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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_H3 Third order 2D C1 Hermite shape functions for triangles.
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_TRI_H3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates C1 Hermite shape functions on quadrilaterals with values defined in the nodes, and cell 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: 1-10 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 [3,4] 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 0 0 1;
3 0 0 0;
3 0 0 0];
xLDof = [repmat([1 0 0;
0 1 0;
0 0 1],1,3) [1/3;1/3;1/3]];
sfun = 'sf_tri_H3';
switch i_eval
case 1
switch i_dof
case 1
vBase = - 2*xi(1)^3 + 3*xi(1)^2 - 7*xi(2)*xi(3)*xi(1);
case 2
vBase = - 2*xi(2)^3 + 3*xi(2)^2 - 7*xi(1)*xi(3)*xi(2);
case 3
vBase = - 2*xi(3)^3 + 3*xi(3)^2 - 7*xi(1)*xi(2)*xi(3);
case 4
vBase = - xi(1)*xi(2)*((-aInvJac(:,6).*aInvJac(:,7)))*(2*xi(1) + xi(2) - 1) - xi(1)*xi(3)*(aInvJac(:,5).*aInvJac(:,7))*(2*xi(1) + xi(3) - 1);
case 5
vBase = xi(1)*xi(2)*((-aInvJac(:,6).*aInvJac(:,7)))*(xi(1) + 2*xi(2) - 1) - xi(2)*xi(3)*((-aInvJac(:,4).*aInvJac(:,7)))*(2*xi(2) + xi(3) - 1);
case 6
vBase = xi(1)*xi(3)*(aInvJac(:,5).*aInvJac(:,7))*(xi(1) + 2*xi(3) - 1) + xi(2)*xi(3)*((-aInvJac(:,4).*aInvJac(:,7)))*(xi(2) + 2*xi(3) - 1);
case 7
vBase = - xi(1)*xi(2)*(aInvJac(:,3).*aInvJac(:,7))*(2*xi(1) + xi(2) - 1) - xi(1)*xi(3)*((-aInvJac(:,2).*aInvJac(:,7)))*(2*xi(1) + xi(3) - 1);
case 8
vBase = xi(1)*xi(2)*(aInvJac(:,3).*aInvJac(:,7))*(xi(1) + 2*xi(2) - 1) - xi(2)*xi(3)*(aInvJac(:,1).*aInvJac(:,7))*(2*xi(2) + xi(3) - 1);
case 9
vBase = xi(1)*xi(3)*((-aInvJac(:,2).*aInvJac(:,7)))*(xi(1) + 2*xi(3) - 1) + xi(2)*xi(3)*(aInvJac(:,1).*aInvJac(:,7))*(xi(2) + 2*xi(3) - 1);
case 10
vBase = 27*xi(1)*xi(2)*xi(3);
end
case {2,3}
switch i_dof
case 1
dNdxi1 = - 6*xi(1)^2 + 6*xi(1) - 7*xi(2)*xi(3);
dNdxi2 = -7*xi(1)*xi(3);
dNdxi3 = -7*xi(1)*xi(2);
case 2
dNdxi1 = -7*xi(2)*xi(3);
dNdxi2 = - 6*xi(2)^2 + 6*xi(2) - 7*xi(1)*xi(3);
dNdxi3 = -7*xi(1)*xi(2);
case 3
dNdxi1 = -7*xi(2)*xi(3);
dNdxi2 = -7*xi(1)*xi(3);
dNdxi3 = - 6*xi(3)^2 + 6*xi(3) - 7*xi(1)*xi(2);
case 4
