CondNumBasis.cpp

// Gmsh - Copyright (C) 1997-2018 C. Geuzaine, J.-F. Remacle
//
// See the LICENSE.txt file for license information. Please report all
// issues on https://gitlab.onelab.info/gmsh/gmsh/issues

#include "CondNumBasis.h"

#include "GmshDefines.h"
#include "GmshMessage.h"

#include <vector>
#include "polynomialBasis.h"
#include "pyramidalBasis.h"
#include "pointsGenerators.h"
#include "BasisFactory.h"
#include "Numeric.h"

namespace {

  // Compute the determinant of a 3x3 matrix
  inline double calcDet3x3(double M11, double M12, double M13, double M21,
                           double M22, double M23, double M31, double M32,
                           double M33)
  {
    return M11 * (M22 * M33 - M23 * M32) - M12 * (M21 * M33 - M23 * M31) +
           M13 * (M21 * M32 - M22 * M31);
  }

  // Compute the squared Frobenius norm of the inverse of a matrix
  template <bool sign>
  inline double calcInvCondNum2D(double dxdX, double dxdY, double dydX,
                                 double dydY, double dzdX, double dzdY,
                                 double nx, double ny, double nz)
  {
    const double dxdXSq = dxdX * dxdX, dydXSq = dydX * dydX,
                 dzdXSq = dzdX * dzdX;
    const double dxdYSq = dxdY * dxdY, dydYSq = dydY * dydY,
                 dzdYSq = dzdY * dzdY;
    const double Dx = dxdXSq - dxdYSq, Dy = dydXSq - dydYSq;
    const double Cx = dxdX * dxdY, Cy = dydX * dydY;
    const double S1 = dzdYSq * dzdYSq;
    const double S2 = (dzdXSq - Dy - Dx) * dzdYSq;
    const double S3 = (Cy + Cx) * dzdX * dzdY;
    const double S4 = dzdXSq * dzdXSq;
    const double S5 = (Dy + Dx) * dzdXSq;
    const double S6 = Cx * Cy;
    const double S7 = dydXSq * dydXSq;
    const double S8 = Dy * dydXSq;
    const double S9 = dxdXSq * dxdXSq;
    const double S10 = Dx * dxdXSq;
    const double S11 = Dy * Dy;
    const double S12 = Dx * Dy;
    const double S13 = Dx * Dx;
    const double S = 2. * (S2 + S5 + S12) + 4. * (S7 - S8 + S9 - S10) +
                     8. * (S3 + S6) + S1 + S4 + S11 + S13;
    const double N = dxdXSq + dxdYSq + dydXSq + dydYSq + dzdXSq + dzdYSq;
    const double sqrtS = (S > 0.0) ? sqrt(S) : 0.0;
    const double sigma1Sq = 0.5 * (N + sqrtS), sigma2Sq = 0.5 * (N - sqrtS);
    const double iCN = 2. * sqrt(sigma1Sq * sigma2Sq) / (sigma1Sq + sigma2Sq);
    if(sign) {
      const double lnx = dydX * dzdY - dzdX * dydY,
                   lny = dzdX * dxdY -
                         dxdX * dzdY, // Local normal from mapping gradients
        lnz = dxdX * dydY - dydX * dxdY;
      const double dp = lnx * nx + lny * ny +
                        lnz * nz; // Dot product to determine element validity
      return (dp >= 0.) ? iCN : -iCN;
    }
    else
      return iCN;
    //  return std::min(sqrt(sigma1Sq), sqrt(sigma2Sq)) /
    //  std::max(sqrt(sigma1Sq), sqrt(sigma2Sq));
  }

