CE/ME 532 · Section 3.5 · Switch between the Q8 serendipity and Q9 Lagrangian element. Drag the red corner nodes, the orange mid-side nodes, or the purple centre node (Q9) to bend the edges and interior. Drag the green probe to sample the shape functions and |J|.
The blue mesh on the right is the image of ξ=const and η=const grid lines. With Q8/Q9 the image lines are parabolic, so each physical edge can curve. Red shading marks the region where |J| < 0. Mid-side nodes push off their chord midpoint more than the ¼–rule allows ⇒ the Jacobian can fold.
Both elements share the same 8 boundary nodes: 4 corners and 4 mid-side nodes, one per edge of the parent square. The Q9 Lagrangian adds a 9th node at the centre (ξ,η) = (0,0).
At the current (ξ,η):
Use the same Ni for geometry and displacement:
With mid-side nodes each physical edge is the quadratic Lagrange interpolation of its three nodes — a parabola (or a straight line when the mid-side node sits on the chord midpoint).
Same definition as the 4-node case, but the derivatives Ni,ξ, Ni,η are now quadratic:
Along one edge (say η=−1) the three active shape functions are the 1-D quadratic Lagrange functions of ξ. The edge in physical space is therefore a parabola through the three edge nodes.
The 1-D quadratic mapping stays monotone iff the mid-side node is within the middle half of the edge (|\Delta| < L/4 from the chord midpoint). Violate this and the edge folds on itself — the element's Jacobian goes negative near that edge.
Q8 has 16 DOFs, Q9 has 18 DOFs. The extra 2 DOFs buy the full biquadratic polynomial space and, with 3×3 Gauss, exact integration on affine meshes.
CPS8, ANSYS PLANE183).
Reduced integration
(2×2) is common to
suppress shear locking, but it introduces one hourglass mode.