NAFEMS LE1: Elliptic Membrane Plane-Stress Benchmark
The problem: an elliptic plate pulled outward
Overview
NAFEMS LE1 is a famous benchmark. If I match the reference, have I verified my code?
One match isn't enough. LE1 tests how well elements approximate curved geometry and capture the stress concentrated at the tip of the inner ellipse. Verification means showing the value settles toward the 92.7 MPa reference as elements are refined. Let's build our own finite element code with linear and quadratic elements and watch how the value moves with the mesh.
Problem definition
Plane stress, thickness 0.1 m, Young's modulus 210 GPa, Poisson's ratio 0.3. Quarter model with x fixed on the x = 0 edge and y fixed on the y = 0 edge; 10 MPa outward pressure on the outer edge. The target is σyy at D (x = 2, y = 0) on the inner ellipse; the reference is 92.7 MPa.
Our own finite element code
A mapped mesh divides the inner and outer ellipses at the same angular parameter and joins them with straight lines. Linear 4-node Q4 (2×2 Gauss) and quadratic 8-node Q8 (3×3 Gauss) elements were used, and the stress at D was taken directly from shape-function derivatives of the element touching D.
Why LE1 was chosen
NAFEMS's linear elastic benchmarks were compiled in the late 1980s, and LE1 is the first. Stress concentration around an elliptic hole is sensitive to how accurately elements represent curved geometry, exposing weaknesses of straight-sided or distorted elements. Yet the shape — bounded by two ellipses — is simple enough for any software to reproduce identically. Being simple yet revealing of element behaviour is why it remains in use.
Worked example: σyy at D versus mesh
| Mesh (hoop × radial) | Q4 (linear) | Error | Q8 (quadratic) | Error |
|---|---|---|---|---|
| 6 × 3 | 86.09 | −7.13% | 90.21 | −2.69% |
| 12 × 6 | 92.82 | +0.12% | 93.00 | +0.32% |
| 24 × 12 | 94.63 | +2.08% | 93.05 | +0.37% |
| 48 × 24 | 94.36 | +1.79% | 92.81 | +0.12% |
| 96 × 48 | 93.73 | +1.11% | 92.71 | +0.01% |
| 192 × 96 | 93.25 | +0.60% | — | — |
Stresses in MPa; errors relative to the 92.7 MPa reference. DOF: about 37,000 for Q4 at 192×96 and 28,000 for Q8 at 96×48.
Q4 at 12×6 matches almost exactly, yet refining makes it drift away.
That's coarse-mesh 'coincidental agreement'. Q4 approximates the curved ellipse with straight edges and represents only simple strain fields inside each element. On the coarse mesh, the geometry error and the stress error happened to have opposite signs and cancel. Refining shrinks the geometry error first, exposing the stress error and pushing the value up to 94.6 MPa; from there it slowly returns toward the reference. Q8's curved edges represent the geometry accurately, staying within 0.4% from 12×6 onward and reaching 0.01% at 96×48. With benchmarks, what matters is not whether one mesh matched, but how the value moves as the mesh changes.
How to run the benchmark
- Solve on both the coarse and fine meshes NAFEMS specifies.
- Further double the mesh repeatedly and check the value settles monotonically.
- Record how stress was obtained (nodal averaging, Gauss-point extrapolation, direct corner evaluation).
- Use quadratic elements with curved edges for curved geometry.
- Tabulate results with mesh, element type and DOF.
“Passed coarse, failed fine”
In one company's software verification, LE1 was solved only on the specified coarse mesh; it matched within 1% and was passed. Later another engineer re-ran it on a fine mesh and got 2% off. A debate followed over which was right; checking many mesh levels showed the coarse match was a coincidence of cancelling errors. Since then, verification always shows the value's trend over at least three mesh levels.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Judging by one mesh | Coincidence missed | Use three or more meshes |
| Pressure reversed | Sign of σyy flipped | Check outward direction |
| Inverted element orientation | Stiffness sign errors | Check Jacobian sign |
| Unstated stress recovery | Results not comparable | State the method |
| Straight elements on curves | Slow convergence | Use quadratic elements |
I'd like to learn related topics.
Try stress concentration with the stress concentration tool. Related pages include LE1 verification results, NAFEMS LE10, NAFEMS T3, three-grid GCI and p-refinement.
Related Topics
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