
    }
 
    void set_weight(number weight) {m_weight = weight;}
 
    template <typename TFVGeom>
    bool update(const TFVGeom* geo,
          const MathVector<dim>* Velocity,
          const MathMatrix<dim, dim>* DiffDisp,
                bool computeDeriv);
 
  private:
  //  weight between no and full upwind (1.0 -> full upwind, 0.0 -> no upwind)
    number m_weight;
    
  private:
  
    struct RegisterElemFunc
    {
      this_type* m_pThis;
      RegisterElemFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TElem> void operator() (TElem &)
      {
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1Geometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, HFV1Geometry<TElem, dim> >();
      }
    };
  
    template <typename TElem, typename TFVGeom>
    void register_func_for_elem_fvgeom()
    {
      typedef bool (this_type::*TFunc) (const TFVGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
    }
    
    struct RegisterRefDimFunc
    {
      this_type* m_pThis;
      RegisterRefDimFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TRefDim> void operator() (TRefDim &) {m_pThis->register_func_for_refDim<TRefDim::value> ();}
    };
 
    template <int refDim>
    void register_func_for_refDim()
    {
      typedef DimFV1Geometry<refDim, dim> TGeom;
      typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
    }
};
 
template <int TDim>
template <typename TFVGeom>
bool ConvectionShapesWeightedUpwind<TDim>::
update(const TFVGeom* geo,
       const MathVector<dim>* Velocity,
       const MathMatrix<dim, dim>* DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
 
//  \todo: think about: this should be something like scvf.num_sh()
  const size_t numSH = geo->num_sh();
 
//  loop subcontrol volume faces
  for(size_t ip = 0; ip < geo->num_scvf(); ++ip)
  {
  //  get subcontrol volume face
    const typename TFVGeom::SCVF& scvf = geo->scvf(ip);
 
  //  Compute flux
    const number flux = VecDot(scvf.normal(), Velocity[ip]);
 
    size_t from = scvf.from();
    size_t to = scvf.to();
 
  //  if e.g. hanging nodes are involved, no upwind can be performed...   
    if((from >= scvf.num_sh()) || (to >= scvf.num_sh())){
    //  No upwind...
    //  Write Shapes
      for(size_t sh = 0; sh < scvf.num_sh(); sh++)
        conv_shape(ip, sh) = flux * scvf.shape(sh);
 
      UG_ASSERT(scvf.num_sh() == numSH, "sh's have to match!");
 
    //  Write Derivatives if wanted
      if(computeDeriv){
        for (size_t sh = 0; sh < scvf.num_sh(); sh++)
          VecScale(D_vel(ip, sh), scvf.normal(), scvf.shape(sh));
      }
      
      continue;
    }
    
  //  Choose Upwind corner
    size_t up = (flux >= 0) ? from : to;
    
  //  write no upwind part of shapes
    const number noUpFlux = (1.-m_weight)*flux;
    for(size_t sh = 0; sh < scvf.num_sh(); ++sh)
      conv_shape(ip, sh) = noUpFlux * scvf.shape(sh);
 
  //  add full upwind part of shapes
    conv_shape(ip, up) += m_weight * flux;
 
  //  Write Derivatives if wanted
    if(computeDeriv)
    {
    //  write no upwind part of derivatives
      for (size_t sh = 0; sh < scvf.num_sh(); sh++)
        VecScale(D_vel(ip, sh), scvf.normal(),
                           (1.-m_weight)*scvf.shape(sh));
    //  see comment above
      for(size_t sh = scvf.num_sh(); sh < numSH; sh++)
        VecSet(D_vel(ip, sh), 0.0);
 
    //  add full upwind part of derivatives
      VecScaleAppend(D_vel(ip, up), m_weight, scvf.normal());
    }
 
