stokes.h

finite element library for mathematic majored research

      const double psitilde_bs_1_1 = 2.5*y + 1.5;
      const double psitilde_bs_2_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.7071067812*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
      const double basisvalue1 = 1.732050808*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
      const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
      const double basisvalue3 = 2.738612788*psitilde_a_2*scalings_y_2*psitilde_bs_2_0;
      const double basisvalue4 = 2.121320344*psitilde_a_1*scalings_y_1*psitilde_bs_1_1;
      const double basisvalue5 = 1.224744871*psitilde_a_0*scalings_y_0*psitilde_bs_0_2;
    
      // Table(s) of coefficients
      const static double coefficients0[6][6] =   \
      {{0, -0.1732050808, -0.1, 0.1217161239, 0.09428090416, 0.0544331054},
      {0, 0.1732050808, -0.1, 0.1217161239, -0.09428090416, 0.0544331054},
      {0, 0, 0.2, 0, 0, 0.1632993162},
      {0.4714045208, 0.2309401077, 0.1333333333, 0, 0.1885618083, -0.1632993162},
      {0.4714045208, -0.2309401077, 0.1333333333, 0, -0.1885618083, -0.1632993162},
      {0.4714045208, 0, -0.2666666667, -0.2434322478, 0, 0.0544331054}};
    
      // 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 coeff0_3 =   coefficients0[dof][3];
      const double coeff0_4 =   coefficients0[dof][4];
      const double coeff0_5 =   coefficients0[dof][5];
    
      // Compute value(s)
      values[0] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2 + coeff0_3*basisvalue3 + coeff0_4*basisvalue4 + coeff0_5*basisvalue5;
    }
    
    if (6 <= i && i <= 11)
    {
      // Map degree of freedom to element degree of freedom
      const unsigned int dof = i - 6;
    
      // Generate scalings
      const double scalings_y_0 = 1;
      const double scalings_y_1 = scalings_y_0*(0.5 - 0.5*y);
      const double scalings_y_2 = scalings_y_1*(0.5 - 0.5*y);
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
      const double psitilde_a_1 = x;
      const double psitilde_a_2 = 1.5*x*psitilde_a_1 - 0.5*psitilde_a_0;
    
      // 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_0_2 = 0.1111111111*psitilde_bs_0_1 + 1.666666667*y*psitilde_bs_0_1 - 0.5555555556*psitilde_bs_0_0;
      const double psitilde_bs_1_0 = 1;
      const double psitilde_bs_1_1 = 2.5*y + 1.5;
      const double psitilde_bs_2_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.7071067812*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
      const double basisvalue1 = 1.732050808*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
      const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
      const double basisvalue3 = 2.738612788*psitilde_a_2*scalings_y_2*psitilde_bs_2_0;
      const double basisvalue4 = 2.121320344*psitilde_a_1*scalings_y_1*psitilde_bs_1_1;
      const double basisvalue5 = 1.224744871*psitilde_a_0*scalings_y_0*psitilde_bs_0_2;
    
      // Table(s) of coefficients
      const static double coefficients0[6][6] =   \
      {{0, -0.1732050808, -0.1, 0.1217161239, 0.09428090416, 0.0544331054},
      {0, 0.1732050808, -0.1, 0.1217161239, -0.09428090416, 0.0544331054},
      {0, 0, 0.2, 0, 0, 0.1632993162},
      {0.4714045208, 0.2309401077, 0.1333333333, 0, 0.1885618083, -0.1632993162},
      {0.4714045208, -0.2309401077, 0.1333333333, 0, -0.1885618083, -0.1632993162},
      {0.4714045208, 0, -0.2666666667, -0.2434322478, 0, 0.0544331054}};
    
      // 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 coeff0_3 =   coefficients0[dof][3];
      const double coeff0_4 =   coefficients0[dof][4];
      const double coeff0_5 =   coefficients0[dof][5];
    
      // Compute value(s)
      values[1] = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2 + coeff0_3*basisvalue3 + coeff0_4*basisvalue4 + coeff0_5*basisvalue5;
    }
    
  }

  /// 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;
    
