CellBasedPdeHandler.cpp

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#include "CellBasedPdeHandler.hpp"
#include "MeshBasedCellPopulationWithGhostNodes.hpp"
#include "NodeBasedCellPopulation.hpp"
#include "PottsBasedCellPopulation.hpp"
#include "VertexBasedCellPopulation.hpp"
#include "CaBasedCellPopulation.hpp"
#include "SimpleLinearEllipticSolver.hpp"
#include "CellBasedPdeSolver.hpp"
#include "Exception.hpp"
#include "VtkMeshWriter.hpp"

template<unsigned DIM>
CellBasedPdeHandler<DIM>::CellBasedPdeHandler(AbstractCellPopulation<DIM>* pCellPopulation,
                                              bool deleteMemberPointersInDestructor)
    : mpCellPopulation(pCellPopulation),
      mWriteAverageRadialPdeSolution(false),
      mWriteDailyAverageRadialPdeSolution(false),
      mAverageRadialSolutionVariableName(""),
      mSetBcsOnCoarseBoundary(true),
      mNumRadialIntervals(UNSIGNED_UNSET),
      mpCoarsePdeMesh(NULL),
      mDeleteMemberPointersInDestructor(deleteMemberPointersInDestructor)
{
    // We must be using a CellPopulation with at least one cell
    assert(mpCellPopulation->GetNumRealCells() != 0);
}

template<unsigned DIM>
CellBasedPdeHandler<DIM>::~CellBasedPdeHandler()
{
    /*
     * Avoid memory leaks. Note that we do not take responsibility for
     * deleting mpCellPopulation, as this object is usually owned by a
     * subclass of AbstractCellBasedSimulation, which deletes the cell
     * population upon destruction if restored from an archive.
     */
    if (mDeleteMemberPointersInDestructor)
    {
        for (unsigned i=0; i<mPdeAndBcCollection.size(); i++)
        {
            delete mPdeAndBcCollection[i];
        }
    }
    if (mpCoarsePdeMesh)
    {
        delete mpCoarsePdeMesh;
    }
}

template<unsigned DIM>
const AbstractCellPopulation<DIM>* CellBasedPdeHandler<DIM>::GetCellPopulation() const
{
    return mpCellPopulation;
}

template<unsigned DIM>
TetrahedralMesh<DIM,DIM>* CellBasedPdeHandler<DIM>::GetCoarsePdeMesh()
{
    return mpCoarsePdeMesh;
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::AddPdeAndBc(PdeAndBoundaryConditions<DIM>* pPdeAndBc)
{
    if (!mPdeAndBcCollection.empty() && (pPdeAndBc->rGetDependentVariableName()==""))
    {
        EXCEPTION("When adding more than one PDE to CellBasedPdeHandler set the dependent variable name using SetDependentVariableName(name).");
    }
    for (unsigned i=0; i< mPdeAndBcCollection.size(); i++)
    {
        if (pPdeAndBc->rGetDependentVariableName() == mPdeAndBcCollection[i]->rGetDependentVariableName())
        {
            EXCEPTION("The name "+ pPdeAndBc->rGetDependentVariableName()+ " has already been used in the PDE collection");
        }
    }
    mPdeAndBcCollection.push_back(pPdeAndBc);
}

template<unsigned DIM>
Vec CellBasedPdeHandler<DIM>::GetPdeSolution(const std::string& rName)
{
    if (rName != "")
    {
        for (unsigned i=0; i<mPdeAndBcCollection.size(); i++)
        {
            if (mPdeAndBcCollection[i]->rGetDependentVariableName() == rName)
            {
                return mPdeAndBcCollection[i]->GetSolution();
            }
        }

        EXCEPTION("The PDE collection does not contain a PDE named " + rName);
    }
    else
    {
        assert(mPdeAndBcCollection.size()==1);
        return mPdeAndBcCollection[0]->GetSolution();
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::InitialiseCellPdeElementMap()
{
    if (mpCoarsePdeMesh == NULL)
    {
        EXCEPTION("InitialiseCellPdeElementMap() should only be called if mpCoarsePdeMesh is set up.");
    }

    mCellPdeElementMap.clear();

