FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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SF_SIMP_P1BUB Linear Lagrange shape function for simplices with bubble (P1+).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_SIMP_P1BUB( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming linear P1 Lagrange shape functions on simplices an additional with bubble function. XI Barycentric coordinates.
Input Value/[Size] Description ----------------------------------------------------------------------------------- i_eval scalar: 1 Evaluate function values >1 Evaluate values of derivatives n_sdim scalar: 1-3 Number of space dimensions n_vert scalar: 2-4 Number of vertices per cell i_dof scalar: 1-n_ldof Local basis function to evaluate xi [n_sdim+1] Local coordinates of evaluation point aInvJac [n,n_sdim+1*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
sfun = 'sf_simp_P1bub'; [~,nLDof,xLDof] = sf_simp_P1( 0, n_sdim, n_vert ); nLDof(4) = 1; switch n_sdim case 1 xLDof = [ xLDof [1;1]/2 ]; case 2 xLDof = [ xLDof [1;1;1]/3 ]; case 3 xLDof = [ xLDof [1;1;1;1]/4 ]; end % Evaluation type flag. if( i_eval==1 ) % Evaluation of function values. if( n_sdim==1 ) vBase = sf_simp_P2( i_eval, n_sdim, n_vert, i_dof, xi, aInvJac, vBase ); elseif( n_sdim==2 ) switch( i_dof ) case 1 vBase = (9*xi(2)*xi(3) - 1)*(xi(2) + xi(3) - 1); case 2 vBase = xi(2) + 9*xi(2)*xi(3)*(xi(2) + xi(3) - 1); case 3 vBase = xi(3) + 9*xi(2)*xi(3)*(xi(2) + xi(3) - 1); case 4 vBase = -27*xi(2)*xi(3)*(xi(2) + xi(3) - 1); end else % 3D. switch( i_dof ) case 1 vBase = (64*xi(2)*xi(3)*xi(4) - 1)*(xi(2) + xi(3) + xi(4) - 1); case 2 vBase = xi(2)*(64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 1); case 3 vBase = xi(3)*(64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 1); case 4 vBase = xi(4)*(64*xi(2)*xi(3)*(xi(2) + xi(3) + xi(4) - 1) + 1); case 5 vBase = -256*xi(2)*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1); end end elseif( i_eval>=2 && i_eval<=n_sdim+1 ) % Evaluation of first derivatives. if( n_sdim==1 ) vBase = sf_simp_P2( i_eval, n_sdim, n_vert, i_dof, xi, aInvJac, vBase ); elseif( n_sdim==2 ) switch i_dof % Basis function to evaluate. case 1 dNdxi1 = 0; dNdxi2 = 18*xi(2)*xi(3) - 9*xi(3) + 9*xi(3)^2 - 1; dNdxi3 = 18*xi(2)*xi(3) - 9*xi(2) + 9*xi(2)^2 - 1; case 2 dNdxi1 = 0; dNdxi2 = 18*xi(2)*xi(3) - 9*xi(3) + 9*xi(3)^2 + 1; dNdxi3 = 9*xi(2)*(xi(2) + 2*xi(3) - 1); case 3 dNdxi1 = 0; dNdxi2 = 9*xi(3)*(2*xi(2) + xi(3) - 1); dNdxi3 = 18*xi(2)*xi(3) - 9*xi(2) + 9*xi(2)^2 + 1; case 4 dNdxi1 = 0; dNdxi2 = -27*xi(3)*(2*xi(2) + xi(3) - 1); dNdxi3 = -27*xi(2)*(xi(2) + 2*xi(3) - 1); end if( i_eval==2 ) vBase = aInvJac(:,1)*dNdxi1 + aInvJac(:,2)*dNdxi2 + aInvJac(:,3)*dNdxi3; else vBase = aInvJac(:,4)*dNdxi1 + aInvJac(:,5)*dNdxi2 + aInvJac(:,6)*dNdxi3; end else % 3D. switch i_dof % Basis function to evaluate. case 1 dNdxi1 = 0; dNdxi2 = 64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1; dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1; dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) - 1; case 2 dNdxi1 = 0; dNdxi2 = 64*xi(3)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1; dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); case 3 dNdxi1 = 0; dNdxi2 = 64*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1); dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1; dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + xi(3) + xi(4) - 1) + 64*xi(2)*xi(3)*xi(4) + 1; case 4 dNdxi1 = 0; dNdxi2 = 64*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1); dNdxi3 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); dNdxi4 = 64*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); case 5 dNdxi1 = 0; dNdxi2 = -256*xi(3)*xi(4)*(2*xi(2) + xi(3) + xi(4) - 1); dNdxi3 = -256*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); dNdxi4 = -256*xi(2)*xi(4)*(xi(2) + 2*xi(3) + xi(4) - 1); end if( i_eval==2 ) vBase = aInvJac(:,1)*dNdxi1 + aInvJac(:,2)*dNdxi2 + aInvJac(:,3)*dNdxi3 + aInvJac(:,4)*dNdxi4; elseif( i_eval==3 ) vBase = aInvJac(:,5)*dNdxi1 + aInvJac(:,6)*dNdxi2 + aInvJac(:,7)*dNdxi3 + aInvJac(:,8)*dNdxi4; else vBase = aInvJac(:,9)*dNdxi1 + aInvJac(:,10)*dNdxi2 + aInvJac(:,11)*dNdxi3 + aInvJac(:,12)*dNdxi4; end end elseif( any(i_eval==[22 23 24 32 33 34 42 43 44]) ) % Evaluation of second derivatives. error('sf_simp_P1bub: second order derivative evaluation not supported.') else vBase = 0; end