Intrepid2_IntegratedLegendreBasis_HGRAD_TRI.hpp

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#ifndef Intrepid2_IntegratedLegendreBasis_HGRAD_TRI_h
#define Intrepid2_IntegratedLegendreBasis_HGRAD_TRI_h

#include <Kokkos_View.hpp>
#include <Kokkos_DynRankView.hpp>

#include <Intrepid2_config.h>

#include "Intrepid2_Polynomials.hpp"
#include "Intrepid2_Utils.hpp"

namespace Intrepid2
{
  template<class ExecutionSpace, class OutputScalar, class PointScalar,
           class OutputFieldType, class InputPointsType>
  struct Hierarchical_HGRAD_TRI_Functor
  {
    using ScratchSpace       = typename ExecutionSpace::scratch_memory_space;
    using OutputScratchView  = Kokkos::View<OutputScalar*,ScratchSpace,Kokkos::MemoryTraits<Kokkos::Unmanaged>>;
    using PointScratchView   = Kokkos::View<PointScalar*, ScratchSpace,Kokkos::MemoryTraits<Kokkos::Unmanaged>>;
    
    using TeamPolicy = Kokkos::TeamPolicy<ExecutionSpace>;
    using TeamMember = typename TeamPolicy::member_type;
    
    EOperator opType_;
    
    OutputFieldType  output_;      // F,P
    InputPointsType  inputPoints_; // P,D
    
    int polyOrder_;
    bool defineVertexFunctions_;
    int numFields_, numPoints_;
    
    size_t fad_size_output_;
    
    static const int numVertices = 3;
    static const int numEdges    = 3;
    const int edge_start_[numEdges] = {0,1,0}; // edge i is from edge_start_[i] to edge_end_[i]
    const int edge_end_[numEdges]   = {1,2,2}; // edge i is from edge_start_[i] to edge_end_[i]
    
    Hierarchical_HGRAD_TRI_Functor(EOperator opType, OutputFieldType output, InputPointsType inputPoints,
                                    int polyOrder, bool defineVertexFunctions)
    : opType_(opType), output_(output), inputPoints_(inputPoints),
      polyOrder_(polyOrder), defineVertexFunctions_(defineVertexFunctions),
      fad_size_output_(getScalarDimensionForView(output))
    {
      numFields_ = output.extent_int(0);
      numPoints_ = output.extent_int(1);
      INTREPID2_TEST_FOR_EXCEPTION(numPoints_ != inputPoints.extent_int(0), std::invalid_argument, "point counts need to match!");
      INTREPID2_TEST_FOR_EXCEPTION(numFields_ != (polyOrder_+1)*(polyOrder_+2)/2, std::invalid_argument, "output field size does not match basis cardinality");
    }
    
    KOKKOS_INLINE_FUNCTION
    void operator()( const TeamMember & teamMember ) const
    {
      auto pointOrdinal = teamMember.league_rank();
      OutputScratchView edge_field_values_at_point, jacobi_values_at_point, other_values_at_point, other_values2_at_point;
      if (fad_size_output_ > 0) {
        edge_field_values_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
        jacobi_values_at_point     = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
        other_values_at_point      = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
        other_values2_at_point     = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1, fad_size_output_);
      }
      else {
        edge_field_values_at_point = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
        jacobi_values_at_point     = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
        other_values_at_point      = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
        other_values2_at_point     = OutputScratchView(teamMember.team_shmem(), polyOrder_ + 1);
      }
      
      const auto & x = inputPoints_(pointOrdinal,0);
      const auto & y = inputPoints_(pointOrdinal,1);
      
