Common infrastructure¶

exception meshmode.Error[source]¶

exception meshmode.DataUnavailable[source]¶

exception meshmode.FileExistsError¶

exception meshmode.InconsistentVerticesError[source]¶

class meshmode.AffineMap(matrix=None, offset=None)[source]¶

An affine map A@x+b represented by a matrix A and an offset vector b.

matrix¶: A numpy.ndarray representing the matrix A, or None.

offset¶: A numpy.ndarray representing the vector b, or None.

__init__(matrix=None, offset=None)[source]¶

Parameters:

matrix – A numpy.ndarray (or something convertible to one), or None.
offset – A numpy.ndarray (or something convertible to one), or None.

inverted()[source]¶: Return the inverse affine map.

__call__(vecs)[source]¶: Apply the affine map to an array vecs whose first axis length matches matrix.shape[1].

__eq__(other)[source]¶: Return self==value.

__ne__(other)[source]¶: Return self!=value.

Mesh management¶

Design of the Data Structure¶

Why does a `Mesh` need to be broken into `MeshElementGroup` instances?¶

Elements can be of different types (e.g. triangle, quadrilateral, tetrahedron, what have you). In addition, elements may vary in the polynomial degree used to represent them (see also below).

All these bits of information could in principle be stored by element, but having large, internally homogeneous groups is a good thing from an efficiency standpoint. (So that you can, e.g., launch one GPU kernel to deal with all order-3 triangles, instead of maybe having to dispatch based on type and size inside the kernel)

What is the difference between ‘vertices’ and ‘nodes’?¶

Nodes exist mainly to represent the (potentially non-affine) deformation of each element, by a one-to-one correspondence with MeshElementGroup.unit_nodes. They are unique to each element. Vertices on the other hand exist to clarify whether or not a point shared by two elements is actually identical (or just happens to be “close”). This is done by assigning (single, globally shared) vertex numbers and having elements refer to them.

Consider the following picture:

Figure made with TikZ

Mesh nodes and vertices

Mesh Data Structure¶

class meshmode.mesh.MeshElementGroup(order: int, vertex_indices: ndarray | None, nodes: ndarray, unit_nodes: ndarray)[source]¶

A group of elements sharing a common reference element.

order¶: The maximum degree used for interpolation. The exact meaning depends on the element type, e.g. for SimplexElementGroup this is the total degree.

dim¶: The number of dimensions spanned by the element. Not the ambient dimension, see Mesh.ambient_dim for that.

nvertices¶: Number of vertices in the reference element.

nfaces¶: Number of faces of the reference element.

nunit_nodes¶: Number of nodes on the reference element.

nelements¶: Number of elements in the group.

nnodes¶: Total number of nodes in the group (equivalent to nelements * nunit_nodes).

vertex_indices¶: An array of shape (nelements, nvertices) of (mesh-wide) vertex indices. This can also be None to support the case where the associated mesh does not have any vertices.

nodes¶: An array of node coordinates with shape (mesh.ambient_dim, nelements, nunit_nodes).

unit_nodes¶: An array with shape (dim, nunit_nodes) of nodes on the reference element. The coordinates nodes are a mapped version of these reference nodes.

is_affine¶: A bool flag that is True if the local-to-global parametrization of all the elements in the group is affine.

Element groups can also be compared for equality using the following methods. Note that these are very expensive, as they compare all the nodes.

__eq__(other)[source]¶: Return self==value.

__ne__(other)[source]¶: Return self!=value.

__init__(order: int, vertex_indices: ndarray | None, nodes: ndarray, unit_nodes: ndarray) → None¶

The following abstract methods must be implemented by subclasses.

abstract classmethod make_group(**kwargs: Any) → MeshElementGroup[source]¶

Instantiate a new group of class cls.

Unlike the constructor, this factory function performs additional consistency checks and should be used instead.

abstract face_vertex_indices() → Tuple[Tuple[int, ...], ...][source]¶

Returns:: a tuple of tuples indicating which vertices (in mathematically positive ordering) make up each face of an element in this group.

abstract vertex_unit_coordinates() → ndarray[source]¶

Returns:: an array of shape (nfaces, dim) with the unit coordinates of each vertex.

class meshmode.mesh.SimplexElementGroup(order: int, vertex_indices: ndarray | None, nodes: ndarray, unit_nodes: ndarray, _modepy_shape: Shape = None, _modepy_space: FunctionSpace = None)[source]¶: Inherits from MeshElementGroup.

class meshmode.mesh.TensorProductElementGroup(order: int, vertex_indices: ndarray | None, nodes: ndarray, unit_nodes: ndarray, _modepy_shape: Shape = None, _modepy_space: FunctionSpace = None)[source]¶: Inherits from MeshElementGroup.

class meshmode.mesh.Mesh(vertices: ndarray | None, groups: Iterable[MeshElementGroup], is_conforming: bool | None = None, vertex_id_dtype: dtype | generic = 'int32', element_id_dtype: dtype | generic = 'int32', face_id_dtype: dtype | generic = 'int8', nodal_adjacency: Literal[False] | Iterable[ndarray] | NodalAdjacency | None = None, facial_adjacency_groups: Literal[False] | Iterable[Iterable[FacialAdjacencyGroup]] | None = None, _nodal_adjacency: Literal[False] | Iterable[ndarray] | NodalAdjacency | None = None, _facial_adjacency_groups: Literal[False] | Iterable[Iterable[FacialAdjacencyGroup]] | None = None, skip_tests: bool = False, node_vertex_consistency_tolerance: float | None = None, skip_element_orientation_test: bool = False, factory_constructed: bool = False)[source]¶

property ambient_dim: int¶: Ambient dimension in which the mesh is embedded.

