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sf_hex_Q2.m File Reference

Description

SF_HEX_Q2 Triquadratic conforming shape function for hexahedrons (Q2).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_HEX_Q2( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming triquadratic Q2 shape functions on hexahedrons with values defined in the nodes, edges, faces, and cell center. XI is [-1..1]^3 reference coordinates.

Input       Value/[Size]           Description
-----------------------------------------------------------------------------------
i_eval      scalar:  1             Evaluate function values
                    >1             Evaluate values of derivatives
n_sdim      scalar:  3             Number of space dimensions
n_vert      scalar:  8             Number of vertices per cell
i_dof       scalar: 1-n_ldof       Local basis function to evaluate
xi          [n_sdim]               Local coordinates of evaluation point
aInvJac     [n,n_sdim*n_sdim]      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       [n_sdim,n_ldof]        Local coordinates of local dofs
sfun        string                 Function name of called shape function
See also
sflag2, sf_hex_Q1

Code listing

 nLDof = [8 12 6 1];
 xLDof = [-1  1  1 -1  -1  1  1 -1   0  1  0 -1  -1  1  1 -1   0  1  0 -1   0  0  1 0 -1 0  0; ...
          -1 -1  1  1  -1 -1  1  1  -1  0  1  0  -1 -1  1  1  -1  0  1  0   0 -1  0 1  0 0  0; ...
          -1 -1 -1 -1   1  1  1  1  -1 -1 -1 -1   0  0  0  0   1  1  1  1  -1  0  0 0  0 1  0];
 sfun  = 'sf_hex_Q2';


 switch i_eval   % Evaluation type flag.

   case 1   % Evaluation of function values.

     switch i_dof   % Basis function to evaluate.

       case 1
         vBase = -xi(1)*(1-xi(1))*xi(2)*(1-xi(2))*xi(3)*(1-xi(3))/8;
       case 2
         vBase =  xi(1)*(1+xi(1))*xi(2)*(1-xi(2))*xi(3)*(1-xi(3))/8;
       case 3
         vBase = -xi(1)*(1+xi(1))*xi(2)*(1+xi(2))*xi(3)*(1-xi(3))/8;
       case 4
         vBase =  xi(1)*(1-xi(1))*xi(2)*(1+xi(2))*xi(3)*(1-xi(3))/8;
       case 5
         vBase =  xi(1)*(1-xi(1))*xi(2)*(1-xi(2))*xi(3)*(1+xi(3))/8;
       case 6
         vBase = -xi(1)*(1+xi(1))*xi(2)*(1-xi(2))*xi(3)*(1+xi(3))/8;
       case 7
         vBase =  xi(1)*(1+xi(1))*xi(2)*(1+xi(2))*xi(3)*(1+xi(3))/8;
       case 8
         vBase = -xi(1)*(1-xi(1))*xi(2)*(1+xi(2))*xi(3)*(1+xi(3))/8;
       case 9
         vBase =  (2-2*xi(1)^2)*xi(2)*(1-xi(2))*xi(3)*(1-xi(3))/8;
       case 10
         vBase = -xi(1)*(1+xi(1))*(2-2*xi(2)^2)*xi(3)*(1-xi(3))/8;
       case 11
         vBase = -(2-2*xi(1)^2)*xi(2)*(1+xi(2))*xi(3)*(1-xi(3))/8;
       case 12
         vBase =  xi(1)*(1-xi(1))*(2-2*xi(2)^2)*xi(3)*(1-xi(3))/8;
       case 13
         vBase =  xi(1)*(1-xi(1))*xi(2)*(1-xi(2))*(2-2*xi(3)^2)/8;
       case 14
         vBase = -xi(1)*(1+xi(1))*xi(2)*(1-xi(2))*(2-2*xi(3)^2)/8;
       case 15
         vBase =  xi(1)*(1+xi(1))*xi(2)*(1+xi(2))*(2-2*xi(3)^2)/8;
       case 16
         vBase = -xi(1)*(1-xi(1))*xi(2)*(1+xi(2))*(2-2*xi(3)^2)/8;
       case 17
         vBase = -(2-2*xi(1)^2)*xi(2)*(1-xi(2))*xi(3)*(1+xi(3))/8;
       case 18
         vBase =  xi(1)*(1+xi(1))*(2-2*xi(2)^2)*xi(3)*(1+xi(3))/8;
       case 19
         vBase =  (2-2*xi(1)^2)*xi(2)*(1+xi(2))*xi(3)*(1+xi(3))/8;
       case 20
         vBase = -xi(1)*(1-xi(1))*(2-2*xi(2)^2)*xi(3)*(1+xi(3))/8;
       case 21
         vBase = -(2-2*xi(1)^2)*(2-2*xi(2)^2)*xi(3)*(1-xi(3))/8;
       case 22
         vBase = -(2-2*xi(1)^2)*xi(2)*(1-xi(2))*(2-2*xi(3)^2)/8;
       case 23
         vBase =  xi(1)*(1+xi(1))*(2-2*xi(2)^2)*(2-2*xi(3)^2)/8;
       case 24
         vBase =  (2-2*xi(1)^2)*xi(2)*(1+xi(2))*(2-2*xi(3)^2)/8;
       case 25
         vBase = -xi(1)*(1-xi(1))*(2-2*xi(2)^2)*(2-2*xi(3)^2)/8;
       case 26
         vBase =  (2-2*xi(1)^2)*(2-2*xi(2)^2)*xi(3)*(1+xi(3))/8;
       case 27
         vBase =  (2-2*xi(1)^2)*(2-2*xi(2)^2)*(2-2*xi(3)^2)/8;

