pygeon.discretizations.fem.h1 module

Module for the discretizations of the H1 space.

class pygeon.discretizations.fem.h1.Lagrange1(keyword='unitary_data')

Bases: Discretization

Class representing the Lagrange1 finite element discretization.

poly_order = 1

Polynomial degree of the basis functions

tensor_order = 0

Scalar-valued discretization

ndof(sd)

Returns the number of degrees of freedom associated to the method. In this case, the number of nodes.

Parameters:

sd – Grid, or a subclass.

Returns:

The number of degrees of freedom.

Return type:

ndof

assemble_mass_matrix(sd, data=None)

Returns the mass matrix for the lowest order Lagrange element

Parameters:

• sd (pg.Grid) – The grid.

• data (dict | None) – Optional data for the assembly process.

Returns:

The mass matrix obtained from the discretization.

Return type:

sps.csc_array

assemble_local_mass(dim)

Compute the local mass matrix on an element with measure 1.

Parameters:

dim (int) – Dimension of the matrix.

Returns:

Local mass matrix of shape (num_nodes_of_cell, num_nodes_of_cell).

Return type:

np.ndarray

assemble_stiff_matrix(sd, data=None)

Assembles the stiffness matrix for the finite element method.

Parameters:

• sd (pg.Grid) – The grid object representing the discretization.

• data (dict) – A dictionary containing the necessary data for assembling the matrix.

Returns:

The assembled stiffness matrix.

Return type:

sps.csc_array

assemble_diff_matrix(sd)

Assembles the differential matrix based on the dimension of the grid.

Parameters:

sd (pg.Grid) – The grid object.

Returns:

The differential matrix.

Return type:

sps.csc_array

local_stiff(K, c_volume, coord, dim)

Compute the local stiffness matrix for P1.

Parameters:

• K (np.ndarray) – Permeability of the cell of (dim, dim) shape.

• c_volume (np.ndarray) – Scalar cell volume.

• coord (np.ndarray) – Coordinates of the cell vertices of (dim+1, dim) shape.

• dim (int) – Dimension of the problem.

Returns:

Local stiffness matrix of (dim+1, dim+1) shape.

Return type:

np.ndarray

static local_grads(coord, dim)

Calculate the local gradients of the finite element basis functions.

Parameters:

• coord (np.ndarray) – The coordinates of the nodes in the element.

• dim (int) – The dimension of the element.

Returns:

The local gradients of the finite element basis functions.

Return type:

np.ndarray

assemble_lumped_matrix(sd, data=None)

Assembles the lumped mass matrix for the finite element method.

Parameters:

• sd (pg.Grid) – The grid object representing the discretization.

• data (dict | None) – Optional data dictionary.

Returns:

The assembled lumped mass matrix.

Return type:

sps.csc_array

proj_to_PwPolynomials(sd)

Construct the matrix for projecting a Lagrangian function to a piecewise linear function.

Parameters:

sd (pg.Grid) – The grid on which to construct the matrix.

Returns:

The matrix representing the projection.

Return type:

sps.csc_array

eval_at_cell_centers(sd)

Construct the matrix for evaluating a Lagrangian function at the cell centers of the given grid.

Parameters:

sd (pg.Grid) – The grid on which to construct the matrix.

Returns:

The matrix representing the projection at the cell centers.

Return type:

sps.csc_array

interpolate(sd, func)

Interpolates a given function over the nodes of a grid.

Parameters:

• sd (pg.Grid) – The grid on which to interpolate the function.

• func (Callable[[np.ndarray], np.ndarray]) – The function to be interpolated.

Returns:

An array containing the interpolated values at each node of the grid.

Return type:

np.ndarray

assemble_nat_bc(sd, func, b_faces)

Assembles the ‘natural’ boundary condition (u, func)_Gamma with u a test function in Lagrange1

Parameters:

• sd (pg.Grid) – The grid object representing the computational domain.

• func (Callable[[np.ndarray], np.ndarray]) – The function used to evaluate the ‘natural’ boundary condition.

• b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled ‘natural’ boundary condition values.

Return type:

np.ndarray

get_range_discr_class(dim)

Returns the appropriate range discretization class based on the dimension.

Parameters:

dim (int) – The dimension of the problem.

Returns:

The range discretization class.

Return type:

object

Raises:

NotImplementedError – If there’s no zero discretization in PyGeoN.

class pygeon.discretizations.fem.h1.Lagrange2(keyword='unitary_data')

Bases: Discretization

Class representing the Lagrange2 finite element discretization.

poly_order = 2

Polynomial degree of the basis functions

tensor_order = 0

Scalar-valued discretization

ndof(sd)

Returns the number of degrees of freedom associated to the method. In this case, the number of nodes plus the number of edges, where edges are one-dimensional mesh entities.

Parameters:

sd – Grid, or a subclass.

Returns:

The number of degrees of freedom.

