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 <[email protected]>
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 <[email protected]>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* clippy

Co-Authored-By: Claude Opus 5.5 <[email protected]>
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 <[email protected]>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

---------

Co-authored-by: jyviko <[email protected]>
Co-authored-by: Claude Opus 5.5 <[email protected]>

crates/solver/src/analysis.rs

@@ -14,7 +14,10 @@ use crate::constraints::{Budget, Reduction, coupling_rows};
 use crate::eigen::{Operator, Wanted, lanczos};
 use crate::ordering::nested_dissection;
 use crate::sparse::{Csr, NodeGraph, add_block, add_diagonal, add_scalar};
-use crate::system::{Mesh, Structural, StructuralKernel, inelastic_warnings, loads, prescribed, structural_body_forces, translation_block};
+use crate::system::{
+    Mesh, Structural, StructuralKernel, add_thermal_loads, inelastic_warnings, loads, prescribed, structural_body_forces, temperature_loads,
+    translation_block,
+};
 use crate::{Error, Limits, Result};
 
 /// Size and effort of a solution.
@@ -99,6 +102,8 @@ pub struct StaticResult {
     pub reaction_total: [f64; 3],
     pub stats: Stats,
     pub warnings: Vec<String>,
+    /// The temperatures of a coupled temperature–displacement step, per node.
+    pub temperature: Option<Vec<f64>>,
 }
 
 /// One natural mode.
@@ -293,17 +298,43 @@ fn budget(limits: &Limits, mesh: &Mesh, graph: &NodeGraph, matrices: usize) -> B
 }
 
 /// Solve a linear static step: the step named `step`, or the model's only static step.
+///
+/// A coupled temperature–displacement step is solved here too: its steady temperatures first
+/// ([`crate::heat`]), then the equilibrium with their thermal strains. The coupling is one-way
+/// (temperature strains the solid; deformation does not heat it), which is the whole of linear
+/// quasi-static thermoelasticity without thermoelastic dissipation, so solving the two in turn
+/// is the monolithic solution exactly.
 pub fn solve_static(model: &Model, step: Option<&str>, limits: &Limits) -> Result<StaticResult> {
     let resolved = model.resolve()?;
-    let step = model.select_step(step, |p| matches!(p, Procedure::Static {}), "static")?;
+    let step = model.select_step(
+        step,
+        |p| matches!(p, Procedure::Static {} | Procedure::CoupledTemperatureDisplacement {}),
+        "static or coupled temperature–displacement",
+    )?;
     let mesh = Mesh::new(model, &resolved, limits.max_dofs)?;
     let mut warnings: Vec<String> = resolved.warnings.iter().map(ToString::to_string).chain(mesh.warnings.iter().cloned()).collect();
     warnings.extend(inelastic_warnings(model));
+    // The temperature field: the coupled step's own heat solution, or the step's temperature
+    // loads.
+    let (field, temperature) = match step.procedure {
+        Procedure::CoupledTemperatureDisplacement {} => {
+            let (labels, t) = crate::heat::temperatures(model, &resolved, step, limits)?;
+            let at = |l: &u32| labels.binary_search(l).ok().and_then(|i| t.get(i)).copied();
+            (Some(mesh.labels.iter().map(at).collect::<Vec<_>>()), Some(mesh.labels.iter().map(|l| at(l).unwrap_or(0.0)).collect()))
+        }
+        _ => (temperature_loads(model, &resolved, &mesh, step)?, None),
+    };
+    if field.is_some() && !mesh.structural.is_empty() {
+        warnings.push("shells and beams take no thermal strain yet; the temperatures strain the solid, 2D and truss elements only".into());
+    }
     let (graph, order) = pattern(&mesh);
     ignored_dashpots(&mesh, &mut warnings);
     let setup = Setup { model, resolved: &resolved, mesh: &mesh, step };
     let red = reduction(&setup, true, &order, budget(limits, &mesh, &graph, 1), &mut warnings)?;
-    let f = loads(model, &resolved, &mesh, step, &red.active)?;
+    let mut f = loads(model, &resolved, &mesh, step, &red.active)?;
+    if let Some(field) = &field {
+        add_thermal_loads(&mut f, &mesh, field)?;
+    }
     let k = assemble(&mesh, &graph, None)?.k;
     if red.neq() == 0 {
         return Err(Error::Invalid("every degree of freedom is prescribed; there is nothing to solve".into()));
@@ -343,7 +374,8 @@ pub fn solve_static(model: &Model, step: Option<&str>, limits: &Limits) -> Resul
         let kernel = e.kernel()?;
         let at = |err| Error::element(e.label, err);
         let ue: Vec<[f64; 3]> = e.nodes.iter().map(|n| displacement.get(*n).copied().unwrap_or_default()).collect();
-        let points = kernel.stresses(&e.d, &ue).map_err(at)?;
+        let thermal = field.as_deref().and_then(|t| e.thermal_strain(t));
+        let points = kernel.stresses_with(&e.d, &ue, thermal.as_ref()).map_err(at)?;
         let values: Vec<[f64; 6]> = points.iter().map(|p| p.stress).collect();
         let (ty, rule) = (kernel.ty(), kernel.rule());
         let cached = extrapolations.iter().position(|(t, r, _)| std::ptr::eq(*t, ty) && *r == rule);
@@ -369,7 +401,8 @@ pub fn solve_static(model: &Model, step: Option<&str>, limits: &Limits) -> Resul
         let truss = t.truss()?;
         let at = |err| Error::element(t.label, err);
         let ue: Vec<[f64; 3]> = t.nodes.iter().map(|n| displacement.get(*n).copied().unwrap_or_default()).collect();
-        let points = truss.stresses(t.young, &ue).map_err(at)?;
+        let thermal = field.as_deref().and_then(|f| t.thermal_strain(f));
+        let points = truss.stresses_with(t.young, &ue, thermal.as_ref()).map_err(at)?;
         let nodal = truss.extrapolate(&points.iter().map(|p| p.stress).collect::<Vec<_>>()).map_err(at)?;
         for (n, s) in t.nodes.iter().zip(&nodal) {
             if let (Some(acc), Some(c)) = (sum.get_mut(*n), count.get_mut(*n)) {
@@ -432,6 +465,7 @@ pub fn solve_static(model: &Model, step: Option<&str>, limits: &Limits) -> Resul
         reaction_total,
         stats: Stats { dofs: mesh.dofs(), equations: red.neq(), factor_entries: chol.entries(), lanczos_vectors: 0 },
         warnings,
+        temperature,
     })
 }
 
