engine: heated-bracket sample, steady heat and thermal stress in steel

file.sample heated-bracket: the corpus filleted bracket in steel, tet10, its
hole bores clamped, the upright's end held at 120 C and the rest cooled by
still air; one coupled temperature-displacement step. Its test checks the
temperatures against the heater and the air, then warms the bracket evenly,
clamps one hole, and matches the free thermal expansion alpha dT times the
distance from the hole along x.

Closes #165

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo

crates/engine/src/sample.rs

@@ -75,6 +75,13 @@ pub const SAMPLES: &[Sample] = &[
         design: Some(UPRIGHT_BRACKET),
         script: |_| flow_past(),
     },
+    Sample {
+        name: "heated-bracket",
+        label: "Heated Bracket",
+        about: "A steel bracket clamped at its holes and heated at one end: steady heat conduction and the thermal stress",
+        design: Some(FILLETED_BRACKET),
+        script: |_| heated_bracket(),
+    },
 ];
 
 /// The sample called `name`.
@@ -367,6 +374,47 @@ pub fn flow_past() -> Value {
     ])
 }
 
+/// The corpus filleted bracket (`tests/corpus/filleted-bracket`): an 8 mm L-bracket, foot along
+/// x (60 mm) and upright along y (50 mm), extruded 40 mm along z, an R6 fillet, an 8 mm hole
+/// through each leg (`F4` through the foot at x = 40, `F5` through the upright at y = 32).
+pub const FILLETED_BRACKET: &str = include_str!("../../../tests/corpus/filleted-bracket/recipe.json");
+
+/// The heated bracket: the heater's and the surrounding air's temperatures (°C, also the stress
+/// free temperature), and steel's expansion coefficient (1/K).
+pub const HEATER: f64 = 120.0;
+pub const AMBIENT: f64 = 20.0;
+pub const STEEL_EXPANSION: f64 = 1.2e-5;
+
+/// [`FILLETED_BRACKET`] in steel (k = 50 W/(m K), α = 12 × 10⁻⁶/K, stress free at 20 °C), tet10
+/// elements of 4 mm: its upright's end (`F2:side:l5`, set `hot`) held at [`HEATER`], the rest of
+/// its outside (set `skin`) cooled by still air at [`AMBIENT`] (h = 10 W/(m² K)), both hole bores
+/// clamped (set `holes`). Step `Heat` (coupled temperature–displacement, steady): the
+/// temperature field and the stress of the expansion the clamps hold back. In mm, t, s, N, MPa:
+/// k in mW/(mm K), h in mW/(mm² K).
+pub fn heated_bracket() -> Value {
+    let face = |f: &str| json!({"body": "Bracket", "face": f});
+    let skin: Vec<Value> = ["F2:start", "F2:top", "F2:side:l1", "F2:side:l2", "F2:side:l3", "F2:side:l4", "F2:side:l6", "F3:blend:F2:side:l3|F2:side:l4"]
+        .into_iter()
+        .map(face)
+        .collect();
+    let holes: Vec<Value> = ["F4:bore:side:0", "F4:bore:side:1", "F5:bore:side:0", "F5:bore:side:1"].into_iter().map(face).collect();
+    json!([
+        {"command": "model.units", "params": {"system": "mm-t-s-N-MPa"}},
+        {"command": "mesh.tet.generate", "params": {"body": "Bracket", "size": 4, "order": "quadratic", "tolerance": 0.05}},
+        {"command": "model.set.create", "params": {"name": "bracket", "geometry": [{"body": "Bracket"}]}},
+        {"command": "model.set.create", "params": {"name": "holes", "geometry": holes}},
+        {"command": "model.set.create", "params": {"name": "hot", "geometry": [face("F2:side:l5")]}},
+        {"command": "model.set.create", "params": {"name": "skin", "geometry": skin}},
+        {"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 210000, "poisson": 0.3}, "density": 7.85e-9,
+            "thermal": {"conductivity": 50, "specific_heat": 4.6e8, "expansion": STEEL_EXPANSION, "reference_temperature": AMBIENT}}},
+        {"command": "model.section.create", "params": {"name": "Bracket", "set": "bracket", "material": "Steel", "kind": {"type": "solid"}}},
+        {"command": "model.boundary_condition.create", "params": {"name": "Clamp", "type": "fixed", "target": {"set": "holes"}}},
+        {"command": "model.boundary_condition.create", "params": {"name": "Heater", "type": "temperature", "target": {"set": "hot"}, "value": HEATER}},
+        {"command": "model.load.create", "params": {"name": "Air", "type": "convection", "set": "skin", "film": 0.01, "sink": AMBIENT}},
+        {"command": "model.step.create", "params": {"name": "Heat", "procedure": {"type": "coupled_temperature_displacement"}, "boundary_conditions": ["Clamp", "Heater"], "loads": ["Air"]}},
+    ])
+}
+
 /// Six times the signed volume of the tetrahedron `p[0..4]` (positive when p3 lies on the side
 /// p0 → p1 → p2 turns counter-clockwise around).
 fn signed_volume(p: &[[f64; 3]]) -> f64 {
@@ -505,6 +553,39 @@ mod tests {
         assert!(r["mass"]["relative_change"].as_f64().unwrap().abs() < 0.01, "{r}");
     }
 
