optimization.h

finite element library for mathematic majored research

    else
      x = 2.0 *x/(1.0 - y) - 1.0;
    y = 2.0*y - 1.0;
    
    // Reset values
    *values = 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);
    const double scalings_y_2 = scalings_y_1*(0.5 - 0.5*y);
    const double scalings_y_3 = scalings_y_2*(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;
    const double psitilde_a_3 = 1.666666667*x*psitilde_a_2 - 0.6666666667*psitilde_a_1;
    
    // 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_0_3 = 0.05*psitilde_bs_0_2 + 1.75*y*psitilde_bs_0_2 - 0.7*psitilde_bs_0_1;
    const double psitilde_bs_1_0 = 1;
    const double psitilde_bs_1_1 = 2.5*y + 1.5;
    const double psitilde_bs_1_2 = 0.54*psitilde_bs_1_1 + 2.1*y*psitilde_bs_1_1 - 0.56*psitilde_bs_1_0;
    const double psitilde_bs_2_0 = 1;
    const double psitilde_bs_2_1 = 3.5*y + 2.5;
    const double psitilde_bs_3_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;
    const double basisvalue6 = 3.741657387*psitilde_a_3*scalings_y_3*psitilde_bs_3_0;
    const double basisvalue7 = 3.16227766*psitilde_a_2*scalings_y_2*psitilde_bs_2_1;
    const double basisvalue8 = 2.449489743*psitilde_a_1*scalings_y_1*psitilde_bs_1_2;
    const double basisvalue9 = 1.414213562*psitilde_a_0*scalings_y_0*psitilde_bs_0_3;
    
    // Table(s) of coefficients
    const static double coefficients0[10][10] = \
    {{0.04714045208, -0.02886751346, -0.01666666667, 0.07824607964, 0.06060915267, 0.03499271061, -0.06013377943, -0.05082231954, -0.03936679944, -0.02272843225},
    {0.04714045208, 0.02886751346, -0.01666666667, 0.07824607964, -0.06060915267, 0.03499271061, 0.06013377943, -0.05082231954, 0.03936679944, -0.02272843225},
    {0.04714045208, 0, 0.03333333333, 0, 0, 0.1049781318, 0, 0, 0, 0.09091372901},
    {0.1060660172, 0.2598076211, -0.15, 0.1173691195, 0.06060915267, -0.07873359888, 0, 0.1016446391, -0.1312226648, 0.09091372901},
    {0.1060660172, 0, 0.3, 0, 0.1515228817, 0.02624453296, 0, 0, 0.1312226648, -0.1363705935},
    {0.1060660172, -0.2598076211, -0.15, 0.1173691195, -0.06060915267, -0.07873359888, 0, 0.1016446391, 0.1312226648, 0.09091372901},
    {0.1060660172, 0, 0.3, 0, -0.1515228817, 0.02624453296, 0, 0, -0.1312226648, -0.1363705935},
    {0.1060660172, -0.2598076211, -0.15, -0.07824607964, 0.09091372901, 0.09622995418, 0.1804013383, 0.05082231954, -0.01312226648, -0.02272843225},
    {0.1060660172, 0.2598076211, -0.15, -0.07824607964, -0.09091372901, 0.09622995418, -0.1804013383, 0.05082231954, 0.01312226648, -0.02272843225},
    {0.6363961031, 0, 0, -0.2347382389, 0, -0.2624453296, 0, -0.2032892782, 0, 0.09091372901}};
    
    // 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];
    const double coeff0_6 = coefficients0[dof][6];
    const double coeff0_7 = coefficients0[dof][7];
    const double coeff0_8 = coefficients0[dof][8];
    const double coeff0_9 = coefficients0[dof][9];
    
    // Compute value(s)
    *values = coeff0_0*basisvalue0 + coeff0_1*basisvalue1 + coeff0_2*basisvalue2 + coeff0_3*basisvalue3 + coeff0_4*basisvalue4 + coeff0_5*basisvalue5 + coeff0_6*basisvalue6 + coeff0_7*basisvalue7 + coeff0_8*basisvalue8 + coeff0_9*basisvalue9;
  }

  /// 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 < 1*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);
    const double scalings_y_2 = scalings_y_1*(0.5 - 0.5*y);
    const double scalings_y_3 = scalings_y_2*(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;
    const double psitilde_a_3 = 1.666666667*x*psitilde_a_2 - 0.6666666667*psitilde_a_1;
    
