# sphere_cavity

Family 5: a round bar with a spherical cavity at its centre, in tension.

Exact anchor: Goodier's solution for a spherical cavity in an infinite body under uniaxial tension, peak stress at the
equator Kt = 3 (9 - 5 nu) / (2 (7 - 5 nu)) (2.045 at nu = 0.3), the one textbook concentration that depends on Poisson's
ratio. A small cavity in the bar must approach it; a large one rises above it (the bar's own section is used up).
An eighth is solved (symmetry on x = 0, y = 0 and on the cavity's equatorial plane z = 0); the far end is pulled.
Lengths are scaled by the bar radius (20 mm in the solves).

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `a_over_R` | 0.1 - 0.6 |
| `nu` | 0.0 - 0.45 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `kt` | 0.4 % / 2.4 % / 2.4 % | max 0.19 % | max 0.076 % |

## Checks

- points: 200 solved, 0 failed
- three-mesh studies: 20 (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): 0.54 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: kt vs a_over_R ok (rho 0.883), kt vs nu ok (rho 0.405)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `kt_over_goodier` | 1.048 | 1.259 | 1.313 |
| `peak_deg` | 0 | 0 | 2 |

## 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.