    // Find the element of mpCoarsePdeMesh that contains each cell and populate mCellPdeElementMap
    for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
         cell_iter != mpCellPopulation->End();
         ++cell_iter)
    {
        const ChastePoint<DIM>& r_position_of_cell = mpCellPopulation->GetLocationOfCellCentre(*cell_iter);
        unsigned elem_index = mpCoarsePdeMesh->GetContainingElementIndex(r_position_of_cell);
        mCellPdeElementMap[*cell_iter] = elem_index;
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::UpdateCellPdeElementMap()
{
    // Find the element of mpCoarsePdeMesh that contains each cell and populate mCellPdeElementMap
    for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
         cell_iter != mpCellPopulation->End();
         ++cell_iter)
    {
        const ChastePoint<DIM>& r_position_of_cell = mpCellPopulation->GetLocationOfCellCentre(*cell_iter);
        unsigned elem_index = mpCoarsePdeMesh->GetContainingElementIndexWithInitialGuess(r_position_of_cell, mCellPdeElementMap[*cell_iter]);
        mCellPdeElementMap[*cell_iter] = elem_index;
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::OpenResultsFiles(std::string outputDirectory)
{
    // If appropriate, make a coarse mesh which exactly overlays the lattice sites of a PottsMesh (used for all OnLattice simulations)
    if ((dynamic_cast<CaBasedCellPopulation<DIM>*>(mpCellPopulation) != NULL) && mpCoarsePdeMesh==NULL)
    {
        assert(DIM ==2);
        ChasteCuboid<DIM> cuboid = mpCellPopulation->rGetMesh().CalculateBoundingBox();

        // Currently only works with square meshes
        assert(cuboid.GetWidth(0) == cuboid.GetWidth(1));

        UseCoarsePdeMesh(1, cuboid, false);
    }

    // If using a NodeBasedCellPopulation a VertexBasedCellPopulation, a CaBasedCellPopulation or a PottsBasedCellPopulation, mpCoarsePdeMesh must be set up
    if (PdeSolveNeedsCoarseMesh() && mpCoarsePdeMesh==NULL)
    {
        EXCEPTION("Trying to solve a PDE on a cell population that doesn't have a mesh. Try calling UseCoarsePdeMesh().");
    }

    if (mpCoarsePdeMesh != NULL)
    {
        InitialiseCellPdeElementMap();

        // Write mesh to file
        TrianglesMeshWriter<DIM,DIM> mesh_writer(outputDirectory+"/coarse_mesh_output", "coarse_mesh",false);
        mesh_writer.WriteFilesUsingMesh(*mpCoarsePdeMesh);
    }

    if (PetscTools::AmMaster())
    {
        OutputFileHandler output_file_handler(outputDirectory+"/", false);

        if (mpCoarsePdeMesh != NULL)
        {
            mpVizPdeSolutionResultsFile = output_file_handler.OpenOutputFile("results.vizcoarsepdesolution");
        }
        else
        {
            mpVizPdeSolutionResultsFile = output_file_handler.OpenOutputFile("results.vizpdesolution");
        }

        if (mWriteAverageRadialPdeSolution)
        {
            mpAverageRadialPdeSolutionResultsFile = output_file_handler.OpenOutputFile("radial_dist.dat");
        }
    }

    mDirPath = outputDirectory; // caching the path to the output directory for VTK
    //double current_time = SimulationTime::Instance()->GetTime();
    //WritePdeSolution(current_time);
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::CloseResultsFiles()
{
    // Close results files
    if (PetscTools::AmMaster())
    {
        mpVizPdeSolutionResultsFile->close();
        if (mWriteAverageRadialPdeSolution)
        {
            WriteAverageRadialPdeSolution(SimulationTime::Instance()->GetTime());
            mpAverageRadialPdeSolutionResultsFile->close();
        }
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::UseCoarsePdeMesh(double stepSize, ChasteCuboid<DIM> meshCuboid, bool centreOnCellPopulation)
{
    if (mPdeAndBcCollection.empty())
    {
        EXCEPTION("mPdeAndBcCollection should be populated prior to calling UseCoarsePdeMesh().");
    }
    // If solving PDEs on a coarse mesh, each PDE must have an averaged source term
    for (unsigned pde_index=0; pde_index<mPdeAndBcCollection.size(); pde_index++)
    {
        if (mPdeAndBcCollection[pde_index]->HasAveragedSourcePde() == false && !dynamic_cast<CaBasedCellPopulation<DIM>*>(mpCellPopulation))
        {
            EXCEPTION("UseCoarsePdeMesh() should only be called if averaged-source PDEs are specified.");
        }
    }

