FEM foundation B: heat transfer and thermal stress (solids, 2D, explicit), coupled temperature–displacement, solve.heat (#45)

* Heat transfer and thermal stress: thermal materials, temperature BCs, heat loads, heat transfer and coupled temperature–displacement steps, solve.heat

Model: Material.thermal (conductivity, specific heat, expansion, isotropic or
per axis; reference temperature), BoundaryCondition::Temperature (dof 11),
loads SurfaceFlux, Convection, HeatSource and Temperature (a field in static
steps), procedures HeatTransfer { transient (θ-method) } and
CoupledTemperatureDisplacement; checks for missing conductivity/specific
heat/expansion. fem: heat conduction kernels (conductivity, capacity,
source, face flux/film, heat flux) on the DofLayout (one T per node), thermal
strain load and stress D(ε − ε_th) for every solid formulation incl.
incompatible modes. Solver: steady and transient heat transfer, thermal loads
in static steps, coupled steps (exact one-way coupling). Engine: solve.heat,
temperature/heat-flux result fields, probes. Decks refuse heat transfer by
name. Analytic verification: slab, source+convection/flux, thick cylinder,
erfc semi-infinite solid, free/constrained expansion, bimetal strip, coupled
step.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* thermal: strains in 2D elements and explicit dynamics

Plane::thermal_load and stresses_with for plane stress (εzz free), plane
strain (σzz = −D₃·ε_th) and axisymmetric elements, incompatible modes
included; the solver's 2D kernels take them. Explicit dynamics turns a
step's temperature loads into a per-element thermal strain rate (reached
linearly over the step) subtracted from each strain increment, so it feeds
J2 and Johnson–Cook updates. Verified: free expansion in every 2D theory
and element, a biaxially held plate, an explicit free bar (ΔL = αΔT L,
stress-free) and a clamped bar at −EαΔT, capped at the yield stress with
perfect plasticity.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* clippy

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* thermal: trusses take the axial thermal strain; keep the roadmap's M7 line

Truss::thermal_load and stresses_with project the thermal strain on the
member's tangent (Σ αₖ tₖ² ΔT); the static solver applies them to rod2/rod3
(#38). Explicit trusses take none yet and a warning says so.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

---------

Co-authored-by: jyviko <2501648+jyviko@users.noreply.github.com>
Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>

crates/fem/src/tests.rs

@@ -208,3 +208,33 @@ fn section_formulations() {
     assert!(Formulation::from_section(ElementKind::Tet4, &hybrid).is_err());
     assert!(Formulation::from_section(ElementKind::Quad4, &full).is_err());
 }
+
+#[test]
+fn uniform_thermal_strain_is_free_expansion_and_its_load_balances_it() {
+    // A uniform ΔT with orthotropic α: the free expansion u = diag(α ΔT) x is stress-free, and
+    // the thermal load equals K u for it (the element is in equilibrium with no other load).
+    let d = orthotropic([1000.0, 800.0, 600.0], [0.3, 0.25, 0.2], [400.0, 300.0, 350.0]).unwrap();
+    let alpha = [1.2e-5, 2.0e-5, 0.7e-5];
+    for (kind, f) in CASES {
+        let xyz = distorted(kind, true);
+        let solid = Solid::new(kind, &xyz, f).unwrap();
+        let th = crate::ThermalStrain { alpha, delta: vec![80.0; xyz.len()] };
+        let u: Vec<[f64; 3]> = xyz.iter().map(|x| [0, 1, 2].map(|k| alpha[k] * 80.0 * x[k])).collect();
+        for p in solid.stresses_with(&d, &u, Some(&th)).unwrap() {
+            assert!(p.stress.iter().all(|s| s.abs() < 1e-9), "{kind:?} {f:?}: {:?}", p.stress);
+            for k in 0..3 {
+                assert!((p.strain[k] - alpha[k] * 80.0).abs() < 1e-12);
+            }
+        }
+        let load = solid.thermal_load(&d, &th).unwrap();
+        let ku = solid.stiffness(&d).unwrap().mul_vec(&u.iter().flatten().copied().collect::<Vec<_>>());
+        let scale = ku.iter().fold(0.0f64, |m, v| m.max(v.abs()));
+        for (a, b) in load.iter().flatten().zip(&ku) {
+            assert!((a - b).abs() < 1e-9 * scale, "{kind:?} {f:?}: {a} vs {b}");
+        }
+        // Without a thermal strain the two stress paths agree bit for bit.
+        assert_eq!(solid.stresses_with(&d, &u, None).unwrap(), solid.stresses(&d, &u).unwrap());
+        assert!(solid.thermal_load(&d, &crate::ThermalStrain { alpha, delta: vec![1.0; 2] }).is_err());
+        assert!(solid.thermal_load(&d, &crate::ThermalStrain { alpha: [f64::NAN; 3], delta: vec![1.0; xyz.len()] }).is_err());
+    }
+}