# rotating_cylinder

Family 20: a long hollow cylinder spinning about its axis, in plane strain (body load from physics.rotation).

Exact at every point, with k = b / a, x = r / a, C1 = (3 - 2 nu) / (8 (1 - nu)), C2 = (1 + 2 nu) / (8 (1 - nu)):
  sigma_t / (rho w^2 a^2) = C1 (1 + k^2 + k^2 / x^2) - C2 x^2
  sigma_r / (rho w^2 a^2) = C1 (1 + k^2 - k^2 / x^2 - x^2)
  u_r E / (rho w^2 a^3)   = x (1 + nu) ((1 - nu) sigma_t - nu sigma_r) / (rho w^2 a^2).
The geometry and the quarter-ring model are the pressure cylinder's (family 3).

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `b_over_a` | 1.1 - 3.0 |
| `nu` | 0.2 - 0.45 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `hoop_star` | 0.18 % / 2 % / 2 % | max 0.035 % | max - |
| `ur_star` | 0.012 % / 0.15 % / 0.15 % | max 0.0031 % | max - |

## Checks

- points: 60 solved, 0 failed
- three-mesh studies: 12 (every 5th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0.5 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: hoop_star vs b_over_a ok (rho 0.997)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `hoop_over_exact` | 0.9997 | 1.001 | 1.003 |
| `ur_over_exact` | 1 | 1 | 1 |

## 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.