lsAdvect.hpp

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#pragma once
 
#include <lsPreCompileMacros.hpp>
 
#include <limits>
#include <vector>
 
#include <hrleSparseIterator.hpp>
#include <hrleSparseStarIterator.hpp>
 
#include <vcLogger.hpp>
#include <vcSmartPointer.hpp>
 
#include <lsBooleanOperation.hpp>
#include <lsDomain.hpp>
#include <lsMarkVoidPoints.hpp>
#include <lsReduce.hpp>
 
// Integration schemes
#include <lsConcepts.hpp>
#include <lsEnquistOsher.hpp>
#include <lsLaxFriedrichs.hpp>
#include <lsLocalLaxFriedrichs.hpp>
#include <lsLocalLaxFriedrichsAnalytical.hpp>
#include <lsLocalLocalLaxFriedrichs.hpp>
#include <lsStencilLocalLaxFriedrichsScalar.hpp>
 
// Velocity accessor
#include <lsVelocityField.hpp>
 
// #define DEBUG_LS_ADVECT_HPP
#ifdef DEBUG_LS_ADVECT_HPP
#include <lsToMesh.hpp>
#include <lsVTKWriter.hpp>
#endif
 
namespace viennals {
 
using namespace viennacore;

enum struct IntegrationSchemeEnum : unsigned {
  ENGQUIST_OSHER_1ST_ORDER = 0,
  ENGQUIST_OSHER_2ND_ORDER = 1,
  LAX_FRIEDRICHS_1ST_ORDER = 2,
  LAX_FRIEDRICHS_2ND_ORDER = 3,
  LOCAL_LAX_FRIEDRICHS_ANALYTICAL_1ST_ORDER = 4,
  LOCAL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER = 5,
  LOCAL_LOCAL_LAX_FRIEDRICHS_2ND_ORDER = 6,
  LOCAL_LAX_FRIEDRICHS_1ST_ORDER = 7,
  LOCAL_LAX_FRIEDRICHS_2ND_ORDER = 8,
  STENCIL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER = 9
};

template <class T, int D> class Advect {
  std::vector<SmartPointer<Domain<T, D>>> levelSets;
  SmartPointer<VelocityField<T>> velocities = nullptr;
  IntegrationSchemeEnum integrationScheme =
      IntegrationSchemeEnum::ENGQUIST_OSHER_1ST_ORDER;
  double timeStepRatio = 0.4999;
  double dissipationAlpha = 1.0;
  bool calculateNormalVectors = true;
  bool ignoreVoids = false;
  double advectionTime = 0.;
  bool performOnlySingleStep = false;
  double advectedTime = 0.;
  unsigned numberOfTimeSteps = 0;
  bool saveAdvectionVelocities = false;
  bool updatePointData = true;
  bool checkDissipation = true;
  static constexpr double wrappingLayerEpsilon = 1e-4;
 
  hrleVectorType<T, 3> findGlobalAlphas() const {
 
    auto &topDomain = levelSets.back()->getDomain();
    auto &grid = levelSets.back()->getGrid();
 
    const T gridDelta = grid.getGridDelta();
    const T deltaPos = gridDelta;
    const T deltaNeg = -gridDelta;
 
    hrleVectorType<T, 3> finalAlphas(0., 0., 0.);
 
#pragma omp parallel num_threads((levelSets.back())->getNumberOfSegments())
    {
      int p = 0;
#ifdef _OPENMP
      p = omp_get_thread_num();
#endif
 
      hrleVectorType<T, 3> localAlphas(0., 0., 0.);
 
      hrleVectorType<hrleIndexType, D> startVector =
          (p == 0) ? grid.getMinGridPoint()
                   : topDomain.getSegmentation()[p - 1];
 
      hrleVectorType<hrleIndexType, D> endVector =
          (p != static_cast<int>(topDomain.getNumberOfSegments() - 1))
              ? topDomain.getSegmentation()[p]
              : grid.incrementIndices(grid.getMaxGridPoint());
 
