inamuroNewtonRaphsonDynamics.hh

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/*  This file is part of the OpenLB library
 *
 *  Copyright (C) 2006, Orestis Malaspinas and Jonas Latt
 *  E-mail contact: info@openlb.net
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 *  <http://www.openlb.net/>
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#ifndef INAMURO_NEWTON_RAPHSON_DYNAMICS_HH
#define INAMURO_NEWTON_RAPHSON_DYNAMICS_HH
 
#include "inamuroNewtonRaphsonDynamics.h"
#include "dynamics/latticeDescriptors.h"
#include "core/util.h"
#include "dynamics/lbm.h"
#include "utilities/omath.h"
 
 
namespace olb {
 
 
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::InamuroNewtonRaphsonDynamics (T omega)
  : legacy::BasicDynamics<T,DESCRIPTOR,MOMENTA>(),
    _boundaryDynamics(omega),
    clout(std::cout,"InamuroNewtonRaphsonDynamics")
{
  this->getName() = "InamuroNewtonRaphsonDynamics";
  _xi[0] = (T)1;
  for (int iDim = 1; iDim < DESCRIPTOR::d; ++iDim) {
    _xi[iDim] = T();
  }
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
T InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::
computeEquilibrium(int iPop, T rho, const T u[DESCRIPTOR::d], T uSqr) const
{
  return _boundaryDynamics.computeEquilibrium(iPop, rho, u, uSqr);
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
CellStatistic<T> InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::collide (
  Cell<T,DESCRIPTOR>& cell,
  LatticeStatistics<T>& statistics )
{
  typedef DESCRIPTOR L;
 
  T rho, u[L::d];
  MOMENTA().computeRhoU(cell,rho,u);
 
  constexpr auto missingIndexes = util::subIndexOutgoing<L,direction,orientation>();
  std::vector<int> knownIndexes;
  bool test[L::q];
  for (int iPop = 0; iPop < L::q; ++iPop) {
    test[iPop] = true;
  }
 
  for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
    test[missingIndexes[iPop]] = false;
  }
  for (int iPop = 0; iPop < L::q; ++iPop) {
    if (test[iPop]) {
      knownIndexes.push_back(iPop);
    }
  }
 
  T approxMomentum[L::d];
 
  computeApproxMomentum(approxMomentum,cell,rho,u,_xi,knownIndexes,missingIndexes);
 
  T error = computeError(rho, u,approxMomentum);
  int counter = 0;
 
  while (error > 1.0e-15) {
    ++counter;
 
    T gradError[L::d], gradGradError[L::d][L::d];
    computeGradGradError(gradGradError,gradError,rho,u,_xi,approxMomentum,missingIndexes);
 
    bool everythingWentFine = newtonRaphson(_xi,gradError,gradGradError);
    if ((counter > 100000) || everythingWentFine == false) {
      // if we need more that 100000 iterations or
      // if we have a problem with the inversion of the
      // jacobian matrix, we stop the program and
      // print this error message on the screen.
      clout << "Failed to converge...." << std::endl;
      clout << "Error = " << error << std::endl;
      clout << "u = (" << rho*u[0];
      for (int iD=1; iD<DESCRIPTOR::d; ++iD) {
        clout << ", " << rho*u[iD];
      }
      clout << ")" << std::endl;
      clout << "uApprox = (" << approxMomentum[0];
      for (int iD=1; iD<DESCRIPTOR::d; ++iD) {
        clout << ", " << approxMomentum[iD];
      }
      clout << ")" << std::endl;
      clout << "xi = (" << _xi[0];
      for (int iD=1; iD<DESCRIPTOR::d; ++iD) {
        clout << ", " << _xi[iD];
      }
      clout << ")" << std::endl;
 
      exit(1);
    }
 
    computeApproxMomentum(approxMomentum,cell,rho,u,_xi,knownIndexes,missingIndexes);
    error = computeError(rho, u,approxMomentum);
 
  }
 
  T uCs[L::d];
  int counterDim = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    if (direction == iDim) {
      ++counterDim;
      uCs[iDim] = u[iDim];
    }
    else {
      uCs[iDim] = u[iDim] + _xi[iDim+1-counterDim];
    }
  }
 
