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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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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
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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
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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
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