TetGeom.cpp

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////////////////////////////////////////////////////////////////////////////////
//
//  File: TetGeom.cpp
//
//  For more information, please see: http://www.nektar.info/
//
//  The MIT License
//
//  Copyright (c) 2006 Division of Applied Mathematics, Brown University (USA),
//  Department of Aeronautics, Imperial College London (UK), and Scientific
//  Computing and Imaging Institute, University of Utah (USA).
//
//  License for the specific language governing rights and limitations under
//  Permission is hereby granted, free of charge, to any person obtaining a
//  copy of this software and associated documentation files (the "Software"),
//  to deal in the Software without restriction, including without limitation
//  the rights to use, copy, modify, merge, publish, distribute, sublicense,
//  and/or sell copies of the Software, and to permit persons to whom the
//  Software is furnished to do so, subject to the following conditions:
//
//  The above copyright notice and this permission notice shall be included
//  in all copies or substantial portions of the Software.
//
//  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
//  OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
//  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
//  THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
//  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
//  FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
//  DEALINGS IN THE SOFTWARE.
//
//  Description: Tetrahedral geometry information.
//
////////////////////////////////////////////////////////////////////////////////

#include <SpatialDomains/TetGeom.h>

#include <SpatialDomains/Geometry1D.h>
#include <StdRegions/StdTetExp.h>
#include <SpatialDomains/SegGeom.h>

namespace Nektar
{
    namespace SpatialDomains
    {
        const unsigned int TetGeom::VertexEdgeConnectivity[4][3] = {
            {0,2,3},{0,1,4},{1,2,5},{3,4,5}};
        const unsigned int TetGeom::VertexFaceConnectivity[4][3] = {
            {0,1,3},{0,1,2},{0,2,3},{1,2,3}};
        const unsigned int TetGeom::EdgeFaceConnectivity  [6][2] = {
            {0,1},{0,2},{0,3},{1,3},{1,2},{2,3}};
        
        TetGeom::TetGeom()
        {
            m_shapeType = LibUtilities::eTetrahedron;
        }
        
        TetGeom::TetGeom(const TriGeomSharedPtr faces[]) :
            Geometry3D(faces[0]->GetEdge(0)->GetVertex(0)->GetCoordim())
        {
            m_shapeType = LibUtilities::eTetrahedron;
            
            /// Copy the face shared pointers
            m_faces.insert(m_faces.begin(), faces, faces+TetGeom::kNfaces);
            
            /// Set up orientation vectors with correct amount of elements.
            m_eorient.resize(kNedges);
            m_forient.resize(kNfaces);
            
            SetUpLocalEdges();
            SetUpLocalVertices();
            SetUpEdgeOrientation();
            SetUpFaceOrientation();
            
            /// Determine necessary order for standard region.
            vector<int> tmp;
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNcoeffs(0));
            int order0 = *max_element(tmp.begin(), tmp.end());
            
            tmp.clear();
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNumPoints(0));
            int points0 = *max_element(tmp.begin(), tmp.end());
            
            tmp.clear();
            tmp.push_back(order0);
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNcoeffs(1));
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNcoeffs(2));
            int order1 = *max_element(tmp.begin(), tmp.end());
            
            tmp.clear();
            tmp.push_back(points0);
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNumPoints(1));
            tmp.push_back(faces[0]->GetXmap()->GetEdgeNumPoints(2));
            int points1 = *max_element(tmp.begin(), tmp.end());

            tmp.clear();
            tmp.push_back(order0);
            tmp.push_back(order1);
            tmp.push_back(faces[1]->GetXmap()->GetEdgeNcoeffs(1));
            tmp.push_back(faces[1]->GetXmap()->GetEdgeNcoeffs(2));
            tmp.push_back(faces[3]->GetXmap()->GetEdgeNcoeffs(1));
            int order2 = *max_element(tmp.begin(), tmp.end());

            tmp.clear();
            tmp.push_back(points0);
            tmp.push_back(points1);
            tmp.push_back(faces[1]->GetXmap()->GetEdgeNumPoints(1));
            tmp.push_back(faces[1]->GetXmap()->GetEdgeNumPoints(2));
            tmp.push_back(faces[3]->GetXmap()->GetEdgeNumPoints(1));
            int points2 = *max_element(tmp.begin(), tmp.end());

