At what rate does the 3D eigenbuckling model converge under uniform mesh refinement?

What it found

A stylised, preliminary study of one short steel bar (24×8×2 mm, tip shear). Three uniformly refined hexahedral meshes give an observed order of about 2.6 and a Richardson limit of 9618 N, 0.12% below the finest mesh. The limit is still 13.7% above the slender-beam closed form, so the gap is not discretisation error (plausibly shear deformation and clamp warping; untested). One geometry, one mesh triple.

Paper, code and results

• README and method
• Experiments and results
• Code
• Tests

More in Simulation-first design

• What does it cost to replace the w/b ≤ 8 rule of the laser-cut bracket with a buckling-factor guard?Replacing the w/b ≤ 8 slenderness rule of the laser-cut bracket with a lateral-torsional buckling factor ≥ 3 lets the optimiser use the 1.5 mm gauge at w ≈ 18.7 mm, 31.3 g instead of 35.9 g (about 12% lighter), and the guard does not bind at that optimum; the result depends on the assumed factor (no feasible design for factors ≥ 5 at that gauge); stylised, preliminary.
• When does the buckling guard of the laser-cut bracket start to bind as load and arm length change?Sweeping tip load (50–400 N) and arm length (60–120 mm) of the laser-cut bracket: the buckling-factor ≥ 3 guard binds only at 120 mm arms (unguarded optimum has factor 1.9; 69 g vs 59 g), while the old w/b ≤ 8 rule costs 14–25% in mass in five of six cases (stylised, two seeds).
• How sharp is the feasible edge of the 2 mm laser-cut bracket web at 0.1 mm resolution, and is it mesh noise?A 0.1 mm scan of the feasible edge of the gauge-2 bracket web (nine mesh/radius cases): peak stress is not monotone in w, with residual jitter of 0.9–2.4 MPa about a quadratic trend, the same size as the 0.1–1.7 MPa gap to the 100 MPa limit just below each edge, so the edge is a band of about 0.5 mm and the earlier refinement shift is not separated from mesh noise; stylised, preliminary.
• Do wider webs of the 2 mm laser-cut bracket stay feasible on fine meshes when the slenderness rule is relaxed?With the assumed slenderness limit raised from 8 to 12.5, gauge-2 bracket webs of 16–17 mm and wider are feasible at h = 1.5, 1.0 and 0.75 mm; refinement from 1.5 to 1.0 mm moves the lightest feasible width up 0.5 mm (about 1 g, 3%); stylised, preliminary.
• Do the isolated feasible pockets of the 2 mm laser-cut bracket survive mesh refinement?The isolated feasible pockets below w = 15 mm at the 2 mm laser-cut gauge vanish at 2.0 and 1.5 mm element size, and the smallest feasible w moves up (14.4 to 15.6 mm at r = 14 mm); stylised, preliminary.
• Is the narrow 2 mm laser-cut bracket region a mesh artefact?The narrow 2 mm laser-cut bracket region is a mesh effect: all 39 designs feasible at h = 2.5 mm fail the stress limit at h ≤ 1.5 mm (peak 1.09–1.27× higher, still rising at 0.75 mm); stylised, negative for the search mesh.

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.