SundanceFeketeQuadrature.cpp

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/* @HEADER@ */
// ************************************************************************
// 
//                              Sundance
//                 Copyright (2005) Sandia Corporation
// 
// Copyright (year first published) Sandia Corporation.  Under the terms 
// of Contract DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government 
// retains certain rights in this software.
// 
// This library is free software; you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 2.1 of the
// License, or (at your option) any later version.
//  
// This library is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
//                                                                                 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
// USA                                                                                
// Questions? Contact Kevin Long (krlong@sandia.gov), 
// Sandia National Laboratories, Livermore, California, USA
// 
// ************************************************************************
/* @HEADER@ */

#include "SundanceFeketeBrickQuadrature.hpp"
#include "SundanceFeketeQuadQuadrature.hpp"
#include "SundanceFeketeQuadrature.hpp"
#include "SundanceFeketeTriangleQuadrature.hpp"
#include "SundanceGaussLobatto1D.hpp"
#include "SundanceOut.hpp"
#include "PlayaTabs.hpp"
#include <stack>

using namespace Sundance;
using namespace Teuchos;

/* declare LAPACK subroutines */
extern "C"
{
/* matrix-vector multiplication */
void dgemv_(char *trans, int *m, int *n, double *alpha, double *a, int *lda,
    double *x, int *incx, double *beta, double *y, int *incy);
}

/* constructor */
FeketeQuadrature::FeketeQuadrature(int order) :
  QuadratureFamilyBase(order)
{
  _hasBasisCoeffs = false;
}

XMLObject FeketeQuadrature::toXML() const
{
  XMLObject rtn("FeketeQuadrature");
  rtn.addAttribute("order", Teuchos::toString(order()));
  return rtn;
}

void FeketeQuadrature::getLineRule(Array<Point>& quadPoints,
    Array<double>& quadWeights) const
{
  int p = order() + 3;
  p = p + (p % 2);
  int n = p / 2;

  quadPoints.resize(n);
  quadWeights.resize(n);

  GaussLobatto1D q1(n, 0.0, 1.0);

  for (int i = 0; i < n; i++)
  {
    quadWeights[i] = q1.weights()[i];
    quadPoints[i] = Point(q1.nodes()[i]);
  }
}

void FeketeQuadrature::getTriangleRule(Array<Point>& quadPoints,
    Array<double>& quadWeights) const
{
  Array<double> x;
  Array<double> y;
  Array<double> w;

  FeketeTriangleQuadrature::getPoints(order(), w, x, y);
  quadPoints.resize(w.length());
  quadWeights.resize(w.length());
  for (int i = 0; i < w.length(); i++)
  {
    quadWeights[i] = 0.5 * w[i];
    quadPoints[i] = Point(x[i], y[i]);
  }
}

void FeketeQuadrature::getQuadRule(Array<Point>& quadPoints,
    Array<double>& quadWeights) const
{
  Array<double> x;
  Array<double> y;
  Array<double> w;

  FeketeQuadQuadrature::getPoints(order(), w, x, y);
  quadPoints.resize(w.length());
  quadWeights.resize(w.length());
  for (int i = 0; i < w.length(); i++)
  {
    quadWeights[i] = w[i];
    quadPoints[i] = Point(x[i], y[i]);
  }
}

void FeketeQuadrature::getBrickRule(Array<Point>& quadPoints,
    Array<double>& quadWeights) const
{
  Array<double> x;
  Array<double> y;
  Array<double> z;
  Array<double> w;

  FeketeBrickQuadrature::getPoints(order(), w, x, y, z);
  quadPoints.resize(w.length());
  quadWeights.resize(w.length());
  for (int i = 0; i < w.length(); i++)
  {
    quadWeights[i] = w[i];
    quadPoints[i] = Point(x[i], y[i], z[i]);
  }
}

void FeketeQuadrature::getAdaptedWeights(const CellType& cellType, int cellDim,
    int cellLID, int facetIndex, const Mesh& mesh, const ParametrizedCurve&
    globalCurve, Array<Point>& quadPoints, Array<double>& quadWeights,
    bool& weightsChanged) const
{
  //int verb = 4;
  Tabs tabs;