            const LibUtilities::BasisKey A(
                LibUtilities::eModified_A, order0,
                LibUtilities::PointsKey(points0,LibUtilities::eGaussLobattoLegendre));
            const LibUtilities::BasisKey B(
                LibUtilities::eModified_B, order1,
                LibUtilities::PointsKey(points1,LibUtilities::eGaussRadauMAlpha1Beta0));
            const LibUtilities::BasisKey C(
                LibUtilities::eModified_C, order2,
                LibUtilities::PointsKey(points2,LibUtilities::eGaussRadauMAlpha2Beta0));

            m_xmap = MemoryManager<StdRegions::StdTetExp>::AllocateSharedPtr(A,B,C);
            SetUpCoeffs(m_xmap->GetNcoeffs());
        }
        
        TetGeom::~TetGeom()
        {
            
        }
        
        /**
         * @brief Determines if a point specified in global coordinates is
         * located within this tetrahedral geometry.
         */
        bool TetGeom::v_ContainsPoint(
            const Array<OneD, const NekDouble> &gloCoord, NekDouble tol)
        {
            Array<OneD,NekDouble> locCoord(GetCoordim(),0.0);
            return v_ContainsPoint(gloCoord,locCoord,tol);
        }

        bool TetGeom::v_ContainsPoint(const Array<OneD, const NekDouble> &gloCoord, 
                                      Array<OneD, NekDouble> &locCoord,
                                      NekDouble tol)
        {
            NekDouble resid; 
            return v_ContainsPoint(gloCoord,locCoord,tol,resid);
        }

        /**
         * @brief Determines if a point specified in global coordinates is
         * located within this tetrahedral geometry and return local caretsian coordinates
         */
        bool TetGeom::v_ContainsPoint(const Array<OneD, const NekDouble> &gloCoord, 
                                      Array<OneD, NekDouble> &locCoord,
                                      NekDouble tol,
                                      NekDouble &resid)
        {
            // Validation checks
            ASSERTL1(gloCoord.num_elements() == 3,
                     "Three dimensional geometry expects three coordinates.");

            // find min, max point and check if within twice this
            // distance other false this is advisable since
            // GetLocCoord is expensive for non regular elements.
            if(GetMetricInfo()->GetGtype() !=  eRegular)
            {
                int i;
                Array<OneD, NekDouble> mincoord(3), maxcoord(3);
                NekDouble diff = 0.0;

                v_FillGeom();

                const int npts = m_xmap->GetTotPoints();
                Array<OneD, NekDouble> pts(npts);
                
                for(i = 0; i < 3; ++i)
                {
                    m_xmap->BwdTrans(m_coeffs[i], pts);

                    mincoord[i] = Vmath::Vmin(pts.num_elements(),pts,1);
                    maxcoord[i] = Vmath::Vmax(pts.num_elements(),pts,1);
                    
                    diff = max(maxcoord[i] - mincoord[i],diff); 
                }

                for(i = 0; i < 3; ++i)
                {
                    if((gloCoord[i] < mincoord[i] - 0.2*diff)||
                       (gloCoord[i] > maxcoord[i] + 0.2*diff))
                    {
                        return false;
                    }
                }
            }
            
            // Convert to the local (eta) coordinates.
            resid = v_GetLocCoords(gloCoord, locCoord);
            
            // Check local coordinate is within cartesian bounds.
            if (locCoord[0] >= -(1+tol) && locCoord[1] >= -(1+tol) &&
                locCoord[2] >= -(1+tol)                            &&
                locCoord[0] + locCoord[1] + locCoord[2] <= -1+tol)
            {
                return true;
            }

            return false;
        }


        /// Get Local cartesian points 
        NekDouble TetGeom::v_GetLocCoords(
            const Array<OneD, const NekDouble>& coords,
                  Array<OneD,       NekDouble>& Lcoords)
        {
            NekDouble resid = 0.0;