     end

   case {2,3,4}   % Evaluation of first order derivatives.

     switch i_dof   % Basis function to evaluate.
       case 1
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)-1)*(xi(2)-1)*(xi(3)-1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)-1)*(xi(1)-1)*(xi(3)-1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)-1)*(xi(1)-1)*(xi(2)-1))/8;
       case 2
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)+1)*(xi(2)-1)*(xi(3)-1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)-1)*(xi(1)+1)*(xi(3)-1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)-1)*(xi(1)+1)*(xi(2)-1))/8;
       case 3
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)+1)*(xi(2)+1)*(xi(3)-1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)+1)*(xi(1)+1)*(xi(3)-1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)-1)*(xi(1)+1)*(xi(2)+1))/8;
       case 4
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)-1)*(xi(2)+1)*(xi(3)-1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)+1)*(xi(1)-1)*(xi(3)-1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)-1)*(xi(1)-1)*(xi(2)+1))/8;
       case 5
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)-1)*(xi(2)-1)*(xi(3)+1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)-1)*(xi(1)-1)*(xi(3)+1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)+1)*(xi(1)-1)*(xi(2)-1))/8;
       case 6
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)+1)*(xi(2)-1)*(xi(3)+1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)-1)*(xi(1)+1)*(xi(3)+1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)+1)*(xi(1)+1)*(xi(2)-1))/8;
       case 7
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)+1)*(xi(2)+1)*(xi(3)+1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)+1)*(xi(1)+1)*(xi(3)+1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)+1)*(xi(1)+1)*(xi(2)+1))/8;
       case 8
         dNdxi1 = (xi(2)*xi(3)*(2*xi(1)-1)*(xi(2)+1)*(xi(3)+1))/8;
         dNdxi2 = (xi(1)*xi(3)*(2*xi(2)+1)*(xi(1)-1)*(xi(3)+1))/8;
         dNdxi3 = (xi(1)*xi(2)*(2*xi(3)+1)*(xi(1)-1)*(xi(2)+1))/8;
       case 9
         dNdxi1 = -(xi(1)*xi(2)*xi(3)*(xi(2)-1)*(xi(3)-1))/2;
         dNdxi2 = -(xi(3)*(2*xi(2)-1)*(xi(1)^2-1)*(xi(3)-1))/4;
         dNdxi3 = -(xi(2)*(2*xi(3)-1)*(xi(1)^2-1)*(xi(2)-1))/4;
       case 10
         dNdxi1 = -(xi(3)*(2*xi(1)+1)*(xi(2)^2-1)*(xi(3)-1))/4;
         dNdxi2 = -(xi(1)*xi(2)*xi(3)*(xi(1)+1)*(xi(3)-1))/2;
         dNdxi3 = -(xi(1)*(2*xi(3)-1)*(xi(2)^2-1)*(xi(1)+1))/4;
       case 11
         dNdxi1 = -(xi(1)*xi(2)*xi(3)*(xi(2)+1)*(xi(3)-1))/2;
         dNdxi2 = -(xi(3)*(2*xi(2)+1)*(xi(1)^2-1)*(xi(3)-1))/4;
         dNdxi3 = -(xi(2)*(2*xi(3)-1)*(xi(1)^2-1)*(xi(2)+1))/4;
       case 12
         dNdxi1 = -(xi(3)*(2*xi(1)-1)*(xi(2)^2-1)*(xi(3)-1))/4;
         dNdxi2 = -(xi(1)*xi(2)*xi(3)*(xi(1)-1)*(xi(3)-1))/2;
         dNdxi3 = -(xi(1)*(2*xi(3)-1)*(xi(2)^2-1)*(xi(1)-1))/4;
