VertexMesh.cpp

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*/
 
#include "VertexMesh.hpp"
#include "MutableMesh.hpp"
#include "RandomNumberGenerator.hpp"
#include "UblasCustomFunctions.hpp"
 
#include "VtkMeshWriter.hpp"
#include "NodesOnlyMesh.hpp"
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexMesh<ELEMENT_DIM, SPACE_DIM>::VertexMesh(std::vector<Node<SPACE_DIM>*> nodes,
                                               std::vector<VertexElement<ELEMENT_DIM, SPACE_DIM>*> vertexElements)
        : mpDelaunayMesh(nullptr)
{
 
    // Reset member variables and clear mNodes and mElements
    Clear();
 
    // Populate mNodes and mElements
    for (unsigned node_index = 0; node_index < nodes.size(); node_index++)
    {
        Node<SPACE_DIM>* p_temp_node = nodes[node_index];
        this->mNodes.push_back(p_temp_node);
    }
    for (unsigned elem_index = 0; elem_index < vertexElements.size(); elem_index++)
    {
        VertexElement<ELEMENT_DIM, SPACE_DIM>* p_temp_vertex_element = vertexElements[elem_index];
        mElements.push_back(p_temp_vertex_element);
    }
 
    // In 3D, populate mFaces
    if (SPACE_DIM == 3)
    {
        // Use a std::set to keep track of which faces have been added to mFaces
        std::set<unsigned> faces_counted;
 
        // Loop over mElements
        for (unsigned elem_index = 0; elem_index < mElements.size(); elem_index++)
        {
            // Loop over faces of this element
            for (unsigned face_index = 0; face_index < mElements[elem_index]->GetNumFaces(); face_index++)
            {
                VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* p_face = mElements[elem_index]->GetFace(face_index);
                unsigned global_index = p_face->GetIndex();
 
                // If this face is not already contained in mFaces, add it, and update faces_counted
                if (faces_counted.find(global_index) == faces_counted.end())
                {
                    mFaces.push_back(p_face);
                    faces_counted.insert(global_index);
                }
            }
        }
    }
 
    // Register elements with nodes
    for (unsigned index = 0; index < mElements.size(); index++)
    {
        VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = mElements[index];
 
        unsigned element_index = p_element->GetIndex();
        unsigned num_nodes_in_element = p_element->GetNumNodes();
 
        for (unsigned node_index = 0; node_index < num_nodes_in_element; node_index++)
        {
            p_element->GetNode(node_index)->AddElement(element_index);
        }
    }
 
    this->GenerateEdgesFromElements(mElements);
 
    this->mMeshChangesDuringSimulation = false;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexMesh<ELEMENT_DIM, SPACE_DIM>::VertexMesh(std::vector<Node<SPACE_DIM>*> nodes,
                                               std::vector<VertexElement<ELEMENT_DIM - 1, SPACE_DIM>*> faces,
                                               std::vector<VertexElement<ELEMENT_DIM, SPACE_DIM>*> vertexElements)
        : mpDelaunayMesh(nullptr)
{
    // Reset member variables and clear mNodes, mFaces and mElements
    Clear();
 
    // Populate mNodes mFaces and mElements
    for (unsigned node_index = 0; node_index < nodes.size(); node_index++)
    {
        Node<SPACE_DIM>* p_temp_node = nodes[node_index];
        this->mNodes.push_back(p_temp_node);
    }
 
    for (unsigned face_index = 0; face_index < faces.size(); face_index++)
    {
        VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* p_temp_face = faces[face_index];
        mFaces.push_back(p_temp_face);
    }
 
    for (unsigned elem_index = 0; elem_index < vertexElements.size(); elem_index++)
    {
        VertexElement<ELEMENT_DIM, SPACE_DIM>* p_temp_vertex_element = vertexElements[elem_index];
        mElements.push_back(p_temp_vertex_element);
    }
 
    // Register elements with nodes
    for (unsigned index = 0; index < mElements.size(); index++)
    {
        VertexElement<ELEMENT_DIM, SPACE_DIM>* p_temp_vertex_element = mElements[index];
        for (unsigned node_index = 0; node_index < p_temp_vertex_element->GetNumNodes(); node_index++)
        {
            Node<SPACE_DIM>* p_temp_node = p_temp_vertex_element->GetNode(node_index);
            p_temp_node->AddElement(p_temp_vertex_element->GetIndex());
        }
    }
 
    this->mMeshChangesDuringSimulation = false;
}
 
template <>
VertexMesh<2, 2>::VertexMesh(TetrahedralMesh<2, 2>& rMesh, 
                             bool isPeriodic, 
                             bool isBounded, 
                             bool scaleBoundByEdgeLength, 
                             double maxDelaunayEdgeLength,
                             bool offsetNewBoundaryNodes)
        : mpDelaunayMesh(&rMesh)
{
    //Note  !isPeriodic is not used except through polymorphic calls in rMesh
 
    // Reset member variables and clear mNodes, mFaces and mElements
    Clear();
 
    if (!isBounded)
    {
        unsigned num_elements = mpDelaunayMesh->GetNumAllNodes();
        unsigned num_nodes = mpDelaunayMesh->GetNumAllElements();
 
        // Allocate memory for mNodes and mElements
        this->mNodes.reserve(num_nodes);
 
        // Create as many elements as there are nodes in the mesh
        mElements.reserve(num_elements);
        for (unsigned elem_index = 0; elem_index < num_elements; elem_index++)
        {
            VertexElement<2, 2>* p_element = new VertexElement<2, 2>(elem_index);
            mElements.push_back(p_element);
        }
 
        // Populate mNodes
        GenerateVerticesFromElementCircumcentres(rMesh);
 
        // Loop over elements of the Delaunay mesh (which are nodes/vertices of this mesh)
        for (unsigned i = 0; i < num_nodes; i++)
        {
            // Loop over nodes owned by this triangular element in the Delaunay mesh
            // Add this node/vertex to each of the 3 vertex elements
            for (unsigned local_index = 0; local_index < 3; local_index++)
            {
                unsigned elem_index = mpDelaunayMesh->GetElement(i)->GetNodeGlobalIndex(local_index);
                unsigned num_nodes_in_elem = mElements[elem_index]->GetNumNodes();
                unsigned end_index = num_nodes_in_elem > 0 ? num_nodes_in_elem - 1 : 0;
 
                mElements[elem_index]->AddNode(this->mNodes[i], end_index);
            }
        }
    }
    else // Is Bounded
    {
        // First create an extended mesh to include points extended from the boundary
        std::vector<Node<2> *> nodes;
        for (typename TetrahedralMesh<2,2>::NodeIterator node_iter = mpDelaunayMesh->GetNodeIteratorBegin();
            node_iter != mpDelaunayMesh->GetNodeIteratorEnd();
            ++node_iter)
        {
            nodes.push_back(new Node<2>(node_iter->GetIndex(), node_iter->rGetLocation(),node_iter->IsBoundaryNode()));
        }
 
        // Add new nodes
        unsigned new_node_index = mpDelaunayMesh->GetNumNodes();
        
        // Lop over elements to work out boundary edges
        for (TetrahedralMesh<2,2>::ElementIterator elem_iter = mpDelaunayMesh->GetElementIteratorBegin();
            elem_iter != mpDelaunayMesh->GetElementIteratorEnd();
            ++elem_iter)
        {
            bool bad_element = false;
            
            for (unsigned j=0; j<3; j++)
            {
                Node<2>* p_node_a = mpDelaunayMesh->GetNode(elem_iter->GetNodeGlobalIndex(j));
                Node<2>* p_node_b = mpDelaunayMesh->GetNode(elem_iter->GetNodeGlobalIndex((j+1)%3));
                if (norm_2(mpDelaunayMesh->GetVectorFromAtoB(p_node_a->rGetLocation(), p_node_b->rGetLocation()))>maxDelaunayEdgeLength)
                {
                    bad_element = true;
                    break;
                }
            }
 
            for (unsigned j=0; j<3; j++)
            {
                Node<2>* p_node_a = mpDelaunayMesh->GetNode(elem_iter->GetNodeGlobalIndex(j));
                Node<2>* p_node_b = mpDelaunayMesh->GetNode(elem_iter->GetNodeGlobalIndex((j+1)%3));
 
                std::set<unsigned> node_a_element_indices = p_node_a->rGetContainingElementIndices();
                std::set<unsigned> node_b_element_indices = p_node_b->rGetContainingElementIndices();
 
                std::set<unsigned> shared_elements;
                std::set_intersection(node_a_element_indices.begin(),
                                      node_a_element_indices.end(),
                                      node_b_element_indices.begin(),
                                      node_b_element_indices.end(),
                                      std::inserter(shared_elements, shared_elements.begin()));
 
                c_vector<double,2> edge = p_node_b->rGetLocation() - p_node_a->rGetLocation();
                double edge_length = norm_2(edge);
 