  // Cache lookup
  /*if (mesh.IsSpecialWeightValid() && mesh.hasSpecialWeight(cellDim, cellLID))
  {
    weightsChanged = true;
    mesh.getSpecialWeight(cellDim, cellLID, quadWeights);
    SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedWeights Found cached weights for cell LID " << cellLID)
    return;
  }*/

  // Maximal cell type
  CellType maxCellType = mesh.cellType(mesh.spatialDim());

  switch (maxCellType)
  {
  // We have a triangle based mesh
  case TriangleCell:
  {
    switch (cellType)
    {
    case TriangleCell:
    {
      getAdaptedTriangleWeights(cellLID, mesh, globalCurve, quadPoints,
          quadWeights, weightsChanged);
      break;
    }
    default:
#ifndef TRILINOS_7
      SUNDANCE_ERROR ("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for submaximal cell " << maxCellType << " in a triangle mesh")
      ;
#else
      SUNDANCE_ERROR7("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for submaximal cell " << maxCellType << " in a triangle mesh");
#endif
    }
    break;
  }
  case QuadCell:
  {
    switch(cellType)
    {
    case QuadCell:
    {
      getAdaptedQuadWeights(cellLID, mesh, globalCurve, quadPoints,
        quadWeights, weightsChanged);
      break;
    }
    default:
#ifndef TRILINOS_7
    SUNDANCE_ERROR ("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for submaximal cell " << maxCellType << " in a triangle mesh")
    ;
#else
    SUNDANCE_ERROR7("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for submaximal cell " << maxCellType << " in a triangle mesh");
#endif
    }
    break;
  }
  default:
#ifndef TRILINOS_7
    SUNDANCE_ERROR("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for cell type " << cellType)
    ;
#else
    SUNDANCE_ERROR7("getAdaptedWeights rule for ACI(Adaptive Cell Integration) not available for cell type " << cellType);
#endif
  }
}

void FeketeQuadrature::getAdaptedTriangleWeights(int cellLID, const Mesh& mesh,
    const ParametrizedCurve& globalCurve, Array<Point>& quadPoints, Array<
        double>& quadWeights, bool& weightsChanged) const
{
  int verb = 0;
  Tabs tabs;

  // ToDo: What tolerance?
  double tol = 1.e-8;
  weightsChanged = true;

  // For pushForward() we need the cellLID in an array
  Array<int> cellLIDs(1);
  cellLIDs[0] = cellLID;

  // Intersection points (parameters) with the edges of triangle
  Array<double> ip01, ip02, ip12;
  int nr_ip01, nr_ip02, nr_ip12;

  Array<double> originalWeights = quadWeights;
  ParametrizedCurve noCurve = new DummyParametrizedCurve();
  Array<double> integrals(originalWeights.size());
  for (int i = 0; i < integrals.size(); i++)
    integrals[i] = 0.0;

  // Create stack and put the initial triangle on stack
  std::stack<Point> s;
  std::stack<double> sa;
  s.push(Point(0.0, 0.0));
  s.push(Point(1.0, 0.0));
  s.push(Point(0.0, 1.0));
  sa.push(0.5);
  Array<Point> nodes;
  Array<Point> nodesref(3);

  // Number of investigated triangles
  unsigned int nTris = 0;

  while (s.size() >= 3)
  {
    // Increment number of triangles investigated
    ++nTris;

    SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedTriangleWeights Current number of triangles on stack: " << sa.size());

    // Integration results
    bool refine = false;
    Array<double> results1(originalWeights.size());
    Array<double> results2(originalWeights.size());

    // Fetch the coordinates of current triangle from stack
    double area = sa.top();
    sa.pop();
    for (int i = 2; i >= 0; i--)
    {
      nodesref[i] = s.top();
      s.pop();
    }
    mesh.pushForward(2, cellLIDs, nodesref, nodes);

    // Intersection points (parameters) with the edges of triangle
    globalCurve.returnIntersect(nodes[0], nodes[1], nr_ip01, ip01);
    globalCurve.returnIntersect(nodes[0], nodes[2], nr_ip02, ip02);
    globalCurve.returnIntersect(nodes[1], nodes[2], nr_ip12, ip12);

    //  Determine kind of intersection
    Point x, xref;
    Point vec1, vec1ref;
    Point vec2, vec2ref;
    bool xInOmega = false;

