FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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SF_QUAD_Q3 Bicubic conforming shape function for quadrilaterals (Q3).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_QUAD_Q3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming bicubic Q3 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: 1-16 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 = [4 8 0 4]; xLDof = [-1 1 1 -1 -1/3 1 1/3 -1 1/3 1 -1/3 -1 -1/3 1/3 1/3 -1/3; -1 -1 1 1 -1 -1/3 1 1/3 -1 1/3 1 -1/3 -1/3 -1/3 1/3 1/3]; sfun = 'sf_quad_Q3'; switch i_eval case 1 switch i_dof case 1 vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1); case 2 vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) - 1); case 3 vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) + 1)*(xi(2) + 1); case 4 vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1); case 5 vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1); case 6 vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1); case 7 vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1); case 8 vBase = (3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); case 9 vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) - 1); case 10 vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) + 1)*(xi(2) - 1); case 11 vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((3*xi(2))/16 + 1/16)*(xi(1) - 1)*(xi(2) + 1); case 12 vBase = -(3*xi(1) - 1)*((3*xi(1))/16 + 1/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); case 13 vBase = (3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); case 14 vBase = -(3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) - 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); case 15 vBase = (3*xi(1) + 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); case 16 vBase = -(3*xi(1) - 1)*((9*xi(1))/16 + 9/16)*(3*xi(2) + 1)*((9*xi(2))/16 + 9/16)*(xi(1) - 1)*(xi(2) - 1); end case {2,3} switch i_dof case 1 dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256; dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256; case 2 dNdxi1 = -((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256; dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256; case 3 dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256; dNdxi2 = ((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256; case 4 dNdxi1 = ((3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256; dNdxi2 = -((3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256; case 5 dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256; dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256; case 6 dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256; dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256; case 7 dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256; dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256; case 8 dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256; dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256; case 9 dNdxi1 = (9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) - 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256; dNdxi2 = -(9*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 27*xi(2)^2 + 18*xi(2) + 1))/256; case 10 dNdxi1 = -(9*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(27*xi(1)^2 + 18*xi(1) - 1))/256; dNdxi2 = -(9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256; case 11 dNdxi1 = -(9*(3*xi(2) - 1)*(3*xi(2) + 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256; dNdxi2 = (9*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(27*xi(2)^2 + 18*xi(2) - 1))/256; case 12 dNdxi1 = (9*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 27*xi(1)^2 + 18*xi(1) + 1))/256; dNdxi2 = (9*(3*xi(1) - 1)*(3*xi(1) + 1)*(xi(1) - 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256; case 13 dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256; dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256; case 14 dNdxi1 = -(81*(3*xi(2) - 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256; dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(- 9*xi(2)^2 + 2*xi(2) + 3))/256; case 15 dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(9*xi(1)^2 + 2*xi(1) - 3))/256; dNdxi2 = (81*(3*xi(1) + 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256; case 16 dNdxi1 = (81*(3*xi(2) + 1)*(xi(2) - 1)*(xi(2) + 1)*(- 9*xi(1)^2 + 2*xi(1) + 3))/256; dNdxi2 = -(81*(3*xi(1) - 1)*(xi(1) - 1)*(xi(1) + 1)*(9*xi(2)^2 + 2*xi(2) - 3))/256; 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_Q3: second order derivative evaluation not supported.') otherwise vBase = 0; end