  //  The shapes do not depend of the diffusion tensor
  }
 
//  we're done
  return true;
}
 
 
// Partial Upwind
 
template <int TDim>
class ConvectionShapesPartialUpwind
  : public IConvectionShapes<TDim>
{
  public:
    typedef IConvectionShapes<TDim> base_type;
 
    typedef ConvectionShapesPartialUpwind<TDim> this_type;
 
    static const int dim = TDim;
 
  protected:
  //  explicitly forward some function
    using base_type::set_non_zero_deriv_diffusion_flag;
    using base_type::conv_shape;
    using base_type::D_vel;
    using base_type::conv_shape_diffusion;
    using base_type::non_zero_deriv_diffusion;
    using base_type::register_update_func;
 
  public:
    ConvectionShapesPartialUpwind()
    {
    //  register evaluation function
      boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
      boost::mpl::for_each< boost::mpl::range_c<int, 1, dim+1> >(RegisterRefDimFunc(this));
    }
 
    template <typename TFVGeom>
    bool update(const TFVGeom* geo,
          const MathVector<dim>* Velocity,
          const MathMatrix<dim, dim>* DiffDisp,
          bool computeDeriv);
 
  private:
    
    struct RegisterElemFunc
    {
      this_type* m_pThis;
      RegisterElemFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TElem> void operator() (TElem &)
      {
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1Geometry<TElem, dim> >();
        m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
      }
    };
  
    template <typename TElem, typename TFVGeom>
    void register_func_for_elem_fvgeom()
    {
      typedef bool (this_type::*TFunc) (const TFVGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
    }
    
    struct RegisterRefDimFunc
    {
      this_type* m_pThis;
      RegisterRefDimFunc(this_type * pThis) : m_pThis(pThis) {}
      template<typename TRefDim> void operator() (TRefDim &) {m_pThis->register_func_for_refDim<TRefDim::value> ();}
    };
 
    template <int refDim>
    void register_func_for_refDim()
    {
      typedef DimFV1Geometry<refDim, dim> TGeom;
      typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
      base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
    }
};
 
template <int TDim>
template <typename TFVGeom>
bool
ConvectionShapesPartialUpwind<TDim>::
update(const TFVGeom* geo,
       const MathVector<dim>* Velocity,
       const MathMatrix<dim, dim>* DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
//  UG_ASSERT(DiffDisp != nullptr, "Null pointer");
 
//  Compute Volume of Element
//  typedef typename TFVGeom::ref_elem_type ref_elem_type;
  const number vol = ElementSize<dim>(geo->roid(), geo->corners());
 
//  loop subcontrol volume faces
  for(size_t i = 0; i < geo->num_scvf(); ++i)
  {
  //  get subcontrol volume face
    const typename TFVGeom::SCVF& scvf = geo->scvf(i);
 
  //  get corners
    const size_t from = scvf.from();
    const size_t to = scvf.to();
 
  //  get gradients
    const MathVector<dim>& gradTo = scvf.global_grad(to);
    const MathVector<dim>& gradFrom = scvf.global_grad(from);
 
  //  set lambda negative as default
    number lambda = -1;
 
  //  if DiffDisp-Tensor passed, compute lambda
    if(DiffDisp != nullptr)
    {
    //  Get Gradients
      MathVector<dim> DiffGrad;
 
    //  Compute DiffGrad = D * Grad Phi_to
      MatVecMult(DiffGrad, DiffDisp[i], gradTo);
 
    //  Compute GradDiffGrad = < Grad Phi_from, DiffGrad >
      const number GradDiffGrad = VecDot(DiffGrad,  gradFrom);
 
    //  Set lambda
      lambda = - GradDiffGrad * vol;
    }
 
    //  Compute flux
  const number flux = VecDot(scvf.normal(), Velocity[i]);
 
  //      reset values
  for (size_t sh = 0; sh < scvf.num_sh(); ++sh)
    conv_shape(i, sh) = 0.0;
  if (computeDeriv)
    for (size_t sh = 0; sh < scvf.num_sh(); ++sh) {
      VecSet(D_vel(i, sh), 0.0);
      MatSet(conv_shape_diffusion(i, sh), 0.0);
    }
 
 
  //  check: Currently hanging nodes not supported.
  //  /todo: add support for hanging nodes
  if (from >= scvf.num_sh() || to >= scvf.num_sh())
  UG_THROW("PartialUpwind: Currently not implemented for hanging nodes.")
 