      // an iterator for each level set
      std::vector<hrleConstSparseIterator<typename Domain<T, D>::DomainType>>
          iterators;
      for (auto it = levelSets.begin(); it != levelSets.end(); ++it) {
        iterators.push_back(
            hrleConstSparseIterator<typename Domain<T, D>::DomainType>(
                (*it)->getDomain()));
      }
 
      // neighborIterator for the top level set
      hrleConstSparseStarIterator<hrleDomain<T, D>, 1> neighborIterator(
          topDomain);
 
      for (hrleSparseIterator<typename Domain<T, D>::DomainType> it(
               topDomain, startVector);
           it.getStartIndices() < endVector; ++it) {
 
        if (!it.isDefined() || std::abs(it.getValue()) > 0.5)
          continue;
 
        const T value = it.getValue();
        const auto indices = it.getStartIndices();
 
        // check if there is any other levelset at the same point:
        // if yes, take the velocity of the lowest levelset
        for (unsigned lowerLevelSetId = 0; lowerLevelSetId < levelSets.size();
             ++lowerLevelSetId) {
          // put iterator to same position as the top levelset
          iterators[lowerLevelSetId].goToIndicesSequential(indices);
 
          // if the lower surface is actually outside, i.e. its LS value
          // is lower or equal
          if (iterators[lowerLevelSetId].getValue() <=
              value + wrappingLayerEpsilon) {
 
            // move neighborIterator to current position
            neighborIterator.goToIndicesSequential(indices);
 
            Vec3D<T> coords;
            for (unsigned i = 0; i < D; ++i) {
              coords[i] = indices[i] * gridDelta;
            }
 
            Vec3D<T> normal = {};
            T normalModulus = 0.;
            for (unsigned i = 0; i < D; ++i) {
              const T phiPos = neighborIterator.getNeighbor(i).getValue();
              const T phiNeg = neighborIterator.getNeighbor(i + D).getValue();
 
              T diffPos = (phiPos - value) / deltaPos;
              T diffNeg = (phiNeg - value) / deltaNeg;
 
              normal[i] = (diffNeg + diffPos) * 0.5;
              normalModulus += normal[i] * normal[i];
            }
            normalModulus = std::sqrt(normalModulus);
            for (unsigned i = 0; i < D; ++i)
              normal[i] /= normalModulus;
 
            T scaVel = velocities->getScalarVelocity(
                coords, lowerLevelSetId, normal,
                neighborIterator.getCenter().getPointId());
            auto vecVel = velocities->getVectorVelocity(
                coords, lowerLevelSetId, normal,
                neighborIterator.getCenter().getPointId());
 
            for (unsigned i = 0; i < D; ++i) {
              T tempAlpha = std::abs((scaVel + vecVel[i]) * normal[i]);
              localAlphas[i] = std::max(localAlphas[i], tempAlpha);
            }
 
            break;
          }
        }
      }
 
#pragma omp critical
      {
        for (unsigned i = 0; i < D; ++i) {
          finalAlphas[i] = std::max(finalAlphas[i], localAlphas[i]);
        }
      }
    } // end of parallel section
 
    return finalAlphas;
  }
 
  void rebuildLS() {
    // TODO: this function uses manhatten distances for renormalisation,
    // since this is the quickest. For visualisation applications, better
    // renormalisation is needed, so it might be good to implement
    // Euler distance renormalisation as an option
    auto &grid = levelSets.back()->getGrid();
    auto newlsDomain = SmartPointer<Domain<T, D>>::New(grid);
    typename Domain<T, D>::DomainType &newDomain = newlsDomain->getDomain();
    typename Domain<T, D>::DomainType &domain = levelSets.back()->getDomain();
 
    newDomain.initialize(domain.getNewSegmentation(),
                         domain.getAllocation() *
                             (2.0 / levelSets.back()->getLevelSetWidth()));
 
    const bool updateData = updatePointData;
    // save how data should be transferred to new level set
    // list of indices into the old pointData vector
    std::vector<std::vector<unsigned>> newDataSourceIds;
    if (updateData)
      newDataSourceIds.resize(newDomain.getNumberOfSegments());
 