+    /// The heated bracket's temperatures lie between the air's and the heater's, falling along
+    /// the bracket away from the heater. Its thermal expansion: with the air at the heater's
+    /// temperature the bracket warms evenly by ΔT, and held at the foot's hole alone (at x = 40)
+    /// it grows freely, so a point moves α ΔT times its distance from the hole's axis along x:
+    /// the upright's top by −0.0432 mm (within 2 %), the foot's end by +0.024 mm (within 6 %:
+    /// the clamped bore cannot grow).
+    #[test]
+    #[ignore = "heavy: cargo xtask test --heavy (CI heavy job)"]
+    fn the_heated_bracket_warms_and_expands_as_steel() {
+        let mut s = open("heated-bracket");
+        let r = s.execute("solve.run", &json!({})).unwrap();
+        let r = &r["steps"][0]["summary"];
+        assert!((r["max_temperature"]["value"].as_f64().unwrap() - HEATER).abs() < 1e-9, "{r}");
+        let probe = |s: &mut Session| s.execute("solve.probe", &json!({"step": "Heat", "at": [[4.0, 50.0, 20.0], [60.0, 4.0, 20.0]]})).unwrap();
+        let p = probe(&mut s);
+        let (upright, foot) = (p["nodes"][0]["temperature"].as_f64().unwrap(), p["nodes"][1]["temperature"].as_f64().unwrap());
+        assert!(upright > foot && foot > AMBIENT && upright <= HEATER, "{upright} {foot}");
+        let foot_hole = json!([{"body": "Bracket", "face": "F4:bore:side:0"}, {"body": "Bracket", "face": "F4:bore:side:1"}]);
+        s.execute("model.set.create", &json!({"name": "foot_hole", "geometry": foot_hole})).unwrap();
+        s.execute("model.boundary_condition.edit", &json!({"name": "Clamp", "type": "fixed", "target": {"set": "foot_hole"}})).unwrap();
+        s.execute("model.load.edit", &json!({"name": "Air", "type": "convection", "set": "skin", "film": 0.01, "sink": HEATER})).unwrap();
+        s.execute("solve.static", &json!({})).unwrap();
+        let p = probe(&mut s);
+        let grow = STEEL_EXPANSION * (HEATER - AMBIENT);
+        // The bore itself cannot grow, which holds back the foot's end, 16 mm from its edge, a
+        // little more than the upright's top, 32 mm away.
+        for (i, tolerance) in [(0, 0.02), (1, 0.06)] {
+            let node = &p["nodes"][i];
+            let (ux, expected) = (node["displacement"][0].as_f64().unwrap(), grow * (node["xyz"][0].as_f64().unwrap() - 40.0));
+            assert!((ux - expected).abs() < tolerance * expected.abs(), "{node}: {ux} against {expected}");
+        }
+    }
+
     #[test]
     fn the_drop_test_opens_ready_to_solve() {
         let s = open("drop-test");