FEATool  v1.8
Finite Element Analysis Toolbox
 All Files Functions Pages
sf_quad_Q1nc.m File Reference

Description

SF_QUAD_Q1NC Bilinear nonconforming shape function for quadrilaterals (Q1~).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q1NC( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates nonconforming rotated bilinear Q1~ shape functions on quadrilaterals with values defined on the edge midpoints. XI is [-1..1]^2 reference coordinates.

Input       Value/[Size]           Description
-----------------------------------------------------------------------------------
i_eval      scalar:  1             Evaluate function values
                    >1             Evaluate values of derivatives
n_sdim      scalar:  2             Number of space dimensions
n_vert      scalar:  4             Number of vertices per cell
i_dof       scalar: 1-n_ldof       Local basis function to evaluate
xi          [n_sdim]               Local coordinates of evaluation point
aInvJac     [n,n_sdim*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
sflag1, sf_quad_Q1

Code listing

 nLDof = [0 4 0 0];
 xLDof = [ 0  1 0 -1; ...
          -1  0 1  0];
 sfun  = 'sf_quad_Q1nc';


 switch i_eval   % Evaluation type flag.

   case 1   % Evaluation of function values.

     switch i_dof   % Basis function to evaluate.

       case 1
         vBase = (-xi(1)^2+xi(2)^2-2*xi(2)+1)/4;
       case 2
         vBase = ( xi(1)^2-xi(2)^2+2*xi(1)+1)/4;
       case 3
         vBase = (-xi(1)^2+xi(2)^2+2*xi(2)+1)/4;
       case 4
         vBase = ( xi(1)^2-xi(2)^2-2*xi(1)+1)/4;
     end

   case {2,3}   % Evaluation of first order derivatives.

     switch i_dof   % Basis function to evaluate.

       case 1
         dNdxi1 = -xi(1)/2;
         dNdxi2 = (xi(2)-1)/2;
       case 2
         dNdxi1 = (xi(1)+1)/2;
         dNdxi2 = -xi(2)/2;
       case 3
         dNdxi1 = -xi(1)/2;
         dNdxi2 = (xi(2)+1)/2;
       case 4
         dNdxi1 = (xi(1)-1)/2;
         dNdxi2 = -xi(2)/2;
     end

     if     ( i_eval==2 )   % x-derivative.

       vBase = aInvJac(:,1)*dNdxi1+aInvJac(:,2)*dNdxi2;

     elseif ( i_eval==3 )   % y-derivative.

       vBase = aInvJac(:,3)*dNdxi1+aInvJac(:,4)*dNdxi2;
     end

   case {22,23,32,33}   % Evaluation of second order derivatives.

     if( any(abs([aInvJac(:,2);aInvJac(:,3)])>eps*1e2) )
       warning('sf_quad_Q1nc: 2nd derivatives for non-rectangular cells shapes not supported.')
     end

     switch i_dof   % Basis function to evaluate.

       case {1,3}
         d2Ndxi12    = -1/2;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi22    = 1/2;
       case {2,4}
         d2Ndxi12    = 1/2;
         d2Ndxi1dxi2 = 0;
         d2Ndxi2dxi1 = 0;
         d2Ndxi22    = -1/2;
     end

     if ( i_eval==22 )   % xx-derivative.

       vBase  = aInvJac(:,1).*( aInvJac(:,1)*d2Ndxi12    + aInvJac(:,2)*d2Ndxi1dxi2 ) + ...
                aInvJac(:,2).*( aInvJac(:,1)*d2Ndxi2dxi1 + aInvJac(:,2)*d2Ndxi22    );

     elseif ( i_eval==23 )   % xy-derivative.

       vBase  = aInvJac(:,3).*( aInvJac(:,1)*d2Ndxi12    + aInvJac(:,2)*d2Ndxi1dxi2 ) + ...
                aInvJac(:,4).*( aInvJac(:,1)*d2Ndxi2dxi1 + aInvJac(:,2)*d2Ndxi22    );

     elseif ( i_eval==32 )   % yx-derivative.

       vBase  = aInvJac(:,1).*( aInvJac(:,3)*d2Ndxi12    + aInvJac(:,4)*d2Ndxi1dxi2 ) + ...
                aInvJac(:,2).*( aInvJac(:,3)*d2Ndxi2dxi1 + aInvJac(:,4)*d2Ndxi22    );

     elseif ( i_eval==33 )   % yy-derivative.

       vBase  = aInvJac(:,3).*( aInvJac(:,3)*d2Ndxi12    + aInvJac(:,4)*d2Ndxi1dxi2 ) + ...
                aInvJac(:,4).*( aInvJac(:,3)*d2Ndxi2dxi1 + aInvJac(:,4)*d2Ndxi22    );
     end

   otherwise
     vBase = 0;

 end