  // Compute the squared Frobenius norm of the inverse of a matrix
  template <bool sign>
  inline double calcInvCondNum3D(double J11, double J12, double J13, double J21,
                                 double J22, double J23, double J31, double J32,
                                 double J33)
  {
    const double D = calcDet3x3(J11, J12, J13, J21, J22, J23, J31, J32, J33);
    if(D == 0.) return 0.;
    const double I11 = J22 * J33 - J23 * J32, I12 = J13 * J32 - J12 * J33,
                 I13 = J12 * J23 - J13 * J22, I21 = J23 * J31 - J21 * J33,
                 I22 = J11 * J33 - J13 * J31, I23 = J13 * J21 - J11 * J23,
                 I31 = J21 * J32 - J22 * J31, I32 = J12 * J31 - J11 * J32,
                 I33 = J11 * J22 - J12 * J21;
    const double nSqJ = J11 * J11 + J12 * J12 + J13 * J13 + J21 * J21 +
                        J22 * J22 + J23 * J23 + J31 * J31 + J32 * J32 +
                        J33 * J33;
    const double nSqDInvJ = I11 * I11 + I12 * I12 + I13 * I13 + I21 * I21 +
                            I22 * I22 + I23 * I23 + I31 * I31 + I32 * I32 +
                            I33 * I33;
    if(sign)
      return 3. * D / sqrt(nSqJ * nSqDInvJ);
    else
      return 3. * std::fabs(D) / sqrt(nSqJ * nSqDInvJ);
  }

  // Compute condition number and its gradients
  // w.r.t. node positions, at one location in a 2D element
  template <bool sign>
  inline void calcGradInvCondNum2D(double dxdX, double dxdY, double dydX,
                                   double dydY, double dzdX, double dzdY,
                                   double nx, double ny, double nz, int i,
                                   int numMapNodes,
                                   const fullMatrix<double> &gSMatX,
                                   const fullMatrix<double> &gSMatY,
                                   fullMatrix<double> &IDI)
  {
    const double EpsDegen = 1.e-6;

    bool posJac = true;
    if(sign) {
      const double lnx = dydX * dzdY - dzdX * dydY,
                   lny = dzdX * dxdY -
                         dxdX * dzdY, // Local normal from mapping gradients
        lnz = dxdX * dydY - dydX * dxdY;
      const double dp = lnx * nx + lny * ny +
                        lnz * nz; // Dot product to determine element validity
      posJac = (dp >= 0.);
    }

    const double dxdXSq = dxdX * dxdX, dydXSq = dydX * dydX,
                 dzdXSq = dzdX * dzdX;
    const double dxdYSq = dxdY * dxdY, dydYSq = dydY * dydY,
                 dzdYSq = dzdY * dzdY;
    const double Dx = dxdXSq - dxdYSq, Dy = dydXSq - dydYSq;
    const double Cx = dxdX * dxdY, Cy = dydX * dydY;
    const double S1 = dzdYSq * dzdYSq;
    const double S2 = (dzdXSq - Dy - Dx) * dzdYSq;
    const double S3 = (Cy + Cx) * dzdX * dzdY;
    const double S4 = dzdXSq * dzdXSq;
    const double S5 = (Dy + Dx) * dzdXSq;
    const double S6 = Cx * Cy;
    const double S7 = dydXSq * dydXSq;
    const double S8 = Dy * dydXSq;
    const double S9 = dxdXSq * dxdXSq;
    const double S10 = Dx * dxdXSq;
    const double S11 = Dy * Dy;
    const double S12 = Dx * Dy;
    const double S13 = Dx * Dx;
    const double S = 2. * (S2 + S5 + S12) + 4. * (S7 - S8 + S9 - S10) +
                     8. * (S3 + S6) + S1 + S4 + S11 + S13;
    if(S == 0.) { // S == 0. -> Ideal element
      for(int j = 0; j < 3 * numMapNodes; j++) IDI(i, j) = 0.;
      IDI(i, 3 * numMapNodes) = posJac ? 1. : -1.;
      return;
    }
    const double N = dxdXSq + dxdYSq + dydXSq + dydYSq + dzdXSq + dzdYSq;
    const double sqrtS = sqrt(S), invSqrtS = 1. / sqrtS;
    const double sigma1Sq = 0.5 * (N + sqrtS), sigma2Sq = 0.5 * (N - sqrtS);
    const bool degen =
      (sigma2Sq < EpsDegen * sigma1Sq); // Check for degenerate element
    const double sum = sigma1Sq + sigma2Sq, invSum = 1. / sum;
    const double prod = sigma1Sq * sigma2Sq;
    const double sqrtProd = sqrt(prod);
    const double halfICN = sqrtProd * invSum;
    IDI(i, 3 * numMapNodes) = posJac ? 2. * halfICN : -2. * halfICN;

