Projection Operators

xpektra.projection_operator.GalerkinProjection

A 'final' class implementing the material-independent Galerkin projection.

This implementation is 'matrix-free'. It does not materialize the full 4th-order Ghat tensor. Instead, it stores the gradient operator (Dξs) and its inverse (Dξ_inv) and computes the projection on the fly. This saves a massive amount of memory.

Parameters:

Name	Type	Description	Default
scheme	DiagonalScheme	The differential scheme providing the gradient operator.	required

Returns:

Type	Description
	The Galerkin projection operator.

Example

projection = GalerkinProjection(scheme) eps_hat = projection.project(field_hat)

project(field_hat: jaxlib._jax.Array) -> jaxlib._jax.Array

Applies the projection on the fly. This is the core of the class.

Computes: eps_hat = [δ_im * Dξ_j * Dξ_inv_l] * sigma_hat_lm

Parameters:

Name	Type	Description	Default
field_hat	Array	The input field in Fourier space, shape (..., dim, dim).	required

Returns: The projected field in Fourier space, shape (..., dim, dim).

The formula is given by:

\[\hat{G}_{ijlm} = \delta_{im} * D\xi_{j} * D\xi^{-1}_{l}\]

where \(\delta_{im}\) is the Kronecker delta, \(D\xi_{j}\) is the gradient operator and \(D\xi^{-1}_{l}\) is the inverse of the gradient operator.

xpektra.projection_operator.MoulinecSuquetProjection

A class implementing the Moulinec-Suquet (MS) Green's operator, which depends on a homogeneous isotropic reference material.

Parameters:

Name	Type	Description	Default
space	SpectralSpace	The spectral space defining the grid and transform.	required
lambda0	Array	The first Lamé parameter of the reference material.	required
mu0	Array	The second Lamé parameter (shear modulus) of the reference material.	required

Returns: The Moulinec-Suquet projection operator.

Example

Ghat = MoulinecSuquetProjection(space, lambda0, mu0).compute_operator()

compute_operator() -> jaxlib._jax.Array

Implements the Moulinec-Suquet projection operator.

Returns:

Type	Description
Array	The Green's operator Ghat, shape (..., dim, dim, dim, dim).