FEATool  v1.9
Finite Element Analysis Toolbox
sf_tri_P5.m File Reference

Description

SF_TRI_P5 Fifth order Lagrange shape functions for triangles (P5).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_TRI_P5( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming fifth order P5 Lagrange shape functions on 2D triangular elements with values defined in the nodes, edges, and center. XI are Barycentric coordinates.

Input       Value/[Size]           Description
-----------------------------------------------------------------------------------
i_eval      scalar:  1             Evaluate function values
                    >1             Evaluate values of derivatives
n_sdim      scalar: 2              Number of space dimensions
n_vert      scalar: 3              Number of vertices per cell
i_dof       scalar: 1-21           Local basis function to evaluate
xi          array [3,1]            Local coordinates of evaluation point
aInvJac     [n,6]                  Inverse of transformation Jacobian
vBase       [n]                    Preallocated output vector
                                                                                  .
Output      Value/[Size]           Description
-----------------------------------------------------------------------------------
vBase       [n]                    Evaluated function values
nLDof       [4]                    Number of local degrees of freedom on
                                   vertices, edges, faces, and cell interiors
xLDof       [3,n_ldof]             Local coordinates of local dofs
sfun        string                 Function name of called shape function
See also
sf_tri_P1

Code listing

 nLDof = [3 12 0 6];
 xLDof = [1 0 0 4/5   0 1/5 3/5   0 2/5 2/5   0 3/5 1/5   0 4/5 3/5 1/5 1/5 2/5 1/5 2/5;
          0 1 0 1/5 4/5   0 2/5 3/5   0 3/5 2/5   0 4/5 1/5   0 1/5 3/5 1/5 2/5 2/5 1/5;
          0 0 1   0 1/5 4/5   0 2/5 3/5   0 3/5 2/5   0 4/5 1/5 1/5 1/5 3/5 1/5 2/5 2/5];
 sfun  = 'sf_tri_P5';


 switch i_eval

   case 1

     switch i_dof
       case 1
         vBase = (xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(1) - 3)*(5*xi(1) - 4))/24;
       case 2
         vBase = (xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(2) - 3)*(5*xi(2) - 4))/24;
       case 3
         vBase = (xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2)*(5*xi(3) - 3)*(5*xi(3) - 4))/24;
       case 4
         vBase = (25*xi(1)*xi(2)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(1) - 3))/24;
       case 5
         vBase = (25*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(2) - 3))/24;
       case 6
         vBase = (25*xi(1)*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2)*(5*xi(3) - 3))/24;
       case 7
         vBase = (25*xi(1)*xi(2)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(2) - 1))/12;
       case 8
         vBase = (25*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(3) - 1))/12;
       case 9
         vBase = (25*xi(1)*xi(3)*(5*xi(1) - 1)*(5*xi(3) - 1)*(5*xi(3) - 2))/12;
       case 10
         vBase = (25*xi(1)*xi(2)*(5*xi(1) - 1)*(5*xi(2) - 1)*(5*xi(2) - 2))/12;
       case 11
         vBase = (25*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(3) - 1)*(5*xi(3) - 2))/12;
       case 12
         vBase = (25*xi(1)*xi(3)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(3) - 1))/12;
       case 13
         vBase = (25*xi(1)*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(2) - 3))/24;
       case 14
         vBase = (25*xi(2)*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2)*(5*xi(3) - 3))/24;
       case 15
         vBase = (25*xi(1)*xi(3)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(1) - 3))/24;
       case 16
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
       case 17
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
       case 18
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
       case 19
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(1) - 1)*(5*xi(2) - 1))/4;
       case 20
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(3) - 1))/4;
       case 21
         vBase = (125*xi(1)*xi(2)*xi(3)*(5*xi(1) - 1)*(5*xi(3) - 1))/4;
     end