    // Create a regular coarse tetrahedral mesh
    mpCoarsePdeMesh = new TetrahedralMesh<DIM,DIM>;
    switch (DIM)
    {
        case 1:
            mpCoarsePdeMesh->ConstructRegularSlabMesh(stepSize, meshCuboid.GetWidth(0));
            break;
        case 2:
            mpCoarsePdeMesh->ConstructRegularSlabMesh(stepSize, meshCuboid.GetWidth(0), meshCuboid.GetWidth(1));
            break;
        case 3:
            mpCoarsePdeMesh->ConstructRegularSlabMesh(stepSize, meshCuboid.GetWidth(0), meshCuboid.GetWidth(1), meshCuboid.GetWidth(2));
            break;
        default:
            NEVER_REACHED;
    }

    if (centreOnCellPopulation)
    {
        // Find the centre of the coarse PDE mesh
        c_vector<double,DIM> centre_of_coarse_mesh = zero_vector<double>(DIM);
        for (unsigned i=0; i<mpCoarsePdeMesh->GetNumNodes(); i++)
        {
            centre_of_coarse_mesh += mpCoarsePdeMesh->GetNode(i)->rGetLocation();
        }
        centre_of_coarse_mesh /= mpCoarsePdeMesh->GetNumNodes();

        // Translate the centre of coarse PDE mesh to the centre of the cell population
        c_vector<double,DIM> centre_of_cell_population = mpCellPopulation->GetCentroidOfCellPopulation();
        mpCoarsePdeMesh->Translate(centre_of_cell_population - centre_of_coarse_mesh);
    }
    else
    {
        // Get centroid of meshCuboid
        ChastePoint<DIM> upper = meshCuboid.rGetUpperCorner();
        ChastePoint<DIM> lower = meshCuboid.rGetLowerCorner();
        c_vector<double,DIM> centre_of_cuboid = 0.5*(upper.rGetLocation() + lower.rGetLocation());

        // Find the centre of the coarse PDE mesh
        c_vector<double,DIM> centre_of_coarse_mesh = zero_vector<double>(DIM);
        for (unsigned i=0; i<mpCoarsePdeMesh->GetNumNodes(); i++)
        {
            centre_of_coarse_mesh += mpCoarsePdeMesh->GetNode(i)->rGetLocation();
        }
        centre_of_coarse_mesh /= mpCoarsePdeMesh->GetNumNodes();

        mpCoarsePdeMesh->Translate(centre_of_cuboid - centre_of_coarse_mesh);
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::SolvePdeAndWriteResultsToFile(unsigned samplingTimestepMultiple)
{
    // Record whether we are solving PDEs on a coarse mesh
    bool using_coarse_pde_mesh = (mpCoarsePdeMesh != NULL);

    // If solving PDEs on a coarse mesh, each PDE should have an averaged source term; otherwise none should
    assert(!mPdeAndBcCollection.empty());
    for (unsigned pde_index=0; pde_index<mPdeAndBcCollection.size(); pde_index++)
    {
        assert(mPdeAndBcCollection[pde_index]);
        assert(mPdeAndBcCollection[pde_index]->HasAveragedSourcePde() == using_coarse_pde_mesh || dynamic_cast<CaBasedCellPopulation<DIM>*>(mpCellPopulation));
    }

    // Make sure the cell population is in a nice state
    mpCellPopulation->Update();

    // Store a pointer to the (population-level or coarse) mesh
    TetrahedralMesh<DIM,DIM>* p_mesh;
    if (using_coarse_pde_mesh)
    {
        p_mesh = mpCoarsePdeMesh;
    }
    else
    {
        // If not using a coarse PDE mesh, we must be using a MeshBasedCellPopulation
        p_mesh = &(static_cast<MeshBasedCellPopulation<DIM>*>(mpCellPopulation)->rGetMesh());
    }

    // Loop over elements of mPdeAndBcCollection
    for (unsigned pde_index=0; pde_index<mPdeAndBcCollection.size(); pde_index++)
    {
        // Get pointer to this PdeAndBoundaryConditions object
        PdeAndBoundaryConditions<DIM>* p_pde_and_bc = mPdeAndBcCollection[pde_index];

