FEATool  v1.9Finite Element Analysis Toolbox

## Description

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming bicubic Q3 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: 1-16           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
```

# Code listing

``` nLDof = [4 8 0 4];
xLDof = [-1  1 1 -1 -1/3  1   1/3 -1    1/3 1   -1/3 -1   -1/3  1/3 1/3 -1/3;
-1 -1 1  1 -1   -1/3 1    1/3 -1   1/3  1   -1/3 -1/3 -1/3 1/3  1/3];

switch i_eval

case 1

switch i_dof
case 1
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 2
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) - 1);
case 3
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) + 1);
case 4
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 5
vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 6
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1);
case 7
vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 8
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 9
vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 10
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1);
case 11
vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 12
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 13
vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 14
vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 15
vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 16
vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
end

case {2,3}

switch i_dof
case 1
dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 2
dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 3
dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 4
dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 5
dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 6
dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 7
dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 8
dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 9
dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 10
dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 11
dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 12
dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 13
dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 14
dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 15
dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 16
dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
end

if( i_eval==2 )
vBase = aInvJac(:,1)*dNdxi1 + aInvJac(:,2)*dNdxi2;
else
vBase = aInvJac(:,3)*dNdxi1 + aInvJac(:,4)*dNdxi2;
end

case {22,23,32,33}   % Evaluation of second order derivatives.
error('sf_quad_Q3: second order derivative evaluation not supported.')

otherwise
vBase = 0;

end```