      // write as barycentric coordinates:
      const PointScalar lambda[3]    = {1. - x - y, x, y};
      const PointScalar lambda_dx[3] = {-1., 1., 0.};
      const PointScalar lambda_dy[3] = {-1., 0., 1.};
      
      const int num1DEdgeFunctions = polyOrder_ - 1;
      
      switch (opType_)
      {
        case OPERATOR_VALUE:
        {
          // vertex functions come first, according to vertex ordering: (0,0), (1,0), (0,1)
          for (int vertexOrdinal=0; vertexOrdinal<numVertices; vertexOrdinal++)
          {
            output_(vertexOrdinal,pointOrdinal) = lambda[vertexOrdinal];
          }
          if (!defineVertexFunctions_)
          {
            // "DG" basis case
            // here, we overwrite the first vertex function with 1:
            output_(0,pointOrdinal) = 1.0;
          }
          
          // edge functions
          int fieldOrdinalOffset = 3;
          for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
          {
            const auto & s0 = lambda[edge_start_[edgeOrdinal]];
            const auto & s1 = lambda[  edge_end_[edgeOrdinal]];
            
            Polynomials::shiftedScaledIntegratedLegendreValues(edge_field_values_at_point, polyOrder_, PointScalar(s1), PointScalar(s0+s1));
            for (int edgeFunctionOrdinal=0; edgeFunctionOrdinal<num1DEdgeFunctions; edgeFunctionOrdinal++)
            {
              // the first two integrated legendre functions are essentially the vertex functions; hence the +2 on on the RHS here:
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal) = edge_field_values_at_point(edgeFunctionOrdinal+2);
            }
            fieldOrdinalOffset += num1DEdgeFunctions;
          }
          
          // face functions
          {
            // these functions multiply the edge functions from the 01 edge by integrated Jacobi functions, appropriately scaled
            const double jacobiScaling = 1.0; // s0 + s1 + s2
            
            for (int i=2; i<polyOrder_; i++)
            {
              const int edgeBasisOrdinal = i+numVertices-2; // i+1: where the value of the edge function is stored in output_
              const auto & edgeValue = output_(edgeBasisOrdinal,pointOrdinal);
              const double alpha = i*2.0;
              
              Polynomials::integratedJacobiValues(jacobi_values_at_point, alpha, polyOrder_-2, lambda[2], jacobiScaling);
              for (int j=1; i+j <= polyOrder_; j++)
              {
                const auto & jacobiValue = jacobi_values_at_point(j);
                output_(fieldOrdinalOffset,pointOrdinal) = edgeValue * jacobiValue;
                fieldOrdinalOffset++;
              }
            }
          }
        } // end OPERATOR_VALUE
          break;
        case OPERATOR_GRAD:
        case OPERATOR_D1:
        {
          // vertex functions
          if (defineVertexFunctions_)
          {
            // standard, "CG" basis case
            // first vertex function is 1-x-y
            output_(0,pointOrdinal,0) = -1.0;
            output_(0,pointOrdinal,1) = -1.0;
          }
          else
          {
            // "DG" basis case
            // here, the first "vertex" function is 1, so the derivative is 0:
            output_(0,pointOrdinal,0) = 0.0;
            output_(0,pointOrdinal,1) = 0.0;
          }
          // second vertex function is x
          output_(1,pointOrdinal,0) = 1.0;
          output_(1,pointOrdinal,1) = 0.0;
          // third vertex function is y
          output_(2,pointOrdinal,0) = 0.0;
          output_(2,pointOrdinal,1) = 1.0;
          
          // edge functions
          int fieldOrdinalOffset = 3;
          /*
           Per Fuentes et al. (see Appendix E.1, E.2), the edge functions, defined for i ≥ 2, are
             [L_i](s0,s1) = L_i(s1; s0+s1)
           and have gradients:
             grad [L_i](s0,s1) = [P_{i-1}](s0,s1) grad s1 + [R_{i-1}](s0,s1) grad (s0 + s1)
           where
             [R_{i-1}](s0,s1) = R_{i-1}(s1; s0+s1) = d/dt L_{i}(s0; s0+s1)
           The P_i we have implemented in shiftedScaledLegendreValues, while d/dt L_{i+1} is
           implemented in shiftedScaledIntegratedLegendreValues_dt.
           */
          // rename the scratch memory to match our usage here:
          auto & P_i_minus_1 = edge_field_values_at_point;
          auto & L_i_dt      = jacobi_values_at_point;
          for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
          {
            const auto & s0 = lambda[edge_start_[edgeOrdinal]];
            const auto & s1 = lambda[  edge_end_[edgeOrdinal]];
            