        // Set up boundary conditions
        std::auto_ptr<BoundaryConditionsContainer<DIM,DIM,1> > p_bcc = ConstructBoundaryConditionsContainer(p_pde_and_bc, p_mesh);

        // If the solution at the previous timestep exists...
        PetscInt previous_solution_size = 0;
        if (p_pde_and_bc->GetSolution())
        {
            VecGetSize(p_pde_and_bc->GetSolution(), &previous_solution_size);
        }

        // ...then record whether it is the correct size...
        bool is_previous_solution_size_correct = (previous_solution_size == (int)p_mesh->GetNumNodes());

        // ...and if it is, store it as an initial guess for the PDE solver
        Vec initial_guess;
        if (is_previous_solution_size_correct)
        {
            // This Vec is copied by the solver's Solve() method, so must be deleted here too
            VecDuplicate(p_pde_and_bc->GetSolution(), &initial_guess);
            VecCopy(p_pde_and_bc->GetSolution(), initial_guess);
            p_pde_and_bc->DestroySolution();
        }
        else
        {
            if (!using_coarse_pde_mesh && p_pde_and_bc->GetSolution())
            {
                assert(previous_solution_size != 0);
                p_pde_and_bc->DestroySolution();
            }
        }

        if (using_coarse_pde_mesh)
        {
            // We update mCellPdeElementMap before setting up source terms to speed up
            // finding cells in the case of AveragedSourcePdes. This is also used with
            // non-AveragedSourcePdes when using a CaBasedCellPopulation.
            this->UpdateCellPdeElementMap();
        }

        // Create a PDE solver and solve the PDE on the (population-level or coarse) mesh
        if (p_pde_and_bc->HasAveragedSourcePde())
        {
            // When using a coarse PDE mesh, we must set up the source terms before solving the PDE.
            // Pass in already updated CellPdeElementMap to speed up finding cells.
            p_pde_and_bc->SetUpSourceTermsForAveragedSourcePde(p_mesh, &mCellPdeElementMap);

            SimpleLinearEllipticSolver<DIM,DIM> solver(p_mesh, p_pde_and_bc->GetPde(), p_bcc.get());

            // If we have an initial guess, use this when solving the system...
            if (is_previous_solution_size_correct)
            {
                p_pde_and_bc->SetSolution(solver.Solve(initial_guess));
                PetscTools::Destroy(initial_guess);
            }
            else // ...otherwise do not supply one
            {
                p_pde_and_bc->SetSolution(solver.Solve());
            }
        }
        else
        {
            CellBasedPdeSolver<DIM> solver(p_mesh, p_pde_and_bc->GetPde(), p_bcc.get());

            // If we have an initial guess, use this...
            if (is_previous_solution_size_correct)
            {
                p_pde_and_bc->SetSolution(solver.Solve(initial_guess));
                PetscTools::Destroy(initial_guess);
            }
            else // ...otherwise do not supply one
            {
                p_pde_and_bc->SetSolution(solver.Solve());
            }
        }

        // Store the PDE solution in an accessible form
        ReplicatableVector solution_repl(p_pde_and_bc->GetSolution());

        // Having solved the PDE, now update CellData
        for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
             cell_iter != mpCellPopulation->End();
             ++cell_iter)
        {
            unsigned node_index = mpCellPopulation->GetLocationIndexUsingCell(*cell_iter);
            double solution_at_node = 0.0;

            if (using_coarse_pde_mesh)
            {
                // When using a coarse PDE mesh, the cells are not nodes of the mesh, so we must interpolate

                // Find the element in the coarse mesh that contains this cell. CellElementMap has been updated so use this.
                unsigned elem_index = mCellPdeElementMap[*cell_iter];
                Element<DIM,DIM>* p_element = mpCoarsePdeMesh->GetElement(elem_index);

                const ChastePoint<DIM>& node_location = mpCellPopulation->GetLocationOfCellCentre(*cell_iter);

                c_vector<double,DIM+1> weights = p_element->CalculateInterpolationWeights(node_location);
                for (unsigned i=0; i<DIM+1; i++)
                {
                    double nodal_value = solution_repl[p_element->GetNodeGlobalIndex(i)];
                    solution_at_node += nodal_value * weights(i);
                }
            }
            else
            {
                solution_at_node = solution_repl[node_index];
            }
            cell_iter->GetCellData()->SetItem(mPdeAndBcCollection[pde_index]->rGetDependentVariableName(), solution_at_node);
        }
    }

