FEATool Multiphysics  v1.10Finite Element Analysis Toolbox

## Description

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q2( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming biquadratic Q2 shape functions on quadrilaterals with values defined in the nodes, edges, and cell center. 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

sflag2

# Code listing

 nLDof = [4 4 0 1];
xLDof = [-1  1 1 -1  0 1 0 -1 0; ...
-1 -1 1  1 -1 0 1  0 0];

switch i_eval   % Evaluation type flag.

case 1   % Evaluation of function values.

switch i_dof   % Basis function to evaluate.
case 1
vBase =  (1-xi(1))*(1-xi(2))*xi(1)*xi(2)/4;
case 2
vBase = -(1+xi(1))*(1-xi(2))*xi(1)*xi(2)/4;
case 3
vBase =  (1+xi(1))*(1+xi(2))*xi(1)*xi(2)/4;
case 4
vBase = -(1-xi(1))*(1+xi(2))*xi(1)*xi(2)/4;
case 5
vBase = -(1-xi(1)*xi(1))*(1-xi(2))*xi(2)/2;
case 6
vBase =  (1+xi(1))*(1-xi(2)*xi(2))*xi(1)/2;
case 7
vBase =  (1-xi(1)*xi(1))*(1+xi(2))*xi(2)/2;
case 8
vBase = -(1-xi(1))*(1-xi(2)*xi(2))*xi(1)/2;
case 9
vBase =  (1-xi(1)*xi(1))*(1-xi(2)*xi(2));
end

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

switch i_dof   % Basis function to evaluate.
case 1
dNdxi1 =  (1-2*xi(1))*(1-xi(2))*xi(2)/4;
dNdxi2 =  (1-xi(1))*(1-2*xi(2))*xi(1)/4;
case 2
dNdxi1 = -(1+2*xi(1))*(1-xi(2))*xi(2)/4;
dNdxi2 = -(1+xi(1))*(1-2*xi(2))*xi(1)/4;
case 3
dNdxi1 =  (1+2*xi(1))*(1+xi(2))*xi(2)/4;
dNdxi2 =  (1+xi(1))*(1+2*xi(2))*xi(1)/4;
case 4
dNdxi1 = -(1-2*xi(1))*(1+xi(2))*xi(2)/4;
dNdxi2 = -(1-xi(1))*(1+2*xi(2))*xi(1)/4;
case 5
dNdxi1 =  (1-xi(2))*xi(1)*xi(2);
dNdxi2 = -(1-xi(1)*xi(1))*(1-2*xi(2))/2;
case 6
dNdxi1 =  (1+2*xi(1))*(1-xi(2)*xi(2))/2;
dNdxi2 = -(1+xi(1))*xi(1)*xi(2);
case 7
dNdxi1 = -(1+xi(2))*xi(1)*xi(2);
dNdxi2 =  (1-xi(1)*xi(1))*(1+2*xi(2))/2;
case 8
dNdxi1 = -(1-2*xi(1))*(1-xi(2)*xi(2))/2;
dNdxi2 =  (1-xi(1))*xi(1)*xi(2);
case 9
dNdxi1 = -2*(1-xi(2)*xi(2))*xi(1);
dNdxi2 = -2*(1-xi(1)*xi(1))*xi(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}

if( any(abs([aInvJac(:,2);aInvJac(:,3)])>eps*1e2) )
warning('sf_quad_Q2: 2nd derivatives for non-rectangular cells shapes not supported.')
end

switch i_dof   % Basis function to evaluate.
case 1
d2Ndxi12    = xi(2)^2/2 - xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) - 1)*(2*xi(2) - 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) - 1)*(2*xi(2) - 1))/4;
d2Ndxi22    = xi(1)^2/2 - xi(1)/2;
case 2
d2Ndxi12    = xi(2)^2/2 - xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) + 1)*(2*xi(2) - 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) + 1)*(2*xi(2) - 1))/4;
d2Ndxi22    = xi(1)^2/2 + xi(1)/2;
case 3
d2Ndxi12    = xi(2)^2/2 + xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) + 1)*(2*xi(2) + 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) + 1)*(2*xi(2) + 1))/4;
d2Ndxi22    = xi(1)^2/2 + xi(1)/2;
case 4
d2Ndxi12    = xi(2)^2/2 + xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) - 1)*(2*xi(2) + 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) - 1)*(2*xi(2) + 1))/4;
d2Ndxi22    = xi(1)^2/2 - xi(1)/2;
case 5
d2Ndxi12    = xi(2) - xi(2)^2;
d2Ndxi1dxi2 = xi(1) - 2*xi(1)*xi(2);
d2Ndxi2dxi1 = xi(1) - 2*xi(1)*xi(2);
d2Ndxi22    = 1 - xi(1)^2;
case 6
d2Ndxi12    = 1 - xi(2)^2;
d2Ndxi1dxi2 = -xi(2)*(2*xi(1) + 1);
d2Ndxi2dxi1 = -xi(2)*(2*xi(1) + 1);
d2Ndxi22    = - xi(1)^2 - xi(1);
case 7
d2Ndxi12    = - xi(2)^2 - xi(2);
d2Ndxi1dxi2 = -xi(1)*(2*xi(2) + 1);
d2Ndxi2dxi1 = -xi(1)*(2*xi(2) + 1);
d2Ndxi22    = 1 - xi(1)^2;
case 8
d2Ndxi12    = 1 - xi(2)^2;
d2Ndxi1dxi2 = xi(2) - 2*xi(1)*xi(2);
d2Ndxi2dxi1 = xi(2) - 2*xi(1)*xi(2);
d2Ndxi22    = xi(1) - xi(1)^2;
case 9
d2Ndxi12    = 2*xi(2)^2 - 2;
d2Ndxi1dxi2 = 4*xi(1)*xi(2);
d2Ndxi2dxi1 = 4*xi(1)*xi(2);
d2Ndxi22    = 2*xi(1)^2 - 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