oomph::TBubbleEnrichedElementShape< 2, 3 > Class Reference

#include <Telements.h>

Public Member Functions
unsigned	n_enriched_nodes ()
	Return the number of nodes required for enrichement. More...
void	local_coordinate_of_node (const unsigned &j, Vector< double > &s) const
	Return local coordinates of node j. More...
void	shape (const Vector< double > &s, Shape &psi) const
	Shape function for specific TBubbleEnrichedElement<2,3> More...
void	dshape_local (const Vector< double > &s, Shape &psi, DShape &dpsids) const
	Derivatives of shape functions for specific TBubbleElement<2,3> More...
void	d2shape_local (const Vector< double > &s, Shape &psi, DShape &dpsids, DShape &d2psids) const

Detailed Description

//////////////////////////////////////////////////////////////////// Specific Enriched TElementShape inline functions /////////////////////////////////////////////////////////////////// Standard quadratic shape functions enriched by the addition of a cubic bubble, which consists of adding a single node at the centroid

Member Function Documentation

d2shape_local()

void oomph::TBubbleEnrichedElementShape< 2, 3 >::d2shape_local ( const Vector< double > &  s, Shape &  psi, DShape &  dpsids, DShape &  d2psids  ) const

inline

Second derivatives of shape functions for specific TBubbleElement<2,3>: d2psids(i,0) = \( \partial^2 \psi_j / \partial s_0^2 \) d2psids(i,1) = \( \partial^2 \psi_j / \partial s_1^2 \) d2psids(i,2) = \( \partial^2 \psi_j / \partial s_0 \partial s_1 \)

    {
      // ALH: Don't know why object qualifier is needed
      this->dshape_local(s, psi, dpsids);
 
      // Calculate derivatives of bubble functions
      const double d2_bubble_ds0 = -2.0 * s[1];
      const double d2_bubble_ds1 = -2.0 * s[0];
      const double d2_bubble_ds2 = 1.0 - 2.0 * s[0] - 2.0 * s[1];
 
      d2psids(0, 0) = 4.0 + 3.0 * d2_bubble_ds0;
      d2psids(0, 1) = 0.0 + 3.0 * d2_bubble_ds1;
      d2psids(0, 2) = 0.0 + 3.0 * d2_bubble_ds2;
 
      d2psids(1, 0) = 0.0 + 3.0 * d2_bubble_ds0;
      d2psids(1, 1) = 4.0 + 3.0 * d2_bubble_ds1;
      d2psids(1, 2) = 0.0 + 3.0 * d2_bubble_ds2;
 
      d2psids(2, 0) = 4.0 + 3.0 * d2_bubble_ds0;
      d2psids(2, 1) = 4.0 + 3.0 * d2_bubble_ds1;
      d2psids(2, 2) = 4.0 + 3.0 * d2_bubble_ds2;
 
      d2psids(3, 0) = 0.0 - 12.0 * d2_bubble_ds0;
      d2psids(3, 1) = 0.0 - 12.0 * d2_bubble_ds1;
      d2psids(3, 2) = 4.0 - 12.0 * d2_bubble_ds2;
 
      d2psids(4, 0) = 0.0 - 12.0 * d2_bubble_ds0;
      d2psids(4, 1) = -8.0 - 12.0 * d2_bubble_ds1;
      d2psids(4, 2) = -4.0 - 12.0 * d2_bubble_ds2;
 
      d2psids(5, 0) = -8.0 - 12.0 * d2_bubble_ds0;
      d2psids(5, 1) = 0.0 - 12.0 * d2_bubble_ds1;
      d2psids(5, 2) = -4.0 - 12.0 * d2_bubble_ds2;
 
      d2psids(6, 0) = 27.0 * d2_bubble_ds0;
      d2psids(6, 1) = 27.0 * d2_bubble_ds1;
      d2psids(6, 2) = 27.0 * d2_bubble_ds2;
    }

References s.

dshape_local()

void oomph::TBubbleEnrichedElementShape< 2, 3 >::dshape_local ( const Vector< double > &  s, Shape &  psi, DShape &  dpsids  ) const

inline

Derivatives of shape functions for specific TBubbleElement<2,3>

    {
      // ALH: Don't know why object qualifier is needed
      this->shape(s, psi);
 
      // Calculate derivatives of bubble functions
      const double d_bubble_ds0 = s[1] * (1.0 - s[1] - 2.0 * s[0]);
      const double d_bubble_ds1 = s[0] * (1.0 - s[0] - 2.0 * s[1]);
 