    // Write results to file if required
    SimulationTime* p_time = SimulationTime::Instance();
    if ((p_time->GetTimeStepsElapsed())%samplingTimestepMultiple == 0)
    {
        WritePdeSolution(p_time->GetTime());
    }
#define COVERAGE_IGNORE
    if (!using_coarse_pde_mesh)
    {
        if (mWriteDailyAverageRadialPdeSolution)
        {
            p_time = SimulationTime::Instance();
            unsigned num_timesteps_per_day = (unsigned) (DBL_EPSILON + 24/SimulationTime::Instance()->GetTimeStep());
            if ((p_time->GetTimeStepsElapsed()) % num_timesteps_per_day == 0)
            {
                WriteAverageRadialPdeSolution(p_time->GetTime());
            }
        }
    }
#undef COVERAGE_IGNORE
}

template<unsigned DIM>
std::auto_ptr<BoundaryConditionsContainer<DIM,DIM,1> > CellBasedPdeHandler<DIM>::ConstructBoundaryConditionsContainer(
        PdeAndBoundaryConditions<DIM>* pPdeAndBc,
        TetrahedralMesh<DIM,DIM>* pMesh)
{
    std::auto_ptr<BoundaryConditionsContainer<DIM,DIM,1> > p_bcc(new BoundaryConditionsContainer<DIM,DIM,1>(false));

    AbstractBoundaryCondition<DIM>* p_bc = pPdeAndBc->GetBoundaryCondition();

    if (pPdeAndBc->IsNeumannBoundaryCondition()) // this BC is of Neumann type
    {
        // Note p_mesh is the coarse mesh or the natural mesh as appropriate
        for (typename TetrahedralMesh<DIM,DIM>::BoundaryElementIterator elem_iter = pMesh->GetBoundaryElementIteratorBegin();
             elem_iter != pMesh->GetBoundaryElementIteratorEnd();
             ++elem_iter)
        {
            p_bcc->AddNeumannBoundaryCondition(*elem_iter, p_bc);
        }
    }
    else // assume that if the BC is not of Neumann type, then it is Dirichlet
    {
        bool using_coarse_pde_mesh = (mpCoarsePdeMesh != NULL);

        if (using_coarse_pde_mesh && !mSetBcsOnCoarseBoundary)
        {
            // Get the set of coarse element indices that contain cells
            std::set<unsigned> coarse_element_indices_in_map;
            for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
                 cell_iter != mpCellPopulation->End();
                 ++cell_iter)
            {
                coarse_element_indices_in_map.insert(mCellPdeElementMap[*cell_iter]);
            }

            // Find the node indices associated with elements whose indices are NOT in the set coarse_element_indices_in_map
            std::set<unsigned> coarse_mesh_boundary_node_indices;
            for (unsigned i=0; i<pMesh->GetNumElements(); i++)
            {
                if (coarse_element_indices_in_map.find(i) == coarse_element_indices_in_map.end())
                {
                    Element<DIM,DIM>* p_element = pMesh->GetElement(i);
                    for (unsigned j=0; j<DIM+1; j++)
                    {
                        unsigned node_index = p_element->GetNodeGlobalIndex(j);
                        coarse_mesh_boundary_node_indices.insert(node_index);
                    }
                }
            }

            // Apply boundary condition to the nodes in the set coarse_mesh_boundary_node_indices
            for (std::set<unsigned>::iterator iter = coarse_mesh_boundary_node_indices.begin();
                 iter != coarse_mesh_boundary_node_indices.end();
                 ++iter)
            {
                p_bcc->AddDirichletBoundaryCondition(pMesh->GetNode(*iter), p_bc, 0, false);
            }
        }
        else // apply BC at boundary nodes of (population-level or coarse) mesh
        {
            for (typename TetrahedralMesh<DIM,DIM>::BoundaryNodeIterator node_iter = pMesh->GetBoundaryNodeIteratorBegin();
                 node_iter != pMesh->GetBoundaryNodeIteratorEnd();
                 ++node_iter)
            {
                p_bcc->AddDirichletBoundaryCondition(*node_iter, p_bc);
            }
        }
    }

    return p_bcc;
}

template<unsigned DIM>
unsigned CellBasedPdeHandler<DIM>::FindCoarseElementContainingCell(CellPtr pCell)
{
    // Get containing element at last timestep from mCellPdeElementMap
    unsigned old_element_index = mCellPdeElementMap[pCell];

