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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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SF_QUAD_Q4 Biquartic conforming shape function for quadrilaterals (Q4).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q4( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming biquartic Q4 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: 1-25 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 12 0 9];
xLDof = [-1 1 1 -1 -1/2 1 1/2 -1 0 1 0 -1 1/2 1 -1/2 -1 -1/2 1/2 1/2 -1/2 0 1/2 0 -1/2 0;
-1 -1 1 1 -1 -1/2 1 1/2 -1 0 1 0 -1 1/2 1 -1/2 -1/2 -1/2 1/2 1/2 -1/2 0 1/2 0 0];
sfun = 'sf_quad_Q4';
switch i_eval
case 1
switch i_dof
case 1
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/36;
case 2
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/36;
case 3
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/36;
case 4
vBase = (xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/36;
case 5
vBase = -(2*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/9;
case 6
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 7
vBase = -(2*xi(1)*xi(2)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 8
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 9
vBase = -(xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1))/6;
case 10
vBase = -(xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1))/6;
case 11
vBase = -(xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/6;
case 12
vBase = -(xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1))/6;
case 13
vBase = -(2*xi(1)*xi(2)*(- 4*xi(2)^3 + 4*xi(2)^2 + xi(2) - 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 14
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 - 4*xi(1)^2 + xi(1) + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 15
vBase = -(2*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 4*xi(2)^3 - 4*xi(2)^2 + xi(2) + 1))/9;
case 16
vBase = -(2*xi(1)*xi(2)*(- 4*xi(1)^3 + 4*xi(1)^2 + xi(1) - 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 17
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/9;
case 18
vBase = (16*xi(1)*xi(2)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/9;
case 19
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 20
vBase = (16*xi(1)*xi(2)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/9;
case 21
vBase = (4*xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 2*xi(2)^3 + xi(2)^2 + 2*xi(2) - 1))/3;
case 22
vBase = (4*xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 2*xi(1)^3 - xi(1)^2 + 2*xi(1) + 1))/3;
case 23
vBase = (4*xi(2)*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 2*xi(2)^3 - xi(2)^2 + 2*xi(2) + 1))/3;
case 24
vBase = (4*xi(1)*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 2*xi(1)^3 + xi(1)^2 + 2*xi(1) - 1))/3;
case 25
vBase = (4*xi(1)^4 - 5*xi(1)^2 + 1)*(4*xi(2)^4 - 5*xi(2)^2 + 1);
end
case {2,3}
switch i_dof
case 1
dNdxi1 = -(xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/36;
dNdxi2 = -(xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/36;
case 2
dNdxi1 = (xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(4*xi(1)^2 + 2*xi(1) - 1))/36;
dNdxi2 = -(xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/36;
case 3
dNdxi1 = (xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/36;
dNdxi2 = (xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/36;
case 4
dNdxi1 = -(xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/36;
dNdxi2 = (xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(4*xi(2)^2 + 2*xi(2) - 1))/36;
case 5
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/9;
case 6
dNdxi1 = -(2*xi(2)*(4*xi(1) + 1)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 7
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(2*xi(1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/9;
case 8
dNdxi1 = (2*xi(2)*(4*xi(1) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) - 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 9
dNdxi1 = (xi(1)*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) - 1))/3;
dNdxi2 = -((2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/6;
case 10
dNdxi1 = ((4*xi(1) + 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/6;
dNdxi2 = (xi(1)*xi(2)*(2*xi(1) - 1)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) + 1))/3;
case 11
dNdxi1 = (xi(1)*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) + 1))/3;
dNdxi2 = ((2*xi(1) - 1)*(2*xi(1) + 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/6;
case 12
dNdxi1 = -((4*xi(1) - 1)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/6;
dNdxi2 = (xi(1)*xi(2)*(2*xi(1) - 1)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) - 1))/3;
case 13
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) - 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) + 1)*(4*xi(2) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 4*xi(2)^2 + 2*xi(2) + 1))/9;
case 14
dNdxi1 = -(2*xi(2)*(4*xi(1) + 1)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(4*xi(1)^2 + 2*xi(1) - 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 15
dNdxi1 = (2*xi(2)*(2*xi(2) - 1)*(2*xi(2) + 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(2*xi(1)*(2*xi(1) - 1)*(4*xi(2) + 1)*(xi(1) - 1)*(xi(1) + 1)*(4*xi(2)^2 + 2*xi(2) - 1))/9;
case 16
dNdxi1 = (2*xi(2)*(4*xi(1) - 1)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 4*xi(1)^2 + 2*xi(1) + 1))/9;
dNdxi2 = (2*xi(1)*(2*xi(1) - 1)*(2*xi(1) + 1)*(xi(1) - 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 17
dNdxi1 = -(16*xi(2)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 18
dNdxi1 = -(16*xi(2)*(2*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/9;
case 19
dNdxi1 = -(16*xi(2)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 20
dNdxi1 = -(16*xi(2)*(2*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/9;
dNdxi2 = -(16*xi(1)*(2*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/9;
case 21
dNdxi1 = -(8*xi(1)*xi(2)*(2*xi(2) - 1)*(8*xi(1)^2 - 5)*(xi(2) - 1)*(xi(2) + 1))/3;
dNdxi2 = (4*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 8*xi(2)^3 + 3*xi(2)^2 + 4*xi(2) - 1))/3;
case 22
dNdxi1 = (4*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 8*xi(1)^3 - 3*xi(1)^2 + 4*xi(1) + 1))/3;
dNdxi2 = -(8*xi(1)*xi(2)*(2*xi(1) + 1)*(8*xi(2)^2 - 5)*(xi(1) - 1)*(xi(1) + 1))/3;
case 23
dNdxi1 = -(8*xi(1)*xi(2)*(2*xi(2) + 1)*(8*xi(1)^2 - 5)*(xi(2) - 1)*(xi(2) + 1))/3;
dNdxi2 = (4*(4*xi(1)^4 - 5*xi(1)^2 + 1)*(- 8*xi(2)^3 - 3*xi(2)^2 + 4*xi(2) + 1))/3;
case 24
dNdxi1 = (4*(4*xi(2)^4 - 5*xi(2)^2 + 1)*(- 8*xi(1)^3 + 3*xi(1)^2 + 4*xi(1) - 1))/3;
dNdxi2 = -(8*xi(1)*xi(2)*(2*xi(1) - 1)*(8*xi(2)^2 - 5)*(xi(1) - 1)*(xi(1) + 1))/3;
case 25
dNdxi1 = 2*xi(1)*(8*xi(1)^2 - 5)*(4*xi(2)^4 - 5*xi(2)^2 + 1);
dNdxi2 = 2*xi(2)*(8*xi(2)^2 - 5)*(4*xi(1)^4 - 5*xi(1)^2 + 1);
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_Q4: second order derivative evaluation not supported.')
otherwise
vBase = 0;
end