                /*
                 * Note using boundary nodes to identify the boundary edges won't work with
                 * triangles which have 3 boundary nodes
                 * if ((p_node_a->IsBoundaryNode() && p_node_b->IsBoundaryNode()))
                 */
                bool is_boundary_edge = false;
                double direction_of_normal = 1.0;
                if ((edge_length < maxDelaunayEdgeLength) && (shared_elements.size() == 1)) // its a boundary edge
                {
                    is_boundary_edge = true;
                }
                if (bad_element && (edge_length < maxDelaunayEdgeLength))
                {
                    /*
                    * Here one or more of the edges in the element is longer than maxDelaunayEdgeLength so the other edges are boundary edges
                    */
                   
                    assert(!is_boundary_edge); // We shouldnt have short edged which are in 2 elements.
                    is_boundary_edge = true;
                    // Here we're pointing in to the element
                    direction_of_normal = -1.0;
                } 
 
                if (is_boundary_edge)
                {  
                    c_vector<double,2> normal_vector;
 
                    normal_vector[0]= edge[1];
                    normal_vector[1]= -edge[0];
 
                    double dij = norm_2(normal_vector);
                    assert(dij>1e-5); //Sanity check
                    normal_vector /= dij;
 
                    double new_node_distance = 1.0;
 
                    if (scaleBoundByEdgeLength)
                    {
                        new_node_distance = edge_length;
                    }
 
                    int num_sections = 1;
                    for (int section=0; section<=num_sections; section++)
                    {
                        double ratio;
                        if (offsetNewBoundaryNodes)
                        {
                            ratio = ((double)section+0.5)/((double)num_sections+1);
                        }
                        else 
                        {
                            ratio = ((double)section)/((double)num_sections);
                        }
                        
                        assert(ratio>=0.0);
                        assert(ratio<=1.0);
                        c_vector<double,2> new_node_location = direction_of_normal * new_node_distance * normal_vector + ratio*p_node_a->rGetLocation() + (1-ratio)*p_node_b->rGetLocation();
                        
                        //Check if near other nodes (could be inefficient)
                        double node_clearance = 0.01;
                        if (!IsNearExistingNodes(new_node_location,nodes,node_clearance))
                        {
                            nodes.push_back(new Node<2>(new_node_index, new_node_location));
                            new_node_index++;
                        }
                    }
                }
            }
        }
        
// // Plot new nodes 
// NodesOnlyMesh<2> temp_mesh;
// temp_mesh.ConstructNodesWithoutMesh(nodes, 1.0);
// VtkMeshWriter<2, 2> mesh_writer("tempMesh", "ExtendedMesh", false);
// mesh_writer.WriteFilesUsingMesh(temp_mesh);
 
        MutableMesh<2,2> extended_mesh(nodes);
 
        unsigned num_elements = mpDelaunayMesh->GetNumAllNodes();
        unsigned num_nodes = extended_mesh.GetNumAllElements();
 
        // Allocate memory for mNodes and mElements
        this->mNodes.reserve(num_nodes);
 
        // Create as many elements as there are nodes in the mesh
        mElements.reserve(num_elements);
        for (unsigned elem_index = 0; elem_index < num_elements; elem_index++)
        {
            VertexElement<2, 2>* p_element = new VertexElement<2, 2>(elem_index);
            mElements.push_back(p_element);
        }
 
        // Populate mNodes
        GenerateVerticesFromElementCircumcentres(extended_mesh);
 
        // Loop over elements of the Delaunay mesh (which are nodes/vertices of this mesh)
        for (unsigned i = 0; i < num_nodes; i++)
        {
            // Loop over nodes owned by this triangular element in the Delaunay mesh
            // Add this node/vertex to each of the 3 vertex elements
            for (unsigned local_index = 0; local_index < 3; local_index++)
            {
                unsigned elem_index = extended_mesh.GetElement(i)->GetNodeGlobalIndex(local_index);
 
                if (elem_index < num_elements)
                {
                    unsigned num_nodes_in_elem = mElements[elem_index]->GetNumNodes();
                    unsigned end_index = num_nodes_in_elem > 0 ? num_nodes_in_elem - 1 : 0;
 
                    mElements[elem_index]->AddNode(this->mNodes[i], end_index);
                }
            }
        }
    }
 
    // Reorder mNodes anticlockwise
    for (unsigned elem_index = 0; elem_index < mElements.size(); elem_index++)
    {
        std::vector<std::pair<double, unsigned> > index_angle_list;
        for (unsigned local_index = 0; local_index < mElements[elem_index]->GetNumNodes(); local_index++)
        {
            c_vector<double, 2> vectorA = mpDelaunayMesh->GetNode(elem_index)->rGetLocation();
            c_vector<double, 2> vectorB = mElements[elem_index]->GetNodeLocation(local_index);
            c_vector<double, 2> centre_to_vertex = mpDelaunayMesh->GetVectorFromAtoB(vectorA, vectorB);
 
            double angle = atan2(centre_to_vertex(1), centre_to_vertex(0));
            unsigned global_index = mElements[elem_index]->GetNodeGlobalIndex(local_index);
 
            std::pair<double, unsigned> pair(angle, global_index);
            index_angle_list.push_back(pair);
        }
 
        // Sort the list in order of increasing angle
        sort(index_angle_list.begin(), index_angle_list.end());
 
        // Create a new Voronoi element and pass in the appropriate Nodes, ordered anticlockwise
        VertexElement<2, 2>* p_new_element = new VertexElement<2, 2>(elem_index);
        for (unsigned count = 0; count < index_angle_list.size(); count++)
        {
            unsigned local_index = count > 1 ? count - 1 : 0;
            p_new_element->AddNode(mNodes[index_angle_list[count].second], local_index);
        }
 
        // Replace the relevant member of mElements with this Voronoi element
        delete mElements[elem_index];
        mElements[elem_index] = p_new_element;
    }
 
    this->mMeshChangesDuringSimulation = false;
}
 
template <>
VertexMesh<3, 3>::VertexMesh(TetrahedralMesh<3, 3>& rMesh)
        : mpDelaunayMesh(&rMesh)
{
    // Reset member variables and clear mNodes, mFaces and mElements
    Clear();
 
    unsigned num_nodes = mpDelaunayMesh->GetNumAllElements();
 
    // Allocate memory for mNodes
    this->mNodes.reserve(num_nodes);
 
    // Populate mNodes
    GenerateVerticesFromElementCircumcentres(rMesh);
 
    std::map<unsigned, VertexElement<3, 3>*> index_element_map;
    unsigned face_index = 0;
    unsigned element_index = 0;
 
    // Loop over each edge of the Delaunay mesh and populate mFaces and mElements
    for (TetrahedralMesh<3, 3>::EdgeIterator edge_iterator = mpDelaunayMesh->EdgesBegin();
         edge_iterator != mpDelaunayMesh->EdgesEnd();
         ++edge_iterator)
    {
        Node<3>* p_node_a = edge_iterator.GetNodeA();
        Node<3>* p_node_b = edge_iterator.GetNodeB();
 
        if (!(p_node_a->IsBoundaryNode() && p_node_b->IsBoundaryNode()))
        {
            std::set<unsigned>& node_a_element_indices = p_node_a->rGetContainingElementIndices();
            std::set<unsigned>& node_b_element_indices = p_node_b->rGetContainingElementIndices();
            std::set<unsigned> edge_element_indices;
 
            std::set_intersection(node_a_element_indices.begin(),
                                  node_a_element_indices.end(),
                                  node_b_element_indices.begin(),
                                  node_b_element_indices.end(),
                                  std::inserter(edge_element_indices, edge_element_indices.begin()));
 
            c_vector<double, 3> edge_vector;
            edge_vector = p_node_b->rGetLocation() - p_node_a->rGetLocation();
 
            c_vector<double, 3> mid_edge;
            mid_edge = edge_vector * 0.5 + p_node_a->rGetLocation();
 
            unsigned element0_index = *(edge_element_indices.begin());
 
            c_vector<double, 3> basis_vector1;
            basis_vector1 = mNodes[element0_index]->rGetLocation() - mid_edge;
 
            c_vector<double, 3> basis_vector2;
            basis_vector2 = VectorProduct(edge_vector, basis_vector1);
 
            std::vector<std::pair<double, unsigned> > index_angle_list;
 
