porepy.numerics.fv.mpsa module

Implementation of the multi-point stress appoximation method.

The implementation is based on the weakly symmetric version of MPSA, described in

Keilegavlen, Nordbotten: Finite volume methods for elasticity with weak symmetry,
IJNME, 2017.

class Mpsa(keyword)[source]

Bases: Discretization

Implementation of the Multi-point stress approximation.

Parameters: keyword (str) –

keyword

Keyword used to identify the parameter dictionary. Defaults to “mechanics”.

Type: str

stress_matrix_key

Keyword used to identify the discretization matrix for the stress. Defaults to “stress”.

Type: str

bound_stress_matrix_key

Keyword used to identify the discretization matrix for the boundary conditions for stress. Defaults to “bound_stress”.

Type: str

bound_displacement_cell_matrix_key

Keyword used to identify the discretization matrix for the cell center displacement contribution to boundary displacement reconstrution. Defaults to “bound_displacement_cell”.

Type: str

bound_displacement_face_matrix_key

Keyword used to identify the discretization matrix for the boundary conditions’ contribution to boundary displacement reconstrution. Defaults to “bound_displacement_face”.

Type: str

assemble_matrix(sd, data)[source]

Return the matrix for a discretization of a second order elliptic vector equation using a FV method.

The name of data in the input dictionary (data) are: fourth_order_tensor: FourthOrderTensor stiffness tensor defined cell-wise. bc: boundary conditions, pp.BoundaryConditionVectorial. bc_values: (sd.dim * sd.num_faces) Values of the boundary conditions.

Parameters

g (Grid) – Computational grid, with geometry fields computed.
data (dictionary) – With data stored.
sd (Grid) –

Returns

System matrix of this discretization. The: size of the matrix will depend on the specific discretization.

Return type

scipy.sparse.csr_matrix

assemble_matrix_rhs(sd, data)[source]

Return the matrix and right-hand side for a discretization of a second order elliptic equation using a FV method with a multi-point stress approximation.

Parameters

g : grid, or a subclass, with geometry fields computed. data: dictionary to store the data. For details on necessary keywords,

see method discretize()

discretize (boolean, optional): default True. Whether to discetize: prior to matrix assembly. If False, data should already contain discretization.

Return

matrix: sparse csr (sd.dim * g_num_cells, sd.dim * g_num_cells): Discretization matrix.
rhs: array (sd.dim * g_num_cells): Right-hand side which contains the boundary conditions and the vector source term.

Parameters

sd (Grid) –
data (dict) –

Return type

tuple[scipy.sparse._base.spmatrix, numpy.ndarray]

assemble_rhs(sd, data)[source]

Return the right-hand side for a discretization of a second order elliptic equation using a finite volume method.

Also discretize the necessary operators if the data dictionary does not contain a discretization of the boundary equation.

Parameters

g (Grid) – Computational grid, with geometry fields computed.
data (dictionary) – With data stored.
sd (Grid) –

Returns

Right hand side vector with representation of boundary: conditions. The size of the vector will depend on the discretization.

Return type

np.ndarray

discretize(sd, data)[source]

Discretize the second order vector elliptic equation using multi-point stress approximation.

The method computes traction over faces in terms of cell center displacements.

It is possible to do a partial discretization via parameters specified_cells, _nodes and _faces. This is considered an advanced, and only partly tested option.

We assume the following two sub-dictionaries to be present in the data dictionary:

parameter_dictionary, storing all parameters.
Stored in data[pp.PARAMETERS][self.keyword].

matrix_dictionary, for storage of discretization matrices.
Stored in data[pp.DISCRETIZATION_MATRICES][self.keyword]

parameter_dictionary contains the entries:

fourth_order_tensor: (pp.FourthOrderTensor) Stiffness tensor defined cell-wise. bc: (BoundaryConditionVectorial) boundary conditions mpsa_eta: (float/np.ndarray) Optional. Range [0, 1). Location of

displacement continuity point: eta. eta = 0 gives cont. pt. at face midpoint, eta = 1 at the vertex. If not given, porepy tries to set an optimal value. If a float is given this value is set to all subfaces, except the boundary (where, 0 is used). If eta is a np.ndarray its size should equal SubcellTopology(sd).num_subfno.

matrix_dictionary will be updated with the following entries:

stress: sps.csc_matrix (sd.dim * sd.num_faces, sd.dim * sd.num_cells): stress discretization, cell center contribution
bound_flux: sps.csc_matrix (sd.dim * sd.num_faces, sd.dim * sd.num_faces): stress discretization, face contribution
bound_displacement_cell: sps.csc_matrix (sd.dim * sd.num_faces,: sd.dim * sd.num_cells)

Operator for reconstructing the displacement trace. Cell center contribution.
bound_displacement_face: sps.csc_matrix (sd.dim * sd.num_faces,: sd.dim * sd.num_faces)

Operator for reconstructing the displacement trace. Face contribution.

Parameters

sd (pp.Grid): grid, or a subclass, with geometry fields computed. data (dict): For entries, see above.

Parameters

sd (Grid) –
data (dict) –

Return type

None

extract_displacement(sd, solution_array, d)[source]

Extract the displacement part of a solution.

Parameters

g (grid) – To which the solution array belongs.
solution_array (np.array) – Solution for this grid.
d (dictionary) – Data dictionary associated with the grid. Not used, but included for consistency reasons.
sd (Grid) –

Returns

Displacement solution vector. Will be identical: to solution_array.

Return type

np.array (sd.num_cells)

extract_stress(sd, solution_array, d)[source]

Extract the stress corresponding to a solution

The stress is composed of contributions from the solution variable and the boundary conditions.

Parameters

g (grid) – To which the solution array belongs.
solution_array (np.array) – Solution for this grid.
d (dictionary) – Data dictionary associated with the grid.
sd (Grid) –

Returns

Vector of stresses on the grid faces.

Return type

np.array (sd.num_cells)

ndof(sd)[source]

Return the number of degrees of freedom associated to the method. In this case number of cells times dimension (stress dof).

Parameter

sd: grid, or a subclass.

Return

dof: the number of degrees of freedom.

Parameters: sd (Grid) –
Return type: int

update_discretization(sd, data)[source]

Update discretization.

The updates can generally come as a combination of two forms:

The discretization on part of the grid should be recomputed.
The old discretization can be used (in parts of the grid), but the numbering of unknowns has changed, and the discretization should be reorder accordingly.

Information on the basis for the update should be stored in a field

data[‘update_discretization’]

This should be a dictionary which could contain keys:

modified_cells, modified_faces

define cells, faces and nodes that have been modified (either parameters, geometry or topology), and should be rediscretized. It is up to the discretization method to implement the change necessary by this modification. Note that depending on the computational stencil of the discretization method, a grid quantity may be rediscretized even if it is not marked as modified.

The dictionary data[‘update_discretization’] should further have keys:

cell_index_map, face_index_map

these should specify sparse matrices that maps old to new indices. If not provided, the cell and face bookkeeping will be assumed constant.

It is up to the caller to specify which parts of the grid to recompute, and how to update the numbering of degrees of freedom. If the discretization method does not provide a tailored implementation for update, it is not necessary to provide this information.

Parameters

g (pp.Grid) – Grid to be rediscretized.
data (dictionary) – With discretization parameters.
sd (Grid) –

Return type

None