WEDGE15 / PYR13 — degenerate-corner serendipity hex#

The 15-node quadratic wedge and the 13-node apex-singular pyramid are collapsed-corner forms of HEX20 — 20-node serendipity hexahedron — HEX20 with one or two pairs of corner nodes coincident. Both ship in femorph_solver.elements._wedge15_pyr13 and are auto-dispatched from the HEX20 entry when the input deck supplies a degenerate connectivity.

  • MAPDL aliases: SOLID186W (wedge), SOLID186P (pyramid)

  • Specs: ELEMENTS.WEDGE15, ELEMENTS.PYR13

WEDGE15#

15 nodes — 6 corners + 9 mid-edges. 45 DOFs per element. Reference shape is a triangle \(\times\) line:

\[\hat\Omega_{\mathrm{wedge}} = \{ (L_0, L_1, L_2, \zeta) : L_0 + L_1 + L_2 = 1, \zeta \in [-1, +1] \}.\]

Shape functions. Quadratic serendipity on the triangle (in volume coordinates \(L_i\)) tensored with a quadratic shape function in \(\zeta\). Bedrosian (1992) gives the explicit basis; ZT §8.8.3 walks the construction.

Integration.

  • Stiffness: 9-point composite — 3-point Dunavant triangle (degree-2 exact) \(\times\) 3-point Gauss-Legendre line.

  • Mass: 21-point — 7-point Dunavant triangle (degree-5 exact) \(\times\) 3-point Gauss-Legendre line. Higher point count tracks the higher polynomial degree of \(\mathbf{N}^{\!\top}\, \mathbf{N}\) vs. \(\mathbf{B}^{\!\top}\, \mathbf{C}\, \mathbf{B}\).

PYR13#

13 nodes — 5 corners (4 base + 1 apex) + 8 mid-edges. 39 DOFs per element. The pyramid is the most awkward of the standard solid shapes because the apex collapse makes the shape functions rational (a removable \((1 - \zeta)^{-1}\) factor) — Bedrosian’s 1992 explicit basis carries the rational form directly.

Shape functions. Apex-singular polynomial / rational basis (Bedrosian 1992). Alternatives: Zgainski-Coulomb-Marechal (IEEE Mag. 1996) and Wachspress (A Rational Finite Element Basis, Academic Press, 1975).

Integration. 2 × 2 × 2 Duffy collapsed-hex rule (Duffy 1982). The Duffy transformation maps the pyramid to a collapsed cube where the rational integrand becomes finite at every Gauss point. 8 points, exact for the degree-3 polynomial limit of the rational basis. Matches MAPDL’s SOLID186-pyramid mass kernel bit-exactly on canonical CDB inputs.

References#

  • 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 (WEDGE15 + PYR13 shape functions).

  • Zienkiewicz, O. C. and Taylor, R. L. (2013) The Finite Element Method: Its Basis and Fundamentals, 7th ed., §8.8.3 (wedge serendipity).

  • Dunavant, D. A. (1985) “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” International Journal for Numerical Methods in Engineering 21 (6), 1129–1148.

  • Duffy, M. G. (1982) “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM Journal on Numerical Analysis 19 (6), 1260–1262.

  • Zgainski, F.-X., Coulomb, J.-L. and Marechal, Y. (1996) “A new family of finite elements: the pyramidal elements,” IEEE Transactions on Magnetics 32 (3), 1393–1396 (alternative pyramid basis).

  • Wachspress, E. L. (1975) A Rational Finite Element Basis, Academic Press (rational-basis foundation).

Implementation: femorph_solver.elements._wedge15_pyr13.