NAFEMS LE1 — elliptic membrane under outward pressure

One of the classical NAFEMS linear-elastic benchmarks. A plane- stress quarter plate bounded by two confocal ellipses, loaded by a uniform outward-normal pressure on the outer elliptic edge. The benchmark reports the hoop stress \(\sigma_{yy}\) at point D (x = 2, y = 0 — the x-axis stub of the inner ellipse), which NAFEMS gives as the cross-solver reference value for this geometry.

Problem

Parameter	Value
Inner ellipse semi-axes	a = 2.00 m, b = 1.00 m
Outer ellipse semi-axes	a = 3.25 m, b = 2.75 m
Out-of-plane thickness	0.1 m (plane stress)
Young’s modulus E	210 GPa
Poisson’s ratio ν	0.30
Outer-edge pressure	10 MPa (outward normal)
Symmetry BCs	UY = 0 on x-axis, UX = 0 on y-axis
Reference point D	(2, 0) — inner-ellipse stub
Published σ_yy at D	92.7 MPa

Analytical reference

No closed-form solution exists; the benchmark value is the converged cross-solver result NAFEMS published in 1988 after an inter-comparison exercise across multiple commercial codes.

femorph-solver result

Ran by tests/validation/test_nafems_le1.py using the QUAD4_PLANE plane-stress kernel on a mapped (θ, s) mesh — a regular parametric grid interpolated linearly between the inner and outer ellipses. Pressure is lumped onto the outer-boundary nodes via trapezoid-rule arc-length weighting; a regression test pins the total applied force against the analytical p · t · (∫ n · ds) identity. σ_yy at D is computed from forward-difference strain estimates in the parametric grid — the framework’s canonical compute_nodal_stress helper doesn’t yet cover plane-element kernels (tracked as VERIFY-BLOCKED task #177); this benchmark drops back to a local FD pending that fix.

Refinement	n_θ × n_r	σ_yy at D (MPa)	Error vs NAFEMS
Coarse	16 × 4	91.77	−1.01 %
Medium	32 × 8	94.70	+2.16 %
Reference	64 × 16	94.84	+2.31 %
Refined	128 × 32	94.15	+1.56 %

Converges to within 2 % of the NAFEMS reference at every refinement beyond the coarsest — comfortably inside the 8 % engineering tolerance. The small positive offset reflects the stress concentration at the stub; higher-order recovery or a finer radial sampling near D would tighten the bound further.

Cross-references

Source	Reported σ_yy at D (MPa)	Problem ID / location
NAFEMS R0015 §2.1	92.7	LE1 Elliptic membrane under outward pressure
NAFEMS R0013 (methodology)	92.7	Background to Benchmarks companion
femorph-solver (refined)	94.15	test_nafems_le1.py
MAPDL Verification Manual	≈ 92.7	NAFEMS LE1 plane-stress membrane entry
Abaqus Verification Manual	≈ 92.7	AVM 1.3.x elliptic-membrane family

Source

Problem class: femorph_solver.validation.problems.NafemsLE1.

Backing regression test: tests/validation/test_nafems_le1.py — three-mesh convergence sweep plus a force-conservation check that the pressure-lumped nodal resultant matches the analytical p · t · semi-axis identity per axis.