#ifdef DEBUG_LS_ADVECT_HPP
    {
      auto mesh = SmartPointer<Mesh<T>>::New();
      ToMesh<T, D>(levelSets.back(), mesh).apply();
      VTKWriter<T>(mesh, "Advect_beforeRebuild.vtk").apply();
    }
#endif
 
#pragma omp parallel num_threads(newDomain.getNumberOfSegments())
    {
      int p = 0;
#ifdef _OPENMP
      p = omp_get_thread_num();
#endif
 
      auto &domainSegment = newDomain.getDomainSegment(p);
 
      hrleVectorType<hrleIndexType, D> startVector =
          (p == 0) ? grid.getMinGridPoint()
                   : newDomain.getSegmentation()[p - 1];
 
      hrleVectorType<hrleIndexType, D> endVector =
          (p != static_cast<int>(newDomain.getNumberOfSegments() - 1))
              ? newDomain.getSegmentation()[p]
              : grid.incrementIndices(grid.getMaxGridPoint());
 
      // reserve a bit more to avoid reallocation
      // would expect number of points to roughly double
      if (updateData)
        newDataSourceIds[p].reserve(2.5 * domainSegment.getNumberOfPoints());
 
      for (hrleSparseStarIterator<typename Domain<T, D>::DomainType, 1> it(
               domain, startVector);
           it.getIndices() < endVector; ++it) {
 
        // if the center is an active grid point
        // <1.0 since it could have been change by 0.5 max
        if (std::abs(it.getCenter().getValue()) <= 1.0) {
 
          int k = 0;
          for (; k < 2 * D; k++)
            if (std::signbit(it.getNeighbor(k).getValue() - 1e-7) !=
                std::signbit(it.getCenter().getValue() + 1e-7))
              break;
 
          // if there is at least one neighbor of opposite sign
          if (k != 2 * D) {
            if (it.getCenter().getDefinedValue() > 0.5) {
              int j = 0;
              for (; j < 2 * D; j++) {
                if (std::abs(it.getNeighbor(j).getValue()) <= 1.0)
                  if (it.getNeighbor(j).getDefinedValue() < -0.5)
                    break;
              }
              if (j == 2 * D) {
                domainSegment.insertNextDefinedPoint(
                    it.getIndices(), it.getCenter().getDefinedValue());
                if (updateData)
                  newDataSourceIds[p].push_back(it.getCenter().getPointId());
                // if there is at least one active grid point, which is < -0.5
              } else {
                domainSegment.insertNextDefinedPoint(it.getIndices(), 0.5);
                if (updateData)
                  newDataSourceIds[p].push_back(it.getNeighbor(j).getPointId());
              }
            } else if (it.getCenter().getDefinedValue() < -0.5) {
              int j = 0;
              for (; j < 2 * D; j++) {
                if (std::abs(it.getNeighbor(j).getValue()) <= 1.0)
                  if (it.getNeighbor(j).getDefinedValue() > 0.5)
                    break;
              }
 
              if (j == 2 * D) {
                domainSegment.insertNextDefinedPoint(
                    it.getIndices(), it.getCenter().getDefinedValue());
                if (updateData)
                  newDataSourceIds[p].push_back(it.getCenter().getPointId());
                // if there is at least one active grid point, which is > 0.5
              } else {
                domainSegment.insertNextDefinedPoint(it.getIndices(), -0.5);
                if (updateData)
                  newDataSourceIds[p].push_back(it.getNeighbor(j).getPointId());
              }
            } else {
              domainSegment.insertNextDefinedPoint(
                  it.getIndices(), it.getCenter().getDefinedValue());
              if (updateData)
                newDataSourceIds[p].push_back(it.getCenter().getPointId());
            }
          } else {
            domainSegment.insertNextUndefinedPoint(
                it.getIndices(), (it.getCenter().getDefinedValue() < 0)
                                     ? Domain<T, D>::NEG_VALUE
                                     : Domain<T, D>::POS_VALUE);
          }
 
