FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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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
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