    // Get coordinates and map to the reference (UFC) element
    double x = (element_coordinates[0][1]*element_coordinates[2][0] -\
                element_coordinates[0][0]*element_coordinates[2][1] +\
                J_11*coordinates[0] - J_01*coordinates[1]) / detJ;
    double y = (element_coordinates[1][1]*element_coordinates[0][0] -\
                element_coordinates[1][0]*element_coordinates[0][1] -\
                J_10*coordinates[0] + J_00*coordinates[1]) / detJ;
    
    // Map coordinates to the reference square
    if (std::abs(y - 1.0) < 1e-09)
      x = -1.0;
    else
      x = 2.0 *x/(1.0 - y) - 1.0;
    y = 2.0*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
    const double Jinv[2][2] =  {{J_11 / detJ, -J_01 / detJ}, {-J_10 / detJ, 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;
    
    if (0 <= i && i <= 5)
    {
      // 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);
      const double scalings_y_2 = scalings_y_1*(0.5 - 0.5*y);
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
      const double psitilde_a_1 = x;
      const double psitilde_a_2 = 1.5*x*psitilde_a_1 - 0.5*psitilde_a_0;
    
      // 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_0_2 = 0.1111111111*psitilde_bs_0_1 + 1.666666667*y*psitilde_bs_0_1 - 0.5555555556*psitilde_bs_0_0;
      const double psitilde_bs_1_0 = 1;
      const double psitilde_bs_1_1 = 2.5*y + 1.5;
      const double psitilde_bs_2_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.7071067812*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
      const double basisvalue1 = 1.732050808*psitilde_a_1*scalings_y_1*psitilde_bs_1_0;
      const double basisvalue2 = psitilde_a_0*scalings_y_0*psitilde_bs_0_1;
      const double basisvalue3 = 2.738612788*psitilde_a_2*scalings_y_2*psitilde_bs_2_0;
      const double basisvalue4 = 2.121320344*psitilde_a_1*scalings_y_1*psitilde_bs_1_1;
      const double basisvalue5 = 1.224744871*psitilde_a_0*scalings_y_0*psitilde_bs_0_2;
    
      // Table(s) of coefficients
      const static double coefficients0[6][6] =   \
      {{0, -0.1732050808, -0.1, 0.1217161239, 0.09428090416, 0.0544331054},
      {0, 0.1732050808, -0.1, 0.1217161239, -0.09428090416, 0.0544331054},
      {0, 0, 0.2, 0, 0, 0.1632993162},
      {0.4714045208, 0.2309401077, 0.1333333333, 0, 0.1885618083, -0.1632993162},
      {0.4714045208, -0.2309401077, 0.1333333333, 0, -0.1885618083, -0.1632993162},
      {0.4714045208, 0, -0.2666666667, -0.2434322478, 0, 0.0544331054}};
    
      // Interesting (new) part
      // Tables of derivatives of the polynomial base (transpose)
      const static double dmats0[6][6] =   \
      {{0, 0, 0, 0, 0, 0},
      {4.898979486, 0, 0, 0, 0, 0},
      {0, 0, 0, 0, 0, 0},
      {0, 9.486832981, 0, 0, 0, 0},
      {4, 0, 7.071067812, 0, 0, 0},
      {0, 0, 0, 0, 0, 0}};
    
      const static double dmats1[6][6] =   \
      {{0, 0, 0, 0, 0, 0},
      {2.449489743, 0, 0, 0, 0, 0},
      {4.242640687, 0, 0, 0, 0, 0},
      {2.581988897, 4.74341649, -0.9128709292, 0, 0, 0},
      {2, 6.123724357, 3.535533906, 0, 0, 0},
      {-2.309401077, 0, 8.164965809, 0, 0, 0}};
    
      // Compute reference derivatives
      // Declare pointer to array of derivatives on FIAT element
      double *derivatives = new double [num_derivatives];
    
      // Declare coefficients
      double coeff0_0 = 0;
      double coeff0_1 = 0;
      double coeff0_2 = 0;
      double coeff0_3 = 0;
      double coeff0_4 = 0;
      double coeff0_5 = 0;
    
      // Declare new coefficients
      double new_coeff0_0 = 0;
      double new_coeff0_1 = 0;
      double new_coeff0_2 = 0;
      double new_coeff0_3 = 0;
      double new_coeff0_4 = 0;
      double new_coeff0_5 = 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_coeff0_3 = coefficients0[dof][3];
        new_coeff0_4 = coefficients0[dof][4];
        new_coeff0_5 = coefficients0[dof][5];
    
