MoveMeshToCriticalLayer.cpp

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///////////////////////////////////////////////////////////////////////////////
//
// File: MoveMeshToCriticalLayer.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).
//
// 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: MoveMesh to critical layer
//
///////////////////////////////////////////////////////////////////////////////
 
#include <cstdio>
#include <cstdlib>
#include <iomanip>
 
#include <LibUtilities/BasicUtils/ParseUtils.h>
#include <MultiRegions/ExpList.h>
#include <SpatialDomains/MeshGraph.h>
#include <tinyxml.h>
 
using namespace std;
using namespace Nektar;
 
#include "ExtractCriticalLayerFunctions.h"
 
enum SolveType
{
    eSolveX,
    eSolveY,
    eSolveXY,
    eNoSolve
};
 
struct MoveVerts
{
    vector<NekDouble> kspring;
    vector<int> springVid;
    SolveType solve;
};
 
void GetInterfaceVerts(const int compositeID,
                       SpatialDomains::MeshGraphSharedPtr &mesh,
                       vector<int> &InterfaceVerts);
 
void GetStreakLocation(LibUtilities::SessionReaderSharedPtr &vSession,
                       SpatialDomains::MeshGraphSharedPtr &mesh,
                       string &fieldfile, Array<OneD, NekDouble> &xc,
                       Array<OneD, NekDouble> &yc);
 
void GetNewVertexLocation(TiXmlElement *doc,
                          SpatialDomains::MeshGraphSharedPtr &mesh,
                          vector<int> &InterfaceVerts,
                          Array<OneD, NekDouble> &xstreak,
                          Array<OneD, NekDouble> &ystreak,
                          Array<OneD, NekDouble> &vertx,
                          Array<OneD, NekDouble> &verty, int maxiter);
 
void TurnOffEdges(TiXmlElement *doc, SpatialDomains::SegGeomMap &meshedges,
                  Array<OneD, MoveVerts> &verts);
 
void RedefineVertices(TiXmlElement *doc, Array<OneD, NekDouble> &dvertx,
                      Array<OneD, NekDouble> &dverty);
 
void EnforceRotationalSymmetry(SpatialDomains::MeshGraphSharedPtr &mesh,
                               Array<OneD, NekDouble> &dvertx,
                               Array<OneD, NekDouble> &dverty);
 
int main(int argc, char *argv[])
{
    if (argc != 4)
    {
        fprintf(
            stderr,
            "Usage: ./MoveMeshToCriticalLayer  meshfile streakfile  outfile\n");
        exit(1);
    }
 
    //------------------------------------------------------------
    // Create Session file by reading only first meshfile
    //-----------------------------------------------------------
    LibUtilities::SessionReaderSharedPtr vSession =
        LibUtilities::SessionReader::CreateInstance(2, argv);
    SpatialDomains::MeshGraphSharedPtr mesh =
        SpatialDomains::MeshGraph::Read(vSession);
 
    //-------------------------------------------------------------
    // Read in mesh from input file
    //------------------------------------------------------------
    TiXmlDocument &meshdoc = vSession->GetDocument();
    TiXmlHandle docHandle(&meshdoc);
    TiXmlElement *doc = docHandle.FirstChildElement("NEKTAR")
                            .FirstChildElement("GEOMETRY")
                            .Element();
 
    //------------------------------------------------------------
    // Get list of vertices along interface
    //------------------------------------------------------------
    vector<int> InterfaceVerts;
    GetInterfaceVerts(300, mesh, InterfaceVerts);
 
    //------------------------------------------------------------
    // Determine location of new interface vertices
    //------------------------------------------------------------
    Array<OneD, NekDouble> xstreak(50), ystreak(50);
    string fieldfile(argv[argc - 2]);
    GetStreakLocation(vSession, mesh, fieldfile, xstreak, ystreak);
 
    //------------------------------------------------------------
    // Move internal mesh using critical layer info and under string analogy
    //------------------------------------------------------------
    Array<OneD, NekDouble> dvertx(mesh->GetNvertices(), 0.0),
        dverty(mesh->GetNvertices(), 0.0);
    int maxiter;
    vSession->LoadParameter("MoveMeshMaxIterations", maxiter, 100);
 
