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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_QUAD_Q3 Bicubic conforming shape function for quadrilaterals (Q3).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming bicubic Q3 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: 1-16 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 8 0 4];
xLDof = [-1 1 1 -1 -1/3 1 1/3 -1 1/3 1 -1/3 -1 -1/3 1/3 1/3 -1/3;
-1 -1 1 1 -1 -1/3 1 1/3 -1 1/3 1 -1/3 -1/3 -1/3 1/3 1/3];
sfun = 'sf_quad_Q3';
switch i_eval
case 1
switch i_dof
case 1
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 2
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) - 1);
case 3
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) + 1);
case 4
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 5
vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 6
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1);
case 7
vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 8
vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 9
vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1);
case 10
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1);
case 11
vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1);
case 12
vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 13
vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 14
vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 15
vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
case 16
vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1);
end
case {2,3}
switch i_dof
case 1
dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 2
dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 3
dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 4
dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 5
dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 6
dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 7
dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 8
dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 9
dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256;
case 10
dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256;
dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 11
dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256;
case 12
dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256;
dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 13
dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 14
dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256;
case 15
dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256;
dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
case 16
dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256;
dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256;
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_Q3: second order derivative evaluation not supported.')
otherwise
vBase = 0;
end