property dim: int¶: Dimension of the elements in the mesh.

property nvertices: int¶: Number of vertices in the mesh, if available.

property nelements: int¶: Number of elements in the mesh (sum over all the groups).

property base_element_nrs: ndarray¶: An array of size (len(groups),) of starting element indices for each group in the mesh.

property base_node_nrs¶: An array of size (len(groups),) of starting node indices for each group in the mesh.

property vertex_dtype¶: The dtype of the vertices array, if any.

groups: Tuple[MeshElementGroup, ...]¶: A tuple of MeshElementGroup instances.

vertices: ndarray | None¶: None or an array of vertex coordinates with shape (ambient_dim, nvertices). If None, vertices are not known for this mesh and no adjacency information can be constructed.

is_conforming: bool | None¶: True if it is known that all element interfaces are conforming. False if it is known that some element interfaces are non-conforming. None if neither of the two is known.

vertex_id_dtype: dtype¶: The dtype used to index into the vertex array.

element_id_dtype: dtype¶: The dtype used to index into the element array (relative to each group).

face_id_dtype: dtype¶: The dtype used to index element faces (relative to each element).

property nodal_adjacency: NodalAdjacency¶

Nodal adjacency of the mesh, if available.

This property gets the Mesh._nodal_adjacency of the mesh. If the attribute value is None, the adjacency is computed and cached.

Raises:: DataUnavailable – if the nodal adjacency cannot be obtained.

property facial_adjacency_groups: Tuple[Tuple[FacialAdjacencyGroup, ...], ...]¶

Facial adjacency of the mesh, if available.

This function gets the Mesh._facial_adjacency_groups of the mesh. If the attribute value is None, the adjacency is computed and cached.

Each facial_adjacency_groups[igrp] gives the facial adjacency relations for group igrp, expressed as a list of FacialAdjacencyGroup instances.

Figure made with TikZ

Facial Adjacency Group

For example for the mesh in the figure, the following data structure could be present:

[
    [...],  # connectivity for group 0
    [...],  # connectivity for group 1
    [...],  # connectivity for group 2
    # connectivity for group 3
    [
        # towards group 1, green
        InteriorAdjacencyGroup(ineighbor_group=1, ...),
        # towards group 2, pink
        InteriorAdjacencyGroup(ineighbor_group=2, ...),
        # towards the boundary, orange
        BoundaryAdjacencyGroup(...)
    ]
]

Note that element groups are not necessarily geometrically contiguous like the figure may suggest.

Raises:: DataUnavailable – if the facial adjacency cannot be obtained.

_nodal_adjacency: None | Literal[False] | NodalAdjacency¶

A description of the nodal adjacency of the mesh. This can be False if no adjacency is known or should be computed, None to compute the adjacency on demand or a given NodalAdjacency instance.

This attribute caches the values of nodal_adjacency.

_facial_adjacency_groups: None | Literal[False] | Tuple[Tuple[FacialAdjacencyGroup, ...], ...]¶

A description of the facial adjacency of the mesh. This can be False if no adjacency is known or should be computed, None to compute the adjacency on demand or a list of FacialAdjacencyGroup instances.

This attribute caches the values of facial_adjacency_groups.

__eq__(other)[source]¶: Compare two meshes for equality.

Warning

This operation is very expensive, as it compares all the vertices and groups between the two meshes. If available, the nodal and facial adjacency information is compared as well.

Warning

Only the (uncached) _nodal_adjacency and _facial_adjacency_groups are compared. This can fail for two meshes if one called nodal_adjacency() and the other one did not, even if they would be equal.

meshmode.mesh.make_mesh(vertices: ndarray | None, groups: Iterable[MeshElementGroup], *, nodal_adjacency: Literal[False] | Iterable[ndarray] | NodalAdjacency | None = None, facial_adjacency_groups: Literal[False] | Iterable[Iterable[FacialAdjacencyGroup]] | None = None, is_conforming: bool | None = None, vertex_id_dtype: dtype | generic = 'int32', element_id_dtype: dtype | generic = 'int32', face_id_dtype: dtype | generic = 'int8', skip_tests: bool = False, node_vertex_consistency_tolerance: float | None = None, skip_element_orientation_test: bool = False) → Mesh[source]¶

Construct a new mesh from a given list of groups.

This constructor performs additional checks on the mesh once constructed and should be preferred to calling the constructor of the Mesh class directly.