    GetNewVertexLocation(doc, mesh, InterfaceVerts, xstreak, ystreak, dvertx,
                         dverty, maxiter);
 
    //------------------------------------------------------------
    // Enforce rotational symmetry on mesh
    //------------------------------------------------------------
    if (vSession->DefinesSolverInfo("EnforceRotationalSymmetry"))
    {
        EnforceRotationalSymmetry(mesh, dvertx, dverty);
    }
 
    //------------------------------------------------------------
    // Redfine vertices in doc
    //------------------------------------------------------------
    RedefineVertices(doc, dvertx, dverty);
 
    //------------------------------------------------------------
    // Write out moved mesh file
    //------------------------------------------------------------
    std::string outfile(argv[argc - 1]);
    meshdoc.SaveFile(outfile);
}
 
void GetInterfaceVerts(const int compositeID,
                       SpatialDomains::MeshGraphSharedPtr &mesh,
                       vector<int> &InterfaceVerts)
{
    SpatialDomains::CompositeSharedPtr composite;
    composite    = mesh->GetComposite(compositeID);
    int compsize = composite->m_geomVec.size();
    map<int, int> vertmap;
 
    for (int i = 0; i < compsize; ++i)
    {
        if (vertmap.count(composite->m_geomVec[i]->GetVid(0)) == 0)
        {
            InterfaceVerts.push_back(composite->m_geomVec[i]->GetVid(0));
            vertmap[composite->m_geomVec[i]->GetVid(0)] = 1;
        }
 
        if (vertmap.count(composite->m_geomVec[i]->GetVid(1)) == 0)
        {
            InterfaceVerts.push_back(composite->m_geomVec[i]->GetVid(1));
            vertmap[composite->m_geomVec[i]->GetVid(1)] = 1;
        }
    }
}
 
void GetStreakLocation(LibUtilities::SessionReaderSharedPtr &vSession,
                       SpatialDomains::MeshGraphSharedPtr &mesh,
                       string &fieldfile, Array<OneD, NekDouble> &xc,
                       Array<OneD, NekDouble> &yc)
{
    //-------------------------------------------------------------
    // Define Streak Expansion
    MultiRegions::ExpListSharedPtr streak;
 
    streak =
        MemoryManager<MultiRegions::ExpList>::AllocateSharedPtr(vSession, mesh);
    //---------------------------------------------------------------
 
    //----------------------------------------------
    // Import field file.
    vector<LibUtilities::FieldDefinitionsSharedPtr> fielddef;
    vector<vector<NekDouble>> fielddata;
    LibUtilities::Import(fieldfile, fielddef, fielddata);
    //----------------------------------------------
 
    //----------------------------------------------
    // Copy data from field file
    string streak_field("w");
    for (unsigned int i = 0; i < fielddata.size(); ++i)
    {
        streak->ExtractDataToCoeffs(fielddef[i], fielddata[i], streak_field,
                                    streak->UpdateCoeffs());
    }
    //----------------------------------------------
 
    NekDouble cr = 0.0;
 
    cerr << "Extracting Critical Layer ";
    Computestreakpositions(streak, xc, yc, cr);
}
 
void GetNewVertexLocation(TiXmlElement *doc,
                          SpatialDomains::MeshGraphSharedPtr &mesh,
                          vector<int> &InterfaceVerts,
                          Array<OneD, NekDouble> &xstreak,
                          Array<OneD, NekDouble> &ystreak,
                          Array<OneD, NekDouble> &dvertx,
                          Array<OneD, NekDouble> &dverty, int maxiter)
{
    int i, j, k;
    int nverts = mesh->GetNvertices();
 
    SpatialDomains::SegGeomMap meshedges = mesh->GetAllSegGeoms();
 
    Array<OneD, MoveVerts> Verts(nverts);
 
    // loop mesh edges and fill in verts info
    SpatialDomains::PointGeomSharedPtr v0, v1;
    SpatialDomains::PointGeom dist;
 
    int vid0, vid1;
    NekDouble kspring;
    NekDouble x, y, x1, y1, z1, x2, y2, z2;
 
