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FEATool Multiphysics
v1.16.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.16.5
Finite Element Analysis Toolbox
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SF_TRI_P3 Third order Lagrange shape functions for triangles (P3).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_TRI_P3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming third order P3 Lagrange shape functions on 2D triangular elements with values defined in the nodes, edges, and center. XI are Barycentric coordinates.
Input Value/[Size] Description
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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: 1-10 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
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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 6 0 1];
xLDof = [1 0 0 2/3 0 1/3 1/3 0 2/3 1/3;
0 1 0 1/3 2/3 0 2/3 1/3 0 1/3;
0 0 1 0 1/3 2/3 0 2/3 1/3 1/3];
sfun = 'sf_tri_P3';
switch i_eval
case 1
switch i_dof
case 1
vBase = (xi(1)*(3*xi(1) - 1)*(3*xi(1) - 2))/2;
case 2
vBase = (xi(2)*(3*xi(2) - 1)*(3*xi(2) - 2))/2;
case 3
vBase = (xi(3)*(3*xi(3) - 1)*(3*xi(3) - 2))/2;
case 4
vBase = (9*xi(1)*xi(2)*(3*xi(1) - 1))/2;
case 5
vBase = (9*xi(2)*xi(3)*(3*xi(2) - 1))/2;
case 6
vBase = (9*xi(1)*xi(3)*(3*xi(3) - 1))/2;
case 7
vBase = (9*xi(1)*xi(2)*(3*xi(2) - 1))/2;
case 8
vBase = (9*xi(2)*xi(3)*(3*xi(3) - 1))/2;
case 9
vBase = (9*xi(1)*xi(3)*(3*xi(1) - 1))/2;
case 10
vBase = 27*xi(1)*xi(2)*xi(3);
end
case {2,3}
switch i_dof
case 1
dNdxi1 = (27*xi(1)^2)/2 - 9*xi(1) + 1;
dNdxi2 = 0;
dNdxi3 = 0;
case 2
dNdxi1 = 0;
dNdxi2 = (27*xi(2)^2)/2 - 9*xi(2) + 1;
dNdxi3 = 0;
case 3
dNdxi1 = 0;
dNdxi2 = 0;
dNdxi3 = (27*xi(3)^2)/2 - 9*xi(3) + 1;
case 4
dNdxi1 = (9*xi(2)*(6*xi(1) - 1))/2;
dNdxi2 = (9*xi(1)*(3*xi(1) - 1))/2;
dNdxi3 = 0;
case 5
dNdxi1 = 0;
dNdxi2 = (9*xi(3)*(6*xi(2) - 1))/2;
dNdxi3 = (9*xi(2)*(3*xi(2) - 1))/2;
case 6
dNdxi1 = (9*xi(3)*(3*xi(3) - 1))/2;
dNdxi2 = 0;
dNdxi3 = (9*xi(1)*(6*xi(3) - 1))/2;
case 7
dNdxi1 = (9*xi(2)*(3*xi(2) - 1))/2;
dNdxi2 = (9*xi(1)*(6*xi(2) - 1))/2;
dNdxi3 = 0;
case 8
dNdxi1 = 0;
dNdxi2 = (9*xi(3)*(3*xi(3) - 1))/2;
dNdxi3 = (9*xi(2)*(6*xi(3) - 1))/2;
case 9
dNdxi1 = (9*xi(3)*(6*xi(1) - 1))/2;
dNdxi2 = 0;
dNdxi3 = (9*xi(1)*(3*xi(1) - 1))/2;
case 10
dNdxi1 = 27*xi(2)*xi(3);
dNdxi2 = 27*xi(1)*xi(3);
dNdxi3 = 27*xi(1)*xi(2);
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 = 27*xi(1) - 9;
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 = 27*xi(2) - 9;
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 = 27*xi(3) - 9;
case 4
d2Ndxi1dxi1 = 27*xi(2);
d2Ndxi2dxi1 = 27*xi(1) - 9/2;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 27*xi(1) - 9/2;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 0;
case 5
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 27*xi(3);
d2Ndxi3dxi2 = 27*xi(2) - 9/2;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 27*xi(2) - 9/2;
d2Ndxi3dxi3 = 0;
case 6
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 27*xi(3) - 9/2;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 27*xi(3) - 9/2;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 27*xi(1);
case 7
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 27*xi(2) - 9/2;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 27*xi(2) - 9/2;
d2Ndxi2dxi2 = 27*xi(1);
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 0;
case 8
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 27*xi(3) - 9/2;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 27*xi(3) - 9/2;
d2Ndxi3dxi3 = 27*xi(2);
case 9
d2Ndxi1dxi1 = 27*xi(3);
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 27*xi(1) - 9/2;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 27*xi(1) - 9/2;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 0;
case 10
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 27*xi(3);
d2Ndxi3dxi1 = 27*xi(2);
d2Ndxi1dxi2 = 27*xi(3);
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 27*xi(1);
d2Ndxi1dxi3 = 27*xi(2);
d2Ndxi2dxi3 = 27*xi(1);
d2Ndxi3dxi3 = 0;
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