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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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SF_SIMP_P2BUB Quadratic Lagrange shape function for simplices with bubble (P2+).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_SIMP_P2BUB( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming quadratic P2 Lagrange shape functions on simplices an additional with bubble function. XI Barycentric coordinates.
Input Value/[Size] Description
-----------------------------------------------------------------------------------
i_eval scalar: 1 Evaluate function values
>1 Evaluate values of derivatives
n_sdim scalar: 1-3 Number of space dimensions
n_vert scalar: 2-4 Number of vertices per cell
i_dof scalar: 1-n_ldof Local basis function to evaluate
xi [n_sdim+1] Local coordinates of evaluation point
aInvJac [n,n_sdim+1*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
[~,nLDof,xLDof] = sf_simp_P2( 0, n_sdim, n_vert );
nLDof(4) = 1;
switch n_sdim
case 2
xLDof = [ xLDof [1;1;1]/3 ];
case 3
xLDof = [ xLDof [1;1;1;1]/4 ];
end
sfun = 'sf_simp_P2bub';
% Evaluation type flag.
vBase = 0;
if( nargin>=4 && i_dof==sum(nLDof) && n_sdim>1 )
if( i_eval==1 ) % Evaluation of function values.
switch n_sdim
case 2
vBase = 27*xi(1)*xi(2)*xi(3);
case 3
vBase = 256*xi(1)*xi(2)*xi(3)*xi(4);
end
elseif( i_eval>=2 && i_eval<=n_sdim+1 ) % Evaluation of first derivatives.
switch n_sdim
case 2
dNdxi1 = 27*xi(2)*xi(3);
dNdxi2 = 27*xi(1)*xi(3);
dNdxi3 = 27*xi(1)*xi(2);
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 3
dNdxi1 = 256*xi(2)*xi(3)*xi(4);
dNdxi2 = 256*xi(1)*xi(3)*xi(4);
dNdxi3 = 256*xi(1)*xi(2)*xi(4);
dNdxi4 = 256*xi(1)*xi(2)*xi(3);
if( i_eval==2 )
vBase = aInvJac(:,1)*dNdxi1 + aInvJac(:, 2)*dNdxi2 + aInvJac(:, 3)*dNdxi3 + aInvJac(:, 4)*dNdxi4;
elseif( i_eval==3 )
vBase = aInvJac(:,5)*dNdxi1 + aInvJac(:, 6)*dNdxi2 + aInvJac(:, 7)*dNdxi3 + aInvJac(:, 8)*dNdxi4;
else
vBase = aInvJac(:,9)*dNdxi1 + aInvJac(:,10)*dNdxi2 + aInvJac(:,11)*dNdxi3 + aInvJac(:,12)*dNdxi4;
end
end
elseif( any(i_eval==[22 23 24 32 33 34 42 43 44]) ) % Evaluation of second derivatives.
switch n_sdim
case 2
d2Ndxi1dxi1 = 0;
d2Ndxi1dxi2 = 27*xi(3);
d2Ndxi1dxi3 = 27*xi(2);
d2Ndxi2dxi1 = 27*xi(1);
d2Ndxi2dxi2 = 0;
d2Ndxi2dxi3 = 27*xi(3);
d2Ndxi3dxi1 = 27*xi(1);
d2Ndxi3dxi2 = 27*xi(2);
d2Ndxi3dxi3 = 0;
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
case 3
d2Ndxi1dxi1 = 0;
d2Ndxi1dxi2 = 256*xi(3)*xi(4);
d2Ndxi1dxi3 = 256*xi(2)*xi(4);
d2Ndxi1dxi4 = 256*xi(2)*xi(3);
d2Ndxi2dxi1 = 256*xi(3)*xi(4);
d2Ndxi2dxi2 = 0;
d2Ndxi2dxi3 = 256*xi(1)*xi(4);
d2Ndxi2dxi4 = 256*xi(1)*xi(3);
d2Ndxi3dxi1 = 256*xi(2)*xi(4);
d2Ndxi3dxi2 = 256*xi(1)*xi(4);
d2Ndxi3dxi3 = 0;
d2Ndxi3dxi4 = 256*xi(1)*xi(2);
d2Ndxi4dxi1 = 256*xi(2)*xi(3);
d2Ndxi4dxi2 = 256*xi(1)*xi(3);
d2Ndxi4dxi3 = 256*xi(1)*xi(2);
d2Ndxi4dxi4 = 0;
switch( i_eval )
case 22
vBase = aInvJac(:,1).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:, 2)*d2Ndxi2dxi1 + aInvJac(:, 3)*d2Ndxi3dxi1 + aInvJac(:, 4)*d2Ndxi4dxi1 ) + ...
aInvJac(:,2).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:, 2)*d2Ndxi2dxi2 + aInvJac(:, 3)*d2Ndxi3dxi2 + aInvJac(:, 4)*d2Ndxi4dxi2 ) + ...
aInvJac(:,3).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:, 2)*d2Ndxi2dxi3 + aInvJac(:, 3)*d2Ndxi3dxi3 + aInvJac(:, 4)*d2Ndxi4dxi3 ) + ...
aInvJac(:,4).*( aInvJac(:,1)*d2Ndxi1dxi4 + aInvJac(:, 2)*d2Ndxi2dxi4 + aInvJac(:, 3)*d2Ndxi3dxi4 + aInvJac(:, 4)*d2Ndxi4dxi4 );
case 33
vBase = aInvJac(:,5).*( aInvJac(:,5)*d2Ndxi1dxi1 + aInvJac(:, 6)*d2Ndxi2dxi1 + aInvJac(:, 7)*d2Ndxi3dxi1 + aInvJac(:, 8)*d2Ndxi4dxi1 ) + ...