    // 'No intersection' is also handled here
    if ((nr_ip01 + nr_ip02 + nr_ip12 == 0) || (nr_ip01 == 0 && nr_ip02 == 0
        && nr_ip12 == 1) || (nr_ip01 == 1 && nr_ip02 == 1 && nr_ip12
        == 0))
    {
      x = nodes[0];
      vec1 = nodes[2] - nodes[0];
      vec2 = nodes[1] - nodes[0];
      xref = nodesref[0];
      vec1ref = nodesref[2] - nodesref[0];
      vec2ref = nodesref[1] - nodesref[0];
    }
    else if ((nr_ip01 == 0 && nr_ip02 == 1 && nr_ip12 == 0) || (nr_ip12
        == 1 && nr_ip01 == 1 && nr_ip02 == 0))
    {
      x = nodes[1];
      vec1 = nodes[0] - nodes[1];
      vec2 = nodes[2] - nodes[1];
      xref = nodesref[1];
      vec1ref = nodesref[0] - nodesref[1];
      vec2ref = nodesref[2] - nodesref[1];
    }
    else if ((nr_ip01 == 1 && nr_ip02 == 0 && nr_ip12 == 0) || (nr_ip02
        == 1 && nr_ip12 == 1 && nr_ip01 == 0))
    {
      x = nodes[2];
      vec1 = nodes[1] - nodes[2];
      vec2 = nodes[0] - nodes[2];
      xref = nodesref[2];
      vec1ref = nodesref[1] - nodesref[2];
      vec2ref = nodesref[0] - nodesref[2];
    }
    else
    {
      refine = true;
    }

    // If we found a case we can deal with, calculate the integrals
    if (!refine)
    {
      if (globalCurve.curveEquation(x) < 0)
        xInOmega = true;
      integrateRegion(TriangleCell, 2, order() + 1, x, xref, vec1, vec2,
          vec1ref, vec2ref, globalCurve, results1);
      if (results1.size() == 0)
      {
        SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedTriangleWeights Unknown error in integration. Refine.")
        refine = true;
      }
      else
      {
        integrateRegion(TriangleCell, 2, order() + 2, x, xref, vec1,
            vec2, vec1ref, vec2ref, globalCurve, results2);

        // Check results
        for (int i = 0; i < results1.size(); i++)
        {
          if (::fabs(results2[i] - results1[i]) > sqrt(area * tol))
          {
            SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedTriangleWeights Insufficient accuracy. Refine.")
            refine = true;
            break;
          }
        }
      }
    }

    // Refine (either because of lacking accuracy or strange intersections)
    if (refine)
    {
      // Divide triangle into 4 smaller ones (introduce new nodes at the
      // center of each edge)
      s.push(nodesref[0]);
      s.push((nodesref[0] + nodesref[1]) / 2);
      s.push((nodesref[0] + nodesref[2]) / 2);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[1]) / 2);
      s.push(nodesref[1]);
      s.push((nodesref[1] + nodesref[2]) / 2);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[2]) / 2);
      s.push((nodesref[1] + nodesref[2]) / 2);
      s.push(nodesref[2]);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[2]) / 2);
      s.push((nodesref[0] + nodesref[1]) / 2);
      s.push((nodesref[1] + nodesref[2]) / 2);
      sa.push(area / 4);
      continue;
    }

    // We found an accurate solution to that part of the triangle!
    if (xInOmega)
    {
      for (int i = 0; i < integrals.size(); i++)
        integrals[i] += results2[i];
    }
    else
    {
      integrateRegion(TriangleCell, 2, order() + 1, x, xref, vec1, vec2,
          vec1ref, vec2ref, noCurve, results1);
      for (int i = 0; i < integrals.size(); i++)
        integrals[i] += (results1[i] - results2[i]);
    }
  } // Stack depleted here! We're done.

  SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedTriangleWeights Calculations in cell " << cellLID << " completed. " << nTris << " triangle(s) used.");

  Array<double> alphas;
  globalCurve.getIntegrationParams(alphas);
  for (int i = 0; i < quadWeights.size(); i++)
    quadWeights[i] = alphas[1] * integrals[i] + alphas[0]
        * (originalWeights[i] - integrals[i]);
  mesh.setSpecialWeight(2, cellLID, quadWeights);
}

void FeketeQuadrature::getAdaptedQuadWeights(int cellLID, const Mesh& mesh,
    const ParametrizedCurve& globalCurve, Array<Point>& quadPoints, Array<
        double>& quadWeights, bool& weightsChanged) const
{
  // Verbosity
  int verb = 0;
  Tabs tabs;

  // ToDo: What tolerance?
  double tol = 1.e-8;

  // Initialization
  weightsChanged = true;
  Array<double> originalWeights = quadWeights;

  ParametrizedCurve noCurve = new DummyParametrizedCurve();
  Array<double> integrals(originalWeights.size());
  for (int i = 0; i < integrals.size(); i++)
    integrals[i] = 0.0;

  // For pushForward() we need the cellLID in an array
  Array<int> cellLIDs(1);
  cellLIDs[0] = cellLID;
  SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedQuadWeights: Current cell LID " << cellLID)

  //        IV
  //   2 __________ 3
  //    |          |
  //    |          |
  // II |          | III
  //    |__________|
  //   0            1
  //         I

  Array<int> nIntEdge(4);
  Array<Array<double> > IntEdgePars(4);
  int edgeIndex[4][2] = { {0,1} , {0,2} , {1,3} , {2,3} };

  // Create stack and put the initial quad
  std::stack<Point> s;
  std::stack<double> sa;
  s.push(Point(0.0, 0.0));
  s.push(Point(1.0, 0.0));
  s.push(Point(0.0, 1.0));
  s.push(Point(1.0, 1.0));
  sa.push(1.);
  Array<Point> nodes;
  Array<Point> nodesref(4);

  // Number of investigated quads
  unsigned int nQuads = 0;

  while (s.size() >= 4)
  {
    // Increment number of quads investigated
    ++nQuads;
    SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedQuadWeights: Current number of quads on stack: " << sa.size())

    // Integration results
    bool refine = false;
    bool xInOmega = false;
    Array<double> results1(originalWeights.size());
    Array<double> results2(originalWeights.size());

    // Fetch the coordinates of current quad from stack
    double area = sa.top();
    sa.pop();
    for (int i = 3; i >= 0; i--)
    {
      nodesref[i] = s.top();
      s.pop();
    }
    mesh.pushForward(2, cellLIDs, nodesref, nodes);

    // Intersection points (parameters) with the edges of triangle
    for (int jj = 0 ; jj < 4 ; jj++ ) {
      globalCurve.returnIntersect(nodes[edgeIndex[jj][0]], nodes[edgeIndex[jj][1]], nIntEdge[jj], IntEdgePars[jj] );
  /*    if ( nIntEdge[jj] == 0 ) {
        x = nodes[edgeIndex[jj][0]];
        vec1 =
        vec2 =
      }
  */  }

    //  Determine kind of intersection
    Point x, xref;
    Point vec1, vec1ref;
    Point vec2, vec2ref;

    // Baseline is not intersected
    if ( nIntEdge[0] == 0 )
    {
      x = nodes[0];
      vec1 = nodes[1] - nodes[0];
      vec2 = nodes[2] - nodes[0];
      xref = nodesref[0];
      vec1ref = nodesref[1] - nodesref[0];
      vec2ref = nodesref[2] - nodesref[0];
    }
    else if ( nIntEdge[1] == 0 )
    {
      x = nodes[2];
      vec1 = nodes[0] - nodes[2];
      vec2 = nodes[3] - nodes[2];
      xref = nodesref[2];
      vec1ref = nodesref[0] - nodesref[2];
      vec2ref = nodesref[3] - nodesref[2];
    }
    else if ( nIntEdge[2] == 0 )
    {
      x = nodes[1];
      vec1 = nodes[3] - nodes[1];
      vec2 = nodes[0] - nodes[1];
      xref = nodesref[1];
      vec1ref = nodesref[3] - nodesref[1];
      vec2ref = nodesref[0] - nodesref[1];
    }
    else
    {
      refine = true;
    }

