Core Functions

core

This file contains the core functions needed for the local update.

For readability we sometimes include the shape of arrays, with the following meaning: N: number of elements in mesh (i.e. triangles) d_e: number of points in one element M: number of points in mesh d: dimension of the underlying space

fimjax.core.Mesh

dataclass for holding everything that belongs to the mesh. Note that this only supports keyword arguments due to Chex. Use it like: mesh = Mesh(points=points, elements=elements)

N: Number of elements in a mesh (i.e. triangles) d_e: Number of points in an element M: Number of points in a mesh d: Dimension of the underlying space

Attributes:

Name	Type	Description
points	np.ndarray	[M, d] array of points
elements	np.ndarray	[N, d_e] array of indices into points that correspond to elements in a mesh
points_triangle	np.ndarray	[N, d_e, d] like elements, but with the points instead of indices

__init__

__init__(points: np.ndarray, elements: np.ndarray)

Constructor.

Parameters:

Name	Type	Description	Default
points	np.ndarray	[M, d] array of points	required
elements	np.ndarray	[N] array of	required

fimjax.core._norm_squared

_norm_squared(A: np.ndarray, x1: np.ndarray, x2: np.ndarray) -> np.ndarray

Custom vectorized implementation of :math:\\left<A \\mathbf{x}_1, \\mathbf{x}_2 \\right>.

Parameters:

Name	Type	Description	Default
A	np.ndarray	[N, d, d] Array of tensors	required
x1	np.ndarray	[N, d] Point 1	required
x2	np.ndarray	[N, d] Point 2	required

Returns:

Type	Description
np.ndarray	[N] array of norm values

fimjax.core._norm

_norm(A: np.ndarray, x1: np.ndarray, x2: np.ndarray) -> np.ndarray

Custom vectorized implementation of:math:\\sqrt{\\left<A \\mathbf{x}_1, \\mathbf{x}_2 \\right>}.

Parameters:

Name	Type	Description	Default
A	np.ndarray	[N, d, d] Array of tensors	required
x1	np.ndarray	[N, d] Point 1	required
x2	np.ndarray	[N, d] Point 2	required

Returns:

Type	Description
np.ndarray	[N] array of norm values

fimjax.core._update_point

_update_point(x1: np.ndarray, x2: np.ndarray, x3: np.ndarray, D: np.ndarray, u1: np.ndarray, u2: np.ndarray) -> np.ndarray

Calculates the minimal solution along the border of a triangle.

Parameters:

Name	Type	Description	Default
x1	np.ndarray	[N, d] array with stack of the first vertex of the triangles	required
x2	np.ndarray	[N, d] array with stack of the second vertex of the triangles	required
x3	np.ndarray	[N, d] array with stack of the third vertex of the triangles	required
D	np.ndarray	[N, d] array with stack of speed tensors	required
u1	np.ndarray	[N] array with current solution at the first vertex	required
u2	np.ndarray	[N] array with current solution at the first vertex	required

Returns:

Type	Description
np.ndarray	[N] array minimal solution along the borders of the triangle

fimjax.core._update_triangle

_update_triangle(x1: np.ndarray, x2: np.ndarray, x3: np.ndarray, D: np.ndarray, u1: np.ndarray, u2: np.ndarray) -> np.ndarray

Update triangle(s) locally.

Calculates all triangle updates in a broadcasted way by solving a constrained optimization problem analytically. See paper on FIM for triangulated meshes for more infos.

Parameters:

Name	Type	Description	Default
x1	np.ndarray	[N, d] array with stack of the first vertex of the triangles	required
x2	np.ndarray	[N, d] array with stack of the second vertex of the triangles	required
x3	np.ndarray	[N, d] array with stack of the third vertex of the triangles	required
D	np.ndarray	[N, d] array with stack of speed tensors	required
u1	np.ndarray	[N] array with current solution at the first vertex	required
u2	np.ndarray	[N] array with current solution at the first vertex	required

Returns:

Type	Description
np.ndarray	[N] minimal solution along the borders of the triangle

fimjax.core._update_all_triangles

_update_all_triangles(mesh: Mesh, D: np.ndarray, solution: np.ndarray) -> np.ndarray

Performs one Jacobi update.

Calculates the solution to all update direction and picks the smallest one for each point.

Parameters:

Name	Type	Description	Default
mesh	Mesh	Mesh	required
D	np.ndarray	[N, d, d] array with metric tensor field	required
solution	np.ndarray	[M] array solution before iteration	required

Returns:

Type	Description
np.ndarray	[M] new solution after one iteration

fimjax.core._update_all_triangles_no_self_update

_update_all_triangles_no_self_update(mesh: Mesh, D: np.ndarray, solution: np.ndarray) -> np.ndarray

Performs one Jacobi update.

Calculates the solution to all update direction and picks the smallest one for each point. This does not include the self update, so a node is always updated from the adjacent nodes, so using this method the iteration may NOT converge.

Parameters:

Name	Type	Description	Default
mesh	Mesh	Mesh	required
D	np.ndarray	[N, d, d] array with metric tensor field	required
solution	np.ndarray	[M] array solution before iteration	required

Returns:

Type	Description
np.ndarray	[M] new solution after one iteration

fimjax.core._update_all_triangles_softmin

_update_all_triangles_softmin(mesh: Mesh, D: np.ndarray, solution: np.ndarray) -> np.ndarray

Performs one Jacobi update.

Calculates the solution to all update direction and picks the smallest one for each point. Instead of the minimum the softmin is used.

Parameters:

Name	Type	Description	Default
mesh	Mesh	Mesh	required
D	np.ndarray	[N, d, d] array with metric tensor field	required
solution	np.ndarray	[M] array solution before iteration	required

Returns:

Type	Description
np.ndarray	[M] new solution after one iteration