   case {2,3}

     switch i_dof

       case 1
         dNdxi1 = (3125*xi(1)^4)/24 - (625*xi(1)^3)/3 + (875*xi(1)^2)/8 - (125*xi(1))/6 + 1;
         dNdxi2 = 0;
         dNdxi3 = 0;
       case 2
         dNdxi1 = 0;
         dNdxi2 = (3125*xi(2)^4)/24 - (625*xi(2)^3)/3 + (875*xi(2)^2)/8 - (125*xi(2))/6 + 1;
         dNdxi3 = 0;
       case 3
         dNdxi1 = 0;
         dNdxi2 = 0;
         dNdxi3 = (3125*xi(3)^4)/24 - (625*xi(3)^3)/3 + (875*xi(3)^2)/8 - (125*xi(3))/6 + 1;
       case 4
         dNdxi1 = (25*xi(2)*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         dNdxi2 = (25*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(1) - 3))/24;
         dNdxi3 = 0;
       case 5
         dNdxi1 = 0;
         dNdxi2 = (25*xi(3)*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         dNdxi3 = (25*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(2) - 3))/24;
       case 6
         dNdxi1 = (25*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2)*(5*xi(3) - 3))/24;
         dNdxi2 = 0;
         dNdxi3 = (25*xi(1)*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
       case 7
         dNdxi1 = (25*xi(2)*(5*xi(2) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         dNdxi2 = (25*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2)*(10*xi(2) - 1))/12;
         dNdxi3 = 0;
       case 8
         dNdxi1 = 0;
         dNdxi2 = (25*xi(3)*(5*xi(3) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         dNdxi3 = (25*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2)*(10*xi(3) - 1))/12;
       case 9
         dNdxi1 = (25*xi(3)*(10*xi(1) - 1)*(5*xi(3) - 1)*(5*xi(3) - 2))/12;
         dNdxi2 = 0;
         dNdxi3 = (25*xi(1)*(5*xi(1) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
       case 10
         dNdxi1 = (25*xi(2)*(10*xi(1) - 1)*(5*xi(2) - 1)*(5*xi(2) - 2))/12;
         dNdxi2 = (25*xi(1)*(5*xi(1) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         dNdxi3 = 0;
       case 11
         dNdxi1 = 0;
         dNdxi2 = (25*xi(3)*(10*xi(2) - 1)*(5*xi(3) - 1)*(5*xi(3) - 2))/12;
         dNdxi3 = (25*xi(2)*(5*xi(2) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
       case 12
         dNdxi1 = (25*xi(3)*(5*xi(3) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         dNdxi2 = 0;
         dNdxi3 = (25*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2)*(10*xi(3) - 1))/12;
       case 13
         dNdxi1 = (25*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2)*(5*xi(2) - 3))/24;
         dNdxi2 = (25*xi(1)*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         dNdxi3 = 0;
       case 14
         dNdxi1 = 0;
         dNdxi2 = (25*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2)*(5*xi(3) - 3))/24;
         dNdxi3 = (25*xi(2)*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
       case 15
         dNdxi1 = (25*xi(3)*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         dNdxi2 = 0;
         dNdxi3 = (25*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2)*(5*xi(1) - 3))/24;
       case 16
         dNdxi1 = (125*xi(2)*xi(3)*(75*xi(1)^2 - 30*xi(1) + 2))/6;
         dNdxi2 = (125*xi(1)*xi(3)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
         dNdxi3 = (125*xi(1)*xi(2)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
       case 17
         dNdxi1 = (125*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
         dNdxi2 = (125*xi(1)*xi(3)*(75*xi(2)^2 - 30*xi(2) + 2))/6;
         dNdxi3 = (125*xi(1)*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
       case 18
         dNdxi1 = (125*xi(2)*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         dNdxi2 = (125*xi(1)*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         dNdxi3 = (125*xi(1)*xi(2)*(75*xi(3)^2 - 30*xi(3) + 2))/6;
       case 19
         dNdxi1 = (125*xi(2)*xi(3)*(10*xi(1) - 1)*(5*xi(2) - 1))/4;
         dNdxi2 = (125*xi(1)*xi(3)*(5*xi(1) - 1)*(10*xi(2) - 1))/4;
         dNdxi3 = (125*xi(1)*xi(2)*(5*xi(1) - 1)*(5*xi(2) - 1))/4;
       case 20
         dNdxi1 = (125*xi(2)*xi(3)*(5*xi(2) - 1)*(5*xi(3) - 1))/4;
         dNdxi2 = (125*xi(1)*xi(3)*(10*xi(2) - 1)*(5*xi(3) - 1))/4;
         dNdxi3 = (125*xi(1)*xi(2)*(5*xi(2) - 1)*(10*xi(3) - 1))/4;
       case 21
         dNdxi1 = (125*xi(2)*xi(3)*(10*xi(1) - 1)*(5*xi(3) - 1))/4;
         dNdxi2 = (125*xi(1)*xi(3)*(5*xi(1) - 1)*(5*xi(3) - 1))/4;
         dNdxi3 = (125*xi(1)*xi(2)*(5*xi(1) - 1)*(10*xi(3) - 1))/4;
     end

