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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.5
Finite Element Analysis Toolbox
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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
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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 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