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41 results

BackgroundMesh.cpp

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    BackgroundMesh.cpp 22.33 KiB
    // Gmsh - Copyright (C) 1997-2017 C. Geuzaine, J.-F. Remacle
    //
    // See the LICENSE.txt file for license information. Please report all
    // bugs and problems to the public mailing list <gmsh@onelab.info>.
    
    #include "GmshMessage.h"
    #include "BackgroundMesh.h"
    #include "Numeric.h"
    #include "Context.h"
    #include "GVertex.h"
    #include "GEdge.h"
    #include "GEdgeCompound.h"
    #include "GFace.h"
    #include "GFaceCompound.h"
    #include "GModel.h"
    #include "OS.h"
    #include "Field.h"
    #include "MElement.h"
    #include "MElementOctree.h"
    #include "MLine.h"
    #include "MTriangle.h"
    #include "MQuadrangle.h"
    #include "MVertex.h"
    
    #if defined(HAVE_SOLVER)
    #include "dofManager.h"
    #include "laplaceTerm.h"
    #include "linearSystemGMM.h"
    #include "linearSystemCSR.h"
    #include "linearSystemFull.h"
    #include "linearSystemPETSc.h"
    #endif
    
    #if defined(HAVE_ANN)
    static int _NBANN = 2;
    #endif
    
    void backgroundMesh::set(GFace *gf)
    {
      if (_current) delete _current;
      _current = new backgroundMesh(gf);
    }
    
    void backgroundMesh::setCrossFieldsByDistance(GFace *gf)
    {
      if (_current) delete _current;
      _current = new backgroundMesh(gf, true);
    }
    
    void backgroundMesh::unset()
    {
      if (_current) delete _current;
      _current = 0;
    }
    
    backgroundMesh::backgroundMesh(GFace *_gf, bool cfd)
    #if defined(HAVE_ANN)
      : _octree(0), uv_kdtree(0), nodes(0), angle_nodes(0), angle_kdtree(0)
    #endif
    {
    
      if (cfd){
        Msg::Info("Building A Cross Field Using Closest Distance");
        propagateCrossFieldByDistance(_gf);
        return;
      }
    
      // create a bunch of triangles on the parametric space
      // those triangles are local to the backgroundMesh so that
      // they do not depend on the actual mesh that can be deleted
    
      std::set<SPoint2> myBCNodes;
      for (unsigned int i = 0; i < _gf->triangles.size(); i++){
        MTriangle *e = _gf->triangles[i];
        MVertex *news[3];
        for (int j=0;j<3;j++){
          MVertex *v = e->getVertex(j);
          std::map<MVertex*,MVertex*>::iterator it = _3Dto2D.find(v);
          MVertex *newv =0;
          if (it == _3Dto2D.end()){
            SPoint2 p;
            reparamMeshVertexOnFace(v, _gf, p);
            newv = new MVertex (p.x(), p.y(), 0.0);
            _vertices.push_back(newv);
            _3Dto2D[v] = newv;
            _2Dto3D[newv] = v;
            if(v->onWhat()->dim()<2) myBCNodes.insert(p);
          }
          else newv = it->second;
          news[j] = newv;
        }
        MTriangle *T2D = new MTriangle(news[0],news[1],news[2]);
        _triangles.push_back(T2D);
      }
    
    #if defined(HAVE_ANN)
      //printf("creating uv kdtree %d \n", myBCNodes.size());
      index = new ANNidx[2];
      dist  = new ANNdist[2];
      nodes = annAllocPts(myBCNodes.size(), 3);
      std::set<SPoint2>::iterator itp = myBCNodes.begin();
      int ind = 0;
      while (itp != myBCNodes.end()){
        SPoint2 pt = *itp;
        //fprintf(of, "SP(%g,%g,%g){%g};\n", pt.x(), pt.y(), 0.0, 10000);
        nodes[ind][0] = pt.x();
        nodes[ind][1] = pt.y();
        nodes[ind][2] = 0.0;
        itp++; ind++;
      }
      uv_kdtree = new ANNkd_tree(nodes, myBCNodes.size(), 3);
    #endif
    
