oomph::TBubbleEnrichedElementShape< 3, 3 > Class Reference

Return local coordinates of node j. More...

#include <Telements.h>

Public Member Functions
unsigned	n_enriched_nodes ()
void	local_coordinate_of_node (const unsigned &j, Vector< double > &s) const
void	shape (const Vector< double > &s, Shape &psi) const
	Shape function for specific TBubbleEnrichedElement<3,3> More...
void	dshape_local (const Vector< double > &s, Shape &psi, DShape &dpsids) const
	Derivatives of shape functions for specific TElement<3,3> More...
void	d2shape_local (const Vector< double > &s, Shape &psi, DShape &dpsids, DShape &d2psids) const

Detailed Description

Return local coordinates of node j.

/////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////// Standard quadratic shape functions enriched by the addition of three cubic "face" bubbles and quartic "volume" bubble, which consists of adding a node at the centroid of each face and a single node at the centroid of the tetrahedron

Member Function Documentation

d2shape_local()

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

inline

Second derivatives of shape functions for specific TElement<3,3>: 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) = \( \partial^2 \psi_j / \partial s_2^2 \) d2psids(i,3) = \( \partial^2 \psi_j / \partial s_0 \partial s_1 \) d2psids(i,4) = \( \partial^2 \psi_j / \partial s_0 \partial s_2 \) d2psids(i,5) = \( \partial^2 \psi_j / \partial s_1 \partial s_2 \)

    {
      // ALH: Don't know why object qualifier is needed
      this->dshape_local(s, psi, dpsids);
 
      // Define s3
      const double s3 = 1.0 - s[0] - s[1] - s[2];
 
      // Calculate second derivatives of the bubble functions
      //(.,3) for mixed derivative s[0]-s[1]
      //(.,4) for mixed derivative s[0]-s[2]
      //(.,5) for mixed derivative s[1]-s[2]
 
 
      const double d2_quartic_bubble_ds0 = -2.0 * s[1] * s[2];
      const double d2_quartic_bubble_ds1 = -2.0 * s[0] * s[2];
      const double d2_quartic_bubble_ds2 = -2.0 * s[0] * s[1];
      const double d2_quartic_bubble_ds3 =
        s[2] * (1.0 - 2.0 * s[0] - 2.0 * s[1] - s[2]);
      const double d2_quartic_bubble_ds4 =
        s[1] * (1.0 - 2.0 * s[0] - 2.0 * s[2] - s[1]);
      const double d2_quartic_bubble_ds5 =
        s[0] * (1.0 - 2.0 * s[1] - 2.0 * s[2] - s[0]);
 
      const double d2_cubic_bubble012_ds0 = 0.0;
      const double d2_cubic_bubble012_ds1 = 0.0;
      const double d2_cubic_bubble012_ds2 = 0.0;
      const double d2_cubic_bubble012_ds3 = s[2];
      const double d2_cubic_bubble012_ds4 = s[1];
      const double d2_cubic_bubble012_ds5 = s[0];
 
      const double d2_cubic_bubble013_ds0 = -2.0 * s[1];
      const double d2_cubic_bubble013_ds1 = -2.0 * s[0];
      const double d2_cubic_bubble013_ds2 = 0.0;
      const double d2_cubic_bubble013_ds3 = s3 - s[0] - s[1];
      const double d2_cubic_bubble013_ds4 = -s[1];
      const double d2_cubic_bubble013_ds5 = -s[0];
 
      const double d2_cubic_bubble023_ds0 = -2.0 * s[2];
      const double d2_cubic_bubble023_ds1 = 0.0;
      const double d2_cubic_bubble023_ds2 = -2.0 * s[0];
      const double d2_cubic_bubble023_ds3 = -s[2];
      const double d2_cubic_bubble023_ds4 = s3 - s[0] - s[2];
      const double d2_cubic_bubble023_ds5 = -s[0];
 