       case 13
         dNdxi1 = -(xi(2)*(2*xi(1)-1)*(xi(3)^2-1)*(xi(2)-1))/4;
         dNdxi2 = -(xi(1)*(2*xi(2)-1)*(xi(3)^2-1)*(xi(1)-1))/4;
         dNdxi3 = -(xi(1)*xi(2)*xi(3)*(xi(1)-1)*(xi(2)-1))/2;
       case 14
         dNdxi1 = -(xi(2)*(2*xi(1)+1)*(xi(3)^2-1)*(xi(2)-1))/4;
         dNdxi2 = -(xi(1)*(2*xi(2)-1)*(xi(3)^2-1)*(xi(1)+1))/4;
         dNdxi3 = -(xi(1)*xi(2)*xi(3)*(xi(1)+1)*(xi(2)-1))/2;
       case 15
         dNdxi1 = -(xi(2)*(2*xi(1)+1)*(xi(3)^2-1)*(xi(2)+1))/4;
         dNdxi2 = -(xi(1)*(2*xi(2)+1)*(xi(3)^2-1)*(xi(1)+1))/4;
         dNdxi3 = -(xi(1)*xi(2)*xi(3)*(xi(1)+1)*(xi(2)+1))/2;
       case 16
         dNdxi1 = -(xi(2)*(2*xi(1)-1)*(xi(3)^2-1)*(xi(2)+1))/4;
         dNdxi2 = -(xi(1)*(2*xi(2)+1)*(xi(3)^2-1)*(xi(1)-1))/4;
         dNdxi3 = -(xi(1)*xi(2)*xi(3)*(xi(1)-1)*(xi(2)+1))/2;
       case 17
         dNdxi1 = -(xi(1)*xi(2)*xi(3)*(xi(2)-1)*(xi(3)+1))/2;
         dNdxi2 = -(xi(3)*(2*xi(2)-1)*(xi(1)^2-1)*(xi(3)+1))/4;
         dNdxi3 = -(xi(2)*(2*xi(3)+1)*(xi(1)^2-1)*(xi(2)-1))/4;
       case 18
         dNdxi1 = -(xi(3)*(2*xi(1)+1)*(xi(2)^2-1)*(xi(3)+1))/4;
         dNdxi2 = -(xi(1)*xi(2)*xi(3)*(xi(1)+1)*(xi(3)+1))/2;
         dNdxi3 = -(xi(1)*(2*xi(3)+1)*(xi(2)^2-1)*(xi(1)+1))/4;
       case 19
         dNdxi1 = -(xi(1)*xi(2)*xi(3)*(xi(2)+1)*(xi(3)+1))/2;
         dNdxi2 = -(xi(3)*(2*xi(2)+1)*(xi(1)^2-1)*(xi(3)+1))/4;
         dNdxi3 = -(xi(2)*(2*xi(3)+1)*(xi(1)^2-1)*(xi(2)+1))/4;
       case 20
         dNdxi1 = -(xi(3)*(2*xi(1)-1)*(xi(2)^2-1)*(xi(3)+1))/4;
         dNdxi2 = -(xi(1)*xi(2)*xi(3)*(xi(1)-1)*(xi(3)+1))/2;
         dNdxi3 = -(xi(1)*(2*xi(3)+1)*(xi(2)^2-1)*(xi(1)-1))/4;
       case 21
         dNdxi1 = xi(1)*xi(3)*(xi(2)^2-1)*(xi(3)-1);
         dNdxi2 = xi(2)*xi(3)*(xi(1)^2-1)*(xi(3)-1);
         dNdxi3 = ((2*xi(3)-1)*(xi(1)^2-1)*(xi(2)^2-1))/2;
       case 22
         dNdxi1 = xi(1)*xi(2)*(xi(3)^2-1)*(xi(2)-1);
         dNdxi2 = ((2*xi(2)-1)*(xi(1)^2-1)*(xi(3)^2-1))/2;
         dNdxi3 = xi(2)*xi(3)*(xi(1)^2-1)*(xi(2)-1);
       case 23
         dNdxi1 = ((2*xi(1)+1)*(xi(2)^2-1)*(xi(3)^2-1))/2;
         dNdxi2 = xi(1)*xi(2)*(xi(3)^2-1)*(xi(1)+1);
         dNdxi3 = xi(1)*xi(3)*(xi(2)^2-1)*(xi(1)+1);
       case 24
         dNdxi1 = xi(1)*xi(2)*(xi(3)^2-1)*(xi(2)+1);
         dNdxi2 = ((2*xi(2)+1)*(xi(1)^2-1)*(xi(3)^2-1))/2;
         dNdxi3 = xi(2)*xi(3)*(xi(1)^2-1)*(xi(2)+1);
       case 25
         dNdxi1 = ((2*xi(1)-1)*(xi(2)^2-1)*(xi(3)^2-1))/2;
         dNdxi2 = xi(1)*xi(2)*(xi(3)^2-1)*(xi(1)-1);
         dNdxi3 = xi(1)*xi(3)*(xi(2)^2-1)*(xi(1)-1);
       case 26
         dNdxi1 = xi(1)*xi(3)*(xi(2)^2-1)*(xi(3)+1);
         dNdxi2 = xi(2)*xi(3)*(xi(1)^2-1)*(xi(3)+1);
         dNdxi3 = ((2*xi(3)+1)*(xi(1)^2-1)*(xi(2)^2-1))/2;
       case 27
         dNdxi1 = -2*xi(1)*(xi(2)^2-1)*(xi(3)^2-1);
         dNdxi2 = -2*xi(2)*(xi(1)^2-1)*(xi(3)^2-1);
         dNdxi3 = -2*xi(3)*(xi(1)^2-1)*(xi(2)^2-1);
     end

