symfem.references

Reference elements.

Module Contents

Classes

Reference	A reference cell on which a finite element can be defined.
Point	A point.
Interval	An interval.
Triangle	A triangle.
Tetrahedron	A tetrahedron.
Quadrilateral	A quadrilateral.
Hexahedron	A hexahedron.
Prism	A (triangular) prism.
Pyramid	A (square-based) pyramid.
DualPolygon	A polygon on a barycentric dual grid.

Functions

_which_side(→ Optional[int])	Check which side of a line or plane a set of points are.
_vsub(→ symfem.geometry.PointType)	Subtract.
_vadd(→ symfem.geometry.PointType)	Add.
_vdot(→ sympy.core.expr.Expr)	Compute the dot product.
_vcross(→ symfem.geometry.PointType)	Compute the cross product.
_vnorm(→ sympy.core.expr.Expr)	Find the norm of a vector.
_vnormalise(→ symfem.geometry.PointType)	Normalise a vector.

Attributes

LatticeWithLines
IntLimits

symfem.references.LatticeWithLines

symfem.references.IntLimits

exception symfem.references.NonDefaultReferenceError

Bases: NotImplementedError

Exception to be thrown when an element can only be created on the default reference.

symfem.references._which_side(vs: symfem.geometry.SetOfPoints, p: symfem.geometry.PointType, q: symfem.geometry.PointType) → int | None

Check which side of a line or plane a set of points are.

Parameters:

• vs – The set of points

• p – A point on the line or plane

• q – Another point on the line (2D) or the normal to the plane (3D)

Returns:

2 if the points are all to the left, 1 if the points are all to the left or on the line, 0 if the points are all on the line, -1 if the points are all to the right or on the line, -1 if the points are all to the right, None if there are some points on either side.

symfem.references._vsub(v: symfem.geometry.PointTypeInput, w: symfem.geometry.PointTypeInput) → symfem.geometry.PointType

Subtract.

Parameters:

• v – A vector

• w – A vector

Returns:

The vector v - w

symfem.references._vadd(v: symfem.geometry.PointTypeInput, w: symfem.geometry.PointTypeInput) → symfem.geometry.PointType

Add.

Parameters:

• v – A vector

• w – A vector

Returns:

The vector v + w

symfem.references._vdot(v: symfem.geometry.PointTypeInput, w: symfem.geometry.PointTypeInput) → sympy.core.expr.Expr

Compute the dot product.

Parameters:

• v – A vector

• w – A vector

Returns:

The scalar v.w

symfem.references._vcross(v_in: symfem.geometry.PointTypeInput, w_in: symfem.geometry.PointTypeInput) → symfem.geometry.PointType

Compute the cross product.

Parameters:

• v – A vector

• w – A vector

Returns:

The vector v x w

symfem.references._vnorm(v_in: symfem.geometry.PointTypeInput) → sympy.core.expr.Expr

Find the norm of a vector.

Parameters:

v_in – A vector

Returns:

The norm of v_in

symfem.references._vnormalise(v_in: symfem.geometry.PointTypeInput) → symfem.geometry.PointType

Normalise a vector.

Parameters:

v_in – A vector

Returns:

A unit vector pointing in the same direction as v_in

class symfem.references.Reference(vertices: symfem.geometry.SetOfPointsInput = ())

Bases: abc.ABC

A reference cell on which a finite element can be defined.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

__build__(tdim: int, name: str, origin: symfem.geometry.PointTypeInput, axes: symfem.geometry.SetOfPointsInput, reference_vertices: symfem.geometry.SetOfPointsInput, vertices: symfem.geometry.SetOfPointsInput, edges: Tuple[Tuple[int, int], Ellipsis], faces: Tuple[Tuple[int, Ellipsis], Ellipsis], volumes: Tuple[Tuple[int, Ellipsis], Ellipsis], sub_entity_types: List[List[str] | str | None], simplex: bool = False, tp: bool = False)

Create a reference cell.