    if(degen) { // Degenerate element: special formula for gradients
      const double nnXx = dzdX * ny - dydX * nz, nnXy = dxdX * nz - dzdX * nx,
                   nnXz = dydX * nx - dxdX * ny;
      const double nnYx = dzdY * ny - dydY * nz, nnYy = dxdY * nz - dzdY * nx,
                   nnYz = dydY * nx - dxdY * ny;
      const double fact = 2. / N;
      for(int j = 0; j < numMapNodes; j++) {
        const double &dPhidX = gSMatX(i, j);
        const double &dPhidY = gSMatY(i, j);
        IDI(i, j) = fact * (dPhidY * nnXx - dPhidX * nnYx);
        IDI(i, j + numMapNodes) = fact * (dPhidY * nnXy - dPhidX * nnYy);
        IDI(i, j + 2 * numMapNodes) = fact * (dPhidY * nnXz - dPhidX * nnYz);
      }
      return;
    }

    const double invSqrtProd = 1. / sqrtProd;

    for(int j = 0; j < numMapNodes; j++) {
      const double &dPhidX = gSMatX(i, j);
      const double &dPhidY = gSMatY(i, j);

      const double ddxdXSqdxj = 2. * dPhidX * dxdX,
                   ddxdYSqdxj = 2. * dPhidY * dxdY;
      const double dDxdxj = ddxdXSqdxj - ddxdYSqdxj;
      const double dCxdxj = dPhidX * dxdY + dxdX * dPhidY;
      const double dS2dxj = -dDxdxj * dzdYSq;
      const double dS3dxj = dCxdxj * dzdX * dzdY;
      const double dS5dxj = dDxdxj * dzdXSq;
      const double dS6dxj = dCxdxj * Cy;
      const double dS9dxj = 2. * ddxdXSqdxj * dxdXSq;
      const double dS10dxj = dDxdxj * dxdXSq + Dx * ddxdXSqdxj;
      const double dS12dxj = dDxdxj * Dy;
      const double dS13dxj = 2. * dDxdxj * Dx;
      const double dSdxj = 2. * (dS2dxj + dS5dxj + dS12dxj) +
                           4. * (dS9dxj - dS10dxj) + 8. * (dS3dxj + dS6dxj) +
                           dS13dxj;
      const double dNdxj = ddxdXSqdxj + ddxdYSqdxj;
      const double dsqrtSdxj = 0.5 * dSdxj * invSqrtS;
      const double dsigma1Sqdxj = 0.5 * (dNdxj + dsqrtSdxj),
                   dsigma2Sqdxj = 0.5 * (dNdxj - dsqrtSdxj);
      const double dSumdxj = dsigma1Sqdxj + dsigma2Sqdxj;
      const double dProddxj = dsigma1Sqdxj * sigma2Sq + sigma1Sq * dsigma2Sqdxj;
      const double diCNdxj =
        (dProddxj * sum - 2. * prod * dSumdxj) * invSum * invSum * invSqrtProd;
      IDI(i, j) = posJac ? diCNdxj : -diCNdxj;

      const double ddydXSqdyj = 2. * dPhidX * dydX,
                   ddydYSqdyj = 2. * dPhidY * dydY;
      const double dDydyj = ddydXSqdyj - ddydYSqdyj;
      const double dCydyj = dPhidX * dydY + dydX * dPhidY;
      const double dS2dyj = -dDydyj * dzdYSq;
      const double dS3dyj = dCydyj * dzdX * dzdY;
      const double dS5dyj = dDydyj * dzdXSq;
      const double dS6dyj = Cx * dCydyj;
      const double dS7dyj = 2. * ddydXSqdyj * dydXSq;
      const double dS8dyj = dDydyj * dydXSq + Dy * ddydXSqdyj;
      const double dS11dyj = 2. * dDydyj * Dy;
      const double dS12dyj = Dx * dDydyj;
      const double dSdyj = 2. * (dS2dyj + dS5dyj + dS12dyj) +
                           4. * (dS7dyj - dS8dyj) + 8. * (dS3dyj + dS6dyj) +
                           dS11dyj;
      const double dNdyj = ddydXSqdyj + ddydYSqdyj;
      const double dsqrtSdyj = 0.5 * dSdyj * invSqrtS;
      const double dsigma1Sqdyj = 0.5 * (dNdyj + dsqrtSdyj),
                   dsigma2Sqdyj = 0.5 * (dNdyj - dsqrtSdyj);
      const double dSumdyj = dsigma1Sqdyj + dsigma2Sqdyj;
      const double dProddyj = dsigma1Sqdyj * sigma2Sq + sigma1Sq * dsigma2Sqdyj;
      const double diCNdyj =
        (dProddyj * sum - 2. * prod * dSumdyj) * invSum * invSum * invSqrtProd;
      IDI(i, j + numMapNodes) = posJac ? diCNdyj : -diCNdyj;