     if     ( i_eval==2 )   % x-derivative.

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

     elseif ( i_eval==3 )   % y-derivative.

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

     elseif ( i_eval==4 )   % z-derivative.

       vBase = aInvJac(:,7)*dNdxi1 + aInvJac(:,8)*dNdxi2 + aInvJac(:,9)*dNdxi3;
     end

   case {22,23,24,32,33,34,42,43,44}   % Evaluation of second order derivatives.

     if( any(any(abs([aInvJac(:,[2 3 4 6 7 8])])>eps*1e2)) )
       warning('sf_hex_Q2: 2nd derivatives for non-rectangular cells shapes not supported.')
     end

     switch i_dof

       case 1
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) - 1)*(xi(3) - 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3) - 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2) - 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3) - 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) - 1)*(xi(3) - 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1) - 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2) - 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1) - 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) - 1)*(xi(2) - 1))/4;

       case 2
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) - 1)*(xi(3) - 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3) - 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2) - 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3) - 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) + 1)*(xi(3) - 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1) + 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2) - 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1) + 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) + 1)*(xi(2) - 1))/4;

       case 3
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) + 1)*(xi(3) - 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3) - 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2) + 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3) - 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) + 1)*(xi(3) - 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1) + 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2) + 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1) + 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) + 1)*(xi(2) + 1))/4;

       case 4
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) + 1)*(xi(3) - 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3) - 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2) + 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3) - 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) - 1)*(xi(3) - 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1) - 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2) + 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1) - 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) - 1)*(xi(2) + 1))/4;

       case 5
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) - 1)*(xi(3) + 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3) + 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2) - 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3) + 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) - 1)*(xi(3) + 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1) - 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2) - 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1) - 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) - 1)*(xi(2) - 1))/4;

       case 6
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) - 1)*(xi(3) + 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3) + 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2) - 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3) + 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) + 1)*(xi(3) + 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1) + 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2) - 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1) + 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) + 1)*(xi(2) - 1))/4;

       case 7
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) + 1)*(xi(3) + 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3) + 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2) + 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3) + 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) + 1)*(xi(3) + 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1) + 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2) + 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1) + 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) + 1)*(xi(2) + 1))/4;