Parameters:

• tdim – The topological dimension of the cell

• name – The name of the cell

• origin – The coordinates of the origin

• axes – Vectors representing the axes of the cell

• reference_vertices – The vertices of the default version of this cell

• vertices – The vertices of this cell

• edges – Pairs of vertex numbers that form the edges of the cell

• faces – Tuples of vertex numbers that form the faces of the cell

• volumes – Tuples of vertex numbers that form the volumes of the cell

• sub_entity_types – The cell types of each sub-entity of the cell

• simplex – Is the cell a simplex (interval/triangle/tetrahedron)?

• tp – Is the cell a tensor product (interval/quadrilateral/hexahedron)?

__eq__(other: object) → bool

Check if two references are equal.

__hash__() → int

Check if two references are equal.

intersection(other: Reference) → Reference | None

Get the intersection of two references.

Returns:

A reference element that is the intersection

abstract default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

abstract make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

make_lattice_float(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

make_lattice_with_lines_float(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

z_ordered_entities() → List[List[Tuple[int, int]]]

Get the subentities of the cell in back-to-front plotting order.

Returns:

List of lists of subentity dimensions and numbers

z_ordered_entities_extra_dim() → List[List[Tuple[int, int]]]

Get the subentities in back-to-front plotting order when using an extra dimension.

Returns:

List of lists of subentity dimensions and numbers

get_point(reference_coords: symfem.geometry.PointType) → Tuple[sympy.core.expr.Expr, Ellipsis]

Get a point in the reference from reference coordinates.

Parameters:

reference_coords – The reference coordinates

Returns:

A point in the cell

abstract integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

abstract get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

abstract get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

get_map_to_self() → symfem.geometry.PointType

Get the map from the canonical reference to this reference.

Returns:

The map

abstract _compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

get_inverse_map_to_self() → symfem.geometry.PointType

Get the inverse map from the canonical reference to this reference.

Returns:

The map

abstract _compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

abstract volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

midpoint() → symfem.geometry.PointType

Calculate the midpoint.

Returns:

The midpoint of the cell

jacobian() → sympy.core.expr.Expr

Calculate the Jacobian.

Returns:

The Jacobian

scaled_axes() → symfem.geometry.SetOfPoints

Return the unit axes of the reference.

Returns:

The axes

tangent() → symfem.geometry.PointType

Calculate the tangent to the element.

Returns:

The tangent

normal() → symfem.geometry.PointType

Calculate the normal to the element.

Returns:

The normal

sub_entities(dim: int | None = None, codim: int | None = None) → Tuple[Tuple[int, Ellipsis], Ellipsis]

Get the sub-entities of a given dimension.

Parameters:

• dim – The dimension of the sub-entity

• codim – The co-dimension of the sub-entity

Returns:

A tuple of tuples of vertex numbers

sub_entity_count(dim: int | None = None, codim: int | None = None) → int

Get the number of sub-entities of a given dimension.

Parameters:

• dim – the dimension of the sub-entity

• codim – the codimension of the sub-entity

Returns:

The number of sub-entities

sub_entity(dim: int, n: int, reference_vertices: bool = False) → Any

Get the sub-entity of a given dimension and number.

Parameters:

• dim – the dimension of the sub-entity

• n – The sub-entity number

• reference_vertices – Should the reference vertices be used?

Returns:

The sub-entity

at_vertex(point: symfem.geometry.PointType) → bool

Check if a point is a vertex of the reference.

Parameters:

point – The point

Returns:

Is the point a vertex?

on_edge(point_in: symfem.geometry.PointType) → bool

Check if a point is on an edge of the reference.

Parameters:

point_in – The point

Returns:

Is the point on an edge?

on_face(point_in: symfem.geometry.PointType) → bool

Check if a point is on a face of the reference.

Parameters:

point_in – The point

Returns:

Is the point on a face?

abstract contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

map_polyset_from_default(poly: List[symfem.functions.FunctionInput]) → List[symfem.functions.FunctionInput]

Map the polynomials from the default reference element to this reference.

plot_entity_diagrams(filename: str | List[str], plot_options: Dict[str, Any] = {}, **kwargs: Any)

Plot diagrams showing the entity numbering of the reference.

class symfem.references.Point(vertices: symfem.geometry.SetOfPointsInput = ((),))

Bases: Reference

A point.

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

abstract make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Interval(vertices: symfem.geometry.SetOfPointsInput = ((0,), (1,)))

Bases: Reference

An interval.