            // calculate local coordinates (eta) for coord
            if(GetMetricInfo()->GetGtype() == eRegular)
            {   
                // Point inside tetrahedron
                PointGeom r(m_coordim, 0, coords[0], coords[1], coords[2]);

                // Edges
                PointGeom er0, e10, e20, e30;
                er0.Sub(r,*m_verts[0]);
                e10.Sub(*m_verts[1],*m_verts[0]);
                e20.Sub(*m_verts[2],*m_verts[0]);
                e30.Sub(*m_verts[3],*m_verts[0]);


                // Cross products (Normal times area)
                PointGeom cp1020, cp2030, cp3010;
                cp1020.Mult(e10,e20);
                cp2030.Mult(e20,e30);
                cp3010.Mult(e30,e10);


                // Barycentric coordinates (relative volume)
                NekDouble V = e30.dot(cp1020); // Tet Volume = {(e30)dot(e10)x(e20)}/6
                NekDouble beta  = er0.dot(cp2030) / V; // volume1 = {(er0)dot(e20)x(e30)}/6
                NekDouble gamma = er0.dot(cp3010) / V; // volume1 = {(er0)dot(e30)x(e10)}/6
                NekDouble delta = er0.dot(cp1020) / V; // volume1 = {(er0)dot(e10)x(e20)}/6

                // Make tet bigger
                Lcoords[0] = 2.0*beta  - 1.0;
                Lcoords[1] = 2.0*gamma - 1.0;
                Lcoords[2] = 2.0*delta - 1.0;
            }
            else
            {
                v_FillGeom();

                // Determine nearest point of coords  to values in m_xmap
                int npts = m_xmap->GetTotPoints();
                Array<OneD, NekDouble> ptsx(npts), ptsy(npts), ptsz(npts);
                Array<OneD, NekDouble> tmp1(npts), tmp2(npts);

                m_xmap->BwdTrans(m_coeffs[0], ptsx);
                m_xmap->BwdTrans(m_coeffs[1], ptsy);
                m_xmap->BwdTrans(m_coeffs[2], ptsz);

                const Array<OneD, const NekDouble> za = m_xmap->GetPoints(0);
                const Array<OneD, const NekDouble> zb = m_xmap->GetPoints(1);
                const Array<OneD, const NekDouble> zc = m_xmap->GetPoints(2);
                
                //guess the first local coords based on nearest point
                Vmath::Sadd(npts, -coords[0], ptsx,1,tmp1,1);
                Vmath::Vmul (npts, tmp1,1,tmp1,1,tmp1,1);
                Vmath::Sadd(npts, -coords[1], ptsy,1,tmp2,1);
                Vmath::Vvtvp(npts, tmp2,1,tmp2,1,tmp1,1,tmp1,1);
                Vmath::Sadd(npts, -coords[2], ptsz,1,tmp2,1);
                Vmath::Vvtvp(npts, tmp2,1,tmp2,1,tmp1,1,tmp1,1);
                          
                int min_i = Vmath::Imin(npts,tmp1,1);
                
                // Get collapsed coordinate
                int qa = za.num_elements(), qb = zb.num_elements();
                Lcoords[2] = zc[min_i/(qa*qb)];
                min_i = min_i%(qa*qb);
                Lcoords[1] = zb[min_i/qa];
                Lcoords[0] = za[min_i%qa];

                // recover cartesian coordinate from collapsed coordinate. 
                Lcoords[1] = (1.0+Lcoords[0])*(1.0-Lcoords[2])/2 -1.0;
                Lcoords[0] = (1.0+Lcoords[0])*(-Lcoords[1]-Lcoords[2])/2 -1.0;

