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sf_simp_P2bub.m File Reference

Description

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
See also
sf_simp_P2

Code listing

 [~,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