@@ -585,6 +619,7 @@ pub enum Location {
 pub enum StepResult {
     Static(StaticResult),
     Frequency(FrequencyResult),
+    Heat(crate::heat::HeatResult),
 }
 
 impl StepResult {
@@ -592,6 +627,7 @@ impl StepResult {
         match self {
             StepResult::Static(r) => &r.step,
             StepResult::Frequency(r) => &r.step,
+            StepResult::Heat(r) => &r.step,
         }
     }
 
@@ -599,6 +635,7 @@ impl StepResult {
         match self {
             StepResult::Static(r) => &r.nodes,
             StepResult::Frequency(r) => &r.nodes,
+            StepResult::Heat(r) => &r.nodes,
         }
     }
 
@@ -611,6 +648,7 @@ impl StepResult {
         match self {
             StepResult::Static(r) => r.reference_points.iter().position(|p| p.label == label),
             StepResult::Frequency(r) => r.reference_points.iter().position(|p| p.1 == label),
+            StepResult::Heat(_) => None,
         }
         .map(Location::Point)
     }

crates/solver/src/heat.rs

@@ -0,0 +1,491 @@
+//! Heat transfer: steady and transient conduction with prescribed temperatures, surface fluxes,
+//! convection and internal heat sources.
+//!
+//! Every node of the solid elements carries one temperature ([`modelcraft_fem::heat::HEAT_DOFS`]).
+//! The conductivity K (with the convection films) and capacity C are assembled on the node
+//! graph in element label order ([`modelcraft_fem::heat`]); prescribed temperatures are
+//! eliminated, the free ones numbered by nested dissection and solved by sparse Cholesky.
+//!
+//! - **Steady**: K T = f.
+//! - **Transient**, the generalized trapezoidal (θ) method with n equal increments Δt over the
+//!   step (T. J. R. Hughes, *The Finite Element Method*, Dover 2000, §8.2; unconditionally stable
+//!   for θ ≥ ½, θ = 1 backward Euler, θ = ½ Crank–Nicolson):
+//!   (C + θΔt K) Tₙ₊₁ = (C − (1 − θ)Δt K) Tₙ + Δt f, from a uniform initial temperature, with
+//!   the prescribed temperatures held from the first increment on. The matrix is factored once.
+//!
+//! The heat flux −k∇T is reported at the integration points and averaged at the nodes; the heat
+//! flow at prescribed temperatures is the reaction K T − f (C Ṫ included in transient steps).
+
+use modelcraft_fem::element::{Extrapolation, extrapolation};
+use modelcraft_fem::heat::{Conduction, PointFlux, face_film, face_weights};
+use modelcraft_fem::{DofLayout, DofSet};
+use modelcraft_model::{ElementKind, Load, Model, Procedure, Resolved, SectionKind, Step, TEMPERATURE_DOF};
+
+use crate::cholesky::{Factor, FactorError, UpperColumns};
+use crate::ordering::nested_dissection;
+use crate::sparse::{Csr, NodeGraph, add_block};
+use crate::system::hold;
+use crate::{Error, Limits, Result, Stats};
+
+/// Most temperature values a transient result keeps (frames × nodes); frames are thinned
+/// evenly to stay below it, the last increment always kept.
+pub const MAX_TEMPERATURE_VALUES: usize = 50_000_000;
+
+/// A solid element conducting heat.
+#[derive(Debug, Clone)]
+pub struct HeatElem {
+    pub label: u32,
+    pub kind: ElementKind,
+    /// Indices into [`HeatMesh::labels`].
+    pub nodes: Vec<usize>,
+    pub xyz: Vec<[f64; 3]>,
+    /// Conductivity along X, Y, Z.
+    pub conductivity: [f64; 3],
+    /// ρc, when the material has a density and a specific heat.
+    pub capacity: Option<f64>,
+}
+
+impl HeatElem {
+    fn kernel(&self) -> Result<Conduction<'_>> {
+        Conduction::new(self.kind, &self.xyz).map_err(|e| Error::element(self.label, e))
+    }
+}
+
+/// The conducting mesh: the nodes of the solid elements by label, and the elements by label.
+#[derive(Debug, Clone)]
+pub struct HeatMesh {
+    pub labels: Vec<u32>,
+    pub xyz: Vec<[f64; 3]>,
+    pub elements: Vec<HeatElem>,
+    pub layout: DofLayout,
+}
+
+impl HeatMesh {
+    pub fn new(model: &Model, resolved: &Resolved, max_dofs: usize) -> Result<HeatMesh> {
+        let mut elements = Vec::new();
+        for (name, section) in model.sections() {
+            if !matches!(section.kind, SectionKind::Solid {}) {
+                return Err(Error::NotSupported(format!(
+                    "heat transfer in {} sections (section `{name}`); it has solid elements so far",
+                    section.kind.name()
+                )));
+            }
+            let material = model
+                .material(&section.material)
+                .ok_or_else(|| Error::Invalid(format!("section `{name}`: material `{}` is missing", section.material)))?;
+            let thermal = material.thermal.unwrap_or_default();
+            let conductivity =
+                thermal.conductivity.ok_or_else(|| Error::Invalid(format!("material `{}` has no conductivity", section.material)))?.axes();