        } else { // if the center is not an active grid point
          if (it.getCenter().getValue() >= 0) {
            int usedNeighbor = -1;
            T distance = Domain<T, D>::POS_VALUE;
            for (int i = 0; i < 2 * D; i++) {
              T value = it.getNeighbor(i).getValue();
              if (std::abs(value) <= 1.0 && (value < 0.)) {
                if (distance > value + 1.0) {
                  distance = value + 1.0;
                  usedNeighbor = i;
                }
              }
            }
 
            if (distance <= 1.) {
              domainSegment.insertNextDefinedPoint(it.getIndices(), distance);
              if (updateData)
                newDataSourceIds[p].push_back(
                    it.getNeighbor(usedNeighbor).getPointId());
            } else {
              domainSegment.insertNextUndefinedPoint(it.getIndices(),
                                                     Domain<T, D>::POS_VALUE);
            }
 
          } else {
            int usedNeighbor = -1;
            T distance = Domain<T, D>::NEG_VALUE;
            for (int i = 0; i < 2 * D; i++) {
              T value = it.getNeighbor(i).getValue();
              if (std::abs(value) <= 1.0 && (value > 0)) {
                if (distance < value - 1.0) {
                  // distance = std::max(distance, value - T(1.0));
                  distance = value - 1.0;
                  usedNeighbor = i;
                }
              }
            }
 
            if (distance >= -1.) {
              domainSegment.insertNextDefinedPoint(it.getIndices(), distance);
              if (updateData)
                newDataSourceIds[p].push_back(
                    it.getNeighbor(usedNeighbor).getPointId());
            } else {
              domainSegment.insertNextUndefinedPoint(it.getIndices(),
                                                     Domain<T, D>::NEG_VALUE);
            }
          }
        }
      }
    }
 
    // now copy old data into new level set
    if (updateData) {
      auto &pointData = levelSets.back()->getPointData();
      newlsDomain->getPointData().translateFromMultiData(pointData,
                                                         newDataSourceIds);
    }
 
    newDomain.finalize();
    newDomain.segment();
    levelSets.back()->deepCopy(newlsDomain);
    levelSets.back()->finalize(2);
  }

  double advect(double maxTimeStep = std::numeric_limits<double>::max()) {
    // check whether a level set and velocites have been given
    if (levelSets.size() < 1) {
      Logger::getInstance()
          .addWarning("No level sets passed to Advect. Not advecting.")
          .print();
      return std::numeric_limits<double>::max();
    }
    if (velocities == nullptr) {
      Logger::getInstance()
          .addWarning("No velocity field passed to Advect. Not advecting.")
          .print();
      return std::numeric_limits<double>::max();
    }
 
    prepareLS();
 
    double currentTime = 0.;
    if (integrationScheme == IntegrationSchemeEnum::ENGQUIST_OSHER_1ST_ORDER) {
      auto is = lsInternal::EnquistOsher<T, D, 1>(levelSets.back(), velocities,
                                                  calculateNormalVectors);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::ENGQUIST_OSHER_2ND_ORDER) {
      auto is = lsInternal::EnquistOsher<T, D, 2>(levelSets.back(), velocities,
                                                  calculateNormalVectors);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LAX_FRIEDRICHS_1ST_ORDER) {
      auto alphas = findGlobalAlphas();
      auto is = lsInternal::LaxFriedrichs<T, D, 1>(levelSets.back(), velocities,
                                                   dissipationAlpha, alphas,
                                                   calculateNormalVectors);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LAX_FRIEDRICHS_2ND_ORDER) {
      auto alphas = findGlobalAlphas();
      auto is = lsInternal::LaxFriedrichs<T, D, 2>(levelSets.back(), velocities,
                                                   dissipationAlpha, alphas,
                                                   calculateNormalVectors);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::
                   LOCAL_LAX_FRIEDRICHS_ANALYTICAL_1ST_ORDER) {
      auto is = lsInternal::LocalLaxFriedrichsAnalytical<T, D, 1>(
          levelSets.back(), velocities);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      auto is = lsInternal::LocalLocalLaxFriedrichs<T, D, 1>(
          levelSets.back(), velocities, dissipationAlpha);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LOCAL_LAX_FRIEDRICHS_2ND_ORDER) {
      auto is = lsInternal::LocalLocalLaxFriedrichs<T, D, 2>(
          levelSets.back(), velocities, dissipationAlpha);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      auto is = lsInternal::LocalLaxFriedrichs<T, D, 1>(
          levelSets.back(), velocities, dissipationAlpha);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LAX_FRIEDRICHS_2ND_ORDER) {
      auto is = lsInternal::LocalLaxFriedrichs<T, D, 2>(
          levelSets.back(), velocities, dissipationAlpha);
      currentTime = integrateTime(is, maxTimeStep);
    } else if (integrationScheme ==
               IntegrationSchemeEnum::STENCIL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      auto is = lsInternal::StencilLocalLaxFriedrichsScalar<T, D, 1>(
          levelSets.back(), velocities, dissipationAlpha);
      currentTime = integrateTime(is, maxTimeStep);
    } else {
      Logger::getInstance()
          .addWarning("Advect: Integration scheme not found. Not advecting.")
          .print();
 
