reference_mapping.h

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/*
 * Copyright (c) 2010-2015:  G-CSC, Goethe University Frankfurt
 * Author: Andreas Vogel
 * 
 * This file is part of UG4.
 * 
 * UG4 is free software: you can redistribute it and/or modify it under the
 * terms of the GNU Lesser General Public License version 3 (as published by the
 * Free Software Foundation) with the following additional attribution
 * requirements (according to LGPL/GPL v3 §7):
 * 
 * (1) The following notice must be displayed in the Appropriate Legal Notices
 * of covered and combined works: "Based on UG4 (www.ug4.org/license)".
 * 
 * (2) The following notice must be displayed at a prominent place in the
 * terminal output of covered works: "Based on UG4 (www.ug4.org/license)".
 * 
 * (3) The following bibliography is recommended for citation and must be
 * preserved in all covered files:
 * "Reiter, S., Vogel, A., Heppner, I., Rupp, M., and Wittum, G. A massively
 *   parallel geometric multigrid solver on hierarchically distributed grids.
 *   Computing and visualization in science 16, 4 (2013), 151-164"
 * "Vogel, A., Reiter, S., Rupp, M., Nägel, A., and Wittum, G. UG4 -- a novel
 *   flexible software system for simulating pde based models on high performance
 *   computers. Computing and visualization in science 16, 4 (2013), 165-179"
 * 
 * This program 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.
 */
 
#ifndef __H__UG__LIB_DISC__REFERENCE_ELEMENT__REFERENCE_ELEMENT_MAPPING__
#define __H__UG__LIB_DISC__REFERENCE_ELEMENT__REFERENCE_ELEMENT_MAPPING__
 
#include <cassert>
#include <iostream>
#include <sstream>
#include "common/common.h"
#include "common/math/ugmath.h"
#include "lib_disc/reference_element/reference_element.h"
 
namespace ug{
 
extern DebugID DID_REFERENCE_MAPPING;
extern DebugID DID_REFERENCE_MAPPING_GLOB_TO_LOC;
 
template <typename TRefElem, int TWorldDim>
class ReferenceMapping
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = TRefElem::dim;
 
    static const bool isLinear = false;
 
  public:
    ReferenceMapping();
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord);
 
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord);
 
    void update(const MathVector<worldDim>* vCornerCoord);
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord);
 
    bool is_linear() const;
 
    void local_to_global(MathVector<worldDim>& globPos,
                         const MathVector<dim>& locPos) const;
 
    void local_to_global(MathVector<worldDim>* vGlobPos,
               const MathVector<dim>* vLocPos, size_t n) const;
 
    void local_to_global(std::vector<MathVector<worldDim> >& vGlobPos,
               const std::vector<MathVector<dim> >& vLocPos) const;
 
    void global_to_local(MathVector<dim>& locPos,
               const MathVector<worldDim>& globPos,
               const size_t maxIter = 1000,
               const number tol = 1e-10) const;
 
    void global_to_local(MathVector<dim>* vLocPos,
               const MathVector<worldDim>* vGlobPos, size_t n,
               const size_t maxIter = 1000,
               const number tol = 1e-10) const;
 
    void global_to_local(std::vector<MathVector<dim> >& vLocPos,
               const std::vector<MathVector<worldDim> >& vGlobPos,
               const size_t maxIter = 1000,
               const number tol = 1e-10) const;
 
    void jacobian(MathMatrix<worldDim, dim>& J,
                  const MathVector<dim>& locPos) const;
 
    void jacobian(MathMatrix<worldDim, dim>* vJ,
            const MathVector<dim>* vLocPos, size_t n) const;
 
    void jacobian(std::vector<MathMatrix<worldDim, dim> >& J,
            const std::vector<MathVector<dim> >& vLocPos) const;
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                             const MathVector<dim>& locPos) const;
 
    void jacobian_transposed(MathMatrix<dim, worldDim>* vJT,
                             const MathVector<dim>* vLocPos, size_t n) const;
 
    void jacobian_transposed(std::vector<MathMatrix<dim, worldDim> >& vJT,
                 const std::vector<MathVector<dim> >& vLocPos) const;
 
    number jacobian_transposed_inverse(MathMatrix<worldDim, dim>& JTInv,
                                       const MathVector<dim>& locPos) const;
 
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv,
                                     const MathVector<dim>* vLocPos, size_t n) const;
 
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv,
                                     number* vDet,
                     const MathVector<dim>* vLocPos, size_t n) const;
 
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv,
                     const std::vector<MathVector<dim> >& vLocPos) const;
 