+            let capacity = material.density.zip(thermal.specific_heat).map(|(rho, c)| rho * c);
+            let set = resolved.sets.get(&section.set).ok_or_else(|| Error::Invalid(format!("set `{}` is not resolved", section.set)))?;
+            for &label in &set.elements {
+                let el = model.element(label).ok_or_else(|| Error::Invalid(format!("element {label} is missing")))?;
+                elements.push((label, el.kind, el.nodes.clone(), conductivity, capacity));
+            }
+        }
+        elements.sort_by_key(|e| e.0);
+        let layout = DofLayout::uniform(elements.iter().flat_map(|e| e.2.iter().copied()), DofSet::TEMPERATURE);
+        if layout.dofs() > max_dofs {
+            return Err(Error::Limit(format!("the model has {} temperatures; the limit is {max_dofs}", layout.dofs())));
+        }
+        let labels: Vec<u32> = (0..layout.nodes()).filter_map(|s| layout.label(s)).collect();
+        let xyz: Vec<[f64; 3]> = labels
+            .iter()
+            .map(|l| model.node(*l).map(|n| n.xyz).ok_or_else(|| Error::Invalid(format!("node {l} is missing"))))
+            .collect::<Result<_>>()?;
+        let mut mesh = HeatMesh { labels, xyz, elements: Vec::with_capacity(elements.len()), layout };
+        for (label, kind, nodes, conductivity, capacity) in elements {
+            let idx: Vec<usize> = nodes.iter().filter_map(|n| mesh.layout.node(*n)).collect();
+            let xyz = idx.iter().filter_map(|i| mesh.xyz.get(*i).copied()).collect();
+            mesh.elements.push(HeatElem { label, kind, nodes: idx, xyz, conductivity, capacity });
+        }
+        Ok(mesh)
+    }
+
+    fn node_of(&self, what: &str, label: u32) -> Result<usize> {
+        self.layout.node(label).ok_or_else(|| Error::Invalid(format!("{what}: node {label} belongs to no solid element")))
+    }
+
+    fn element(&self, label: u32) -> Option<&HeatElem> {
+        self.elements.binary_search_by_key(&label, |e| e.label).ok().and_then(|i| self.elements.get(i))
+    }
+}
+
+/// The heat flux in one element.
+#[derive(Debug, Clone, PartialEq)]
+pub struct ElementFlux {
+    pub label: u32,
+    pub kind: ElementKind,
+    pub points: Vec<PointFlux>,
+}
+
+/// The result of a heat transfer step.
+#[derive(Debug, Clone, PartialEq)]
+pub struct HeatResult {
+    pub step: String,
+    /// Node labels with a temperature, sorted; per-node vectors follow this order.
+    pub nodes: Vec<u32>,
+    /// The time of each frame (one frame at time 0 for a steady step).
+    pub times: Vec<f64>,
+    /// Nodal temperatures per frame; the last frame is the end of the step.
+    pub frames: Vec<Vec<f64>>,
+    /// Heat flux averaged at the nodes, at the end of the step.
+    pub flux: Vec<[f64; 3]>,
+    /// Heat flux at the integration points, at the end of the step.
+    pub elements: Vec<ElementFlux>,
+    /// The heat each prescribed temperature supplies to the body at the end of the step
+    /// (negative where it takes heat away; zero at free nodes).
+    pub reaction: Vec<f64>,
+    /// Heat supplied by fluxes, convection and sources (at the end temperatures), and the total
+    /// reaction heat flow; in a steady state they cancel.
+    pub applied_total: f64,
+    pub reaction_total: f64,
+    pub stats: Stats,
+    pub warnings: Vec<String>,
+}
+
+impl HeatResult {
+    /// The temperatures at the end of the step.
+    pub fn temperature(&self) -> &[f64] {
+        self.frames.last().map(Vec::as_slice).unwrap_or_default()
+    }
+}
+
+/// The assembled heat system of a step: conductivity (with films), capacity when transient,
+/// and the heat supplied.
+struct HeatSystem {
+    k: Csr,
+    c: Option<Csr>,
+    f: Vec<f64>,
+}
+
+fn assemble(model: &Model, resolved: &Resolved, mesh: &HeatMesh, graph: &NodeGraph, step: &Step, transient: bool) -> Result<HeatSystem> {
+    let layout = &mesh.layout;
+    let mut k = graph.zero_matrix(layout);
+    let mut c = transient.then(|| graph.zero_matrix(layout));
+    for e in &mesh.elements {
+        let kernel = e.kernel()?;
+        let at = |err| Error::element(e.label, err);
+        add_block(&mut k, layout, &e.nodes, DofSet::TEMPERATURE, &kernel.conductivity(e.conductivity).map_err(at)?.data)?;
+        if let Some(c) = c.as_mut() {
+            let rho_c = e.capacity.ok_or_else(|| Error::Invalid(format!("element {}: its material needs a density and a specific heat", e.label)))?;
+            add_block(c, layout, &e.nodes, DofSet::TEMPERATURE, &kernel.capacity(rho_c).map_err(at)?.data)?;
+        }
+    }
+    let mut f = vec![0.0; layout.dofs()];
+    let add = |f: &mut [f64], nodes: &[usize], v: &[f64]| {
+        for (n, x) in nodes.iter().zip(v) {
+            if let Some(y) = f.get_mut(*n) {
+                *y += x;
+            }
+        }
+    };
+    for name in &step.loads {