      // build a search structure
      _octree = new MElementOctree(_triangles);
    
      // compute the mesh sizes at nodes
      if (CTX::instance()->mesh.lcFromPoints){
        propagate1dMesh(_gf);
      }
      else {
        std::map<MVertex*, MVertex*>::iterator itv2 = _2Dto3D.begin();
        for ( ; itv2 != _2Dto3D.end(); ++itv2){
          _sizes[itv2->first] = CTX::instance()->mesh.lcMax;
        }
      }
      // ensure that other criteria are fullfilled
      updateSizes(_gf);
    
      // compute optimal mesh orientations
      propagateCrossField(_gf);
    
      _3Dto2D.clear();
      _2Dto3D.clear();
    }
    
    backgroundMesh::~backgroundMesh()
    {
      for (unsigned int i = 0; i < _vertices.size(); i++) delete _vertices[i];
      for (unsigned int i = 0; i < _triangles.size(); i++) delete _triangles[i];
      if (_octree)delete _octree;
    #if defined(HAVE_ANN)
      if(uv_kdtree) delete uv_kdtree;
      if(angle_kdtree) delete angle_kdtree;
      if(nodes) annDeallocPts(nodes);
      if(angle_nodes) annDeallocPts(angle_nodes);
      delete[]index;
      delete[]dist;
    #endif
    }
    
    static void propagateValuesOnFace(GFace *_gf,
                                      std::map<MVertex*,double> &dirichlet,
    				  simpleFunction<double> *ONE,
    				  bool in_parametric_plane = false)
    {
    #if defined(HAVE_SOLVER)
      linearSystem<double> *_lsys = 0;
    #if defined(HAVE_PETSC) && !defined(HAVE_TAUCS)
      _lsys = new linearSystemPETSc<double>;
    #elif defined(HAVE_GMM) && !defined(HAVE_TAUCS)
      linearSystemGmm<double> *_lsysb = new linearSystemGmm<double>;
      _lsysb->setGmres(1);
      _lsys = _lsysb;
    #elif defined(HAVE_TAUCS)
      _lsys = new linearSystemCSRTaucs<double>;
    #else
      _lsys = new linearSystemFull<double>;
    #endif
    
      dofManager<double> myAssembler(_lsys);
    
      // fix boundary conditions
      std::map<MVertex*, double>::iterator itv = dirichlet.begin();
      for ( ; itv != dirichlet.end(); ++itv){
        myAssembler.fixVertex(itv->first, 0, 1, itv->second);
      }
    
      // Number vertices
      std::set<MVertex*> vs;
      for (unsigned int k = 0; k < _gf->triangles.size(); k++)
        for (int j=0;j<3;j++)vs.insert(_gf->triangles[k]->getVertex(j));
      for (unsigned int k = 0; k < _gf->quadrangles.size(); k++)
        for (int j=0;j<4;j++)vs.insert(_gf->quadrangles[k]->getVertex(j));
    
      std::map<MVertex*,SPoint3> theMap;
      if ( in_parametric_plane) {
        for (std::set<MVertex*>::iterator it = vs.begin(); it != vs.end(); ++it){
          SPoint2 p;
          reparamMeshVertexOnFace ( *it, _gf, p);
          theMap[*it] = SPoint3((*it)->x(),(*it)->y(),(*it)->z());
          (*it)->setXYZ(p.x(),p.y(),0.0);
        }
      }
    
      for (std::set<MVertex*>::iterator it = vs.begin(); it != vs.end(); ++it)
        myAssembler.numberVertex(*it, 0, 1);
    
      // Assemble
      laplaceTerm l(0, 1, ONE);
      for (unsigned int k = 0; k < _gf->triangles.size(); k++){
        MTriangle *t = _gf->triangles[k];
        SElement se(t);
        l.addToMatrix(myAssembler, &se);
      }
    
