getfem_interpolation.h

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/* -*- c++ -*- (enables emacs c++ mode) */
/*===========================================================================
 
 Copyright (C) 2001-2020 Yves Renard, Julien Pommier
 
 This file is a part of GetFEM
 
 GetFEM  is  free software;  you  can  redistribute  it  and/or modify it
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===========================================================================*/
 
/**@file getfem_interpolation.h
   @author Yves Renard    <Yves.Renard@insa-lyon.fr>
   @author Julien Pommier <Julien.Pommier@insa-toulouse.fr>
   @date October 15, 2001.
   @brief Interpolation of fields from a mesh_fem onto another.
*/
 
#ifndef GETFEM_INTERPOLATION_H__
#define GETFEM_INTERPOLATION_H__
 
#include "getfem_mesh_fem.h"
#include "bgeot_torus.h"
#include "dal_tree_sorted.h"
#include "getfem_im_data.h"
#include "getfem_torus.h"
 
namespace getfem {
 
  /* ********************************************************************* */
  /*                                                                       */
  /*        I. Distribution of a set of points on a mesh.                  */
  /*                                                                       */
  /* ********************************************************************* */
 
  class mesh_trans_inv : public bgeot::geotrans_inv {
 
  protected :
    typedef std::set<size_type>::const_iterator set_iterator;
    typedef std::map<size_type,size_type>::const_iterator map_iterator;
 
    const mesh &msh;
    std::vector<std::set<size_type> > pts_cvx;
    std::vector<base_node> ref_coords;
    std::map<size_type,size_type> ids;
 
  public :
 
    size_type nb_points_on_convex(size_type i) const
    { return pts_cvx[i].size(); }
    void points_on_convex(size_type i, std::vector<size_type> &itab) const;
    size_type point_on_convex(size_type cv, size_type i) const;
    const std::vector<base_node> &reference_coords(void) const { return ref_coords; }
 
    void add_point_with_id(base_node n, size_type id)
    { size_type ipt = add_point(n); ids[ipt] = id; }
    size_type id_of_point(size_type ipt) const;
    const mesh &linked_mesh(void) const { return msh; }
 
    /* extrapolation = 0 : Only the points inside the mesh are distributed.
     * extrapolation = 1 : Try to extrapolate the exterior points near the
     *                     boundary.
     * extrapolation = 2 : Extrapolate all the exterior points. Could be
     *                     expensive.
     *
     * if rg_source is provided only the corresponding part of the mesh is
     * taken into account and extrapolation is done with respect to the
     * boundary of the specified region. rg_source must contain only convexes.
     */
    void distribute(int extrapolation = 0,
                    mesh_region rg_source=mesh_region::all_convexes());
    mesh_trans_inv(const mesh &m, double EPS_ = 1E-12)
      : bgeot::geotrans_inv(EPS_), msh(m) {}
  };
 
 
  /* ********************************************************************* */
  /*                                                                       */
  /*        II. Interpolation of functions.                                */
  /*                                                                       */
  /* ********************************************************************* */
 
 
  template <typename VECT, typename F, typename M>
  inline void interpolation_function__(const mesh_fem &mf, VECT &V,
                                       F &f, const dal::bit_vector &dofs,
                                       const M &, gmm::abstract_null_type) {
    size_type Q = mf.get_qdim();
    GMM_ASSERT1(gmm::vect_size(V) == mf.nb_basic_dof() && Q == 1,
                "Dof vector has not the right size");
    for (dal::bv_visitor i(dofs); !i.finished(); ++i)
      V[i] = f(mf.point_of_basic_dof(i));
  }
 
  template <typename VECT, typename F, typename M>
  inline void interpolation_function__(const mesh_fem &mf, VECT &V,
                                       F &f, const dal::bit_vector &dofs,
                                       const M &v, gmm::abstract_vector) {
    size_type N = gmm::vect_size(v),  Q = mf.get_qdim();
    GMM_ASSERT1(gmm::vect_size(V) == mf.nb_basic_dof()*N/Q,
                "Dof vector has not the right size");
    for (dal::bv_visitor i(dofs); !i.finished(); ++i)
      if (i % Q == 0)
        gmm::copy(f(mf.point_of_basic_dof(i)),
                  gmm::sub_vector(V, gmm::sub_interval(i*N/Q, N)));
  }
 
  template <typename VECT, typename F, typename M>
  inline void interpolation_function__(const mesh_fem &mf, VECT &V,
                                       F &f, const dal::bit_vector &dofs,
                                       const M &mm, gmm::abstract_matrix) {
    // typedef typename gmm::linalg_traits<VECT>::value_type T;
    size_type Nr = gmm::mat_nrows(mm), Nc = gmm::mat_ncols(mm), N = Nr*Nc;
    size_type Q = mf.get_qdim();
    base_matrix m(Nr, Nc);
    GMM_ASSERT1(gmm::vect_size(V) == mf.nb_basic_dof()*N/Q,
                "Dof vector has not the right size");
    for (dal::bv_visitor i(dofs); !i.finished(); ++i)
      if (i % Q == 0) {
        gmm::copy(f(mf.point_of_basic_dof(i)), m);
        for (size_type j = 0; j < Nc; ++j)
          gmm::copy(gmm::mat_col(m, j),
                    gmm::sub_vector(V, gmm::sub_interval(i*N/Q+j*Nr, Nr)));
      }
 