            // Loop over each element containing this edge (i.e. those containing both nodes of the edge)
            for (std::set<unsigned>::iterator index_iter = edge_element_indices.begin();
                 index_iter != edge_element_indices.end();
                 ++index_iter)
            {
                // Calculate angle
                c_vector<double, 3> vertex_vector = mNodes[*index_iter]->rGetLocation() - mid_edge;
 
                double local_vertex_dot_basis_vector1 = inner_prod(vertex_vector, basis_vector1);
                double local_vertex_dot_basis_vector2 = inner_prod(vertex_vector, basis_vector2);
 
                double angle = atan2(local_vertex_dot_basis_vector2, local_vertex_dot_basis_vector1);
 
                std::pair<double, unsigned> pair(angle, *index_iter);
                index_angle_list.push_back(pair);
            }
 
            // Sort the list in order of increasing angle
            sort(index_angle_list.begin(), index_angle_list.end());
 
            // Create face
            VertexElement<2, 3>* p_face = new VertexElement<2, 3>(face_index);
            face_index++;
            for (unsigned count = 0; count < index_angle_list.size(); count++)
            {
                unsigned local_index = count > 1 ? count - 1 : 0;
                p_face->AddNode(mNodes[index_angle_list[count].second], local_index);
            }
 
            // Add face to list of faces
            mFaces.push_back(p_face);
 
            // Add face to appropriate elements
            if (!p_node_a->IsBoundaryNode())
            {
                unsigned node_a_index = p_node_a->GetIndex();
                if (index_element_map[node_a_index])
                {
                    // If there is already an element with this index, add the face to it...
                    index_element_map[node_a_index]->AddFace(p_face);
                }
                else
                {
                    // ...otherwise create an element, add the face to it, and add to the map
                    mVoronoiElementIndexMap[node_a_index] = element_index;
                    VertexElement<3, 3>* p_element = new VertexElement<3, 3>(element_index);
                    element_index++;
                    p_element->AddFace(p_face);
                    index_element_map[node_a_index] = p_element;
                }
            }
            if (!p_node_b->IsBoundaryNode())
            {
                unsigned node_b_index = p_node_b->GetIndex();
                if (index_element_map[node_b_index])
                {
                    // If there is already an element with this index, add the face to it...
                    index_element_map[node_b_index]->AddFace(p_face);
                }
                else
                {
                    // ...otherwise create an element, add the face to it, and add to the map
                    mVoronoiElementIndexMap[node_b_index] = element_index;
                    VertexElement<3, 3>* p_element = new VertexElement<3, 3>(element_index);
                    element_index++;
                    p_element->AddFace(p_face);
                    index_element_map[node_b_index] = p_element;
                }
            }
        }
    }
 
    // Populate mElements
    unsigned elem_count = 0;
    for (std::map<unsigned, VertexElement<3, 3>*>::iterator element_iter = index_element_map.begin();
         element_iter != index_element_map.end();
         ++element_iter)
    {
        mElements.push_back(element_iter->second);
        mVoronoiElementIndexMap[element_iter->first] = elem_count;
        elem_count++;
    }
 
    this->mMeshChangesDuringSimulation = false;
}
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM>
void VertexMesh<ELEMENT_DIM, SPACE_DIM>::GenerateEdgesFromElements(
    std::vector<VertexElement<ELEMENT_DIM, SPACE_DIM>*>& rElements)
{
    // Build a list of unique edges from nodes within all the elements
    for (auto elem : rElements)
    {
        elem->SetEdgeHelper(&mEdgeHelper);
        elem->BuildEdges();
    }
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNumEdges() const
{
    return mEdgeHelper.GetNumEdges();
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM>
Edge<SPACE_DIM>* VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetEdge(unsigned index) const
{
    return mEdgeHelper.GetEdge(index);
}
 
template<unsigned ELEMENT_DIM, unsigned SPACE_DIM>
const EdgeHelper<SPACE_DIM>& VertexMesh<ELEMENT_DIM, SPACE_DIM>::rGetEdgeHelper() const
{
    return mEdgeHelper;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
void VertexMesh<ELEMENT_DIM, SPACE_DIM>::GenerateVerticesFromElementCircumcentres(TetrahedralMesh<ELEMENT_DIM, SPACE_DIM>& rMesh)
{
    c_matrix<double, SPACE_DIM, ELEMENT_DIM> jacobian;
    c_matrix<double, ELEMENT_DIM, SPACE_DIM> inverse_jacobian;
    double jacobian_det;
 
    // Loop over elements of the Delaunay mesh and populate mNodes
    for (unsigned i = 0; i < rMesh.GetNumElements(); i++)
    {
        // Calculate the circumcentre of this element in the Delaunay mesh
        rMesh.GetInverseJacobianForElement(i, jacobian, jacobian_det, inverse_jacobian);
        c_vector<double, SPACE_DIM + 1> circumsphere = rMesh.GetElement(i)->CalculateCircumsphere(jacobian, inverse_jacobian);
 
        c_vector<double, SPACE_DIM> circumcentre;
        for (unsigned j = 0; j < SPACE_DIM; j++)
        {
            circumcentre(j) = circumsphere(j);
        }
 
        // Create a node in the Voronoi mesh at the location of this circumcentre
        this->mNodes.push_back(new Node<SPACE_DIM>(i, circumcentre));
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetEdgeLength(unsigned elementIndex1, unsigned elementIndex2)
{
    assert(SPACE_DIM == 2); // LCOV_EXCL_LINE - code will be removed at compile time
 
    std::set<unsigned> node_indices_1;
    for (unsigned i = 0; i < mElements[elementIndex1]->GetNumNodes(); i++)
    {
        node_indices_1.insert(mElements[elementIndex1]->GetNodeGlobalIndex(i));
    }
    std::set<unsigned> node_indices_2;
    for (unsigned i = 0; i < mElements[elementIndex2]->GetNumNodes(); i++)
    {
        node_indices_2.insert(mElements[elementIndex2]->GetNodeGlobalIndex(i));
    }
 
    std::set<unsigned> shared_nodes;
    std::set_intersection(node_indices_1.begin(), node_indices_1.end(),
                          node_indices_2.begin(), node_indices_2.end(),
                          std::inserter(shared_nodes, shared_nodes.begin()));
 
    if (shared_nodes.size() == 1)
    {
        // It's possible that these two elements are actually infinite but are on the edge of the domain
        EXCEPTION("Elements " << elementIndex1 << " and " << elementIndex2 << " share only one node.");
    }
    assert(shared_nodes.size() == 2);
 
    unsigned index1 = *(shared_nodes.begin());
    unsigned index2 = *(++(shared_nodes.begin()));
 
    double edge_length = this->GetDistanceBetweenNodes(index1, index2);
    return edge_length;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetElongationShapeFactorOfElement(unsigned index)
{
    assert(SPACE_DIM == 2); // LCOV_EXCL_LINE - code will be removed at compile time
 
    c_vector<double, 3> moments = CalculateMomentsOfElement(index);
 
    double discriminant = sqrt((moments(0) - moments(1)) * (moments(0) - moments(1)) + 4.0 * moments(2) * moments(2));
 