      // Solve
      if (myAssembler.sizeOfR()){
        _lsys->systemSolve();
      }
    
      // save solution
      for (std::set<MVertex*>::iterator it = vs.begin(); it != vs.end(); ++it){
        myAssembler.getDofValue(*it, 0, 1, dirichlet[*it]);
      }
    
      if ( in_parametric_plane) {
        for (std::set<MVertex*>::iterator it = vs.begin(); it != vs.end(); ++it){
          SPoint3 p = theMap[(*it)];
          (*it)->setXYZ(p.x(),p.y(),p.z());
        }
      }
      delete _lsys;
    #endif
    }
    
    void backgroundMesh::propagate1dMesh(GFace *_gf)
    {
      std::list<GEdge*> e;// = _gf->edges();
      replaceMeshCompound(_gf, e);
      std::list<GEdge*>::const_iterator it = e.begin();
      std::map<MVertex*,double> sizes;
    
      for( ; it != e.end(); ++it ){
        if (!(*it)->isSeam(_gf)){
          for(unsigned int i = 0; i < (*it)->lines.size(); i++ ){
            MVertex *v1 = (*it)->lines[i]->getVertex(0);
            MVertex *v2 = (*it)->lines[i]->getVertex(1);
            if (v1 != v2){
              double d = sqrt((v1->x() - v2->x()) * (v1->x() - v2->x()) +
                  (v1->y() - v2->y()) * (v1->y() - v2->y()) +
                  (v1->z() - v2->z()) * (v1->z()  -v2->z()));
              for (int k=0;k<2;k++){
                MVertex *v = (*it)->lines[i]->getVertex(k);
                std::map<MVertex*, double>::iterator itv = sizes.find(v);
                if (itv == sizes.end())
                  sizes[v] = log(d);
                else
                  itv->second = 0.5 * (itv->second + log(d));
              }
            }
          }
        }
      }
    
      simpleFunction<double> ONE(1.0);
      propagateValuesOnFace(_gf, sizes,&ONE);
    
      std::map<MVertex*,MVertex*>::iterator itv2 = _2Dto3D.begin();
      for ( ; itv2 != _2Dto3D.end(); ++itv2){
        MVertex *v_2D = itv2->first;
        MVertex *v_3D = itv2->second;
        _sizes[v_2D] = exp(sizes[v_3D]);
      }
    }
    
    crossField2d::crossField2d(MVertex* v, GEdge* ge)
    {
      double p;
      bool success = reparamMeshVertexOnEdge(v, ge, p);
      if (!success){
        Msg::Warning("cannot reparametrize a point in crossField");
        _angle = 0;
        return;
      }
      SVector3 t = ge->firstDer (p);
      t.normalize();
      _angle = atan2 (t.y(),t.x());
      crossField2d::normalizeAngle (_angle);
    }
    
    void backgroundMesh::propagateCrossFieldByDistance(GFace *_gf)
    {
      std::list<GEdge*> e;
      replaceMeshCompound(_gf, e);
    
      std::list<GEdge*>::const_iterator it = e.begin();
      std::map<MVertex*,double> _cosines4,_sines4;
      std::map<MVertex*,SPoint2> _param;
    