      const double d2_cubic_bubble123_ds0 = 0.0;
      const double d2_cubic_bubble123_ds1 = -2.0 * s[2];
      const double d2_cubic_bubble123_ds2 = -2.0 * s[1];
      const double d2_cubic_bubble123_ds3 = -s[2];
      const double d2_cubic_bubble123_ds4 = -s[1];
      const double d2_cubic_bubble123_ds5 = s3 - s[1] - s[2];
 
 
      d2psids(0, 0) = 4.0 +
                      3.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble013_ds0 +
                             d2_cubic_bubble023_ds0) -
                      4.0 * d2_quartic_bubble_ds0;
      d2psids(0, 1) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble013_ds1 +
                             d2_cubic_bubble023_ds1) -
                      4.0 * d2_quartic_bubble_ds1;
      d2psids(0, 2) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble013_ds2 +
                             d2_cubic_bubble023_ds2) -
                      4.0 * d2_quartic_bubble_ds2;
      d2psids(0, 3) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble013_ds3 +
                             d2_cubic_bubble023_ds3) -
                      4.0 * d2_quartic_bubble_ds3;
      d2psids(0, 4) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble013_ds4 +
                             d2_cubic_bubble023_ds4) -
                      4.0 * d2_quartic_bubble_ds4;
      d2psids(0, 5) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble013_ds5 +
                             d2_cubic_bubble023_ds5) -
                      4.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(1, 0) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble013_ds0 +
                             d2_cubic_bubble123_ds0) -
                      4.0 * d2_quartic_bubble_ds0;
      d2psids(1, 1) = 4.0 +
                      3.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble013_ds1 +
                             d2_cubic_bubble123_ds1) -
                      4.0 * d2_quartic_bubble_ds1;
      d2psids(1, 2) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble013_ds2 +
                             d2_cubic_bubble123_ds2) -
                      4.0 * d2_quartic_bubble_ds2;
      d2psids(1, 3) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble013_ds3 +
                             d2_cubic_bubble123_ds3) -
                      4.0 * d2_quartic_bubble_ds3;
      d2psids(1, 4) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble013_ds4 +
                             d2_cubic_bubble123_ds4) -
                      4.0 * d2_quartic_bubble_ds4;
      d2psids(1, 5) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble013_ds5 +
                             d2_cubic_bubble123_ds5) -
                      4.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(2, 0) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble023_ds0 +
                             d2_cubic_bubble123_ds0) -
                      4.0 * d2_quartic_bubble_ds0;
      d2psids(2, 1) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble023_ds1 +
                             d2_cubic_bubble123_ds1) -
                      4.0 * d2_quartic_bubble_ds1;
      d2psids(2, 2) = 4.0 +
                      3.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble023_ds2 +
                             d2_cubic_bubble123_ds2) -
                      4.0 * d2_quartic_bubble_ds2;
      d2psids(2, 3) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble023_ds3 +
                             d2_cubic_bubble123_ds3) -
                      4.0 * d2_quartic_bubble_ds3;
      d2psids(2, 4) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble023_ds4 +
                             d2_cubic_bubble123_ds4) -
                      4.0 * d2_quartic_bubble_ds4;
      d2psids(2, 5) = 0.0 +
                      3.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble023_ds5 +
                             d2_cubic_bubble123_ds5) -
                      4.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(3, 0) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds0 + d2_cubic_bubble023_ds0 +
                             d2_cubic_bubble123_ds0) -
                      4.0 * d2_quartic_bubble_ds0;
      d2psids(3, 1) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds1 + d2_cubic_bubble023_ds1 +
                             d2_cubic_bubble123_ds1) -
                      4.0 * d2_quartic_bubble_ds1;
      d2psids(3, 2) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds2 + d2_cubic_bubble023_ds2 +
                             d2_cubic_bubble123_ds2) -
                      4.0 * d2_quartic_bubble_ds2;
      d2psids(3, 3) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds3 + d2_cubic_bubble023_ds3 +
                             d2_cubic_bubble123_ds3) -
                      4.0 * d2_quartic_bubble_ds3;
      d2psids(3, 4) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds4 + d2_cubic_bubble023_ds4 +
                             d2_cubic_bubble123_ds4) -
                      4.0 * d2_quartic_bubble_ds4;
      d2psids(3, 5) = 4.0 +
                      3.0 * (d2_cubic_bubble013_ds5 + d2_cubic_bubble023_ds5 +
                             d2_cubic_bubble123_ds5) -
                      4.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(4, 0) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble013_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(4, 1) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble013_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(4, 2) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble013_ds2) +
                      32.0 * d2_quartic_bubble_ds2;
      d2psids(4, 3) = 4.0 -
                      12.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble013_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(4, 4) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble013_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(4, 5) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble013_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(5, 0) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble023_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(5, 1) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble023_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(5, 2) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble023_ds2) +
                      32.0 * d2_quartic_bubble_ds2;
      d2psids(5, 3) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble023_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(5, 4) = 4.0 -
                      12.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble023_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(5, 5) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble023_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
 