        // 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;
          coeff0_3 = new_coeff0_3;
          coeff0_4 = new_coeff0_4;
          coeff0_5 = new_coeff0_5;
    
          if(combinations[deriv_num][j] == 0)
          {
            new_coeff0_0 = coeff0_0*dmats0[0][0] + coeff0_1*dmats0[1][0] + coeff0_2*dmats0[2][0] + coeff0_3*dmats0[3][0] + coeff0_4*dmats0[4][0] + coeff0_5*dmats0[5][0];
            new_coeff0_1 = coeff0_0*dmats0[0][1] + coeff0_1*dmats0[1][1] + coeff0_2*dmats0[2][1] + coeff0_3*dmats0[3][1] + coeff0_4*dmats0[4][1] + coeff0_5*dmats0[5][1];
            new_coeff0_2 = coeff0_0*dmats0[0][2] + coeff0_1*dmats0[1][2] + coeff0_2*dmats0[2][2] + coeff0_3*dmats0[3][2] + coeff0_4*dmats0[4][2] + coeff0_5*dmats0[5][2];
            new_coeff0_3 = coeff0_0*dmats0[0][3] + coeff0_1*dmats0[1][3] + coeff0_2*dmats0[2][3] + coeff0_3*dmats0[3][3] + coeff0_4*dmats0[4][3] + coeff0_5*dmats0[5][3];
            new_coeff0_4 = coeff0_0*dmats0[0][4] + coeff0_1*dmats0[1][4] + coeff0_2*dmats0[2][4] + coeff0_3*dmats0[3][4] + coeff0_4*dmats0[4][4] + coeff0_5*dmats0[5][4];
            new_coeff0_5 = coeff0_0*dmats0[0][5] + coeff0_1*dmats0[1][5] + coeff0_2*dmats0[2][5] + coeff0_3*dmats0[3][5] + coeff0_4*dmats0[4][5] + coeff0_5*dmats0[5][5];
          }
          if(combinations[deriv_num][j] == 1)
          {
            new_coeff0_0 = coeff0_0*dmats1[0][0] + coeff0_1*dmats1[1][0] + coeff0_2*dmats1[2][0] + coeff0_3*dmats1[3][0] + coeff0_4*dmats1[4][0] + coeff0_5*dmats1[5][0];
            new_coeff0_1 = coeff0_0*dmats1[0][1] + coeff0_1*dmats1[1][1] + coeff0_2*dmats1[2][1] + coeff0_3*dmats1[3][1] + coeff0_4*dmats1[4][1] + coeff0_5*dmats1[5][1];
            new_coeff0_2 = coeff0_0*dmats1[0][2] + coeff0_1*dmats1[1][2] + coeff0_2*dmats1[2][2] + coeff0_3*dmats1[3][2] + coeff0_4*dmats1[4][2] + coeff0_5*dmats1[5][2];
            new_coeff0_3 = coeff0_0*dmats1[0][3] + coeff0_1*dmats1[1][3] + coeff0_2*dmats1[2][3] + coeff0_3*dmats1[3][3] + coeff0_4*dmats1[4][3] + coeff0_5*dmats1[5][3];
            new_coeff0_4 = coeff0_0*dmats1[0][4] + coeff0_1*dmats1[1][4] + coeff0_2*dmats1[2][4] + coeff0_3*dmats1[3][4] + coeff0_4*dmats1[4][4] + coeff0_5*dmats1[5][4];
            new_coeff0_5 = coeff0_0*dmats1[0][5] + coeff0_1*dmats1[1][5] + coeff0_2*dmats1[2][5] + coeff0_3*dmats1[3][5] + coeff0_4*dmats1[4][5] + coeff0_5*dmats1[5][5];
          }
    
        }
        // 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 + new_coeff0_3*basisvalue3 + new_coeff0_4*basisvalue4 + new_coeff0_5*basisvalue5;
      }
    
      // 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];
        }
      }
      // Delete pointer to array of derivatives on FIAT element
      delete [] derivatives;
    
      // Delete pointer to array of combinations of derivatives and transform
      for (unsigned int row = 0; row < num_derivatives; row++)
      {
        delete [] combinations[row];
        delete [] transform[row];
      }
    
      delete [] combinations;
      delete [] transform;