HEX20 — 20-node serendipity hexahedron#

The quadratic-hex serendipity kernel. Eight corner nodes plus twelve mid-edge nodes — 20 total, 60 DOFs per element. The serendipity basis omits the body / face / mid-element nodes of the full Lagrange family while preserving exact reproduction of quadratic polynomials (Ergatoudis, Irons & Zienkiewicz 1968).

Dispatches collapsed-corner input to WEDGE15 / PYR13 — degenerate-corner serendipity hex automatically.

Spec: ELEMENTS.HEX20

Shape functions#

Two families on the reference cube \(\hat\Omega = [-1, +1]^{3}\):

Corner nodes \((\xi_i, \eta_i, \zeta_i) \in \{-1, +1\}^{3}\):

\[N^{c}_{i}(\xi, \eta, \zeta) = \tfrac{1}{8} (1 + \xi_i \xi)(1 + \eta_i \eta)(1 + \zeta_i \zeta) (\xi_i \xi + \eta_i \eta + \zeta_i \zeta - 2).\]

Mid-edge nodes with one of \(\xi_j, \eta_j, \zeta_j\) zero (the axis along which the edge runs). For \(\xi_j = 0\) mid-edges:

\[N^{m}_{j}(\xi, \eta, \zeta) = \tfrac{1}{4} (1 - \xi^{2})(1 + \eta_j \eta)(1 + \zeta_j \zeta),\]

with analogous expressions for \(\eta_j = 0\) and \(\zeta_j = 0\) mid-edges (the missing-variable axis carries the \((1 - \cdot^{2})\) factor).

The basis satisfies Kronecker-delta interpolation at all 20 nodes and partition-of-unity on \(\hat\Omega\). See HEX20 shape-function slices on the bottom face for slice plots of the eight functions that don’t vanish on the \(\zeta = -1\) face.

Integration#

Stiffness: 2 × 2 × 2 reduced Gauss-Legendre (8 points) by default — Bedrosian (1992) and the standard practitioner choice for serendipity hex. Bending-mode response stays exact; spurious hourglass modes are absent because the 8 points still couple all 60 stiffness rows.
Mass: 14-point Irons rule (Irons 1971). Integrates polynomials of degree ≤ 5 exactly on the cube — a non-tensor- product cubature constructed by hand for the cube’s symmetry group with the minimum point count for this degree. Preferred over the 27-point \(3 \times 3 \times 3\) rule because it preserves the consistent-mass sparsity pattern without overshooting the integrand polynomial degree.

The full 27-point \(3 \times 3 \times 3\) rule is also available via ELEMENTS.HEX20(integration="full"); useful only for cross-checking the reduced rule on a sensitive bending mode.

Verification cross-references#

HEX20 — uniaxial tension on a 20-node hex — single-hex uniaxial via mid-edge consistent loads.
HEX20 reference geometry — corner + mid-edge nodes, Gauss points — corner + mid-edge node layout, stiffness and mass Gauss points.

Implementation: femorph_solver.elements.hex20; collapsed-corner forms in femorph_solver.elements._wedge15_pyr13.

References#

Ergatoudis, J. G., Irons, B. M. and Zienkiewicz, O. C. (1968) “Curved isoparametric quadrilateral elements for finite element analysis,” International Journal of Solids and Structures 4 (1), 31–42.
Hughes, T. J. R. (2000) The Finite Element Method — Linear Static and Dynamic Finite Element Analysis, Dover, §3.7 (serendipity family).
Zienkiewicz, O. C. and Taylor, R. L. (2013) The Finite Element Method: Its Basis and Fundamentals, 7th ed., Butterworth-Heinemann, §8.7.3.
Irons, B. M. (1971) “Quadrature rules for brick based finite elements,” International Journal for Numerical Methods in Engineering 3 (2), 293–294 (14-point degree-5 rule for the consistent mass integral).
Bedrosian, G. (1992) “Shape functions and integration formulas for three-dimensional finite element analysis,” International Journal for Numerical Methods in Engineering 35 (1), 95–108 (reduced 2 × 2 × 2 stiffness).
Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, §6.5 (quadratic hex).