            const auto & s0_dx = lambda_dx[edge_start_[edgeOrdinal]];
            const auto & s0_dy = lambda_dy[edge_start_[edgeOrdinal]];
            const auto & s1_dx = lambda_dx[  edge_end_[edgeOrdinal]];
            const auto & s1_dy = lambda_dy[  edge_end_[edgeOrdinal]];
            
            Polynomials::shiftedScaledLegendreValues             (P_i_minus_1, polyOrder_-1, PointScalar(s1), PointScalar(s0+s1));
            Polynomials::shiftedScaledIntegratedLegendreValues_dt(L_i_dt,      polyOrder_,   PointScalar(s1), PointScalar(s0+s1));
            for (int edgeFunctionOrdinal=0; edgeFunctionOrdinal<num1DEdgeFunctions; edgeFunctionOrdinal++)
            {
              // the first two (integrated) Legendre functions are essentially the vertex functions; hence the +2 here:
              const int i = edgeFunctionOrdinal+2;
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal,0) = P_i_minus_1(i-1) * s1_dx + L_i_dt(i) * (s1_dx + s0_dx);
              output_(edgeFunctionOrdinal+fieldOrdinalOffset,pointOrdinal,1) = P_i_minus_1(i-1) * s1_dy + L_i_dt(i) * (s1_dy + s0_dy);
            }
            fieldOrdinalOffset += num1DEdgeFunctions;
          }
          
          /*
           Fuentes et al give the face functions as phi_{ij}, with gradient:
             grad phi_{ij}(s0,s1,s2) = [L^{2i}_j](s0+s1,s2) grad [L_i](s0,s1) + [L_i](s0,s1) grad [L^{2i}_j](s0+s1,s2)
           where:
           - grad [L_i](s0,s1) is the edge function gradient we computed above
           - [L_i](s0,s1) is the edge function which we have implemented above (in OPERATOR_VALUE)
           - L^{2i}_j is a Jacobi polynomial with:
               [L^{2i}_j](s0,s1) = L^{2i}_j(s1;s0+s1)
             and the gradient for j ≥ 1 is
               grad [L^{2i}_j](s0,s1) = [P^{2i}_{j-1}](s0,s1) grad s1 + [R^{2i}_{j-1}(s0,s1)] grad (s0 + s1)
           Here,
             [P^{2i}_{j-1}](s0,s1) = P^{2i}_{j-1}(s1,s0+s1)
           and
             [R^{2i}_{j-1}(s0,s1)] = d/dt L^{2i}_j(s1,s0+s1)
           We have implemented P^{alpha}_{j} as shiftedScaledJacobiValues,
           and d/dt L^{alpha}_{j} as integratedJacobiValues_dt.
           */
          // rename the scratch memory to match our usage here:
          auto & P_2i_j_minus_1 = edge_field_values_at_point;
          auto & L_2i_j_dt      = jacobi_values_at_point;
          auto & L_i            = other_values_at_point;
          auto & L_2i_j         = other_values2_at_point;
          {
            // face functions multiply the edge functions from the 01 edge by integrated Jacobi functions, appropriately scaled
            const double jacobiScaling = 1.0; // s0 + s1 + s2

            for (int i=2; i<polyOrder_; i++)
            {
              // the edge function here is for edge 01, in the first set of edge functions.
              const int edgeBasisOrdinal = i+numVertices-2; // i+1: where the value of the edge function is stored in output_
              const auto & grad_L_i_dx = output_(edgeBasisOrdinal,pointOrdinal,0);
              const auto & grad_L_i_dy = output_(edgeBasisOrdinal,pointOrdinal,1);
              
              const double alpha = i*2.0;

