mixedpoisson.h

Dolfin provide a high-performance linear algebra library

    // Compute determinant of Jacobian
    const double detJ = J_00*J_11 - J_01*J_10;
    
    // Compute constants
    const double C0 = element_coordinates[1][0] + element_coordinates[2][0];
    const double C1 = element_coordinates[1][1] + element_coordinates[2][1];
    
    // Get coordinates and map to the reference (FIAT) element
    double x = (J_01*C1 - J_11*C0 + 2.0*J_11*coordinates[0] - 2.0*J_01*coordinates[1]) / detJ;
    double y = (J_10*C0 - J_00*C1 - 2.0*J_10*coordinates[0] + 2.0*J_00*coordinates[1]) / detJ;
    
    // Map coordinates to the reference square
    if (std::abs(y - 1.0) < 1e-14)
      x = -1.0;
    else
      x = 2.0 * (1.0 + x)/(1.0 - y) - 1.0;
    
    // Reset values
    values[0] = 0;
    values[1] = 0;
    
    // Map degree of freedom to element degree of freedom
    const unsigned int dof = i;
    
    // Generate scalings
    const double scalings_y_0 = 1;
    const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
    
    // Compute psitilde_a
    const double psitilde_a_0 = 1;
    const double psitilde_a_1 = x;
    
    // Compute psitilde_bs
    const double psitilde_bs_0_0 = 1;
    const double psitilde_bs_0_1 = 1.5*y + 0.5;
    const double psitilde_bs_1_0 = 1;
    
    // Compute basisvalues
    const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
    const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
    const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
    
    // Table(s) of coefficients
    const static double coefficients0[6][3] = \
    {{0.942809041582063, 0.577350269189626, -0.333333333333333},
    {-0.471404520791032, -0.288675134594813, 0.166666666666667},
    {0.471404520791032, -0.577350269189626, -0.666666666666667},
    {0.471404520791032, 0.288675134594813, 0.833333333333333},
    {-0.471404520791032, -0.288675134594813, 0.166666666666667},
    {0.942809041582063, 0.577350269189626, -0.333333333333333}};
    
    const static double coefficients1[6][3] = \
    {{-0.471404520791032, 0, -0.333333333333333},
    {0.942809041582063, 0, 0.666666666666667},
    {0.471404520791032, 0, 0.333333333333333},
    {-0.942809041582063, 0, -0.666666666666667},
    {-0.471404520791032, 0.866025403784439, 0.166666666666667},
    {-0.471404520791032, -0.866025403784439, 0.166666666666667}};
    
    // Extract relevant coefficients
    const double coeff0_0 = coefficients0[dof][0];
    const double coeff0_1 = coefficients0[dof][1];
    const double coeff0_2 = coefficients0[dof][2];
    const double coeff1_0 = coefficients1[dof][0];
    const double coeff1_1 = coefficients1[dof][1];
    const double coeff1_2 = coefficients1[dof][2];
    
    // Compute value(s)
    values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2;
    values[1] = coeff1_0*basisvalue0 + coeff1_1*basisvalue1 + coeff1_2*basisvalue2;
  }

  /// Evaluate order n derivatives of basis function i at given point in cell
  virtual void evaluate_basis_derivatives(unsigned int i,
                                          unsigned int n,
                                          double* values,
                                          const double* coordinates,
                                          const ufc::cell& c) const
  {
    // Extract vertex coordinates
    const double * const * element_coordinates = c.coordinates;
    
    // Compute Jacobian of affine map from reference cell
    const double J_00 = element_coordinates[1][0] - element_coordinates[0][0];
    const double J_01 = element_coordinates[2][0] - element_coordinates[0][0];
    const double J_10 = element_coordinates[1][1] - element_coordinates[0][1];
    const double J_11 = element_coordinates[2][1] - element_coordinates[0][1];
      
    // Compute determinant of Jacobian
    const double detJ = J_00*J_11 - J_01*J_10;
    
    // Compute constants
    const double C0 = element_coordinates[1][0] + element_coordinates[2][0];
    const double C1 = element_coordinates[1][1] + element_coordinates[2][1];
    
    // Get coordinates and map to the reference (FIAT) element
    double x = (J_01*C1 - J_11*C0 + 2.0*J_11*coordinates[0] - 2.0*J_01*coordinates[1]) / detJ;
    double y = (J_10*C0 - J_00*C1 - 2.0*J_10*coordinates[0] + 2.0*J_00*coordinates[1]) / detJ;
    
    // Map coordinates to the reference square
    if (std::abs(y - 1.0) < 1e-14)
      x = -1.0;
    else
      x = 2.0 * (1.0 + x)/(1.0 - y) - 1.0;
    
    // Compute number of derivatives
    unsigned int num_derivatives = 1;
    
    for (unsigned int j = 0; j < n; j++)
      num_derivatives *= 2;
    