Parameters:

vertices – an array of vertices that match the given element groups. These can be None for meshes where adjacency is not required (e.g. non-conforming meshes).
nodal_adjacency –
a definition of the nodal adjacency of the mesh. This argument can take one of four values:
- False, in which case the information is marked as unavailable for this mesh and will not be computed at all. This should be used if the vertex adjacency does not convey the full picture, e.g if there are hanging nodes in the geometry.
- None, in which case the nodal adjacency will be deduced from the vertex adjacency on demand (this requires the vertices).
- a tuple of (element_neighbors_starts, element_neighbors) from which a NodalAdjacency object can be constructed.
- a NodalAdjacency object.
facial_adjacency_groups –
a definition of the facial adjacency for each group in the mesh. This argument can take one of three values:
- False, in which case the information is marked as unavailable for this mesh and will not be computed.
- None, in which case the facial adjacency will be deduced from the vertex adjacency on demand (this requires vertices).
- an iterable of FacialAdjacencyGroup objects.
is_conforming – True if the mesh is known to be conforming.
vertex_id_dtype – an integer dtype for the vertex indices.
element_id_dtype – an integer dtype for the element indices (relative to an element group).
face_id_dtype – an integer dtype for the face indices (relative to an element).
skip_tests – a flag used to skip any mesh consistency checks. This can be set to True in special situation, e.g. when loading a broken mesh that will be fixed later.
node_vertex_consistency_tolerance – If False, do not check for consistency between vertex and nodal data. If None, a default tolerance based on the dtype of the vertices array will be used.
skip_element_orientation_test – If False, check that element orientation is positive in volume meshes (i.e. ones where ambient and topological dimension match).

meshmode.mesh.check_mesh_consistency(mesh: Mesh, *, node_vertex_consistency_tolerance: Literal[False] | float | None = None, skip_element_orientation_test: bool = False) → None[source]¶

Check the mesh for consistency between the vertices, nodes, and their adjacency.

This function checks:

The node to vertex consistency, by interpolation.
The dtype of the various arrays matching the ones in Mesh.
The nodal adjacency shapes and dtypes.
The facial adjacency shapes and dtypes.
The mesh orientation using find_volume_mesh_element_orientations().

class meshmode.mesh.NodalAdjacency(neighbors_starts: ndarray, neighbors: ndarray)[source]¶

Describes nodal element adjacency information, i.e. information about elements that touch in at least one point.

neighbors_starts¶

element_id_t [nelements+1]

Use together with neighbors. neighbors_starts[iel] and neighbors_starts[iel+1] together indicate a ranges of element indices neighbors which are adjacent to iel.

neighbors¶

element_id_t []

See neighbors_starts.

__eq__(other)[source]¶: Return self==value.

__ne__(other)[source]¶: Return self!=value.

class meshmode.mesh.FacialAdjacencyGroup(igroup: int)[source]¶

Describes facial element adjacency information for one MeshElementGroup, i.e. information about elements that share (part of) a face or elements that lie on a boundary.

Figure made with TikZ

Facial Adjacency Group

Represents (for example) one of the (colored) interfaces between MeshElementGroup instances, or an interface between MeshElementGroup and a boundary. (Note that element groups are not necessarily contiguous like the figure may suggest.)

igroup¶

class meshmode.mesh.InteriorAdjacencyGroup(igroup: int, ineighbor_group: int, elements: ndarray, element_faces: ndarray, neighbors: ndarray, neighbor_faces: ndarray, aff_map: AffineMap)[source]¶

Describes interior facial element adjacency information for one MeshElementGroup.

igroup¶: The mesh element group number of this group.

ineighbor_group¶: ID of neighboring group, or None for boundary faces. If identical to igroup, then this contains the self-connectivity in this group.

elements¶: element_id_t [nfagrp_elements]. elements[i] gives the element number within igroup of the interior face.

element_faces¶: face_id_t [nfagrp_elements]. element_faces[i] gives the face index of the interior face in element elements[i].

neighbors¶: element_id_t [nfagrp_elements]. neighbors[i] gives the element number within ineighbor_group of the element opposite elements[i].

neighbor_faces¶: face_id_t [nfagrp_elements]. neighbor_faces[i] gives the face index of the opposite face in element neighbors[i]

aff_map¶: An AffineMap representing the mapping from the group’s faces to their corresponding neighbor faces.

__eq__(other)[source]¶: Return self==value.

__ne__(other)[source]¶: Return self!=value.

class meshmode.mesh.BoundaryAdjacencyGroup(igroup: int, boundary_tag: Hashable, elements: ndarray, element_faces: ndarray)[source]¶

Describes boundary adjacency information for one MeshElementGroup.

igroup¶: The mesh element group number of this group.

boundary_tag¶: The boundary tag identifier of this group.

elements¶: element_id_t [nfagrp_elements]. elements[i] gives the element number within igroup of the boundary face.

element_faces¶: face_id_t [nfagrp_elements]. element_faces[i] gives the face index of the boundary face in element elements[i].

class meshmode.mesh.InterPartAdjacencyGroup(igroup: int, boundary_tag: Hashable, elements: ndarray, element_faces: ndarray, part_id: Hashable, neighbors: ndarray, neighbor_faces: ndarray, aff_map: AffineMap)[source]¶

Describes inter-part adjacency information for one MeshElementGroup.

igroup¶: The mesh element group number of this group.

boundary_tag¶: The boundary tag identifier of this group. Will be an instance of BTAG_PARTITION.

part_id¶: The identifier of the neighboring part.

elements¶: Group-local element numbers. Element element_id_dtype elements[i] and face face_id_dtype element_faces[i] is connected to neighbor element element_id_dtype neighbors[i] with face face_id_dtype neighbor_faces[i].

element_faces¶: face_id_dtype element_faces[i] gives the face of element_id_dtype elements[i] that is connected to neighbors[i].

neighbors¶: element_id_dtype neighbors[i] gives the volume element number within the neighboring part of the element connected to element_id_dtype elements[i] (which is a boundary element index). Use ~meshmode.mesh.processing.find_group_indices to find the group that the element belongs to, then subtract element_nr_base to find the element of the group.

neighbor_faces¶: face_id_dtype global_neighbor_faces[i] gives face index within the neighboring part of the face connected to element_id_dtype elements[i]

aff_map¶: An AffineMap representing the mapping from the group’s faces to their corresponding neighbor faces.