dNdxi1 = - xi(2)*((-aInvJac(:,6).*aInvJac(:,7)))*(2*xi(1) + xi(2) - 1) - xi(3)*(aInvJac(:,5).*aInvJac(:,7))*(2*xi(1) + xi(3) - 1) - 2*xi(1)*xi(2)*((-aInvJac(:,6).*aInvJac(:,7))) - 2*xi(1)*xi(3)*(aInvJac(:,5).*aInvJac(:,7));
dNdxi2 = -xi(1)*((-aInvJac(:,6).*aInvJac(:,7)))*(2*xi(1) + 2*xi(2) - 1);
dNdxi3 = -xi(1)*(aInvJac(:,5).*aInvJac(:,7))*(2*xi(1) + 2*xi(3) - 1);
case 5
dNdxi1 = xi(2)*((-aInvJac(:,6).*aInvJac(:,7)))*(2*xi(1) + 2*xi(2) - 1);
dNdxi2 = xi(1)*((-aInvJac(:,6).*aInvJac(:,7)))*(xi(1) + 2*xi(2) - 1) - xi(3)*((-aInvJac(:,4).*aInvJac(:,7)))*(2*xi(2) + xi(3) - 1) + 2*xi(1)*xi(2)*((-aInvJac(:,6).*aInvJac(:,7))) - 2*xi(2)*xi(3)*((-aInvJac(:,4).*aInvJac(:,7)));
dNdxi3 = -xi(2)*((-aInvJac(:,4).*aInvJac(:,7)))*(2*xi(2) + 2*xi(3) - 1);
case 6
dNdxi1 = xi(3)*(aInvJac(:,5).*aInvJac(:,7))*(2*xi(1) + 2*xi(3) - 1);
dNdxi2 = xi(3)*((-aInvJac(:,4).*aInvJac(:,7)))*(2*xi(2) + 2*xi(3) - 1);
dNdxi3 = xi(1)*(aInvJac(:,5).*aInvJac(:,7))*(xi(1) + 2*xi(3) - 1) + xi(2)*((-aInvJac(:,4).*aInvJac(:,7)))*(xi(2) + 2*xi(3) - 1) + 2*xi(1)*xi(3)*(aInvJac(:,5).*aInvJac(:,7)) + 2*xi(2)*xi(3)*((-aInvJac(:,4).*aInvJac(:,7)));
case 7
dNdxi1 = - 2*xi(1)*xi(2)*(aInvJac(:,3).*aInvJac(:,7)) - 2*xi(1)*xi(3)*((-aInvJac(:,2).*aInvJac(:,7))) - xi(2)*(aInvJac(:,3).*aInvJac(:,7))*(2*xi(1) + xi(2) - 1) - xi(3)*((-aInvJac(:,2).*aInvJac(:,7)))*(2*xi(1) + xi(3) - 1);
dNdxi2 = -xi(1)*(aInvJac(:,3).*aInvJac(:,7))*(2*xi(1) + 2*xi(2) - 1);
dNdxi3 = -xi(1)*((-aInvJac(:,2).*aInvJac(:,7)))*(2*xi(1) + 2*xi(3) - 1);
case 8
dNdxi1 = xi(2)*(aInvJac(:,3).*aInvJac(:,7))*(2*xi(1) + 2*xi(2) - 1);
dNdxi2 = 2*xi(1)*xi(2)*(aInvJac(:,3).*aInvJac(:,7)) - 2*xi(2)*xi(3)*(aInvJac(:,1).*aInvJac(:,7)) + xi(1)*(aInvJac(:,3).*aInvJac(:,7))*(xi(1) + 2*xi(2) - 1) - xi(3)*(aInvJac(:,1).*aInvJac(:,7))*(2*xi(2) + xi(3) - 1);
dNdxi3 = -xi(2)*(aInvJac(:,1).*aInvJac(:,7))*(2*xi(2) + 2*xi(3) - 1);
case 9
dNdxi1 = xi(3)*((-aInvJac(:,2).*aInvJac(:,7)))*(2*xi(1) + 2*xi(3) - 1);
dNdxi2 = xi(3)*(aInvJac(:,1).*aInvJac(:,7))*(2*xi(2) + 2*xi(3) - 1);
dNdxi3 = 2*xi(1)*xi(3)*((-aInvJac(:,2).*aInvJac(:,7))) + 2*xi(2)*xi(3)*(aInvJac(:,1).*aInvJac(:,7)) + xi(1)*((-aInvJac(:,2).*aInvJac(:,7)))*(xi(1) + 2*xi(3) - 1) + xi(2)*(aInvJac(:,1).*aInvJac(:,7))*(xi(2) + 2*xi(3) - 1);
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
otherwise
vBase = 0;
end