      d2psids(6, 0) = -8.0 -
                      12.0 * (d2_cubic_bubble013_ds0 + d2_cubic_bubble023_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(6, 1) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds1 + d2_cubic_bubble023_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(6, 2) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds2 + d2_cubic_bubble023_ds2) +
                      32.0 * d2_quartic_bubble_ds2;
      d2psids(6, 3) = -4.0 -
                      12.0 * (d2_cubic_bubble013_ds3 + d2_cubic_bubble023_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(6, 4) = -4.0 -
                      12.0 * (d2_cubic_bubble013_ds4 + d2_cubic_bubble023_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(6, 5) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds5 + d2_cubic_bubble023_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
      d2psids(7, 0) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds0 + d2_cubic_bubble123_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(7, 1) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds1 + d2_cubic_bubble123_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(7, 2) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds2 + d2_cubic_bubble123_ds2) +
                      32.0 * d2_quartic_bubble_ds2;
      d2psids(7, 3) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds3 + d2_cubic_bubble123_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(7, 4) = 0.0 -
                      12.0 * (d2_cubic_bubble012_ds4 + d2_cubic_bubble123_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(7, 5) = 4.0 -
                      12.0 * (d2_cubic_bubble012_ds5 + d2_cubic_bubble123_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
      d2psids(8, 0) = 0.0 -
                      12.0 * (d2_cubic_bubble023_ds0 + d2_cubic_bubble123_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(8, 1) = 0.0 -
                      12.0 * (d2_cubic_bubble023_ds1 + d2_cubic_bubble123_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(8, 2) = -8.0 -
                      12.0 * (d2_cubic_bubble023_ds2 + d2_cubic_bubble123_ds2) +
                      32.0 * d2_quartic_bubble_ds2;
      d2psids(8, 3) = 0.0 -
                      12.0 * (d2_cubic_bubble023_ds3 + d2_cubic_bubble123_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(8, 4) = -4.0 -
                      12.0 * (d2_cubic_bubble023_ds4 + d2_cubic_bubble123_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(8, 5) = -4.0 -
                      12.0 * (d2_cubic_bubble023_ds5 + d2_cubic_bubble123_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
      d2psids(9, 0) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds0 + d2_cubic_bubble123_ds0) +
                      32.0 * d2_quartic_bubble_ds0;
      d2psids(9, 1) = -8.0 -
                      12.0 * (d2_cubic_bubble013_ds1 + d2_cubic_bubble123_ds1) +
                      32.0 * d2_quartic_bubble_ds1;
      d2psids(9, 2) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds2 + d2_cubic_bubble123_ds2) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(9, 3) = -4.0 -
                      12.0 * (d2_cubic_bubble013_ds3 + d2_cubic_bubble123_ds3) +
                      32.0 * d2_quartic_bubble_ds3;
      d2psids(9, 4) = 0.0 -
                      12.0 * (d2_cubic_bubble013_ds4 + d2_cubic_bubble123_ds4) +
                      32.0 * d2_quartic_bubble_ds4;
      d2psids(9, 5) = -4.0 -
                      12.0 * (d2_cubic_bubble013_ds5 + d2_cubic_bubble123_ds5) +
                      32.0 * d2_quartic_bubble_ds5;
 
