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Isotropic Conductivity 1D

Bases: IsotropicConductivity3D

Isotropic heat conductivity material in 1D.

Parameters:

  • kappa (Tensor | float) –

    Thermal conductivity along the bar. If a float is provided, it is converted. Shape: () for a scalar or (N,) for a batch of materials.

  • rho (Tensor | float, default: 1.0 ) –

    Mass density. If a float is provided, it is converted. Shape: () for a scalar or (N,) for a batch of materials.

Notes
  • No internal state variables (n_state = 0).
  • Supports batched/vectorized material parameters.
  • The inherited step method operates on thermal tensors with shapes (..., 1, 1) and returns an algorithmic tangent of shape (..., 1, 1).
Definition of axial conductivity

Same constitutive law as IsotropicConductivity3D, reduced to the single component along the bar, $$ q = -\kappa \frac{\partial T}{\partial s}. $$ Heat conduction transverse to the bar is not modelled.

vectorize(n_elem)

Returns the material batched over n_elem elements.

Parameters:

  • n_elem (int) –

    Number of elements to vectorize the material for.

Returns:

  • Material ( T ) –

    A material of the same type carrying one entry per element, or itself if it is vectorized already.

step(grad_inc, grad, flux, state, cl, iter)

Performs an incremental step in the isotropic heat conduction model.

Fourier's law, \(\Delta \mathbf{q} = -\pmb{\kappa} \cdot \Delta \nabla T\), with a constant conductivity.

Parameters:

  • grad_inc (Tensor) –

    Incremental temperature gradient. Shape: (..., 1, 3), where ... represents batch dimensions.

  • grad (Tensor) –

    Current temperature gradient. Unused. Shape: (..., 1, 3), same as grad_inc.

  • flux (Tensor) –

    Current heat flux. Shape: (..., 1, 3).

  • state (Tensor) –

    Internal state variables (unused in heat conductivity). Shape: Arbitrary, remains unchanged.

  • cl (Tensor) –

    Characteristic lengths. Shape: (...).

  • iter (int) –

    Current iteration number.

Returns:

  • flux_new ( Tensor ) –

    Updated heat flux. Shape: (..., 1, 3).

  • state_new ( Tensor ) –

    Updated internal state (unchanged). Shape: same as state.

  • dqdg ( Tensor ) –

    Algorithmic tangent conductivity. Shape: (..., 3, 3).