+        let load = model.load(name).ok_or_else(|| Error::Invalid(format!("load `{name}` is missing")))?;
+        let faces = || -> Result<Vec<(u32, Vec<usize>, Vec<[f64; 3]>)>> {
+            let set = load.set().unwrap_or_default();
+            resolved
+                .sets
+                .get(set)
+                .map(|s| s.faces.as_slice())
+                .unwrap_or_default()
+                .iter()
+                .map(|fc| {
+                    let labels = model
+                        .face_nodes(*fc)
+                        .ok_or_else(|| Error::Invalid(format!("load `{name}`: element {} face {} is missing", fc.element, fc.face)))?;
+                    let idx: Vec<usize> = labels.iter().map(|l| mesh.node_of(&format!("load `{name}`"), *l)).collect::<Result<_>>()?;
+                    let xyz = idx.iter().filter_map(|i| mesh.xyz.get(*i).copied()).collect();
+                    Ok((fc.element, idx, xyz))
+                })
+                .collect()
+        };
+        match load {
+            Load::SurfaceFlux { flux, .. } => {
+                for (label, idx, xyz) in faces()? {
+                    let w = face_weights(&xyz).map_err(|e| Error::element(label, e))?;
+                    add(&mut f, &idx, &w.iter().map(|w| w * flux).collect::<Vec<_>>());
+                }
+            }
+            Load::Convection { film, sink, .. } => {
+                for (label, idx, xyz) in faces()? {
+                    let at = |e| Error::element(label, e);
+                    add_block(&mut k, layout, &idx, DofSet::TEMPERATURE, &face_film(&xyz, *film).map_err(at)?.data)?;
+                    let w = face_weights(&xyz).map_err(at)?;
+                    add(&mut f, &idx, &w.iter().map(|w| w * film * sink).collect::<Vec<_>>());
+                }
+            }
+            Load::HeatSource { set, power } => {
+                for label in resolved.sets.get(set).map(|s| s.elements.as_slice()).unwrap_or_default() {
+                    let e = mesh.element(*label).ok_or_else(|| Error::Invalid(format!("load `{name}`: element {label} is not a solid element")))?;
+                    let q = e.kernel()?.source(*power).map_err(|err| Error::element(e.label, err))?;
+                    add(&mut f, &e.nodes, &q);
+                }
+            }
+            // Mechanical loads and temperature fields do not act on heat conduction.
+            _ => {}
+        }
+    }
+    Ok(HeatSystem { k, c, f })
+}
+
+/// The prescribed temperature of every node (`None` = free) under the step's boundary
+/// conditions.
+fn prescribed(model: &Model, resolved: &Resolved, mesh: &HeatMesh, step: &Step) -> Result<Vec<Option<f64>>> {
+    let mut out = vec![None; mesh.layout.dofs()];
+    for name in &step.boundary_conditions {
+        let bc = model.boundary_condition(name).ok_or_else(|| Error::Invalid(format!("boundary condition `{name}` is missing")))?;
+        let Some(v) = bc.value(TEMPERATURE_DOF) else { continue };
+        for &label in resolved.target_nodes(bc.target()) {
+            let Some(n) = mesh.layout.node(label) else { continue };
+            hold(out.get_mut(n), v, |old| format!("node {label} is held at both {old} and {v} (boundary condition `{name}` and another)"))?;
+        }
+    }
+    Ok(out)
+}
+
+/// The free temperatures numbered in nested-dissection order.
+struct Equations {
+    /// Per node, its equation or `usize::MAX` when prescribed.
+    eq: Vec<usize>,
+    /// Per equation, its node.
+    node: Vec<usize>,
+}
+
+impl Equations {
+    fn new(mesh: &HeatMesh, graph: &NodeGraph, fixed: &[Option<f64>]) -> Equations {
+        let mut eq = vec![usize::MAX; mesh.labels.len()];
+        let mut node = Vec::new();
+        for n in nested_dissection(&mesh.xyz, &graph.adjacency) {
+            if fixed.get(n).is_some_and(Option::is_none)
+                && let Some(e) = eq.get_mut(n)
+            {
+                *e = node.len();
+                node.push(n);
+            }
+        }
+        Equations { eq, node }
+    }
+
+    /// The free-free block of a node-level matrix, rows and columns in equation order.
+    fn reduce(&self, a: &Csr) -> Csr {
+        let mut ptr = vec![0];
+        let (mut idx, mut val) = (Vec::new(), Vec::new());
+        for &n in &self.node {
+            let (cols, vals) = a.row(n);
+            let mut row: Vec<(usize, f64)> =
+                cols.iter().zip(vals).filter_map(|(c, v)| self.eq.get(*c).filter(|e| **e != usize::MAX).map(|e| (*e, *v))).collect();
+            row.sort_unstable_by_key(|t| t.0);
+            for (c, v) in row {
+                idx.push(c);
+                val.push(v);
+            }
+            ptr.push(idx.len());
+        }
+        Csr { n: self.node.len(), ptr, idx, val }
+    }