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv,
                     std::vector<number>& vDet,
                     const std::vector<MathVector<dim> >& vLocPos) const;
 
    number sqrt_gram_det(const MathVector<dim>& locPos) const;
 
    void sqrt_gram_det(number* vDet, const MathVector<dim>* vLocPos, size_t n) const;
 
    void sqrt_gram_det(std::vector<number> vDet,
              const std::vector<MathVector<dim> >& vLocPos) const;
};
class ReferenceMapping {…};
 
 
// Concrete Reference Mappings
 
template <int dim, int worldDim, bool isLinear, typename TImpl>
class BaseReferenceMapping
{
  public:
    bool is_linear() const {return isLinear;}
 
    void local_to_global(MathVector<worldDim>* vGlobPos,
               const MathVector<dim>* vLocPos, size_t n) const
    {
      for(size_t ip = 0; ip < n; ++ip)
        getImpl().local_to_global(vGlobPos[ip], vLocPos[ip]);
    }
    void local_to_global(MathVector<worldDim>* vGlobPos, {…}
 
    void local_to_global(std::vector<MathVector<worldDim> >& vGlobPos,
               const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vGlobPos.resize(n);
      local_to_global(&vGlobPos[0], &vLocPos[0], n);
    }
    void local_to_global(std::vector<MathVector<worldDim> >& vGlobPos, {…}
 
    void global_to_local
    (
      MathVector<dim>& locPos, 
      const MathVector<worldDim>& globPos, 
      const size_t maxIter = 1000, 
      const number tol = 1e-10 
    ) const
    {
      // We apply the Newton's method for the transformation. We assume here that
      // the Newton's method converges without the linesearch, and the Jacobian is
      // non-singular in the whole iteration process. This is true in particular for
      // the bilinear transformation of convex quadrilaterals if the initial guess
      // is inside of the element.
      MathMatrix<worldDim, dim> J;
      MathMatrix<dim, worldDim> JInv;
      MathVector<worldDim> dist, compGlobPos, minDist;
      MathVector<dim> corr;
      
      VecScale(minDist, globPos, tol);
 
      for (size_t i = 0; i < maxIter; ++i) {
 
      //  f(x) := \phi(x) - x_{glob}
        getImpl().local_to_global(compGlobPos, locPos);
        VecSubtract(dist, compGlobPos, globPos);
        
      //  Floating-point cancellation protection:
        if(VecAbsIsLess(dist, minDist))
          return;
        // REMARK: Note that the tolerance is specified not only to provide the
        // sufficient accuracy for the solution but also to protect the iteration
        // from the cancellation phenomena in the floating-point arithmetics.
        // Computation of the distance involves subtraction which is a reason for
        // the loss of precision phenomena. If a small element is located very far
        // from the coordinate origin, the accuracy of the local->global transform
        // is restricted and this cannot be overcome. After we reach this bound,
        // the iterates will oscillate and the defect will not be reduced. We use
        // globPos for the reference values.
        //      Note that this check may not be used as the only stopping criterion:
        // globPos may be 0 by specification. 
 
        UG_DLOG(DID_REFERENCE_MAPPING_GLOB_TO_LOC, 2,
            "reference_mapping.h: global_to_local() Newton iteration: Iter# "
            << i << "; fabs(VecTwoNorm(dist)) = " << fabs(VecTwoNorm(dist)) <<
            "; dist = " << dist << "; locPos: " << locPos << std::endl);
 
      //  compute jacobian df/dx = d \phi(x) / dx =: J
        getImpl().jacobian(J, locPos);
 
      //  solve c -= J^{-1} f
        LeftInverse(JInv, J);
        MatVecMult(corr, JInv, dist);
        
      //  check if tol reached
        if(VecAbsIsLess(corr, tol))
          return;
        // REMARK: Note that using the Euclidean or maximum norm directly to dist
        // would need tuning of the tolerance for every particular grid: For big
        // elements with the diameter of order 1, the tolerance 1e-10 is good accuracy,
        // but for a refined grid with the elements of the size of order 1e-5 it
        // can be pour. But ||corr|| = ||J^{-1} dist|| is also a norm of dist, and
        // it is rescaled according to the element dimensions.
        //      Besides that, ||corr|| measures whether we can get any further progress
        // in the iterations.
 
