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Orthotropic Conductivity 3D

Bases: IsotropicConductivity3D

Orthotropic heat conductivity material in 3D.

Parameters:

  • kappa_1 (Tensor | float) –

    Conductivity along local axis 1.

  • kappa_2 (Tensor | float) –

    Conductivity along local axis 2.

  • kappa_3 (Tensor | float) –

    Conductivity along local axis 3.

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

    Mass density. Default is 1.0.

Notes
  • No internal state variables (n_state = 0).
  • Supports batched/vectorized material parameters.
  • Supports rotation of the material coordinate system via rotate().
Orthotropic conductivity tensor

The principal conductivities are aligned with the material axes, so the conductivity tensor is diagonal in the material frame $$ \pmb{\kappa} = \begin{bmatrix} \kappa_1 & 0 & 0 \cr 0 & \kappa_2 & 0 \cr 0 & 0 & \kappa_3 \end{bmatrix} = \sum_{i=1}^{3} \kappa_i \, \mathbf{e}_i \otimes \mathbf{e}_i $$ with the material axes \(\mathbf{e}_i\). rotate() maps it into the global frame as \(\kappa_{ij} \mapsto R_{ik} R_{jl} \kappa_{kl}\).

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).

rotate(R)

Returns a copy with its conductivity tensor rotated by R.