    // 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_0_3 = 0.05*psitilde_bs_0_2 + 1.75*y*psitilde_bs_0_2 - 0.7*psitilde_bs_0_1;
    const double psitilde_bs_1_0 = 1;
    const double psitilde_bs_1_1 = 2.5*y + 1.5;
    const double psitilde_bs_1_2 = 0.54*psitilde_bs_1_1 + 2.1*y*psitilde_bs_1_1 - 0.56*psitilde_bs_1_0;
    const double psitilde_bs_2_0 = 1;
    const double psitilde_bs_2_1 = 3.5*y + 2.5;
    const double psitilde_bs_3_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;
    const double basisvalue6 = 3.741657387*psitilde_a_3*scalings_y_3*psitilde_bs_3_0;
    const double basisvalue7 = 3.16227766*psitilde_a_2*scalings_y_2*psitilde_bs_2_1;
    const double basisvalue8 = 2.449489743*psitilde_a_1*scalings_y_1*psitilde_bs_1_2;
    const double basisvalue9 = 1.414213562*psitilde_a_0*scalings_y_0*psitilde_bs_0_3;
    
    // Table(s) of coefficients
    const static double coefficients0[10][10] = \
    {{0.04714045208, -0.02886751346, -0.01666666667, 0.07824607964, 0.06060915267, 0.03499271061, -0.06013377943, -0.05082231954, -0.03936679944, -0.02272843225},
    {0.04714045208, 0.02886751346, -0.01666666667, 0.07824607964, -0.06060915267, 0.03499271061, 0.06013377943, -0.05082231954, 0.03936679944, -0.02272843225},
    {0.04714045208, 0, 0.03333333333, 0, 0, 0.1049781318, 0, 0, 0, 0.09091372901},
    {0.1060660172, 0.2598076211, -0.15, 0.1173691195, 0.06060915267, -0.07873359888, 0, 0.1016446391, -0.1312226648, 0.09091372901},
    {0.1060660172, 0, 0.3, 0, 0.1515228817, 0.02624453296, 0, 0, 0.1312226648, -0.1363705935},
    {0.1060660172, -0.2598076211, -0.15, 0.1173691195, -0.06060915267, -0.07873359888, 0, 0.1016446391, 0.1312226648, 0.09091372901},
    {0.1060660172, 0, 0.3, 0, -0.1515228817, 0.02624453296, 0, 0, -0.1312226648, -0.1363705935},
    {0.1060660172, -0.2598076211, -0.15, -0.07824607964, 0.09091372901, 0.09622995418, 0.1804013383, 0.05082231954, -0.01312226648, -0.02272843225},
    {0.1060660172, 0.2598076211, -0.15, -0.07824607964, -0.09091372901, 0.09622995418, -0.1804013383, 0.05082231954, 0.01312226648, -0.02272843225},
    {0.6363961031, 0, 0, -0.2347382389, 0, -0.2624453296, 0, -0.2032892782, 0, 0.09091372901}};
    
    // Interesting (new) part
    // Tables of derivatives of the polynomial base (transpose)
    const static double dmats0[10][10] = \
    {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {4.898979486, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {0, 9.486832981, 0, 0, 0, 0, 0, 0, 0, 0},
    {4, 0, 7.071067812, 0, 0, 0, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {5.291502622, 0, -2.993325909, 13.66260102, 0, 0.6110100927, 0, 0, 0, 0},
    {0, 4.38178046, 0, 0, 12.52198067, 0, 0, 0, 0, 0},
    {3.464101615, 0, 7.838367177, 0, 0, 8.4, 0, 0, 0, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}};
    
    const static double dmats1[10][10] = \
    {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {2.449489743, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {4.242640687, 0, 0, 0, 0, 0, 0, 0, 0, 0},
    {2.581988897, 4.74341649, -0.9128709292, 0, 0, 0, 0, 0, 0, 0},
    {2, 6.123724357, 3.535533906, 0, 0, 0, 0, 0, 0, 0},
    {-2.309401077, 0, 8.164965809, 0, 0, 0, 0, 0, 0, 0},
    {2.645751311, 5.184592559, -1.496662955, 6.831300511, -1.058300524, 0.3055050463, 0, 0, 0, 0},
    {2.236067977, 2.19089023, 2.529822128, 8.082903769, 6.260990337, -1.807392228, 0, 0, 0, 0},
    {1.732050808, -5.091168825, 3.919183588, 0, 9.699484522, 4.2, 0, 0, 0, 0},
    {5, 0, -2.828427125, 0, 0, 12.12435565, 0, 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;
    double coeff0_6 = 0;
    double coeff0_7 = 0;
    double coeff0_8 = 0;
    double coeff0_9 = 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;
    double new_coeff0_6 = 0;
    double new_coeff0_7 = 0;
    double new_coeff0_8 = 0;
    double new_coeff0_9 = 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];
      new_coeff0_6 = coefficients0[dof][6];
      new_coeff0_7 = coefficients0[dof][7];
      new_coeff0_8 = coefficients0[dof][8];
      new_coeff0_9 = coefficients0[dof][9];