FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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SF_HEX_Q1NC Trilinear nonconforming shape function for hexahedrons (Q1~).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_HEX_Q1NC( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates nonconforming trilinear Q1~ shape functions on hexahedrons with values defined in the face centers. 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
nLDof = [0 0 6 0]; xLDof = [ 0 0 1 0 -1 0; ... 0 -1 0 1 0 0; ... -1 0 0 0 0 1]; sfun = 'sf_hex_Q1nc'; switch i_eval % Evaluation type flag. case 1 % Evaluation of function values. switch i_dof % Basis function to evaluate. case 1 vBase = (xi(1)^2-xi(2)^2)/6-(xi(1)^2-xi(3)^2)/3-xi(3)/2+1/6; case 2 vBase =-(xi(1)^2-xi(2)^2)/3+(xi(1)^2-xi(3)^2)/6-xi(2)/2+1/6; case 3 vBase = (xi(1)^2-xi(2)^2)/6+(xi(1)^2-xi(3)^2)/6+xi(1)/2+1/6; case 4 vBase =-(xi(1)^2-xi(2)^2)/3+(xi(1)^2-xi(3)^2)/6+xi(2)/2+1/6; case 5 vBase = (xi(1)^2-xi(2)^2)/6+(xi(1)^2-xi(3)^2)/6-xi(1)/2+1/6; case 6 vBase = (xi(1)^2-xi(2)^2)/6-(xi(1)^2-xi(3)^2)/3+xi(3)/2+1/6; end case {2,3,4} % Evaluation of first order derivatives. switch i_dof % Basis function to evaluate. case 1 dNdxi1 = -xi(1)/3; dNdxi2 = -xi(2)/3; dNdxi3 = 2/3*xi(3)-1/2; case 2 dNdxi1 = -xi(1)/3; dNdxi2 = 2/3*xi(2)-1/2; dNdxi3 = -xi(3)/3; case 3 dNdxi1 = 2/3*xi(1)+1/2; dNdxi2 = -xi(2)/3; dNdxi3 = -xi(3)/3; case 4 dNdxi1 = -xi(1)/3; dNdxi2 = 2/3*xi(2)+1/2; dNdxi3 = -xi(3)/3; case 5 dNdxi1 = 2/3*xi(1)-1/2; dNdxi2 = -xi(2)/3; dNdxi3 = -xi(3)/3; case 6 dNdxi1 = -xi(1)/3; dNdxi2 = -xi(2)/3; dNdxi3 = 2/3*xi(3)+1/2; 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_Q1nc: 2nd derivatives for non-rectangular cells shapes not supported.') end switch i_dof case 1 d2Ndxi1dxi1 = -1/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = -1/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = 2/3; case 2 d2Ndxi1dxi1 = -1/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = 2/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = -1/3; case 3 d2Ndxi1dxi1 = 2/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = -1/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = -1/3; case 4 d2Ndxi1dxi1 = -1/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = 2/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = -1/3; case 5 d2Ndxi1dxi1 = 2/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = -1/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = -1/3; case 6 d2Ndxi1dxi1 = -1/3; d2Ndxi2dxi1 = 0; d2Ndxi3dxi1 = 0; d2Ndxi1dxi2 = 0; d2Ndxi2dxi2 = -1/3; d2Ndxi3dxi2 = 0; d2Ndxi1dxi3 = 0; d2Ndxi2dxi3 = 0; d2Ndxi3dxi3 = 2/3; 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