# thick_cylinder

Family 3: a thick-walled cylinder (bore radius a, outer radius b) under internal pressure, in plane strain.

Lame's solution is exact, so every point of this family is checked against theory, not only a subset: it is the
anchor that proves the solver across the whole range before the families without a closed form are trusted.
  hoop stress at the bore     sigma_t(a) / p = (b^2 + a^2) / (b^2 - a^2)
  radial displacement at a    u(a) E / (p a) = (1 + nu) ((1 - 2 nu) a^2 + b^2) / (b^2 - a^2)
A quarter of a ring of length 0.5 a is solved: symmetry on x = 0 and y = 0, and on both ends (plane strain).

## 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` | 0.15 % / 0.24 % / 0.27 % | max 0.054 % | max 0.15 % |
| `ur_star` | 0.0045 % / 0.0061 % / 0.0069 % | max 0.0019 % | max 0.00028 % |

## Checks

- points: 100 solved, 0 failed
- three-mesh studies: 10 (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): 0.17 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: hoop_star vs b_over_a ok (rho -1.0), hoop_star vs nu ok (rho 0.015), ur_star vs b_over_a ok (rho -0.982)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `hoop_over_lame` | 0.9995 | 1 | 1.001 |
| `ur_over_lame` | 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.