      for( ; it != e.end(); ++it ){
        if (!(*it)->isSeam(_gf)){
          for(unsigned int i = 0; i < (*it)->lines.size(); i++ ){
            MVertex *v[2];
            v[0] = (*it)->lines[i]->getVertex(0);
            v[1] = (*it)->lines[i]->getVertex(1);
            SPoint2 p1,p2;
            reparamMeshEdgeOnFace(v[0],v[1],_gf,p1,p2);
            /* a correct way of computing angles  */
            Pair<SVector3, SVector3> der = _gf->firstDer((p1+p2)*.5);
            SVector3 t1 = der.first();
            SVector3 t2 (v[1]->x()-v[0]->x(),v[1]->y()-v[0]->y(),v[1]->z()-v[0]->z());
            t1.normalize();
            t2.normalize();
            double _angle = angle (t1,t2);
            //        double angle = atan2 ( p1.y()-p2.y() , p1.x()-p2.x() );
            crossField2d::normalizeAngle (_angle);
            for (int i=0;i<2;i++){
              std::map<MVertex*,double>::iterator itc = _cosines4.find(v[i]);
              std::map<MVertex*,double>::iterator its = _sines4.find(v[i]);
              if (itc != _cosines4.end()){
                itc->second  = 0.5*(itc->second + cos(4*_angle));
                its->second  = 0.5*(its->second + sin(4*_angle));
              }
              else {
                _param[v[i]] = (i==0) ? p1 : p2;
                _cosines4[v[i]] = cos(4*_angle);
                _sines4[v[i]] = sin(4*_angle);
              }
            }
          }
        }
      }
    
    #if defined(HAVE_ANN)
      index = new ANNidx[_NBANN];
      dist  = new ANNdist[_NBANN];
      angle_nodes = annAllocPts(_cosines4.size(), 3);
      std::map<MVertex*,double>::iterator itp = _cosines4.begin();
      int ind = 0;
      _sin.clear();
      _cos.clear();
      while (itp !=  _cosines4.end()){
        MVertex *v = itp->first;
        double c = itp->second;
        SPoint2 pt = _param[v];
        double s = _sines4[v];
        angle_nodes[ind][0] = pt.x();
        angle_nodes[ind][1] = pt.y();
        angle_nodes[ind][2] = 0.0;
        _cos.push_back(c);
        _sin.push_back(s);
        itp++;ind++;
      }
      angle_kdtree = new ANNkd_tree(angle_nodes, _cosines4.size(), 3);
    #endif
    }
    
    inline double myAngle (const SVector3 &a, const SVector3 &b, const SVector3 &d)
    {
      double cosTheta = dot(a,b);
      double sinTheta = dot(crossprod(a,b),d);
      return atan2 (sinTheta,cosTheta);
    }
    
    // smoothness = h * (|grad (cos 4 a)| + |grad (sin 4 a)|)
    // smoothness is of order 1 if not smooth
    // smoothness is of order h/L if smooth
    // h --> mesh size
    // L --> domain size
    double backgroundMesh::getSmoothness(MElement *e)
    {
      MVertex *v0 = _3Dto2D[e->getVertex(0)];
      MVertex *v1 = _3Dto2D[e->getVertex(1)];
      MVertex *v2 = _3Dto2D[e->getVertex(2)];
      std::map<MVertex*,double> :: const_iterator i0 = _angles.find (v0);
      std::map<MVertex*,double> :: const_iterator i1 = _angles.find (v1);
      std::map<MVertex*,double> :: const_iterator i2 = _angles.find (v2);
      double a[3] = {cos(4*i0->second),cos(4*i1->second),cos(4*i2->second)};
      double b[3] = {sin(4*i0->second),sin(4*i1->second),sin(4*i2->second)};
      //      printf("coucou\n");
      double f[3];
      e->interpolateGrad(a,0,0,0,f);
      const double gradcos = sqrt (f[0]*f[0]+f[1]*f[1]+f[2]*f[2]);
      e->interpolateGrad(b,0,0,0,f);
      //const double gradsin = sqrt (f[0]*f[0]+f[1]*f[1]+f[2]*f[2]);
      const double h = e->maxEdge();
      return (gradcos /*+ gradsin*/) * h;
    }
    