      const double ddzdXSqdzj = 2. * dPhidX * dzdX,
                   ddzdYSqdzj = 2. * dPhidY * dzdY;
      const double dS1dzj = 2. * ddzdYSqdzj * dzdYSq;
      const double dS2dzj = (dzdXSq - Dy - Dx) * ddzdYSqdzj;
      const double dS3dzj = (Cy + Cx) * (ddzdXSqdzj * dzdY + dzdX * ddzdYSqdzj);
      const double dS4dzj = 2. * ddzdXSqdzj * dzdXSq;
      const double dS5dzj = (Dy + Dx) * ddzdXSqdzj;
      const double dSdzj =
        2. * (dS2dzj + dS5dzj) + 8. * dS3dzj + dS1dzj + dS4dzj;
      const double dNdzj = ddzdXSqdzj + ddzdYSqdzj;
      const double dsqrtSdzj = 0.5 * dSdzj * invSqrtS;
      const double dsigma1Sqdzj = 0.5 * (dNdzj + dsqrtSdzj),
                   dsigma2Sqdzj = 0.5 * (dNdzj - dsqrtSdzj);
      const double dSumdzj = dsigma1Sqdzj + dsigma2Sqdzj;
      const double dProddzj = dsigma1Sqdzj * sigma2Sq + sigma1Sq * dsigma2Sqdzj;
      const double diCNdzj =
        (dProddzj * sum - 2. * prod * dSumdzj) * invSum * invSum * invSqrtProd;
      IDI(i, j + 2 * numMapNodes) = posJac ? diCNdzj : -diCNdzj;
    }
  }

  // Compute condition number and its gradients
  // w.r.t. node positions, at one location in a 3D element
  template <bool sign>
  inline void calcGradInvCondNum3D(
    double dxdX, double dxdY, double dxdZ, double dydX, double dydY,
    double dydZ, double dzdX, double dzdY, double dzdZ, int i, int numMapNodes,
    const fullMatrix<double> &gSMatX, const fullMatrix<double> &gSMatY,
    const fullMatrix<double> &gSMatZ, fullMatrix<double> &IDI)
  {
    const double normJSq = dxdX * dxdX + dxdY * dxdY + dxdZ * dxdZ +
                           dydX * dydX + dydY * dydY + dydZ * dydZ +
                           dzdX * dzdX + dzdY * dzdY + dzdZ * dzdZ;
    const double I11 = dydY * dzdZ - dydZ * dzdY,
                 I12 = dxdZ * dzdY - dxdY * dzdZ,
                 I13 = dxdY * dydZ - dxdZ * dydY,
                 I21 = dydZ * dzdX - dydX * dzdZ,
                 I22 = dxdX * dzdZ - dxdZ * dzdX,
                 I23 = dxdZ * dydX - dxdX * dydZ,
                 I31 = dydX * dzdY - dydY * dzdX,
                 I32 = dxdY * dzdX - dxdX * dzdY,
                 I33 = dxdX * dydY - dxdY * dydX;
    const double normISq = I11 * I11 + I12 * I12 + I13 * I13 + I21 * I21 +
                           I22 * I22 + I23 * I23 + I31 * I31 + I32 * I32 +
                           I33 * I33;
    const double invProd = 1. / (normJSq * normISq),
                 invSqrtProd = sqrt(invProd);
    const double D =
      calcDet3x3(dxdX, dxdY, dxdZ, dydX, dydY, dydZ, dzdX, dzdY, dzdZ);
    const bool reverse = (!sign && (D < 0.));
    const double sICN = 3. * D * invSqrtProd;
    IDI(i, 3 * numMapNodes) = reverse ? -sICN : sICN;