    // If we found a case we can deal with, calculate the integrals
    if (!refine)
    {
      if (globalCurve.curveEquation(x) < 0)
        xInOmega = true;
      integrateRegion(QuadCell, 2, order() + 1, x, xref, vec1, vec2,
          vec1ref, vec2ref, globalCurve, results1);
      if (results1.size() == 0)
      {
        SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedQuadWeights Unknown error in integration. Refine.")
        refine = true;
      }
      else
      {
        integrateRegion(QuadCell, 2, order() + 2, x, xref, vec1,
            vec2, vec1ref, vec2ref, globalCurve, results2);

        // Check results
        double reltol = ::sqrt(area * tol);
        for (int i = 0; i < results1.size(); i++)
        {
          if (::fabs(results2[i] - results1[i]) > reltol)
          {
            SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedQuadWeights Insufficient accuracy. Refine.")
            refine = true;
            break;
          }
        }
      }
    }

    // Refine (either because of lacking accuracy or strange intersections)
    if (refine)
    {
      // Divide quad into 4 smaller ones (introduce new nodes at the
      // center of each edge)
      s.push(nodesref[0]);
      s.push((nodesref[0] + nodesref[1]) / 2);
      s.push((nodesref[0] + nodesref[2]) / 2);
      s.push((nodesref[0] + nodesref[3]) / 2);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[1]) / 2);
      s.push(nodesref[1]);
      s.push((nodesref[1] + nodesref[2]) / 2);
      s.push((nodesref[1] + nodesref[3]) / 2);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[2]) / 2);
      s.push((nodesref[1] + nodesref[2]) / 2);
      s.push(nodesref[2]);
      s.push((nodesref[2] + nodesref[3]) / 2);
      sa.push(area / 4);
      s.push((nodesref[0] + nodesref[3]) / 2);
      s.push((nodesref[1] + nodesref[3]) / 2);
      s.push((nodesref[2] + nodesref[3]) / 2);
      s.push(nodesref[3]);
      sa.push(area / 4);
      continue;
    }

    // We found an accurate solution to that part of the triangle!
    if (xInOmega)
    {
      for (int i = 0; i < integrals.size(); i++)
        integrals[i] += results2[i];
    }
    else
    {
      integrateRegion(QuadCell, 2, order() + 1, x, xref, vec1, vec2,
          vec1ref, vec2ref, noCurve, results1);
      for (int i = 0; i < integrals.size(); i++)
        integrals[i] += (results1[i] - results2[i]);
    }
  } // Stack depleted here! We're done.

  SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::getAdaptedQuadWeights Calculations in cell " << cellLID << " completed. " << nQuads << " quad(s) used.");

  Array<double> alphas;
  globalCurve.getIntegrationParams(alphas);
  for (int i = 0; i < quadWeights.size(); i++)
    quadWeights[i] = alphas[1] * integrals[i] + alphas[0]
        * (originalWeights[i] - integrals[i]);
  if (verb > 1 ) {
      writeTable(Out::os(), tabs, quadWeights, quadWeights.size());
  }
  mesh.setSpecialWeight(2, cellLID, quadWeights);
}

void FeketeQuadrature::integrateRegion(const CellType& cellType,
    const int cellDim, const int innerOrder, const Point& x,
    const Point& xref, const Point& vec1, const Point& vec2,
    const Point& vec1ref, const Point& vec2ref,
    const ParametrizedCurve& curve, Array<double>& integrals) const
{
  int verb = 0;
  Tabs tabs;

  switch (cellType)
  {
  case TriangleCell:
  {
    // Prepare output array
    integrals.resize((order() + 1) * (order() + 2) / 2);
    for (int i = 0; i < integrals.size(); i++)
      integrals[i] = 0.0;

    // Get 'inner' integration points (along a line)
    int nrInnerQuadPts = innerOrder + 3;
    nrInnerQuadPts = (nrInnerQuadPts + nrInnerQuadPts % 2) / 2;
    GaussLobatto1D innerQuad(nrInnerQuadPts, 0.0, 1.0);

    Array<double> innerQuadPts(nrInnerQuadPts);
    Array<double> innerQuadWgts(nrInnerQuadPts);
    innerQuadPts = innerQuad.nodes();
    innerQuadWgts = innerQuad.weights();

