lagrange.polyddg

Classes

DifferentialOperators

Polygonal mesh discrete differential operators

Module Contents

class lagrange.polyddg.DifferentialOperators(mesh)

Polygonal mesh discrete differential operators

Parameters:

mesh (lagrange.core.SurfaceMesh)

connection_laplacian(*, beta: float = 1) → scipy.sparse.csc_matrix[float]

connection_laplacian(fid: int, *, beta: float = 1) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete connection Laplacian operator for a single facet.

Parameters:

• fid – Facet index.

• beta – Weight of projection term (default: 1).

Returns:

A dense matrix representing the per-facet connection Laplacian operator.

connection_laplacian_nrosy(*, n: int, beta: float = 1) → scipy.sparse.csc_matrix[float]

connection_laplacian_nrosy(fid: int, *, n: int, beta: float = 1) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete connection Laplacian operator for a single facet for n-rosy fields.

Parameters:

• fid – Facet index.

• n – Number of times to apply the connection.

• beta – Weight of projection term (default: 1).

Returns:

A dense matrix representing the per-facet connection Laplacian operator.

covariant_derivative() → scipy.sparse.csc_matrix[float]

covariant_derivative(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=4, None, order='F')]

Compute the discrete covariant derivative operator for a single facet.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet covariant derivative operator.

covariant_derivative_nrosy(*, n: int) → scipy.sparse.csc_matrix[float]

covariant_derivative_nrosy(fid: int, n: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=4, None, order='F')]

Compute the discrete covariant derivative operator for a single facet for n-rosy fields.

Parameters:

• fid – Facet index.

• n – Number of times to apply the connection.

Returns:

A dense matrix representing the per-facet covariant derivative operator.

covariant_projection(fid)

Compute the discrete covariant projection operator for a single facet.

Parameters:

fid (int) – Facet index.

Returns:

A dense matrix representing the per-facet covariant projection operator.

Return type:

Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=(None, None), order=’F’)]

covariant_projection_nrosy(fid, n)

Compute the discrete covariant projection operator for a single facet for n-rosy fields.

Parameters:

• fid (int) – Facet index.

• n (int) – Number of times to apply the connection.

Returns:

A dense matrix representing the per-facet covariant projection operator.

Return type:

Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=(None, None), order=’F’)]

curl()

Compute the discrete polygonal curl operator.

Returns:

A sparse matrix representing the curl operator.

Return type:

scipy.sparse.csc_matrix[float]

d0() → scipy.sparse.csc_matrix[float]

d0(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete d0 operator for a single facet.

The discrete d0 operator for a single facet is a n by n matrix, where n is the number vertices/edges in the facet. It maps a scalar functions defined on the vertices to a 1-form defined on the edges.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet d0 operator.

d1() → scipy.sparse.csc_matrix[float]

d1(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, order='C')]

Compute the discrete d1 operator for a single facet.

The discrete d1 operator for a single facet is a row vector of size 1 by n, where n is the number edges in the facet. It maps a 1-form defined on the edges to a 2-form defined on the facet.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet d1 operator.

divergence(*, beta=1)

Compute the discrete polygonal divergence operator.

Parameters:

beta (float) – Weight of projection term for the 1-form inner product (default: 1).

Returns:

A sparse matrix representing the divergence operator.

Return type:

scipy.sparse.csc_matrix[float]

facet_tangent_coordinates()

Compute the coordinate transformation that maps a per-facet tangent vector field expressed in the global 3D coordinate to the local tangent basis at each facet.

Returns:

A sparse matrix representing the coordinate transformation.

Return type:

scipy.sparse.csc_matrix[float]

flat() → scipy.sparse.csc_matrix[float]

flat(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, 3, order='F')]

Compute the discrete flat operator for a single facet.

The discrete flat operator for a single facet is a n by 3 matrix, where n is the number of edges of the facet. It maps a vector field defined on the facet to a 1-form defined on the edges of the facet.

Parameters:

fid – Facet index.

Returns:

A Nx3 dense matrix representing the per-facet flat operator.

gradient() → scipy.sparse.csc_matrix[float]

gradient(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=3, None, order='F')]

Compute the discrete gradient operator for a single facet.