Return type:

ndof

assemble_mass_matrix(sd, data=None)

Returns the mass matrix for the second order Lagrange element

Parameters:

• sd (pg.Grid) – The grid.

• data (dict | None) – Optional data for the assembly process.

Returns:

The mass matrix.

Return type:

sps.csc_array

assemble_local_mass(dim)

Computes the local mass matrix of the basis functions on a d-simplex with measure 1.

Parameters:

dim (int) – The dimension of the simplex.

Returns:

The local mass matrix.

Return type:

np.ndarray

assemble_barycentric_mass(expnts)

Compute the inner products of all monomials up to degree 2

Parameters:

expnts (np.ndarray) – Each column is an array of exponents alpha_i of the monomial expressed as prod_i lambda_i ^ alpha_i.

Returns:

The inner products of the monomials on a simplex with measure 1.

Return type:

np.ndarray

integrate_monomial(alphas)

Exact integration of products of monomials based on Vermolen and Segal (2018).

Parameters:

alphas (np.ndarray) – Array of exponents alpha_i of the monomial expressed as prod_i lambda_i ^ alpha_i.

Returns:

The integral of the monomial on a simplex with measure 1.

Return type:

float

num_edges_per_cell(dim)

Compute the number of edges of a simplex of a given dimension.

Parameters:

dim (int) – Dimension.

Returns:

The number of adjacent edges.

Return type:

int

get_local_edge_nodes(dim)

Lists the local edge-node connectivity in the cell

Parameters:

dim (int) – Dimension.

Returns:

Row i contains the local indices of the nodes connected to the edge with local index i.

Return type:

np.ndarray

eval_grads_at_nodes(dphi, e_nodes)

Evaluates the gradients of the basis functions at the nodes

Parameters:

• dphi (np.ndarray) – Gradients of the P1 basis functions.

• e_nodes (np.ndarray) – The local edge-node connectivity.

Returns:

The gradient of basis function i at node j is in elements [i, 3 * (j:J + 1)].

Return type:

np.ndarray

get_edge_dof_indices(sd, cell, faces)

Finds the indices for the edge degrees of freedom that correspond to the local numbering of the edges.

Parameters:

• sd (pg.Grid) – The grid.

• cell (int) – The cell index.

• faces (np.ndarray) – Face indices of the cell.

Returns:

Indices of the edge degrees of freedom.

Return type:

np.ndarray

assemble_stiff_matrix(sd, data=None)

Assembles the stiffness matrix for the P2 finite element method.

Parameters:

• sd (pg.Grid) – The grid object representing the discretization.

• data (dict) – A dictionary containing the necessary data for assembling the matrix.

Returns:

The stiffness matrix.

Return type:

sps.csc_array

assemble_diff_matrix(sd)

Assembles the differential matrix based on the dimension of the grid.

Parameters:

sd (pg.Grid) – The grid object.

Returns:

The differential matrix.

Return type:

sps.csc_array

eval_at_cell_centers(sd)

Construct the matrix for evaluating a P2 function at the cell centers of the given grid.

Parameters:

sd (pg.Grid) – The grid on which to construct the matrix.

Returns:

The matrix representing the projection at the cell centers.

Return type:

sps.csc_array

assemble_lumped_matrix(sd, data=None)

Assembles the lumped mass matrix for the quadratic Lagrange space. This is based on the integration rule by Eggers and Radu, and is not block-diagonal for this space.

Parameters:

• sd (pg.Grid) – The grid object representing the discretization.

• data (dict | None) – A dictionary containing the necessary data for assembling the matrix.

Returns:

The lumped mass matrix.

Return type:

sps.csc_array

proj_to_PwPolynomials(sd)

Construct the matrix for projecting a quadratic Lagrangian function to a piecewise quadratic function.

Parameters:

sd (pg.Grid) – The grid on which to construct the matrix.

Returns:

The matrix representing the projection.

Return type:

sps.csc_array

interpolate(sd, func)

Interpolates a given function over the nodes of a grid.

Parameters:

• sd (pg.Grid) – The grid on which to interpolate the function.

• func (Callable[[np.ndarray], np.ndarray]) – The function to be interpolated.

Returns:

An array containing the interpolated values at each node of the grid.

Return type:

np.ndarray

assemble_nat_bc(sd, func, b_faces)

Assembles the ‘natural’ boundary condition (func, u)_Gamma with u a test function in Lagrange2

Parameters:

• sd (pg.Grid) – The grid object representing the computational domain.

• func (Callable[[np.ndarray], np.ndarray]) – The function used to evaluate the ‘natural’ boundary condition.

• b_faces (np.ndarray) – The array of boundary faces.

Returns:

The assembled ‘natural’ boundary condition values.

Return type:

np.ndarray

get_range_discr_class(dim)

Returns the appropriate range discretization class based on the dimension.

Parameters:

dim (int) – The dimension of the problem.

Returns:

The range discretization class.

Return type:

object

Raises:

NotImplementedError – There is no zero-dimensional discretization in PyGeoN.