              Polynomials::shiftedScaledIntegratedLegendreValues(L_i, polyOrder_, lambda[1], lambda[0]+lambda[1]);
              Polynomials::integratedJacobiValues_dt(     L_2i_j_dt, alpha, polyOrder_,   lambda[2], jacobiScaling);
              Polynomials::integratedJacobiValues   (        L_2i_j, alpha, polyOrder_,   lambda[2], jacobiScaling);
              Polynomials::shiftedScaledJacobiValues(P_2i_j_minus_1, alpha, polyOrder_-1, lambda[2], jacobiScaling);
              
              const auto & s0_dx = lambda_dx[0];
              const auto & s0_dy = lambda_dy[0];
              const auto & s1_dx = lambda_dx[1];
              const auto & s1_dy = lambda_dy[1];
              const auto & s2_dx = lambda_dx[2];
              const auto & s2_dy = lambda_dy[2];
              
              for (int j=1; i+j <= polyOrder_; j++)
              {
                const OutputScalar basisValue_dx = L_2i_j(j) * grad_L_i_dx + L_i(i) * (P_2i_j_minus_1(j-1) * s2_dx + L_2i_j_dt(j) * (s0_dx + s1_dx + s2_dx));
                const OutputScalar basisValue_dy = L_2i_j(j) * grad_L_i_dy + L_i(i) * (P_2i_j_minus_1(j-1) * s2_dy + L_2i_j_dt(j) * (s0_dy + s1_dy + s2_dy));
                
                output_(fieldOrdinalOffset,pointOrdinal,0) = basisValue_dx;
                output_(fieldOrdinalOffset,pointOrdinal,1) = basisValue_dy;
                fieldOrdinalOffset++;
              }
            }
          }
        }
          break;
        case OPERATOR_D2:
        case OPERATOR_D3:
        case OPERATOR_D4:
        case OPERATOR_D5:
        case OPERATOR_D6:
        case OPERATOR_D7:
        case OPERATOR_D8:
        case OPERATOR_D9:
        case OPERATOR_D10:
          INTREPID2_TEST_FOR_ABORT( true,
                                   ">>> ERROR: (Intrepid2::Basis_HGRAD_TRI_Cn_FEM_ORTH::OrthPolynomialTri) Computing of second and higher-order derivatives is not currently supported");
        default:
          // unsupported operator type
          device_assert(false);
      }
    }
    
    // Provide the shared memory capacity.
    // This function takes the team_size as an argument,
    // which allows team_size-dependent allocations.
    size_t team_shmem_size (int team_size) const
    {
      // we will use shared memory to create a fast buffer for basis computations
      size_t shmem_size = 0;
      if (fad_size_output_ > 0)
        shmem_size += 4 * OutputScratchView::shmem_size(polyOrder_ + 1, fad_size_output_);
      else
        shmem_size += 4 * OutputScratchView::shmem_size(polyOrder_ + 1);
      
      return shmem_size;
    }
  };
  
  template<typename ExecutionSpace=Kokkos::DefaultExecutionSpace,
           typename OutputScalar = double,
           typename PointScalar  = double,
           bool defineVertexFunctions = true>            // if defineVertexFunctions is true, first three basis functions are 1-x-y, x, and y.  Otherwise, they are 1, x, and y.
  class IntegratedLegendreBasis_HGRAD_TRI
  : public Basis<ExecutionSpace,OutputScalar,PointScalar>
  {
  public:
    using OrdinalTypeArray1DHost = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OrdinalTypeArray1DHost;
    using OrdinalTypeArray2DHost = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OrdinalTypeArray2DHost;
    
