Does a patch-averaged stress give a mesh-converged stress concentration for a bolt hole?

What it found

A strip in tension with a central circular hole is solved by finite elements on meshes of 48 to 768 points around the hole, and the pointwise peak of σ_xx is compared with its average over a disc of 0.1 to 0.5 hole radii, for d/W from 0.05 to 0.4, and with Kirsch’s analytic field. The peak is already nearly settled for a smooth hole (0.3% change over the last refinement; the patch measures 0.0–0.1%), and at d/W = 0.05 the patch values match Kirsch’s disc averages to within 0.4%. The patch reads 65–91% of the peak, so it needs its own limit. One coarse mesh (d/W = 0.4, 96 points) is an unexplained outlier for every measure. Stylised; linear plane stress, no allowable calibrated.

Paper, code and results

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

More in Simulation-first design

• How far below the exact stress concentration does a finite-element peak stress sit, and how fast does it close?Checks the finite-element peak stress at a circular hole against Kirsch's exact 3× concentration: the element-centroid peak sits 27% (coarse) to 1.1% (fine) below it and closes only at first order, while hole-edge vertex values are within 1% (stylised).
• What causes the peak-stress price scatter of laser-cutting rules in the bracket search?The peak-stress price scatter of laser-cutting rules is a missed search at the narrow 2 mm gauge (1 seed in 20), not a cost of the rules; stylised, mesh effects untested.
• Do search restarts remove the missed 2 mm sub-search in the laser-cut bracket problem?A second DE restart or a doubled population removes the missed 2 mm sub-search (0 misses in 40 runs against 1 in 20 at baseline) and shifts results to the lighter of two mass groups at about twice the cost; stylised, 2.5 mm search mesh only.
• Are the two mass groups of the 2 mm laser-cut bracket search two separate basins?The two gauge-2 mass groups differ in web width w (about 14.3–14.9 against 15.2–15.6 mm), not fillet radius, and the lighter group matches ragged isolated feasible pockets on the 2.5 mm mesh rather than a second broad basin; stylised, preliminary.
• Do the isolated feasible pockets of the 2 mm laser-cut bracket survive mesh refinement?The isolated feasible pockets of the gauge-2 bracket scan (web width below 15.4 mm) vanish when the mesh is refined from 2.5 to 1.5 mm and the smallest feasible width rises by about 1 mm; stylised, preliminary.
• Does the feasible boundary of the 2 mm laser-cut bracket converge when the mesh is refined to 0.5 mm?On meshes of 1.0 and 0.5 mm no width in the usable 15.0–16.0 mm window of the gauge-2 bracket scan is feasible (one point at 0.75 mm), and the lowest peak stress rises 11–14% from 1.5 to 0.5 mm; negative, stylised, preliminary.

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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.