  }
 
  template <typename VECT, typename F, typename M>
  inline void interpolation_function_(const mesh_fem &mf, VECT &V,
                                      F &f, const dal::bit_vector &dofs,
                                      const M &m) {
    interpolation_function__(mf, V, f, dofs, m,
                             typename gmm::linalg_traits<M>::linalg_type());
  }
 
#if GETFEM_PARA_LEVEL > 0
  template <typename T>
  void take_one_op(void *a, void *b, int *len, MPI_Datatype *) {
    T aa = *((T*)a);
    return aa ? aa : *((T*)b);
  }
 
  template <typename T>
  inline MPI_Op mpi_take_one_op(T) {
    static bool isinit = false;
    static MPI_Op op;
    if (!isinit) {
      MPI_Op_create(take_one_op<T>, true, &op);
      isinit = true;
    }
    return op;
  }
#endif
 
  // TODO : verify that rhs is a lagrange fem
  /**
     @brief interpolation of a function f on mf_target.
     - mf_target must be of lagrange type.
     - mf_target's qdim should be equal to the size of the return value of f,
       or equal to 1
     - V should have the right size
     CAUTION: with the parallized version (GETFEM_PARA_LEVEL >= 2) the
     resulting vector V is distributed.
  */
  template <typename VECT, typename F>
  void interpolation_function(mesh_fem &mf_target, const VECT &VV, F &f,
                              mesh_region rg=mesh_region::all_convexes()) {
    typedef typename gmm::linalg_traits<VECT>::value_type T;
    size_type qqdimt = gmm::vect_size(VV) / mf_target.nb_dof();
    std::vector<T> V(mf_target.nb_basic_dof()*qqdimt);
    mf_target.linked_mesh().intersect_with_mpi_region(rg);
    dal::bit_vector dofs = mf_target.basic_dof_on_region(rg);
    if (dofs.card() > 0)
      interpolation_function_(mf_target, V, f, dofs,
                              f(mf_target.point_of_basic_dof(dofs.first())));
 
    if (mf_target.is_reduced()) {
      for (size_type k = 0; k < qqdimt; ++k)
        gmm::mult(mf_target.reduction_matrix(),
                  gmm::sub_vector(V,
                                  gmm::sub_slice(k, mf_target.nb_basic_dof(),
                                                 qqdimt)),
                  gmm::sub_vector(const_cast<VECT &>(VV),
                                  gmm::sub_slice(k, mf_target.nb_dof(),
                                                 qqdimt)));
    }
    else
      gmm::copy(V, const_cast<VECT &>(VV));
  }
 
  /* ********************************************************************* */
  /*                                                                       */
  /*        III. Interpolation between two meshes.                         */
  /*                                                                       */
  /* ********************************************************************* */
 
  /* ------------------------------ Interface -----------------------------*/
 
  /**
     @brief interpolation/extrapolation of (mf_source, U) on mf_target.
     - mf_target must be of lagrange type.
     - mf_target's qdim should be equal to mf_source qdim, or equal to 1
     - U.size() >= mf_source.get_qdim()
     - V.size() >= (mf_target.nb_dof() / mf_target.get_qdim())
                   * mf_source.get_qdim()
 
     With extrapolation = 0 a strict interpolation is done, with extrapolation = 1
     an extrapolation of the exterior points near the boundary is done (if any)
     and with extrapolation = 2 all  exterior points are extrapolated (could be expensive).
 
     If both mesh_fem shared the same mesh object, a fast interpolation
     will be used.
 
     If rg_source and rg_target are provided the operation is restricted to
     these regions. rg_source must contain only convexes.
  */
  template<typename VECTU, typename VECTV>
  void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target,
                     const VECTU &U, VECTV &V, int extrapolation = 0,
                     double EPS = 1E-10,
                     mesh_region rg_source=mesh_region::all_convexes(),
                     mesh_region rg_target=mesh_region::all_convexes());
 
  /**
     @brief Build the interpolation matrix of mf_source on mf_target.
     the matrix M is
     such that (V = M*U) == interpolation(mf_source, mf_target, U, V).
 