                // Perform newton iteration to find local coordinates 
                NewtonIterationForLocCoord(coords, ptsx, ptsy, ptsz, Lcoords,resid);
            }
            return resid;
        }
        
        int TetGeom::v_GetNumVerts() const
        {
            return 4;
        }
        
        int TetGeom::v_GetNumEdges() const
        {
            return 6;
        }

        int TetGeom::v_GetNumFaces() const
        {
            return 4;
        }

        int TetGeom::v_GetDir(const int faceidx, const int facedir) const
        {
            if (faceidx == 0)
            {
                return facedir;
            }
            else if (faceidx == 1)
            {
                return 2 * facedir;
            }
            else
            {
                return 1 + facedir;
            }
        }

        int TetGeom::v_GetVertexEdgeMap(const int i, const int j) const
    {
        return VertexEdgeConnectivity[i][j];
    }

        int TetGeom::v_GetVertexFaceMap(const int i, const int j) const
    {
        return VertexFaceConnectivity[i][j];
    }

        int TetGeom::v_GetEdgeFaceMap(const int i, const int j) const
    {
        return EdgeFaceConnectivity[i][j];
    }

        void TetGeom::SetUpLocalEdges()
        {
            
            // find edge 0
            int i,j;
            unsigned int check;

            SegGeomSharedPtr edge;

            // First set up the 3 bottom edges

            if(m_faces[0]->GetEid(0) != m_faces[1]->GetEid(0))
            {
                std::ostringstream errstrm;
                errstrm << "Local edge 0 (eid=" << m_faces[0]->GetEid(0);
                errstrm  << ") on face " <<  m_faces[0]->GetFid(); 
                errstrm << " must be the same as local edge 0 (eid="<<m_faces[1]->GetEid(0);
                errstrm << ") on face " <<   m_faces[1]->GetFid(); 
                ASSERTL0(false, errstrm.str());
            }

            int faceConnected;
            for(faceConnected = 1; faceConnected < 4 ; faceConnected++)
            {
                check = 0;
                for(i = 0; i < 3; i++)
                {
                    if( (m_faces[0])->GetEid(i) == (m_faces[faceConnected])->GetEid(0) )
                    {
                        edge = boost::dynamic_pointer_cast<SegGeom>((m_faces[0])->GetEdge(i));
                        m_edges.push_back(edge);
                        check++;
                    }
                }

                if( check < 1 )
                {
                    std::ostringstream errstrm;
                    errstrm << "Face 0 does not share an edge with first edge of adjacent face. Faces ";
                    errstrm << (m_faces[0])->GetFid() << ", " << (m_faces[faceConnected])->GetFid();
                    ASSERTL0(false, errstrm.str());
                }
                else if( check > 1)
                {
                    std::ostringstream errstrm;
                    errstrm << "Connected faces share more than one edge. Faces ";
                    errstrm << (m_faces[0])->GetFid() << ", " << (m_faces[faceConnected])->GetFid();
                    ASSERTL0(false, errstrm.str());
                }
            }


            // Then, set up the 3 vertical edges
            check = 0;
            for(i = 0; i < 3; i++) //Set up the vertical edge :face(1) and face(3)
            {
                for(j = 0; j < 3; j++)
                {
                    if( (m_faces[1])->GetEid(i) == (m_faces[3])->GetEid(j) )
                    {
                        edge = boost::dynamic_pointer_cast<SegGeom>((m_faces[1])->GetEdge(i));
                        m_edges.push_back(edge);
                        check++;
                    }
                }
            }
            if( check < 1 )
            {
                std::ostringstream errstrm;
                errstrm << "Connected faces do not share an edge. Faces ";
                errstrm << (m_faces[1])->GetFid() << ", " << (m_faces[3])->GetFid();
                ASSERTL0(false, errstrm.str());
            }
            else if( check > 1)
            {
                std::ostringstream errstrm;
                errstrm << "Connected faces share more than one edge. Faces ";
                errstrm << (m_faces[1])->GetFid() << ", " << (m_faces[3])->GetFid();
                ASSERTL0(false, errstrm.str());
            }
            // Set up vertical edges: face(1) through face(3)
            for(faceConnected = 1; faceConnected < 3 ; faceConnected++)
            {
                check = 0;
                for(i = 0; i < 3; i++)
                {
                    for(j = 0; j < 3; j++)
                    {
                        if( (m_faces[faceConnected])->GetEid(i) == (m_faces[faceConnected+1])->GetEid(j) )
                        {
                            edge = boost::dynamic_pointer_cast<SegGeom>((m_faces[faceConnected])->GetEdge(i));
                            m_edges.push_back(edge);
                            check++;
                        }
                    }
                }