    // Setup intiial spring and verts
    for (auto &segIter : meshedges)
    {
        vid0 = (segIter.second)->GetVid(0);
        vid1 = (segIter.second)->GetVid(1);
 
        v0 = (segIter.second)->GetVertex(0);
        v1 = (segIter.second)->GetVertex(1);
 
        kspring = 1.0 / v0->dist(*v1);
 
        Verts[vid0].kspring.push_back(kspring);
        Verts[vid0].springVid.push_back(vid1);
        Verts[vid1].kspring.push_back(kspring);
        Verts[vid1].springVid.push_back(vid0);
    }
 
    // Scale spring by total around vertex and turn on all vertices.
    for (i = 0; i < nverts; ++i)
    {
        NekDouble invktot = 0.0;
        for (j = 0; j < Verts[i].kspring.size(); ++j)
        {
            invktot += Verts[i].kspring[j];
        }
        invktot = 1.0 / invktot;
        for (j = 0; j < Verts[i].kspring.size(); ++j)
        {
            Verts[i].kspring[j] *= invktot;
        }
 
        Verts[i].solve = eSolveXY;
    }
 
    // Turn off all edges defined by composite lists of correct dimension
    TurnOffEdges(doc, meshedges, Verts);
 
    NekDouble z, h0, h1, h2;
    // Set interface vertices to lie on critical layer
    for (i = 0; i < InterfaceVerts.size(); ++i)
    {
        Verts[InterfaceVerts[i]].solve = eNoSolve;
        mesh->GetVertex(InterfaceVerts[i])->GetCoords(x, y, z);
 
        for (j = 0; j < xstreak.size() - 1; ++j)
        {
            if ((x >= xstreak[j]) && (x <= xstreak[j + 1]))
            {
                break;
            }
        }
 
        ASSERTL0(j != xstreak.size(),
                 "Did not find x location along critical layer");
 
        k = (j == 0) ? 1 : j; // offset index at beginning
 
        // quadraticalling interpolate points
        h0 =
            (x - xstreak[k]) * (x - xstreak[k + 1]) /
            ((xstreak[k - 1] - xstreak[k]) * (xstreak[k - 1] - xstreak[k + 1]));
        h1 = (x - xstreak[k - 1]) * (x - xstreak[k + 1]) /
             ((xstreak[k] - xstreak[k - 1]) * (xstreak[k] - xstreak[k + 1]));
        h2 =
            (x - xstreak[k - 1]) * (x - xstreak[k]) /
            ((xstreak[k + 1] - xstreak[k - 1]) * (xstreak[k + 1] - xstreak[k]));
 
        dvertx[InterfaceVerts[i]] =
            (xstreak[k - 1] * h0 + xstreak[k] * h1 + xstreak[k + 1] * h2) - x;
        dverty[InterfaceVerts[i]] =
            (ystreak[k - 1] * h0 + ystreak[k] * h1 + ystreak[k + 1] * h2) - y;
    }
 
    // shift quads in critical layer to move more or less rigidly
    SpatialDomains::QuadGeomMap quadgeom = mesh->GetAllQuadGeoms();
    for (auto &quadIter : quadgeom)
    {
        for (i = 0; i < 4; ++i)
        {
            vid0 = (quadIter.second)->GetVid(i);
 
            switch (Verts[vid0].solve)
            {
                case eSolveXY:
                {
                    mesh->GetVertex(vid0)->GetCoords(x, y, z);
 
                    // find nearest interface vert
                    mesh->GetVertex(InterfaceVerts[0])->GetCoords(x1, y1, z1);
                    for (j = 0; j < InterfaceVerts.size() - 1; ++j)
                    {
                        mesh->GetVertex(InterfaceVerts[j + 1])
                            ->GetCoords(x2, y2, z2);
                        if ((x >= x1) && (x < x2))
                        {
                            break;
                        }
                        x1 = x2;
                        y1 = y2;
                    }
 