     Useful for repeated interpolations.
     For performance reasons the matrix M is recommended to be either
     a row or a row and column matrix.
 
     If rg_source and rg_target are provided the operation is restricted to
     these regions. rg_source must contain only convexes.
   */
  template<typename MAT>
  void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target,
                     MAT &M, int extrapolation = 0, double EPS = 1E-10,
                     mesh_region rg_source=mesh_region::all_convexes(),
                     mesh_region rg_target=mesh_region::all_convexes());
 
 
  /* --------------------------- Implementation ---------------------------*/
 
  /*
     interpolation of a solution on same mesh.
     - &mf_target.linked_mesh() == &mf_source.linked_mesh()
     - mf_target must be of lagrange type.
     - mf_target's qdim should be equal to mf_source qdim, or equal to 1
     - U.size() >= mf_source.get_qdim()
     - V.size() >= (mf_target.nb_dof() / mf_target.get_qdim())
                   * mf_source.get_qdim()
  */
  template<typename VECTU, typename VECTV, typename MAT>
    void interpolation_same_mesh(const mesh_fem &mf_source,
                                 const mesh_fem &mf_target,
                                 const VECTU &UU, VECTV &VV,
                                 MAT &MM, int version) {
 
    typedef typename gmm::linalg_traits<VECTU>::value_type T;
    base_matrix G;
    dim_type qdim = mf_source.get_qdim();
    dim_type qqdim = dim_type(gmm::vect_size(UU)/mf_source.nb_dof());
    std::vector<T> val(qdim);
    std::vector<std::vector<T> > coeff;
    std::vector<size_type> dof_source;
    GMM_ASSERT1(qdim == mf_target.get_qdim() || mf_target.get_qdim() == 1,
                "Attempt to interpolate a field of dimension "
                << qdim << " on a mesh_fem whose Qdim is " <<
                int(mf_target.get_qdim()));
    size_type qmult = mf_source.get_qdim()/mf_target.get_qdim();
    size_type qqdimt = qqdim * mf_source.get_qdim()/mf_target.get_qdim();
    fem_precomp_pool fppool;
    std::vector<size_type> dof_t_passes(mf_target.nb_basic_dof());
    std::vector<T> U(mf_source.nb_basic_dof()*qqdim);
    std::vector<T> V(mf_target.nb_basic_dof()*qqdimt);
    gmm::row_matrix<gmm::rsvector<scalar_type> >
      M(mf_target.nb_basic_dof(), mf_source.nb_basic_dof());
 
    if (version == 0) mf_source.extend_vector(UU, U);
 
    /* we should sort convexes by their fem */
    for (dal::bv_visitor cv(mf_source.convex_index()); !cv.finished(); ++cv) {
      bgeot::pgeometric_trans pgt=mf_source.linked_mesh().trans_of_convex(cv);
      pfem pf_s = mf_source.fem_of_element(cv);
      if (!mf_target.convex_index().is_in(cv))
        continue;
      pfem pf_t = mf_target.fem_of_element(cv);
      size_type nbd_s = pf_s->nb_dof(cv);
      size_type nbd_t = pf_t->nb_dof(cv);
      mesh_fem::ind_dof_ct::const_iterator itdof;
      size_type cvnbdof = mf_source.nb_basic_dof_of_element(cv);
 
      bool discontinuous_source = false;
      for (size_type dof=0; dof < nbd_s; ++dof)
        if (!dof_linkable(pf_s->dof_types()[dof])) {
          discontinuous_source = true;
          break;
        }
 
      if (version == 0) {
        coeff.resize(qqdim);
        for (size_type qq=0; qq < qqdim; ++qq) {
          coeff[qq].resize(cvnbdof);
          itdof = mf_source.ind_basic_dof_of_element(cv).begin();
          for (size_type k = 0; k < cvnbdof; ++k, ++itdof) {
            coeff[qq][k] = U[(*itdof)*qqdim+qq];
          }
        }
      }
      if (pf_s->need_G())
        bgeot::vectors_to_base_matrix
          (G, mf_source.linked_mesh().points_of_convex(cv));
 