     if( i_eval==2 )

       vBase = aInvJac(:,1).*dNdxi1 + aInvJac(:,2).*dNdxi2 + aInvJac(:,3).*dNdxi3;

     else

       vBase = aInvJac(:,4).*dNdxi1 + aInvJac(:,5).*dNdxi2 + aInvJac(:,6).*dNdxi3;

     end

   case {22,23,32,33}   % Evaluation of second derivatives.

     switch i_dof

       case 1
         d2Ndxi1dxi1 = (125*(5*xi(1) - 2)*(10*xi(1)^2 - 8*xi(1) + 1))/12;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 2
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = (125*(5*xi(2) - 2)*(10*xi(2)^2 - 8*xi(2) + 1))/12;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 3
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = (125*(5*xi(3) - 2)*(10*xi(3)^2 - 8*xi(3) + 1))/12;

       case 4
         d2Ndxi1dxi1 = (125*xi(2)*(150*xi(1)^2 - 90*xi(1) + 11))/12;
         d2Ndxi2dxi1 = (25*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = (25*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 5
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = (125*xi(3)*(150*xi(2)^2 - 90*xi(2) + 11))/12;
         d2Ndxi3dxi2 = (25*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = (25*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         d2Ndxi3dxi3 = 0;

       case 6
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = (25*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = (25*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = (125*xi(1)*(150*xi(3)^2 - 90*xi(3) + 11))/12;

       case 7
         d2Ndxi1dxi1 = (125*xi(2)*(5*xi(1) - 1)*(5*xi(2) - 1))/2;
         d2Ndxi2dxi1 = (25*(10*xi(2) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = (25*(10*xi(2) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         d2Ndxi2dxi2 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 8
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = (125*xi(3)*(5*xi(2) - 1)*(5*xi(3) - 1))/2;
         d2Ndxi3dxi2 = (25*(10*xi(3) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = (25*(10*xi(3) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         d2Ndxi3dxi3 = (125*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;

       case 9
         d2Ndxi1dxi1 = (125*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = (25*(10*xi(1) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = (25*(10*xi(1) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(3) - 1))/2;

       case 10
         d2Ndxi1dxi1 = (125*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
         d2Ndxi2dxi1 = (25*(10*xi(1) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = (25*(10*xi(1) - 1)*(75*xi(2)^2 - 30*xi(2) + 2))/12;
         d2Ndxi2dxi2 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(2) - 1))/2;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 11
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = (125*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         d2Ndxi3dxi2 = (25*(10*xi(2) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = (25*(10*xi(2) - 1)*(75*xi(3)^2 - 30*xi(3) + 2))/12;
         d2Ndxi3dxi3 = (125*xi(2)*(5*xi(2) - 1)*(5*xi(3) - 1))/2;

       case 12
         d2Ndxi1dxi1 = (125*xi(3)*(5*xi(1) - 1)*(5*xi(3) - 1))/2;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = (25*(10*xi(3) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = (25*(10*xi(3) - 1)*(75*xi(1)^2 - 30*xi(1) + 2))/12;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;

       case 13
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = (25*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = (25*(10*xi(2) - 3)*(25*xi(2)^2 - 15*xi(2) + 1))/12;
         d2Ndxi2dxi2 = (125*xi(1)*(150*xi(2)^2 - 90*xi(2) + 11))/12;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 14
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = 0;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = (25*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
         d2Ndxi1dxi3 = 0;
         d2Ndxi2dxi3 = (25*(10*xi(3) - 3)*(25*xi(3)^2 - 15*xi(3) + 1))/12;
         d2Ndxi3dxi3 = (125*xi(2)*(150*xi(3)^2 - 90*xi(3) + 11))/12;