    using OutputViewType = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::OutputViewType;
    using PointViewType  = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::PointViewType;
    using ScalarViewType = typename Basis<ExecutionSpace,OutputScalar,PointScalar>::ScalarViewType;
  protected:
    int polyOrder_; // the maximum order of the polynomial
    bool defineVertexFunctions_; // if true, first and second basis functions are x and 1-x.  Otherwise, they are 1 and x.
  public:
    IntegratedLegendreBasis_HGRAD_TRI(int polyOrder)
    :
    polyOrder_(polyOrder)
    {
      this->basisCardinality_  = ((polyOrder+2) * (polyOrder+1)) / 2;
      this->basisDegree_       = polyOrder;
      this->basisCellTopology_ = shards::CellTopology(shards::getCellTopologyData<shards::Triangle<> >() );
      this->basisType_         = BASIS_FEM_HIERARCHICAL;
      this->basisCoordinates_  = COORDINATES_CARTESIAN;
      this->functionSpace_     = FUNCTION_SPACE_HGRAD;
      
      const int degreeLength = 1;
      this->fieldOrdinalPolynomialDegree_ = OrdinalTypeArray2DHost("Integrated Legendre H(grad) triangle polynomial degree lookup", this->basisCardinality_, degreeLength);
      
      int fieldOrdinalOffset = 0;
      // **** vertex functions **** //
      const int numVertices = this->basisCellTopology_.getVertexCount();
      const int numFunctionsPerVertex = 1;
      const int numVertexFunctions = numVertices * numFunctionsPerVertex;
      for (int i=0; i<numVertexFunctions; i++)
      {
        // for H(grad) on triangle, if defineVertexFunctions is false, first three basis members are linear
        // if not, then the only difference is that the first member is constant
        this->fieldOrdinalPolynomialDegree_(i,0) = 1;
      }
      if (!defineVertexFunctions)
      {
        this->fieldOrdinalPolynomialDegree_(0,0) = 0;
      }
      fieldOrdinalOffset += numVertexFunctions;
      
      // **** edge functions **** //
      const int numFunctionsPerEdge = polyOrder - 1; // bubble functions: all but the vertices
      const int numEdges            = this->basisCellTopology_.getEdgeCount();
      for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
      {
        for (int i=0; i<numFunctionsPerEdge; i++)
        {
          this->fieldOrdinalPolynomialDegree_(i+fieldOrdinalOffset,0) = i+2; // vertex functions are 1st order; edge functions start at order 2
        }
        fieldOrdinalOffset += numFunctionsPerEdge;
      }
      
      // **** face functions **** //
      const int max_ij_sum = polyOrder;
      for (int i=2; i<max_ij_sum; i++)
      {
        for (int j=1; i+j<=max_ij_sum; j++)
        {
          this->fieldOrdinalPolynomialDegree_(fieldOrdinalOffset,0) = i+j;
          fieldOrdinalOffset++;
        }
      }
      const int numFaces = 1;
      const int numFunctionsPerFace = ((polyOrder-1)*(polyOrder-2))/2;
      INTREPID2_TEST_FOR_EXCEPTION(fieldOrdinalOffset != this->basisCardinality_, std::invalid_argument, "Internal error: basis enumeration is incorrect");
      
      // initialize tags
      {
        const auto & cardinality = this->basisCardinality_;
        
        // Basis-dependent initializations
        const ordinal_type tagSize  = 4;        // size of DoF tag, i.e., number of fields in the tag
        const ordinal_type posScDim = 0;        // position in the tag, counting from 0, of the subcell dim
        const ordinal_type posScOrd = 1;        // position in the tag, counting from 0, of the subcell ordinal
        const ordinal_type posDfOrd = 2;        // position in the tag, counting from 0, of DoF ordinal relative to the subcell
        