      GMM_ASSERT1(pf_t->target_dim() == 1,
                  "won't interpolate on a vector FEM... ");
      pfem_precomp pfp = fppool(pf_s, pf_t->node_tab(cv));
      fem_interpolation_context ctx(pgt,pfp,size_type(-1), G, cv,
                                    short_type(-1));
      itdof = mf_target.ind_basic_dof_of_element(cv).begin();
      if (version != 0) {
        const mesh_fem::ind_dof_ct &idct
          = mf_source.ind_basic_dof_of_element(cv);
        dof_source.assign(idct.begin(), idct.end());
      }
      for (size_type i = 0; i < nbd_t; ++i, itdof+=mf_target.get_qdim()) {
        size_type dof_t = *itdof*qmult;
        if (!discontinuous_source && dof_t_passes[*itdof] > 0) continue;
        dof_t_passes[*itdof] += 1;
        ctx.set_ii(i);
        if (version == 0) {
          for (size_type qq=0; qq < qqdim; ++qq) {
            pf_s->interpolation(ctx, coeff[qq], val, qdim);
            for (size_type k=0; k < qdim; ++k)
              V[(dof_t + k)*qqdim+qq] += val[k];
          }
        }
        else {
          base_matrix Mloc(qdim, mf_source.nb_basic_dof_of_element(cv));
          pf_s->interpolation(ctx, Mloc, qdim);
          for (size_type k=0; k < qdim; ++k) {
            for (size_type j=0; j < dof_source.size(); ++j) {
              M(dof_t + k, dof_source[j]) += Mloc(k, j);
            }
          }
        }
      }
    }
 
    // calculate averages for discontinuous source and continuous target
    for (size_type i = 0; i < mf_target.nb_basic_dof(); ++i) {
      size_type dof_t = i*qmult;
      scalar_type passes = scalar_type(dof_t_passes[i]);
      if (version == 0 && passes > scalar_type(0))
        for (size_type qq=0; qq < qqdim; ++qq)
          for (size_type k=0; k < qdim; ++k)
            V[(dof_t + k)*qqdim+qq] /= passes;
      else if (passes > scalar_type(0))
        for (size_type k=0; k < qdim; ++k)
          for (size_type j=0; j < dof_source.size(); ++j)
            gmm::scale(gmm::mat_row(M, dof_t + k), scalar_type(1)/passes);
    }
 
    if (version == 0)
      mf_target.reduce_vector(V, VV);
    else {
      if (mf_target.is_reduced())
        if (mf_source.is_reduced()) {
          gmm::row_matrix<gmm::rsvector<scalar_type> >
            MMM(mf_target.nb_dof(), mf_source.nb_basic_dof());
          gmm::mult(mf_target.reduction_matrix(), M, MMM);
          gmm::mult(MMM, mf_source.extension_matrix(), MM);
        }
        else
          gmm::mult(mf_target.reduction_matrix(), M, MM);
      else
        if (mf_source.is_reduced())
          gmm::mult(M, mf_source.extension_matrix(), MM);
        else
          gmm::copy(M, MM);
    }
  }
 
 
  /*
     interpolation of a solution on another mesh.
     - mti contains the points where to interpolate.
     - the solution should be continuous.
   */
  template<typename VECTU, typename VECTV, typename MAT>
  void interpolation(const mesh_fem &mf_source,
                     mesh_trans_inv &mti,
                     const VECTU &UU, VECTV &V, MAT &MM,
                     int version, int extrapolation = 0,
                     dal::bit_vector *dof_untouched = 0,
                     mesh_region rg_source=mesh_region::all_convexes()) {
 
    typedef typename gmm::linalg_traits<VECTU>::value_type T;
    const mesh &msh(mf_source.linked_mesh());
    dim_type qdim_s = mf_source.get_qdim();
    dim_type qqdim = dim_type(gmm::vect_size(UU)/mf_source.nb_dof());
 
    std::vector<T> U(mf_source.nb_basic_dof()*qqdim);
    gmm::row_matrix<gmm::rsvector<scalar_type> > M;
    if (version != 0) M.resize(gmm::mat_nrows(MM), mf_source.nb_basic_dof());
 
    if (version == 0) mf_source.extend_vector(UU, U);
 
    mti.distribute(extrapolation, rg_source);
    std::vector<size_type> itab;
    base_matrix G;
 
    /* interpolation */
    dal::bit_vector points_to_do; points_to_do.add(0, mti.nb_points());
    std::vector<T> val(qdim_s);
    std::vector<std::vector<T> > coeff;
    base_tensor Z;
    std::vector<size_type> dof_source;
 
    for (dal::bv_visitor cv(mf_source.convex_index()); !cv.finished(); ++cv) {
      bgeot::pgeometric_trans pgt = msh.trans_of_convex(cv);
      mti.points_on_convex(cv, itab);
      if (itab.size() == 0) continue;
 
      pfem pf_s = mf_source.fem_of_element(cv);
      if (pf_s->need_G())
        bgeot::vectors_to_base_matrix(G, msh.points_of_convex(cv));
 