  
  //  Case 1:
  //  full upwind is used
    if(lambda <= 0 || DiffDisp == nullptr)
    {
    //  Choose Upwind corner
      const size_t up = (flux >= 0) ? scvf.from() : scvf.to();
 
    //  Write Shapes
      conv_shape(i, up) = flux;
 
    //  Write Derivatives if wanted
      if(computeDeriv)
      {
      //  set derivative
        D_vel(i, up) = scvf.normal();
 
      //  does not depend on diffusion
        set_non_zero_deriv_diffusion_flag(false);
      }
 
    //  everything done
      continue;
    }
 
  //  Case 2:
  //  The case of the diffusion dominance (central differences)
    if (2 * lambda > fabs(flux))
    {
      conv_shape(i, from) = flux / 2.0;
      conv_shape(i, to) = flux / 2.0;
 
      if(computeDeriv)
      {
        set_non_zero_deriv_diffusion_flag(false);
 
        VecScale(D_vel(i,from), scvf.normal(), 1.0/2.0);
        VecScale(D_vel(i, to), scvf.normal(), 1.0/2.0);
      }
 
    //  everything done
      continue;
    }
 
  //  Case 3:
  //  The cases of the convection dominance
    set_non_zero_deriv_diffusion_flag(true);
    if (flux >= 0)
    {
      conv_shape(i, from) = flux - lambda;
      conv_shape(i, to) = lambda;
 
      if(computeDeriv)
        D_vel(i,from) = scvf.normal();
    }
    else
    {
      conv_shape(i, from) = - lambda;
      conv_shape(i, to) = flux + lambda;
 
      if(computeDeriv)
        D_vel(i,to) = scvf.normal();
    }
 
    if (computeDeriv)
    {
      for (size_t k = 0; k < (size_t)dim; k++)
        for (size_t l = 0; l < (size_t)dim; l++)
        {
          conv_shape_diffusion(i, from)(k,l) = gradFrom[k]*gradTo[l]*vol;
          conv_shape_diffusion(i, to)(k,l) = - gradFrom[k]*gradTo[l]*vol;
        }
    }
  }
 
//  we're done
  return true;
}
 
// Skewed Upwind
 
template <int TDim>
class ConvectionShapesSkewedUpwind
  : public IConvectionShapes<TDim>
{
public:
  typedef IConvectionShapes<TDim> base_type;
 
  typedef ConvectionShapesSkewedUpwind<TDim> this_type;
 
  static const int dim = TDim;
 
protected:
  //  explicitly forward some function
  using base_type::conv_shape;
  using base_type::conv_shape_diffusion;
  using base_type::D_vel;
  using base_type::non_zero_deriv_diffusion;
  using base_type::register_update_func;
  using base_type::set_non_zero_deriv_diffusion_flag;
 
public:
  ConvectionShapesSkewedUpwind()
  {
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
    set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
    boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
    boost::mpl::for_each<boost::mpl::range_c<int, 1, dim + 1>>(RegisterRefDimFunc(this));
  }
 
  template <typename TFVGeom>
  bool update(const TFVGeom *geo,
        const MathVector<dim> *Velocity,
        const MathMatrix<dim, dim> *DiffDisp,
        bool computeDeriv);
 
private:
  struct RegisterElemFunc
  {
    this_type *m_pThis;
    RegisterElemFunc(this_type *pThis) : m_pThis(pThis) {}
    template <typename TElem>
    void operator()(TElem &)
    {
      m_pThis->template register_func_for_elem_fvgeom<TElem, FV1Geometry<TElem, dim>>();
      //m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
      //m_pThis->template register_func_for_elem_fvgeom< TElem, HFV1Geometry<TElem, dim> >();
    }
  };
 
  template <typename TElem, typename TFVGeom>
  void register_func_for_elem_fvgeom()
  {
    typedef bool (this_type::*TFunc)(const TFVGeom *, const MathVector<dim> *, const MathMatrix<dim, dim> *, bool);
    base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
  }
 