    // Note that as the matrix of second moments of area is symmetric, both its eigenvalues are real
    double largest_eigenvalue = (moments(0) + moments(1) + discriminant) * 0.5;
    double smallest_eigenvalue = (moments(0) + moments(1) - discriminant) * 0.5;
 
    double elongation_shape_factor = sqrt(largest_eigenvalue / smallest_eigenvalue);
    return elongation_shape_factor;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexMesh<ELEMENT_DIM, SPACE_DIM>::VertexMesh()
{
    mpDelaunayMesh = nullptr;
    this->mMeshChangesDuringSimulation = false;
    Clear();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexMesh<ELEMENT_DIM, SPACE_DIM>::~VertexMesh()
{
 Clear();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::SolveNodeMapping(unsigned index) const
{
    assert(index < this->mNodes.size());
    return index;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::SolveElementMapping(unsigned index) const
{
    assert(index < this->mElements.size());
    return index;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::SolveBoundaryElementMapping(unsigned index) const
{
    //    assert(index < this->mBoundaryElements.size() );
    return index;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetDelaunayNodeIndexCorrespondingToVoronoiElementIndex(unsigned elementIndex)
{
    unsigned node_index = UNSIGNED_UNSET;
 
    if (mVoronoiElementIndexMap.empty())
    {
        node_index = elementIndex;
    }
    else
    {
        for (std::map<unsigned, unsigned>::iterator iter = mVoronoiElementIndexMap.begin();
             iter != mVoronoiElementIndexMap.end();
             ++iter)
        {
            if (iter->second == elementIndex)
            {
                node_index = iter->first;
                break;
            }
        }
    }
    assert(node_index != UNSIGNED_UNSET);
    return node_index;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetVoronoiElementIndexCorrespondingToDelaunayNodeIndex(unsigned nodeIndex)
{
    unsigned element_index = UNSIGNED_UNSET;
 
    if (mVoronoiElementIndexMap.empty())
    {
        element_index = nodeIndex;
    }
    else
    {
        std::map<unsigned, unsigned>::iterator iter = mVoronoiElementIndexMap.find(nodeIndex);
 
        if (iter == mVoronoiElementIndexMap.end())
        {
            EXCEPTION("This index does not correspond to a VertexElement");
        }
        else
        {
            element_index = iter->second;
        }
    }
    assert(element_index != UNSIGNED_UNSET);
    return element_index;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetRosetteRankOfElement(unsigned index)
{
    assert(SPACE_DIM == 2 || SPACE_DIM == 3); // LCOV_EXCL_LINE - code will be removed at compile time
 
    // Get pointer to this element
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = GetElement(index);
 
    // Loop over nodes in the current element and find which is contained in the most elements
    unsigned rosette_rank = 0;
    for (unsigned node_idx = 0; node_idx < p_element->GetNumNodes(); node_idx++)
    {
        unsigned num_elems_this_node = p_element->GetNode(node_idx)->rGetContainingElementIndices().size();
 
        if (num_elems_this_node > rosette_rank)
        {
            rosette_rank = num_elems_this_node;
        }
    }
 
    // Return the rosette rank
    return rosette_rank;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
void VertexMesh<ELEMENT_DIM, SPACE_DIM>::Clear()
{
    // Delete elements
    for (unsigned i = 0; i < mElements.size(); i++)
    {
        delete mElements[i];
    }
    mElements.clear();
 
    // Delete faces
    for (unsigned i = 0; i < mFaces.size(); i++)
    {
        delete mFaces[i];
    }
    mFaces.clear();
 
 
    // Delete nodes
    for (unsigned i = 0; i < this->mNodes.size(); i++)
    {
        delete this->mNodes[i];
    }
    this->mNodes.clear();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNumNodes() const
{
    return this->mNodes.size();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNumElements() const
{
    return mElements.size();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNumAllElements() const
{
    return mElements.size();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNumFaces() const
{
    return mFaces.size();
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexElement<ELEMENT_DIM, SPACE_DIM>* VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetElement(unsigned index) const
{
    assert(index < mElements.size());
    return mElements[index];
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetFace(unsigned index) const
{
    assert(index < mFaces.size());
    return mFaces[index];
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetCentroidOfElement(unsigned index)
{
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = GetElement(index);
    unsigned num_nodes = p_element->GetNumNodes();
 
    c_vector<double, SPACE_DIM> centroid = zero_vector<double>(SPACE_DIM);
 
    switch (SPACE_DIM)
    {
        case 1:
        {
            centroid = 0.5 * (p_element->GetNodeLocation(0) + p_element->GetNodeLocation(1));
        }
        break;
        case 2:
        {
            double centroid_x = 0;
            double centroid_y = 0;
 
            // Note that we cannot use GetVolumeOfElement() below as it returns the absolute, rather than signed, area
            double element_signed_area = 0.0;
 
            // Map the first vertex to the origin and employ GetVectorFromAtoB() to allow for periodicity
            c_vector<double, SPACE_DIM> first_node_location = p_element->GetNodeLocation(0);
            c_vector<double, SPACE_DIM> pos_1 = zero_vector<double>(SPACE_DIM);
 
            // Loop over vertices
            for (unsigned local_index = 0; local_index < num_nodes; local_index++)
            {
                c_vector<double, SPACE_DIM> next_node_location = p_element->GetNodeLocation((local_index + 1) % num_nodes);
                c_vector<double, SPACE_DIM> pos_2 = GetVectorFromAtoB(first_node_location, next_node_location);
 
                double this_x = pos_1[0];
                double this_y = pos_1[1];
                double next_x = pos_2[0];
                double next_y = pos_2[1];
 
                double signed_area_term = this_x * next_y - this_y * next_x;
 
                centroid_x += (this_x + next_x) * signed_area_term;
                centroid_y += (this_y + next_y) * signed_area_term;
                element_signed_area += 0.5 * signed_area_term;
 
                pos_1 = pos_2;
            }
 
            assert(element_signed_area != 0.0);
 
            // Finally, map back and employ GetVectorFromAtoB() to allow for periodicity
            centroid = first_node_location;
            centroid(0) += centroid_x / (6.0 * element_signed_area);
            centroid(1) += centroid_y / (6.0 * element_signed_area);
        }
        break;
        case 3:
        {
            for (unsigned local_index = 0; local_index < num_nodes; local_index++)
            {
                centroid += p_element->GetNodeLocation(local_index);
            }
            centroid /= ((double)num_nodes);
        }
        break;
        default:
            NEVER_REACHED;
    }
    return centroid;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
std::set<unsigned> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNeighbouringNodeIndices(unsigned nodeIndex)
{
    // Create a set of neighbouring node indices
    std::set<unsigned> neighbouring_node_indices;
 
    // Find the indices of the elements owned by this node
    std::set<unsigned> containing_elem_indices = this->GetNode(nodeIndex)->rGetContainingElementIndices();
 
    // Iterate over these elements
    for (std::set<unsigned>::iterator elem_iter = containing_elem_indices.begin();
         elem_iter != containing_elem_indices.end();
         ++elem_iter)
    {
        // Find the local index of this node in this element
        unsigned local_index = GetElement(*elem_iter)->GetNodeLocalIndex(nodeIndex);
 
        // Find the global indices of the preceding and successive nodes in this element
        unsigned num_nodes = GetElement(*elem_iter)->GetNumNodes();
        unsigned previous_local_index = (local_index + num_nodes - 1) % num_nodes;
        unsigned next_local_index = (local_index + 1) % num_nodes;
 
        // Add the global indices of these two nodes to the set of neighbouring node indices
        neighbouring_node_indices.insert(GetElement(*elem_iter)->GetNodeGlobalIndex(previous_local_index));
        neighbouring_node_indices.insert(GetElement(*elem_iter)->GetNodeGlobalIndex(next_local_index));
    }
 
    return neighbouring_node_indices;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
std::set<unsigned> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNeighbouringNodeNotAlsoInElement(unsigned nodeIndex, unsigned elemIndex)
{
    // Get a pointer to this element
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = this->GetElement(elemIndex);
 
    // Get the indices of the nodes contained in this element
    std::set<unsigned> node_indices_in_this_element;
    for (unsigned local_index = 0; local_index < p_element->GetNumNodes(); local_index++)
    {
        unsigned global_index = p_element->GetNodeGlobalIndex(local_index);
        node_indices_in_this_element.insert(global_index);
    }
 
    // Check that the node is in fact contained in the element
    if (node_indices_in_this_element.find(nodeIndex) == node_indices_in_this_element.end())
    {
        EXCEPTION("The given node is not contained in the given element.");
    }
 
    // Create a set of neighbouring node indices
    std::set<unsigned> neighbouring_node_indices_not_in_this_element;
 