      // Add the bubble function on the face spanned by 0 1 3
      d2psids(10, 0) =
        27.0 * d2_cubic_bubble013_ds0 - 108.0 * d2_quartic_bubble_ds0;
      d2psids(10, 1) =
        27.0 * d2_cubic_bubble013_ds1 - 108.0 * d2_quartic_bubble_ds1;
      d2psids(10, 2) =
        27.0 * d2_cubic_bubble013_ds2 - 108.0 * d2_quartic_bubble_ds2;
      d2psids(10, 3) =
        27.0 * d2_cubic_bubble013_ds3 - 108.0 * d2_quartic_bubble_ds3;
      d2psids(10, 4) =
        27.0 * d2_cubic_bubble013_ds4 - 108.0 * d2_quartic_bubble_ds4;
      d2psids(10, 5) =
        27.0 * d2_cubic_bubble013_ds5 - 108.0 * d2_quartic_bubble_ds5;
 
      // Add the bubble function on the face spanned by 0 1 2
      d2psids(11, 0) =
        27.0 * d2_cubic_bubble012_ds0 - 108.0 * d2_quartic_bubble_ds0;
      d2psids(11, 1) =
        27.0 * d2_cubic_bubble012_ds1 - 108.0 * d2_quartic_bubble_ds1;
      d2psids(11, 2) =
        27.0 * d2_cubic_bubble012_ds2 - 108.0 * d2_quartic_bubble_ds2;
      d2psids(11, 3) =
        27.0 * d2_cubic_bubble012_ds3 - 108.0 * d2_quartic_bubble_ds3;
      d2psids(11, 4) =
        27.0 * d2_cubic_bubble012_ds4 - 108.0 * d2_quartic_bubble_ds4;
      d2psids(11, 5) =
        27.0 * d2_cubic_bubble012_ds5 - 108.0 * d2_quartic_bubble_ds5;
 
      // Add the bubble function on the face spanned by 0 2 3
      d2psids(12, 0) =
        27.0 * d2_cubic_bubble023_ds0 - 108.0 * d2_quartic_bubble_ds0;
      d2psids(12, 1) =
        27.0 * d2_cubic_bubble023_ds1 - 108.0 * d2_quartic_bubble_ds1;
      d2psids(12, 2) =
        27.0 * d2_cubic_bubble023_ds2 - 108.0 * d2_quartic_bubble_ds2;
      d2psids(12, 3) =
        27.0 * d2_cubic_bubble023_ds3 - 108.0 * d2_quartic_bubble_ds3;
      d2psids(12, 4) =
        27.0 * d2_cubic_bubble023_ds4 - 108.0 * d2_quartic_bubble_ds4;
      d2psids(12, 5) =
        27.0 * d2_cubic_bubble023_ds5 - 108.0 * d2_quartic_bubble_ds5;
 
      // Add the bubble function on the face spanned by 1 2 3
      d2psids(13, 0) =
        27.0 * d2_cubic_bubble123_ds0 - 108.0 * d2_quartic_bubble_ds0;
      d2psids(13, 1) =
        27.0 * d2_cubic_bubble123_ds1 - 108.0 * d2_quartic_bubble_ds1;
      d2psids(13, 2) =
        27.0 * d2_cubic_bubble123_ds2 - 108.0 * d2_quartic_bubble_ds2;
      d2psids(13, 3) =
        27.0 * d2_cubic_bubble123_ds3 - 108.0 * d2_quartic_bubble_ds3;
      d2psids(13, 4) =
        27.0 * d2_cubic_bubble123_ds4 - 108.0 * d2_quartic_bubble_ds4;
      d2psids(13, 5) =
        27.0 * d2_cubic_bubble123_ds5 - 108.0 * d2_quartic_bubble_ds5;
 