       case 15
         d2Ndxi1dxi1 = (125*xi(3)*(150*xi(1)^2 - 90*xi(1) + 11))/12;
         d2Ndxi2dxi1 = 0;
         d2Ndxi3dxi1 = (25*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = 0;
         d2Ndxi1dxi3 = (25*(10*xi(1) - 3)*(25*xi(1)^2 - 15*xi(1) + 1))/12;
         d2Ndxi2dxi3 = 0;
         d2Ndxi3dxi3 = 0;

       case 16
         d2Ndxi1dxi1 = 625*xi(2)*xi(3)*(5*xi(1) - 1);
         d2Ndxi2dxi1 = (125*xi(3)*(75*xi(1)^2 - 30*xi(1) + 2))/6;
         d2Ndxi3dxi1 = (125*xi(2)*(75*xi(1)^2 - 30*xi(1) + 2))/6;
         d2Ndxi1dxi2 = (125*xi(3)*(75*xi(1)^2 - 30*xi(1) + 2))/6;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
         d2Ndxi1dxi3 = (125*xi(2)*(75*xi(1)^2 - 30*xi(1) + 2))/6;
         d2Ndxi2dxi3 = (125*xi(1)*(5*xi(1) - 1)*(5*xi(1) - 2))/6;
         d2Ndxi3dxi3 = 0;

       case 17
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = (125*xi(3)*(75*xi(2)^2 - 30*xi(2) + 2))/6;
         d2Ndxi3dxi1 = (125*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
         d2Ndxi1dxi2 = (125*xi(3)*(75*xi(2)^2 - 30*xi(2) + 2))/6;
         d2Ndxi2dxi2 = 625*xi(1)*xi(3)*(5*xi(2) - 1);
         d2Ndxi3dxi2 = (125*xi(1)*(75*xi(2)^2 - 30*xi(2) + 2))/6;
         d2Ndxi1dxi3 = (125*xi(2)*(5*xi(2) - 1)*(5*xi(2) - 2))/6;
         d2Ndxi2dxi3 = (125*xi(1)*(75*xi(2)^2 - 30*xi(2) + 2))/6;
         d2Ndxi3dxi3 = 0;

       case 18
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = (125*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         d2Ndxi3dxi1 = (125*xi(2)*(75*xi(3)^2 - 30*xi(3) + 2))/6;
         d2Ndxi1dxi2 = (125*xi(3)*(5*xi(3) - 1)*(5*xi(3) - 2))/6;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = (125*xi(1)*(75*xi(3)^2 - 30*xi(3) + 2))/6;
         d2Ndxi1dxi3 = (125*xi(2)*(75*xi(3)^2 - 30*xi(3) + 2))/6;
         d2Ndxi2dxi3 = (125*xi(1)*(75*xi(3)^2 - 30*xi(3) + 2))/6;
         d2Ndxi3dxi3 = 625*xi(1)*xi(2)*(5*xi(3) - 1);

       case 19
         d2Ndxi1dxi1 = (625*xi(2)*xi(3)*(5*xi(2) - 1))/2;
         d2Ndxi2dxi1 = (125*xi(3)*(10*xi(1) - 1)*(10*xi(2) - 1))/4;
         d2Ndxi3dxi1 = (125*xi(2)*(10*xi(1) - 1)*(5*xi(2) - 1))/4;
         d2Ndxi1dxi2 = (125*xi(3)*(10*xi(1) - 1)*(10*xi(2) - 1))/4;
         d2Ndxi2dxi2 = (625*xi(1)*xi(3)*(5*xi(1) - 1))/2;
         d2Ndxi3dxi2 = (125*xi(1)*(5*xi(1) - 1)*(10*xi(2) - 1))/4;
         d2Ndxi1dxi3 = (125*xi(2)*(10*xi(1) - 1)*(5*xi(2) - 1))/4;
         d2Ndxi2dxi3 = (125*xi(1)*(5*xi(1) - 1)*(10*xi(2) - 1))/4;
         d2Ndxi3dxi3 = 0;