    // Get the indices of this node's neighbours
    std::set<unsigned> node_neighbours = GetNeighbouringNodeIndices(nodeIndex);
 
    // Check if each neighbour is also in this element; if not, add it to the set
    for (std::set<unsigned>::iterator iter = node_neighbours.begin();
         iter != node_neighbours.end();
         ++iter)
    {
        if (node_indices_in_this_element.find(*iter) == node_indices_in_this_element.end())
        {
            neighbouring_node_indices_not_in_this_element.insert(*iter);
        }
    }
 
    return neighbouring_node_indices_not_in_this_element;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
std::set<unsigned> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNeighbouringElementIndices(unsigned elementIndex)
{
    // Get a pointer to this element
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = this->GetElement(elementIndex);
 
    // Create a set of neighbouring element indices
    std::set<unsigned> neighbouring_element_indices;
 
    // Loop over nodes owned by this element
    for (unsigned local_index = 0; local_index < p_element->GetNumNodes(); local_index++)
    {
        // Get a pointer to this node
        Node<SPACE_DIM>* p_node = p_element->GetNode(local_index);
 
        // Find the indices of the elements owned by this node
        std::set<unsigned> containing_elem_indices = p_node->rGetContainingElementIndices();
 
        // Form the union of this set with the current set of neighbouring element indices
        std::set<unsigned> all_elements;
        std::set_union(neighbouring_element_indices.begin(), neighbouring_element_indices.end(),
                       containing_elem_indices.begin(), containing_elem_indices.end(),
                       std::inserter(all_elements, all_elements.begin()));
 
        // Update the set of neighbouring element indices
        neighbouring_element_indices = all_elements;
    }
 
    // Lastly remove this element's index from the set of neighbouring element indices
    neighbouring_element_indices.erase(elementIndex);
 
    return neighbouring_element_indices;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
VertexMesh<ELEMENT_DIM, SPACE_DIM>* VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetMeshForVtk()
{
    return this;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
bool VertexMesh<ELEMENT_DIM, SPACE_DIM>::IsNearExistingNodes(c_vector<double,SPACE_DIM> newNodeLocation, std::vector<Node<SPACE_DIM> *> nodesToCheck, double minClearance)
{
    bool node_near = false;
 
    for (unsigned i=0; i<nodesToCheck.size(); i++)
    {
        double distance = norm_2(mpDelaunayMesh->GetVectorFromAtoB(nodesToCheck[i]->rGetLocation(), newNodeLocation));
        if (distance < minClearance)
        {
            node_near = true;
            break;
        }
    }
 
    return node_near;
}
 
template <>
void VertexMesh<1, 1>::ConstructFromMeshReader(AbstractMeshReader<1, 1>& rMeshReader)
{
    EXCEPTION("VertexMesh<1,1>::ConstructFromMeshReader() is not implemented");
}
 
template <>
void VertexMesh<1, 2>::ConstructFromMeshReader(AbstractMeshReader<1, 2>& rMeshReader)
{
    EXCEPTION("VertexMesh<1,2>::ConstructFromMeshReader() is not implemented");
}
 
template <>
void VertexMesh<1, 3>::ConstructFromMeshReader(AbstractMeshReader<1, 3>& rMeshReader)
{
    EXCEPTION("VertexMesh<1,3>::ConstructFromMeshReader() is not implemented");
}
 
template <>
void VertexMesh<2, 3>::ConstructFromMeshReader(AbstractMeshReader<2, 3>& rMeshReader)
{
    EXCEPTION("VertexMesh<2,3>::ConstructFromMeshReader() is not implemented");
}
 
template <>
void VertexMesh<2, 2>::ConstructFromMeshReader(AbstractMeshReader<2, 2>& rMeshReader)
{
    assert(rMeshReader.HasNodePermutation() == false);
    // Store numbers of nodes and elements
    unsigned num_nodes = rMeshReader.GetNumNodes();
    unsigned num_elements = rMeshReader.GetNumElements();
 
    // Reserve memory for nodes
    this->mNodes.reserve(num_nodes);
 
    rMeshReader.Reset();
 
    // Add nodes
    std::vector<double> node_data;
    for (unsigned i = 0; i < num_nodes; i++)
    {
        node_data = rMeshReader.GetNextNode();
        unsigned is_boundary_node = (bool)node_data[2];
        node_data.pop_back();
        this->mNodes.push_back(new Node<2>(i, node_data, is_boundary_node));
    }
 
    rMeshReader.Reset();
 
    // Reserve memory for elements
    mElements.reserve(rMeshReader.GetNumElements());
 
    // Add elements
    for (unsigned elem_index = 0; elem_index < num_elements; elem_index++)
    {
        // Get the data for this element
        ElementData element_data = rMeshReader.GetNextElementData();
 
        // Get the nodes owned by this element
        std::vector<Node<2>*> nodes;
        unsigned num_nodes_in_element = element_data.NodeIndices.size();
        for (unsigned j = 0; j < num_nodes_in_element; j++)
        {
            assert(element_data.NodeIndices[j] < this->mNodes.size());
            nodes.push_back(this->mNodes[element_data.NodeIndices[j]]);
        }
 
        // Use nodes and index to construct this element
        VertexElement<2, 2>* p_element = new VertexElement<2, 2>(elem_index, nodes);
        mElements.push_back(p_element);
 
        if (rMeshReader.GetNumElementAttributes() > 0)
        {
            assert(rMeshReader.GetNumElementAttributes() == 1);
            unsigned attribute_value = (unsigned)element_data.AttributeValue;
            p_element->SetAttribute(attribute_value);
        }
    }
    GenerateEdgesFromElements(mElements);
}
 
template <>
void VertexMesh<3, 3>::ConstructFromMeshReader(AbstractMeshReader<3, 3>& rMeshReader)
{
    assert(rMeshReader.HasNodePermutation() == false);
 
    // Store numbers of nodes and elements
    unsigned num_nodes = rMeshReader.GetNumNodes();
    unsigned num_elements = rMeshReader.GetNumElements();
 
    // Reserve memory for nodes
    this->mNodes.reserve(num_nodes);
 
    rMeshReader.Reset();
 
    // Add nodes
    std::vector<double> node_data;
    for (unsigned i = 0; i < num_nodes; i++)
    {
        node_data = rMeshReader.GetNextNode();
        unsigned is_boundary_node = (bool)node_data[3];
        node_data.pop_back();
        this->mNodes.push_back(new Node<3>(i, node_data, is_boundary_node));
    }
 
    rMeshReader.Reset();
 
    // Reserve memory for nodes
    mElements.reserve(rMeshReader.GetNumElements());
 
    // Use a std::set to keep track of which faces have been added to mFaces
    std::set<unsigned> faces_counted;
 
    // Add elements
    for (unsigned elem_index = 0; elem_index < num_elements; elem_index++)
    {
        typedef VertexMeshReader<3, 3> VERTEX_MESH_READER;
        assert(dynamic_cast<VERTEX_MESH_READER*>(&rMeshReader) != nullptr);
 
        // Get the data for this element
        VertexElementData element_data = static_cast<VERTEX_MESH_READER*>(&rMeshReader)->GetNextElementDataWithFaces();
 
        // Get the nodes owned by this element
        std::vector<Node<3>*> nodes;
        unsigned num_nodes_in_element = element_data.NodeIndices.size();
        for (unsigned j = 0; j < num_nodes_in_element; j++)
        {
            assert(element_data.NodeIndices[j] < this->mNodes.size());
            nodes.push_back(this->mNodes[element_data.NodeIndices[j]]);
        }
 
        // Get the faces owned by this element
        std::vector<VertexElement<2, 3>*> faces;
        unsigned num_faces_in_element = element_data.Faces.size();
        for (unsigned i = 0; i < num_faces_in_element; i++)
        {
            // Get the data for this face
            ElementData face_data = element_data.Faces[i];
 
            // Get the face index
            unsigned face_index = (unsigned)face_data.AttributeValue;
 
            // Get the nodes owned by this face
            std::vector<Node<3>*> nodes_in_face;
            unsigned num_nodes_in_face = face_data.NodeIndices.size();
            for (unsigned j = 0; j < num_nodes_in_face; j++)
            {
                assert(face_data.NodeIndices[j] < this->mNodes.size());
                nodes_in_face.push_back(this->mNodes[face_data.NodeIndices[j]]);
            }
 
