puncturedfem.MeshCell

Represent a mesh cell, which may be multiply connected.

Contains: - parameterization of the boundary - methods to evaluate and integrate functions on the boundary - interior points - relative position of the cell in the mesh (topological info)

Attributes:

Name	Type	Description
idx	int	The cell id as it appears in the mesh.
components	list[ClosedContour]	The boundary components of the cell.
num_holes	int	The number of holes in the cell.
num_edges	int	The number of edges on the cell boundary.
num_pts	int	The number of sampled points on the cell boundary.
component_start_idx	list[int]	The index of the first sampled point on each boundary component.
closest_vert_idx	np.ndarray	The index of the closest vertex in the mesh to each sampled point.
edge_orients	list[int]	The orientation of each Edge in the cell (+/- 1).
int_mesh_size	tuple[int, int]

__init__(idx, edges, int_mesh_size=(33, 33), rtol=0.05, atol=0.05)

Initialize a MeshCell object.

Parameters:

Name	Type	Description	Default
idx	int	The cell id.	required
edges	list[Edge]	The edges in the cell.	required
int_mesh_size	tuple[int, int], optional	The size of the interior mesh, by default (101, 101)	(33, 33)
rtol	float, optional	The relative tolerance for interior points, by default 0.02	0.05
atol	float, optional	The absolute tolerance for interior points, by default 0.02	0.05

deparameterize()

Remove parameterization of each edge.

dot_with_normal(comp1, comp2)

Return the dot product (comp1, comp2) * unit_normal.

Parameters:

Name	Type	Description	Default
comp1	np.ndarray	The first component of the vector.	required
comp2	np.ndarray	The second component of the vector.	required

Returns:

Type	Description
np.ndarray	The dot product (comp1, comp2) * unit_normal.

dot_with_tangent(comp1, comp2)

Return the dot product (comp1, comp2) * unit_tangent.

Parameters:

Name	Type	Description	Default
comp1	np.ndarray	The first component of the vector.	required
comp2	np.ndarray	The second component of the vector.	required

Returns:

Type	Description
np.ndarray	The dot product (comp1, comp2) * unit_tangent.

evaluate_function_on_boundary(fun)

Return fun(x) for each sampled point on the cell boundary.

Parameters:

Name	Type	Description	Default
fun	Func_R2_R	The function to evaluate.	required

Returns:

Type	Description
np.ndarray	The values of fun(x) at each sampled point on the boundary.

find_boundary_components(edges)

Find the boundary components of the cell.

Parameters:

Name	Type	Description	Default
edges	list[Edge]	The edges in the cell.	required

Notes

The following attributes are set: - components: the boundary components of the cell - num_holes: the number of holes in the cell

find_edge_orientations(edges)

Find the orientation of each Edge in the cell.

The orientation is +1 if the Edge starts at the cell, -1 if it ends at the cell, and 0 if the Edge is not incident to the cell.

Parameters:

Name	Type	Description	Default
edges	list[Edge]	The edges in the cell.	required

find_num_edges()

Find the number of edges in the cell.

The following attributes are set: - num_edges: the number of edges in the cell

generate_interior_points()

Generate interior points for the cell.

The following attributes are set: - int_x1: the x-coordinates of the interior points - int_x2: the y-coordinates of the interior points - is_inside: a boolean array indicating whether each point is in the interior

get_boundary_points()

Return the boundary points.

Returns:

Name	Type	Description
x1	np.ndarray	The x-coordinates of the boundary points.
x2	np.ndarray	The y-coordinates of the boundary points.

get_bounding_box()

Return the bounding box of the cell.

Returns:

Name	Type	Description
xmin	float	The minimum x-coordinate of the cell.
xmax	float	The maximum x-coordinate of the cell.
ymin	float	The minimum y-coordinate of the cell.
ymax	float	The maximum y-coordinate of the cell.

get_distance_to_boundary(x, y)

Return the distance to the boundary at each point.

Parameters:

Name	Type	Description	Default
x	float	The x-coordinate of the point.	required
y	float	The y-coordinate of the point.	required

Returns:

Type	Description
float	The distance to the boundary at the point.

get_dx_norm()

Return the norm of the first derivative.

Returns:

Type	Description
np.ndarray	The norm of the first derivative.

get_edge_endpoint_incidence(edges)

Get the incidence array for the edges in the cell.

Return incidence array: for each Edge i, point to an Edge j whose starting point is the terminal point of Edge i Edge i vertex Edge j --->--->--->--- o --->--->--->---

get_edges()

Return a list of all edges in the cell.

Returns:

Type	Description
list[Edge]	A list of all edges in the cell.

get_unit_normal()

Return the components of the unit normal vector.

Returns:

Type	Description
tuple[np.ndarray, np.ndarray]	The components of the unit normal vector.

get_unit_tangent()

Return the components of the unit tangent vector.

Returns:

Type	Description
tuple[np.ndarray, np.ndarray]	The components of the unit tangent vector.

integrate_over_boundary(vals)

Integrate vals over the boundary.

Parameters:

Name	Type	Description	Default
vals	np.ndarray	The values of f at each sampled point on the boundary.	required

Returns:

Type	Description
float	The integral of f over the boundary.

integrate_over_boundary_preweighted(vals_dx_norm)

Integrate vals over the boundary without multiplying by dx_norm.

Parameters:

Name	Type	Description	Default
vals_dx_norm	np.ndarray	The values of f * dx_norm at each sampled point on the boundary.	required

Returns:

Type	Description
float	The integral of f over the boundary.

is_in_interior(x, y)

Return a boolean array indicating whether each point is in the interior.

Parameters:

Name	Type	Description	Default
x	np.ndarray	The x-coordinates of the points.	required
y	np.ndarray	The y-coordinates of the points, same shape as x.	required

Returns:

Type	Description
np.ndarray	A boolean array indicating whether each point is in the interior, same shape as x and y.

is_parameterized()

Return whether all edges of the cell are parameterized.

Returns:

Type	Description
bool	True if the cell is parameterized, False otherwise.

multiply_by_dx_norm(vals)

Return vals * dx_norm for each sampled point on the cell boundary.

Parameters:

Name	Type	Description	Default
vals	np.ndarray	The values of f at each sampled point on the boundary.	required

Returns:

Type	Description
np.ndarray	The values of f * dx_norm at each sampled point on the boundary.

parameterize(quad_dict, compute_interior_points=True)

Sample the boundary on each edge.

Parameters:

Name	Type	Description	Default
quad_dict	QuadDict	The quadrature dictionary, which specifies how the edges are to be sampled.	required
compute_interior_points	bool, optional	Whether to generate the interior points and triangulation. True by default.	True

set_idx(idx)

Set the global cell index.

Parameters:

Name	Type	Description	Default
idx	int	The cell index.	required

set_interior_mesh_size(rows, cols)

Set the size of the mesh of interior points.

Parameters:

Name	Type	Description	Default
rows	int	The number of rows in the mesh.	required
cols	int	The number of columns in the mesh.	required

set_interior_point_tolerance(rtol=0.02, atol=0.02)

Set the minimum distance to the boundary for sampled interior points.

Parameters:

Name	Type	Description	Default
rtol	float, optional	The relative tolerance, by default 0.02	0.02
atol	float, optional	The absolute tolerance, by default 0.02	0.02