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Isotropic Conductivity 3D

Bases: HeatMaterial

Isotropic heat conductivity material in 3D.

Parameters:

  • kappa (Tensor | float) –

    Thermal conductivity. 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.
Definition of conductivity tensor

This class represents a 3D isotropic heat conductivity material, defined by the (batched) thermal conductivity \(\kappa\). The constructor derives the (batched) conductivity tensor $$ \pmb{\kappa} = \kappa \mathbf{I} $$ which relates the heat flux to the temperature gradient via Fourier's law \(\mathbf{q} = -\pmb{\kappa} \cdot \nabla T\).

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).