    // Calculate intersection points of 'integration ray' with curve
    // Here one needs physical coordinates because of the curve
    for (int edgePos = 0; edgePos < nrInnerQuadPts; edgePos++)
    {
      // Direction of current integration
      Point ray = innerQuadPts[edgePos] * vec1 + (1.0
          - innerQuadPts[edgePos]) * vec2;
      Point rayref = innerQuadPts[edgePos] * vec1ref + (1.0
          - innerQuadPts[edgePos]) * vec2ref;

      int nrIntersect = 0;
      Array<double> t;
      Point rayEnd = x + ray;
      curve.returnIntersect(x, rayEnd, nrIntersect, t);

      double mu = 1.0; // w/o an intersection we integrate the whole line
      if (nrIntersect == 1)
      {
        mu = t[0];
      }
      else if (nrIntersect >= 2)
      {
        // More than one intersection -> We have to refine!
        integrals.resize(0);
        return;
      }

      // Array for values of basis functions at inner integration points
      Array<double> innerIntegrals(integrals.size());
      for (int i = 0; i < innerIntegrals.size(); i++)
        innerIntegrals[i] = 0.0;

      // We follow one of the rays (in reference coordinates)
      for (int rayPos = 0; rayPos < nrInnerQuadPts; rayPos++)
      {
        Point q = xref + mu * innerQuadPts[rayPos] * rayref;

        // Evaluate all basis functions at quadrature point
        Array<double> funcVals;
        evaluateAllBasisFunctions(TriangleCell, q, funcVals);

        // Do quadrature for all basis functions
        for (int i = 0; i < funcVals.size(); i++)
          innerIntegrals[i] += funcVals[i] * innerQuadWgts[rayPos]
              * innerQuadPts[rayPos];
      }
      for (int j = 0; j < integrals.size(); j++)
        integrals[j] += innerQuadWgts[edgePos] * mu * mu
            * innerIntegrals[j];
    }

    // We calculated on parts of the triangle -> Scale to right size
    double det = ::fabs(vec1ref[0] * vec2ref[1] - vec2ref[0] * vec1ref[1]);
    for (int j = 0; j < integrals.size(); j++)
      integrals[j] *= det;
  } break;

  case QuadCell:
  {
    // Prepare output array
    integrals.resize((order() + 1) * (order() + 1));
    for (int i = 0; i < integrals.size(); i++)
      integrals[i] = 0.0;
    SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Initializing quadrature on a QuadCell")

    // Get 'inner' integration points (along a line)
    int nrInnerQuadPts = innerOrder + 3;
    nrInnerQuadPts = (nrInnerQuadPts + nrInnerQuadPts % 2) / 2;
    GaussLobatto1D innerQuad(nrInnerQuadPts, 0.0, 1.0);

    Array<double> innerQuadPts(nrInnerQuadPts);
    Array<double> innerQuadWgts(nrInnerQuadPts);
    innerQuadPts = innerQuad.nodes();
    innerQuadWgts = innerQuad.weights();

    // We assume: no intersection along first axis
    for (int xPos = 0; xPos < nrInnerQuadPts; xPos++)
    {
      // Direction of current integration
      Point x_cur = x + innerQuadPts[xPos] * vec1;
      Point x_curref = xref + innerQuadPts[xPos] * vec1ref;

      SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Integrating at xPos " << xPos << " Ref point " << x_curref)

      // Intersection along second axis?
      int nrIntersect = 0;
      Array<double> t;
      Point x_curEnd = x_cur + vec2;
      curve.returnIntersect(x_cur, x_curEnd, nrIntersect, t);
      SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Nr. of intersections along y: " << t.size())

      double mu = 1.0; // w/o an intersection we integrate the whole line
      if (nrIntersect == 1)
      {
        mu = t[0];
        SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Intersection parameter is " << mu)
      }
      else if (nrIntersect >= 2)
      {
        // More than one intersection -> We have to refine!
        integrals.resize(0);
        SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: More than one intersection -> Abort!")

        return;
      }

      // Array for values of basis functions at inner integration points
      Array<double> innerIntegrals(integrals.size());
      for (int jj = 0; jj < innerIntegrals.size(); jj++)
        innerIntegrals[jj] = 0.0;

      for (int yPos = 0; yPos < nrInnerQuadPts; yPos++)
      {
        Point y_curref = x_curref + mu * innerQuadPts[yPos] * vec2ref;
        SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Integrating at yPos " << yPos << " Ref point " << y_curref)

