FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox
sf_hex_Q1nc.m File Reference

## Description

SF_HEX_Q1NC Trilinear nonconforming shape function for hexahedrons (Q1~).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_HEX_Q1NC( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates nonconforming trilinear Q1~ shape functions on hexahedrons with values defined in the face centers. XI is [-1..1]^3 reference coordinates.

Input       Value/[Size]           Description
-----------------------------------------------------------------------------------
i_eval      scalar:  1             Evaluate function values
>1             Evaluate values of derivatives
n_sdim      scalar:  3             Number of space dimensions
n_vert      scalar:  8             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

sflag1, sf_hex_Q1

# Code listing

 nLDof = [0 0 6 0];
xLDof = [ 0  0  1 0 -1 0; ...
0 -1  0 1  0 0; ...
-1  0  0 0  0 1];
sfun  = 'sf_hex_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)/6-(xi(1)^2-xi(3)^2)/3-xi(3)/2+1/6;
case 2
vBase =-(xi(1)^2-xi(2)^2)/3+(xi(1)^2-xi(3)^2)/6-xi(2)/2+1/6;
case 3
vBase = (xi(1)^2-xi(2)^2)/6+(xi(1)^2-xi(3)^2)/6+xi(1)/2+1/6;
case 4
vBase =-(xi(1)^2-xi(2)^2)/3+(xi(1)^2-xi(3)^2)/6+xi(2)/2+1/6;
case 5
vBase = (xi(1)^2-xi(2)^2)/6+(xi(1)^2-xi(3)^2)/6-xi(1)/2+1/6;
case 6
vBase = (xi(1)^2-xi(2)^2)/6-(xi(1)^2-xi(3)^2)/3+xi(3)/2+1/6;
end

case {2,3,4}   % Evaluation of first order derivatives.

switch i_dof   % Basis function to evaluate.

case 1
dNdxi1 = -xi(1)/3;
dNdxi2 = -xi(2)/3;
dNdxi3 = 2/3*xi(3)-1/2;
case 2
dNdxi1 = -xi(1)/3;
dNdxi2 = 2/3*xi(2)-1/2;
dNdxi3 = -xi(3)/3;
case 3
dNdxi1 = 2/3*xi(1)+1/2;
dNdxi2 = -xi(2)/3;
dNdxi3 = -xi(3)/3;
case 4
dNdxi1 = -xi(1)/3;
dNdxi2 = 2/3*xi(2)+1/2;
dNdxi3 = -xi(3)/3;
case 5
dNdxi1 = 2/3*xi(1)-1/2;
dNdxi2 = -xi(2)/3;
dNdxi3 = -xi(3)/3;
case 6
dNdxi1 = -xi(1)/3;
dNdxi2 = -xi(2)/3;
dNdxi3 = 2/3*xi(3)+1/2;
end

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

vBase = aInvJac(:,1)*dNdxi1 + aInvJac(:,2)*dNdxi2 + aInvJac(:,3)*dNdxi3;

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

vBase = aInvJac(:,4)*dNdxi1 + aInvJac(:,5)*dNdxi2 + aInvJac(:,6)*dNdxi3;

elseif ( i_eval==4 )   % z-derivative.

vBase = aInvJac(:,7)*dNdxi1 + aInvJac(:,8)*dNdxi2 + aInvJac(:,9)*dNdxi3;
end

case {22,23,24,32,33,34,42,43,44}   % Evaluation of second order derivatives.

if( any(any(abs([aInvJac(:,[2 3 4 6 7 8])])>eps*1e2)) )
warning('sf_hex_Q1nc: 2nd derivatives for non-rectangular cells shapes not supported.')
end

switch i_dof

case 1
d2Ndxi1dxi1 = -1/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = -1/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 2/3;

case 2
d2Ndxi1dxi1 = -1/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 2/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = -1/3;

case 3
d2Ndxi1dxi1 = 2/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = -1/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = -1/3;

case 4
d2Ndxi1dxi1 = -1/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = 2/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = -1/3;

case 5
d2Ndxi1dxi1 = 2/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = -1/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = -1/3;