    double backgroundMesh::getSmoothness(double u, double v, double w)
    {
      MElement *e = _octree->find(u, v, w, 2, true);
      if (!e) return -1.0;
      MVertex *v0 = e->getVertex(0);
      MVertex *v1 = e->getVertex(1);
      MVertex *v2 = e->getVertex(2);
      std::map<MVertex*,double> :: const_iterator i0 = _angles.find (v0);
      std::map<MVertex*,double> :: const_iterator i1 = _angles.find (v1);
      std::map<MVertex*,double> :: const_iterator i2 = _angles.find (v2);
      double a[3] = {cos(4*i0->second),cos(4*i1->second),cos(4*i2->second)};
      double b[3] = {sin(4*i0->second),sin(4*i1->second),sin(4*i2->second)};
      //      printf("coucou\n");
      double f[3];
      e->interpolateGrad(a,0,0,0,f);
      const double gradcos = sqrt (f[0]*f[0]+f[1]*f[1]+f[2]*f[2]);
      e->interpolateGrad(b,0,0,0,f);
      //const double gradsin = sqrt (f[0]*f[0]+f[1]*f[1]+f[2]*f[2]);
      const double h = e->maxEdge();
      return (gradcos /*+ gradsin*/) * h;
    }
    
    void backgroundMesh::propagateCrossField(GFace *_gf)
    {
      propagateCrossFieldHJ (_gf);
      // solve the non liear problem
      constantPerElement<double> C;
      int ITER = 0;
      //  int NSMOOTH = _gf->triangles.size();
      while(0){
        //    int NSMOOTH_NOW = 0;
        for (unsigned int i = 0; i < _gf->triangles.size(); i++){
          double smoothness = getSmoothness (_gf->triangles[i]);
          double val = smoothness < .5 ? 1.0 : 1.e-3 ;//exp(-absf/10);
          C.set(_gf->triangles[i],val);
        }
        //    if (NSMOOTH_NOW == NSMOOTH) break;
        //    NSMOOTH = NSMOOTH_NOW;
        //    break;
        _angles.clear();
        propagateCrossField (_gf,&C);
        if (++ITER > 0)break;
      }
      //  printf("converged in %d iterations\n", ITER);
    #if 0 // debug print
      char name[256];
      sprintf(name, "cross-%d-%d.pos", _gf->tag(), ITER);
      print(name, 0, 1);
      sprintf(name, "smooth-%d-%d.pos", _gf->tag(), ITER);
      print(name, _gf, 2);
    #endif
    }
    
    void backgroundMesh::propagateCrossFieldHJ(GFace *_gf)
    {
      simpleFunction<double> ONE(1.0);
      propagateCrossField (_gf, &ONE);
    
    }
    
    void backgroundMesh::propagateCrossField(GFace *_gf, simpleFunction<double> *ONE)
    {
      std::map<MVertex*,double> _cosines4,_sines4;
    
      std::list<GEdge*> e;
      replaceMeshCompound(_gf, e);
    
      std::list<GEdge*>::const_iterator it = e.begin();
    
      for( ; it != e.end(); ++it ){
        if (!(*it)->isSeam(_gf)){
          for(unsigned int i = 0; i < (*it)->lines.size(); i++ ){
            MVertex *v[2];
            v[0] = (*it)->lines[i]->getVertex(0);
            v[1] = (*it)->lines[i]->getVertex(1);
            SPoint2 p1,p2;
            reparamMeshEdgeOnFace(v[0],v[1],_gf,p1,p2);
            Pair<SVector3, SVector3> der = _gf->firstDer((p1+p2)*.5);
            SVector3 t1 = der.first();
            SVector3 t2 = der.second();
            SVector3 n = crossprod(t1,t2);
            n.normalize();
            SVector3 d1(v[1]->x()-v[0]->x(),v[1]->y()-v[0]->y(),v[1]->z()-v[0]->z());
            t1.normalize();
            d1.normalize();
            double _angle = myAngle (t1,d1,n);
            crossField2d::normalizeAngle (_angle);
            for (int i=0;i<2;i++){
              std::map<MVertex*,double>::iterator itc = _cosines4.find(v[i]);
              std::map<MVertex*,double>::iterator its = _sines4.find(v[i]);
              if (itc != _cosines4.end()){
                itc->second  = 0.5*(itc->second + cos(4*_angle));
                its->second  = 0.5*(its->second + sin(4*_angle));
              }
              else {
                _cosines4[v[i]] = cos(4*_angle);
                _sines4[v[i]] = sin(4*_angle);
              }
            }
          }
        }
      }
    
