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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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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
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