      //  apply the correction
        VecSubtract(locPos, locPos, corr);
      }
 
      // compiler does not know that maxIter will never be 0
      // therefore it warns that dist may be uninit'ed;
      // as we will throw here anyway, we may as well check that
      UG_COND_THROW(!maxIter, "Without a single iteration, local-to-global "
              "mapping can never converge.");
 
      UG_DLOG(DID_REFERENCE_MAPPING_GLOB_TO_LOC, 2, "Last JInv:" << std::endl);
      for(int i = 0; i < 3; ++i)
      {
        UG_DLOG(DID_REFERENCE_MAPPING_GLOB_TO_LOC, 2,
            JInv(i, 0) << "; " << JInv(i, 1) << "; " << JInv(i, 2) << std::endl);
      }
 
      UG_THROW("ReferenceMapping::global_to_local: Newton method did not"
          " reach a tolerance "<<tol<<" after "<<maxIter<<" steps. "
          "Global Pos: "<<globPos<<", dim: "<<dim<<", worldDim: "<<
          worldDim<<", last newton defect: "<<fabs(VecTwoNorm(dist)));
    }
    void global_to_local {…}
 
    void global_to_local(MathVector<dim>* vLocPos,
               const MathVector<worldDim>* vGlobPos, size_t n,
               const size_t maxIter = 1000,
               const number tol = 1e-10) const
    {
      if(isLinear){
        if(n == 0) return;
 
        MathMatrix<worldDim, dim> J;
        MathMatrix<dim, worldDim> JInv;
        MathVector<worldDim> dist, compGlobPos;
 
      //  compute jacobian df/dx = d \phi(x) / dx =: J
        getImpl().jacobian(J, vLocPos[0]);
 
      //  solve x -= J^{-1} f
        LeftInverse(JInv, J);
 
        for(size_t ip = 0; ip < n; ++ip)
        {
        //  f(x) := \phi(x) - x_{glob}
          getImpl().local_to_global(compGlobPos, vLocPos[ip]);
          VecSubtract(dist, compGlobPos, vGlobPos[ip]);
 
          MatVecScaleMultAppend(vLocPos[ip], -1.0, JInv, dist);
        }
      }
      else{
        for(size_t ip = 0; ip < n; ++ip)
          getImpl().global_to_local(vLocPos[ip], vGlobPos[ip], maxIter, tol);
      }
    }
    void global_to_local(MathVector<dim>* vLocPos, {…}
 
    void global_to_local(std::vector<MathVector<dim> >& vLocPos,
               const std::vector<MathVector<worldDim> >& vGlobPos,
               const size_t maxIter = 1000,
               const number tol = 1e-10) const
    {
      const size_t n = vGlobPos.size();
      vLocPos.resize(n);
      global_to_local(&vLocPos[0], &vGlobPos[0], n, maxIter, tol);
    }
    void global_to_local(std::vector<MathVector<dim> >& vLocPos, {…}
 
    void jacobian(MathMatrix<worldDim, dim>& J,
            const MathVector<dim>& locPos) const
    {
      MathMatrix<dim, worldDim> JT;
      getImpl().jacobian_transposed(JT, locPos);
      Transpose(J, JT);
    }
    void jacobian(MathMatrix<worldDim, dim>& J, {…}
 
    void jacobian(MathMatrix<worldDim, dim>* vJ,
            const MathVector<dim>* vLocPos, size_t n) const
    {
      if(isLinear){
        if(n == 0) return;
        getImpl().jacobian(vJ[0], vLocPos[0]);
        for(size_t ip = 1; ip < n; ++ip) vJ[ip] = vJ[0];
      }
      else {
        for(size_t ip = 0; ip < n; ++ip)
          getImpl().jacobian(vJ[ip], vLocPos[ip]);
      }
    }
    void jacobian(MathMatrix<worldDim, dim>* vJ, {…}
 
    void jacobian(std::vector<MathMatrix<worldDim, dim> >& vJ,
            const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vJ.resize(n);
      jacobian(&vJ[0], &vLocPos[0], n);
    }
    void jacobian(std::vector<MathMatrix<worldDim, dim> >& vJ, {…}
 
 
    void jacobian_transposed(MathMatrix<dim, worldDim>* vJT,
                 const MathVector<dim>* vLocPos, size_t n) const
    {
      if(isLinear){
        if(n == 0) return;
        getImpl().jacobian_transposed(vJT[0], vLocPos[0]);
        for(size_t ip = 1; ip < n; ++ip) vJT[ip] = vJT[0];
      }
      else {
        for(size_t ip = 0; ip < n; ++ip)
          getImpl().jacobian_transposed(vJT[ip], vLocPos[ip]);
      }
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>* vJT, {…}
 
    void jacobian_transposed(std::vector<MathMatrix<dim, worldDim> >& vJT,
                 const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vJT.resize(n);
      jacobian_transposed(&vJT[0], &vLocPos[0], n);
    }
    void jacobian_transposed(std::vector<MathMatrix<dim, worldDim> >& vJT, {…}
 