                    // currently just shift vert as average of two sides
                    dvertx[vid0] =
                        (x2 - x) / (x2 - x1) * dvertx[InterfaceVerts[j]] +
                        (x - x1) / (x2 - x1) * dvertx[InterfaceVerts[j + 1]];
                    dverty[vid0] =
                        (x2 - x) / (x2 - x1) * dverty[InterfaceVerts[j]] +
                        (x - x1) / (x2 - x1) * dverty[InterfaceVerts[j + 1]];
                }
                break;
                case eSolveY:
                {
                    mesh->GetVertex(vid0)->GetCoords(x, y, z);
                    mesh->GetVertex(InterfaceVerts[0])->GetCoords(x1, y1, z1);
 
                    if (fabs(x - x1) < 1e-6)
                    {
                        dverty[vid0] = dverty[InterfaceVerts[0]];
                    }
                    else
                    {
                        dverty[vid0] =
                            dverty[InterfaceVerts[InterfaceVerts.size() - 1]];
                    }
                }
                break;
                default:
                    break;
            }
            Verts[vid0].solve = eNoSolve;
        }
    }
 
    // Iterate internal vertices
    bool ContinueToIterate = true;
    int cnt                = 0;
    int nsum               = 0;
    NekDouble dsum, dx, dy, sum, prev_sum = 0.0;
    NekDouble tol = 1e-3, fac;
    int blend     = 50;
 
    while (ContinueToIterate)
    {
 
        sum  = 0.0;
        nsum = 0;
 
        // use a ramping function to help move interior slowly
        fac = (cnt < blend) ? 1.0 / (blend + 1.0) * (cnt + 1) : 1.0;
 
        for (i = 0; i < nverts; ++i)
        {
            if (Verts[i].solve != eNoSolve)
            {
                dx = dy = 0.0;
 
                if ((Verts[i].solve == eSolveX) || (Verts[i].solve == eSolveXY))
                {
                    for (j = 0; j < Verts[i].kspring.size(); ++j)
                    {
                        dx += fac * Verts[i].kspring[j] *
                              dvertx[Verts[i].springVid[j]];
                    }
                }
 
                if ((Verts[i].solve == eSolveY) || (Verts[i].solve == eSolveXY))
                {
                    for (j = 0; j < Verts[i].kspring.size(); ++j)
                    {
                        dy += fac * Verts[i].kspring[j] *
                              dverty[Verts[i].springVid[j]];
                    }
                }
 
                dsum = (dx * dx + dy * dy);
 
                dvertx[i] = dx;
                dverty[i] = dy;
 
                if (dsum > 1e-16)
                {
                    // sum  += dsum/(dvertx[i]*dvertx[i] + dverty[i]*dverty[i]);
                    sum += dsum;
                    nsum += 1;
                }
            }
        }
 
        if (nsum)
        {
            sum = sqrt(sum / (NekDouble)nsum);
 
            NekDouble chg = sum - prev_sum;
            prev_sum      = sum;
 
            cerr << "Iteration " << cnt << " : " << chg << endl;
 
            if ((chg < tol) && (cnt > blend))
            {
                ContinueToIterate = false;
            }
        }
        else if (cnt > blend)
        {
            ContinueToIterate = false;
        }
 
        if (cnt++ > maxiter)
        {
            ContinueToIterate = false;
        }
    }
}
 
// Read Composites from xml document and turn off verts that are along edge
// composites.
void TurnOffEdges(TiXmlElement *doc, SpatialDomains::SegGeomMap &meshedges,
                  Array<OneD, MoveVerts> &Verts)
{
    TiXmlElement *field = doc->FirstChildElement("COMPOSITE");
    ASSERTL0(field, "Unable to find COMPOSITE tag in file.");
 
    int nextCompositeNumber = -1;
 
    /// All elements are of the form: "<C ID = "N"> ... </C>".
    /// Read the ID field first.
    TiXmlElement *composite = field->FirstChildElement("C");
 
    while (composite)
    {
        nextCompositeNumber++;
 
        int indx;
        int err = composite->QueryIntAttribute("ID", &indx);
        ASSERTL0(err == TIXML_SUCCESS, "Unable to read attribute ID.");
 
