stokes.h

finite element library for mathematic majored research

    }
    
    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}};
    
      // 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[num_derivatives + 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;
    }
    
  }

  /// 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
  {
    double values[2];
    double coordinates[2];
    
    // Nodal coordinates on reference cell
    static double X[12][2] = {{0, 0}, {1, 0}, {0, 1}, {0.5, 0.5}, {0, 0.5}, {0.5, 0}, {0, 0}, {1, 0}, {0, 1}, {0.5, 0.5}, {0, 0.5}, {0.5, 0}};
    
    // Components for each dof
    static unsigned int components[12] = {0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1};
    
    // Extract vertex coordinates
    const double * const * x = c.coordinates;
    
    // Evaluate basis functions for affine mapping
    const double w0 = 1.0 - X[i][0] - X[i][1];
    const double w1 = X[i][0];
    const double w2 = X[i][1];
    
    // Compute affine mapping x = F(X)
    coordinates[0] = w0*x[0][0] + w1*x[1][0] + w2*x[2][0];
    coordinates[1] = w0*x[0][1] + w1*x[1][1] + w2*x[2][1];
    
    // Evaluate function at coordinates
    f.evaluate(values, coordinates, c);
    
    // Pick component for evaluation
    return values[components[i]];
  }

  /// Interpolate vertex values from dof values
  virtual void interpolate_vertex_values(double* vertex_values,
                                         const double* dof_values,
                                         const ufc::cell& c) const
  {
    // Evaluate at vertices and use affine mapping
    vertex_values[0] = dof_values[0];
    vertex_values[1] = dof_values[1];
    vertex_values[2] = dof_values[2];
    // Evaluate at vertices and use affine mapping
    vertex_values[3] = dof_values[6];
    vertex_values[4] = dof_values[7];
    vertex_values[5] = dof_values[8];
  }

  /// Return the number of sub elements (for a mixed element)
  virtual unsigned int num_sub_elements() const
  {
    return 2;
  }

  /// Create a new finite element for sub element i (for a mixed element)
  virtual ufc::finite_element* create_sub_element(unsigned int i) const
  {
    switch ( i )
    {
    case 0:
      return new StokesBilinearForm_finite_element_0_0_0();
      break;
    case 1:
      return new StokesBilinearForm_finite_element_0_0_1();
      break;
    }
    return 0;
  }

};

/// This class defines the interface for a finite element.

class StokesBilinearForm_finite_element_0_1: public ufc::finite_element
{
public:

  /// Constructor
  StokesBilinearForm_finite_element_0_1() : ufc::finite_element()
  {
    // Do nothing
  }

  /// Destructor
  virtual ~StokesBilinearForm_finite_element_0_1()
  {
    // Do nothing
  }

  /// Return a string identifying the finite element
  virtual const char* signature() const
  {
    return "Lagrange finite element of degree 1 on a triangle";
  }

  /// Return the cell shape
  virtual ufc::shape cell_shape() const
  {
    return ufc::triangle;
  }

  /// Return the dimension of the finite element function space
  virtual unsigned int space_dimension() const
  {
    return 3;
  }

  /// Return the rank of the value space
  virtual unsigned int value_rank() const
  {
    return 0;
  }

  /// Return the dimension of the value space for axis i
  virtual unsigned int value_dimension(unsigned int i) const
  {
    return 1;
  }

  /// Evaluate basis function i at given point in cell
  virtual void evaluate_basis(unsigned int i,
                              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;
    
    // 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);
    
    // 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 = 0.7071067812*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
    const double basisvalu