  T uCsSqr = util::normSqr<T,L::d>(uCs);
 
  for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
    cell[missingIndexes[iPop]] = computeEquilibrium(missingIndexes[iPop],_xi[0],uCs,uCsSqr);
  }
 
  _boundaryDynamics.collide(cell, statistics);
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
T InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::getOmega() const
{
  return _boundaryDynamics.getOmega();
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
void InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::setOmega(T omega)
{
  _boundaryDynamics.setOmega(omega);
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
void InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::
computeApproxMomentum(T approxMomentum[DESCRIPTOR::d], ConstCell<T,DESCRIPTOR>& cell,
                      const T &rho, const T u[DESCRIPTOR::d], const T xi[DESCRIPTOR::d],
                      const std::vector<int> knownIndexes,const std::vector<int> missingIndexes)
{
  typedef DESCRIPTOR L;
 
  T csVel[L::d];
  int counter = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    if (direction == iDim) {
      ++counter;
      csVel[iDim] = u[iDim];
    }
    else {
      csVel[iDim] = u[iDim] + xi[iDim+1-counter];
    }
  }
 
  T csVelSqr = util::normSqr<T,L::d>(csVel);
 
  for (int iDim = 0; iDim < L::d; ++iDim) {
    approxMomentum[iDim] = T();
    for (unsigned iPop = 0; iPop < knownIndexes.size(); ++iPop) {
      approxMomentum[iDim] += descriptors::c<L>(knownIndexes[iPop],iDim) *
                              cell[knownIndexes[iPop]];
    }
    for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
      approxMomentum[iDim] += descriptors::c<L>(missingIndexes[iPop],iDim) *
                              computeEquilibrium(missingIndexes[iPop],xi[0],csVel,csVelSqr);
    }
  }
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
T InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::
computeError(const T &rho, const T u[DESCRIPTOR::d], const T approxMomentum[DESCRIPTOR::d])
{
  typedef DESCRIPTOR L;
 
  T err = T();
  for (int iDim = 0; iDim < L::d; ++iDim) {
    err += (rho * u[iDim]-approxMomentum[iDim]) * (rho * u[iDim]-approxMomentum[iDim]);
  }
  return util::sqrt(err);
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
void InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::computeGradGradError(
  T gradGradError[DESCRIPTOR::d][DESCRIPTOR::d],T gradError[DESCRIPTOR::d],
  const T &rho, const T u[DESCRIPTOR::d],const T xi[DESCRIPTOR::d],
  const T approxMomentum[DESCRIPTOR::d],
  const std::vector<int> missingIndexes)
{
  typedef DESCRIPTOR L;
 
  T csVel[L::d];
  int counter = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    if (direction == iDim) {
      ++counter;
      csVel[iDim] = u[iDim];
    }
    else {
      csVel[iDim] = u[iDim] + xi[iDim+1-counter];
    }
  }
  T csVelSqr = util::normSqr<T,L::d>(csVel);
 
  std::vector<T> df[L::d];
 
  for (int iXi = 0; iXi < L::d; ++iXi) {
    for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
      df[iXi].push_back(T());
    }
  }
 
  // all the null terms are allready contained in df (no need for else in the ifs)
 
  for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
    T cu = T();
    for (int iDim = 0; iDim < L::d; ++iDim) {
      cu += descriptors::c<L>(missingIndexes[iPop],iDim) * csVel[iDim];
    }
    df[0][iPop] = descriptors::t<T,L>(missingIndexes[iPop])
                  * ((T)1+descriptors::invCs2<T,L>()*cu
                     + 0.5 * descriptors::invCs2<T,L>() * descriptors::invCs2<T,L>() * cu * cu
                     - 0.5 * descriptors::invCs2<T,L>() * csVelSqr);
  }
 
  counter = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    if (direction == iDim) {
      ++counter;
    }
    else {
      for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
        T temp = T();
        for (int jDim = 0; jDim < L::d; ++jDim) {
          temp += descriptors::c<L>(missingIndexes[iPop],jDim) * csVel[jDim];
        }
 
