pygeon.discretizations.vem.vec_h1 module
Module for the discretizations of the H1 space.
- class pygeon.discretizations.vem.vec_h1.VecVLagrange1(keyword='unitary_data')[source]
Bases:
VecDiscretizationVector Lagrange virtual element discretization for H1 space in 2d.
This class represents a virtual element discretization for the H1 space using vector virtual Lagrange elements. It provides methods for assembling various matrices and operators, such as the mass matrix, divergence matrix, symmetric gradient matrix, and more.
Convention for the ordering is first all the x then all the y.
The stress tensor and strain tensor are represented as vectors unrolled row-wise. In 2D, the stress tensor has a length of 4.
We are considering the following structure of the stress tensor in 2D:
\[\begin{split}\sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{bmatrix}\end{split}\]which is represented in the code unrolled row-wise as a vector of length 4:
\[\sigma = [\sigma_{xx}, \sigma_{xy}, \sigma_{yx}, \sigma_{yy}]\]The strain tensor follows the same approach.
- poly_order = 1
Polynomial degree of the basis functions
- tensor_order = 1
Vector-valued discretization
- __init__(keyword='unitary_data')[source]
Initialize the vector discretization class. The base discretization class is pg.Lagrange1.
- Parameters:
keyword (str) – The keyword for the vector discretization class. Default is pg.UNITARY_DATA.
- Returns:
None
- assemble_div_matrix(sd)[source]
Returns the div matrix operator for the lowest order vector Lagrange element
- Parameters:
sd (pg.Grid) – The grid object.
- Returns:
The div matrix obtained from the discretization.
- Return type:
sps.csc_array
- local_div(sd, cell, diam, nodes)[source]
Compute the local div matrix for vector P1.
- assemble_div_div_matrix(sd, data=None)[source]
Returns the div-div matrix operator for the lowest order vector Lagrange element. The matrix is multiplied by the Lame’ parameter lambda.
- Parameters:
sd (pg.Grid) – The grid object.
data (dict | None) – Additional data, the Lame’ parameter lambda. Defaults to None.
- Returns:
Sparse (sd.num_nodes, sd.num_nodes) Div-div matrix obtained from the discretization.
- Return type:
sps.csc_array
- assemble_symgrad_matrix(sd)[source]
Returns the symmetric gradient matrix operator for the lowest order vector Lagrange element
- Parameters:
sd (pg.Grid) – The grid object representing the domain.
- Returns:
The sparse symmetric gradient matrix operator.
- Return type:
sps.csc_array
Notes
If a 0-dimensional grid is given, a zero matrix is returned.
The method maps the domain to a reference geometry.
The method allocates data to store matrix entries efficiently.
The symmetrization matrix is constructed differently for 2D and 3D cases.
The method computes the symgrad local matrix for each cell and saves the values in the global structure.
Finally, the method constructs the global matrices using the saved values.
- local_symgrad(sd, cell, diam, nodes, sym)[source]
Compute the local symgrad matrix for vector virtual Lagrangian.
- Parameters:
- Returns:
Local symmetric gradient matrix.
- Return type:
np.ndarray
- assemble_symgrad_symgrad_matrix(sd, data=None)[source]
Returns the symgrad-symgrad matrix operator for the lowest order vector Lagrange element. The matrix is multiplied by twice the Lame’ parameter mu.
- Parameters:
sd (pg.Grid) – The grid.
data (dict | None) – Additional data, the Lame’ parameter mu. Defaults to None.
- Returns:
Sparse symgrad-symgrad matrix of shape (sd.num_nodes, sd.num_nodes). The matrix obtained from the discretization.
- Return type:
sps.csc_array
Notes
Duplicate of pg.VecLagrange1.assemble_symgrad_symgrad_matrix
- assemble_penalisation_matrix(sd, data=None)[source]
Assembles and returns the penalisation matrix.
- Parameters:
sd (pg.Grid) – The grid.
data (dict | None) – Optional data for the assembly process.
- Returns:
The penalisation matrix obtained from the discretization.
- Return type:
sps.csc_array
- assemble_loc_penalisation_matrix(sd, cell, diam, nodes)[source]
Computes the local penalisation VEM matrix on a given cell according to the Hitchhiker’s (6.5)
- Parameters:
- Returns:
The computed local VEM mass matrix.
- Return type:
np.ndarray
- assemble_diff_matrix(sd)[source]
Assembles the matrix corresponding to the differential operator.
- Parameters:
sd (pg.Grid) – Grid object or a subclass.
- Returns:
The differential matrix.
- Return type:
sps.csc_array
Notes
Duplicate of pg.VecLagrange1.assemble_diff_matrix
- assemble_stiff_matrix(sd, data=None)[source]
Assembles the global stiffness matrix for the finite element method.
- Parameters:
sd (pg.Grid) – The grid on which the finite element method is defined.
data (dict | None) – Additional data required for the assembly process.
- Returns:
The assembled global stiffness matrix.
- Return type:
sps.csc_array
- get_range_discr_class(dim)[source]
Returns the discretization class that contains the range of the differential.
- Parameters:
dim (int) – The dimension of the range.
- Returns:
The discretization class that contains the range of the differential.
- Return type:
Discretization
- Raises:
NotImplementedError – There is no range discretization for the vector Lagrangian 1 in PyGeoN.
- compute_stress(sd, u, data)[source]
Compute the stress tensor for a given displacement field.
- Parameters:
sd (pg.Grid) – The spatial discretization object.
u (ndarray) – The displacement field.
data (dict) – Data for the computation including the Lame parameters accessed with the keys pg.LAME_LAMBDA and pg.LAME_MU. Both float and np.ndarray are accepted.
- Returns:
The stress tensor.
- Return type:
ndarray
Notes
Duplicate of pg.VecLagrange1.compute_stress