      // if no correct scheme was found return infinity
      // to stop advection
      return std::numeric_limits<double>::max();
    }
 
    rebuildLS();
 
    // Adjust all level sets below the advected one
    // This means, that when the top level-set and one below
    // are etched, the lower one is moved with the top level-set
    // TODO: Adjust lower layers also when they have grown,
    // to allow for two different growth rates of materials
    if (integrationScheme !=
        IntegrationSchemeEnum::STENCIL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      for (unsigned i = 0; i < levelSets.size() - 1; ++i) {
        BooleanOperation<T, D>(levelSets[i], levelSets.back(),
                               BooleanOperationEnum::INTERSECT)
            .apply();
      }
    }
 
    return currentTime;
  }

  template <class IntegrationSchemeType>
  double
  integrateTime(IntegrationSchemeType IntegrationScheme,
                double maxTimeStep = std::numeric_limits<double>::max()) {
    if (timeStepRatio >= 0.5) {
      Logger::getInstance()
          .addWarning("Integration time step ratio should be smaller than 0.5. "
                      "Advection might fail!")
          .print();
    }
 
    auto &topDomain = levelSets.back()->getDomain();
    auto &grid = levelSets.back()->getGrid();
 
    std::vector<std::vector<std::pair<std::pair<T, T>, T>>> totalTempRates;
    totalTempRates.resize((levelSets.back())->getNumberOfSegments());
 
    typename PointData<T>::ScalarDataType *voidMarkerPointer;
    if (ignoreVoids) {
      MarkVoidPoints<T, D>(levelSets.back()).apply();
      auto &pointData = levelSets.back()->getPointData();
      voidMarkerPointer =
          pointData.getScalarData(MarkVoidPoints<T, D>::voidPointLabel, true);
      if (voidMarkerPointer == nullptr) {
        Logger::getInstance()
            .addWarning("Advect: Cannot find void point markers. Not "
                        "ignoring void points.")
            .print();
        ignoreVoids = false;
      }
    }
 
    const bool ignoreVoidPoints = ignoreVoids;
 
#pragma omp parallel num_threads((levelSets.back())->getNumberOfSegments())
    {
      int p = 0;
#ifdef _OPENMP
      p = omp_get_thread_num();
#endif
 
      hrleVectorType<hrleIndexType, D> startVector =
          (p == 0) ? grid.getMinGridPoint()
                   : topDomain.getSegmentation()[p - 1];
 
      hrleVectorType<hrleIndexType, D> endVector =
          (p != static_cast<int>(topDomain.getNumberOfSegments() - 1))
              ? topDomain.getSegmentation()[p]
              : grid.incrementIndices(grid.getMaxGridPoint());
 
      double tempMaxTimeStep = maxTimeStep;
      auto &tempRates = totalTempRates[p];
      tempRates.reserve(topDomain.getNumberOfPoints() /
                            double((levelSets.back())->getNumberOfSegments()) +
                        10);
 
