mixedpoisson.h

Dolfin provide a high-performance linear algebra library

      // 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;
    
    }
    
    if (6 <= i and i <= 6)
    {
      // Map degree of freedom to element degree of freedom
      const unsigned int dof = i - 6;
    
      // Generate scalings
      const double scalings_y_0 = 1;
    
      // Compute psitilde_a
      const double psitilde_a_0 = 1;
    
      // Compute psitilde_bs
      const double psitilde_bs_0_0 = 1;
    
      // Compute basisvalues
      const double basisvalue0 = 0.707106781186548*psitilde_a_0*scalings_y_0*psitilde_bs_0_0;
    
      // Table(s) of coefficients
      const static double coefficients0[1][1] =   \
      {{1.41421356237309}};
    
      // Interesting (new) part
      // Tables of derivatives of the polynomial base (transpose)
      const static double dmats0[1][1] =   \
      {{0}};
    
      const static double dmats1[1][1] =   \
      {{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;
    
      // Declare new coefficients
      double new_coeff0_0 = 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];
    
        // Loop derivative order
        for (unsigned int j = 0; j < n; j++)
        {
          // Update old coefficients
          coeff0_0 = new_coeff0_0;
    
          if(combinations[deriv_num][j] == 0)
          {
            new_coeff0_0 = coeff0_0*dmats0[0][0];
          }
          if(combinations[deriv_num][j] == 1)
          {
            new_coeff0_0 = coeff0_0*dmats1[0][0];
          }
    
        }
        // Compute derivatives on reference element as dot product of coefficients and basisvalues
        derivatives[deriv_num] = new_coeff0_0*basisvalue0;
      }
    
      // 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[2*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
      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) + sign_facet2*dof_values[5]*J_01);
    vertex_values[1] = (1.0/detJ)*(sign_facet0*dof_values[0]*2*J_00 + sign_facet0*dof_values[1]*J_00 + sign_facet2*dof_values[4]*(J_00 + J_01) + sign_facet2*dof_values[5]*(2*J_00 - 2*J_01));
    vertex_values[2] = (1.0/detJ)*(sign_facet0*dof_values[0]*J_01 + sign_facet0*dof_values[1]*2*J_01 + sign_facet1*dof_values[2]*(J_00 + J_01) + sign_facet1*dof_values[3]*(2*J_00 - 2*J_01));
    vertex_values[3] = (1.0/detJ)*(sign_facet1*dof_values[2]*2*J_10 + sign_facet1*dof_values[3]*J_10 + sign_facet2*dof_values[4]*(-2*J_11) + sign_facet2*dof_values[5]*J_11);
    vertex_values[4] = (1.0/detJ)*(sign_facet0*dof_values[0]*2*J_10 + sign_facet0*dof_values[1]*J_10 + sign_facet2*dof_values[4]*(J_10 + J_11) + sign_facet2*dof_values[5]*(2*J_10 - 2*J_11));
    vertex_values[5] = (1.0/detJ)*(sign_facet0*dof_values[0]*J_11 + sign_facet0*dof_values[1]*2*J_11 + sign_facet1*dof_values[2]*(J_10 + J_11) + sign_facet1*dof_values[3]*(2*J_10 - 2*J_11));
    // Evaluate at vertices and use affine mapping
    vertex_values[6] = dof_values[6];
    vertex_values[7] = dof_values[6];
    vertex_values[8] = dof_values[6];
  }

  /// 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 UFC_MixedPoissonBilinearForm_finite_element_0_0();
      break;
    case 1:
      return new UFC_MixedPoissonBilinearForm_finite_element_0_1();
      break;
    }
    return 0;
  }

};

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

class UFC_MixedPoissonBilinearForm_finite_element_1_0: public ufc::finite_element
{
public:

  /// Constructor
  UFC_MixedPoissonBilinearForm_finite_element_1_0() : ufc::finite_element()
  {
    // Do nothing
  }

  /// Destructor
  virtual ~UFC_MixedPoissonBilinearForm_finite_element_1_0()
  {
    // Do nothing
  }

  /// Return a string identifying the finite element
  virtual const char* signature() const
  {
    return "Brezzi-Douglas-Marini 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 6;
  }

  /// Return the rank of the value space
  virtual unsigned int value_rank() const
  {
    return 1;
  }

  /// Return the dimension of the value space for axis i
  virtual unsigned int value_dimension(unsigned int i) const
  {
    return 2;
  }

  /// 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];