+
+    /// Solve for the free temperatures with right-hand side `rhs` (over every node; its free
+    /// rows are used) and write them into `t`, whose prescribed entries are kept.
+    fn solve(&self, chol: &Factor, rhs: &[f64], q: &mut Vec<f64>, t: &mut [f64]) -> Result<()> {
+        q.clear();
+        q.extend(self.node.iter().map(|n| rhs.get(*n).copied().unwrap_or(0.0)));
+        chol.solve(q)?;
+        for (n, v) in self.node.iter().zip(q.iter()) {
+            if let Some(x) = t.get_mut(*n) {
+                *x = *v;
+            }
+        }
+        if t.iter().any(|x| !x.is_finite()) {
+            return Err(Error::Singular("the temperatures are not finite; the model is ill-conditioned".into()));
+        }
+        Ok(())
+    }
+}
+
+fn factor(a: &Csr, limits: &Limits, mesh: &HeatMesh, eqs: &Equations) -> Result<Factor> {
+    Factor::new(&UpperColumns::from_symmetric(a), limits.max_factor_entries).map_err(|e| match e {
+        FactorError::TooLarge { entries, limit } => {
+            Error::Limit(format!("the factorization needs {entries} entries; the limit is {limit} (refine less or raise the limit)"))
+        }
+        FactorError::NotPositiveDefinite { column } => {
+            let label = eqs.node.get(column).and_then(|n| mesh.labels.get(*n)).copied().unwrap_or(0);
+            Error::Singular(format!(
+                "the temperature is undetermined at node {label}: no temperature boundary condition or convection reaches its part of the model"
+            ))
+        }
+    })
+}
+
+/// y = A x.
+fn mul(a: &Csr, x: &[f64]) -> Vec<f64> {
+    let mut y = vec![0.0; a.n];
+    a.mul(x, &mut y);
+    y
+}
+
+/// The temperature history of a heat transfer or coupled step and what produced it.
+struct Solution {
+    mesh: HeatMesh,
+    eqs: Equations,
+    k: Csr,
+    c: Option<Csr>,
+    f: Vec<f64>,
+    fixed: Vec<Option<f64>>,
+    times: Vec<f64>,
+    frames: Vec<Vec<f64>>,
+    /// Ṫ over the last increment (zero when steady).
+    rate: Vec<f64>,
+    factor_entries: usize,
+}
+
+impl Solution {
+    fn temperature(&self) -> &[f64] {
+        self.frames.last().map(Vec::as_slice).unwrap_or_default()
+    }
+}
+
+/// Solve the step's temperatures (steady unless the step is transient).
+fn run(model: &Model, resolved: &Resolved, step: &Step, limits: &Limits) -> Result<Solution> {
+    let transient = match step.procedure {
+        Procedure::HeatTransfer { transient } => transient,
+        _ => None,
+    };
+    let mesh = HeatMesh::new(model, resolved, limits.max_dofs)?;
+    let graph = NodeGraph::new(mesh.labels.len(), mesh.elements.iter().map(|e| e.nodes.as_slice()));
+    let fixed = prescribed(model, resolved, &mesh, step)?;
+    let eqs = Equations::new(&mesh, &graph, &fixed);
+    if eqs.node.is_empty() {
+        return Err(Error::Invalid("every temperature is prescribed; there is nothing to solve".into()));
+    }
+    let HeatSystem { k, c, f } = assemble(model, resolved, &mesh, &graph, step, transient.is_some())?;
+    let n = mesh.labels.len();
+    let held: Vec<f64> = fixed.iter().map(|v| v.unwrap_or(0.0)).collect();
+    let mut q = Vec::with_capacity(eqs.node.len());
+    let (times, frames, rate, factor_entries) = match transient.zip(c.as_ref()) {
+        Some((tr, c)) => {
+            let steps = tr.increments();
+            let dt = tr.end_time / steps as f64;
+            // (C + θΔt K) Tₙ₊₁ = (C − (1 − θ)Δt K) Tₙ + Δt f, the prescribed values held from the
+            // first increment on (their part A_fp T_p moves to the right-hand side).
+            let a = c.add_scaled(tr.theta * dt, &k)?;
+            let b = c.add_scaled(-(1.0 - tr.theta) * dt, &k)?;
+            let chol = factor(&eqs.reduce(&a), limits, &mesh, &eqs)?;
+            let a_held = mul(&a, &held);
+            let every = (steps.saturating_mul(n) / MAX_TEMPERATURE_VALUES.max(1)).saturating_add(1);
+            let mut t = vec![tr.initial_temperature; n];
+            let (mut frames, mut times) = (vec![t.clone()], vec![0.0]);
+            let mut rhs = vec![0.0; n];
+            let mut next = held.clone();
+            for i in 1..=steps {
+                next.copy_from_slice(&held);
+                b.mul(&t, &mut rhs);
+                for ((r, fj), aj) in rhs.iter_mut().zip(&f).zip(&a_held) {
+                    *r += dt * fj - aj;
+                }
+                eqs.solve(&chol, &rhs, &mut q, &mut next)?;
+                std::mem::swap(&mut t, &mut next);
+                if i % every == 0 || i == steps {
+                    frames.push(t.clone());
+                    times.push(if i == steps { tr.end_time } else { i as f64 * dt });
+                }
+            }
+            // After the swap `next` holds the state before the last increment.