                if( check < 1 )
                {
                    std::ostringstream errstrm;
                    errstrm << "Connected faces do not share an edge. Faces ";
                    errstrm << (m_faces[faceConnected])->GetFid() << ", " << (m_faces[faceConnected+1])->GetFid();
                    ASSERTL0(false, errstrm.str());
                }
                else if( check > 1)
                {
                    std::ostringstream errstrm;
                    errstrm << "Connected faces share more than one edge. Faces ";
                    errstrm << (m_faces[faceConnected])->GetFid() << ", " << (m_faces[faceConnected+1])->GetFid();
                    ASSERTL0(false, errstrm.str());
                }
            }
        };

        void TetGeom::SetUpLocalVertices(){

            // Set up the first 2 vertices (i.e. vertex 0,1)
            if( ( m_edges[0]->GetVid(0) == m_edges[1]->GetVid(0) ) ||
                ( m_edges[0]->GetVid(0) == m_edges[1]->GetVid(1) ) )
            {
                m_verts.push_back(m_edges[0]->GetVertex(1));
                m_verts.push_back(m_edges[0]->GetVertex(0));
            }
            else if( ( m_edges[0]->GetVid(1) == m_edges[1]->GetVid(0) ) ||
                     ( m_edges[0]->GetVid(1) == m_edges[1]->GetVid(1) ) )
            {
                m_verts.push_back(m_edges[0]->GetVertex(0));
                m_verts.push_back(m_edges[0]->GetVertex(1));
            }
            else
            {
                std::ostringstream errstrm;
                errstrm << "Connected edges do not share a vertex. Edges ";
                errstrm << m_edges[0]->GetEid() << ", " << m_edges[1]->GetEid();
                ASSERTL0(false, errstrm.str());
            }

            // set up the other bottom vertices (i.e. vertex 2)
            for(int i = 1; i < 2; i++)
            {
                if( m_edges[i]->GetVid(0) == m_verts[i]->GetVid() )
                {
                    m_verts.push_back(m_edges[i]->GetVertex(1));
                }
                else if( m_edges[i]->GetVid(1) == m_verts[i]->GetVid() )
                {
                    m_verts.push_back(m_edges[i]->GetVertex(0));
                }
                else
                {
                    std::ostringstream errstrm;
                    errstrm << "Connected edges do not share a vertex. Edges ";
                    errstrm << m_edges[i]->GetEid() << ", " << m_edges[i-1]->GetEid();
                    ASSERTL0(false, errstrm.str());
                }
            }

            // set up top vertex
            if (m_edges[3]->GetVid(0) == m_verts[0]->GetVid())
            {
                m_verts.push_back(m_edges[3]->GetVertex(1));
            }
            else {
                m_verts.push_back(m_edges[3]->GetVertex(0));
            }

            // Check the other edges match up.
            int check = 0;
            for (int i = 4; i < 6; ++i)
            {
                if( (m_edges[i]->GetVid(0) == m_verts[i-3]->GetVid()
                    && m_edges[i]->GetVid(1) == m_verts[3]->GetVid())
                    ||(m_edges[i]->GetVid(1) == m_verts[i-3]->GetVid()
                    && m_edges[i]->GetVid(0) == m_verts[3]->GetVid()))
                {
                    check++;
                }
            }
            if (check != 2) {
                std::ostringstream errstrm;
                errstrm << "Connected edges do not share a vertex. Edges ";
                errstrm << m_edges[3]->GetEid() << ", " << m_edges[2]->GetEid();
                ASSERTL0(false, errstrm.str());
            }
        };

        void TetGeom::SetUpEdgeOrientation(){

            // This 2D array holds the local id's of all the vertices
            // for every edge. For every edge, they are ordered to what we
            // define as being Forwards
            const unsigned int edgeVerts[kNedges][2] =
                { {0,1},
                  {1,2},
                  {0,2},
                  {0,3},
                  {1,3},
                  {2,3} };

            int i;
            for(i = 0; i < kNedges; i++)
            {
                if( m_edges[i]->GetVid(0) == m_verts[ edgeVerts[i][0] ]->GetVid() )
                {
                    m_eorient[i] = StdRegions::eForwards;
                }
                else if( m_edges[i]->GetVid(0) == m_verts[ edgeVerts[i][1] ]->GetVid() )
                {
                    m_eorient[i] = StdRegions::eBackwards;
                }
                else
                {
                    ASSERTL0(false,"Could not find matching vertex for the edge");
                }
            }