      // an iterator for each level set
      std::vector<hrleSparseIterator<typename Domain<T, D>::DomainType>>
          iterators;
      for (auto it = levelSets.begin(); it != levelSets.end(); ++it) {
        iterators.push_back(
            hrleSparseIterator<typename Domain<T, D>::DomainType>(
                (*it)->getDomain()));
      }
 
      IntegrationSchemeType scheme(IntegrationScheme);
 
      for (hrleSparseIterator<typename Domain<T, D>::DomainType> it(
               topDomain, startVector);
           it.getStartIndices() < endVector; ++it) {
 
        if (!it.isDefined() || std::abs(it.getValue()) > 0.5)
          continue;
 
        T value = it.getValue();
        double maxStepTime = 0;
        double cfl = timeStepRatio;
 
        for (int currentLevelSetId = levelSets.size() - 1;
             currentLevelSetId >= 0; --currentLevelSetId) {
 
          std::pair<T, T> gradNDissipation;
 
          if (!(ignoreVoidPoints && (*voidMarkerPointer)[it.getPointId()])) {
            // check if there is any other levelset at the same point:
            // if yes, take the velocity of the lowest levelset
            for (unsigned lowerLevelSetId = 0;
                 lowerLevelSetId < levelSets.size(); ++lowerLevelSetId) {
              // put iterator to same position as the top levelset
              iterators[lowerLevelSetId].goToIndicesSequential(
                  it.getStartIndices());
 
              // if the lower surface is actually outside, i.e. its LS value
              // is lower or equal
              if (iterators[lowerLevelSetId].getValue() <=
                  value + wrappingLayerEpsilon) {
                gradNDissipation =
                    scheme(it.getStartIndices(), lowerLevelSetId);
                break;
              }
            }
          }
 
          T valueBelow;
          // get value of material below (before in levelSets list)
          if (currentLevelSetId > 0) {
            iterators[currentLevelSetId - 1].goToIndicesSequential(
                it.getStartIndices());
            valueBelow = iterators[currentLevelSetId - 1].getValue();
          } else {
            valueBelow = std::numeric_limits<T>::max();
          }
 
          // if velocity is positive, set maximum time step possible without
          // violating the cfl condition
          T velocity = gradNDissipation.first - gradNDissipation.second;
          if (velocity > 0.) {
            maxStepTime += cfl / velocity;
            tempRates.push_back(std::make_pair(gradNDissipation,
                                               -std::numeric_limits<T>::max()));
            break;
            // if velocity is 0, maximum time step is infinite
          } else if (velocity == 0.) {
            maxStepTime = std::numeric_limits<T>::max();
            tempRates.push_back(std::make_pair(gradNDissipation,
                                               std::numeric_limits<T>::max()));
            break;
            // if the velocity is negative apply the velocity for as long as
            // possible without infringing on material below
          } else {
            T difference = std::abs(valueBelow - value);
 
            if (difference >= cfl) {
              maxStepTime -= cfl / velocity;
              tempRates.push_back(std::make_pair(
                  gradNDissipation, std::numeric_limits<T>::max()));
              break;
            } else {
              maxStepTime -= difference / velocity;
              // the second part of the pair indicates how far we can move
              // in this time step until the end of the material is reached
              tempRates.push_back(std::make_pair(gradNDissipation, valueBelow));
              cfl -= difference;
              // use new LS value for next calculations
              value = valueBelow;
            }
          }
        }
 
        if (maxStepTime < tempMaxTimeStep)
          tempMaxTimeStep = maxStepTime;
      }
 
#pragma omp critical
      {
        // If a Lax Friedrichs scheme is selected the time step is
        // reduced depending on the dissipation coefficients
        // For Engquist Osher scheme this function is empty.
        scheme.reduceTimeStepHamiltonJacobi(
            tempMaxTimeStep, levelSets.back()->getGrid().getGridDelta());
 
        // set global timestep maximum
        if (tempMaxTimeStep < maxTimeStep)
          maxTimeStep = tempMaxTimeStep;
      }
    } // end of parallel section
 