    for(int j = 0; j < numMapNodes; j++) {
      const double &dPhidX = gSMatX(i, j);
      const double &dPhidY = gSMatY(i, j);
      const double &dPhidZ = gSMatZ(i, j);

      const double dNormJSqdxj =
        2. * (dPhidX * dxdX + dPhidY * dxdY + dPhidZ * dxdZ);
      const double dNormISqdxj = 2. * ((dPhidZ * dzdY - dPhidY * dzdZ) * I12 +
                                       (dPhidY * dydZ - dPhidZ * dydY) * I13 +
                                       (dPhidX * dzdZ - dPhidZ * dzdX) * I22 +
                                       (dPhidZ * dydX - dPhidX * dydZ) * I23 +
                                       (dPhidY * dzdX - dPhidX * dzdY) * I32 +
                                       (dPhidX * dydY - dPhidY * dydX) * I33);
      const double dProddxj = dNormJSqdxj * normISq + dNormISqdxj * normJSq;
      const double dDdxj = dPhidX * dydY * dzdZ + dzdX * dPhidY * dydZ +
                           dydX * dzdY * dPhidZ - dzdX * dydY * dPhidZ -
                           dPhidX * dzdY * dydZ - dydX * dPhidY * dzdZ;
      const double dsICNdxj =
        3. * (dDdxj * invSqrtProd - 0.5 * D * dProddxj * invProd * invSqrtProd);
      IDI(i, j) = reverse ? -dsICNdxj : dsICNdxj;

      const double dNormJSqdyj =
        2. * (dPhidX * dydX + dPhidY * dydY + dPhidZ * dydZ);
      const double dNormISqdyj = 2. * ((dPhidY * dzdZ - dPhidZ * dzdY) * I11 +
                                       (dxdY * dPhidZ - dxdZ * dPhidY) * I13 +
                                       (dPhidZ * dzdX - dPhidX * dzdZ) * I21 +
                                       (dxdZ * dPhidX - dxdX * dPhidZ) * I23 +
                                       (dPhidX * dzdY - dPhidY * dzdX) * I31 +
                                       (dxdX * dPhidY - dxdY * dPhidX) * I33);
      const double dProddyj = dNormJSqdyj * normISq + dNormISqdyj * normJSq;
      const double dDdyj = dxdX * dPhidY * dzdZ + dzdX * dxdY * dPhidZ +
                           dPhidX * dzdY * dxdZ - dzdX * dPhidY * dxdZ -
                           dxdX * dzdY * dPhidZ - dPhidX * dxdY * dzdZ;
      const double dsICNdyj =
        3. * (dDdyj * invSqrtProd - 0.5 * D * dProddyj * invProd * invSqrtProd);
      IDI(i, j + numMapNodes) = reverse ? -dsICNdyj : dsICNdyj;

      const double dNormJSqdzj =
        2. * (dPhidX * dzdX + dPhidY * dzdY + dPhidZ * dzdZ);
      const double dNormISqdzj = 2. * ((dydY * dPhidZ - dydZ * dPhidY) * I11 +
                                       (dxdZ * dPhidY - dxdY * dPhidZ) * I12 +
                                       (dydZ * dPhidX - dydX * dPhidZ) * I21 +
                                       (dxdX * dPhidZ - dxdZ * dPhidX) * I22 +
                                       (dydX * dPhidY - dydY * dPhidX) * I31 +
                                       (dxdY * dPhidX - dxdX * dPhidY) * I32);
      const double dProddzj = dNormJSqdzj * normISq + dNormISqdzj * normJSq;
      const double dDdzj = dxdX * dydY * dPhidZ + dPhidX * dxdY * dydZ +
                           dydX * dPhidY * dxdZ - dPhidX * dydY * dxdZ -
                           dxdX * dPhidY * dydZ - dydX * dxdY * dPhidZ;
      const double dsICNdzj =
        3. * (dDdzj * invSqrtProd - 0.5 * D * dProddzj * invProd * invSqrtProd);
      IDI(i, j + 2 * numMapNodes) = reverse ? -dsICNdzj : dsICNdzj;
    }
  }