      fem_interpolation_context ctx(pgt, pf_s, base_node(), G, cv,
                                    short_type(-1));
      if (version == 0) {
        coeff.resize(qqdim);
        size_type cvnbdof = mf_source.nb_basic_dof_of_element(cv);
        mesh_fem::ind_dof_ct::const_iterator itdof;
        for (size_type qq=0; qq < qqdim; ++qq) {
          coeff[qq].resize(cvnbdof);
          itdof = mf_source.ind_basic_dof_of_element(cv).begin();
          for (size_type k = 0; k < cvnbdof; ++k, ++itdof) {
            coeff[qq][k] = U[(*itdof)*qqdim+qq];
          }
        }
      }
      if (version != 0) {
        const mesh_fem::ind_dof_ct &idct
          = mf_source.ind_basic_dof_of_element(cv);
        dof_source.assign(idct.begin(), idct.end());
      }
      for (size_type i = 0; i < itab.size(); ++i) {
        size_type ipt = itab[i];
        if (points_to_do.is_in(ipt)) {
          points_to_do.sup(ipt);
          ctx.set_xref(mti.reference_coords()[ipt]);
          size_type dof_t = mti.id_of_point(ipt);
          size_type pos = dof_t * qdim_s;
          if (version == 0) {
            for (size_type qq=0; qq < qqdim; ++qq) {
              pf_s->interpolation(ctx, coeff[qq], val, qdim_s);
              for (size_type k=0; k < qdim_s; ++k)
                V[(pos + k)*qqdim+qq] = val[k];
            }
            // Part to be improved if one wants in option to be able to
            // interpolate the gradient.
            //          if (PVGRAD) {
            // base_matrix grad(mdim, qdim);
            // pf_s->interpolation_grad(ctx,coeff,gmm::transposed(grad), qdim);
            // std::copy(grad.begin(), grad.end(), V.begin()+dof_t*qdim*mdim);
            // }
          } else {
            base_matrix Mloc(qdim_s, mf_source.nb_basic_dof_of_element(cv));
            pf_s->interpolation(ctx, Mloc, qdim_s);
            for (size_type k=0; k < qdim_s; ++k) {
              for (size_type j=0; j < gmm::mat_ncols(Mloc); ++j)
                M(pos+k, dof_source[j]) = Mloc(k,j);
                /* does not work with col matrices
                   gmm::clear(gmm::mat_row(M, pos+k));
                   gmm::copy(gmm::mat_row(Mloc, k),
                   gmm::sub_vector(gmm::mat_row(M, pos+k), isrc));
                */
            }
          }
        }
      }
    }
    if (points_to_do.card() != 0) {
      if (dof_untouched) {
        dof_untouched->clear();
        for (dal::bv_visitor ipt(points_to_do); !ipt.finished(); ++ipt)
          dof_untouched->add(mti.id_of_point(ipt));
      }
      else {
        dal::bit_vector dofs_to_do;
        for (dal::bv_visitor ipt(points_to_do); !ipt.finished(); ++ipt)
          dofs_to_do.add(mti.id_of_point(ipt));
        GMM_WARNING2("in interpolation (different meshes),"
                     << dofs_to_do.card() << " dof of target mesh_fem have "
                     << " been missed\nmissing dofs : " << dofs_to_do);
      }
    }
 
    if (version != 0) {
      if (mf_source.is_reduced())
        gmm::mult(M, mf_source.extension_matrix(), MM);
      else
        gmm::copy(M, MM);
    }
 
  }
 
  template<typename VECTU, typename VECTV>
  void interpolation(const mesh_fem &mf_source, mesh_trans_inv &mti,
                     const VECTU &U, VECTV &V, int extrapolation = 0,
                     dal::bit_vector *dof_untouched = 0,
                     mesh_region rg_source=mesh_region::all_convexes()) {
    base_matrix M;
    GMM_ASSERT1((gmm::vect_size(U) % mf_source.nb_dof()) == 0 &&
                gmm::vect_size(V)!=0, "Dimension of vector mismatch");
    interpolation(mf_source, mti, U, V, M, 0, extrapolation, dof_untouched, rg_source);
  }
 
 
 
  /*
     interpolation of a solution on another mesh.
     - mf_target must be of lagrange type.
     - the solution should be continuous..
   */
  template<typename VECTU, typename VECTV, typename MAT>
  void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target,
                     const VECTU &U, VECTV &VV, MAT &MM,
                     int version, int extrapolation,
                     double EPS,
                     mesh_region rg_source=mesh_region::all_convexes(),
                     mesh_region rg_target=mesh_region::all_convexes()) {
 