    number jacobian_transposed_inverse(MathMatrix<worldDim, dim>& JTInv,
                     const MathVector<dim>& locPos) const
    {
      MathMatrix<dim, worldDim> JT;
      getImpl().jacobian_transposed(JT, locPos);
      UG_DLOG(DID_REFERENCE_MAPPING, 2, ">>OCT_DISC_DEBUG:: " << "reference_mapping.h: " << "jacobian_transposed_inverse(): JT " << std::endl);
      for(int i = 0; i < 3; ++i)
        UG_DLOG(DID_REFERENCE_MAPPING, 2, " JT:" << JT(i, 0) << "; " << JT(i, 1) << "; " << JT(i, 2) << std::endl);
      return RightInverse(JTInv, JT);
    }
    number jacobian_transposed_inverse(MathMatrix<worldDim, dim>& JTInv, {…}
 
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv,
                                     number* vDet,
                     const MathVector<dim>* vLocPos, size_t n) const
    {
      if(isLinear){
        if(n == 0) return;
        vDet[0] = getImpl().jacobian_transposed_inverse(vJTInv[0], vLocPos[0]);
        for(size_t ip = 1; ip < n; ++ip) vJTInv[ip] = vJTInv[0];
        for(size_t ip = 1; ip < n; ++ip) vDet[ip] = vDet[0];
      }
      else {
        for(size_t ip = 0; ip < n; ++ip)
          vDet[ip] = getImpl().jacobian_transposed_inverse(vJTInv[ip], vLocPos[ip]);
      }
    }
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv, {…}
 
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv,
                     const MathVector<dim>* vLocPos, size_t n) const
    {
      if(isLinear){
        if(n == 0) return;
        getImpl().jacobian_transposed_inverse(vJTInv[0], vLocPos[0]);
        for(size_t ip = 1; ip < n; ++ip) vJTInv[ip] = vJTInv[0];
      }
      else {
        for(size_t ip = 0; ip < n; ++ip)
          getImpl().jacobian_transposed_inverse(vJTInv[ip], vLocPos[ip]);
      }
    }
    void jacobian_transposed_inverse(MathMatrix<worldDim, dim>* vJTInv, {…}
 
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv,
                                     std::vector<number>& vDet,
                     const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vJTInv.resize(n); vDet.resize(n);
      jacobian_transposed_inverse(&vJTInv[0], &vDet[0], &vLocPos[0], n);
    }
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv, {…}
 
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv,
                     const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vJTInv.resize(n);
      jacobian_transposed_inverse(&vJTInv[0], &vLocPos[0], n);
    }
    void jacobian_transposed_inverse(std::vector<MathMatrix<worldDim, dim> >& vJTInv, {…}
 
    number sqrt_gram_det(const MathVector<dim>& locPos) const
    {
      MathMatrix<dim, worldDim> JT;
      getImpl().jacobian_transposed(JT, locPos);
      return SqrtGramDeterminant(JT);
    }
    number sqrt_gram_det(const MathVector<dim>& locPos) const {…}
 
    void sqrt_gram_det(number* vDet, const MathVector<dim>* vLocPos, size_t n) const
    {
      if(isLinear){
        if(n == 0) return;
        vDet[0] = sqrt_gram_det(vLocPos[0]);
        for(size_t ip = 1; ip < n; ++ip) vDet[ip] = vDet[0];
      }
      else {
        for(size_t ip = 0; ip < n; ++ip)
          vDet[ip] = sqrt_gram_det(vLocPos[ip]);
      }
    }
    void sqrt_gram_det(number* vDet, const MathVector<dim>* vLocPos, size_t n) const {…}
 
    void sqrt_gram_det(std::vector<number>& vDet,
                      const std::vector<MathVector<dim> >& vLocPos) const
    {
      const size_t n = vLocPos.size();
      vDet.resize(n);
      sqrt_gram_det(&vDet[0], &vLocPos[0], n);
    }
    void sqrt_gram_det(std::vector<number>& vDet, {…}
 
  protected:
    TImpl& getImpl() {return static_cast<TImpl&>(*this);}
 
    const TImpl& getImpl() const {return static_cast<const TImpl&>(*this);}
};
class BaseReferenceMapping {…};
 
// Reference Mapping RegularVertex
 
template <int TWorldDim>
class ReferenceMapping<ReferenceVertex, TWorldDim>
  : public BaseReferenceMapping<ReferenceVertex::dim, TWorldDim, true,
                                ReferenceMapping<ReferenceVertex, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceVertex::dim;
 
    static const bool isLinear = true;
 
  public:
    typedef BaseReferenceMapping<ReferenceVertex::dim, TWorldDim, true,
                    ReferenceMapping<ReferenceVertex, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceEdge::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      co0 = vCornerCoord[0];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      globPos = co0;
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
    {
      //for(int i = 0; i < worldDim; ++i) JT(0,i) = 1;
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> co0;
};
class ReferenceMapping<ReferenceVertex, TWorldDim> {…};
 