The discrete gradient operator for a single facet is a 3 by n vector, where n is the number vertices in the facet. It maps a scalar functions defined on the vertices to a gradient vector defined on the facet.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet gradient operator.

inner_product_0_form() → scipy.sparse.csc_matrix[float]

inner_product_0_form(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete inner product operator for 0-forms for a single facet.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet inner product operator for 0-forms.

inner_product_1_form(*, beta: float = 1) → scipy.sparse.csc_matrix[float]

inner_product_1_form(fid: int, beta: float = 1) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete inner product operator for 1-forms for a single facet.

Parameters:

• fid – Facet index.

• beta – Weight of projection term (default: 1).

Returns:

A dense matrix representing the per-facet inner product operator for 1-forms.

inner_product_2_form() → scipy.sparse.csc_matrix[float]

inner_product_2_form(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=1, order='C')]

Compute the discrete inner product operator for 2-forms for a single facet.

Parameters:

fid – Facet index.

Returns:

A 1x1 dense matrix representing the per-facet inner product operator for 2-forms.

laplacian(*, beta: float = 1) → scipy.sparse.csc_matrix[float]

laplacian(fid: int, *, beta: float = 1) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete Laplacian operator for a single facet.

Parameters:

• fid – Facet index.

• beta – Weight of projection term (default: 1).

Returns:

A dense matrix representing the per-facet Laplacian operator.

levi_civita() → scipy.sparse.csc_matrix[float]

levi_civita(fid: int, lv: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=2, 2, order='F')]

levi_civita(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete Levi-Civita connection for a single facet.

Parameters:

fid – Facet index.

Returns:

A dense matrix representing the per-facet Levi-Civita connection.

levi_civita_nrosy(*, n: int) → scipy.sparse.csc_matrix[float]

levi_civita_nrosy(fid: int, lv: int, *, n: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=2, 2, order='F')]

levi_civita_nrosy(fid: int, *, n: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=None, None, order='F')]

Compute the discrete Levi-Civita connection for a single facet for n-rosy fields.

Parameters:

• fid – Facet index.

• n – Number of times to apply the connection.

Returns:

A dense matrix representing the per-facet Levi-Civita connection.

projection(fid)

Compute the discrete projection operator for a single facet.

Parameters:

fid (int) – Facet index.

Returns:

A dense matrix representing the per-facet projection operator.

Return type:

Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=(None, None), order=’F’)]

sharp() → scipy.sparse.csc_matrix[float]

sharp(fid: int) → Annotated[numpy.typing.NDArray[numpy.float64], dict(shape=3, None, order='F')]

Compute the discrete sharp operator for a single facet.

Parameters:

fid – Facet index.

Returns:

A 3xN dense matrix representing the per-facet sharp operator.

star0()

Compute the discrete Hodge star operator for 0-forms.

The Hodge star operator maps a k-form to a dual (n-k)-form, where n is the dimension of the manifold.

Returns:

A sparse matrix representing the discrete Hodge star operator for 0-forms.

Return type:

scipy.sparse.csc_matrix[float]

star1()

Compute the discrete Hodge star operator for 1-forms.

The Hodge star operator maps a k-form to a dual (n-k)-form, where n is the dimension of the manifold.

Returns:

A sparse matrix representing the discrete Hodge star operator for 1-forms.

Return type:

scipy.sparse.csc_matrix[float]

star2()

Compute the discrete Hodge star operator for 2-forms.

The Hodge star operator maps a k-form to a dual (n-k)-form, where n is the dimension of the manifold.

Returns:

A sparse matrix representing the discrete Hodge star operator for 2-forms.

Return type:

scipy.sparse.csc_matrix[float]

vertex_tangent_coordinates()

Compute the coordinate transformation that maps a per-vertex tangent vector field expressed in the global 3D coordinate to the local tangent basis at each vertex.

Returns:

A sparse matrix representing the coordinate transformation.

Return type:

scipy.sparse.csc_matrix[float]

property centroid_attribute_id: int

Get the attribute ID of the per-facet centroid attribute used in the differential operators.

Return type:

int

property vector_area_attribute_id: int

Get the attribute ID of the per-facet vector area attribute used in the differential operators.

Return type:

int

property vertex_normal_attribute_id: int

Get the attribute ID of the per-vertex normal attribute used in the differential operators.

Return type:

int