        OrdinalTypeArray1DHost tagView("tag view", cardinality*tagSize);
        const int vertexDim = 0, edgeDim = 1, faceDim = 2;

        if (defineVertexFunctions) {
          {
            int tagNumber = 0;
            for (int vertexOrdinal=0; vertexOrdinal<numVertices; vertexOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerVertex; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = vertexDim;             // vertex dimension
                tagView(tagNumber*tagSize+1) = vertexOrdinal;         // vertex id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerVertex; // total number of dofs in this vertex
                tagNumber++;
              }
            }
            for (int edgeOrdinal=0; edgeOrdinal<numEdges; edgeOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerEdge; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = edgeDim;               // edge dimension
                tagView(tagNumber*tagSize+1) = edgeOrdinal;           // edge id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerEdge;   // total number of dofs on this edge
                tagNumber++;
              }
            }
            for (int faceOrdinal=0; faceOrdinal<numFaces; faceOrdinal++)
            {
              for (int functionOrdinal=0; functionOrdinal<numFunctionsPerFace; functionOrdinal++)
              {
                tagView(tagNumber*tagSize+0) = faceDim;               // face dimension
                tagView(tagNumber*tagSize+1) = faceOrdinal;           // face id
                tagView(tagNumber*tagSize+2) = functionOrdinal;       // local dof id
                tagView(tagNumber*tagSize+3) = numFunctionsPerFace;   // total number of dofs on this face
                tagNumber++;
              }
            }
          }
        } else {
          for (ordinal_type i=0;i<cardinality;++i) {
            tagView(i*tagSize+0) = faceDim;     // face dimension
            tagView(i*tagSize+1) = 0;           // face id
            tagView(i*tagSize+2) = i;           // local dof id
            tagView(i*tagSize+3) = cardinality; // total number of dofs on this face
          }
        }
        
        // Basis-independent function sets tag and enum data in tagToOrdinal_ and ordinalToTag_ arrays:
        // tags are constructed on host
        this->setOrdinalTagData(this->tagToOrdinal_,
                                this->ordinalToTag_,
                                tagView,
                                this->basisCardinality_,
                                tagSize,
                                posScDim,
                                posScOrd,
                                posDfOrd);
      }
    }
    
    const char* getName() const override {
      return "Intrepid2_IntegratedLegendreBasis_HGRAD_TRI";
    }
    
    virtual bool requireOrientation() const override {
      return (this->getDegree() > 2);
    }

    // since the getValues() below only overrides the FEM variant, we specify that
    // we use the base class's getValues(), which implements the FVD variant by throwing an exception.
    // (It's an error to use the FVD variant on this basis.)
    using Basis<ExecutionSpace,OutputScalar,PointScalar>::getValues;
    
    virtual void getValues( OutputViewType outputValues, const PointViewType inputPoints,
                           const EOperator operatorType = OPERATOR_VALUE ) const override
    {
      auto numPoints = inputPoints.extent_int(0);
      
      using FunctorType = Hierarchical_HGRAD_TRI_Functor<ExecutionSpace, OutputScalar, PointScalar, OutputViewType, PointViewType>;
      
      FunctorType functor(operatorType, outputValues, inputPoints, polyOrder_, defineVertexFunctions);
      
      const int outputVectorSize = getVectorSizeForHierarchicalParallelism<OutputScalar>();
      const int pointVectorSize  = getVectorSizeForHierarchicalParallelism<PointScalar>();
      const int vectorSize = std::max(outputVectorSize,pointVectorSize);
      const int teamSize = 1; // because of the way the basis functions are computed, we don't have a second level of parallelism...
      
      auto policy = Kokkos::TeamPolicy<ExecutionSpace>(numPoints,teamSize,vectorSize);
      Kokkos::parallel_for( policy , functor, "Hierarchical_HGRAD_TRI_Functor");
    }
  };
} // end namespace Intrepid2

#endif /* Intrepid2_IntegratedLegendreBasis_HGRAD_TRI_h */