    //Check if it is a torus mesh_fem
    const torus_mesh_fem * pmf_torus = dynamic_cast<const torus_mesh_fem *>(&mf_target);
    if(pmf_torus != 0){
      interpolation_to_torus_mesh_fem(mf_source, *pmf_torus,
                                      U, VV, MM, version, extrapolation, EPS, rg_source, rg_target);
      return;
    }
 
    typedef typename gmm::linalg_traits<VECTU>::value_type T;
    dim_type qqdim = dim_type(gmm::vect_size(U)/mf_source.nb_dof());
    size_type qqdimt = qqdim * mf_source.get_qdim()/mf_target.get_qdim();
    std::vector<T> V(mf_target.nb_basic_dof()*qqdimt);
    mf_target.extend_vector(VV,V);
    gmm::row_matrix<gmm::rsvector<scalar_type> > M;
    if (version != 0) M.resize(mf_target.nb_basic_dof(), mf_source.nb_dof());
 
    const mesh &msh(mf_source.linked_mesh());
    getfem::mesh_trans_inv mti(msh, EPS);
    size_type qdim_s = mf_source.get_qdim(), qdim_t = mf_target.get_qdim();
    GMM_ASSERT1(qdim_s == qdim_t || qdim_t == 1,
                "Attempt to interpolate a field of dimension "
                << qdim_s << " on a mesh_fem whose Qdim is " << qdim_t);
 
    /* test if the target mesh_fem is really of Lagrange type.         */
    for (dal::bv_visitor cv(mf_target.convex_index()); !cv.finished();++cv) {
      pfem pf_t = mf_target.fem_of_element(cv);
      GMM_ASSERT1(pf_t->target_dim() == 1 && pf_t->is_lagrange(),
                  "Target fem not convenient for interpolation");
    }
    /* initialisation of the mesh_trans_inv */
    bool is_target_torus = dynamic_cast<const torus_mesh *>(&mf_target.linked_mesh());
    if (rg_target.id() == mesh_region::all_convexes().id()) {
      size_type nbpts = mf_target.nb_basic_dof() / qdim_t;
      for (size_type i = 0; i < nbpts; ++i){
        if (is_target_torus){
          auto p = mf_target.point_of_basic_dof(i * qdim_t);
          p.resize(msh.dim());
          mti.add_point(p);
        }
        else mti.add_point(mf_target.point_of_basic_dof(i * qdim_t));
      }
    }
    else {
      for (dal::bv_visitor_c dof(mf_target.basic_dof_on_region(rg_target));
           !dof.finished(); ++dof)
        if (dof % qdim_t == 0){
          if (is_target_torus){
            auto p = mf_target.point_of_basic_dof(dof);
            p.resize(msh.dim());
            mti.add_point_with_id(p, dof/qdim_t);
          }
          else
            mti.add_point_with_id(mf_target.point_of_basic_dof(dof),dof/qdim_t);
        }
    }
    interpolation(mf_source, mti, U, V, M, version, extrapolation, 0,rg_source);
 
    if (version == 0)
      mf_target.reduce_vector(V, VV);
    else {
      if (mf_target.is_reduced())
        gmm::mult(mf_target.reduction_matrix(), M, MM);
      else
        gmm::copy(M, MM);
    }
 
  }
 
  /*
     interpolation of a solution on another torus mesh.
     - the solution should be continuous.
   */
  template<typename VECTU, typename VECTV, typename MAT>
  void interpolation_to_torus_mesh_fem(const mesh_fem &mf_source, const torus_mesh_fem &mf_target,
                                       const VECTU &U, VECTV &VV, MAT &MM,
                                       int version, int extrapolation,
                                       double EPS,
                                       mesh_region rg_source=mesh_region::all_convexes(),
                                       mesh_region rg_target=mesh_region::all_convexes()) {
 
    typedef typename gmm::linalg_traits<VECTU>::value_type T;
    dim_type qqdim = dim_type(gmm::vect_size(U)/mf_source.nb_dof());
    size_type qqdimt = qqdim * mf_source.get_qdim()/mf_target.get_qdim();
    std::vector<T> V(mf_target.nb_basic_dof()*qqdimt);
    mf_target.extend_vector(VV,V);
    gmm::row_matrix<gmm::rsvector<scalar_type> >
      M(mf_target.nb_basic_dof(), mf_source.nb_dof());
 
    const mesh &msh(mf_source.linked_mesh());
    getfem::mesh_trans_inv mti(msh, EPS);
    size_type qdim_s = mf_source.get_qdim(), qdim_t = mf_target.get_qdim();
    GMM_ASSERT1(qdim_s == qdim_t || qdim_t == 1,
                "Attempt to interpolate a field of dimension "
                << qdim_s << " on a mesh_fem whose Qdim is " << qdim_t);
 
    /* test if the target mesh_fem is convenient for interpolation.         */
    for (dal::bv_visitor cv(mf_target.convex_index()); !cv.finished();++cv) {
      pfem pf_t = mf_target.fem_of_element(cv);
 
