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Orthotropic Conductivity 2D

Bases: IsotropicConductivity2D

Orthotropic heat conductivity material in 2D.

Parameters:

  • kappa_1 (Tensor | float) –

    Conductivity along local axis 1.

  • kappa_2 (Tensor | float) –

    Conductivity along local axis 2.

  • 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 plane conductivity tensor

The two principal in-plane conductivities are aligned with the material axes, so the conductivity tensor is diagonal in the material frame $$ \pmb{\kappa} = \begin{bmatrix} \kappa_1 & 0 \cr 0 & \kappa_2 \end{bmatrix} = \sum_{i=1}^{2} \kappa_i \, \mathbf{e}_i \otimes \mathbf{e}_i $$ 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.