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sf_simp_P2.m File Reference

Description

SF_SIMP_P2 Quadratic Lagrange shape function for simplices (P2).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_SIMP_P2( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming quadratic P2 Lagrange shape functions on simplices with values defined in the nodes. 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
sflag2

Code listing

 switch n_sdim

   case 1
     nLDof = [ 2 0 0 1 ];
     xLDof = [ 1 0 0.5; ...
               0 1 0.5 ];
   case 2
     nLDof = [ 3 3 0 0 ];
     xLDof = [ 1 0 0 0.5 0   0.5; ...
               0 1 0 0.5 0.5 0  ; ...
               0 0 1 0   0.5 0.5 ];
   case 3
     nLDof = [ 4 6 0 0 ];
     xLDof = [ 1 0 0 0 0.5 0   0.5 0.5 0   0  ; ...
               0 1 0 0 0.5 0.5 0   0   0.5 0  ; ...
               0 0 1 0 0   0.5 0.5 0   0   0.5; ...
               0 0 0 1 0   0   0   0.5 0.5 0.5 ];
 end
 sfun = 'sf_simp_P2';


% Evaluation type flag.
 if( i_eval==1 )    % Evaluation of function values.

     if( i_dof<=n_vert )   % Node dofs.

       vBase = xi(i_dof)*(2*xi(i_dof)-1);

     else   % Edge dofs.

       i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
       j = mod(i,min(n_vert,3))+1;
       if ( i_dof>7 )
         j = 4;
       end

       vBase = 4*xi(i)*xi(j);

     end

 elseif( i_eval>=2 && i_eval<=n_sdim+1 )   % Evaluation of first derivatives.

     if( i_dof<=n_vert )   % Node dofs.

       i_col = i_dof + (i_eval-2)*n_vert;
       dNdxi = 4*xi(i_dof)-1;

       vBase = aInvJac(:,i_col)*dNdxi;

     else   % Edge dofs.

       i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
       j = mod(i,min(n_vert,3))+1;
       if ( i_dof>7 )
         j = 4;
       end
       i_col  = i + (i_eval-2)*n_vert;
       j_col  = j + (i_eval-2)*n_vert;
       dNdxii = 4*xi(j);
       dNdxij = 4*xi(i);

       vBase  = aInvJac(:,i_col)*dNdxii + aInvJac(:,j_col)*dNdxij;

     end

 elseif( any(i_eval==[22 23 24 32 33 34 42 43 44]) )   % Evaluation of second derivatives.

     i_eval1 = floor(i_eval/10);
     i_eval2 = i_eval - 10*i_eval1;

     if( i_dof<=n_vert )   % Node dofs.

       d2Ndxi2 = 4;

       i_eval = floor(i_eval/10);
       i_col  = i_dof + (i_eval1-2)*n_vert;
       j_col  = i_dof + (i_eval2-2)*n_vert;
       vBase  = aInvJac(:,i_col).*aInvJac(:,j_col)*d2Ndxi2;

     else   % Edge dofs.

       d2Ndxii2    = 0;
       d2Ndxiidxij = 4;
       d2Ndxijdxii = d2Ndxiidxij;
       d2Ndxij2    = 0;

       i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
       j = mod(i,min(n_vert,3))+1;
       if( i_dof>7 )
         j = 4;
       end

       i_col = i + (i_eval1-2)*n_vert;
       j_col = j + (i_eval1-2)*n_vert;

       k_col = i + (i_eval2-2)*n_vert;
       l_col = j + (i_eval2-2)*n_vert;

       vBase  = aInvJac(:,k_col).*( aInvJac(:,i_col)*d2Ndxii2    + aInvJac(:,j_col)*d2Ndxiidxij ) + ...
                aInvJac(:,l_col).*( aInvJac(:,i_col)*d2Ndxijdxii + aInvJac(:,j_col)*d2Ndxij2    );
     end

 else

   vBase = 0;

 end