Added in version 2017.1.

meshmode.mesh.as_python(mesh, function_name='make_mesh')[source]¶: Return a snippet of Python code (as a string) that will recreate the mesh given as an input parameter.

meshmode.mesh.is_true_boundary(boundary_tag)[source]¶

meshmode.mesh.mesh_has_boundary(mesh, boundary_tag)[source]¶

meshmode.mesh.check_bc_coverage(mesh, boundary_tags, incomplete_ok=False, true_boundary_only=True)[source]¶

Verify boundary condition coverage.

Given a list of boundary tags as boundary_tags, this function verifies that

1. the union of all these boundaries gives the complete boundary, 1. all these boundaries are disjoint.

Parameters:

incomplete_ok – Do not report an error if some faces are not covered by the boundary conditions.
true_boundary_only – only verify for faces whose tags do not inherit from BTAG_NO_BOUNDARY.

meshmode.mesh.is_boundary_tag_empty(mesh, boundary_tag)[source]¶: Return True if the corresponding boundary tag does not occur as part of mesh.

Predefined Boundary tags¶

class meshmode.mesh.BTAG_NONE[source]¶: A boundary tag representing an empty boundary or volume.

class meshmode.mesh.BTAG_ALL[source]¶

A boundary tag representing the entire boundary or volume.

In the case of the boundary, BTAG_ALL does not include rank boundaries, or, more generally, anything tagged with BTAG_NO_BOUNDARY.

In the case of a mesh representing an element-wise subset of another, BTAG_ALL does not include boundaries induced by taking the subset. Instead, these boundaries will be tagged with BTAG_INDUCED_BOUNDARY.

class meshmode.mesh.BTAG_REALLY_ALL[source]¶

A boundary tag representing the entire boundary.

Unlike BTAG_ALL, this includes rank boundaries, or, more generally, everything tagged with BTAG_NO_BOUNDARY.

In the case of a mesh representing an element-wise subset of another, this tag includes boundaries induced by taking the subset, or, more generally, everything tagged with BTAG_INDUCED_BOUNDARY

class meshmode.mesh.BTAG_NO_BOUNDARY[source]¶: A boundary tag indicating that this edge should not fall under BTAG_ALL. Among other things, this is used to keep rank boundaries out of BTAG_ALL.

class meshmode.mesh.BTAG_PARTITION(part_id: Hashable)[source]¶

A boundary tag indicating that this edge is adjacent to an element of another Mesh. The part identifier of the adjacent mesh is given by part_id.

part_id¶

Added in version 2017.1.

class meshmode.mesh.BTAG_INDUCED_BOUNDARY[source]¶: When a Mesh is created as an element-by-element subset of another (as, for example, when using the Firedrake interop features while passing restrict_to_boundary), boundaries may arise where there were none in the original mesh. This boundary tag is used to indicate such boundaries.

Mesh generation¶

Curves¶

meshmode.mesh.generation.make_curve_mesh(curve_f: Callable[[ndarray], ndarray], element_boundaries: ndarray, order: int, *, unit_nodes: ndarray | None = None, node_vertex_consistency_tolerance: float | bool | None = None, closed: bool = True, return_parametrization_points: bool = False) → Mesh[source]¶

Parameters:

curve_f – parametrization for a curve, accepting a vector of point locations and returning an array of shape (2, npoints).
element_boundaries – a vector of element boundary locations in $[0, 1]$, in order. $0$ must be the first entry, $1$ the last one.
order – order of the (simplex) elements. If unit_nodes is also provided, the orders should match.
unit_nodes – if given, the unit nodes to use. Must have shape (2, nnodes).
node_vertex_consistency_tolerance – passed to the Mesh constructor. If False, no checks are performed.
closed – if True, the curve is assumed closed and the first and last of the element_boundaries must match.
return_parametrization_points – if True, the parametrization points at which all the nodes in the mesh were evaluated are also returned.

Returns:

a Mesh, or if return_parametrization_points is True, a tuple (mesh, par_points), where par_points is an array of parametrization points.

Curve parametrizations¶

meshmode.mesh.generation.circle(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.ellipse(aspect_ratio: float, t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.cloverleaf(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.drop(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.n_gon(n_corners: int, t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.qbx_peanut(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.dumbbell(gamma: float, beta: float, t: ndarray)[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.wobbly_dumbbell(gamma: float, beta: float, p: int, wavenumber: int, t: ndarray)[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.apple(a: float, t: ndarray) → ndarray[source]¶

Parameters:

a – roundness parameter in $[0, 1/2]$, where $0$ gives a circle and $1/2$ gives a cardioid.
t – the parametrization, runs from $[0, 1]$.

Returns:

an array of shape (2, t.size).

meshmode.mesh.generation.clamp_piecewise(r_major: float, r_minor: float, gap: float, t: ndarray) → ndarray[source]¶

Parameters:

r_major – radius of the outer shell.
r_minor – radius of the inner shell.
gap – half-angle (in radians) of the right-hand side gap.