            // If this face index is not already encountered, create a new face and update faces_counted...
            if (faces_counted.find(face_index) == faces_counted.end())
            {
                // Use nodes and index to construct this face
                VertexElement<2, 3>* p_face = new VertexElement<2, 3>(face_index, nodes_in_face);
                mFaces.push_back(p_face);
                faces_counted.insert(face_index);
                faces.push_back(p_face);
            }
            else
            {
                // ... otherwise use the member of mFaces with this index
                bool face_added = false;
                for (unsigned k = 0; k < mFaces.size(); k++)
                {
                    if (mFaces[k]->GetIndex() == face_index)
                    {
                        faces.push_back(mFaces[k]);
                        face_added = true;
                        break;
                    }
                }
                UNUSED_OPT(face_added);
                assert(face_added == true);
            }
        }
 
        std::vector<bool> orientations = std::vector<bool>(num_faces_in_element, true);
 
        // Use faces and index to construct this element
        VertexElement<3, 3>* p_element = new VertexElement<3, 3>(elem_index, faces, orientations, nodes);
        mElements.push_back(p_element);
 
        if (rMeshReader.GetNumElementAttributes() > 0)
        {
            assert(rMeshReader.GetNumElementAttributes() == 1);
            unsigned attribute_value = element_data.AttributeValue;
            p_element->SetAttribute(attribute_value);
        }
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetVectorFromAtoB(
    const c_vector<double, SPACE_DIM>& rLocationA, const c_vector<double, SPACE_DIM>& rLocationB)
{
    c_vector<double, SPACE_DIM> vector;
    if (mpDelaunayMesh)
    {
        vector = mpDelaunayMesh->GetVectorFromAtoB(rLocationA, rLocationB);
    }
    else
    {
        vector = AbstractMesh<ELEMENT_DIM, SPACE_DIM>::GetVectorFromAtoB(rLocationA, rLocationB);
    }
    return vector;
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetVolumeOfElement(unsigned index)
{
    assert(SPACE_DIM == 2 || SPACE_DIM == 3); // LCOV_EXCL_LINE - code will be removed at compile time
 
    // Get pointer to this element
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = GetElement(index);
 
    double element_volume = 0.0;
    if (SPACE_DIM == 2)
    {
        // We map the first vertex to the origin and employ GetVectorFromAtoB() to allow for periodicity
        c_vector<double, SPACE_DIM> first_node_location = p_element->GetNodeLocation(0);
        c_vector<double, SPACE_DIM> pos_1 = zero_vector<double>(SPACE_DIM);
 
        unsigned num_nodes = p_element->GetNumNodes();
        for (unsigned local_index = 0; local_index < num_nodes; local_index++)
        {
            c_vector<double, SPACE_DIM> next_node_location = p_element->GetNodeLocation((local_index + 1) % num_nodes);
            c_vector<double, SPACE_DIM> pos_2 = GetVectorFromAtoB(first_node_location, next_node_location);
 
            double this_x = pos_1[0];
            double this_y = pos_1[1];
            double next_x = pos_2[0];
            double next_y = pos_2[1];
 
            element_volume += 0.5 * (this_x * next_y - next_x * this_y);
 
            pos_1 = pos_2;
        }
    }
    else
    {
        // Loop over faces and add up pyramid volumes
        c_vector<double, SPACE_DIM> pyramid_apex = p_element->GetNodeLocation(0);
        for (unsigned face_index = 0; face_index < p_element->GetNumFaces(); face_index++)
        {
            // Get pointer to face
            VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* p_face = p_element->GetFace(face_index);
 
            // Calculate the area of the face and get unit normal to this face
            c_vector<double, SPACE_DIM> unit_normal;
            double face_area = CalculateUnitNormalToFaceWithArea(p_face, unit_normal);
 
            // Calculate the perpendicular distance from the plane of the face to the chosen apex
            c_vector<double, SPACE_DIM> base_to_apex = GetVectorFromAtoB(p_face->GetNodeLocation(0), pyramid_apex);
            double perpendicular_distance = fabs(inner_prod(base_to_apex, unit_normal));
 
            // Use these to calculate the volume of the pyramid formed by the face and the point pyramid_apex
            element_volume += face_area * perpendicular_distance / 3;
        }
    }
 
    // We take the absolute value just in case the nodes were really oriented clockwise
    return fabs(element_volume);
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetSurfaceAreaOfElement(unsigned index)
{
    assert(SPACE_DIM == 2 || SPACE_DIM == 3); // LCOV_EXCL_LINE - code will be removed at compile time
 
    // Get pointer to this element
    VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = GetElement(index);
 
    double surface_area = 0.0;
    if (SPACE_DIM == 2)
    {
        unsigned num_nodes = p_element->GetNumNodes();
        unsigned this_node_index = p_element->GetNodeGlobalIndex(0);
        for (unsigned local_index = 0; local_index < num_nodes; local_index++)
        {
            unsigned next_node_index = p_element->GetNodeGlobalIndex((local_index + 1) % num_nodes);
 
            surface_area += this->GetDistanceBetweenNodes(this_node_index, next_node_index);
            this_node_index = next_node_index;
        }
    }
    else
    {
        // Loop over faces and add up areas
        for (unsigned face_index = 0; face_index < p_element->GetNumFaces(); face_index++)
        {
            surface_area += CalculateAreaOfFace(p_element->GetFace(face_index));
        }
    }
    return surface_area;
}
 
//                        2D-specific methods                       //
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
bool VertexMesh<ELEMENT_DIM, SPACE_DIM>::ElementIncludesPoint(
    [[maybe_unused]] const c_vector<double, SPACE_DIM>& rTestPoint, [[maybe_unused]] unsigned elementIndex)
{
    if constexpr (ELEMENT_DIM == 2 && SPACE_DIM == 2)
    {
        // Get the element
        VertexElement<2, 2>* p_element = GetElement(elementIndex);
        const unsigned num_nodes = p_element->GetNumNodes();
 
        // Initialise boolean
        bool element_includes_point = false;
 
        //    // Initialise boolean
        //    bool element_includes_point = true;
        //
        //    unsigned winding_number = 0;
        //
        //    c_vector<double, SPACE_DIM> first_node_location = p_element->GetNodeLocation(0);
        //    c_vector<double, SPACE_DIM> test_point = this->GetVectorFromAtoB(first_node_location, rTestPoint);
        //    c_vector<double, SPACE_DIM> this_node_location = zero_vector<double>(SPACE_DIM);
        //
        //    // Loop over edges of the element
        //    for (unsigned local_index=0; local_index<num_nodes; local_index++)
        //    {
        //        c_vector<double, SPACE_DIM> untransformed_vector = p_element->GetNodeLocation((local_index+1)%num_nodes);
        //        c_vector<double, SPACE_DIM> next_node_location = this->GetVectorFromAtoB(first_node_location, untransformed_vector);
        //
        //        // If this edge is crossing upward...
        //        if (this_node_location[1] <= test_point[1])
        //        {
        //            if (next_node_location[1] > test_point[1])
        //            {
        //                double is_left =  (next_node_location[0] - this_node_location[0])*(test_point[1] - this_node_location[1])
        //                                 - (test_point[0] - this_node_location[0])*(next_node_location[1] - this_node_location[1]);
        //
        //                // ...and the test point is to the left of the edge...
        //                if (is_left > DBL_EPSILON)
        //                {
        //                    // ...then there is a valid upward edge-ray intersection to the right of the test point
        //                    winding_number++;
        //                }
        //            }
        //        }
        //        else
        //        {
        //            // ...otherwise, if the edge is crossing downward
        //            if (next_node_location[1] <= test_point[1])
        //            {
        //                double is_left =  (next_node_location[0] - this_node_location[0])*(test_point[1] - this_node_location[1])
        //                                 - (test_point[0] - this_node_location[0])*(next_node_location[1] - this_node_location[1]);
        //
        //                // ...and the test point is to the right of the edge...
        //                if (is_left < -DBL_EPSILON)
        //                {
        //                    // ...then there is a valid downward edge-ray intersection to the right of the test point
        //                    winding_number--;
        //                }
        //            }
        //        }
        //        this_node_location = next_node_location;
        //    }
        //
        //    if (winding_number == 0)
        //    {
        //        element_includes_point = false;
        //    }
 