      GMM_ASSERT1(pf_t->target_dim() == 1 ||
                  (mf_target.get_qdim() == mf_target.linked_mesh().dim()),
                  "Target fem not convenient for interpolation");
    }
    /* initialisation of the mesh_trans_inv */
    if (rg_target.id() == mesh_region::all_convexes().id()) {
      size_type nbpts = mf_target.nb_basic_dof() / qdim_t;
      for (size_type i = 0; i < nbpts; ++i)
      {
        base_node p(msh.dim());
        for (size_type k=0; k < msh.dim(); ++k) p[k] = mf_target.point_of_basic_dof(i * qdim_t)[k];
        mti.add_point(p);
      }
      interpolation(mf_source, mti, U, V, M, version, extrapolation);
    }
    else {
      for (dal::bv_visitor_c dof(mf_target.basic_dof_on_region(rg_target)); !dof.finished(); ++dof)
        if (dof % qdim_t == 0)
        {
           base_node p(msh.dim());
          for (size_type k=0; k < msh.dim(); ++k) p[k] = mf_target.point_of_basic_dof(dof)[k];
          mti.add_point_with_id(p, dof/qdim_t);
        }
      interpolation(mf_source, mti, U, V, M, version, extrapolation, 0, rg_source);
    }
 
    if (version == 0)
      mf_target.reduce_vector(V, VV);
    else {
      if (mf_target.is_reduced())
        gmm::mult(mf_target.reduction_matrix(), M, MM);
      else
        gmm::copy(M, MM);
    }
  }
 
 
 
  template<typename VECTU, typename VECTV>
  void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target,
                     const VECTU &U, VECTV &V, int extrapolation,
                     double EPS,
                     mesh_region rg_source, mesh_region rg_target) {
    base_matrix M;
    GMM_ASSERT1((gmm::vect_size(U) % mf_source.nb_dof()) == 0
                && (gmm::vect_size(V) % mf_target.nb_dof()) == 0
                && gmm::vect_size(V) != 0, "Dimensions mismatch");
    if (&mf_source.linked_mesh() == &mf_target.linked_mesh() &&
        rg_source.id() == mesh_region::all_convexes().id() &&
        rg_target.id() == mesh_region::all_convexes().id())
      interpolation_same_mesh(mf_source, mf_target, U, V, M, 0);
    else {
      omp_distribute<VECTV> V_distributed;
      auto partitioning_allowed = rg_source.is_partitioning_allowed();
      rg_source.prohibit_partitioning();
      GETFEM_OMP_PARALLEL(
          auto &V_thrd = V_distributed.thrd_cast();
          gmm::resize(V_thrd, V.size());
          interpolation(
            mf_source, mf_target, U, V_thrd, M, 0, extrapolation, EPS,
            rg_source, rg_target);
      )
      for (size_type thread=0; thread != V_distributed.num_threads(); ++thread){
        auto &V_thrd2 = V_distributed(thread);
        for (size_type i = 0; i < V_thrd2.size(); ++i) {
          if (gmm::abs(V_thrd2[i]) > EPS) V[i] = V_thrd2[i];
        }
      }
      if (partitioning_allowed) rg_source.allow_partitioning();
    }
  }
 
  template<typename MAT>
  void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target,
                     MAT &M, int extrapolation, double EPS,
                     mesh_region rg_source, mesh_region rg_target) {
    GMM_ASSERT1(mf_source.nb_dof() == gmm::mat_ncols(M)
                && (gmm::mat_nrows(M) % mf_target.nb_dof()) == 0
                && gmm::mat_nrows(M) != 0, "Dimensions mismatch");
    std::vector<scalar_type> U, V;
    if (&mf_source.linked_mesh() == &mf_target.linked_mesh() &&
        rg_source.id() == mesh_region::all_convexes().id() &&
        rg_target.id() ==mesh_region::all_convexes().id())
      interpolation_same_mesh(mf_source, mf_target, U, V, M, 1);
    else
      interpolation(mf_source, mf_target, U, V, M, 1, extrapolation, EPS,
                    rg_source, rg_target);
  }
 
 
  /**Interpolate mesh_fem data to im_data.
   The qdim of mesh_fem must be equal to im_data nb_tensor_elem.
   Both im_data and mesh_fem must reside in the same mesh.
   Only convexes defined with both mesh_fem and im_data will be interpolated.
   The use_im_data_filter flag controls the use of the filtered region of
   im_data (default) or the use the full mesh.
  */
  template <typename VECT>
  void interpolation_to_im_data(const mesh_fem &mf_source, const im_data &im_target,
                                const VECT &nodal_data, VECT &int_pt_data,
                                bool use_im_data_filter = true) {
    // typedef typename gmm::linalg_traits<const VECT>::value_type T;
 
    dim_type qdim = mf_source.get_qdim();
    size_type nb_dof = mf_source.nb_dof();
    size_type nb_basic_dof = mf_source.nb_basic_dof();
    size_type nodal_data_size = gmm::vect_size(nodal_data);
    dim_type data_qdim = nodal_data_size / nb_dof;
 