       case 8
         d2Ndxi1dxi1 = (xi(2)*xi(3)*(xi(2) + 1)*(xi(3) + 1))/4;
         d2Ndxi2dxi1 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3) + 1))/8;
         d2Ndxi3dxi1 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2) + 1))/8;
         d2Ndxi1dxi2 = (xi(3)*(2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3) + 1))/8;
         d2Ndxi2dxi2 = (xi(1)*xi(3)*(xi(1) - 1)*(xi(3) + 1))/4;
         d2Ndxi3dxi2 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1) - 1))/8;
         d2Ndxi1dxi3 = (xi(2)*(2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2) + 1))/8;
         d2Ndxi2dxi3 = (xi(1)*(2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1) - 1))/8;
         d2Ndxi3dxi3 = (xi(1)*xi(2)*(xi(1) - 1)*(xi(2) + 1))/4;

       case 9
         d2Ndxi1dxi1 = -(xi(2)*xi(3)*(xi(2) - 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi1 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi1 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(2) - 1))/2;
         d2Ndxi1dxi2 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi2 = -(xi(3)*(xi(1)^2 - 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi2 = -((2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1)^2 - 1))/4;
         d2Ndxi1dxi3 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi3 = -((2*xi(2) - 1)*(2*xi(3) - 1)*(xi(1)^2 - 1))/4;
         d2Ndxi3dxi3 = -(xi(2)*(xi(1)^2 - 1)*(xi(2) - 1))/2;

       case 10
         d2Ndxi1dxi1 = -(xi(3)*(xi(2)^2 - 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi1 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi1 = -((2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2)^2 - 1))/4;
         d2Ndxi1dxi2 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi2 = -(xi(1)*xi(3)*(xi(1) + 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(1) + 1))/2;
         d2Ndxi1dxi3 = -((2*xi(1) + 1)*(2*xi(3) - 1)*(xi(2)^2 - 1))/4;
         d2Ndxi2dxi3 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*(xi(2)^2 - 1)*(xi(1) + 1))/2;

       case 11
         d2Ndxi1dxi1 = -(xi(2)*xi(3)*(xi(2) + 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi1 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi1 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(2) + 1))/2;
         d2Ndxi1dxi2 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi2 = -(xi(3)*(xi(1)^2 - 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi2 = -((2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1)^2 - 1))/4;
         d2Ndxi1dxi3 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi3 = -((2*xi(2) + 1)*(2*xi(3) - 1)*(xi(1)^2 - 1))/4;
         d2Ndxi3dxi3 = -(xi(2)*(xi(1)^2 - 1)*(xi(2) + 1))/2;

       case 12
         d2Ndxi1dxi1 = -(xi(3)*(xi(2)^2 - 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi1 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi1 = -((2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2)^2 - 1))/4;
         d2Ndxi1dxi2 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(3) - 1))/2;
         d2Ndxi2dxi2 = -(xi(1)*xi(3)*(xi(1) - 1)*(xi(3) - 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(1) - 1))/2;
         d2Ndxi1dxi3 = -((2*xi(1) - 1)*(2*xi(3) - 1)*(xi(2)^2 - 1))/4;
         d2Ndxi2dxi3 = -(xi(1)*xi(2)*(2*xi(3) - 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*(xi(2)^2 - 1)*(xi(1) - 1))/2;

       case 13
         d2Ndxi1dxi1 = -(xi(2)*(xi(3)^2 - 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi1 = -((2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3)^2 - 1))/4;
         d2Ndxi3dxi1 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(2) - 1))/2;
         d2Ndxi1dxi2 = -((2*xi(1) - 1)*(2*xi(2) - 1)*(xi(3)^2 - 1))/4;
         d2Ndxi2dxi2 = -(xi(1)*(xi(3)^2 - 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(1) - 1))/2;
         d2Ndxi1dxi3 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi3 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*xi(2)*(xi(1) - 1)*(xi(2) - 1))/2;

       case 14
         d2Ndxi1dxi1 = -(xi(2)*(xi(3)^2 - 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi1 = -((2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3)^2 - 1))/4;
         d2Ndxi3dxi1 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(2) - 1))/2;
         d2Ndxi1dxi2 = -((2*xi(1) + 1)*(2*xi(2) - 1)*(xi(3)^2 - 1))/4;
         d2Ndxi2dxi2 = -(xi(1)*(xi(3)^2 - 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(1) + 1))/2;
         d2Ndxi1dxi3 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi3 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*xi(2)*(xi(1) + 1)*(xi(2) - 1))/2;

