Element kernels#

femorph-solver ships a structural element library. Each kernel is independently implemented from the open literature; every non-trivial algorithm (shape function, integration rule, formulation variant) carries a References block in its source module. The pre-processing pages under Element library are the user-facing how-to; this section is the engineering reference — topology, shape functions, integration rules, and cites for every element.

HEX8#

8-node trilinear hexahedron. 24 DOFs / element.

  • Shape functions. Trilinear Lagrange on the reference cube, \(N_i = \tfrac{1}{8}(1+\xi_i\xi)(1+\eta_i\eta)(1+\zeta_i\zeta)\) — Zienkiewicz & Taylor §6; Cook §6.2.

  • Integration. 2×2×2 Gauss-Legendre (product rule). K uses Hughes’ volumetric B-bar (default); Simo–Rifai 9-parameter enhanced-assumed-strain variant available via the _SOLID185_FORMULATION = "enhanced_strain" flag.

  • Mass. Consistent at 2×2×2; row-sum (HRZ) lumping optional.

  • MAPDL alias. SOLID185. KEYOPT(2)=0 / 1 / 3 supported.

Full cites in femorph_solver.elements.solid185.

HEX20#

20-node quadratic serendipity hexahedron. 60 DOFs / element. Dispatches collapsed-corner input to the WEDGE15 / PYR13 kernels automatically.

  • Shape functions. Corner-plus-mid-edge serendipity basis from Ergatoudis, Irons, Zienkiewicz (IJSS 1968); Zienkiewicz & Taylor §8.7.3.

  • Stiffness integration. 2×2×2 uniform reduced by default (Bedrosian 1992) — matches MAPDL KEYOPT(2)=0. Full 3×3×3 rule (27 points) available via _SOLID186_FORMULATION = "full".

  • Mass integration. 14-point Irons rule (Irons, IJNME 1971) — integrates polynomials through degree 5 exactly and matches the MAPDL KEYOPT(2)=0 mass kernel bit-exactly.

  • MAPDL alias. SOLID186 (plus SOLID186W / SOLID186P for collapsed-corner forms).

Full cites in femorph_solver.elements.solid186 and femorph_solver.elements.solid186_degenerate.

TET10#

10-node quadratic tetrahedron. 30 DOFs / element.

  • Shape functions. Lagrange in volume coordinates, Nᵢ = Lᵢ(2Lᵢ − 1) on corners, N_{ij} = 4LᵢLⱼ on mid-edges — Zienkiewicz & Taylor §8.8.2; Cook Table 6.5-1.

  • Integration. 4-point Keast Gauss rule (Keast, CMAME 1986, degree-2 exact, weights 1/24) — used for both K and M.

  • MAPDL alias. SOLID187.

Full cites in femorph_solver.elements.solid187.

WEDGE15 / PYR13#

15-node quadratic wedge and 13-node apex-singular pyramid — degenerate serendipity-hex forms.

  • WEDGE15 shape functions. Serendipity on the triangular base × quadratic through thickness (Bedrosian 1992; ZT §8.8.3).

  • WEDGE15 integration. 9-point (triangle × line) for K; 21-point for consistent mass. Triangle rules from Dunavant 1985.

  • PYR13 shape functions. Apex-singular Peterson / Bedrosian 1992 polynomial basis with a removable 1/(1−ζ) factor. Alternatives cited: Zgainski-Coulomb-Marechal (IEEE Mag. 1996), Wachspress, A Rational Finite Element Basis, 1975.

  • PYR13 integration. 2×2×2 Duffy collapsed-hex rule (Duffy, SIAM J. Numer. Anal. 1982) — matches MAPDL’s SOLID186-pyramid mass kernel bit-exactly on ptr.cdb.

  • MAPDL alias. SOLID186W / SOLID186P (auto-dispatched from the HEX20 entry).

BEAM2#

3-D 2-node Euler–Bernoulli beam. 6 DOFs / node, 12 total.

  • Kinematics. Slender-beam limit of Timoshenko-beam theory (Φ = 12EI/(κAGL²) → 0); shear neglected. Timoshenko-beam form is roadmap.

  • Shape functions. Hermite cubic for transverse displacement + slope; linear for axial and torsion. ZT §2.5.1; Cook §2.4–§2.6.

  • Mass. Consistent from Hermite cubics (Cook Table 16.3-1); row-sum lumped available.

  • MAPDL alias. BEAM188.

Full cites in femorph_solver.elements.beam188.

QUAD4_SHELL#

4-node flat Mindlin–Reissner shell. 6 DOFs / node, 24 total.

  • Kinematics. First-order shear-deformation theory (Mindlin JAM 1951; Reissner JAM 1945) with independent bending rotation.

  • Integration. 2×2 Gauss for membrane + bending; 1×1 selective-reduced Gauss on transverse shear (Malkus & Hughes CMAME 1978) to suppress shear locking on thin shells.

  • Drilling DOF stabilisation. α·G·t·A per-node penalty (Allman 1984; Hughes-Brezzi CMAME 1989) keeps the local 24×24 non-singular when θ_z is free.

  • MAPDL alias. SHELL181.

Full cites in femorph_solver.elements.shell181. MITC4 variant (Bathe-Dvorkin IJNME 1985) is roadmap.

QUAD4_PLANE#

2-D 4-node bilinear quadrilateral, plane stress / plane strain.

  • Shape functions. Bilinear Q4 Lagrange (ZT §6.3.2).

  • Integration. 2×2 Gauss-Legendre.

  • Constitutive. Plane-stress / plane-strain selected by _PLANE_MODE material flag; Cook §3.5–§3.6.

  • MAPDL alias. PLANE182.

TRUSS2#

2-node 3-D axial bar. 3 translational DOFs / node. Analytic stiffness + consistent mass (Cook §2.3 / Table 16.3-1). MAPDL alias: LINK180.

SPRING#

2-node 3-D longitudinal spring. 3 translational DOFs / node. Direction-cosine-rotated 6×6 stiffness block. No mass contribution. MAPDL alias: COMBIN14.

POINT_MASS#

Single-node lumped translational mass. Diagonal 3×3 contribution to global M. MAPDL alias: MASS21 (KEYOPT(3)=2 variant).

Full cites in femorph_solver.elements.mass21.

See also

Element library — per-element KEYOPT tables, material-property slots, and runnable demos.