  struct RegisterRefDimFunc
  {
    this_type *m_pThis;
    RegisterRefDimFunc(this_type *pThis) : m_pThis(pThis) {}
    template <typename TRefDim>
    void operator()(TRefDim &) { m_pThis->register_func_for_refDim<TRefDim::value>(); }
  };
 
  template <int refDim>
  void register_func_for_refDim()
  {
    //typedef DimFV1Geometry<refDim, dim> TGeom;
    //typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
    //base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
  }
};
 
template <typename TRefElem, int TWorldDim>
void GetNodeNextToCut(size_t &coOut,
            const MathVector<TWorldDim> &IP,
            const MathVector<TWorldDim> &IPVel,
            const MathVector<TWorldDim> *vCornerCoords)
{
  //  help variables
  size_t side = 0;
  MathVector<TWorldDim> globalIntersection;
  MathVector<TRefElem::dim> localIntersection;
 
  //  compute intersection of ray in direction of ip velocity with elem side
  //  we search the ray only in upwind direction
  if (!ElementSideRayIntersection<TRefElem, TWorldDim>(side, globalIntersection, localIntersection,
                             IP, IPVel, false /* i.e. search upwind */, vCornerCoords))
    UG_THROW("GetNodeNextToCut: ElementSideRayIntersection failed");
 
  //  get reference element
  static const TRefElem &rRefElem = Provider<TRefElem>::get();
  const int dim = TRefElem::dim;
 
  //  reset minimum
  number min = std::numeric_limits<number>::max();
 
  //  loop corners of side
  for (size_t i = 0; i < rRefElem.num(dim - 1, side, 0); ++i)
  {
    //  get corner
    const size_t co = rRefElem.id(dim - 1, side, 0, i);
 
    //  Compute Distance to intersection
    number dist = VecDistanceSq(globalIntersection, vCornerCoords[co]);
 
    //  if distance is smaller, choose this node
    if (dist < min)
    {
      min = dist;
      coOut = co;
    }
  }
}
 
template <int TDim>
template <typename TFVGeom>
bool ConvectionShapesSkewedUpwind<TDim>::
	update(const TFVGeom *geo,
       const MathVector<dim> *Velocity,
       const MathMatrix<dim, dim> *DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
 
  //  \todo: think about: this should be something like scvf.num_sh()
  const size_t numSH = geo->num_sh();
 
  //  loop subcontrol volume faces
  for (size_t ip = 0; ip < geo->num_scvf(); ++ip)
  {
    //  get subcontrol volume face
    const typename TFVGeom::SCVF &scvf = geo->scvf(ip);
 
    //  Compute flux
    const number flux = VecDot(scvf.normal(), Velocity[ip]);
 
    // Switch to no upwind in case of small velocity
    if (VecTwoNorm(Velocity[ip]) < 1e-14)
    {
      // No upwind!
      for (size_t sh = 0; sh < scvf.num_sh(); sh++)
        conv_shape(ip, sh) = flux * scvf.shape(sh);
      for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
        conv_shape(ip, sh) = 0.0;
      if (computeDeriv)
      {
        for (size_t sh = 0; sh < scvf.num_sh(); sh++)
          VecScale(D_vel(ip, sh), scvf.normal(), scvf.shape(sh));
        // temporary: case of hanging nodes
        for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
          VecSet(D_vel(ip, sh), 0.0);
      }
      continue;
    }
 
    //  initialize shapes with zeroes
    for (size_t sh = 0; sh < scvf.num_sh(); sh++)
      conv_shape(ip, sh) = 0.0;
 
    //  Hanging nodes
    //  this is introduced here, hopefully temporarily: The problem is, that
    //  for hanging nodes the number of shape function is not the number of
    //  corners, but scvf.num_sh() currently returns the number of corners.
    //  this is actually enough to interpolate the function, but still we
    //  should reset the interpolation adding for hanging dofs to zero
    for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
      conv_shape(ip, sh) = 0.0;
 
    const MathVector<dim> *vCornerCoords = geo->corners();
    size_t up = 0;
    try
    {
      GetNodeNextToCut<typename TFVGeom::ref_elem_type, dim>(up, scvf.global_ip(), Velocity[ip], vCornerCoords);
    }
    UG_CATCH_THROW("Skewed upwind: Cannot find node next to cut");
 
    conv_shape(ip, up) = flux;
 