      // Add the volumetric bubble function derivatives
      d2psids(14, 0) = 256.0 * d2_quartic_bubble_ds0;
      d2psids(14, 1) = 256.0 * d2_quartic_bubble_ds1;
      d2psids(14, 2) = 256.0 * d2_quartic_bubble_ds2;
      d2psids(14, 3) = 256.0 * d2_quartic_bubble_ds3;
      d2psids(14, 4) = 256.0 * d2_quartic_bubble_ds4;
      d2psids(14, 5) = 256.0 * d2_quartic_bubble_ds5;
    }

References s.

dshape_local()

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

inline

Derivatives of shape functions for specific TElement<3,3>

    {
      // ALH: Don't know why object qualifier is needed
      this->shape(s, psi);
 
      // Define s3 the fourth volume coordinate
      const double s3 = 1.0 - s[0] - s[1] - s[2];
 
      // Calculate derivatives of the bubble function
      const double d_quartic_bubble_ds0 =
        s[1] * s[2] * (1.0 - s[1] - s[2] - 2.0 * s[0]);
      const double d_quartic_bubble_ds1 =
        s[0] * s[2] * (1.0 - s[0] - s[2] - 2.0 * s[1]);
      const double d_quartic_bubble_ds2 =
        s[0] * s[1] * (1.0 - s[0] - s[1] - 2.0 * s[2]);
 
      const double d_cubic_bubble012_ds0 = s[1] * s[2];
      const double d_cubic_bubble012_ds1 = s[0] * s[2];
      const double d_cubic_bubble012_ds2 = s[0] * s[1];
 
      const double d_cubic_bubble013_ds0 = s[1] * (s3 - s[0]);
      const double d_cubic_bubble013_ds1 = s[0] * (s3 - s[1]);
      const double d_cubic_bubble013_ds2 = -s[0] * s[1];
 
      const double d_cubic_bubble023_ds0 = s[2] * (s3 - s[0]);
      const double d_cubic_bubble023_ds1 = -s[0] * s[2];
      const double d_cubic_bubble023_ds2 = s[0] * (s3 - s[2]);
 
      const double d_cubic_bubble123_ds0 = -s[1] * s[2];
      const double d_cubic_bubble123_ds1 = s[2] * (s3 - s[1]);
      const double d_cubic_bubble123_ds2 = s[1] * (s3 - s[2]);
 
 
      // Add the appropriate dervatives of the bubble function to the
      // shape function derivatives
      dpsids(0, 0) = 4.0 * s[0] - 1.0 +
                     3.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble013_ds0 +
                            d_cubic_bubble023_ds0) -
                     4.0 * d_quartic_bubble_ds0;
      dpsids(0, 1) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble013_ds1 +
                            d_cubic_bubble023_ds1) -
                     4.0 * d_quartic_bubble_ds1;
      dpsids(0, 2) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble013_ds2 +
                            d_cubic_bubble023_ds2) -
                     4.0 * d_quartic_bubble_ds2;
 
      dpsids(1, 0) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble013_ds0 +
                            d_cubic_bubble123_ds0) -
                     4.0 * d_quartic_bubble_ds0;
      dpsids(1, 1) = 4.0 * s[1] - 1.0 +
                     3.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble013_ds1 +
                            d_cubic_bubble123_ds1) -
                     4.0 * d_quartic_bubble_ds1;
      dpsids(1, 2) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble013_ds2 +
                            d_cubic_bubble123_ds2) -
                     4.0 * d_quartic_bubble_ds2;
 