       case 15
         d2Ndxi1dxi1 = -(xi(2)*(xi(3)^2 - 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi1 = -((2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3)^2 - 1))/4;
         d2Ndxi3dxi1 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(2) + 1))/2;
         d2Ndxi1dxi2 = -((2*xi(1) + 1)*(2*xi(2) + 1)*(xi(3)^2 - 1))/4;
         d2Ndxi2dxi2 = -(xi(1)*(xi(3)^2 - 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(1) + 1))/2;
         d2Ndxi1dxi3 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi3 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*xi(2)*(xi(1) + 1)*(xi(2) + 1))/2;

       case 16
         d2Ndxi1dxi1 = -(xi(2)*(xi(3)^2 - 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi1 = -((2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3)^2 - 1))/4;
         d2Ndxi3dxi1 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(2) + 1))/2;
         d2Ndxi1dxi2 = -((2*xi(1) - 1)*(2*xi(2) + 1)*(xi(3)^2 - 1))/4;
         d2Ndxi2dxi2 = -(xi(1)*(xi(3)^2 - 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(1) - 1))/2;
         d2Ndxi1dxi3 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi3 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*xi(2)*(xi(1) - 1)*(xi(2) + 1))/2;

       case 17
         d2Ndxi1dxi1 = -(xi(2)*xi(3)*(xi(2) - 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi1 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi1 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(2) - 1))/2;
         d2Ndxi1dxi2 = -(xi(1)*xi(3)*(2*xi(2) - 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi2 = -(xi(3)*(xi(1)^2 - 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi2 = -((2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1)^2 - 1))/4;
         d2Ndxi1dxi3 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(2) - 1))/2;
         d2Ndxi2dxi3 = -((2*xi(2) - 1)*(2*xi(3) + 1)*(xi(1)^2 - 1))/4;
         d2Ndxi3dxi3 = -(xi(2)*(xi(1)^2 - 1)*(xi(2) - 1))/2;

       case 18
         d2Ndxi1dxi1 = -(xi(3)*(xi(2)^2 - 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi1 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi1 = -((2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2)^2 - 1))/4;
         d2Ndxi1dxi2 = -(xi(2)*xi(3)*(2*xi(1) + 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi2 = -(xi(1)*xi(3)*(xi(1) + 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(1) + 1))/2;
         d2Ndxi1dxi3 = -((2*xi(1) + 1)*(2*xi(3) + 1)*(xi(2)^2 - 1))/4;
         d2Ndxi2dxi3 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(1) + 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*(xi(2)^2 - 1)*(xi(1) + 1))/2;

       case 19
         d2Ndxi1dxi1 = -(xi(2)*xi(3)*(xi(2) + 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi1 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi1 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(2) + 1))/2;
         d2Ndxi1dxi2 = -(xi(1)*xi(3)*(2*xi(2) + 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi2 = -(xi(3)*(xi(1)^2 - 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi2 = -((2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1)^2 - 1))/4;
         d2Ndxi1dxi3 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(2) + 1))/2;
         d2Ndxi2dxi3 = -((2*xi(2) + 1)*(2*xi(3) + 1)*(xi(1)^2 - 1))/4;
         d2Ndxi3dxi3 = -(xi(2)*(xi(1)^2 - 1)*(xi(2) + 1))/2;

       case 20
         d2Ndxi1dxi1 = -(xi(3)*(xi(2)^2 - 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi1 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi1 = -((2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2)^2 - 1))/4;
         d2Ndxi1dxi2 = -(xi(2)*xi(3)*(2*xi(1) - 1)*(xi(3) + 1))/2;
         d2Ndxi2dxi2 = -(xi(1)*xi(3)*(xi(1) - 1)*(xi(3) + 1))/2;
         d2Ndxi3dxi2 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(1) - 1))/2;
         d2Ndxi1dxi3 = -((2*xi(1) - 1)*(2*xi(3) + 1)*(xi(2)^2 - 1))/4;
         d2Ndxi2dxi3 = -(xi(1)*xi(2)*(2*xi(3) + 1)*(xi(1) - 1))/2;
         d2Ndxi3dxi3 = -(xi(1)*(xi(2)^2 - 1)*(xi(1) - 1))/2;