       case 20
         d2Ndxi1dxi1 = 0;
         d2Ndxi2dxi1 = (125*xi(3)*(10*xi(2) - 1)*(5*xi(3) - 1))/4;
         d2Ndxi3dxi1 = (125*xi(2)*(5*xi(2) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi1dxi2 = (125*xi(3)*(10*xi(2) - 1)*(5*xi(3) - 1))/4;
         d2Ndxi2dxi2 = (625*xi(1)*xi(3)*(5*xi(3) - 1))/2;
         d2Ndxi3dxi2 = (125*xi(1)*(10*xi(2) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi1dxi3 = (125*xi(2)*(5*xi(2) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi2dxi3 = (125*xi(1)*(10*xi(2) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi3dxi3 = (625*xi(1)*xi(2)*(5*xi(2) - 1))/2;

       case 21
         d2Ndxi1dxi1 = (625*xi(2)*xi(3)*(5*xi(3) - 1))/2;
         d2Ndxi2dxi1 = (125*xi(3)*(10*xi(1) - 1)*(5*xi(3) - 1))/4;
         d2Ndxi3dxi1 = (125*xi(2)*(10*xi(1) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi1dxi2 = (125*xi(3)*(10*xi(1) - 1)*(5*xi(3) - 1))/4;
         d2Ndxi2dxi2 = 0;
         d2Ndxi3dxi2 = (125*xi(1)*(5*xi(1) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi1dxi3 = (125*xi(2)*(10*xi(1) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi2dxi3 = (125*xi(1)*(5*xi(1) - 1)*(10*xi(3) - 1))/4;
         d2Ndxi3dxi3 = (625*xi(1)*xi(2)*(5*xi(1) - 1))/2;
     end

     if( i_eval==22 )
       vBase = aInvJac(:,1).*( aInvJac(:,1).*d2Ndxi1dxi1 + aInvJac(:,2).*d2Ndxi2dxi1 + aInvJac(:,3).*d2Ndxi3dxi1 ) + ...
               aInvJac(:,2).*( aInvJac(:,1).*d2Ndxi1dxi2 + aInvJac(:,2).*d2Ndxi2dxi2 + aInvJac(:,3).*d2Ndxi3dxi2 ) + ...
               aInvJac(:,3).*( aInvJac(:,1).*d2Ndxi1dxi3 + aInvJac(:,2).*d2Ndxi2dxi3 + aInvJac(:,3).*d2Ndxi3dxi3 );
     elseif( i_eval==33 )
       vBase = aInvJac(:,4).*( aInvJac(:,4).*d2Ndxi1dxi1 + aInvJac(:,5).*d2Ndxi2dxi1 + aInvJac(:,6).*d2Ndxi3dxi1 ) + ...
               aInvJac(:,5).*( aInvJac(:,4).*d2Ndxi1dxi2 + aInvJac(:,5).*d2Ndxi2dxi2 + aInvJac(:,6).*d2Ndxi3dxi2 ) + ...
               aInvJac(:,6).*( aInvJac(:,4).*d2Ndxi1dxi3 + aInvJac(:,5).*d2Ndxi2dxi3 + aInvJac(:,6).*d2Ndxi3dxi3 );
     else
       vBase = aInvJac(:,4).*( aInvJac(:,1).*d2Ndxi1dxi1 + aInvJac(:,2).*d2Ndxi2dxi1 + aInvJac(:,3).*d2Ndxi3dxi1 ) + ...
               aInvJac(:,5).*( aInvJac(:,1).*d2Ndxi1dxi2 + aInvJac(:,2).*d2Ndxi2dxi2 + aInvJac(:,3).*d2Ndxi3dxi2 ) + ...
               aInvJac(:,6).*( aInvJac(:,1).*d2Ndxi1dxi3 + aInvJac(:,2).*d2Ndxi2dxi3 + aInvJac(:,3).*d2Ndxi3dxi3 );
     end

   otherwise
     vBase = 0;

 end