        // Remap the origin to the first vertex to allow alternative distance metrics to be used in subclasses
        const c_vector<double, 2> first_vertex = p_element->GetNodeLocation(0);
        c_vector<double, 2> test_point = GetVectorFromAtoB(first_vertex, rTestPoint);
 
        // Loop over edges of the element
        c_vector<double, 2> vertexA = zero_vector<double>(2);
        for (unsigned local_index = 0; local_index < num_nodes; local_index++)
        {
            // Check if this edge crosses the ray running out horizontally (increasing x, fixed y) from the test point
            c_vector<double, 2> vector_a_to_point = GetVectorFromAtoB(vertexA, test_point);
 
            // Pathological case - test point coincides with vertexA
            // (we will check vertexB next time we go through the for loop)
            if (norm_2(vector_a_to_point) < DBL_EPSILON)
            {
                return false;
            }
 
            c_vector<double, 2> vertexB = GetVectorFromAtoB(first_vertex, p_element->GetNodeLocation((local_index + 1) % num_nodes));
            c_vector<double, 2> vector_b_to_point = GetVectorFromAtoB(vertexB, test_point);
            c_vector<double, 2> vector_a_to_b = GetVectorFromAtoB(vertexA, vertexB);
 
            // Pathological case - ray coincides with horizontal edge
            if ((fabs(vector_a_to_b[1]) < DBL_EPSILON) && (fabs(vector_a_to_point[1]) < DBL_EPSILON) && (fabs(vector_b_to_point[1]) < DBL_EPSILON))
            {
                if ((vector_a_to_point[0] > 0) != (vector_b_to_point[0] > 0))
                {
                    return false;
                }
            }
 
            // Non-pathological case
            // A and B on different sides of the line y = test_point[1]
            if ((vertexA[1] > test_point[1]) != (vertexB[1] > test_point[1]))
            {
                // Intersection of y=test_point[1] and vector_a_to_b is on the right of test_point
                if (test_point[0] < vertexA[0] + vector_a_to_b[0] * vector_a_to_point[1] / vector_a_to_b[1])
                {
                    element_includes_point = !element_includes_point;
                }
            }
 
            vertexA = vertexB;
        }
        return element_includes_point;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
unsigned VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetLocalIndexForElementEdgeClosestToPoint(
    [[maybe_unused]] const c_vector<double, SPACE_DIM>& rTestPoint, [[maybe_unused]] unsigned elementIndex)
{
    if constexpr (ELEMENT_DIM == 2 && SPACE_DIM == 2)
    {
        // Get the element
        VertexElement<2, 2>* p_element = GetElement(elementIndex);
        const unsigned num_nodes = p_element->GetNumNodes();
 
        double min_squared_normal_distance = DBL_MAX;
        unsigned min_distance_edge_index = UINT_MAX;
 
        // Loop over edges of the element
        for (unsigned local_index = 0; local_index < num_nodes; local_index++)
        {
            // Get the end points of this edge
            const c_vector<double, 2> vertexA = p_element->GetNodeLocation(local_index);
            const c_vector<double, 2> vertexB = p_element->GetNodeLocation((local_index + 1) % num_nodes);
 
            const c_vector<double, 2> vector_a_to_point = this->GetVectorFromAtoB(vertexA, rTestPoint);
            const c_vector<double, 2> vector_a_to_b = this->GetVectorFromAtoB(vertexA, vertexB);
            const double distance_a_to_b = norm_2(vector_a_to_b);
 
            const c_vector<double, 2> edge_ab_unit_vector = vector_a_to_b / norm_2(vector_a_to_b);
            double distance_parallel_to_edge = inner_prod(vector_a_to_point, edge_ab_unit_vector);
 
            double squared_distance_normal_to_edge = SmallPow(norm_2(vector_a_to_point), 2) - SmallPow(distance_parallel_to_edge, 2);
 
            /*
             * If the point lies almost bang on the supporting line of the edge, then snap to the line.
             * This allows us to do floating point tie-breaks when line is exactly at a node.
             * We adopt a similar approach if the point is at the same position as a point in the
             * element.
             */
            if (squared_distance_normal_to_edge < DBL_EPSILON)
            {
                squared_distance_normal_to_edge = 0.0;
            }
 
            if (fabs(distance_parallel_to_edge) < DBL_EPSILON)
            {
                distance_parallel_to_edge = 0.0;
            }
            else if (fabs(distance_parallel_to_edge - distance_a_to_b) < DBL_EPSILON)
            {
                distance_parallel_to_edge = distance_a_to_b;
            }
 
            // Make sure node is within the confines of the edge and is the nearest edge to the node \this breaks for convex elements
            if (squared_distance_normal_to_edge < min_squared_normal_distance && distance_parallel_to_edge >= 0 && distance_parallel_to_edge <= distance_a_to_b)
            {
                min_squared_normal_distance = squared_distance_normal_to_edge;
                min_distance_edge_index = local_index;
            }
        }
 
        assert(min_distance_edge_index < num_nodes);
        return min_distance_edge_index;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, 3> VertexMesh<ELEMENT_DIM, SPACE_DIM>::CalculateMomentsOfElement([[maybe_unused]] unsigned index)
{
    if constexpr (SPACE_DIM == 2)
    {
        // Define helper variables
        VertexElement<ELEMENT_DIM, SPACE_DIM>* p_element = GetElement(index);
        unsigned num_nodes = p_element->GetNumNodes();
        c_vector<double, 3> moments = zero_vector<double>(3);
 
        // Since we compute I_xx, I_yy and I_xy about the centroid, we must shift each vertex accordingly
        c_vector<double, SPACE_DIM> centroid = GetCentroidOfElement(index);
 
        c_vector<double, SPACE_DIM> this_node_location = p_element->GetNodeLocation(0);
        c_vector<double, SPACE_DIM> pos_1 = this->GetVectorFromAtoB(centroid, this_node_location);
 
        for (unsigned local_index = 0; local_index < num_nodes; local_index++)
        {
            unsigned next_index = (local_index + 1) % num_nodes;
            c_vector<double, SPACE_DIM> next_node_location = p_element->GetNodeLocation(next_index);
            c_vector<double, SPACE_DIM> pos_2 = this->GetVectorFromAtoB(centroid, next_node_location);
 
            double signed_area_term = pos_1(0) * pos_2(1) - pos_2(0) * pos_1(1);
            // Ixx
            moments(0) += (pos_1(1) * pos_1(1) + pos_1(1) * pos_2(1) + pos_2(1) * pos_2(1)) * signed_area_term;
 
            // Iyy
            moments(1) += (pos_1(0) * pos_1(0) + pos_1(0) * pos_2(0) + pos_2(0) * pos_2(0)) * signed_area_term;
 
            // Ixy
            moments(2) += (pos_1(0) * pos_2(1) + 2 * pos_1(0) * pos_1(1) + 2 * pos_2(0) * pos_2(1) + pos_2(0) * pos_1(1)) * signed_area_term;
 
            pos_1 = pos_2;
        }
 
        moments(0) /= 12;
        moments(1) /= 12;
        moments(2) /= 24;
 
        /*
         * If the nodes owned by the element were supplied in a clockwise rather
         * than anticlockwise manner, or if this arose as a result of enforcing
         * periodicity, then our computed quantities will be the wrong sign, so
         * we need to fix this.
         */
        if (moments(0) < 0.0)
        {
            moments(0) = -moments(0);
            moments(1) = -moments(1);
            moments(2) = -moments(2);
        }
        return moments;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetShortAxisOfElement([[maybe_unused]] unsigned index)
{
    if constexpr (SPACE_DIM == 2)
    {
 
        c_vector<double, SPACE_DIM> short_axis = zero_vector<double>(SPACE_DIM);
 
        // Calculate the moments of the element about its centroid (recall that I_xx and I_yy must be non-negative)
        c_vector<double, 3> moments = CalculateMomentsOfElement(index);
 
        // Normalise the moments vector to remove problem of a very small discriminant (see #2874)
        moments /= norm_2(moments);
 