      // Add the appropriate derivatives to the shape functions
      dpsids(0, 0) = 4.0 * s[0] - 1.0 + 3.0 * d_bubble_ds0;
      dpsids(0, 1) = 0.0 + 3.0 * d_bubble_ds1;
      dpsids(1, 0) = 0.0 + 3.0 * d_bubble_ds0;
      dpsids(1, 1) = 4.0 * s[1] - 1.0 + 3.0 * d_bubble_ds1;
      dpsids(2, 0) = 2.0 * (2.0 * s[0] - 1.5 + 2.0 * s[1]) + 3.0 * d_bubble_ds0;
      dpsids(2, 1) = 2.0 * (2.0 * s[0] - 1.5 + 2.0 * s[1]) + 3.0 * d_bubble_ds1;
      dpsids(3, 0) = 4.0 * s[1] - 12.0 * d_bubble_ds0;
      dpsids(3, 1) = 4.0 * s[0] - 12.0 * d_bubble_ds1;
      dpsids(4, 0) = -4.0 * s[1] - 12.0 * d_bubble_ds0;
      dpsids(4, 1) = 4.0 * (1.0 - s[0] - 2.0 * s[1]) - 12.0 * d_bubble_ds1;
      dpsids(5, 0) = 4.0 * (1.0 - 2.0 * s[0] - s[1]) - 12.0 * d_bubble_ds0;
      dpsids(5, 1) = -4.0 * s[0] - 12.0 * d_bubble_ds1;
      dpsids(6, 0) = 27.0 * d_bubble_ds0;
      dpsids(6, 1) = 27.0 * d_bubble_ds1;
    }

References s, and oomph::OneDimLagrange::shape().

local_coordinate_of_node()

void oomph::TBubbleEnrichedElementShape< 2, 3 >::local_coordinate_of_node ( const unsigned &  j, Vector< double > &  s  ) const

inline

Return local coordinates of node j.

    {
      s.resize(2);
 
      switch (j)
      {
        case 0:
          s[0] = 1.0;
          s[1] = 0.0;
          break;
 
        case 1:
          s[0] = 0.0;
          s[1] = 1.0;
          break;
 
        case 2:
          s[0] = 0.0;
          s[1] = 0.0;
          break;
 
        case 3:
          s[0] = 0.5;
          s[1] = 0.5;
          break;
 
        case 4:
          s[0] = 0.0;
          s[1] = 0.5;
          break;
 
        case 5:
          s[0] = 0.5;
          s[1] = 0.0;
          break;
 
          // Add the centroid as the enriched node
        case 6:
          s[0] = 1.0 / 3.0;
          s[1] = 1.0 / 3.0;
          break;
 
        default:
          std::ostringstream error_message;
          error_message
            << "Element only has seven nodes; called with node number " << j
            << std::endl;
 
          throw OomphLibError(error_message.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
      }
    }

References j, OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION, and s.

n_enriched_nodes()

unsigned oomph::TBubbleEnrichedElementShape< 2, 3 >::n_enriched_nodes ( )

inline

Return the number of nodes required for enrichement.

    {
      return 1;
    }

shape()

void oomph::TBubbleEnrichedElementShape< 2, 3 >::shape ( const Vector< double > &  s, Shape &  psi  ) const

inline

Shape function for specific TBubbleEnrichedElement<2,3>

    {
      // Reconstruct the third area coordinate
      const double s_2 = 1.0 - s[0] - s[1];
 
      // Calculate the enrichment function
      const double cubic_bubble = s[0] * s[1] * s_2;
      // The appropriate amount of the cubic bubble function is
      // added/subtracted to each original quadratic shape function to ensure
      // that it is zero at the centroid (1/3,1/3).
 
      // note that s[2] needs replacing by s_2 everywhere since only
      // two independent variables s[0],s[1] and s_2 is expressed in terms of
      // those later.
      psi[0] = 2.0 * s[0] * (s[0] - 0.5) + 3.0 * cubic_bubble;
      psi[1] = 2.0 * s[1] * (s[1] - 0.5) + 3.0 * cubic_bubble;
      psi[2] = 2.0 * s_2 * (s_2 - 0.5) + 3.0 * cubic_bubble;
      psi[3] = 4.0 * s[0] * s[1] - 12.0 * cubic_bubble;
      psi[4] = 4.0 * s[1] * s_2 - 12.0 * cubic_bubble;
      psi[5] = 4.0 * s_2 * s[0] - 12.0 * cubic_bubble;
      // The bubble function scaled to have magnitude one at (1/3,1/3)
      psi[6] = 27.0 * cubic_bubble;
    }

References s.

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The documentation for this class was generated from the following file:

• Telements.h