# thermal_cylinder

Family 19: thermal stress in a thick cylinder whose bore is hotter than its outside, in plane strain (temperature solved
first, then the stress: thermal_load mode "solve").

Exact at every point (Timoshenko and Goodier), with k = b / a, x = r / a and K = 1 / (2 (1 - nu) ln k):
  sigma_t / (E alpha dT) = K (1 - ln(k / x) - (1 + k^2 / x^2) ln k / (k^2 - 1))
  sigma_r / (E alpha dT) = K (  - ln(k / x) - (1 - k^2 / x^2) ln k / (k^2 - 1))
  sigma_z / (E alpha dT) = nu (sigma_r + sigma_t) / (E alpha dT) - T / dT,  T = dT ln(k / x) / ln k above the outside.
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` | 0.74 % / 2.8 % / 2.8 % | max 0.067 % | max - |
| `axial_star` | 0.14 % / 0.66 % / 0.66 % | max 0.025 % | 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): 0.71 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: 

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `hoop_over_exact` | 0.9981 | 1 | 1.002 |
| `axial_over_exact` | 0.9999 | 1.001 | 1.003 |

## 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.