        // If the principal moments are equal...
        double discriminant = (moments(0) - moments(1)) * (moments(0) - moments(1)) + 4.0 * moments(2) * moments(2);
        if (fabs(discriminant) < DBL_EPSILON)
        {
            // ...then every axis through the centroid is a principal axis, so return a random unit vector
            short_axis(0) = RandomNumberGenerator::Instance()->ranf();
            short_axis(1) = sqrt(1.0 - short_axis(0) * short_axis(0));
        }
        else
        {
            // If the product of inertia is zero, then the coordinate axes are the principal axes
            if (fabs(moments(2)) < DBL_EPSILON)
            {
                if (moments(0) < moments(1))
                {
                    short_axis(0) = 0.0;
                    short_axis(1) = 1.0;
                }
                else
                {
                    short_axis(0) = 1.0;
                    short_axis(1) = 0.0;
                }
            }
            else
            {
                // Otherwise we find the eigenvector of the inertia matrix corresponding to the largest eigenvalue
                double lambda = 0.5 * (moments(0) + moments(1) + sqrt(discriminant));
 
                short_axis(0) = 1.0;
                short_axis(1) = (moments(0) - lambda) / moments(2);
 
                // Normalise the short axis before returning it
                short_axis /= norm_2(short_axis);
            }
        }
 
        return short_axis;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetAreaGradientOfElementAtNode(
    [[maybe_unused]] VertexElement<ELEMENT_DIM, SPACE_DIM>* pElement, [[maybe_unused]] unsigned localIndex)
{
    if constexpr (SPACE_DIM == 2)
    {
        unsigned num_nodes_in_element = pElement->GetNumNodes();
        unsigned next_local_index = (localIndex + 1) % num_nodes_in_element;
 
        // We add an extra num_nodes_in_element in the line below as otherwise this term can be negative, which breaks the % operator
        unsigned previous_local_index = (num_nodes_in_element + localIndex - 1) % num_nodes_in_element;
 
        c_vector<double, SPACE_DIM> previous_node_location = pElement->GetNodeLocation(previous_local_index);
        c_vector<double, SPACE_DIM> next_node_location = pElement->GetNodeLocation(next_local_index);
        c_vector<double, SPACE_DIM> difference_vector = this->GetVectorFromAtoB(previous_node_location, next_node_location);
 
        c_vector<double, SPACE_DIM> area_gradient;
 
        area_gradient[0] = 0.5 * difference_vector[1];
        area_gradient[1] = -0.5 * difference_vector[0];
 
        return area_gradient;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetPreviousEdgeGradientOfElementAtNode(
    [[maybe_unused]] VertexElement<ELEMENT_DIM, SPACE_DIM>* pElement, [[maybe_unused]] unsigned localIndex)
{
    if constexpr (SPACE_DIM == 2)
    {
        assert(SPACE_DIM == 2); // LCOV_EXCL_LINE - code will be removed at compile time
 
        const unsigned num_nodes_in_element = pElement->GetNumNodes();
 
        // We add an extra num_nodes_in_element-1 in the line below as otherwise this term can be negative, which breaks the % operator
        const unsigned previous_local_index = (num_nodes_in_element + localIndex - 1) % num_nodes_in_element;
 
        const unsigned this_global_index = pElement->GetNodeGlobalIndex(localIndex);
        const unsigned previous_global_index = pElement->GetNodeGlobalIndex(previous_local_index);
 
        const double previous_edge_length = this->GetDistanceBetweenNodes(this_global_index, previous_global_index);
        assert(previous_edge_length > DBL_EPSILON);
 
        const c_vector<double, SPACE_DIM> previous_edge_gradient = this->GetVectorFromAtoB(pElement->GetNodeLocation(previous_local_index), pElement->GetNodeLocation(localIndex)) / previous_edge_length;
 
        return previous_edge_gradient;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetNextEdgeGradientOfElementAtNode(
    [[maybe_unused]] VertexElement<ELEMENT_DIM, SPACE_DIM>* pElement, [[maybe_unused]] unsigned localIndex)
{
    if constexpr (SPACE_DIM == 2)
    {
        const unsigned next_local_index = (localIndex + 1) % (pElement->GetNumNodes());
 
        const unsigned this_global_index = pElement->GetNodeGlobalIndex(localIndex);
        const unsigned next_global_index = pElement->GetNodeGlobalIndex(next_local_index);
 
        const double next_edge_length = this->GetDistanceBetweenNodes(this_global_index, next_global_index);
        assert(next_edge_length > DBL_EPSILON);
 
        const c_vector<double, SPACE_DIM> next_edge_gradient = this->GetVectorFromAtoB(pElement->GetNodeLocation(next_local_index), pElement->GetNodeLocation(localIndex)) / next_edge_length;
 
        return next_edge_gradient;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
c_vector<double, SPACE_DIM> VertexMesh<ELEMENT_DIM, SPACE_DIM>::GetPerimeterGradientOfElementAtNode(
    [[maybe_unused]] VertexElement<ELEMENT_DIM, SPACE_DIM>* pElement, [[maybe_unused]] unsigned localIndex)
{
    if constexpr (SPACE_DIM == 2)
    {
 
        c_vector<double, SPACE_DIM> previous_edge_gradient = GetPreviousEdgeGradientOfElementAtNode(pElement, localIndex);
        c_vector<double, SPACE_DIM> next_edge_gradient = GetNextEdgeGradientOfElementAtNode(pElement, localIndex);
 
        return previous_edge_gradient + next_edge_gradient;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
//                        3D-specific methods                       //
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::CalculateUnitNormalToFaceWithArea(
    [[maybe_unused]] VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* pFace,
    [[maybe_unused]] c_vector<double, SPACE_DIM>& rNormal)
{
    if constexpr (SPACE_DIM == 3)
    {
        assert(SPACE_DIM == 3); // LCOV_EXCL_LINE - code will be removed at compile time
 
        // As we are in 3D, the face must have at least three vertices
        assert(pFace->GetNumNodes() >= 3u);
 
        // Reset the answer
        rNormal = zero_vector<double>(SPACE_DIM);
 
        c_vector<double, SPACE_DIM> v_minus_v0 = this->GetVectorFromAtoB(pFace->GetNode(0)->rGetLocation(), pFace->GetNode(1)->rGetLocation());
        for (unsigned local_index = 2; local_index < pFace->GetNumNodes(); local_index++)
        {
            c_vector<double, SPACE_DIM> vnext_minus_v0 = this->GetVectorFromAtoB(pFace->GetNode(0)->rGetLocation(), pFace->GetNode(local_index)->rGetLocation());
            rNormal += VectorProduct(v_minus_v0, vnext_minus_v0);
            v_minus_v0 = vnext_minus_v0;
        }
        double magnitude = norm_2(rNormal);
        if (magnitude != 0.0)
        {
            // Normalize the normal vector
            rNormal /= magnitude;
            // If all points are co-located, then magnitude==0.0 and there is potential for a floating point exception
            // here if we divide by zero, so we'll move on.
        }
        return magnitude / 2.0;
    }
    else
    {
        NEVER_REACHED;
    }
}
 
template <unsigned ELEMENT_DIM, unsigned SPACE_DIM>
double VertexMesh<ELEMENT_DIM, SPACE_DIM>::CalculateAreaOfFace(
    [[maybe_unused]] VertexElement<ELEMENT_DIM - 1, SPACE_DIM>* pFace)
{
    if constexpr (SPACE_DIM == 3)
    {
        // Get the unit normal to the plane of this face
        c_vector<double, SPACE_DIM> unit_normal;
        return CalculateUnitNormalToFaceWithArea(pFace, unit_normal);
    }
    else
    {
        NEVER_REACHED;
    }
}
 
 
 
#if defined(__xlC__)
template <>
double VertexMesh<1, 1>::CalculateAreaOfFace(VertexElement<0, 1>* pFace)
{
    NEVER_REACHED;
}
#endif
 
// Explicit instantiation
template class VertexMesh<1, 1>;
template class VertexMesh<1, 2>;
template class VertexMesh<1, 3>;
template class VertexMesh<2, 2>;
template class VertexMesh<2, 3>;
template class VertexMesh<3, 3>;
 
// Serialization for Boost >= 1.36
#include "SerializationExportWrapperForCpp.hpp"
EXPORT_TEMPLATE_CLASS_ALL_DIMS(VertexMesh)