flow.solve_temperature

flow.solve_temperature(
    chi,
    velocity,
    config,
    *,
    density=1.0,
    wall=None,
    source=0.0,
    fixed_mask=None,
    fixed_value=0.0,
)

Solve the conjugate energy equation on the lattice.

Differentiable in chi, velocity, source and fixed_value: the linear solve carries :func:jax.lax.custom_linear_solve’s implicit adjoint, so reverse mode is one transposed solve rather than a tape through the Krylov iterations. That is what lets this compose with the momentum solve’s fixed-point adjoint without either knowing the other exists.

Parameters

Name	Type	Description	Default
chi	jax.Array	Solid fraction, `(NX, NY, NZ)`.	required
velocity	jax.Array	Velocity in lattice units, `(3, NX, NY, NZ)`. Pass zeros for a pure-conduction solve.	required
config	EnergyConfig	The energy configuration.	required
density	Any	Density, `(NX, NY, NZ)` or a scalar. Multiplied into `velocity` to form the convective flux; the default 1.0 is the incompressible reading, and a solve fed a lattice Boltzmann field should pass that field’s own density so the flux is the one the momentum solve actually conserves (see the module docstring).	`1.0`
wall	np.ndarray \| None	Boolean duct-wall mask; `None` builds :func:`~cadjoint.flow.lbm.duct_walls` for the shape.	`None`
source	Any	Volumetric power per cell — a scalar or an `(NX, NY, NZ)` array.	`0.0`
fixed_mask	np.ndarray \| None	Cells held at a prescribed temperature (concrete boolean array), or `None` for none.	`None`
fixed_value	Any	The prescribed temperatures, scalar or `(NX, NY, NZ)`.	`0.0`

Returns

Name	Type	Description
	jax.Array	`(NX, NY, NZ)` temperature, zero on duct-wall cells (which the
	jax.Array	system does not solve for), or all-`NaN` when `velocity` is not
	jax.Array	finite – see the note in the body on why that is better than the
	jax.Array	zeros GMRES would otherwise return.

Raises

Name	Type	Description
	ValueError	If `chi` or `velocity` does not match the configured shape.