       case 21
         d2Ndxi1dxi1 = xi(3)*(xi(2)^2 - 1)*(xi(3) - 1);
         d2Ndxi2dxi1 = 2*xi(1)*xi(2)*xi(3)*(xi(3) - 1);
         d2Ndxi3dxi1 = xi(1)*(2*xi(3) - 1)*(xi(2)^2 - 1);
         d2Ndxi1dxi2 = 2*xi(1)*xi(2)*xi(3)*(xi(3) - 1);
         d2Ndxi2dxi2 = xi(3)*(xi(1)^2 - 1)*(xi(3) - 1);
         d2Ndxi3dxi2 = xi(2)*(2*xi(3) - 1)*(xi(1)^2 - 1);
         d2Ndxi1dxi3 = xi(1)*(2*xi(3) - 1)*(xi(2)^2 - 1);
         d2Ndxi2dxi3 = xi(2)*(2*xi(3) - 1)*(xi(1)^2 - 1);
         d2Ndxi3dxi3 = (xi(1)^2 - 1)*(xi(2)^2 - 1);

       case 22
         d2Ndxi1dxi1 = xi(2)*(xi(3)^2 - 1)*(xi(2) - 1);
         d2Ndxi2dxi1 = xi(1)*(2*xi(2) - 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi1 = 2*xi(1)*xi(2)*xi(3)*(xi(2) - 1);
         d2Ndxi1dxi2 = xi(1)*(2*xi(2) - 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi2 = (xi(1)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi2 = xi(3)*(2*xi(2) - 1)*(xi(1)^2 - 1);
         d2Ndxi1dxi3 = 2*xi(1)*xi(2)*xi(3)*(xi(2) - 1);
         d2Ndxi2dxi3 = xi(3)*(2*xi(2) - 1)*(xi(1)^2 - 1);
         d2Ndxi3dxi3 = xi(2)*(xi(1)^2 - 1)*(xi(2) - 1);

       case 23
         d2Ndxi1dxi1 = (xi(2)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi1 = xi(2)*(2*xi(1) + 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi1 = xi(3)*(2*xi(1) + 1)*(xi(2)^2 - 1);
         d2Ndxi1dxi2 = xi(2)*(2*xi(1) + 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi2 = xi(1)*(xi(3)^2 - 1)*(xi(1) + 1);
         d2Ndxi3dxi2 = 2*xi(1)*xi(2)*xi(3)*(xi(1) + 1);
         d2Ndxi1dxi3 = xi(3)*(2*xi(1) + 1)*(xi(2)^2 - 1);
         d2Ndxi2dxi3 = 2*xi(1)*xi(2)*xi(3)*(xi(1) + 1);
         d2Ndxi3dxi3 = xi(1)*(xi(2)^2 - 1)*(xi(1) + 1);

       case 24
         d2Ndxi1dxi1 = xi(2)*(xi(3)^2 - 1)*(xi(2) + 1);
         d2Ndxi2dxi1 = xi(1)*(2*xi(2) + 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi1 = 2*xi(1)*xi(2)*xi(3)*(xi(2) + 1);
         d2Ndxi1dxi2 = xi(1)*(2*xi(2) + 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi2 = (xi(1)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi2 = xi(3)*(2*xi(2) + 1)*(xi(1)^2 - 1);
         d2Ndxi1dxi3 = 2*xi(1)*xi(2)*xi(3)*(xi(2) + 1);
         d2Ndxi2dxi3 = xi(3)*(2*xi(2) + 1)*(xi(1)^2 - 1);
         d2Ndxi3dxi3 = xi(2)*(xi(1)^2 - 1)*(xi(2) + 1);

       case 25
         d2Ndxi1dxi1 = (xi(2)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi1 = xi(2)*(2*xi(1) - 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi1 = xi(3)*(2*xi(1) - 1)*(xi(2)^2 - 1);
         d2Ndxi1dxi2 = xi(2)*(2*xi(1) - 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi2 = xi(1)*(xi(3)^2 - 1)*(xi(1) - 1);
         d2Ndxi3dxi2 = 2*xi(1)*xi(2)*xi(3)*(xi(1) - 1);
         d2Ndxi1dxi3 = xi(3)*(2*xi(1) - 1)*(xi(2)^2 - 1);
         d2Ndxi2dxi3 = 2*xi(1)*xi(2)*xi(3)*(xi(1) - 1);
         d2Ndxi3dxi3 = xi(1)*(xi(2)^2 - 1)*(xi(1) - 1);