    //  Write Derivatives if wanted
    if (computeDeriv)
    {
      for (size_t sh = 0; sh < numSH; ++sh)
        VecSet(D_vel(ip, sh), 0.0);
      D_vel(ip, up) = scvf.normal();
    }
 
    //  The shapes do not depend of the diffusion tensor
  }
 
  //  we're done
  return true;
}
 
// Linear Profile Skewed Upwind
 
template <int TDim>
class ConvectionShapesLinearProfileSkewedUpwind
  : public IConvectionShapes<TDim>
{
public:
  typedef IConvectionShapes<TDim> base_type;
 
  typedef ConvectionShapesLinearProfileSkewedUpwind<TDim> this_type;
 
  static const int dim = TDim;
 
protected:
  //  explicitly forward some function
  using base_type::conv_shape;
  using base_type::conv_shape_diffusion;
  using base_type::D_vel;
  using base_type::non_zero_deriv_diffusion;
  using base_type::register_update_func;
  using base_type::set_non_zero_deriv_diffusion_flag;
 
public:
  ConvectionShapesLinearProfileSkewedUpwind()
  {
    //  the shapes do not depend on the DiffDisp. Thus, we can set the
    //  derivative to be always zero w.r.t. the DiffDisp for all shapes
    set_non_zero_deriv_diffusion_flag(false);
 
    //  register evaluation function
    boost::mpl::for_each<typename domain_traits<dim>::AllElemList>(RegisterElemFunc(this));
    boost::mpl::for_each<boost::mpl::range_c<int, 1, dim + 1>>(RegisterRefDimFunc(this));
  }
 
  template <typename TFVGeom>
  bool update(const TFVGeom *geo,
        const MathVector<dim> *Velocity,
        const MathMatrix<dim, dim> *DiffDisp,
        bool computeDeriv);
 
private:
  struct RegisterElemFunc
  {
    this_type *m_pThis;
    RegisterElemFunc(this_type *pThis) : m_pThis(pThis) {}
    template <typename TElem>
    void operator()(TElem &)
    {
      m_pThis->template register_func_for_elem_fvgeom<TElem, FV1Geometry<TElem, dim>>();
      //m_pThis->template register_func_for_elem_fvgeom< TElem, FV1CondensedGeometry<TElem, dim> >();
      //m_pThis->template register_func_for_elem_fvgeom< TElem, HFV1Geometry<TElem, dim> >();
    }
  };
 
  template <typename TElem, typename TFVGeom>
  void register_func_for_elem_fvgeom()
  {
    typedef bool (this_type::*TFunc)(const TFVGeom *, const MathVector<dim> *, const MathMatrix<dim, dim> *, bool);
    base_type::template register_update_func<TFVGeom, TFunc>(&this_type::template update<TFVGeom>);
  }
 
  struct RegisterRefDimFunc
  {
    this_type *m_pThis;
    RegisterRefDimFunc(this_type *pThis) : m_pThis(pThis) {}
    template <typename TRefDim>
    void operator()(TRefDim &) { m_pThis->register_func_for_refDim<TRefDim::value>(); }
  };
 
  template <int refDim>
  void register_func_for_refDim()
  {
    //typedef DimFV1Geometry<refDim, dim> TGeom;
    //typedef bool (this_type::*TFunc) (const TGeom*, const MathVector<dim>*, const MathMatrix<dim, dim>*, bool);
    //base_type::template register_update_func<TGeom, TFunc>(&this_type::template update<TGeom>);
  }
};
 
template <int TDim>
template <typename TFVGeom>
bool ConvectionShapesLinearProfileSkewedUpwind<TDim>::
	update(const TFVGeom *geo,
       const MathVector<dim> *Velocity,
       const MathMatrix<dim, dim> *DiffDisp,
       bool computeDeriv)
{
  UG_ASSERT(geo != nullptr, "Null pointer");
  UG_ASSERT(Velocity != nullptr, "Null pointer");
 