Returns:

an array of shape (2, t.size).

class meshmode.mesh.generation.WobblyCircle(coeffs: ndarray, phase: float = 0.0)[source]¶

static random(ncoeffs: int, seed: int) → WobblyCircle[source]¶

__call__(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

class meshmode.mesh.generation.NArmedStarfish(n_arms: int, amplitude: float, phase: float = 0.0)[source]¶

Inherits from WobblyCircle.

__call__(t: ndarray) → ndarray[source]¶

Parameters:: t – the parametrization, runs from $[0, 1]$.
Returns:: an array of shape (2, t.size).

meshmode.mesh.generation.starfish3¶

meshmode.mesh.generation.starfish5¶

Surfaces¶

meshmode.mesh.generation.generate_icosahedron(r: float, order: int, *, node_vertex_consistency_tolerance: float | bool | None = None, unit_nodes: ndarray | None = None) → Mesh[source]¶

meshmode.mesh.generation.generate_cube_surface(r: float, order: int, *, node_vertex_consistency_tolerance: float | bool | None = None, unit_nodes: ndarray | None = None) → Mesh[source]¶

meshmode.mesh.generation.generate_sphere(r: float, order: int, *, uniform_refinement_rounds: int = 0, node_vertex_consistency_tolerance: float | bool | None = None, unit_nodes: ndarray | None = None, group_cls: type | None = None) → Mesh[source]¶

Parameters:

r – radius of the sphere.
order – order of the group elements. If unit_nodes is also provided, the orders should match.
uniform_refinement_rounds – number of uniform refinement rounds to perform after the initial mesh was created.
node_vertex_consistency_tolerance – passed to the Mesh constructor. If False, no checks are performed.
unit_nodes – if given, the unit nodes to use. Must have shape (3, nnodes).
group_cls – a MeshElementGroup subclass. Based on the class, a different polyhedron is used to construct the sphere: simplices use generate_icosahedron() and tensor products use a generate_cube_surface().

meshmode.mesh.generation.generate_torus(r_major: float, r_minor: float, n_major: int = 20, n_minor: int = 10, order: int = 1, node_vertex_consistency_tolerance: float | bool | None = None, unit_nodes: ndarray | None = None, group_cls: type | None = None) → Mesh[source]¶

Generate a torus.

Figure made with TikZ

A torus with major circle (magenta) and minor circle (red).

The torus is obtained as the image of the parameter domain $(u, v) \in [0, 2\pi) \times [0, 2 \pi)$ under the map

\[\begin{split}\begin{aligned} x &= \cos(u) (r_\text{major} + r_\text{minor} \cos(v)) \\ y &= \sin(u) (r_\text{major} + r_\text{minor} \sin(v)) \\ z &= r_\text{minor} \sin(v) \end{aligned}\end{split}\]

where $r_\text{major}$ and $r_\text{minor}$ are the radii of the major and minor circles, respectively. The parameter domain is tiled with $n_\text{major} \times n_\text{minor}$ contiguous rectangles, and then each rectangle is subdivided into two triangles.

Parameters:

r_major – radius of the major circle.
r_minor – radius of the minor circle.
n_major – number of rectangles along major circle.
n_minor – number of rectangles along minor circle.
order – order of the (simplex) elements. If unit_nodes is also provided, the orders should match.
node_vertex_consistency_tolerance – passed to the Mesh constructor. If False, no checks are performed.
unit_nodes – if given, the unit nodes to use. Must have shape (3, nnodes).

Returns:

a Mesh of a torus.

meshmode.mesh.generation.refine_mesh_and_get_urchin_warper(order: int, m: int, n: int, est_rel_interp_tolerance: float, min_rad: float = 0.2, uniform_refinement_rounds: int = 0) → Tuple[Refiner, Callable[[Mesh], Mesh]][source]¶

Parameters:

order – order of the (simplex) elements.
m – order of the spherical harmonic $Y^m_n$.
n – order of the spherical harmonic $Y^m_n$.
est_rel_interp_tolerance – a tolerance for the relative interpolation error estimates on the warped version of the mesh.

Returns:

a tuple (refiner, warp_mesh), where refiner is a RefinerWithoutAdjacency (from which the unwarped mesh may be obtained), and whose get_current_mesh() returns a locally-refined Mesh of a sphere and warp_mesh is a callable taking and returning a mesh that warps the unwarped mesh into a smooth shape covered by a spherical harmonic of order $(m, n)$.

meshmode.mesh.generation.generate_urchin(order: int, m: int, n: int, est_rel_interp_tolerance: float, min_rad: float = 0.2) → Mesh[source]¶

Parameters:

order – order of the (simplex) elements. If unit_nodes is also provided, the orders should match.
m – order of the spherical harmonic $Y^m_n$.
n – order of the spherical harmonic $Y^m_n$.
est_rel_interp_tolerance – a tolerance for the relative interpolation error estimates on the warped version of the mesh.

Returns:

a refined Mesh of a smooth shape covered by a spherical harmonic of order $(m, n)$.

meshmode.mesh.generation.generate_surface_of_revolution(get_radius: Callable[[ndarray, ndarray], ndarray], height_discr: ndarray, angle_discr: ndarray, order: int, *, node_vertex_consistency_tolerance: float | bool | None = None, unit_nodes: ndarray | None = None) → Mesh[source]¶

Return a cylinder aligned with the “height” axis aligned with the Z axis.

