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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_QUAD_Q2 Biquadratic conforming shape function for quadrilaterals (Q2).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q2( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming biquadratic Q2 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
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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-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 = [4 4 0 1];
xLDof = [-1 1 1 -1 0 1 0 -1 0; ...
-1 -1 1 1 -1 0 1 0 0];
sfun = 'sf_quad_Q2';
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))*xi(1)*xi(2)/4;
case 2
vBase = -(1+xi(1))*(1-xi(2))*xi(1)*xi(2)/4;
case 3
vBase = (1+xi(1))*(1+xi(2))*xi(1)*xi(2)/4;
case 4
vBase = -(1-xi(1))*(1+xi(2))*xi(1)*xi(2)/4;
case 5
vBase = -(1-xi(1)*xi(1))*(1-xi(2))*xi(2)/2;
case 6
vBase = (1+xi(1))*(1-xi(2)*xi(2))*xi(1)/2;
case 7
vBase = (1-xi(1)*xi(1))*(1+xi(2))*xi(2)/2;
case 8
vBase = -(1-xi(1))*(1-xi(2)*xi(2))*xi(1)/2;
case 9
vBase = (1-xi(1)*xi(1))*(1-xi(2)*xi(2));
end
case {2,3} % Evaluation of first order derivatives.
switch i_dof % Basis function to evaluate.
case 1
dNdxi1 = (1-2*xi(1))*(1-xi(2))*xi(2)/4;
dNdxi2 = (1-xi(1))*(1-2*xi(2))*xi(1)/4;
case 2
dNdxi1 = -(1+2*xi(1))*(1-xi(2))*xi(2)/4;
dNdxi2 = -(1+xi(1))*(1-2*xi(2))*xi(1)/4;
case 3
dNdxi1 = (1+2*xi(1))*(1+xi(2))*xi(2)/4;
dNdxi2 = (1+xi(1))*(1+2*xi(2))*xi(1)/4;
case 4
dNdxi1 = -(1-2*xi(1))*(1+xi(2))*xi(2)/4;
dNdxi2 = -(1-xi(1))*(1+2*xi(2))*xi(1)/4;
case 5
dNdxi1 = (1-xi(2))*xi(1)*xi(2);
dNdxi2 = -(1-xi(1)*xi(1))*(1-2*xi(2))/2;
case 6
dNdxi1 = (1+2*xi(1))*(1-xi(2)*xi(2))/2;
dNdxi2 = -(1+xi(1))*xi(1)*xi(2);
case 7
dNdxi1 = -(1+xi(2))*xi(1)*xi(2);
dNdxi2 = (1-xi(1)*xi(1))*(1+2*xi(2))/2;
case 8
dNdxi1 = -(1-2*xi(1))*(1-xi(2)*xi(2))/2;
dNdxi2 = (1-xi(1))*xi(1)*xi(2);
case 9
dNdxi1 = -2*(1-xi(2)*xi(2))*xi(1);
dNdxi2 = -2*(1-xi(1)*xi(1))*xi(2);
end
if ( i_eval==2 ) % x-derivative.
vBase = aInvJac(:,1)*dNdxi1+aInvJac(:,2)*dNdxi2;
elseif ( i_eval==3 ) % y-derivative.
vBase = aInvJac(:,3)*dNdxi1+aInvJac(:,4)*dNdxi2;
end
case {22,23,32,33}
if( any(abs([aInvJac(:,2);aInvJac(:,3)])>eps*1e2) )
warning('sf_quad_Q2: 2nd derivatives for non-rectangular cells shapes not supported.')
end
switch i_dof % Basis function to evaluate.