// Reference Mapping Edge
template <int TWorldDim>
class ReferenceMapping<ReferenceEdge, TWorldDim>
  : public BaseReferenceMapping<ReferenceEdge::dim, TWorldDim, true,
                                ReferenceMapping<ReferenceEdge, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceEdge::dim;
 
    static const bool isLinear = true;
 
  public:
    typedef BaseReferenceMapping<ReferenceEdge::dim, TWorldDim, true,
                    ReferenceMapping<ReferenceEdge, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceEdge::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      co0 = vCornerCoord[0];
      VecSubtract(a10, vCornerCoord[1], vCornerCoord[0]);
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      VecScaleAdd(globPos, 1.0, co0, locPos[0], a10);
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
    {
      for(int i = 0; i < worldDim; ++i) JT(0,i) = a10[i];
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> co0, a10;
};
class ReferenceMapping<ReferenceEdge, TWorldDim> {…};
 
// Reference Mapping Triangle
template <int TWorldDim>
class ReferenceMapping<ReferenceTriangle, TWorldDim>
  : public BaseReferenceMapping<ReferenceTriangle::dim, TWorldDim, true,
                                ReferenceMapping<ReferenceTriangle, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceTriangle::dim;
 
    static const bool isLinear = true;
 
  public:
    typedef BaseReferenceMapping<ReferenceTriangle::dim, TWorldDim, true,
                ReferenceMapping<ReferenceTriangle, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceTriangle::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      co0 = vCornerCoord[0];
      VecSubtract(a10, vCornerCoord[1], vCornerCoord[0]);
      VecSubtract(a20, vCornerCoord[2], vCornerCoord[0]);
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      VecScaleAdd(globPos, 1.0, co0, locPos[0], a10, locPos[1], a20);
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
    {
      for(int i = 0; i < worldDim; ++i) JT(0, i) = a10[i];
      for(int i = 0; i < worldDim; ++i) JT(1, i) = a20[i];
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> co0, a10, a20;
 
};
class ReferenceMapping<ReferenceTriangle, TWorldDim> {…};
// Reference Mapping Quadrilateral
 
 
template <int TWorldDim>
class ReferenceMapping<ReferenceQuadrilateral, TWorldDim>
  : public BaseReferenceMapping<ReferenceQuadrilateral::dim, TWorldDim, false,
                  ReferenceMapping<ReferenceQuadrilateral, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceQuadrilateral::dim;
 
    static const bool isLinear = false;
 
  public:
    typedef BaseReferenceMapping<ReferenceQuadrilateral::dim, TWorldDim, false,
          ReferenceMapping<ReferenceQuadrilateral, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceQuadrilateral::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      for(int co = 0; co < ReferenceQuadrilateral::numCorners; ++co)
        x[co] = vCornerCoord[co];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      const number a = (1.-locPos[0]);
      const number b = (1.-locPos[1]);
 
      VecScaleAdd(globPos,            a*b, x[0],
                          locPos[0]*b, x[1],
                locPos[0]*locPos[1], x[2],
                    a*locPos[1], x[3]);
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
    {
      const number a = 1. - locPos[1];
      const number b = 1. - locPos[0];
 
      for(int i = 0; i < worldDim; ++i)
        JT(0, i) = a*(x[1][i] - x[0][i]) + locPos[1]*(x[2][i] - x[3][i]);
 
      for(int i = 0; i < worldDim; ++i)
        JT(1, i) = b*(x[3][i] - x[0][i]) + locPos[0]*(x[2][i] - x[1][i]);
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> x[ReferenceQuadrilateral::numCorners];
};
class ReferenceMapping<ReferenceQuadrilateral, TWorldDim> {…};
 