} // namespace

CondNumBasis::CondNumBasis(int tag, int cnOrder)
  : _tag(tag), _dim(ElementType::getDimension(tag)),
    _condNumOrder(cnOrder >= 0 ? cnOrder : condNumOrder(tag))
{
  if(ElementType::getParentType(tag) == TYPE_TRIH) {
    _nCondNumNodes = 1;
    _nMapNodes = 4;
    _nPrimMapNodes = 4;
    return;
  }
  const bool serendip = false;

  FuncSpaceData data =
    (ElementType::getParentType(tag) == TYPE_PYR) ?
      FuncSpaceData(true, tag, true, 1, _condNumOrder - 1, &serendip) :
      FuncSpaceData(true, tag, _condNumOrder, &serendip);

  fullMatrix<double> lagPoints; // Sampling points
  gmshGeneratePoints(data, lagPoints);
  _nCondNumNodes = lagPoints.size1();
  _nMapNodes = BasisFactory::getNodalBasis(tag)->getNumShapeFunctions();

  // Store shape function gradients of mapping at condition number nodes
  _gradBasis = BasisFactory::getGradientBasis(data);

  // Compute shape function gradients of primary mapping at barycenter,
  // in order to compute normal to straight element
  const int parentType = ElementType::getParentType(tag);
  const int primMapType = ElementType::getType(parentType, 1, false);
  const nodalBasis *primMapBasis = BasisFactory::getNodalBasis(primMapType);
  _nPrimMapNodes = primMapBasis->getNumShapeFunctions();

  double xBar = 0., yBar = 0., zBar = 0.;
  double barycenter[3] = {0., 0., 0.};
  for(int i = 0; i < _nPrimMapNodes; i++) {
    for(int j = 0; j < primMapBasis->points.size2(); ++j) {
      barycenter[j] += primMapBasis->points(i, j);
    }
  }
  barycenter[0] /= _nPrimMapNodes;
  barycenter[1] /= _nPrimMapNodes;
  barycenter[2] /= _nPrimMapNodes;

  double(*barDPsi)[3] = new double[_nPrimMapNodes][3];
  primMapBasis->df(xBar, yBar, zBar, barDPsi);

  // TODO: Make primGradShape from ideal element
  primGradShapeBarycenterX.resize(_nPrimMapNodes);
  primGradShapeBarycenterY.resize(_nPrimMapNodes);
  primGradShapeBarycenterZ.resize(_nPrimMapNodes);
  for(int j = 0; j < _nPrimMapNodes; j++) {
    primGradShapeBarycenterX(j) = barDPsi[j][0];
    primGradShapeBarycenterY(j) = barDPsi[j][1];
    primGradShapeBarycenterZ(j) = barDPsi[j][2];
  }

  delete[] barDPsi;
}

int CondNumBasis::condNumOrder(int tag)
{
  const int parentType = ElementType::getParentType(tag);
  const int order = ElementType::getOrder(tag);
  return condNumOrder(parentType, order);
}

int CondNumBasis::condNumOrder(int parentType, int order)
{
  switch(parentType) {
  case TYPE_PNT: return 0;
  case TYPE_LIN: return order - 1;
  case TYPE_TRI: return (order == 1) ? 0 : order;
  case TYPE_QUA: return order;
  case TYPE_TET: return (order == 1) ? 0 : order;
  case TYPE_PRI: return order;
  case TYPE_HEX: return order;
  case TYPE_PYR: return order;
  case TYPE_TRIH: return 0;
  default:
    Msg::Error("Unknown element type %d, return order 0", parentType);
    return 0;
  }
}

// Calculate the inverse condition number in Frobenius norm for one element,
// with normal vectors to straight element for regularization. Evaluation points
// depend on the given matrices for shape function gradients.
template <bool sign>
inline void CondNumBasis::getInvCondNumGeneral(
  int nCondNumNodes, const fullMatrix<double> &gSMatX,
  const fullMatrix<double> &gSMatY, const fullMatrix<double> &gSMatZ,
  const fullMatrix<double> &nodesXYZ, const fullMatrix<double> &normals,
  fullVector<double> &condNum) const
{
  switch(_dim) {
  case 0: {
    for(int i = 0; i < nCondNumNodes; i++) condNum(i) = 1.;
    break;
  }

  case 1: {
    Msg::Warning("Inverse condition number not implemented in 1D");
    condNum.setAll(0.);
    break;
  }

