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sf_quad_Q4.m File Reference

Description

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
-----------------------------------------------------------------------------------
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
See also
sf_quad_Q1

Code listing

 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