    // Create a std::set of guesses for the current containing element
    std::set<unsigned> test_elements;
    test_elements.insert(old_element_index);

    Element<DIM,DIM>* p_element = mpCoarsePdeMesh->GetElement(old_element_index);
    for (unsigned local_index=0; local_index<DIM+1; local_index++)
    {
        std::set<unsigned> element_indices = p_element->GetNode(local_index)->rGetContainingElementIndices();
        for (std::set<unsigned>::iterator iter = element_indices.begin();
             iter != element_indices.end();
             ++iter)
        {
            test_elements.insert(*iter);
        }
    }

    // Find new element, using the previous one as a guess
    const ChastePoint<DIM>& r_cell_position = mpCellPopulation->GetLocationOfCellCentre(pCell);
    unsigned new_element_index = mpCoarsePdeMesh->GetContainingElementIndex(r_cell_position, false, test_elements);

    // Update mCellPdeElementMap
    mCellPdeElementMap[pCell] = new_element_index;

    return new_element_index;
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::WritePdeSolution(double time)
{
    if (PetscTools::AmMaster())
    {
        (*mpVizPdeSolutionResultsFile) << time << "\t";

#ifdef CHASTE_VTK
        // Note that this mesh writer is only constructed and used if mpCoarsePdeMesh exists
        VtkMeshWriter<DIM,DIM>* p_vtk_mesh_writer = NULL;
        if (DIM>1 && mpCoarsePdeMesh != NULL )
        {
            std::ostringstream time_string;
            time_string << SimulationTime::Instance()->GetTimeStepsElapsed()+1;
            std::string results_file = "pde_results_"+time_string.str();
            // Note that this mesh writer is always constructed, but is only used if mpCoarsePdeMesh exists
            p_vtk_mesh_writer = new VtkMeshWriter<DIM,DIM>(mDirPath, results_file, false);
        }
#endif //CHASTE_VTK
        for (unsigned pde_index=0; pde_index<mPdeAndBcCollection.size(); pde_index++)
        {
            if (mpCoarsePdeMesh != NULL)
            {
                PdeAndBoundaryConditions<DIM>* p_pde_and_bc = mPdeAndBcCollection[pde_index];
                assert(p_pde_and_bc->rGetDependentVariableName() != "");

#ifdef CHASTE_VTK
                if (p_pde_and_bc->GetSolution())
                {
                    if (DIM>1)
                    {
                        ReplicatableVector solution_repl(p_pde_and_bc->GetSolution());
                        std::vector<double> pde_solution;
                        for (unsigned i=0; i<mpCoarsePdeMesh->GetNumNodes(); i++)
                        {
                            pde_solution.push_back(solution_repl[i]);
                        }

                        p_vtk_mesh_writer->AddPointData(p_pde_and_bc->rGetDependentVariableName(),pde_solution);
                    }
                }

#endif //CHASTE_VTK

                for (unsigned i=0; i<mpCoarsePdeMesh->GetNumNodes(); i++)
                {
                    (*mpVizPdeSolutionResultsFile) << i << " ";
                    c_vector<double,DIM> location = mpCoarsePdeMesh->GetNode(i)->rGetLocation();
                    for (unsigned k=0; k<DIM; k++)
                    {
                        (*mpVizPdeSolutionResultsFile) << location[k] << " ";
                    }