      propagateValuesOnFace(_gf,_cosines4,ONE,false);
      propagateValuesOnFace(_gf,_sines4,ONE,false);
    
      std::map<MVertex*,MVertex*>::iterator itv2 = _2Dto3D.begin();
      for ( ; itv2 != _2Dto3D.end(); ++itv2){
        MVertex *v_2D = itv2->first;
        MVertex *v_3D = itv2->second;
        double angle = atan2(_sines4[v_3D],_cosines4[v_3D]) / 4.0;
        crossField2d::normalizeAngle (angle);
        _angles[v_2D] = angle;
      }
    }
    
    void backgroundMesh::updateSizes(GFace *_gf)
    {
      std::map<MVertex*,double>::iterator itv = _sizes.begin();
      for ( ; itv != _sizes.end(); ++itv){
        SPoint2 p;
        MVertex *v = _2Dto3D[itv->first];
        double lc;
        if (v->onWhat()->dim() == 0){
          lc = BGM_MeshSize(v->onWhat(), 0,0,v->x(),v->y(),v->z());
        }
        else if (v->onWhat()->dim() == 1){
          double u;
          v->getParameter(0, u);
          lc = BGM_MeshSize(v->onWhat(), u, 0, v->x(), v->y(), v->z());
        }
        else{
          reparamMeshVertexOnFace(v, _gf, p);
          lc = BGM_MeshSize(_gf, p.x(), p.y(), v->x(), v->y(), v->z());
        }
        // printf("2D -- %g %g 3D -- %g %g\n",p.x(),p.y(),v->x(),v->y());
        itv->second = std::min(lc,itv->second);
        itv->second = std::max(itv->second,  CTX::instance()->mesh.lcMin);
        itv->second = std::min(itv->second,  CTX::instance()->mesh.lcMax);
      }
      // do not allow large variations in the size field
      // (Int. J. Numer. Meth. Engng. 43, 1143-1165 (1998) MESH GRADATION
      // CONTROL, BOROUCHAKI, HECHT, FREY)
      std::set<MEdge,Less_Edge> edges;
      for (unsigned int i = 0; i < _triangles.size(); i++){
        for (int j = 0; j < _triangles[i]->getNumEdges(); j++){
          edges.insert(_triangles[i]->getEdge(j));
        }
      }
      const double _beta = 1.3;
      for (int i=0;i<3;i++){
        std::set<MEdge,Less_Edge>::iterator it = edges.begin();
        for ( ; it != edges.end(); ++it){
          MVertex *v0 = it->getVertex(0);
          MVertex *v1 = it->getVertex(1);
          MVertex *V0 = _2Dto3D[v0];
          MVertex *V1 = _2Dto3D[v1];
          std::map<MVertex*,double>::iterator s0 = _sizes.find(V0);
          std::map<MVertex*,double>::iterator s1 = _sizes.find(V1);
          if (s0->second < s1->second)s1->second = std::min(s1->second,_beta*s0->second);
          else s0->second = std::min(s0->second,_beta*s1->second);
        }
      }
    }
    
    bool backgroundMesh::inDomain (double u, double v, double w) const
    {
      return _octree->find(u, v, w, 2, true) != 0;
    }
    