        TiXmlNode *compositeChild = composite->FirstChild();
        // This is primarily to skip comments that may be present.
        // Comments appear as nodes just like elements.
        // We are specifically looking for text in the body
        // of the definition.
        while (compositeChild &&
               compositeChild->Type() != TiXmlNode::TINYXML_TEXT)
        {
            compositeChild = compositeChild->NextSibling();
        }
 
        ASSERTL0(compositeChild, "Unable to read composite definition body.");
        std::string compositeStr = compositeChild->ToText()->ValueStr();
 
        /// Parse out the element components corresponding to type of element.
        std::istringstream compositeDataStrm(compositeStr.c_str());
 
        try
        {
            std::string compositeElementStr;
            compositeDataStrm >> compositeElementStr;
 
            std::istringstream tokenStream(compositeElementStr);
            char type;
 
            tokenStream >> type;
 
            // in what follows we are assuming there is only one block of data
            std::string::size_type indxBeg =
                compositeElementStr.find_first_of('[') + 1;
            std::string::size_type indxEnd =
                compositeElementStr.find_last_of(']') - 1;
 
            ASSERTL0(indxBeg <= indxEnd,
                     (std::string("Error reading index definition:") +
                      compositeElementStr)
                         .c_str());
 
            std::string indxStr =
                compositeElementStr.substr(indxBeg, indxEnd - indxBeg + 1);
            std::vector<unsigned int> seqVector;
 
            bool err = ParseUtils::GenerateSeqVector(indxStr, seqVector);
 
            ASSERTL0(err, (std::string("Error reading composite elements: ") +
                           indxStr)
                              .c_str());
 
            switch (type)
            {
                case 'E': // Turn off vertices along composite edges
                {
                    int seqlen = seqVector.size();
                    NekDouble x0, y0, z0, x1, y1, z1;
                    int vid0, vid1;
 
                    for (int i = 0; i < seqlen; ++i)
                    {
                        meshedges[seqVector[i]]->GetVertex(0)->GetCoords(x0, y0,
                                                                         z0);
                        meshedges[seqVector[i]]->GetVertex(1)->GetCoords(x1, y1,
                                                                         z1);
                        vid0 = meshedges[seqVector[i]]->GetVid(0);
                        vid1 = meshedges[seqVector[i]]->GetVid(1);
 
                        if (fabs(x0 - x1) < 1e-8)
                        {
                            // identify corners by double visit
                            if (Verts[vid0].solve == eSolveX)
                            {
                                Verts[vid0].solve = eNoSolve;
                            }
                            else
                            {
                                Verts[vid0].solve = eSolveY;
                            }
 
                            if (Verts[vid1].solve == eSolveX)
                            {
                                Verts[vid1].solve = eNoSolve;
                            }
                            else
                            {
                                Verts[vid1].solve = eSolveY;
                            }
                        }
 
                        if (fabs(y0 - y1) < 1e-8)
                        {
                            // identify corners by double visit
                            if (Verts[vid0].solve == eSolveY)
                            {
                                Verts[vid0].solve = eNoSolve;
                            }
                            else
                            {
                                Verts[vid0].solve = eSolveX;
                            }
 
                            if (Verts[vid1].solve == eSolveY)
                            {
                                Verts[vid1].solve = eNoSolve;
                            }
                            else
                            {
                                Verts[vid1].solve = eSolveX;
                            }
                        }
                    }
                }
                break;
 
                case 'T':
                case 'Q': // do nothing
                {
                    break;
                }
                default:
                    NEKERROR(ErrorUtil::efatal,
                             (std::string("Unrecognized composite token: ") +
                              compositeElementStr)
                                 .c_str());
            }
        }
        catch (...)
        {
            NEKERROR(
                ErrorUtil::efatal,
                (std::string("Unable to read COMPOSITE data for composite: ") +
                 compositeStr)
                    .c_str());
        }
 