    // reduce to one layer thickness and apply new values directly to the
    // domain segments --> DO NOT CHANGE SEGMENTATION HERE (true parameter)
    Reduce<T, D>(levelSets.back(), 1, true).apply();
 
    const bool saveVelocities = saveAdvectionVelocities;
    std::vector<std::vector<double>> dissipationVectors(
        levelSets.back()->getNumberOfSegments());
    std::vector<std::vector<double>> velocityVectors(
        levelSets.back()->getNumberOfSegments());
 
    const bool checkDiss = checkDissipation;
 
#pragma omp parallel num_threads((levelSets.back())->getNumberOfSegments())
    {
      int p = 0;
#ifdef _OPENMP
      p = omp_get_thread_num();
#endif
 
      auto itRS = totalTempRates[p].cbegin();
      auto &segment = topDomain.getDomainSegment(p);
      unsigned maxId = segment.getNumberOfPoints();
 
      if (saveVelocities) {
        velocityVectors[p].resize(maxId);
        dissipationVectors[p].resize(maxId);
      }
 
      for (unsigned localId = 0; localId < maxId; ++localId) {
        T &value = segment.definedValues[localId];
        double time = maxTimeStep;
 
        // if there is a change in materials during one time step, deduct
        // the time taken to advect up to the end of the top material and
        // set the LS value to the one below
        T velocity = itRS->first.first - itRS->first.second;
        if (checkDiss && (itRS->first.first < 0 && velocity > 0) ||
            (itRS->first.first > 0 && velocity < 0)) {
          velocity = 0;
        }
        T rate = time * velocity;
        while (std::abs(itRS->second - value) < std::abs(rate)) {
          time -= std::abs((itRS->second - value) / velocity);
          value = itRS->second;
          ++itRS;
          // recalculate velocity and rate
          velocity = itRS->first.first - itRS->first.second;
          if (checkDiss && (itRS->first.first < 0 && velocity > 0) ||
              (itRS->first.first > 0 && velocity < 0)) {
            velocity = 0;
          }
          rate = time * velocity;
        }
 
        // now deduct the velocity times the time step we take
        value -= rate;
 
        if (saveVelocities) {
          velocityVectors[p][localId] = rate;
          dissipationVectors[p][localId] = itRS->first.second;
        }
 
        // this
        // is run when two materials are close but the velocity is too slow
        // to actually reach the second material, to get rid of the extra
        // entry in the TempRatesStop
        while (std::abs(itRS->second) != std::numeric_limits<T>::max())
          ++itRS;
 
        // advance the TempStopRates iterator by one
        ++itRS;
      }
    } // end of parallel section
 
    if (saveVelocities) {
      auto &pointData = levelSets.back()->getPointData();
 
      typename PointData<T>::ScalarDataType vels;
      typename PointData<T>::ScalarDataType diss;
 
      for (unsigned i = 0; i < velocityVectors.size(); ++i) {
        vels.insert(vels.end(),
                    std::make_move_iterator(velocityVectors[i].begin()),
                    std::make_move_iterator(velocityVectors[i].end()));
        diss.insert(diss.end(),
                    std::make_move_iterator(dissipationVectors[i].begin()),
                    std::make_move_iterator(dissipationVectors[i].end()));
      }
      pointData.insertReplaceScalarData(std::move(vels), velocityLabel);
      pointData.insertReplaceScalarData(std::move(diss), dissipationLabel);
    }
 
    return maxTimeStep;
  }
 
public:
  static constexpr char velocityLabel[] = "AdvectionVelocities";
  static constexpr char dissipationLabel[] = "Dissipation";
 
  Advect() {}
 
  Advect(SmartPointer<Domain<T, D>> passedlsDomain) {
    levelSets.push_back(passedlsDomain);
  }
 
  Advect(SmartPointer<Domain<T, D>> passedlsDomain,
         SmartPointer<VelocityField<T>> passedVelocities) {
    levelSets.push_back(passedlsDomain);
    velocities = passedVelocities;
  }
 