    
    // Declare pointer to two dimensional array that holds combinations of derivatives and initialise
    unsigned int **combinations = new unsigned int *[num_derivatives];
        
    for (unsigned int j = 0; j < num_derivatives; j++)
    {
      combinations[j] = new unsigned int [n];
      for (unsigned int k = 0; k < n; k++)
        combinations[j][k] = 0;
    }
        
    // Generate combinations of derivatives
    for (unsigned int row = 1; row < num_derivatives; row++)
    {
      for (unsigned int num = 0; num < row; num++)
      {
        for (unsigned int col = n-1; col+1 > 0; col--)
        {
          if (combinations[row][col] + 1 > 1)
            combinations[row][col] = 0;
          else
          {
            combinations[row][col] += 1;
            break;
          }
        }
      }
    }
    
    // Compute inverse of Jacobian, components are scaled because of difference in FFC/FIAT reference elements
    const double Jinv[2][2] =  {{2*J_11 / detJ, -2*J_01 / detJ}, {-2*J_10 / detJ, 2*J_00 / detJ}};
    
    // Declare transformation matrix
    // Declare pointer to two dimensional array and initialise
    double **transform = new double *[num_derivatives];
        
    for (unsigned int j = 0; j < num_derivatives; j++)
    {
      transform[j] = new double [num_derivatives];
      for (unsigned int k = 0; k < num_derivatives; k++)
        transform[j][k] = 1;
    }
    
    // Construct transformation matrix
    for (unsigned int row = 0; row < num_derivatives; row++)
    {
      for (unsigned int col = 0; col < num_derivatives; col++)
      {
        for (unsigned int k = 0; k < n; k++)
          transform[row][col] *= Jinv[combinations[col][k]][combinations[row][k]];
      }
    }
    
    // Reset values
    for (unsigned int j = 0; j < 2*num_derivatives; j++)
      values[j] = 0;
    
    // Map degree of freedom to element degree of freedom
    const unsigned int dof = i;
    
    // Generate scalings
    const double scalings_y_0 = 1;
    const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
    
    // Compute psitilde_a
    const double psitilde_a_0 = 1;
    const double psitilde_a_1 = x;
    
    // Compute psitilde_bs
    const double psitilde_bs_0_0 = 1;
    const double psitilde_bs_0_1 = 1.5*y + 0.5;
    const double psitilde_bs_1_0 = 1;
    
    // Compute basisvalues
    const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
    const double basisvalue1 = 1.73205080756888*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
    const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
    
    // Table(s) of coefficients
    const static double coefficients0[6][3] = \
    {{0.942809041582063, 0.577350269189626, -0.333333333333333},
    {-0.471404520791032, -0.288675134594813, 0.166666666666667},
    {0.471404520791032, -0.577350269189626, -0.666666666666667},
    {0.471404520791032, 0.288675134594813, 0.833333333333333},
    {-0.471404520791032, -0.288675134594813, 0.166666666666667},
    {0.942809041582063, 0.577350269189626, -0.333333333333333}};
    
    const static double coefficients1[6][3] = \
    {{-0.471404520791032, 0, -0.333333333333333},
    {0.942809041582063, 0, 0.666666666666667},
    {0.471404520791032, 0, 0.333333333333333},
    {-0.942809041582063, 0, -0.666666666666667},
    {-0.471404520791032, 0.866025403784439, 0.166666666666667},
    {-0.471404520791032, -0.866025403784439, 0.166666666666667}};
    
    // Interesting (new) part
    // Tables of derivatives of the polynomial base (transpose)
    const static double dmats0[3][3] = \
    {{0, 0, 0},
    {2.44948974278318, 0, 0},
    {0, 0, 0}};
    
    const static double dmats1[3][3] = \
    {{0, 0, 0},
    {1.22474487139159, 0, 0},
    {2.12132034355964, 0, 0}};
    
    // Compute reference derivatives
    // Declare pointer to array of derivatives on FIAT element
    double *derivatives = new double [2*num_derivatives];
    
    // Declare coefficients
    double coeff0_0 = 0;
    double coeff0_1 = 0;
    double coeff0_2 = 0;
    double coeff1_0 = 0;
    double coeff1_1 = 0;
    double coeff1_2 = 0;
    
    // Declare new coefficients
    double new_coeff0_0 = 0;
    double new_coeff0_1 = 0;
    double new_coeff0_2 = 0;
    double new_coeff1_0 = 0;
    double new_coeff1_1 = 0;
    double new_coeff1_2 = 0;
    
    // Loop possible derivatives
    for (unsigned int deriv_num = 0; deriv_num < num_derivatives; deriv_num++)
    {
      // Get values from coefficients array
      new_coeff0_0 = coefficients0[dof][0];
      new_coeff0_1 = coefficients0[dof][1];
      new_coeff0_2 = coefficients0[dof][2];
      new_coeff1_0 = coefficients1[dof][0];
      new_coeff1_1 = coefficients1[dof][1];
      new_coeff1_2 = coefficients1[dof][2];
    
