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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_P1BUB Linear Lagrange shape function for simplices with bubble (P1+).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_SIMP_P1BUB( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming linear P1 Lagrange shape functions on simplices an additional with bubble function. XI 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: 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
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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 [n_sdim,n_ldof] Local coordinates of local dofs
sfun string Function name of called shape function
sfun = 'sf_simp_P1bub';
[~,nLDof,xLDof] = sf_simp_P1( 0, n_sdim, n_vert );
nLDof(4) = 1;
switch n_sdim
case 1
xLDof = [ xLDof [1;1]/2 ];
case 2
xLDof = [ xLDof [1;1;1]/3 ];
case 3
xLDof = [ xLDof [1;1;1;1]/4 ];
end
% Evaluation type flag.
if( i_eval==1 ) % Evaluation of function values.
if( n_sdim==1 )
vBase = sf_simp_P2( i_eval, n_sdim, n_vert, i_dof, xi, aInvJac, vBase );
elseif( n_sdim==2 )
switch( i_dof )
case 1
vBase = (9*xi(2)*xi(3) - 1)*(xi(2) + xi(3) - 1);
case 2
vBase = xi(2) + 9*xi(2)*xi(3)*(xi(2) + xi(3) - 1);
case 3
vBase = xi(3) + 9*xi(2)*xi(3)*(xi(2) + xi(3) - 1);
case 4
vBase = -27*xi(2)*xi(3)*(xi(2) + xi(3) - 1);
end
else % 3D.
switch( i_dof )
case 1
vBase = (64*xi(2)*xi(3)*xi(4) - 1)*(xi(2) + xi(3) + xi(4) - 1);
case 2
vBase = xi(2)*(64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 1);
case 3
vBase = xi(3)*(64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 1);
case 4
vBase = xi(4)*(64*xi(2)*xi(3)*(xi(2) + xi(3) + xi(4) - 1) + 1);
case 5
vBase = -256*xi(2)*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1);
end
end
elseif( i_eval>=2 && i_eval<=n_sdim+1 ) % Evaluation of first derivatives.
if( n_sdim==1 )
vBase = sf_simp_P2( i_eval, n_sdim, n_vert, i_dof, xi, aInvJac, vBase );
elseif( n_sdim==2 )
switch i_dof % Basis function to evaluate.
case 1
dNdxi1 = 0;
dNdxi2 = 18*xi(2)*xi(3) - 9*xi(3) + 9*xi(3)^2 - 1;
dNdxi3 = 18*xi(2)*xi(3) - 9*xi(2) + 9*xi(2)^2 - 1;
case 2
dNdxi1 = 0;
dNdxi2 = 18*xi(2)*xi(3) - 9*xi(3) + 9*xi(3)^2 + 1;
dNdxi3 = 9*xi(2)*(xi(2) + 2*xi(3) - 1);
case 3
dNdxi1 = 0;
dNdxi2 = 9*xi(3)*(2*xi(2) + xi(3) - 1);
dNdxi3 = 18*xi(2)*xi(3) - 9*xi(2) + 9*xi(2)^2 + 1;
case 4
dNdxi1 = 0;
dNdxi2 = -27*xi(3)*(2*xi(2) + xi(3) - 1);
dNdxi3 = -27*xi(2)*(xi(2) + 2*xi(3) - 1);
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
else % 3D.
switch i_dof % Basis function to evaluate.
case 1
dNdxi1 = 0;
dNdxi2 = 64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1;
dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1;
dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1;
case 2
dNdxi1 = 0;
dNdxi2 = 64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1;
dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
case 3
dNdxi1 = 0;
dNdxi2 = 64*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1);
dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1;
dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1;
case 4
dNdxi1 = 0;
dNdxi2 = 64*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1);
dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
case 5
dNdxi1 = 0;
dNdxi2 = -256*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1);
dNdxi3 = -256*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
dNdxi4 = -256*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1);
end
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.
error('sf_simp_P1bub: second order derivative evaluation not supported.')
else
vBase = 0;
end