Parameters:

get_radius – A callable function that takes in a 1D array of heights and a 1D array of angles and returns a 1D array of radii.
height_discr – A discretization of [0, 2*pi).
angle_discr – A discretization of [0, 2*pi).
order – order of the (simplex) elements. If unit_nodes is also provided, the orders should match.
node_vertex_consistency_tolerance – passed to the Mesh constructor. If False, no checks are performed.
unit_nodes – if given, the unit nodes to use. Must have shape (3, nnodes).

Volumes¶

meshmode.mesh.generation.generate_box_mesh(axis_coords: ~typing.Tuple[~numpy.ndarray, ...], order: int = 1, *, coord_dtype: ~typing.Any = <class 'numpy.float64'>, periodic: bool | None = None, group_cls: ~typing.Type[~meshmode.mesh.MeshElementGroup] | None = None, boundary_tag_to_face: ~typing.Dict[~typing.Any, str] | None = None, mesh_type: str | None = None, unit_nodes: ~numpy.ndarray | None = None) → Mesh[source]¶

Create a semi-structured mesh.

Parameters:

axis_coords – a tuple with a number of entries corresponding to the number of dimensions, with each entry a numpy array specifying the coordinates to be used along that axis. The coordinates for a given axis must define a nonnegative number of subintervals (in other words, the length can be 0 or a number greater than or equal to 2).
periodic – an optional tuple of bool indicating whether the mesh is periodic along each axis. Acts as a shortcut for calling meshmode.mesh.processing.glue_mesh_boundaries().
group_cls – One of meshmode.mesh.SimplexElementGroup or meshmode.mesh.TensorProductElementGroup.
boundary_tag_to_face –
an optional dictionary for tagging boundaries. The keys correspond to custom boundary tags, with the values giving a list of the faces on which they should be applied in terms of coordinate directions (+x, -x, +y, -y, +z, -z, +w, -w).

For example:
```
boundary_tag_to_face = {"bdry_1": ["+x", "+y"], "bdry_2": ["-x"]}
```
mesh_type –
In two dimensions with non-tensor-product elements, mesh_type may be set to "X" to generate this type of mesh:
```
_______
|\   /|
| \ / |
|  X  |
| / \ |
|/   \|
^^^^^^^
```
instead of the default:
```
_______
|\    |
| \   |
|  \  |
|   \ |
|    \|
^^^^^^^
```
Specifying a value other than None for all other mesh dimensionalities and element types is an error.

Changed in version 2017.1: group_factory parameter added.

Changed in version 2020.1: boundary_tag_to_face parameter added.

Changed in version 2020.3: group_factory deprecated and renamed to group_cls.

meshmode.mesh.generation.generate_regular_rect_mesh(a: Sequence[float] = (0, 0), b: Sequence[float] = (1, 1), *, nelements_per_axis: int | None = None, npoints_per_axis: int | None = None, periodic: bool | None = None, order: int = 1, boundary_tag_to_face: Dict[Any, str] | None = None, group_cls: Type[MeshElementGroup] | None = None, mesh_type: str | None = None, n: int | None = None) → Mesh[source]¶

Create a semi-structured rectangular mesh with equispaced elements.

Parameters:

a – the lower left hand point of the rectangle.
b – the upper right hand point of the rectangle.
nelements_per_axis – an optional tuple of integers indicating the number of elements along each axis.
npoints_per_axis – an optional tuple of integers indicating the number of points along each axis.
periodic – an optional tuple of bool indicating whether the mesh is periodic along each axis. Acts as a shortcut for calling meshmode.mesh.processing.glue_mesh_boundaries().
order – the mesh element order.
boundary_tag_to_face – an optional dictionary for tagging boundaries. See generate_box_mesh().
group_cls – see generate_box_mesh().
mesh_type – see generate_box_mesh().

Note

Specify only one of nelements_per_axis and npoints_per_axis.

meshmode.mesh.generation.generate_warped_rect_mesh(dim: int, order: int, *, nelements_side: int | None = None, npoints_side: int | None = None, group_cls: Type[MeshElementGroup] | None = None, n: int | None = None) → Mesh[source]¶: Generate a mesh of a warped square/cube. Mainly useful for testing functionality with curvilinear meshes.

meshmode.mesh.generation.generate_annular_cylinder_slice_mesh(n: int, center: ndarray, inner_radius: float, outer_radius: float, periodic: bool = False) → Mesh[source]¶: Generate a slice of a 3D annular cylinder for $\theta \in [-\frac{\pi}{4}, \frac{\pi}{4}]$. Optionally periodic in $theta$.

Mesh input/output¶

class meshmode.mesh.io.ScriptSource(source, extension)[source]¶: Added in version 2016.1.

class meshmode.mesh.io.FileSource(filename)[source]¶: Added in version 2014.1.

class meshmode.mesh.io.ScriptWithFilesSource(source, filenames, source_name='temp.geo')[source]¶

Added in version 2016.1.

source¶: The script code to be fed to gmsh.

filenames¶: The names of files to be copied to the temporary directory where gmsh is run.

meshmode.mesh.io.read_gmsh(filename, force_ambient_dim=None, mesh_construction_kwargs=None, return_tag_to_elements_map=False)[source]¶

Read a gmsh mesh file from filename and return a meshmode.mesh.Mesh.