      dpsids(2, 0) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble023_ds0 +
                            d_cubic_bubble123_ds0) -
                     4.0 * d_quartic_bubble_ds0;
      dpsids(2, 1) = 0.0 +
                     3.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble023_ds1 +
                            d_cubic_bubble123_ds1) -
                     4.0 * d_quartic_bubble_ds1;
      dpsids(2, 2) = 4.0 * s[2] - 1.0 +
                     3.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble023_ds2 +
                            d_cubic_bubble123_ds2) -
                     4.0 * d_quartic_bubble_ds2;
 
      dpsids(3, 0) = -4.0 * s3 + 1.0 +
                     3.0 * (d_cubic_bubble013_ds0 + d_cubic_bubble023_ds0 +
                            d_cubic_bubble123_ds0) -
                     4.0 * d_quartic_bubble_ds0;
      dpsids(3, 1) = -4.0 * s3 + 1.0 +
                     3.0 * (d_cubic_bubble013_ds1 + d_cubic_bubble023_ds1 +
                            d_cubic_bubble123_ds1) -
                     4.0 * d_quartic_bubble_ds1;
      dpsids(3, 2) = -4.0 * s3 + 1.0 +
                     3.0 * (d_cubic_bubble013_ds2 + d_cubic_bubble023_ds2 +
                            d_cubic_bubble123_ds2) -
                     4.0 * d_quartic_bubble_ds2;
 
      dpsids(4, 0) = 4.0 * s[1] -
                     12.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble013_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(4, 1) = 4.0 * s[0] -
                     12.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble013_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(4, 2) = 0.0 -
                     12.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble013_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      dpsids(5, 0) = 4.0 * s[2] -
                     12.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble023_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(5, 1) = 0.0 -
                     12.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble023_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(5, 2) = 4.0 * s[0] -
                     12.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble023_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      dpsids(6, 0) = 4.0 * (s3 - s[0]) -
                     12.0 * (d_cubic_bubble013_ds0 + d_cubic_bubble023_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(6, 1) = -4.0 * s[0] -
                     12.0 * (d_cubic_bubble013_ds1 + d_cubic_bubble023_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(6, 2) = -4.0 * s[0] -
                     12.0 * (d_cubic_bubble013_ds2 + d_cubic_bubble023_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      dpsids(7, 0) = 0.0 -
                     12.0 * (d_cubic_bubble012_ds0 + d_cubic_bubble123_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(7, 1) = 4.0 * s[2] -
                     12.0 * (d_cubic_bubble012_ds1 + d_cubic_bubble123_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(7, 2) = 4.0 * s[1] -
                     12.0 * (d_cubic_bubble012_ds2 + d_cubic_bubble123_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      dpsids(8, 0) = -4.0 * s[2] -
                     12.0 * (d_cubic_bubble023_ds0 + d_cubic_bubble123_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(8, 1) = -4.0 * s[2] -
                     12.0 * (d_cubic_bubble023_ds1 + d_cubic_bubble123_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(8, 2) = 4.0 * (s3 - s[2]) -
                     12.0 * (d_cubic_bubble023_ds2 + d_cubic_bubble123_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      dpsids(9, 0) = -4.0 * s[1] -
                     12.0 * (d_cubic_bubble013_ds0 + d_cubic_bubble123_ds0) +
                     32.0 * d_quartic_bubble_ds0;
      dpsids(9, 1) = 4.0 * (s3 - s[1]) -
                     12.0 * (d_cubic_bubble013_ds1 + d_cubic_bubble123_ds1) +
                     32.0 * d_quartic_bubble_ds1;
      dpsids(9, 2) = -4.0 * s[1] -
                     12.0 * (d_cubic_bubble013_ds2 + d_cubic_bubble123_ds2) +
                     32.0 * d_quartic_bubble_ds2;
 
      // Add the bubble function on the face spanned by 0 1 3
      dpsids(10, 0) =
        27.0 * d_cubic_bubble013_ds0 - 108.0 * d_quartic_bubble_ds0;
      dpsids(10, 1) =
        27.0 * d_cubic_bubble013_ds1 - 108.0 * d_quartic_bubble_ds1;
      dpsids(10, 2) =
        27.0 * d_cubic_bubble013_ds2 - 108.0 * d_quartic_bubble_ds2;
 