  //  \todo: think about: this should be something like scvf.num_sh()
  const size_t numSH = geo->num_sh();
 
  //  loop subcontrol volume faces
  for (size_t ip = 0; ip < geo->num_scvf(); ++ip)
  {
    //  get subcontrol volume face
    const typename TFVGeom::SCVF &scvf = geo->scvf(ip);
 
    //  Compute flux
    const number flux = VecDot(scvf.normal(), Velocity[ip]);
 
    // Switch to no upwind in case of small velocity
    if (VecTwoNorm(Velocity[ip]) < 1e-14)
    {
      // No upwind!
      for (size_t sh = 0; sh < scvf.num_sh(); sh++)
        conv_shape(ip, sh) = flux * scvf.shape(sh);
      for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
        conv_shape(ip, sh) = 0.0;
      if (computeDeriv)
      {
        for (size_t sh = 0; sh < scvf.num_sh(); sh++)
          VecScale(D_vel(ip, sh), scvf.normal(), scvf.shape(sh));
        // temporary, see above
        for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
          VecSet(D_vel(ip, sh), 0.0);
      }
      continue;
    }
 
    //  initialize shapes with zeroes
    for (size_t sh = 0; sh < scvf.num_sh(); sh++)
      conv_shape(ip, sh) = 0.0;
 
    //  Hanging nodes
    //  this is introduced here, hopefully temporarily: The problem is, that
    //  for hanging nodes the number of shape function is not the number of
    //  corners, but scvf.num_sh() currently returns the number of corners.
    //  this is actually enough to interpolate the function, but still we
    //  should reset the interpolation adding for hanging dofs to zero
    for (size_t sh = scvf.num_sh(); sh < numSH; sh++)
      conv_shape(ip, sh) = 0.0;
 
    const MathVector<dim> *vCornerCoords = geo->corners();
    // Reference element type
    typedef typename TFVGeom::ref_elem_type TRefElem;
    size_t side = 0;
    MathVector<dim> globalIntersection;
    MathVector<TRefElem::dim> localIntersection;
 
    try
    {
      ElementSideRayIntersection<TRefElem, dim>(side, globalIntersection, localIntersection,
                            scvf.global_ip(), Velocity[ip], false /* search upwind */, vCornerCoords);
    }
    UG_CATCH_THROW("GetLinearProfileSkewedUpwindShapes: Cannot find cut side.");
 
    //  get linear trial space
    static const ReferenceObjectID roid = TRefElem::REFERENCE_OBJECT_ID;
    const LocalShapeFunctionSet<TRefElem::dim> &TrialSpace =
      LocalFiniteElementProvider::get<TRefElem::dim>(roid, LFEID(LFEID::LAGRANGE, dim, 1));
 
    //  get Reference Element
    static const TRefElem &rRefElem = Provider<TRefElem>::get();
 
        // Shape function values 
    std::vector<number> vShape;
    // Values of shape function gradients
    std::vector<MathVector<TRefElem::dim>> vShapeGrad;
    // Get values at the intersection point
    TrialSpace.shapes(vShape, localIntersection);
    TrialSpace.grads(vShapeGrad, localIntersection);
 
    size_t num_corners_inters_side = rRefElem.num(dim - 1, side, 0);
 
    //  loop corners of side
    for (size_t j = 0; j < num_corners_inters_side; ++j)
    {
      //  get corner
      const size_t co = rRefElem.id(dim-1, side, 0, j);
 
      //  write shape
      conv_shape(ip, co) = flux * vShape[co];
    }
 
    //  Write Derivatives if wanted
    if (computeDeriv)
    {
      // No derivatives yet!
    }
 
    //  The shapes do not depend of the diffusion tensor
  }
 
  //  we're done
  return true;
}
 
 
} // end namespace ug
 
#endif /* __H__UG__LIB_DISC__SPATIAL_DISC__ELEM_DISC__DENSITY_DRIVEN_FLOW__FV1__CONV_SHAPE__ */