  Advect(std::vector<SmartPointer<Domain<T, D>>> passedlsDomains,
         SmartPointer<VelocityField<T>> passedVelocities)
      : levelSets(passedlsDomains) {
    velocities = passedVelocities;
  }

  void insertNextLevelSet(SmartPointer<Domain<T, D>> passedlsDomain) {
    levelSets.push_back(passedlsDomain);
  }

  void setVelocityField(SmartPointer<VelocityField<T>> passedVelocities) {
    velocities = passedVelocities;
  }

  void setAdvectionTime(double time) { advectionTime = time; }

  void setSingleStep(bool singleStep) { performOnlySingleStep = singleStep; }

  void setTimeStepRatio(const double &cfl) { timeStepRatio = cfl; }

  void setCalculateNormalVectors(bool cnv) { calculateNormalVectors = cnv; }

  void setIgnoreVoids(bool iV) { ignoreVoids = iV; }

  void setSaveAdvectionVelocities(bool sAV) { saveAdvectionVelocities = sAV; }

  double getAdvectedTime() { return advectedTime; }

  unsigned getNumberOfTimeSteps() { return numberOfTimeSteps; }

  double getTimeStepRatio() { return timeStepRatio; }

  bool getCalculateNormalVectors() { return calculateNormalVectors; }

  void setIntegrationScheme(IntegrationSchemeEnum scheme) {
    integrationScheme = scheme;
  }

  void setDissipationAlpha(const double &a) { dissipationAlpha = a; }
 
  // Sets the velocity to 0 if the dissipation is too high
  void setCheckDissipation(bool check) { checkDissipation = check; }

  void setUpdatePointData(bool update) { updatePointData = update; }
 
  // Prepare the levelset for advection, based on the provided integration
  // scheme.
  void prepareLS() {
    // check whether a level set and velocities have been given
    if (levelSets.size() < 1) {
      Logger::getInstance()
          .addWarning("No level sets passed to Advect.")
          .print();
      return;
    }
 
    if (integrationScheme == IntegrationSchemeEnum::ENGQUIST_OSHER_1ST_ORDER) {
      lsInternal::EnquistOsher<T, D, 1>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::ENGQUIST_OSHER_2ND_ORDER) {
      lsInternal::EnquistOsher<T, D, 2>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LAX_FRIEDRICHS_1ST_ORDER) {
      lsInternal::LaxFriedrichs<T, D, 1>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LAX_FRIEDRICHS_2ND_ORDER) {
      lsInternal::LaxFriedrichs<T, D, 2>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::
                   LOCAL_LAX_FRIEDRICHS_ANALYTICAL_1ST_ORDER) {
      lsInternal::LocalLaxFriedrichsAnalytical<T, D, 1>::prepareLS(
          levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      lsInternal::LocalLocalLaxFriedrichs<T, D, 1>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LOCAL_LAX_FRIEDRICHS_2ND_ORDER) {
      lsInternal::LocalLocalLaxFriedrichs<T, D, 2>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      lsInternal::LocalLaxFriedrichs<T, D, 1>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::LOCAL_LAX_FRIEDRICHS_2ND_ORDER) {
      lsInternal::LocalLaxFriedrichs<T, D, 2>::prepareLS(levelSets.back());
    } else if (integrationScheme ==
               IntegrationSchemeEnum::STENCIL_LOCAL_LAX_FRIEDRICHS_1ST_ORDER) {
      lsInternal::StencilLocalLaxFriedrichsScalar<T, D, 1>::prepareLS(
          levelSets.back());
    } else {
      Logger::getInstance()
          .addWarning("Advect: Integration scheme not found. Not advecting.")
          .print();
    }
  }

  void apply() {
    if (advectionTime == 0.) {
      advectedTime = advect();
      numberOfTimeSteps = 1;
    } else {
      double currentTime = 0.0;
      numberOfTimeSteps = 0;
      while (currentTime < advectionTime) {
        currentTime += advect(advectionTime - currentTime);
        ++numberOfTimeSteps;
        if (performOnlySingleStep)
          break;
      }
      advectedTime = currentTime;
    }
  }
};
 
// add all template specializations for this class
PRECOMPILE_PRECISION_DIMENSION(Advect)
 
} // namespace viennals