case 1
d2Ndxi12 = xi(2)^2/2 - xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) - 1)*(2*xi(2) - 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) - 1)*(2*xi(2) - 1))/4;
d2Ndxi22 = xi(1)^2/2 - xi(1)/2;
case 2
d2Ndxi12 = xi(2)^2/2 - xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) + 1)*(2*xi(2) - 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) + 1)*(2*xi(2) - 1))/4;
d2Ndxi22 = xi(1)^2/2 + xi(1)/2;
case 3
d2Ndxi12 = xi(2)^2/2 + xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) + 1)*(2*xi(2) + 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) + 1)*(2*xi(2) + 1))/4;
d2Ndxi22 = xi(1)^2/2 + xi(1)/2;
case 4
d2Ndxi12 = xi(2)^2/2 + xi(2)/2;
d2Ndxi1dxi2 = ((2*xi(1) - 1)*(2*xi(2) + 1))/4;
d2Ndxi2dxi1 = ((2*xi(1) - 1)*(2*xi(2) + 1))/4;
d2Ndxi22 = xi(1)^2/2 - xi(1)/2;
case 5
d2Ndxi12 = xi(2) - xi(2)^2;
d2Ndxi1dxi2 = xi(1) - 2*xi(1)*xi(2);
d2Ndxi2dxi1 = xi(1) - 2*xi(1)*xi(2);
d2Ndxi22 = 1 - xi(1)^2;
case 6
d2Ndxi12 = 1 - xi(2)^2;
d2Ndxi1dxi2 = -xi(2)*(2*xi(1) + 1);
d2Ndxi2dxi1 = -xi(2)*(2*xi(1) + 1);
d2Ndxi22 = - xi(1)^2 - xi(1);
case 7
d2Ndxi12 = - xi(2)^2 - xi(2);
d2Ndxi1dxi2 = -xi(1)*(2*xi(2) + 1);
d2Ndxi2dxi1 = -xi(1)*(2*xi(2) + 1);
d2Ndxi22 = 1 - xi(1)^2;
case 8
d2Ndxi12 = 1 - xi(2)^2;
d2Ndxi1dxi2 = xi(2) - 2*xi(1)*xi(2);
d2Ndxi2dxi1 = xi(2) - 2*xi(1)*xi(2);
d2Ndxi22 = xi(1) - xi(1)^2;
case 9
d2Ndxi12 = 2*xi(2)^2 - 2;
d2Ndxi1dxi2 = 4*xi(1)*xi(2);
d2Ndxi2dxi1 = 4*xi(1)*xi(2);
d2Ndxi22 = 2*xi(1)^2 - 2;
end
if ( i_eval==22 ) % xx-derivative.
vBase = aInvJac(:,1).*( aInvJac(:,1)*d2Ndxi12 + aInvJac(:,2)*d2Ndxi1dxi2 ) + ...
aInvJac(:,2).*( aInvJac(:,1)*d2Ndxi2dxi1 + aInvJac(:,2)*d2Ndxi22 );
elseif ( i_eval==23 ) % xy-derivative.
vBase = aInvJac(:,3).*( aInvJac(:,1)*d2Ndxi12 + aInvJac(:,2)*d2Ndxi1dxi2 ) + ...
aInvJac(:,4).*( aInvJac(:,1)*d2Ndxi2dxi1 + aInvJac(:,2)*d2Ndxi22 );
elseif ( i_eval==32 ) % yx-derivative.
vBase = aInvJac(:,1).*( aInvJac(:,3)*d2Ndxi12 + aInvJac(:,4)*d2Ndxi1dxi2 ) + ...
aInvJac(:,2).*( aInvJac(:,3)*d2Ndxi2dxi1 + aInvJac(:,4)*d2Ndxi22 );
elseif ( i_eval==33 ) % yy-derivative.
vBase = aInvJac(:,3).*( aInvJac(:,3)*d2Ndxi12 + aInvJac(:,4)*d2Ndxi1dxi2 ) + ...
aInvJac(:,4).*( aInvJac(:,3)*d2Ndxi2dxi1 + aInvJac(:,4)*d2Ndxi22 );
end
otherwise
vBase = 0;
end