      // Loop derivative order
      for (unsigned int j = 0; j < n; j++)
      {
        // Update old coefficients
        coeff0_0 = new_coeff0_0;
        coeff0_1 = new_coeff0_1;
        coeff0_2 = new_coeff0_2;
        coeff1_0 = new_coeff1_0;
        coeff1_1 = new_coeff1_1;
        coeff1_2 = new_coeff1_2;
    
        if(combinations[deriv_num][j] == 0)
        {
          new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0];
          new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1];
          new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2];
          new_coeff1_0 = coeff1_0*dmats0[0][0] + coeff1_1*dmats0[1][0] + coeff1_2*dmats0[2][0];
          new_coeff1_1 = coeff1_0*dmats0[0][1] + coeff1_1*dmats0[1][1] + coeff1_2*dmats0[2][1];
          new_coeff1_2 = coeff1_0*dmats0[0][2] + coeff1_1*dmats0[1][2] + coeff1_2*dmats0[2][2];
        }
        if(combinations[deriv_num][j] == 1)
        {
          new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0];
          new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1];
          new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2];
          new_coeff1_0 = coeff1_0*dmats1[0][0] + coeff1_1*dmats1[1][0] + coeff1_2*dmats1[2][0];
          new_coeff1_1 = coeff1_0*dmats1[0][1] + coeff1_1*dmats1[1][1] + coeff1_2*dmats1[2][1];
          new_coeff1_2 = coeff1_0*dmats1[0][2] + coeff1_1*dmats1[1][2] + coeff1_2*dmats1[2][2];
        }
    
      }
      // Compute derivatives on reference element as dot product of coefficients and basisvalues
      derivatives[deriv_num] = new_coeff0_0*basisvalue0 + new_coeff0_1*basisvalue1 + new_coeff0_2*basisvalue2;
      derivatives[num_derivatives + deriv_num] = new_coeff1_0*basisvalue0 + new_coeff1_1*basisvalue1 + new_coeff1_2*basisvalue2;
    }
    
    // Transform derivatives back to physical element
    for (unsigned int row = 0; row < num_derivatives; row++)
    {
      for (unsigned int col = 0; col < num_derivatives; col++)
      {
        values[row] += transform[row][col]*derivatives[col];
        values[num_derivatives + row] += transform[row][col]*derivatives[num_derivatives + col];
      }
    }
    // Delete pointer to array of derivatives on FIAT element
    delete [] derivatives;
    
    // Delete pointer to array of combinations of derivatives
    delete [] combinations;
    
  }

  /// Evaluate linear functional for dof i on the function f
  virtual double evaluate_dof(unsigned int i,
                              const ufc::function& f,
                              const ufc::cell& c) const
  {
    throw std::runtime_error("evaluate_dof not implemented for this type of element");
  }

  /// Interpolate vertex values from dof values
  virtual void interpolate_vertex_values(double* vertex_values,
                                         const double* dof_values,
                                         const ufc::cell& c) const
  {
    // Extract vertex coordinates
    const double * const * x = c.coordinates;
    
    // Compute Jacobian of affine map from reference cell
    const double J_00 = x[1][0] - x[0][0];
    const double J_01 = x[2][0] - x[0][0];
    const double J_10 = x[1][1] - x[0][1];
    const double J_11 = x[2][1] - x[0][1];
      
    // Compute determinant of Jacobian
    double detJ = J_00*J_11 - J_01*J_10;
      
    // Compute inverse of Jacobian
    
    // Take absolute value of determinant
    detJ = std::abs(detJ);
    
    // Compute signs of edges (need to flip edge degrees of freedom)
    
    // Compute the edges
    const double e0_0 = x[2][0] - x[1][0];
    const double e0_1 = x[2][1] - x[1][1];
    const double e1_0 = x[2][0] - x[0][0];
    const double e1_1 = x[2][1] - x[0][1];
    const double e2_0 = x[1][0] - x[0][0];
    const double e2_1 = x[1][1] - x[0][1];
    
    // Compute edges normals by rotating edges 90 degrees clockwise
    const double n0_0 = e0_1;
    const double n0_1 = -e0_0;
    const double n1_0 = e1_1;
    const double n1_1 = -e1_0;
    const double n2_0 = e2_1;
    const double n2_1 = -e2_0;
    
    // Compute the orientation of the normals relative to the cell
    int sign_facet0 = n0_0*e2_0 + n0_1*e2_1 > 0 ? 1 : -1;
    int sign_facet1 = n1_0*e0_0 + n1_1*e0_1 > 0 ? 1 : -1;
    int sign_facet2 = n2_0*e1_0 + n2_1*e1_1 < 0 ? 1 : -1;
    
    // Evaluate at vertices and use Piola mapping
    vertex_values[0] = (1.0/detJ)*(sign_facet1*dof_values[2]*2*J_00 + sign_facet1*dof_values[3]*J_00 + sign_facet2*dof_values[4]*(-2*J_01)