        };

        void TetGeom::SetUpFaceOrientation(){

            int f,i;

            // These arrays represent the vector of the A and B
            // coordinate of the local elemental coordinate system
            // where A corresponds with the coordinate direction xi_i
            // with the lowest index i (for that particular face)
            // Coordinate 'B' then corresponds to the other local
            // coordinate (i.e. with the highest index)
            Array<OneD,NekDouble> elementAaxis(m_coordim);
            Array<OneD,NekDouble> elementBaxis(m_coordim);

            // These arrays correspond to the local coordinate
            // system of the face itself (i.e. the Geometry2D)
            // faceAaxis correspond to the xi_0 axis
            // faceBaxis correspond to the xi_1 axis
            Array<OneD,NekDouble> faceAaxis(m_coordim);
            Array<OneD,NekDouble> faceBaxis(m_coordim);

            // This is the base vertex of the face (i.e. the Geometry2D)
            // This corresponds to thevertex with local ID 0 of the
            // Geometry2D
            unsigned int baseVertex;

            // The lenght of the vectors above
            NekDouble elementAaxis_length;
            NekDouble elementBaxis_length;
            NekDouble faceAaxis_length;
            NekDouble faceBaxis_length;

            // This 2D array holds the local id's of all the vertices
            // for every face. For every face, they are ordered in such
            // a way that the implementation below allows a unified approach
            // for all faces.
            const unsigned int faceVerts[kNfaces][TriGeom::kNverts] =
                { {0,1,2} ,
                  {0,1,3} ,
                  {1,2,3} ,
                  {0,2,3} };

            NekDouble dotproduct1 = 0.0;
            NekDouble dotproduct2 = 0.0;

            unsigned int orientation;

            // Loop over all the faces to set up the orientation
            for(f = 0; f < kNqfaces + kNtfaces; f++)
            {
                // initialisation
                elementAaxis_length = 0.0;
                elementBaxis_length = 0.0;
                faceAaxis_length = 0.0;
                faceBaxis_length = 0.0;

                dotproduct1 = 0.0;
                dotproduct2 = 0.0;

                baseVertex = m_faces[f]->GetVid(0);

                // We are going to construct the vectors representing the A and B axis
                // of every face. These vectors will be constructed as a vector-representation
                // of the edges of the face. However, for both coordinate directions, we can
                // represent the vectors by two different edges. That's why we need to make sure that
                // we pick the edge to which the baseVertex of the Geometry2D-representation of the face
                // belongs...

                // Compute the length of edges on a base-face
                if( baseVertex == m_verts[ faceVerts[f][0] ]->GetVid() )
                {
                    for(i = 0; i < m_coordim; i++)
                    {
                        elementAaxis[i] = (*m_verts[ faceVerts[f][1] ])[i] - (*m_verts[ faceVerts[f][0] ])[i];
                        elementBaxis[i] = (*m_verts[ faceVerts[f][2] ])[i] - (*m_verts[ faceVerts[f][0] ])[i];
                    }
                }
                else if( baseVertex == m_verts[ faceVerts[f][1] ]->GetVid() )
                {
                    for(i = 0; i < m_coordim; i++)
                    {
                        elementAaxis[i] = (*m_verts[ faceVerts[f][1] ])[i] - (*m_verts[ faceVerts[f][0] ])[i];
                        elementBaxis[i] = (*m_verts[ faceVerts[f][2] ])[i] - (*m_verts[ faceVerts[f][1] ])[i];
                    }
                }
                else if( baseVertex == m_verts[ faceVerts[f][2] ]->GetVid() )
                {
                    for(i = 0; i < m_coordim; i++)
                    {
                        elementAaxis[i] = (*m_verts[ faceVerts[f][1] ])[i] - (*m_verts[ faceVerts[f][2] ])[i];
                        elementBaxis[i] = (*m_verts[ faceVerts[f][2] ])[i] - (*m_verts[ faceVerts[f][0] ])[i];
                    }
                }
                else
                {
                    ASSERTL0(false, "Could not find matching vertex for the face");
                }