      // Add the bubble function on the face spanned by 0 1 2
      dpsids(11, 0) =
        27.0 * d_cubic_bubble012_ds0 - 108.0 * d_quartic_bubble_ds0;
      dpsids(11, 1) =
        27.0 * d_cubic_bubble012_ds1 - 108.0 * d_quartic_bubble_ds1;
      dpsids(11, 2) =
        27.0 * d_cubic_bubble012_ds2 - 108.0 * d_quartic_bubble_ds2;
 
      // Add the bubble function on the face spanned by 0 2 3
      dpsids(12, 0) =
        27.0 * d_cubic_bubble023_ds0 - 108.0 * d_quartic_bubble_ds0;
      dpsids(12, 1) =
        27.0 * d_cubic_bubble023_ds1 - 108.0 * d_quartic_bubble_ds1;
      dpsids(12, 2) =
        27.0 * d_cubic_bubble023_ds2 - 108.0 * d_quartic_bubble_ds2;
 
      // Add the bubble function on the face spanned by 1 2 3
      dpsids(13, 0) =
        27.0 * d_cubic_bubble123_ds0 - 108.0 * d_quartic_bubble_ds0;
      dpsids(13, 1) =
        27.0 * d_cubic_bubble123_ds1 - 108.0 * d_quartic_bubble_ds1;
      dpsids(13, 2) =
        27.0 * d_cubic_bubble123_ds2 - 108.0 * d_quartic_bubble_ds2;
 
      // Add the volumetric bubble function derivatives
      dpsids(14, 0) = 256.0 * d_quartic_bubble_ds0;
      dpsids(14, 1) = 256.0 * d_quartic_bubble_ds1;
      dpsids(14, 2) = 256.0 * d_quartic_bubble_ds2;
    }

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

local_coordinate_of_node()

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

inline

    {
      s.resize(3);
 
      switch (j)
      {
        case 0:
          s[0] = 1.0;
          s[1] = 0.0;
          s[2] = 0.0;
          break;
 
        case 1:
          s[0] = 0.0;
          s[1] = 1.0;
          s[2] = 0.0;
          break;
 
        case 2:
          s[0] = 0.0;
          s[1] = 0.0;
          s[2] = 1.0;
          break;
 
        case 3:
          s[0] = 0.0;
          s[1] = 0.0;
          s[2] = 0.0;
          break;
 
        case 4:
          s[0] = 0.5;
          s[1] = 0.5;
          s[2] = 0.0;
          break;
 
        case 5:
          s[0] = 0.5;
          s[1] = 0.0;
          s[2] = 0.5;
          break;
 
        case 6:
          s[0] = 0.5;
          s[1] = 0.0;
          s[2] = 0.0;
          break;
 
        case 7:
          s[0] = 0.0;
          s[1] = 0.5;
          s[2] = 0.5;
          break;
 
        case 8:
          s[0] = 0.0;
          s[1] = 0.0;
          s[2] = 0.5;
          break;
 
        case 9:
          s[0] = 0.0;
          s[1] = 0.5;
          s[2] = 0.0;
          break;
 
          // Node at centroid of face spanned by nodes 0, 1, 3
        case 10:
          s[0] = 1.0 / 3.0;
          s[1] = 1.0 / 3.0;
          s[2] = 0.0;
          break;
 
          // Node at centroid of face spanned by nodes 0, 1, 2
        case 11:
          s[0] = 1.0 / 3.0;
          s[1] = 1.0 / 3.0;
          s[2] = 1.0 / 3.0;
          break;
 
          // Node at centroid of face spanned by nodes 0, 2, 3
        case 12:
          s[0] = 1.0 / 3.0;
          s[1] = 0.0;
          s[2] = 1.0 / 3.0;
          break;
 