        // Evaluate all basis functions at quadrature point
        Array<double> funcVals;
        evaluateAllBasisFunctions(QuadCell, y_curref, funcVals);

        // Do quadrature for all basis functions
        for (int jj = 0; jj < funcVals.size(); jj++)
          innerIntegrals[jj] += funcVals[jj] * innerQuadWgts[yPos] * mu;
      }
      for (int ii = 0; ii < integrals.size(); ii++)
        integrals[ii] += innerQuadWgts[xPos] * innerIntegrals[ii];
    }

    // Scale to right size
    double det = ::fabs(vec1ref[0] * vec2ref[1] - vec2ref[0] * vec1ref[1]);
    for (int i = 0; i < integrals.size(); i++)
      integrals[i] *= det;
    SUNDANCE_MSG3(verb, tabs << "FeketeQuadrature::integrateRegion: Integration of region finished. Integral values " << integrals)
  } break;

  default:
#ifndef TRILINOS_7
    SUNDANCE_ERROR("FeketeQuadrature::integrateRegion() not available for cell type " << cellType)
    ;
#else
    SUNDANCE_ERROR7("FeketeQuadrature::integrateRegion() not available for cell type " << cellType);
#endif
  }

}

void FeketeQuadrature::evaluateAllBasisFunctions(const CellType cellType,
    const Point& q, Array<double>& result) const
{

  switch (cellType)
  {
  case TriangleCell:
  {
    // Coefficients in _basisCoeffs together with PKD polynomials form
    // a Lagrange basis at Fekete points
    if (!_hasBasisCoeffs)
    {
      FeketeTriangleQuadrature::computeBasisCoeffs(order(), _basisCoeffs);
      _hasBasisCoeffs = true;
    }

    // KL added explicit cast of sqrt argument to double to allow
    // compilation on windows.
    // One has nFeketePts basis functions
    int nFeketePts = sqrt((double) _basisCoeffs.size());
    result.resize(nFeketePts);

    // Determine values of all PKD polynomials at evaluation point q
    Array<double> pkdVals(nFeketePts);
    FeketeTriangleQuadrature::evalPKDpolynomials(order(), q[0], q[1],
        &(pkdVals[0]));

    // For each Lagrange basis function, sum up weighted PKD polynomials
    // (The weights are found in _basisCoeffs)
    char trans = 'N';
    double alpha = 1.;
    double beta = 0.;
    int inc = 1;
		::dgemv_(&trans, &nFeketePts, &nFeketePts, &alpha, &(_basisCoeffs[0]),
        &nFeketePts, &(pkdVals[0]), &inc, &beta, &(result[0]), &inc);
    break;
  }
  case QuadCell:
  {
    // Coefficients in _basisCoeffs form a Lagrange basis at Fekete points
    if (!_hasBasisCoeffs)
    {
      FeketeQuadQuadrature::computeBasisCoeffs(order(), _basisCoeffs);
      _hasBasisCoeffs = true;
    }

    // KL added explicit cast of sqrt argument to double to allow
    // compilation on windows.
    // One has nFeketePts basis functions
    int nFeketePts = sqrt((double) _basisCoeffs.size());
    result.resize(nFeketePts);

    // Determine values of all polynomials at evaluation point q
    Array<double> polVals(nFeketePts);
    FeketeQuadQuadrature::evalPolynomials(nFeketePts, q[0], q[1],
        &(polVals[0]));

    // For each Lagrange basis function, sum up weighted polynomials
    // (The weights are found in _basisCoeffs)
    char trans = 'N';
    double alpha = 1.;
    double beta = 0.;
    int inc = 1;
		::dgemv_(&trans, &nFeketePts, &nFeketePts, &alpha, &(_basisCoeffs[0]),
        &nFeketePts, &(polVals[0]), &inc, &beta, &(result[0]), &inc);
    break;
  }
  default:
#ifndef TRILINOS_7
    SUNDANCE_ERROR("FeketeQuadrature::evaluateAllBasisFunctions() not available for cell type " << cellType)
    ;
#else
    SUNDANCE_ERROR7("FeketeQuadrature::evaluateAllBasisFunctions() not available for cell type " << cellType);
#endif
  }
}

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