  case 2: {
    fullMatrix<double> dxyzdX(nCondNumNodes, 3), dxyzdY(nCondNumNodes, 3);
    gSMatX.mult(nodesXYZ, dxyzdX);
    gSMatY.mult(nodesXYZ, dxyzdY);
    for(int i = 0; i < nCondNumNodes; i++) {
      const double &dxdX = dxyzdX(i, 0), &dydX = dxyzdX(i, 1),
                   &dzdX = dxyzdX(i, 2);
      const double &dxdY = dxyzdY(i, 0), &dydY = dxyzdY(i, 1),
                   &dzdY = dxyzdY(i, 2);
      const double &nx = normals(0, 0), &ny = normals(0, 1),
                   &nz = normals(0, 2);
      condNum(i) =
        calcInvCondNum2D<sign>(dxdX, dxdY, dydX, dydY, dzdX, dzdY, nx, ny, nz);
    }
    break;
  }

  case 3: {
    if(ElementType::getParentType(_tag) == TYPE_TRIH) {
      for(int i = 0; i < nCondNumNodes; i++) condNum(i) = 1.;
      break;
    }
    fullMatrix<double> dxyzdX(nCondNumNodes, 3), dxyzdY(nCondNumNodes, 3),
      dxyzdZ(nCondNumNodes, 3);
    gSMatX.mult(nodesXYZ, dxyzdX);
    gSMatY.mult(nodesXYZ, dxyzdY);
    gSMatZ.mult(nodesXYZ, dxyzdZ);
    for(int i = 0; i < nCondNumNodes; i++) {
      const double &dxdX = dxyzdX(i, 0), &dydX = dxyzdX(i, 1),
                   &dzdX = dxyzdX(i, 2);
      const double &dxdY = dxyzdY(i, 0), &dydY = dxyzdY(i, 1),
                   &dzdY = dxyzdY(i, 2);
      const double &dxdZ = dxyzdZ(i, 0), &dydZ = dxyzdZ(i, 1),
                   &dzdZ = dxyzdZ(i, 2);
      condNum(i) = calcInvCondNum3D<sign>(dxdX, dxdY, dxdZ, dydX, dydY, dydZ,
                                          dzdX, dzdY, dzdZ);
    }
    break;
  }
  }
}

void CondNumBasis::getInvCondNumGeneral(int nCondNumNodes,
                                        const fullMatrix<double> &gSMatX,
                                        const fullMatrix<double> &gSMatY,
                                        const fullMatrix<double> &gSMatZ,
                                        const fullMatrix<double> &nodesXYZ,
                                        fullVector<double> &invCond) const
{
  fullMatrix<double> dumNormals;
  getInvCondNumGeneral<false>(nCondNumNodes, gSMatX, gSMatY, gSMatZ, nodesXYZ,
                              dumNormals, invCond);
}

void CondNumBasis::getSignedInvCondNumGeneral(
  int nCondNumNodes, const fullMatrix<double> &gSMatX,
  const fullMatrix<double> &gSMatY, const fullMatrix<double> &gSMatZ,
  const fullMatrix<double> &nodesXYZ, const fullMatrix<double> &normals,
  fullVector<double> &invCond) const
{
  getInvCondNumGeneral<true>(nCondNumNodes, gSMatX, gSMatY, gSMatZ, nodesXYZ,
                             normals, invCond);
}

// Calculate the inverse condition number in Frobenius norm and its gradients
// w.r.t. node position, with normal vectors to straight element  for
// regularization. Evaluation points depend on the given matrices for shape
// function gradients.
template <bool sign>
inline void CondNumBasis::getInvCondNumAndGradientsGeneral(
  int nCondNumNodes, const fullMatrix<double> &gSMatX,
  const fullMatrix<double> &gSMatY, const fullMatrix<double> &gSMatZ,
  const fullMatrix<double> &nodesXYZ, const fullMatrix<double> &normals,
  fullMatrix<double> &IDI) const
{
  fullMatrix<double> JDJ(nCondNumNodes, 3 * _nMapNodes + 1);