        df[iDim+1-counter][iPop] = xi[0]*descriptors::t<T,L>(missingIndexes[iPop]) *
                                   (descriptors::invCs2<T,L>()*descriptors::c<L>(missingIndexes[iPop],iDim)
                                    + descriptors::invCs2<T,L>()*descriptors::invCs2<T,L>()*descriptors::c<L>(missingIndexes[iPop],iDim)*temp
                                    - descriptors::invCs2<T,L>()*csVel[iDim]);
      }
    }
  }
 
  std::vector<T> ddf[L::d][L::d];
 
  for (int iAlpha = 0; iAlpha < L::d; ++iAlpha) {
    for (int iBeta = 0; iBeta < L::d; ++iBeta) {
      for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
        ddf[iAlpha][iBeta].push_back(T());
      }
    }
  }
 
  // ddf contains allready all the zero terms
 
  counter = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    if (direction == iDim) {
      ++counter;
    }
    else {
      for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
        T temp = T();
        for (int jDim = 0; jDim < L::d; ++jDim) {
          temp += descriptors::c<L>(missingIndexes[iPop],jDim) * csVel[jDim];
        }
 
        T d_rho_sa = descriptors::t<T,L>(missingIndexes[iPop]) *
                     (descriptors::invCs2<T,L>()*descriptors::c<L>(missingIndexes[iPop],iDim)
                      + descriptors::invCs2<T,L>()*descriptors::invCs2<T,L>()*descriptors::c<L>(missingIndexes[iPop],iDim)*temp
                      - descriptors::invCs2<T,L>()*csVel[iDim]);
 
        ddf[iDim+1-counter][0][iPop] = d_rho_sa;
        ddf[0][iDim+1-counter][iPop] = d_rho_sa;
      }
    }
  }
 
  for (int iAlpha = 1; iAlpha < L::d; ++iAlpha) {
    for (int iBeta = 1; iBeta < L::d; ++iBeta) {
      for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
        ddf[iAlpha][iBeta][iPop] = descriptors::t<T,L>(missingIndexes[iPop])*xi[0] *
                                   descriptors::invCs2<T,L>()*descriptors::invCs2<T,L>()*descriptors::c<L>(missingIndexes[iPop],iAlpha)*descriptors::c<L>(missingIndexes[iPop],iBeta);
        if (iAlpha == iBeta) {
          ddf[iAlpha][iBeta][iPop] -= descriptors::t<T,L>(missingIndexes[iPop])*xi[0] * descriptors::invCs2<T,L>();
        }
      }
    }
  }
 
  // computation of the vector gradError
  counter = 0;
  for (int iDim = 0; iDim < L::d; ++iDim) {
    T du[L::d];
    gradError[iDim] = T();
    for (int jDim = 0; jDim < L::d; ++jDim) {
      du[jDim] = T();
      for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
        du[jDim] += descriptors::c<L>(missingIndexes[iPop],jDim) * df[iDim][iPop];
      }
      gradError[iDim] += (approxMomentum[jDim]- rho * u[jDim]) * du[jDim];
    }
    gradError[iDim] *= (T)2;
  }
 
  // computation of the matrix gradGradError
 
  for (int iAlpha = 0; iAlpha < L::d; ++iAlpha) {
    for (int iBeta = 0; iBeta < L::d; ++iBeta) {
      gradGradError[iAlpha][iBeta] = T();
 
      T duAlpha[L::d], duBeta[L::d], ddu[L::d];
      for (int iDim = 0; iDim < L::d; ++iDim) {
        duAlpha[iDim] = T();
        duBeta[iDim] = T();
        ddu[iDim] = T();
        for (unsigned iPop = 0; iPop < missingIndexes.size(); ++iPop) {
          duAlpha[iDim] += descriptors::c<L>(missingIndexes[iPop],iDim)
                           * df[iAlpha][iPop];
 
          duBeta[iDim] += descriptors::c<L>(missingIndexes[iPop],iDim)
                          * df[iBeta][iPop];
 
          ddu[iDim] += descriptors::c<L>(missingIndexes[iPop],iDim)
                       * ddf[iAlpha][iBeta][iPop];
        }
        gradGradError[iAlpha][iBeta] += duAlpha[iDim] * duBeta[iDim]
                                        + (approxMomentum[iDim]-rho * u[iDim]) * ddu[iDim];
      }
      gradGradError[iAlpha][iBeta] *= (T)2;
      // gradGradError = 2*sum_alpha(du_alpha/dxi_beta*du_alpha/dxi_gamma +
      //                  approxMomentum_alpha-u_alpha*d^2u_alpha/(dxi_gamma*dxi_beta))
    }
  }
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
bool InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::
newtonRaphson(T xi[DESCRIPTOR::d],
              const T gradError[DESCRIPTOR::d],
              const T gradGradError[DESCRIPTOR::d][DESCRIPTOR::d])
{
  typedef DESCRIPTOR L;
 