aInvJac(:,6).*( aInvJac(:,5)*d2Ndxi1dxi2 + aInvJac(:, 6)*d2Ndxi2dxi2 + aInvJac(:, 7)*d2Ndxi3dxi2 + aInvJac(:, 8)*d2Ndxi4dxi2 ) + ...
aInvJac(:,7).*( aInvJac(:,5)*d2Ndxi1dxi3 + aInvJac(:, 6)*d2Ndxi2dxi3 + aInvJac(:, 7)*d2Ndxi3dxi3 + aInvJac(:, 8)*d2Ndxi4dxi3 ) + ...
aInvJac(:,8).*( aInvJac(:,5)*d2Ndxi1dxi4 + aInvJac(:, 6)*d2Ndxi2dxi4 + aInvJac(:, 7)*d2Ndxi3dxi4 + aInvJac(:, 8)*d2Ndxi4dxi4 );
case 44
vBase = aInvJac(:, 9).*( aInvJac(:,9)*d2Ndxi1dxi1 + aInvJac(:,10)*d2Ndxi2dxi1 + aInvJac(:,11)*d2Ndxi3dxi1 + aInvJac(:,12)*d2Ndxi4dxi1 ) + ...
aInvJac(:,10).*( aInvJac(:,9)*d2Ndxi1dxi2 + aInvJac(:,10)*d2Ndxi2dxi2 + aInvJac(:,11)*d2Ndxi3dxi2 + aInvJac(:,12)*d2Ndxi4dxi2 ) + ...
aInvJac(:,11).*( aInvJac(:,9)*d2Ndxi1dxi3 + aInvJac(:,10)*d2Ndxi2dxi3 + aInvJac(:,11)*d2Ndxi3dxi3 + aInvJac(:,12)*d2Ndxi4dxi3 ) + ...
aInvJac(:,12).*( aInvJac(:,9)*d2Ndxi1dxi4 + aInvJac(:,10)*d2Ndxi2dxi4 + aInvJac(:,11)*d2Ndxi3dxi4 + aInvJac(:,12)*d2Ndxi4dxi4 );
case {23,32}
vBase = aInvJac(:,5).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:, 2)*d2Ndxi2dxi1 + aInvJac(:, 3)*d2Ndxi3dxi1 + aInvJac(:, 4)*d2Ndxi4dxi1 ) + ...
aInvJac(:,6).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:, 2)*d2Ndxi2dxi2 + aInvJac(:, 3)*d2Ndxi3dxi2 + aInvJac(:, 4)*d2Ndxi4dxi2 ) + ...
aInvJac(:,7).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:, 2)*d2Ndxi2dxi3 + aInvJac(:, 3)*d2Ndxi3dxi3 + aInvJac(:, 4)*d2Ndxi4dxi3 ) + ...
aInvJac(:,8).*( aInvJac(:,1)*d2Ndxi1dxi4 + aInvJac(:, 2)*d2Ndxi2dxi4 + aInvJac(:, 3)*d2Ndxi3dxi4 + aInvJac(:, 4)*d2Ndxi4dxi4 );
case {24,42}
vBase = aInvJac(:, 9).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:, 2)*d2Ndxi2dxi1 + aInvJac(:, 3)*d2Ndxi3dxi1 + aInvJac(:, 4)*d2Ndxi4dxi1 ) + ...
aInvJac(:,10).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:, 2)*d2Ndxi2dxi2 + aInvJac(:, 3)*d2Ndxi3dxi2 + aInvJac(:, 4)*d2Ndxi4dxi2 ) + ...
aInvJac(:,11).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:, 2)*d2Ndxi2dxi3 + aInvJac(:, 3)*d2Ndxi3dxi3 + aInvJac(:, 4)*d2Ndxi4dxi3 ) + ...
aInvJac(:,12).*( aInvJac(:,1)*d2Ndxi1dxi4 + aInvJac(:, 2)*d2Ndxi2dxi4 + aInvJac(:, 3)*d2Ndxi3dxi4 + aInvJac(:, 4)*d2Ndxi4dxi4 );
case {34,43}
vBase = aInvJac(:, 9).*( aInvJac(:,5)*d2Ndxi1dxi1 + aInvJac(:, 6)*d2Ndxi2dxi1 + aInvJac(:, 7)*d2Ndxi3dxi1 + aInvJac(:, 8)*d2Ndxi4dxi1 ) + ...
aInvJac(:,10).*( aInvJac(:,5)*d2Ndxi1dxi2 + aInvJac(:, 6)*d2Ndxi2dxi2 + aInvJac(:, 7)*d2Ndxi3dxi2 + aInvJac(:, 8)*d2Ndxi4dxi2 ) + ...
aInvJac(:,11).*( aInvJac(:,5)*d2Ndxi1dxi3 + aInvJac(:, 6)*d2Ndxi2dxi3 + aInvJac(:, 7)*d2Ndxi3dxi3 + aInvJac(:, 8)*d2Ndxi4dxi3 ) + ...
aInvJac(:,12).*( aInvJac(:,5)*d2Ndxi1dxi4 + aInvJac(:, 6)*d2Ndxi2dxi4 + aInvJac(:, 7)*d2Ndxi3dxi4 + aInvJac(:, 8)*d2Ndxi4dxi4 );
end
end
end
elseif( nargin>=4 )
vBase = sf_simp_P2( i_eval, n_sdim, n_vert, i_dof, xi, aInvJac, vBase );
end