porepy.numerics.ad.discretizations module

For any discretization class compatible with PorePy, wrap_discretization associates a discretization with all attributes of the class’ attributes that end with ‘_matrix_key’.

Example

# Generate grid >>> g = pp.CartGrid([2, 2]) # Associate an Ad representation of an Mpfa method, aimed this grid >>> discr = MpfaAd(keyword=’flow’, grids=[g]) # The flux discretization of Mpfa can now be accesed by >>> discr.flux # While the discretization of boundary conditions is available by >>> discr.bound_flux.

The representation of different discretization objects can be combined with other Ad objects into an operator tree, using lazy evaluation.

It is assumed that the actual action of discretization (creation of the discretization matrices) is performed before the operator tree is parsed.

class BiotAd(keyword, subdomains, flow_keyword='flow')[source]

Bases: Discretization

Ad wrapper around the Biot discretization class.

For description of the method, we refer to the standard Biot class.

Parameters

keyword (str) –
subdomains (List[pp.Grid]) –
flow_keyword (str) –

class BiotStabilizationAd(keyword, subdomains)[source]

Bases: Discretization

Parameters

keyword (str) –
subdomains (List[pp.Grid]) –

class ColoumbContactAd(keyword, interfaces)[source]

Bases: Discretization

Parameters

keyword (str) –
interfaces (List[pp.MortarGrid]) –

class ContactTractionAd(keyword, interfaces, low_dim_subdomains)[source]

Bases: Discretization

Parameters

keyword (str) –
interfaces (List[pp.MortarGrid]) –
low_dim_subdomains (List[pp.Grid]) –

class DifferentiableFVAd(subdomains, mdg, base_discr, system_manager, permeability_function, permeability_argument, potential, keyword)[source]

Bases: object

This class represents the application of the product and chain rule of the flux expression

q = T(k(u)) * p

Where the transmissibility matrix T is a function of the cell permeability k, which again is a function of a primary variable u, while p is the potential (pressure). The chain rule applied to this expression reads

dq = p * dT/dk * dk/du * du + T * dp

The transmissibility matrix can be computed from a Tpfa or Mpfa discretization, or (in principle) any other finite volume method. The derivative of the transmissibilities, dT/dk, is approximated with a two-point flux stencil.

If vector sources are included, p should be replaced by (p - dist * vector_source), with dist the distance roughly corresponding to the inverse of the gradient.

Parameters

subdomains (List[pp.Grid]) –
mdg (pp.MixedDimensionalGrid) –
base_discr (Union[pp.ad.MpfaAd, pp.ad.TpfaAd]) –
system_manager (pp.ad.EquationSystem) –
permeability_function (Callable[[pp.ad.Variable], pp.ad.Ad_array]) –
permeability_argument (pp.ad.Variable) –
potential (pp.ad.Variable) –
keyword (str) –

bound_pressure_face()[source]

Boundary face contribution to pressure reconstruction.

Returns

pp.ad.Operator representing the value and jacobian of bound_pressure_face,: which is used to reconstruct the pressure trace on the boundary (see FV discretizations).

Return type

Operator

flux()[source]

Flux from potential, BCs and vector sources.

Returns: pp.ad.Operator representing the flux.
Return type: Operator

class Discretization[source]

Bases: ABC

General/utility methods for AD discretization classes.

The init of the children classes below typically calls wrap_discretization and has arguments including subdomains or interfaces and keywords for parameter and possibly matrix storage.

class DivUAd(keyword, subdomains, mat_dict_keyword)[source]