  T invGradGradError[L::d][L::d];
  bool inversion = invert(gradGradError,invGradGradError);
 
  for (int iXi = 0; iXi < L::d; ++iXi) {
    T correction = T();
    for (int iAlpha = 0; iAlpha < L::d; ++iAlpha) {
      correction += invGradGradError[iXi][iAlpha] * gradError[iAlpha];
    }
 
    xi[iXi] -= correction;
  }
 
  return inversion;
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
bool InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::
invert(const T a[2][2],T invA[2][2])
{
  T detA = a[0][0]*a[1][1] - a[0][1]*a[1][0];
 
  if (util::fabs(detA) < 1.0e-13) {
    clout << "error detA too small! = " << detA << std::endl;
    for (int iAlpha = 0; iAlpha < 2; ++iAlpha) {
      for (int iBeta = 0; iBeta < 2; ++iBeta) {
        clout << a[iAlpha][iBeta] << "\t";
      }
      clout << std::endl;
    }
    return false;
  }
  else {
    invA[0][0] = a[1][1];
    invA[1][1] = a[0][0];
 
    invA[0][1] = -a[1][0];
    invA[1][0] = -a[0][1];
 
    for (int iPop = 0; iPop < 2; ++iPop) {
      for (int jPop = 0; jPop < 2; ++jPop) {
        invA[iPop][jPop] /= detA;
      }
    }
    return true;
  }
}
 
template<typename T, typename DESCRIPTOR, typename Dynamics, typename MOMENTA, int direction, int orientation>
bool InamuroNewtonRaphsonDynamics<T,DESCRIPTOR,Dynamics,MOMENTA,direction,orientation>::invert(const T a[3][3],T invA[3][3])
{
  T detA = a[0][0]*a[1][1]*a[2][2] + a[1][0]*a[2][1]*a[0][2] + a[2][0]*a[0][1]*a[1][2]
           - a[0][0]*a[2][1]*a[1][2] - a[2][0]*a[1][1]*a[0][2] - a[1][0]*a[0][1]*a[2][2];
 
 
  if (util::fabs(detA) < 1.0e-13) {
    clout << "Error: detA too small! = " << detA << std::endl;
    for (int iAlpha = 0; iAlpha < 3; ++iAlpha) {
      for (int iBeta = 0; iBeta < 3; ++iBeta) {
        clout << a[iAlpha][iBeta] << "\t";
      }
      clout << std::endl;
    }
    return false;
  }
  else {
    invA[0][0] = a[1][1]*a[2][2]-a[1][2]*a[2][1];
    invA[0][1] = a[0][2]*a[2][1]-a[0][1]*a[2][2];
    invA[0][2] = a[0][1]*a[1][2]-a[0][2]*a[1][1];
 
    invA[1][0] = a[1][2]*a[2][0]-a[1][0]*a[2][2];
    invA[1][1] = a[0][0]*a[2][2]-a[0][2]*a[2][0];
    invA[1][2] = a[0][2]*a[1][0]-a[0][0]*a[1][2];
 
    invA[2][0] = a[1][0]*a[2][1]-a[1][1]*a[2][0];
    invA[2][1] = a[0][1]*a[2][0]-a[0][0]*a[2][1];
    invA[2][2] = a[0][0]*a[1][1]-a[0][1]*a[1][0];
 
    for (int iPop = 0; iPop < 3; ++iPop) {
      for (int jPop = 0; jPop < 3; ++jPop) {
        invA[iPop][jPop] /= detA;
      }
    }
    return true;
  }
}
 
}  // namespace olb
 
 
#endif