FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox
sf_simp_P1bub.m File Reference

## Description

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
-----------------------------------------------------------------------------------
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

sf_simp_P1

# Code listing

 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