# grooved_shaft

Family 6: a round bar of diameter D with a U-shaped circumferential groove (bottom radius r, minimum diameter d), in
tension.

Reference: Peterson's U-groove curve fit (as given by amesweb.info's calculator, not checked against the book), a trend
check only, not a certificate. The certificate is the mesh study, CalculiX and the axisymmetry check.
A quarter is solved (symmetry on x = 0 and y = 0) and half the length: the groove's mid-plane z = 0 is a symmetry plane,
the far end is pulled. Lengths are scaled by d (d = 20 mm in the solves).

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `D_over_d` | 1.2 - 2.0 |
| `r_over_h` | 0.25 - 0.85 |
| `nu` | 0.25 - 0.35 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `kt` | 1 % / 1.3 % / 1.3 % | max 0.36 % | max 0.38 % |

## Checks

- points: 150 solved, 0 failed
- three-mesh studies: 15 (every 10th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0.76 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: kt vs r_over_h ok (rho -0.632), kt vs nu ok (rho -0.001)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `kt_over_peterson` | 1.072 | 1.091 | 1.096 |
| `peak_angle_deg` | 0 | 0 | 4.5 |

## Columns of points.csv

parameters, the answers, `<answer>_band` (relative), the hand-theory ratios, `mirror_err`, `equilibrium_err` (where a single reaction exists), `<answer>_loo_err` / `_neighbour_flag` (leave-one-out prediction from all other points), `<answer>_recheck_diff` / `_flag_resolved` (finer re-solve of a flagged point), `has_gci` / `has_similarity` / `has_calculix`, `nodes`, `solve_s`, `studio` (version).

Solved with NovaSolver Studio through the same commands an AI client uses over MCP; the first section says which solver and mesh. The claim is verification (the equations are solved within the band), not validation against tests.

## License

This data is licensed under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/
Credit: NovaSolver, https://novasolver.jp/studio/verified/
Provided as is, without warranty. Design decisions remain the user's responsibility.