// Reference Mapping Tetrahedron
 
template <int TWorldDim>
class ReferenceMapping<ReferenceTetrahedron, TWorldDim>
  : public BaseReferenceMapping<ReferenceTetrahedron::dim, TWorldDim, true,
                  ReferenceMapping<ReferenceTetrahedron, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceTetrahedron::dim;
 
    static const bool isLinear = true;
 
  public:
    typedef BaseReferenceMapping<ReferenceTetrahedron::dim, TWorldDim, true,
          ReferenceMapping<ReferenceTetrahedron, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceTetrahedron::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      co0 = vCornerCoord[0];
      VecSubtract(a10, vCornerCoord[1], vCornerCoord[0]);
      VecSubtract(a20, vCornerCoord[2], vCornerCoord[0]);
      VecSubtract(a30, vCornerCoord[3], vCornerCoord[0]);
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      VecScaleAdd(globPos,       1.0, co0,
                       locPos[0], a10,
                 locPos[1], a20,
                 locPos[2], a30);
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
    {
      for(int i = 0; i < worldDim; ++i) JT[0][i] = a10[i];
      for(int i = 0; i < worldDim; ++i) JT[1][i] = a20[i];
      for(int i = 0; i < worldDim; ++i) JT[2][i] = a30[i];
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> co0, a10, a20, a30;
};
class ReferenceMapping<ReferenceTetrahedron, TWorldDim> {…};
 
// Reference Mapping Pyramid
template <int TWorldDim>
class ReferenceMapping<ReferencePyramid, TWorldDim>
  : public BaseReferenceMapping<ReferencePyramid::dim, TWorldDim, false,
                  ReferenceMapping<ReferencePyramid, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferencePyramid::dim;
 
    static const bool isLinear = false;
 
  public:
    typedef BaseReferenceMapping<ReferencePyramid::dim, TWorldDim, false,
          ReferenceMapping<ReferencePyramid, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferencePyramid::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      for(int co = 0; co < ReferencePyramid::numCorners; ++co)
        x[co] = vCornerCoord[co];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      const number a = 1.0 - locPos[0];
      const number b = 1.0 - locPos[1];
      if (locPos[0] > locPos[1])
      {
        const number a0 = a * b - locPos[2] * b;
        const number a1 = locPos[0] * b - locPos[2]*locPos[1];
        const number a2 = locPos[0] * locPos[1] + locPos[2]*locPos[1];
        const number a3 = a * locPos[1] - locPos[2] * locPos[1];
 
        for(int d = 0; d < worldDim; ++d)
          globPos[d] = a0*x[0][d]+a1*x[1][d]+a2*x[2][d]
                +a3*x[3][d]+locPos[2]*x[4][d];
      }
      else
      {
        const number a0 = a * b - locPos[2] * a;
        const number a1 = locPos[0] * b - locPos[2]*locPos[0];
        const number a2 = locPos[0] * locPos[1] + locPos[2]*locPos[0];
        const number a3 = a * locPos[1] - locPos[2] * locPos[0];
        for(int d = 0; d < worldDim; ++d)
          globPos[d] = a0*x[0][d]+a1*x[1][d]+
                 a2*x[2][d]+a3*x[3][d]+locPos[2]*x[4][d];
      }
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
     {
      number a[3];
      for(int d = 0; d < worldDim; ++d)
        a[d] = x[0][d]-x[1][d]+x[2][d]-x[3][d];
 
      if (locPos[0] > locPos[1])
      {
        for(int d = 0; d < worldDim; ++d)
          JT(0,d) = x[1][d]-x[0][d]+locPos[1]*a[d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(1,d) = x[3][d]-x[0][d]+(locPos[0]+locPos[2])*a[d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(2,d) = x[4][d]-x[0][d]+locPos[1]*a[d];
      }
      else
      {
        for(int d = 0; d < worldDim; ++d)
          JT(0,d) = x[1][d]-x[0][d]+(locPos[1]+locPos[2])*a[d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(1,d) = x[3][d]-x[0][d]+locPos[0]*a[d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(2,d) = x[4][d]-x[0][d]+locPos[0]*a[d];
      }
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> x[ReferencePyramid::numCorners];
};
class ReferenceMapping<ReferencePyramid, TWorldDim> {…};
 
// Reference Mapping Prism
 
template <int TWorldDim>
class ReferenceMapping<ReferencePrism, TWorldDim>
  : public BaseReferenceMapping<ReferencePrism::dim, TWorldDim, false,
                  ReferenceMapping<ReferencePrism, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferencePrism::dim;
 
    static const bool isLinear = false;
 
  public:
    typedef BaseReferenceMapping<ReferencePrism::dim, TWorldDim, false,
          ReferenceMapping<ReferencePrism, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferencePrism::numCorners,
            "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      for(int co = 0; co < ReferencePrism::numCorners; ++co)
        x[co] = vCornerCoord[co];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      const number a = 1.0 - locPos[0] - locPos[1];
      const number b = 1.0 - locPos[2];
      const number a0 = a * b;
      const number a1 = locPos[0] * b;
      const number a2 = locPos[1] * b;
      const number a3 = a * locPos[2];
      const number a4 = locPos[0] * locPos[2];
      const number a5 = locPos[1] * locPos[2];
 