          // Node at centroid of face spannd by nodes 1, 2, 3
        case 13:
          s[0] = 0.0;
          s[1] = 1.0 / 3.0;
          s[2] = 1.0 / 3.0;
          break;
 
          // Node at centroid of volume
        case 14:
          s[0] = 0.25;
          s[1] = 0.25;
          s[2] = 0.25;
          break;
 
 
        default:
          std::ostringstream error_message;
          error_message
            << "Element only has fifteen 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< 3, 3 >::n_enriched_nodes ( )

inline

    {
      return 5;
    }

shape()

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

inline

Shape function for specific TBubbleEnrichedElement<3,3>

    {
      // Constructe the fourth volume coordinate
      const double s3 = 1.0 - s[0] - s[1] - s[2];
      // calculate the enrichment functions
      const double quartic_bubble = s[0] * s[1] * s[2] * s3;
      const double cubic_bubble012 = s[0] * s[1] * s[2];
      const double cubic_bubble013 = s[0] * s[1] * s3;
      const double cubic_bubble023 = s[0] * s[2] * s3;
      const double cubic_bubble123 = s[1] * s[2] * s3;
 
      // The appropriate "amount" of cubic and quartic bubble functions are
      // added/subtracted
      // to each original quadratic shape function to ensure that the new
      // shape function is zero at the centroid (0.25,0.25,0.25)
      // and at the face centroids
      psi[0] = (2.0 * s[0] - 1.0) * s[0] +
               3.0 * (cubic_bubble012 + cubic_bubble013 + cubic_bubble023) -
               4.0 * quartic_bubble;
      psi[1] = (2.0 * s[1] - 1.0) * s[1] +
               3.0 * (cubic_bubble012 + cubic_bubble013 + cubic_bubble123) -
               4.0 * quartic_bubble;
      psi[2] = (2.0 * s[2] - 1.0) * s[2] +
               3.0 * (cubic_bubble012 + cubic_bubble023 + cubic_bubble123) -
               4.0 * quartic_bubble;
      psi[3] = (2.0 * s3 - 1.0) * s3 +
               3.0 * (cubic_bubble013 + cubic_bubble023 + cubic_bubble123) -
               4.0 * quartic_bubble;
      psi[4] = 4.0 * s[0] * s[1] - 12.0 * (cubic_bubble012 + cubic_bubble013) +
               32.0 * quartic_bubble;
      psi[5] = 4.0 * s[0] * s[2] - 12.0 * (cubic_bubble012 + cubic_bubble023) +
               32.0 * quartic_bubble;
      psi[6] = 4.0 * s[0] * s3 - 12.0 * (cubic_bubble013 + cubic_bubble023) +
               32.0 * quartic_bubble;
      psi[7] = 4.0 * s[1] * s[2] - 12.0 * (cubic_bubble012 + cubic_bubble123) +
               32.0 * quartic_bubble;
      psi[8] = 4.0 * s[2] * s3 - 12.0 * (cubic_bubble023 + cubic_bubble123) +
               32.0 * quartic_bubble;
      psi[9] = 4.0 * s[1] * s3 - 12.0 * (cubic_bubble013 + cubic_bubble123) +
               32.0 * quartic_bubble;
      // Add the bubble function on the face spanned by 0 1 3
      psi[10] = 27.0 * cubic_bubble013 - 108.0 * quartic_bubble;
      // Add the bubble function on the face spanned by 0 1 2
      psi[11] = 27.0 * cubic_bubble012 - 108.0 * quartic_bubble;
      // Add the bubble function on the face spanned by 0 2 3
      psi[12] = 27.0 * cubic_bubble023 - 108.0 * quartic_bubble;
      // Add the bubble function on the face spanned by 1 2 3
      psi[13] = 27.0 * cubic_bubble123 - 108.0 * quartic_bubble;
      // Add the volume bubble function, scaled to have value one
      psi[14] = 256.0 * quartic_bubble;
    }

References s.

---

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

• Telements.h