                // Now, construct the edge-vectors of the local coordinates of
                // the Geometry2D-representation of the face
                for(i = 0; i < m_coordim; i++)
                {
                    faceAaxis[i] = (*m_faces[f]->GetVertex(1))[i] - (*m_faces[f]->GetVertex(0))[i];
                    faceBaxis[i] = (*m_faces[f]->GetVertex(2))[i] - (*m_faces[f]->GetVertex(0))[i];

                    elementAaxis_length += pow(elementAaxis[i],2);
                    elementBaxis_length += pow(elementBaxis[i],2);
                    faceAaxis_length += pow(faceAaxis[i],2);
                    faceBaxis_length += pow(faceBaxis[i],2);
                }

                elementAaxis_length = sqrt(elementAaxis_length);
                elementBaxis_length = sqrt(elementBaxis_length);
                faceAaxis_length = sqrt(faceAaxis_length);
                faceBaxis_length = sqrt(faceBaxis_length);

                // Calculate the inner product of both the A-axis
                // (i.e. Elemental A axis and face A axis)
                for(i = 0 ; i < m_coordim; i++)
                {
                    dotproduct1 += elementAaxis[i]*faceAaxis[i];
                }

                orientation = 0;
                // if the innerproduct is equal to the (absolute value of the ) products of the lengths
                // of both vectors, then, the coordinate systems will NOT be transposed
                if( fabs(elementAaxis_length*faceAaxis_length - fabs(dotproduct1)) < NekConstants::kNekZeroTol )
                {
                    // if the inner product is negative, both A-axis point
                    // in reverse direction
                    if(dotproduct1 < 0.0)
                    {
                        orientation += 2;
                    }

                    // calculate the inner product of both B-axis
                    for(i = 0 ; i < m_coordim; i++)
                    {
                        dotproduct2 += elementBaxis[i]*faceBaxis[i];
                    }

                    // check that both these axis are indeed parallel
                    ASSERTL1(fabs(elementBaxis_length*faceBaxis_length - fabs(dotproduct2)) <
                             NekConstants::kNekZeroTol,
                             "These vectors should be parallel");

                    // if the inner product is negative, both B-axis point
                    // in reverse direction
                    if( dotproduct2 < 0.0 )
                    {
                        orientation++;
                    }
                }
                // The coordinate systems are transposed
                else
                {
                    orientation = 4;

                    // Calculate the inner product between the elemental A-axis
                    // and the B-axis of the face (which are now the corresponding axis)
                    dotproduct1 = 0.0;
                    for(i = 0 ; i < m_coordim; i++)
                    {
                        dotproduct1 += elementAaxis[i]*faceBaxis[i];
                    }

                    // check that both these axis are indeed parallel
                    ASSERTL1(fabs(elementAaxis_length*faceBaxis_length - fabs(dotproduct1)) <
                             NekConstants::kNekZeroTol,
                             "These vectors should be parallel");

                    // if the result is negative, both axis point in reverse
                    // directions
                    if(dotproduct1 < 0.0)
                    {
                        orientation += 2;
                    }

                    // Do the same for the other two corresponding axis
                    dotproduct2 = 0.0;
                    for(i = 0 ; i < m_coordim; i++)
                    {
                        dotproduct2 += elementBaxis[i]*faceAaxis[i];
                    }

                    // check that both these axis are indeed parallel
                    ASSERTL1(fabs(elementBaxis_length*faceAaxis_length - fabs(dotproduct2)) <
                             NekConstants::kNekZeroTol,
                             "These vectors should be parallel");

                    if( dotproduct2 < 0.0 )
                    {
                        orientation++;
                    }
                }
                
                orientation = orientation + 5;

                // Fill the m_forient array
                m_forient[f] = (StdRegions::Orientation) orientation;
            }
        }
    }; //end of namespace
}; //end of namespace