    double backgroundMesh::operator() (double u, double v, double w) const
    {
      double uv[3] = {u, v, w};
      double uv2[3];
      MElement *e = _octree->find(u, v, w, 2, true);
      if (!e) {
    #if defined(HAVE_ANN)
        //printf("BGM octree not found --> find in kdtree \n");
        if(uv_kdtree->nPoints() < 2) return -1000.;
        double pt[3] = {u, v, 0.0};
        uv_kdtree->annkSearch(pt, 2, index, dist);
        SPoint3  p1(nodes[index[0]][0], nodes[index[0]][1], nodes[index[0]][2]);
        SPoint3  p2(nodes[index[1]][0], nodes[index[1]][1], nodes[index[1]][2]);
        SPoint3 pnew; double d;
        signedDistancePointLine(p1, p2, SPoint3(u, v, 0.), d, pnew);
        e = _octree->find(pnew.x(), pnew.y(), 0.0, 2, true);
    #endif
        if(!e){
          Msg::Error("BGM octree: cannot find UVW=%g %g %g", u, v, w);
          return -1000.0;//0.4;
        }
      }
      e->xyz2uvw(uv, uv2);
      std::map<MVertex*,double>::const_iterator itv1 = _sizes.find(e->getVertex(0));
      std::map<MVertex*,double>::const_iterator itv2 = _sizes.find(e->getVertex(1));
      std::map<MVertex*,double>::const_iterator itv3 = _sizes.find(e->getVertex(2));
      return itv1->second * (1-uv2[0]-uv2[1]) + itv2->second * uv2[0] + itv3->second * uv2[1];
    }
    
    double backgroundMesh::getAngle(double u, double v, double w) const
    {
      // JFR :
      // we can use closest point for computing
      // cross field angles : this allow NOT to
      // generate a spurious mesh and solve a PDE
      if (!_octree){
    #if defined(HAVE_ANN)
        double angle = 0.;
        if(angle_kdtree->nPoints() >= _NBANN){
          double pt[3] = {u,v,0.0};
          angle_kdtree->annkSearch(pt, _NBANN, index, dist);
          double SINE = 0.0 , COSINE = 0.0;
          for (int i=0;i<_NBANN;i++){
            SINE += _sin[index[i]];
            COSINE += _cos[index[i]];
            //      printf("%2d %2d %12.5E %12.5E\n",i,index[i],_sin[index[i]],_cos[index[i]]);
          }
          angle = atan2(SINE,COSINE)/4.0;
        }
        crossField2d::normalizeAngle (angle);
        return angle;
    #endif
      }
    
      // HACK FOR LEWIS
      // h = 1+30(y-x^2)^2  + (1-x)^2
      //  double x = u;
      //  double y = v;
      //  double dhdx = 30 * 2 * (y-x*x) * (-2*x) - 2 * (1-x);
      //  double dhdy = 30 * 2 * (y-x*x);
      //  double angles = atan2(y,x*x);
      //  crossField2d::normalizeAngle (angles);
      //  return angles;
    
      double uv[3] = {u, v, w};
      double uv2[3];
      MElement *e = _octree->find(u, v, w, 2, true);
      if (!e) {
    #if defined(HAVE_ANN)
        //printf("BGM octree not found --> find in kdtree \n");
        if(uv_kdtree->nPoints() < 2) return -1000.0;
        double pt[3] = {u,v,0.0};
        uv_kdtree->annkSearch(pt, 2, index, dist);
        SPoint3  p1(nodes[index[0]][0], nodes[index[0]][1], nodes[index[0]][2]);
        SPoint3  p2(nodes[index[1]][0], nodes[index[1]][1], nodes[index[1]][2]);
        SPoint3 pnew; double d;
        signedDistancePointLine(p1, p2, SPoint3(u, v, 0.), d, pnew);
        e = _octree->find(pnew.x(), pnew.y(), 0., 2, true);
    #endif
        if(!e){
          Msg::Error("BGM octree angle: cannot find UVW=%g %g %g", u, v, w);
          return -1000.0;
        }
      }
      e->xyz2uvw(uv, uv2);
      std::map<MVertex*,double>::const_iterator itv1 = _angles.find(e->getVertex(0));
      std::map<MVertex*,double>::const_iterator itv2 = _angles.find(e->getVertex(1));
      std::map<MVertex*,double>::const_iterator itv3 = _angles.find(e->getVertex(2));
    