+            let rate = t.iter().zip(&next).map(|(a, b)| (a - b) / dt).collect();
+            (times, frames, rate, chol.entries())
+        }
+        None => {
+            let rhs: Vec<f64> = f.iter().zip(mul(&k, &held)).map(|(a, b)| a - b).collect();
+            let chol = factor(&eqs.reduce(&k), limits, &mesh, &eqs)?;
+            let mut t = held;
+            eqs.solve(&chol, &rhs, &mut q, &mut t)?;
+            (vec![0.0], vec![t], vec![0.0; n], chol.entries())
+        }
+    };
+    Ok(Solution { mesh, eqs, k, c, f, fixed, times, frames, rate, factor_entries })
+}
+
+/// The steady temperatures of a coupled temperature–displacement step, by node label (sorted).
+pub(crate) fn temperatures(model: &Model, resolved: &Resolved, step: &Step, limits: &Limits) -> Result<(Vec<u32>, Vec<f64>)> {
+    let mut s = run(model, resolved, step, limits)?;
+    let t = s.frames.pop().unwrap_or_default();
+    Ok((s.mesh.labels, t))
+}
+
+/// Solve a heat transfer step: the step named `step`, or the model's only one.
+pub fn solve_heat(model: &Model, step: Option<&str>, limits: &Limits) -> Result<HeatResult> {
+    let resolved = model.resolve()?;
+    let step = model.select_step(step, |p| matches!(p, Procedure::HeatTransfer { .. }), "heat transfer")?;
+    let mut warnings: Vec<String> = resolved.warnings.iter().map(ToString::to_string).collect();
+    let s = run(model, &resolved, step, limits)?;
+    let t = s.temperature();
+    // Reactions: the heat each prescribed node supplies, K T + C Ṫ − f.
+    let kt = mul(&s.k, t);
+    let ct = s.c.as_ref().map(|c| mul(c, &s.rate));
+    let reaction: Vec<f64> = kt
+        .iter()
+        .zip(&s.f)
+        .zip(&s.fixed)
+        .enumerate()
+        .map(|(i, ((kt, f), fixed))| if fixed.is_some() { kt + ct.as_ref().and_then(|c| c.get(i)).copied().unwrap_or(0.0) - f } else { 0.0 })
+        .collect();
+    // Heat supplied: fluxes and sources, and convection at the end temperatures, Σ (f − K_film T)
+    // = Σ (f − K T) since conduction alone has zero row sums.
+    let applied_total: f64 = kt.iter().zip(&s.f).map(|(a, b)| b - a).sum();
+    let reaction_total: f64 = reaction.iter().sum();
+    let (flux, elements) = fluxes(&s.mesh, t)?;
+    // Measured against the gross heat flows, which cancel in the net sums.
+    let scale = reaction.iter().chain(&s.f).map(|v| v.abs()).sum::<f64>();
+    if s.c.is_none() && scale > 0.0 && (applied_total + reaction_total).abs() > 1e-6 * scale {
+        warnings.push(format!("heat supplied and removed differ by {:.3e}", (applied_total + reaction_total).abs()));
+    }
+    let stats = Stats { dofs: s.mesh.layout.dofs(), equations: s.eqs.node.len(), factor_entries: s.factor_entries, lanczos_vectors: 0 };
+    Ok(HeatResult {
+        step: step.name.clone(),
+        nodes: s.mesh.labels,
+        times: s.times,
+        frames: s.frames,
+        flux,
+        elements,
+        reaction,
+        applied_total,
+        reaction_total,
+        stats,
+        warnings,
+    })
+}
+
+/// The heat flux at the integration points and averaged at the nodes (one extrapolation per
+/// element kind).
+fn fluxes(mesh: &HeatMesh, t: &[f64]) -> Result<(Vec<[f64; 3]>, Vec<ElementFlux>)> {
+    let mut sum = vec![[0.0; 3]; mesh.labels.len()];
+    let mut count = vec![0u32; mesh.labels.len()];
+    let mut out = Vec::with_capacity(mesh.elements.len());
+    let mut extrapolations: Vec<(ElementKind, Extrapolation)> = Vec::new();
+    for e in &mesh.elements {
+        let kernel = e.kernel()?;
+        let at = |err| Error::element(e.label, err);
+        let te: Vec<f64> = e.nodes.iter().map(|n| t.get(*n).copied().unwrap_or(0.0)).collect();
+        let points = kernel.flux(e.conductivity, &te).map_err(at)?;
+        let padded: Vec<[f64; 6]> = points.iter().map(|p| [p.flux[0], p.flux[1], p.flux[2], 0.0, 0.0, 0.0]).collect();
+        let index = match extrapolations.iter().position(|(k, _)| *k == e.kind) {
+            Some(i) => i,
+            None => {
+                extrapolations.push((e.kind, extrapolation(kernel.ty, kernel.ty.full).map_err(at)?));
+                extrapolations.len() - 1
+            }
+        };
+        let nodal = extrapolations.get(index).map(|(_, x)| x.apply(&padded)).unwrap_or_default();
+        for (n, v) in e.nodes.iter().zip(&nodal) {
+            if let (Some(s), Some(c)) = (sum.get_mut(*n), count.get_mut(*n)) {
+                for k in 0..3 {
+                    s[k] += v[k];
+                }
+                *c += 1;
+            }
+        }
+        out.push(ElementFlux { label: e.label, kind: e.kind, points });
+    }
+    let flux = sum.iter().zip(&count).map(|(s, c)| s.map(|v| if *c > 0 { v / f64::from(*c) } else { 0.0 })).collect();
+    Ok((flux, out))
+}

