FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox

## Description

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q4( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming biquartic Q4 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: 1-25           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 12 0 9];
xLDof = [-1  1 1 -1  -1/2  1   1/2 -1     0  1 0 -1   1/2 1   -1/2 -1    -1/2  1/2 1/2 -1/2  0   1/2 0   -1/2 0;
-1 -1 1  1  -1   -1/2 1    1/2  -1  0 1  0  -1   1/2  1   -1/2  -1/2 -1/2 1/2  1/2 -1/2 0   1/2  0   0];

switch i_eval

case 1

switch i_dof
case 1
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/36;
case 2
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/36;
case 3
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/36;
case 4
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/36;
case 5
vBase = -(2*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/9;
case 6
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 7
vBase = -(2*xi(1)*xi(2)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 8
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 9
vBase = -(xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/6;
case 10
vBase = -(xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1))/6;
case 11
vBase = -(xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/6;
case 12
vBase = -(xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1))/6;
case 13
vBase = -(2*xi(1)*xi(2)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 14
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 15
vBase = -(2*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/9;
case 16
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 17
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 18
vBase = (16*xi(1)*xi(2)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 19
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 20
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 21
vBase = (4*xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/3;
case 22
vBase = (4*xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/3;
case 23
vBase = (4*xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/3;
case 24
vBase = (4*xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1))/3;
case 25
vBase = (4*xi(1)^4 - 5*xi(1)^2 + 1)*(4*xi(2)^4 - 5*xi(2)^2 + 1);
end

case {2,3}

switch i_dof
case 1
dNdxi1 = -(xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/36;
dNdxi2 = -(xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/36;
case 2
dNdxi1 = (xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(4*xi(1)^2 + 2*xi(1) - 1))/36;
dNdxi2 = -(xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/36;
case 3
dNdxi1 = (xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/36;
dNdxi2 = (xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/36;
case 4
dNdxi1 = -(xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/36;
dNdxi2 = (xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(4*xi(2)^2 + 2*xi(2) - 1))/36;
case 5
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/9;
case 6
dNdxi1 = -(2*xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 7
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(2*xi(1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/9;
case 8
dNdxi1 = (2*xi(2)*(4*xi(1) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) - 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 9
dNdxi1 = (xi(1)*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) - 1))/3;
dNdxi2 = -((2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/6;
case 10
dNdxi1 = ((4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/6;
dNdxi2 = (xi(1)*xi(2)*(2*xi(1) - 1)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) + 1))/3;
case 11
dNdxi1 = (xi(1)*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) + 1))/3;
dNdxi2 = ((2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/6;
case 12
dNdxi1 = -((4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/6;
dNdxi2 = (xi(1)*xi(2)*(2*xi(1) - 1)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) - 1))/3;
case 13
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/9;
case 14
dNdxi1 = -(2*xi(2)*(4*xi(1) + 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 15
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(2*xi(1)*(2*xi(1) - 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/9;
case 16
dNdxi1 = (2*xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) - 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 17
dNdxi1 = -(16*xi(2)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 18
dNdxi1 = -(16*xi(2)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 19
dNdxi1 = -(16*xi(2)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 20
dNdxi1 = -(16*xi(2)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 21
dNdxi1 = -(8*xi(1)*xi(2)*(2*xi(2) - 1)*(8*xi(1)^2 - 5)*(xi(2) - 1)*(xi(2) + 1))/3;
dNdxi2 = (4*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/3;
case 22
dNdxi1 = (4*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/3;
dNdxi2 = -(8*xi(1)*xi(2)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) - 1)*(xi(1) + 1))/3;
case 23
dNdxi1 = -(8*xi(1)*xi(2)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) - 1)*(xi(2) + 1))/3;
dNdxi2 = (4*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/3;
case 24
dNdxi1 = (4*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/3;
dNdxi2 = -(8*xi(1)*xi(2)*(2*xi(1) - 1)*(8*xi(2)^2 - 5)*(xi(1) - 1)*(xi(1) + 1))/3;
case 25
dNdxi1 = 2*xi(1)*(8*xi(1)^2 - 5)*(4*xi(2)^4 - 5*xi(2)^2 + 1);
dNdxi2 = 2*xi(2)*(8*xi(2)^2 - 5)*(4*xi(1)^4 - 5*xi(1)^2 + 1);
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_Q4: second order derivative evaluation not supported.')

otherwise
vBase = 0;

end