Module steady_conduction 

Layer 5 (small solver) — steady heat conduction with a volumetric source, the steady form of laplacianFoam, with radial and voxel meshers, a closed-form verification reference and an energy balance (gh:#949).

Steady heat conduction with a volumetric source (gh:#949)

A small steady conduction solver: the steady form of OpenFOAM’s laplacianFoam,

  -div( k(T) grad T ) = q'''          in the solid

with a per-zone uniform heat source q''' (W/m^3), a per-zone conductivity k(T) (W/(m K)) supplied by the caller, and one thermal condition per boundary patch: adiabatic, fixed temperature, or convective (-k dT/dn = h (T - T_ambient)).

It exists for the beginner thermal profile of outram-mc-libs (gh:#949, under #945): a coarse temperature field for any shape, with an energy balance (heat generated against heat leaving through the boundary) reported beside it. Serious thermal-hydraulics is nee_soon’s.

What is ported, and what is new

piece	source
the equation, fvm::laplacian(k, T) == source	laplacianFoam.C (TEqn), run steady (ddt dropped, as with ddtSchemes steadyState)
the convective boundary as a mixed (Robin) condition, valueFraction = h/(h + k/d), refValue = T_a	externalWallHeatFluxTemperatureFvPatchScalarField.C (hp without radiation or solid layers)
k(T) re-evaluated between outer passes (Picard)	the outer simple.loop() of laplacianFoam, here with k updated each pass (upstream’s DT is constant)
face conductivity as series resistance, A / (d_P/k_P + d_N/k_N)	OpenFOAM’s harmonic interpolation (schemes/harmonic/harmonic.H: 1/reverseLinear(1/k)), chosen over the default Gauss linear, which averages k linearly and is wrong at a material interface
energy balance from the boundary condition definitions, not from the matrix	new here
radial (slab / cylinder / sphere) and voxel meshers	new here (mesh)

The energy balance, and what it can catch

The heat leaving each boundary face is recomputed from that face’s own condition (h, T_a, k of its cell, the face distance) and the solved cell temperature, independently of the assembled matrix. In a correct finite-volume solve the interior fluxes cancel pairwise, so generated minus removed equals the sum of the linear-solver residuals: it is small by construction. Its job is to catch an assembly that drops or doubles a boundary contribution (a zero boundary conductivity was such a defect elsewhere in this workspace; see outram-foam-appbuilder-lib’s thermo_mechanics::mesh_solve) and an unconverged solve. It does not measure discretisation error; the closed-form tests in this module do.

Verification

closed_form is the exact solution for uniform q''' in nested slab, cylinder and sphere shells with a fixed-temperature or convective surface. The tests in tests.rs compare the solve against it and measure the convergence order; the methodology and results are in their doc comments.

Status: untrusted AI-generated draft (2026-10-17) per RESPONSIBLE_USE.md. For research, education and V&V only.

Re-exports

pub use closed_form::RadialClosedForm;

pub use closed_form::RadialLayer;

pub use mesh::radial_mesh;

pub use mesh::voxel_mesh;

pub use mesh::ConductionMesh;

pub use mesh::RadialGeometry;

pub use mesh::RadialShell;

pub use mesh::VoxelGrid;

pub use mesh::VoxelMesh;

Modules

closed_form

Exact steady temperature for uniform heat generation in nested slab, cylinder and sphere shells: the verification reference for the finite-volume solve (gh:#949).

mesh

Meshes for the steady conduction solver (gh:#949).

Structs

EnergyBalance

Heat generated in the body against heat leaving through its boundary.

PatchCondition

A ThermalBoundary applied to one mesh patch, with an optional per-face film-area fraction.

SteadyConductionProblem

A steady conduction problem: a mesh, the heat source of each zone and the condition on each patch.

SteadyConductionSettings

Iteration controls. The defaults converge the energy balance to about 1e-10 of the generated power on the meshes in this module’s tests.

SteadyConductionSolution

The solved steady temperature and its energy balance.

Enums

SteadyConductionError

What can go wrong setting up or solving a steady conduction problem.

ThermalBoundary

The thermal condition on one boundary patch.