Parameters:

force_ambient_dim – if not None, truncate point coordinates to this many dimensions.
mesh_construction_kwargs – None or a dictionary of keyword arguments passed to the meshmode.mesh.Mesh constructor.
return_tag_to_elements_map – If True, return in addition to the mesh a dict that maps each volume tag in the gmsh file to a numpy.ndarray containing meshwide indices of the elements that belong to that volume.

meshmode.mesh.io.generate_gmsh(source, dimensions=None, order=None, other_options=None, extension='geo', gmsh_executable='gmsh', force_ambient_dim=None, output_file_name='output.msh', mesh_construction_kwargs=None, target_unit=None, return_tag_to_elements_map=False)[source]¶

Run gmsh on the input given by source, and return a meshmode.mesh.Mesh based on the result.

Parameters:

source – an instance of either gmsh_interop.reader.FileSource or gmsh_interop.reader.ScriptSource
force_ambient_dim – if not None, truncate point coordinates to this many dimensions.
mesh_construction_kwargs – None or a dictionary of keyword arguments passed to the meshmode.mesh.Mesh constructor.
target_unit – Value of the option Geometry.OCCTargetUnit. Supported values are the strings ‘M’ or ‘MM’.

meshmode.mesh.io.from_meshpy(mesh_info, order=1)[source]¶: Imports a mesh from a meshpy mesh_info data structure, which may be generated by either meshpy.triangle or meshpy.tet.

meshmode.mesh.io.from_vertices_and_simplices(vertices, simplices, order=1, fix_orientation=False)[source]¶

Imports a mesh from a numpy array of vertices and an array of simplices.

Parameters:

vertices – An array of vertex coordinates with shape (ambient_dim, nvertices)
simplices – An array (nelements, nvertices) of (mesh-wide) vertex indices.

meshmode.mesh.io.to_json(mesh)[source]¶: Return a JSON-able Python data structure for mesh. The structure directly reflects the meshmode.mesh.Mesh data structure.

Mesh processing¶

class meshmode.mesh.processing.BoundaryPairMapping(from_btag: int, to_btag: int, aff_map: AffineMap)[source]¶

Represents an affine mapping from one boundary to another.

from_btag¶: The tag of one boundary.

to_btag¶: The tag of the other boundary.

aff_map¶: An meshmode.AffineMap that maps points on boundary from_btag into points on boundary to_btag.

meshmode.mesh.processing.find_group_indices(groups: List[MeshElementGroup], meshwide_elems: ndarray) → ndarray[source]¶

Parameters:

groups – A list of MeshElementGroup instances that contain meshwide_elems.
meshwide_elems – A numpy.ndarray of mesh-wide element numbers. Usually computed by elem + base_element_nr.

Returns:

A numpy.ndarray of group numbers that meshwide_elem belongs to.

meshmode.mesh.processing.partition_mesh(mesh: Mesh, part_id_to_elements: Mapping[Hashable, ndarray], return_parts: Sequence[Hashable] | None = None) → Mapping[Hashable, Mesh][source]¶

Parameters:

mesh – A Mesh to be partitioned.
part_id_to_elements – A dict mapping a part identifier to a sorted numpy.ndarray of elements.
return_parts – An optional list of parts to return. By default, returns all parts.

Returns:

A dict mapping part identifiers to instances of Mesh that represent the corresponding part of mesh.

meshmode.mesh.processing.find_volume_mesh_element_orientations(mesh: Mesh, *, tolerate_unimplemented_checks: bool = False) → ndarray[source]¶

Return a positive floating point number for each positively oriented element, and a negative floating point number for each negatively oriented element.

Parameters:: tolerate_unimplemented_checks – If True, elements for which no check is available will return NaN.

meshmode.mesh.processing.flip_simplex_element_group(vertices: ndarray, grp: MeshElementGroup, grp_flip_flags: ndarray) → MeshElementGroup[source]¶

meshmode.mesh.processing.perform_flips(mesh: Mesh, flip_flags: ndarray, skip_tests: bool = False) → Mesh[source]¶

Parameters:: flip_flags – A numpy.ndarray with meshmode.mesh.Mesh.nelements entries indicating by their Boolean value whether the element is to be flipped.

meshmode.mesh.processing.find_bounding_box(mesh: Mesh) → Tuple[ndarray, ndarray][source]¶

Returns:: a tuple (min, max), each consisting of a numpy.ndarray indicating the minimal and maximal extent of the geometry along each axis.

meshmode.mesh.processing.merge_disjoint_meshes(meshes: Sequence[Mesh], *, skip_tests: bool = False, single_group: bool = False) → Mesh[source]¶

meshmode.mesh.processing.make_mesh_grid(mesh: Mesh, *, shape: Tuple[int, ...], offset: Tuple[ndarray, ...] | None = None, skip_tests: bool = False) → Mesh[source]¶

Constructs a grid of copies of mesh, with shape copies in each dimensions at the given offset.

Returns:: a merged mesh representing the grid.

meshmode.mesh.processing.split_mesh_groups(mesh: Mesh, element_flags: ndarray, return_subgroup_mapping: bool = False) → Mesh | Tuple[Mesh, Dict[Tuple[int, int], int]][source]¶

Split all the groups in mesh according to the values of element_flags. The element flags are expected to be integers defining, for each group, how the elements are to be split into subgroups. For example, a single-group mesh with flags:

element_flags = [0, 0, 0, 42, 42, 42, 0, 0, 0, 41, 41, 41]

will create three subgroups. The integer flags need not be increasing or contiguous and can repeat across different groups (i.e. they are group-local).