                    if (p_pde_and_bc->GetSolution())
                    {
                        ReplicatableVector solution_repl(p_pde_and_bc->GetSolution());
                        (*mpVizPdeSolutionResultsFile) << solution_repl[i] << " ";
                    }
                    else
                    {
                        // should only come into this method AFTER solving the PDE
                        NEVER_REACHED;
                    }
                }
            }
            else // Not coarse mesh
            {
                for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
                     cell_iter != mpCellPopulation->End();
                     ++cell_iter)
                {
                    unsigned node_index = mpCellPopulation->GetLocationIndexUsingCell(*cell_iter);
                    (*mpVizPdeSolutionResultsFile) << node_index << " ";
                    const c_vector<double,DIM>& position = mpCellPopulation->GetLocationOfCellCentre(*cell_iter);
                    for (unsigned i=0; i<DIM; i++)
                    {
                        (*mpVizPdeSolutionResultsFile) << position[i] << " ";
                    }
                    double solution = cell_iter->GetCellData()->GetItem(mPdeAndBcCollection[pde_index]->rGetDependentVariableName());
                    (*mpVizPdeSolutionResultsFile) << solution << " ";
                }
            }
        }
        (*mpVizPdeSolutionResultsFile) << "\n";
#ifdef CHASTE_VTK
        if (p_vtk_mesh_writer != NULL)
        {
            p_vtk_mesh_writer->WriteFilesUsingMesh(*mpCoarsePdeMesh);
            delete p_vtk_mesh_writer;
        }
#endif //CHASTE_VTK
    }
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::SetWriteAverageRadialPdeSolution(const std::string& rName, unsigned numRadialIntervals, bool writeDailyResults)
{
    mAverageRadialSolutionVariableName = rName;
    mWriteAverageRadialPdeSolution = true;
    mNumRadialIntervals = numRadialIntervals;
    mWriteDailyAverageRadialPdeSolution = writeDailyResults;
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::SetImposeBcsOnCoarseBoundary(bool setBcsOnCoarseBoundary)
{
    mSetBcsOnCoarseBoundary = setBcsOnCoarseBoundary;
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::WriteAverageRadialPdeSolution(double time)
{
    (*mpAverageRadialPdeSolutionResultsFile) << time << " ";

    // Calculate the centre of the cell population
    c_vector<double,DIM> centre = mpCellPopulation->GetCentroidOfCellPopulation();

    // Calculate the distance between each node and the centre of the cell population, as well as the maximum of these
    std::map<double, CellPtr> radius_cell_map;
    double max_distance_from_centre = 0.0;
    for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = mpCellPopulation->Begin();
         cell_iter != mpCellPopulation->End();
         ++cell_iter)
    {
        double distance = norm_2(mpCellPopulation->GetLocationOfCellCentre(*cell_iter) - centre);
        radius_cell_map[distance] = *cell_iter;

        if (distance > max_distance_from_centre)
        {
            max_distance_from_centre = distance;
        }
    }

    // Create vector of radius intervals
    std::vector<double> radius_intervals;
    for (unsigned i=0; i<mNumRadialIntervals; i++)
    {
        double upper_radius = max_distance_from_centre*((double) i+1)/((double) mNumRadialIntervals);
        radius_intervals.push_back(upper_radius);
    }

    // Calculate PDE solution in each radial interval
    double lower_radius = 0.0;
    for (unsigned i=0; i<mNumRadialIntervals; i++)
    {
        unsigned counter = 0;
        double average_solution = 0.0;

        for (std::map<double, CellPtr>::iterator iter = radius_cell_map.begin(); iter != radius_cell_map.end(); ++iter)
        {
            if (iter->first > lower_radius && iter->first <= radius_intervals[i])
            {
                average_solution += (iter->second)->GetCellData()->GetItem(mAverageRadialSolutionVariableName);
                counter++;
            }
        }
        if (counter != 0)
        {
            average_solution /= (double) counter;
        }

        // Write results to file
        (*mpAverageRadialPdeSolutionResultsFile) << radius_intervals[i] << " " << average_solution << " ";
        lower_radius = radius_intervals[i];
    }
    (*mpAverageRadialPdeSolutionResultsFile) << "\n";
}

template<unsigned DIM>
double CellBasedPdeHandler<DIM>::GetPdeSolutionAtPoint(const c_vector<double,DIM>& rPoint, const std::string& rVariable)
{
    double solution_at_point = 0.0;

    unsigned pde_index = UINT_MAX;

    // Loop over elements of mPdeAndBcCollection to find correct PDE
    for (unsigned i=0; i<mPdeAndBcCollection.size(); i++)
    {
        if (mPdeAndBcCollection[i]->rGetDependentVariableName() == rVariable)
        {
            pde_index = i;
            break;
        }
    }
    if (pde_index == UINT_MAX)
    {
        EXCEPTION("Tried to get the solution of a variable name: " + rVariable + ". There is no PDE with that variable.");
    }
    PdeAndBoundaryConditions<DIM>* p_pde_and_bc = mPdeAndBcCollection[pde_index];