        /// Keep looking
        composite = composite->NextSiblingElement("C");
    }
}
 
void RedefineVertices(TiXmlElement *doc, Array<OneD, NekDouble> &dvertx,
                      Array<OneD, NekDouble> &dverty)
{
 
    TiXmlElement *element = doc->FirstChildElement("VERTEX");
    ASSERTL0(element, "Unable to find mesh VERTEX tag in file.");
 
    TiXmlElement *vertex = element->FirstChildElement("V");
 
    int indx;
    int nextVertexNumber = -1;
    int err; /// Error value returned by TinyXML.
 
    vector<NekDouble> xpts, ypts, zpts;
    NekDouble xval, yval, zval;
 
    while (vertex)
    {
        nextVertexNumber++;
 
        TiXmlAttribute *vertexAttr = vertex->FirstAttribute();
        std::string attrName(vertexAttr->Name());
 
        ASSERTL0(attrName == "ID",
                 (std::string("Unknown attribute name: ") + attrName).c_str());
 
        err = vertexAttr->QueryIntValue(&indx);
        ASSERTL0(err == TIXML_SUCCESS, "Unable to read attribute ID.");
 
        // Now read body of vertex
        std::string vertexBodyStr;
 
        TiXmlNode *vertexBody = vertex->FirstChild();
 
        while (vertexBody)
        {
            // Accumulate all non-comment body data.
            if (vertexBody->Type() == TiXmlNode::TINYXML_TEXT)
            {
                vertexBodyStr += vertexBody->ToText()->Value();
                vertexBodyStr += " ";
            }
 
            vertexBody = vertexBody->NextSibling();
        }
 
        ASSERTL0(!vertexBodyStr.empty(),
                 "Vertex definitions must contain vertex data.");
 
        // Get vertex data from the data string.
        std::istringstream vertexDataStrm(vertexBodyStr.c_str());
 
        try
        {
            while (!vertexDataStrm.fail())
            {
                vertexDataStrm >> xval >> yval >> zval;
            }
 
            xval += dvertx[indx];
            yval += dverty[indx];
 
            stringstream s;
            s << scientific << setprecision(8) << xval << " " << yval << " "
              << zval;
 
            vertex->ReplaceChild(vertex->FirstChild(), TiXmlText(s.str()));
        }
        catch (...)
        {
            ASSERTL0(false, "Unable to read VERTEX data.");
        }
        vertex = vertex->NextSiblingElement("V");
    }
}
 
void EnforceRotationalSymmetry(SpatialDomains::MeshGraphSharedPtr &mesh,
                               Array<OneD, NekDouble> &dvertx,
                               Array<OneD, NekDouble> &dverty)
{
    int i, j;
    int nverts = mesh->GetNvertices();
    Array<OneD, NekDouble> x(nverts), y(nverts);
    NekDouble xval, yval, zval;
 
    for (i = 0; i < nverts; ++i)
    {
        mesh->GetVertex(i)->GetCoords(xval, yval, zval);
        x[i] = xval + dvertx[i];
        y[i] = yval + dverty[i];
    }
 
    NekDouble xmax = Vmath::Vmax(nverts, x, 1);
    NekDouble tol  = 1e-5, dist2, xrot, yrot;
    Array<OneD, int> index(nverts);
    // find nearest
    for (i = 0; i < nverts; ++i)
    {
        xrot = -x[i] + xmax;
        yrot = -y[i];
        tol  = 1.0;
 
        for (j = 0; j < nverts; ++j)
        {
            dist2 =
                (x[j] - xrot) * (x[j] - xrot) + (y[j] - yrot) * (y[j] - yrot);
            if (dist2 < tol)
            {
                index[i] = j;
                tol      = dist2;
            }
        }
    }
 
    // average points and recalcualte dvertx, dverty
    for (i = 0; i < nverts; ++i)
    {
        mesh->GetVertex(i)->GetCoords(xval, yval, zval);
 
        xrot = 0.5 * (-x[index[i]] + xmax + x[i]);
        yrot = 0.5 * (-y[index[i]] + y[i]);
 
        dvertx[i] = xrot - xval;
        dverty[i] = yrot - yval;
    }
}