    GMM_ASSERT1(data_qdim * mf_source.nb_dof() == nodal_data_size,
                "Incompatible size of mesh fem " << mf_source.nb_dof() * data_qdim <<
                " with the data " << nodal_data_size);
 
    GMM_ASSERT1(qdim * data_qdim == im_target.nb_tensor_elem(),
                "Incompatible size of qdim for mesh_fem " << qdim
                << " and im_data " << im_target.nb_tensor_elem());
    GMM_ASSERT1(&mf_source.linked_mesh() == &im_target.linked_mesh(),
                "mf_source and im_data do not share the same mesh.");
 
    GMM_ASSERT1(nb_dof * data_qdim == nodal_data_size,
                "Provided nodal data size is " << nodal_data_size
                << " but expecting vector size of " << nb_dof);
 
    size_type size_im_data = im_target.nb_index(use_im_data_filter)
                           * im_target.nb_tensor_elem();
    GMM_ASSERT1(size_im_data == gmm::vect_size(int_pt_data),
                "Provided im data size is " << gmm::vect_size(int_pt_data)
                << " but expecting vector size of " << size_im_data);
 
    VECT extended_nodal_data_((nb_dof != nb_basic_dof) ? nb_basic_dof * data_qdim : 0);
    if (nb_dof != nb_basic_dof)
      mf_source.extend_vector(nodal_data, extended_nodal_data_);
    const VECT &extended_nodal_data = (nb_dof == nb_basic_dof) ? nodal_data : extended_nodal_data_;
 
    dal::bit_vector im_data_convex_index(im_target.convex_index(use_im_data_filter));
 
    base_matrix G;
    base_vector coeff;
    bgeot::base_tensor tensor_int_point(im_target.tensor_size());
    fem_precomp_pool fppool;
    for (dal::bv_visitor cv(im_data_convex_index); !cv.finished(); ++cv) {
 
      bgeot::pgeometric_trans pgt = mf_source.linked_mesh().trans_of_convex(cv);
      pfem pf_source = mf_source.fem_of_element(cv);
      if (pf_source == NULL)
        continue;
 
      mesh_fem::ind_dof_ct::const_iterator it_dof;
      size_type cv_nb_dof = mf_source.nb_basic_dof_of_element(cv);
      size_type nb_nodal_pt = cv_nb_dof / qdim;
      coeff.resize(cv_nb_dof);
      getfem::slice_vector_on_basic_dof_of_element(mf_source, extended_nodal_data, cv, coeff);
 
      const getfem::papprox_integration pim(im_target.approx_int_method_of_element(cv));
      if (pf_source->need_G())
        bgeot::vectors_to_base_matrix(G, *(pim->pintegration_points()));
 
      pfem_precomp pfp = fppool(pf_source, pim->pintegration_points());
 
      // interior of the convex
      size_type nb_int_pts;
      size_type int_pt_id = im_target.index_of_first_point(cv, short_type(-1),
                                                           use_im_data_filter);
      if (int_pt_id != size_type(-1)) {
        nb_int_pts = im_target.nb_points_of_element(cv, short_type(-1));
        fem_interpolation_context ctx(pgt, pfp, size_type(-1), G, cv);
        for (size_type i = 0; i < nb_int_pts; ++i, ++int_pt_id) {
          ctx.set_ii(i);
          ctx.pf()->interpolation(ctx, coeff, tensor_int_point.as_vector(), qdim * data_qdim);
          im_target.set_tensor(int_pt_data, int_pt_id, tensor_int_point);
        }
      }
 
      // convex faces
      for (short_type f=0, nb_faces=im_target.nb_faces_of_element(cv);
           f < nb_faces; ++f) {
        int_pt_id = im_target.index_of_first_point(cv, f, use_im_data_filter);
        if (int_pt_id != size_type(-1)) {
          nb_int_pts = im_target.nb_points_of_element(cv, f);
          fem_interpolation_context ctx(pgt, pfp, size_type(-1), G, cv, f);
          size_type i0 = pim->ind_first_point_on_face(f);
          for (size_type i = 0; i < nb_int_pts; ++i, ++int_pt_id) {
            ctx.set_ii(i+i0);
            ctx.pf()->interpolation(ctx, coeff, tensor_int_point.as_vector(), qdim * data_qdim);
            im_target.set_tensor(int_pt_data, int_pt_id, tensor_int_point);
          }
        }
      }
    }//end of convex loop
  }
 
}  /* end of namespace getfem.                                             */
 
 
#endif /* GETFEM_INTERPOLATION_H__  */