    Element<DIM,DIM>* p_containing_element;

    if (mpCoarsePdeMesh != NULL)
    {
        // Find PDE element containing point
        unsigned elem_index = mpCoarsePdeMesh->GetContainingElementIndex(ChastePoint<DIM>(rPoint));
        p_containing_element = mpCoarsePdeMesh->GetElement(elem_index);
    }
    else // Tetrahedral mesh
    {
        // If not using a coarse PDE mesh, we must be using a MeshBasedCellPopulation
        TetrahedralMesh<DIM,DIM>* p_tetrahedral_mesh = &(static_cast<MeshBasedCellPopulation<DIM>*>(mpCellPopulation)->rGetMesh());

        unsigned elem_index = p_tetrahedral_mesh->GetContainingElementIndex(ChastePoint<DIM>(rPoint));
        p_containing_element = p_tetrahedral_mesh->GetElement(elem_index);
    }

    // Interpolate solution
    if (p_pde_and_bc->GetSolution())
    {
        ReplicatableVector solution_repl(p_pde_and_bc->GetSolution());
        c_vector<double,DIM+1> weights = p_containing_element->CalculateInterpolationWeights(rPoint);
        for (unsigned i=0; i<DIM+1; i++)
        {
            double nodal_value = solution_repl[p_containing_element->GetNodeGlobalIndex(i)];
            solution_at_point += nodal_value * weights(i);
        }
    }
    else
    {
        // should only come into this method AFTER solving the PDE
        NEVER_REACHED;
    }

    return solution_at_point;
}

template<unsigned DIM>
bool CellBasedPdeHandler<DIM>::GetWriteAverageRadialPdeSolution()
{
    return mWriteAverageRadialPdeSolution;
}

template<unsigned DIM>
bool CellBasedPdeHandler<DIM>::GetWriteDailyAverageRadialPdeSolution()
{
    return mWriteDailyAverageRadialPdeSolution;
}

template<unsigned DIM>
bool CellBasedPdeHandler<DIM>::GetImposeBcsOnCoarseBoundary()
{
    return mSetBcsOnCoarseBoundary;
}

template<unsigned DIM>
unsigned CellBasedPdeHandler<DIM>::GetNumRadialIntervals()
{
    return mNumRadialIntervals;
}

template<unsigned DIM>
void CellBasedPdeHandler<DIM>::OutputParameters(out_stream& rParamsFile)
{
    std::string type = GetIdentifier();

    *rParamsFile << "\t\t<" << type << ">\n";
    *rParamsFile << "\t\t<WriteAverageRadialPdeSolution>" << mWriteAverageRadialPdeSolution << "</WriteAverageRadialPdeSolution>\n";
    *rParamsFile << "\t\t<WriteDailyAverageRadialPdeSolution>" << mWriteDailyAverageRadialPdeSolution << "</WriteDailyAverageRadialPdeSolution>\n";
    *rParamsFile << "\t\t<SetBcsOnCoarseBoundary>" << mSetBcsOnCoarseBoundary << "</SetBcsOnCoarseBoundary>\n";
    *rParamsFile << "\t\t<NumRadialIntervals>" << mNumRadialIntervals << "</NumRadialIntervals>\n";
    *rParamsFile << "\t\t</" << type << ">\n";
}

template<unsigned DIM>
bool CellBasedPdeHandler<DIM>::PdeSolveNeedsCoarseMesh()
{
    return ((dynamic_cast<NodeBasedCellPopulation<DIM>*>(mpCellPopulation) != NULL)
            || (dynamic_cast<PottsBasedCellPopulation<DIM>*>(mpCellPopulation) != NULL)
            || (dynamic_cast<CaBasedCellPopulation<DIM>*>(mpCellPopulation) != NULL)
            || (dynamic_cast<VertexBasedCellPopulation<DIM>*>(mpCellPopulation) != NULL));
}

// Serialization for Boost >= 1.36
#include "SerializationExportWrapperForCpp.hpp"
EXPORT_TEMPLATE_CLASS_SAME_DIMS(CellBasedPdeHandler)

template class CellBasedPdeHandler<1>;
template class CellBasedPdeHandler<2>;
template class CellBasedPdeHandler<3>;