crates/solver/src/lib.rs

@@ -27,6 +27,7 @@ pub mod cholesky;
 pub mod constraints;
 pub mod discrete;
 pub mod eigen;
+pub mod heat;
 pub mod ordering;
 pub mod sparse;
 pub mod system;
@@ -35,6 +36,7 @@ pub use analysis::{
     ElementStress, FrequencyOptions, FrequencyResult, Location, MassKind, Mode, PointResult, StaticResult, Stats, StepResult, solve_frequency,
     solve_static,
 };
+pub use heat::{HeatResult, solve_heat};
 
 #[derive(Debug, Clone, PartialEq, thiserror::Error)]
 pub enum Error {

crates/solver/src/system.rs

@@ -9,7 +9,7 @@
 
 use modelcraft_fem::beam::{Beam, BeamSection};
 use modelcraft_fem::dense::Matrix;
-use modelcraft_fem::element::{Formulation, Solid};
+use modelcraft_fem::element::{Formulation, Solid, ThermalStrain};
 use modelcraft_fem::plane::{Plane, Theory};
 use modelcraft_fem::shell::{Shell, ShellSection};
 use modelcraft_fem::truss::{Truss, young_modulus};
@@ -35,6 +35,9 @@ pub struct Elem {
     pub density: Option<f64>,
     /// The 2D theory of an element of a 2D section; `None` for solids.
     pub theory: Option<Theory>,
+    /// The expansion coefficients along X, Y, Z and the reference temperature, when the
+    /// material expands.
+    pub expansion: Option<([f64; 3], f64)>,
 }
 
 /// An element's kernel: a solid or a 2D element.
@@ -90,11 +93,20 @@ impl Kernel<'_> {
         }
     }
 
-    /// Integration-point stresses for nodal displacements given as 3-vectors.
-    pub fn stresses(&self, d: &Elasticity, u: &[[f64; 3]]) -> modelcraft_fem::Result<Vec<PointStress>> {
+    /// Integration-point stresses for nodal displacements given as 3-vectors, with a thermal
+    /// strain when given.
+    pub fn stresses_with(&self, d: &Elasticity, u: &[[f64; 3]], thermal: Option<&ThermalStrain>) -> modelcraft_fem::Result<Vec<PointStress>> {
         match self {
-            Kernel::Solid(s) => s.stresses(d, u),
-            Kernel::Plane(p) => p.stresses(d, &u.iter().map(|v| [v[0], v[1]]).collect::<Vec<_>>()),
+            Kernel::Solid(s) => s.stresses_with(d, u, thermal),
+            Kernel::Plane(p) => p.stresses_with(d, &u.iter().map(|v| [v[0], v[1]]).collect::<Vec<_>>(), thermal),
+        }
+    }
+
+    /// The equivalent nodal forces of a thermal strain, as 3-vectors (Z zero for 2D).
+    pub fn thermal_load(&self, d: &Elasticity, thermal: &ThermalStrain) -> modelcraft_fem::Result<Vec<[f64; 3]>> {
+        match self {
+            Kernel::Solid(s) => s.thermal_load(d, thermal),
+            Kernel::Plane(p) => Ok(p.thermal_load(d, thermal)?.into_iter().map(|v| [v[0], v[1], 0.0]).collect()),
         }
     }
 
@@ -121,6 +133,12 @@ impl Elem {
         };
         k.map_err(|e| Error::element(self.label, e))
     }
+
+    /// The element's thermal strain in a temperature field (nodes the field leaves out are at
+    /// the reference temperature); `None` when the material does not expand or nothing changes.
+    pub fn thermal_strain(&self, field: &[Option<f64>]) -> Option<ThermalStrain> {
+        thermal_strain(self.expansion, &self.nodes, field)
+    }
 }
 
 /// A truss element (rod2, rod3) ready for the kernel.
@@ -134,12 +152,29 @@ pub struct Bar {
     pub young: f64,
     pub area: f64,
     pub density: Option<f64>,
+    /// The expansion coefficients along X, Y, Z and the reference temperature, when the
+    /// material expands.
+    pub expansion: Option<([f64; 3], f64)>,
 }
 
 impl Bar {
     pub fn truss(&self) -> Result<Truss<'_>> {
         Truss::new(self.kind, &self.xyz).map_err(|e| Error::element(self.label, e))
     }
+
+    /// The member's thermal strain in a temperature field, as for [`Elem::thermal_strain`].
+    pub fn thermal_strain(&self, field: &[Option<f64>]) -> Option<ThermalStrain> {
+        thermal_strain(self.expansion, &self.nodes, field)
+    }
+}
+
+/// The thermal strain of an element of nodes `nodes` in a temperature field (nodes the field
+/// leaves out are at the reference temperature); `None` when the material does not expand or
+/// nothing changes.
+fn thermal_strain(expansion: Option<([f64; 3], f64)>, nodes: &[usize], field: &[Option<f64>]) -> Option<ThermalStrain> {
+    let (alpha, reference) = expansion?;
+    let delta: Vec<f64> = nodes.iter().map(|n| field.get(*n).copied().flatten().map_or(0.0, |t| t - reference)).collect();
+    delta.iter().any(|d| *d != 0.0).then_some(ThermalStrain { alpha, delta })
 }
 