      for(int d = 0; d < worldDim; ++d)
        globPos[d] = a0*x[0][d]+a1*x[1][d]+a2*x[2][d]+a3*x[3][d]+a4*x[4][d]+a5*x[5][d];
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
     {
          number a[worldDim], b[worldDim];
 
      for(int d = 0; d < worldDim; ++d)
            a[d] = x[0][d]-x[1][d]-x[3][d]+x[4][d];
 
      for(int d = 0; d < worldDim; ++d)
        b[d] = x[0][d]-x[2][d]-x[3][d]+x[5][d];
 
      for(int d = 0; d < worldDim; ++d){
            JT(0,d) = x[1][d]-x[0][d]+locPos[2]*a[d];
            JT(1,d) = x[2][d]-x[0][d]+locPos[2]*b[d];
            JT(2,d) = x[3][d]-x[0][d]+locPos[0]*a[d]+locPos[1]*b[d];
      }
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> x[ReferencePrism::numCorners];
};
class ReferenceMapping<ReferencePrism, TWorldDim> {…};
 
 
// Reference Mapping Hexahedron
 
template <int TWorldDim>
class ReferenceMapping<ReferenceHexahedron, TWorldDim>
  : public BaseReferenceMapping<ReferenceHexahedron::dim, TWorldDim, false,
                  ReferenceMapping<ReferenceHexahedron, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceHexahedron::dim;
 
    static const bool isLinear = false;
 
  public:
    typedef BaseReferenceMapping<ReferenceHexahedron::dim, TWorldDim, false,
          ReferenceMapping<ReferenceHexahedron, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceHexahedron::numCorners,
           "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      for(int co = 0; co < ReferenceHexahedron::numCorners; ++co)
        x[co] = vCornerCoord[co];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
      const number a = 1.0 - locPos[0];
      const number b = 1.0 - locPos[1];
      const number c = 1.0 - locPos[2];
      const number a0 = a * b * c;
      const number a1 = locPos[0] * b * c;
      const number a2 = locPos[0] * locPos[1] * c;
      const number a3 = a * locPos[1] * c;
      const number a4 = a * b * locPos[2];
      const number a5 = locPos[0] * b * locPos[2];
      const number a6 = locPos[0] * locPos[1] * locPos[2];
      const number a7 = a * locPos[1] * locPos[2];
 
      for(int d = 0; d < worldDim; ++d){
        globPos[d] = a0*x[0][d]+a1*x[1][d]+a2*x[2][d]+a3*x[3][d]+
               a4*x[4][d]+a5*x[5][d]+a6*x[6][d]+a7*x[7][d];
      }
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
     {
      number a0,a1,a2,a3;
      const number a = 1.0 - locPos[0];
      const number b = 1.0 - locPos[1];
      const number c = 1.0 - locPos[2];
      a0 = b * c;
      a1 = locPos[1] * c;
      a2 = locPos[1] * locPos[2];
      a3 = b * locPos[2];
      for(int d = 0; d < worldDim; ++d)
        JT(0,d) = a0*(x[1][d]-x[0][d])+a1*(x[2][d]-x[3][d])
            + a2*(x[6][d]-x[7][d])+a3*(x[5][d]-x[4][d]);
 
          a0 = a * c;
          a1 = locPos[0] * c;
          a2 = locPos[0] * locPos[2];
          a3 = a * locPos[2];
      for(int d = 0; d < worldDim; ++d)
        JT(1,d) = a0*(x[3][d]-x[0][d])+a1*(x[2][d]-x[1][d])
            + a2*(x[6][d]-x[5][d])+a3*(x[7][d]-x[4][d]);
 
          a0 = a * b;
          a1 = locPos[0] * b;
          a2 = locPos[0] * locPos[1];
          a3 = a * locPos[1];
      for(int d = 0; d < worldDim; ++d)
        JT(2,d) = a0*(x[4][d]-x[0][d])+a1*(x[5][d]-x[1][d])
            + a2*(x[6][d]-x[2][d])+a3*(x[7][d]-x[3][d]);
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> x[ReferenceHexahedron::numCorners];
};
class ReferenceMapping<ReferenceHexahedron, TWorldDim> {…};
 