      double cos4 = cos (4*itv1->second) * (1-uv2[0]-uv2[1]) +
        cos (4*itv2->second) * uv2[0] +
        cos (4*itv3->second) * uv2[1] ;
      double sin4 = sin (4*itv1->second) * (1-uv2[0]-uv2[1]) +
        sin (4*itv2->second) * uv2[0] +
        sin (4*itv3->second) * uv2[1] ;
      double angle = atan2(sin4,cos4)/4.0;
      crossField2d::normalizeAngle (angle);
    
      return angle;
    }
    
    void backgroundMesh::print(const std::string &filename, GFace *gf,
                               const std::map<MVertex*,double> &_whatToPrint, int smooth)
    {
      FILE *f = Fopen(filename.c_str(), "w");
      if(!f){
        Msg::Error("Could not open file '%s'", filename.c_str());
        return;
      }
      fprintf(f, "View \"Background Mesh\"{\n");
      if (smooth){
        for(unsigned int i = 0; i < gf->triangles.size(); i++){
          MVertex *v1 = gf->triangles[i]->getVertex(0);
          MVertex *v2 = gf->triangles[i]->getVertex(1);
          MVertex *v3 = gf->triangles[i]->getVertex(2);
          double x = getSmoothness (gf->triangles[i]);
          fprintf(f,"ST(%g,%g,%g,%g,%g,%g,%g,%g,%g) {%g,%g,%g};\n",
    	      v1->x(), v1->y(), v1->z(),
    	      v2->x(), v2->y(), v2->z(),
    	      v3->x(), v3->y(), v3->z(), x, x, x);
        }
      }
      else {
        for(unsigned int i = 0; i < _triangles.size(); i++){
          MVertex *v1 = _triangles[i]->getVertex(0);
          MVertex *v2 = _triangles[i]->getVertex(1);
          MVertex *v3 = _triangles[i]->getVertex(2);
          std::map<MVertex*,double>::const_iterator itv1 = _whatToPrint.find(v1);
          std::map<MVertex*,double>::const_iterator itv2 = _whatToPrint.find(v2);
          std::map<MVertex*,double>::const_iterator itv3 = _whatToPrint.find(v3);
          if (!gf){
    	fprintf(f,"ST(%g,%g,%g,%g,%g,%g,%g,%g,%g) {%g,%g,%g};\n",
    		v1->x(), v1->y(), v1->z(),
    		v2->x(), v2->y(), v2->z(),
    		v3->x(), v3->y(), v3->z(), itv1->second, itv2->second, itv3->second);
          }
          else {
    	GPoint p1 = gf->point(SPoint2(v1->x(),v1->y()));
    	GPoint p2 = gf->point(SPoint2(v2->x(),v2->y()));
    	GPoint p3 = gf->point(SPoint2(v3->x(),v3->y()));
    	fprintf(f,"ST(%g,%g,%g,%g,%g,%g,%g,%g,%g) {%g,%g,%g};\n",
    		p1.x(), p1.y(), p1.z(),
    		p2.x(), p2.y(), p2.z(),
    		p3.x(), p3.y(), p3.z(), itv1->second, itv2->second, itv3->second);
          }
        }
      }
      fprintf(f, "};\n");
      fclose(f);
    }
    
    MElementOctree* backgroundMesh::get_octree(){
    
      return _octree;
    }
    
    MElement *backgroundMesh::getMeshElementByCoord(double u, double v, double w, bool strict)
    {
      if(!_octree){
        Msg::Debug("Rebuilding BackgroundMesh element octree");
        _octree = new MElementOctree(_triangles);
      }
      return _octree->find(u,v,w, 2, strict);
    }
    
    backgroundMesh* backgroundMesh::_current = 0;