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sf_hex_Q1.m File Reference

Description

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
-----------------------------------------------------------------------------------
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
-----------------------------------------------------------------------------------
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
sflag1, sf_hex_Q1nc

Code listing

 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