case 6
d2Ndxi1dxi1 = -1/3;
d2Ndxi2dxi1 = 0;
d2Ndxi3dxi1 = 0;
d2Ndxi1dxi2 = 0;
d2Ndxi2dxi2 = -1/3;
d2Ndxi3dxi2 = 0;
d2Ndxi1dxi3 = 0;
d2Ndxi2dxi3 = 0;
d2Ndxi3dxi3 = 2/3;

end

switch( i_eval )
case 22
vBase = aInvJac(:,1).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:,2)*d2Ndxi2dxi1 + aInvJac(:,3)*d2Ndxi3dxi1 ) + ...
aInvJac(:,2).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:,2)*d2Ndxi2dxi2 + aInvJac(:,3)*d2Ndxi3dxi2 ) + ...
aInvJac(:,3).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:,2)*d2Ndxi2dxi3 + aInvJac(:,3)*d2Ndxi3dxi3 );
case 33
vBase = aInvJac(:,4).*( aInvJac(:,4)*d2Ndxi1dxi1 + aInvJac(:,5)*d2Ndxi2dxi1 + aInvJac(:,6)*d2Ndxi3dxi1 ) + ...
aInvJac(:,5).*( aInvJac(:,4)*d2Ndxi1dxi2 + aInvJac(:,5)*d2Ndxi2dxi2 + aInvJac(:,6)*d2Ndxi3dxi2 ) + ...
aInvJac(:,6).*( aInvJac(:,4)*d2Ndxi1dxi3 + aInvJac(:,5)*d2Ndxi2dxi3 + aInvJac(:,6)*d2Ndxi3dxi3 );

case 44
vBase = aInvJac(:,7).*( aInvJac(:,7)*d2Ndxi1dxi1 + aInvJac(:,8)*d2Ndxi2dxi1 + aInvJac(:,9)*d2Ndxi3dxi1 ) + ...
aInvJac(:,8).*( aInvJac(:,7)*d2Ndxi1dxi2 + aInvJac(:,8)*d2Ndxi2dxi2 + aInvJac(:,9)*d2Ndxi3dxi2 ) + ...
aInvJac(:,9).*( aInvJac(:,7)*d2Ndxi1dxi3 + aInvJac(:,8)*d2Ndxi2dxi3 + aInvJac(:,9)*d2Ndxi3dxi3 );

case {23,32}
vBase = aInvJac(:,4).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:,2)*d2Ndxi2dxi1 + aInvJac(:,3)*d2Ndxi3dxi1 ) + ...
aInvJac(:,5).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:,2)*d2Ndxi2dxi2 + aInvJac(:,3)*d2Ndxi3dxi2 ) + ...
aInvJac(:,6).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:,2)*d2Ndxi2dxi3 + aInvJac(:,3)*d2Ndxi3dxi3 );

case {24,42}
vBase = aInvJac(:,7).*( aInvJac(:,1)*d2Ndxi1dxi1 + aInvJac(:,2)*d2Ndxi2dxi1 + aInvJac(:,3)*d2Ndxi3dxi1 ) + ...
aInvJac(:,8).*( aInvJac(:,1)*d2Ndxi1dxi2 + aInvJac(:,2)*d2Ndxi2dxi2 + aInvJac(:,3)*d2Ndxi3dxi2 ) + ...
aInvJac(:,9).*( aInvJac(:,1)*d2Ndxi1dxi3 + aInvJac(:,2)*d2Ndxi2dxi3 + aInvJac(:,3)*d2Ndxi3dxi3 );

case {34,43}
vBase = aInvJac(:,7).*( aInvJac(:,4)*d2Ndxi1dxi1 + aInvJac(:,5)*d2Ndxi2dxi1 + aInvJac(:,6)*d2Ndxi3dxi1 ) + ...
aInvJac(:,8).*( aInvJac(:,4)*d2Ndxi1dxi2 + aInvJac(:,5)*d2Ndxi2dxi2 + aInvJac(:,6)*d2Ndxi3dxi2 ) + ...
aInvJac(:,9).*( aInvJac(:,4)*d2Ndxi1dxi3 + aInvJac(:,5)*d2Ndxi2dxi3 + aInvJac(:,6)*d2Ndxi3dxi3 );
end

otherwise
vBase = 0;

end