// Reference Mapping Octahedron
template <int TWorldDim>
class ReferenceMapping<ReferenceOctahedron, TWorldDim>
  : public BaseReferenceMapping<ReferenceOctahedron::dim, TWorldDim, false,
                  ReferenceMapping<ReferenceOctahedron, TWorldDim> >
{
  public:
    static const int worldDim = TWorldDim;
 
    static const int dim = ReferenceOctahedron::dim;
 
    static const bool isLinear = false;
 
  public:
    typedef BaseReferenceMapping<ReferenceOctahedron::dim, TWorldDim, false,
          ReferenceMapping<ReferenceOctahedron, TWorldDim> > base_type;
    using base_type::local_to_global;
    using base_type::jacobian;
    using base_type::jacobian_transposed;
    using base_type::jacobian_transposed_inverse;
    using base_type::sqrt_gram_det;
 
  public:
    ReferenceMapping() {}
 
    ReferenceMapping(const MathVector<worldDim>* vCornerCoord) {update(vCornerCoord);}
    ReferenceMapping(const std::vector<MathVector<worldDim> >& vCornerCoord) {update(vCornerCoord);}
 
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord)
    {
      UG_ASSERT((int)vCornerCoord.size() >= ReferenceOctahedron::numCorners,
                "ReferenceMapping: to few Corner Coordinates.");
      update(&vCornerCoord[0]);
    }
    void update(const std::vector<MathVector<worldDim> >& vCornerCoord) {…}
 
    void update(const MathVector<worldDim>* vCornerCoord)
    {
      for(int co = 0; co < ReferenceOctahedron::numCorners; ++co)
        x[co] = vCornerCoord[co];
    }
    void update(const MathVector<worldDim>* vCornerCoord) {…}
 
    void local_to_global(MathVector<worldDim>& globPos,
               const MathVector<dim>& locPos) const
    {
    //  the octahedral shape functions correspond to the tetrahedral ones,
    //  but locally piecewise linear on a subdivision of the octahedron
    //  into 4 sub-tetrahedrons.
      const number z_sgn  = (locPos[2] < 0) ? -1.0 : 1.0;
      const number a0   = (locPos[2] < 0) ? -locPos[2] : 0.0;
      const number a5   = (locPos[2] < 0) ?  0.0 : locPos[2];
 
      if (locPos[0] > locPos[1])
      {
        const number a1 = 1.0 - locPos[0] - z_sgn * locPos[2];
        const number a2 = locPos[0] - locPos[1];
        const number a3 = locPos[1];
        const number a4 = 0.0;
 
        for(int d = 0; d < worldDim; ++d)
          globPos[d] = a0*x[0][d]+a1*x[1][d]+a2*x[2][d]
                +a3*x[3][d]+a4*x[4][d]+a5*x[5][d];
      }
      else
      {
        const number a1 = 1.0 - locPos[1] - z_sgn * locPos[2];
        const number a2 = 0.0;
        const number a3 = locPos[0];
        const number a4 = locPos[1] - locPos[0];
 
        for(int d = 0; d < worldDim; ++d)
          globPos[d] = a0*x[0][d]+a1*x[1][d]+a2*x[2][d]
                +a3*x[3][d]+a4*x[4][d]+a5*x[5][d];
      }
    }
    void local_to_global(MathVector<worldDim>& globPos, {…}
 
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT,
                 const MathVector<dim>& locPos) const
     {
    //  the octahedral shape functions correspond to the tetrahedral ones,
    //  but locally piecewise linear on a subdivision of the octahedron
    //  into 4 sub-tetrahedrons.
      const number z_sgn  = (locPos[2] < 0) ? -1.0 : 1.0;
      const number Da0  = (locPos[2] < 0) ? -1.0 : 0.0;
      const number Da5  = (locPos[2] < 0) ?  0.0 : 1.0;
 
      if (locPos[0] > locPos[1])
      {
        for(int d = 0; d < worldDim; ++d)
          JT(0,d) = -x[1][d]+x[2][d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(1,d) = -x[2][d]+x[3][d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(2,d) = Da0*x[0][d] - z_sgn*x[1][d] + Da5*x[5][d];
      }
      else
      {
        for(int d = 0; d < worldDim; ++d)
          JT(0,d) = x[3][d]-x[4][d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(1,d) = -x[1][d]+x[4][d];
 
        for(int d = 0; d < worldDim; ++d)
          JT(2,d) = Da0*x[0][d] - z_sgn*x[1][d] + Da5*x[5][d];
      }
    }
    void jacobian_transposed(MathMatrix<dim, worldDim>& JT, {…}
 
  private:
    MathVector<worldDim> x[ReferenceOctahedron::numCorners];
};
class ReferenceMapping<ReferenceOctahedron, TWorldDim> {…};
 
 
} // end namespace ug
 
#endif /* __H__UG__LIB_DISC__REFERENCE_ELEMENT__REFERENCE_ELEMENT_MAPPING__ */