Parameters:: element_flags – a numpy.ndarray with nelements entries indicating how the elements in a group are to be split.
Returns:: a Mesh where each group has been split according to flags in element_flags. If return_subgroup_mapping is True, it also returns a mapping of (group_index, subgroup) -> new_group_index.

meshmode.mesh.processing.glue_mesh_boundaries(mesh: Mesh, bdry_pair_mappings_and_tols: List[Tuple[BoundaryPairMapping, float]], *, use_tree: bool | None = None) → Mesh[source]¶

Create a new mesh from mesh in which one or more pairs of boundaries are “glued” together such that the boundary surfaces become part of the interior of the mesh. This can be used to construct, e.g., periodic boundaries.

Corresponding boundaries’ vertices must map into each other via an affine transformation (though the vertex ordering need not be the same). Currently operates only on facial adjacency; any existing nodal adjacency in mesh is ignored/invalidated.

Parameters:

bdry_pair_mappings_and_tols – a list of tuples (mapping, tol), where mapping is a BoundaryPairMapping instance that specifies a mapping between two boundaries in mesh that should be glued together, and tol is the allowed tolerance between the transformed vertex coordinates of the first boundary and the vertex coordinates of the second boundary when attempting to match the two. Pass at most one mapping for each unique (order-independent) pair of boundaries.
use_tree – Optional argument indicating whether to use a spatial binary search tree or a (quadratic) numpy algorithm when matching vertices.

meshmode.mesh.processing.map_mesh(mesh: Mesh, f: Callable[[ndarray], ndarray]) → Mesh[source]¶: Apply the map f to the mesh. f needs to accept and return arrays of shape (ambient_dim, npoints).

meshmode.mesh.processing.affine_map(mesh: Mesh, A: generic | ndarray | None = None, b: generic | ndarray | None = None) → Mesh[source]¶: Apply the affine map $f(x) = A x + b$ to the geometry of mesh.

meshmode.mesh.processing.rotate_mesh_around_axis(mesh: Mesh, *, theta: float, axis: ndarray | None = None) → Mesh[source]¶

Rotate the mesh by theta radians around the axis axis.

Parameters:: axis – a (not necessarily unit) vector. By default, the rotation is performed around the $z$ axis.

Mesh visualization¶

meshmode.mesh.visualization.draw_2d_mesh(mesh: Mesh, *, draw_vertex_numbers: bool = True, draw_element_numbers: bool = True, draw_nodal_adjacency: bool = False, draw_face_numbers: bool = False, set_bounding_box: bool = False, **kwargs: Any) → None[source]¶

Draw the mesh and its connectivity using matplotlib.

Parameters:

set_bounding_box – if True, the plot limits are set to the mesh bounding box. This can help if some of the actors are not visible.
kwargs – additional arguments passed to PathPatch when drawing the mesh group elements.

meshmode.mesh.visualization.draw_curve(mesh: Mesh, *, el_bdry_style: str = 'o', el_bdry_kwargs: Dict[str, Any] | None = None, node_style: str = 'x-', node_kwargs: Dict[str, Any] | None = None) → None[source]¶

Draw a curve mesh.

Parameters:

el_bdry_kwargs – passed to plot when drawing elements.
node_kwargs – passed to plot when drawing group nodes.

meshmode.mesh.visualization.write_vertex_vtk_file(mesh: Mesh, file_name: str, *, compressor: str | None = None, overwrite: bool = False) → None[source]¶

meshmode.mesh.visualization.mesh_to_tikz(mesh: Mesh) → str[source]¶

meshmode.mesh.visualization.vtk_visualize_mesh(actx: ArrayContext, mesh: Mesh, filename: str, *, vtk_high_order: bool = True, overwrite: bool = False) → None[source]¶

meshmode.mesh.visualization.write_stl_file(mesh: Mesh, stl_name: str, *, overwrite: bool = False) → None[source]¶: Writes a STL file from a triangular mesh in 3D. Requires the numpy-stl package.

meshmode.mesh.visualization.visualize_mesh_vertex_resampling_error(actx: ArrayContext, mesh: Mesh, filename: str, *, overwrite: bool = False) → None[source]¶

Common infrastructure¶

Mesh management¶

Design of the Data Structure¶

Why does a `Mesh` need to be broken into `MeshElementGroup` instances?¶

What is the difference between ‘vertices’ and ‘nodes’?¶

Mesh Data Structure¶

Predefined Boundary tags¶

Mesh generation¶

Curves¶

Curve parametrizations¶

Surfaces¶

Volumes¶

Tools for Iterative Refinement¶

Mesh input/output¶

Mesh processing¶

Mesh refinement¶

Mesh visualization¶

Common infrastructure¶

Mesh management¶

Design of the Data Structure¶

Why does a Mesh need to be broken into MeshElementGroup instances?¶

What is the difference between ‘vertices’ and ‘nodes’?¶

Mesh Data Structure¶

Predefined Boundary tags¶

Mesh generation¶

Curves¶

Curve parametrizations¶

Surfaces¶

Volumes¶

Tools for Iterative Refinement¶

Mesh input/output¶

Mesh processing¶

Mesh refinement¶

Mesh visualization¶

Why does a `Mesh` need to be broken into `MeshElementGroup` instances?¶