  switch(_dim) {
  case 0: {
    for(int i = 0; i < nCondNumNodes; i++) {
      for(int j = 0; j < _nMapNodes; j++) {
        IDI(i, j) = 0.;
        IDI(i, j + 1 * _nMapNodes) = 0.;
        IDI(i, j + 2 * _nMapNodes) = 0.;
      }
      IDI(i, 3 * _nMapNodes) = 1.;
    }
    break;
  }

  case 1: {
    Msg::Warning("Inverse condition number not implemented in 1D");
    IDI.setAll(0.);
    break;
  }

  case 2: {
    fullMatrix<double> dxyzdX(nCondNumNodes, 3), dxyzdY(nCondNumNodes, 3);
    gSMatX.mult(nodesXYZ, dxyzdX);
    gSMatY.mult(nodesXYZ, dxyzdY);
    for(int i = 0; i < nCondNumNodes; i++) {
      const double &dxdX = dxyzdX(i, 0), &dydX = dxyzdX(i, 1),
                   &dzdX = dxyzdX(i, 2);
      const double &dxdY = dxyzdY(i, 0), &dydY = dxyzdY(i, 1),
                   &dzdY = dxyzdY(i, 2);
      const double &nx = normals(0, 0), &ny = normals(0, 1),
                   &nz = normals(0, 2);
      calcGradInvCondNum2D<sign>(dxdX, dxdY, dydX, dydY, dzdX, dzdY, nx, ny, nz,
                                 i, _nMapNodes, gSMatX, gSMatY, IDI);
    }
    break;
  }

  case 3: {
    if(ElementType::getParentType(_tag) == TYPE_TRIH) {
      for(int i = 0; i < nCondNumNodes; i++) {
        for(int j = 0; j < _nMapNodes; j++) {
          IDI(i, j) = 0.;
          IDI(i, j + 1 * _nMapNodes) = 0.;
          IDI(i, j + 2 * _nMapNodes) = 0.;
        }
        IDI(i, 3 * _nMapNodes) = 1.;
      }
      break;
    }
    fullMatrix<double> dxyzdX(nCondNumNodes, 3), dxyzdY(nCondNumNodes, 3),
      dxyzdZ(nCondNumNodes, 3);
    gSMatX.mult(nodesXYZ, dxyzdX);
    gSMatY.mult(nodesXYZ, dxyzdY);
    gSMatZ.mult(nodesXYZ, dxyzdZ);
    for(int i = 0; i < nCondNumNodes; i++) {
      const double &dxdX = dxyzdX(i, 0), &dydX = dxyzdX(i, 1),
                   &dzdX = dxyzdX(i, 2);
      const double &dxdY = dxyzdY(i, 0), &dydY = dxyzdY(i, 1),
                   &dzdY = dxyzdY(i, 2);
      const double &dxdZ = dxyzdZ(i, 0), &dydZ = dxyzdZ(i, 1),
                   &dzdZ = dxyzdZ(i, 2);
      calcGradInvCondNum3D<sign>(dxdX, dxdY, dxdZ, dydX, dydY, dydZ, dzdX, dzdY,
                                 dzdZ, i, _nMapNodes, gSMatX, gSMatY, gSMatZ,
                                 IDI);
    }
    break;
  }
  }
}

void CondNumBasis::getInvCondNumAndGradientsGeneral(
  int nCondNumNodes, const fullMatrix<double> &gSMatX,
  const fullMatrix<double> &gSMatY, const fullMatrix<double> &gSMatZ,
  const fullMatrix<double> &nodesXYZ, fullMatrix<double> &IDI) const
{
  fullMatrix<double> dumNormals;
  getInvCondNumAndGradientsGeneral<false>(nCondNumNodes, gSMatX, gSMatY, gSMatZ,
                                          nodesXYZ, dumNormals, IDI);
}

void CondNumBasis::getSignedInvCondNumAndGradientsGeneral(
  int nCondNumNodes, const fullMatrix<double> &gSMatX,
  const fullMatrix<double> &gSMatY, const fullMatrix<double> &gSMatZ,
  const fullMatrix<double> &nodesXYZ, const fullMatrix<double> &normals,
  fullMatrix<double> &IDI) const
{
  getInvCondNumAndGradientsGeneral<true>(nCondNumNodes, gSMatX, gSMatY, gSMatZ,
                                         nodesXYZ, normals, IDI);
}