 /// What a shell or beam element is made of.
@@ -394,7 +429,8 @@ impl Mesh {
                     young_modulus(elastic).map_err(|e| Error::Invalid(format!("material `{}` of section `{name}`: {e}", section.material)))?;
                 for &label in &set.elements {
                     let el = model.element(label).ok_or_else(|| Error::Invalid(format!("element {label} is missing")))?;
-                    trusses.push((label, el.kind, el.nodes.clone(), young, area, material.density));
+                    let expansion = material.thermal.and_then(|t| t.expansion.map(|a| (a.axes(), t.reference_temperature)));
+                    trusses.push((label, el.kind, el.nodes.clone(), young, area, material.density, expansion));
                 }
                 continue;
             }
@@ -406,7 +442,8 @@ impl Mesh {
                     Some(_) => plane::formulation(el.kind, &section.formulation).map(|f| (f, ElementDofs::new(DofSet::PLANE))),
                 }
                 .map_err(|e| Error::element(label, e))?;
-                elements.push((label, el.kind, el.nodes.clone(), formulation, d, material.density, theory, dofs));
+                let expansion = material.thermal.and_then(|t| t.expansion.map(|a| (a.axes(), t.reference_temperature)));
+                elements.push((label, el.kind, el.nodes.clone(), formulation, d, material.density, theory, dofs, expansion));
             }
         }
         if let (Some(_), Some(name)) = (planar, model.couplings().keys().next()) {
@@ -472,15 +509,15 @@ impl Mesh {
             warnings,
             discrete: crate::discrete::Discrete::default(),
         };
-        for (label, kind, nodes, formulation, d, density, theory, _) in elements {
+        for (label, kind, nodes, formulation, d, density, theory, _, expansion) in elements {
             let idx: Vec<usize> = nodes.iter().filter_map(|n| mesh.node(*n)).collect();
             let exyz: Vec<[f64; 3]> = idx.iter().filter_map(|i| mesh.xyz.get(*i).copied()).collect();
-            mesh.elements.push(Elem { label, kind, nodes: idx, xyz: exyz, formulation, d, density, theory });
+            mesh.elements.push(Elem { label, kind, nodes: idx, xyz: exyz, formulation, d, density, theory, expansion });
         }
-        for (label, kind, nodes, young, area, density) in trusses {
+        for (label, kind, nodes, young, area, density, expansion) in trusses {
             let idx: Vec<usize> = nodes.iter().filter_map(|n| mesh.node(*n)).collect();
             let exyz: Vec<[f64; 3]> = idx.iter().filter_map(|i| mesh.xyz.get(*i).copied()).collect();
-            mesh.trusses.push(Bar { label, kind, nodes: idx, xyz: exyz, young, area, density });
+            mesh.trusses.push(Bar { label, kind, nodes: idx, xyz: exyz, young, area, density, expansion });
         }
         for (label, kind, nodes, density, section) in structural {
             let idx: Vec<usize> = nodes.iter().filter_map(|n| mesh.node(*n)).collect();
@@ -713,6 +750,9 @@ pub fn loads(model: &Model, resolved: &Resolved, mesh: &Mesh, step: &Step, activ
                     mesh.discrete.add_gravity(&mut f, *acceleration, mesh);
                 }
             }
+            // Heat loads act in heat transfer; temperature fields through their thermal strain
+            // (add_thermal_loads).
+            Load::SurfaceFlux { .. } | Load::Convection { .. } | Load::HeatSource { .. } | Load::Temperature { .. } => {}
             Load::BodyForce { set, vector } => {
                 for label in resolved.sets.get(set).map(|r| r.elements.as_slice()).unwrap_or_default() {
                     if let Some(t) = mesh.truss(*label) {
@@ -779,6 +819,52 @@ fn structural_body(f: &mut [f64], mesh: &Mesh, e: &Structural, b: [f64; 3]) -> R
     Ok(())
 }
 
+/// The temperature field of a static step's temperature loads, per mesh node; `None` when the
+/// step has none.
+pub fn temperature_loads(model: &Model, resolved: &Resolved, mesh: &Mesh, step: &Step) -> Result<Option<Vec<Option<f64>>>> {
+    let mut field: Option<Vec<Option<f64>>> = None;
+    for name in &step.loads {
+        let Some(Load::Temperature { target, value }) = model.load(name) else { continue };
+        let field = field.get_or_insert_with(|| vec![None; mesh.labels.len()]);
+        for &label in resolved.target_nodes(target) {
+            let Some(n) = mesh.node(label) else { continue };
+            hold(field.get_mut(n), *value, |old| format!("node {label} is given both {old} and {value} (temperature load `{name}` and another)"))?;
+        }
+    }
+    Ok(field)
+}
+
+/// Set a node's value, refusing a different one already there (`conflict` names both).
+pub(crate) fn hold(slot: Option<&mut Option<f64>>, value: f64, conflict: impl FnOnce(f64) -> String) -> Result<()> {
+    let slot = slot.ok_or_else(|| Error::Internal("node index".into()))?;
+    match *slot {
+        Some(old) if old != value => Err(Error::Invalid(conflict(old))),
+        _ => {
+            *slot = Some(value);
+            Ok(())
+        }
+    }
+}
+
+/// Add the equivalent nodal forces of the thermal strains of a temperature field.
+pub fn add_thermal_loads(f: &mut [f64], mesh: &Mesh, field: &[Option<f64>]) -> Result<()> {
+    for e in &mesh.elements {
+        let Some(th) = e.thermal_strain(field) else { continue };
+        let forces = e.kernel()?.thermal_load(&e.d, &th).map_err(|err| Error::element(e.label, err))?;
+        for (node, v) in e.nodes.iter().zip(forces) {
+            add(f, &translations(mesh, *node)?, v);
+        }
+    }
+    for t in &mesh.trusses {
+        let Some(th) = t.thermal_strain(field) else { continue };
+        let forces = t.truss()?.thermal_load(t.young, t.area, &th).map_err(|err| Error::element(t.label, err))?;
+        for (node, v) in t.nodes.iter().zip(forces) {
+            add(f, &translations(mesh, *node)?, v);
+        }
+    }
+    Ok(())
+}
+
 /// The uniform force per volume a step's gravity and body forces put on each shell and beam
 /// element ([`Mesh::structural`] order): beam end forces are recovered with it.
 pub fn structural_body_forces(model: &Model, resolved: &Resolved, mesh: &Mesh, step: &Step) -> Result<Vec<[f64; 3]>> {