FEATool Multiphysics  v1.11
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
See also
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