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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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SF_HEX_Q1 Trilinear conforming shape function for hexahedrons (Q1).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_HEX_Q1( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming trilinear Q1 shape functions on hexahedrons with values defined in the nodes. XI is [-1..1]^3 reference coordinates.
Input Value/[Size] Description
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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 = [8 0 0 0];
xLDof = [-1 1 1 -1 -1 1 1 -1; ...
-1 -1 1 1 -1 -1 1 1; ...
-1 -1 -1 -1 1 1 1 1];
sfun = 'sf_hex_Q1';
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))*(1-xi(3))/8;
case 2
vBase = (1+xi(1))*(1-xi(2))*(1-xi(3))/8;
case 3
vBase = (1+xi(1))*(1+xi(2))*(1-xi(3))/8;
case 4
vBase = (1-xi(1))*(1+xi(2))*(1-xi(3))/8;
case 5
vBase = (1-xi(1))*(1-xi(2))*(1+xi(3))/8;
case 6
vBase = (1+xi(1))*(1-xi(2))*(1+xi(3))/8;
case 7
vBase = (1+xi(1))*(1+xi(2))*(1+xi(3))/8;
case 8
vBase = (1-xi(1))*(1+xi(2))*(1+xi(3))/8;
end
case {2,3,4} % Evaluation of first order derivatives.
switch i_dof % Basis function to evaluate.
case 1
dNdxi1 = -(1-xi(2))*(1-xi(3))/8;
dNdxi2 = -(1-xi(1))*(1-xi(3))/8;
dNdxi3 = -(1-xi(1))*(1-xi(2))/8;
case 2
dNdxi1 = (1-xi(2))*(1-xi(3))/8;
dNdxi2 = -(1+xi(1))*(1-xi(3))/8;
dNdxi3 = -(1+xi(1))*(1-xi(2))/8;
case 3
dNdxi1 = (1+xi(2))*(1-xi(3))/8;
dNdxi2 = (1+xi(1))*(1-xi(3))/8;
dNdxi3 = -(1+xi(1))*(1+xi(2))/8;
case 4
dNdxi1 = -(1+xi(2))*(1-xi(3))/8;
dNdxi2 = (1-xi(1))*(1-xi(3))/8;
dNdxi3 = -(1-xi(1))*(1+xi(2))/8;
case 5
dNdxi1 = -(1-xi(2))*(1+xi(3))/8;
dNdxi2 = -(1-xi(1))*(1+xi(3))/8;
dNdxi3 = (1-xi(1))*(1-xi(2))/8;
case 6
dNdxi1 = (1-xi(2))*(1+xi(3))/8;
dNdxi2 = -(1+xi(1))*(1+xi(3))/8;
dNdxi3 = (1+xi(1))*(1-xi(2))/8;
case 7
dNdxi1 = (1+xi(2))*(1+xi(3))/8;
dNdxi2 = (1+xi(1))*(1+xi(3))/8;
dNdxi3 = (1+xi(1))*(1+xi(2))/8;
case 8
dNdxi1 = -(1+xi(2))*(1+xi(3))/8;
dNdxi2 = (1-xi(1))*(1+xi(3))/8;
dNdxi3 = (1-xi(1))*(1+xi(2))/8;
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_Q1: 2nd derivatives for non-rectangular cells shapes not supported.')
end
switch i_dof
case 1
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 1/8 - xi(3)/8;
d2Ndxi3dxi1 = 1/8 - xi(2)/8;
d2Ndxi1dxi2 = 1/8 - xi(3)/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 1/8 - xi(1)/8;
d2Ndxi1dxi3 = 1/8 - xi(2)/8;
d2Ndxi2dxi3 = 1/8 - xi(1)/8;
d2Ndxi3dxi3 = 0;
case 2
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = xi(3)/8 - 1/8;
d2Ndxi3dxi1 = xi(2)/8 - 1/8;
d2Ndxi1dxi2 = xi(3)/8 - 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = xi(1)/8 + 1/8;
d2Ndxi1dxi3 = xi(2)/8 - 1/8;
d2Ndxi2dxi3 = xi(1)/8 + 1/8;
d2Ndxi3dxi3 = 0;
case 3
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = 1/8 - xi(3)/8;
d2Ndxi3dxi1 = - xi(2)/8 - 1/8;
d2Ndxi1dxi2 = 1/8 - xi(3)/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = - xi(1)/8 - 1/8;
d2Ndxi1dxi3 = - xi(2)/8 - 1/8;
d2Ndxi2dxi3 = - xi(1)/8 - 1/8;
d2Ndxi3dxi3 = 0;
case 4
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = xi(3)/8 - 1/8;
d2Ndxi3dxi1 = xi(2)/8 + 1/8;
d2Ndxi1dxi2 = xi(3)/8 - 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = xi(1)/8 - 1/8;
d2Ndxi1dxi3 = xi(2)/8 + 1/8;
d2Ndxi2dxi3 = xi(1)/8 - 1/8;
d2Ndxi3dxi3 = 0;
case 5
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = xi(3)/8 + 1/8;
d2Ndxi3dxi1 = xi(2)/8 - 1/8;
d2Ndxi1dxi2 = xi(3)/8 + 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = xi(1)/8 - 1/8;
d2Ndxi1dxi3 = xi(2)/8 - 1/8;
d2Ndxi2dxi3 = xi(1)/8 - 1/8;
d2Ndxi3dxi3 = 0;
case 6
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = - xi(3)/8 - 1/8;
d2Ndxi3dxi1 = 1/8 - xi(2)/8;
d2Ndxi1dxi2 = - xi(3)/8 - 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = - xi(1)/8 - 1/8;
d2Ndxi1dxi3 = 1/8 - xi(2)/8;
d2Ndxi2dxi3 = - xi(1)/8 - 1/8;
d2Ndxi3dxi3 = 0;
case 7
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = xi(3)/8 + 1/8;
d2Ndxi3dxi1 = xi(2)/8 + 1/8;
d2Ndxi1dxi2 = xi(3)/8 + 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = xi(1)/8 + 1/8;
d2Ndxi1dxi3 = xi(2)/8 + 1/8;
d2Ndxi2dxi3 = xi(1)/8 + 1/8;
d2Ndxi3dxi3 = 0;
case 8
d2Ndxi1dxi1 = 0;
d2Ndxi2dxi1 = - xi(3)/8 - 1/8;
d2Ndxi3dxi1 = - xi(2)/8 - 1/8;
d2Ndxi1dxi2 = - xi(3)/8 - 1/8;
d2Ndxi2dxi2 = 0;
d2Ndxi3dxi2 = 1/8 - xi(1)/8;
d2Ndxi1dxi3 = - xi(2)/8 - 1/8;
d2Ndxi2dxi3 = 1/8 - xi(1)/8;
d2Ndxi3dxi3 = 0;
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