       case 26
         d2Ndxi1dxi1 = xi(3)*(xi(2)^2 - 1)*(xi(3) + 1);
         d2Ndxi2dxi1 = 2*xi(1)*xi(2)*xi(3)*(xi(3) + 1);
         d2Ndxi3dxi1 = xi(1)*(2*xi(3) + 1)*(xi(2)^2 - 1);
         d2Ndxi1dxi2 = 2*xi(1)*xi(2)*xi(3)*(xi(3) + 1);
         d2Ndxi2dxi2 = xi(3)*(xi(1)^2 - 1)*(xi(3) + 1);
         d2Ndxi3dxi2 = xi(2)*(2*xi(3) + 1)*(xi(1)^2 - 1);
         d2Ndxi1dxi3 = xi(1)*(2*xi(3) + 1)*(xi(2)^2 - 1);
         d2Ndxi2dxi3 = xi(2)*(2*xi(3) + 1)*(xi(1)^2 - 1);
         d2Ndxi3dxi3 = (xi(1)^2 - 1)*(xi(2)^2 - 1);

       case 27
         d2Ndxi1dxi1 = -2*(xi(2)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi2dxi1 = -4*xi(1)*xi(2)*(xi(3)^2 - 1);
         d2Ndxi3dxi1 = -4*xi(1)*xi(3)*(xi(2)^2 - 1);
         d2Ndxi1dxi2 = -4*xi(1)*xi(2)*(xi(3)^2 - 1);
         d2Ndxi2dxi2 = -2*(xi(1)^2 - 1)*(xi(3)^2 - 1);
         d2Ndxi3dxi2 = -4*xi(2)*xi(3)*(xi(1)^2 - 1);
         d2Ndxi1dxi3 = -4*xi(1)*xi(3)*(xi(2)^2 - 1);
         d2Ndxi2dxi3 = -4*xi(2)*xi(3)*(xi(1)^2 - 1);
         d2Ndxi3dxi3 = -2*(xi(1)^2 - 1)*(xi(2)^2 - 1);

     end

     switch( i_eval )
       case 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 );
       case 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 );

       case 44
         vBase = aInvJac(:,7).*( aInvJac(:,7)*d2Ndxi1dxi1 + aInvJac(:,8)*d2Ndxi2dxi1 + aInvJac(:,9)*d2Ndxi3dxi1 ) + ...
                 aInvJac(:,8).*( aInvJac(:,7)*d2Ndxi1dxi2 + aInvJac(:,8)*d2Ndxi2dxi2 + aInvJac(:,9)*d2Ndxi3dxi2 ) + ...
                 aInvJac(:,9).*( aInvJac(:,7)*d2Ndxi1dxi3 + aInvJac(:,8)*d2Ndxi2dxi3 + aInvJac(:,9)*d2Ndxi3dxi3 );

       case {23,32}
         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 );

       case {24,42}
         vBase = aInvJac(:,7).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:,2)*d2Ndxi2dxi1 + aInvJac(:,3)*d2Ndxi3dxi1 ) + ...
                 aInvJac(:,8).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:,2)*d2Ndxi2dxi2 + aInvJac(:,3)*d2Ndxi3dxi2 ) + ...
                 aInvJac(:,9).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:,2)*d2Ndxi2dxi3 + aInvJac(:,3)*d2Ndxi3dxi3 );

       case {34,43}
         vBase = aInvJac(:,7).*( aInvJac(:,4)*d2Ndxi1dxi1 + aInvJac(:,5)*d2Ndxi2dxi1 + aInvJac(:,6)*d2Ndxi3dxi1 ) + ...
                 aInvJac(:,8).*( aInvJac(:,4)*d2Ndxi1dxi2 + aInvJac(:,5)*d2Ndxi2dxi2 + aInvJac(:,6)*d2Ndxi3dxi2 ) + ...
                 aInvJac(:,9).*( aInvJac(:,4)*d2Ndxi1dxi3 + aInvJac(:,5)*d2Ndxi2dxi3 + aInvJac(:,6)*d2Ndxi3dxi3 );
     end

   otherwise
     vBase = 0;

 end