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Triangle(vertices: symfem.geometry.SetOfPointsInput = ((0, 0), (1, 0), (0, 1)))

Bases: Reference

A triangle.

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

z_ordered_entities_extra_dim() → List[List[Tuple[int, int]]]

Get the subentities in back-to-front plotting order when using an extra dimension.

Returns:

List of lists of subentity dimensions and numbers

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Tetrahedron(vertices: symfem.geometry.SetOfPointsInput = ((0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)))

Bases: Reference

A tetrahedron.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

z_ordered_entities() → List[List[Tuple[int, int]]]

Get the subentities of the cell in back-to-front plotting order.

Returns:

List of lists of subentity dimensions and numbers

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Quadrilateral(vertices: symfem.geometry.SetOfPointsInput = ((0, 0), (1, 0), (0, 1), (1, 1)))

Bases: Reference

A quadrilateral.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

z_ordered_entities_extra_dim() → List[List[Tuple[int, int]]]

Get the subentities in back-to-front plotting order when using an extra dimension.

Returns:

List of lists of subentity dimensions and numbers

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Hexahedron(vertices: symfem.geometry.SetOfPointsInput = ((0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), (1, 1, 1)))

Bases: Reference

A hexahedron.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

z_ordered_entities() → List[List[Tuple[int, int]]]

Get the subentities of the cell in back-to-front plotting order.

Returns:

List of lists of subentity dimensions and numbers

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Prism(vertices: symfem.geometry.SetOfPointsInput = ((0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1)))

Bases: Reference

A (triangular) prism.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

z_ordered_entities() → List[List[Tuple[int, int]]]

Get the subentities of the cell in back-to-front plotting order.

Returns:

List of lists of subentity dimensions and numbers

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.Pyramid(vertices: symfem.geometry.SetOfPointsInput = ((0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1)))

Bases: Reference

A (square-based) pyramid.

property clockwise_vertices: symfem.geometry.SetOfPoints

Get list of vertices in clockwise order.

Returns:

A list of vertices

z_ordered_entities() → List[List[Tuple[int, int]]]

Get the subentities of the cell in back-to-front plotting order.

Returns:

List of lists of subentity dimensions and numbers

default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

abstract make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

_compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

_compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

class symfem.references.DualPolygon(number_of_triangles: int, vertices: symfem.geometry.SetOfPointsInput | None = None)

Bases: Reference

A polygon on a barycentric dual grid.

reference_origin: symfem.geometry.PointType

abstract contains(point: symfem.geometry.PointType) → bool

Check if a point is contained in the reference.

Parameters:

point – The point

Returns:

Is the point contained in the reference?

abstract integration_limits(vars: symfem.symbols.AxisVariablesNotSingle = t) → IntLimits

Get the limits for an integral over this reference.

Parameters:

vars – The variables to use for each direction

Returns:

Integration limits that can be passed into sympy.integrate

abstract get_map_to(vertices: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the map from the reference to a cell.

Parameters:

vertices – The vertices of a call

Returns:

The map

abstract get_inverse_map_to(vertices_in: symfem.geometry.SetOfPointsInput) → symfem.geometry.PointType

Get the inverse map from a cell to the reference.

Parameters:

vertices_in – The vertices of a cell

Returns:

The inverse map

abstract _compute_map_to_self() → symfem.geometry.PointType

Compute the map from the canonical reference to this reference.

Returns:

The map

abstract _compute_inverse_map_to_self() → symfem.geometry.PointType

Compute the inverse map from the canonical reference to this reference.

Returns:

The map

abstract volume() → sympy.core.expr.Expr

Calculate the volume.

Returns:

The volume of the cell

abstract default_reference() → Reference

Get the default reference for this cell type.

Returns:

The default reference

make_lattice(n: int) → symfem.geometry.SetOfPoints

Make a lattice of points.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points offset from the edge of the cell

abstract make_lattice_with_lines(n: int) → LatticeWithLines

Make a lattice of points, and a list of lines connecting them.

Parameters:

n – The number of points along each edge

Returns:

A lattice of points including the edges of the cell Pairs of point numbers that make a mesh of lines across the cell

z_ordered_entities_extra_dim() → List[List[Tuple[int, int]]]

Get the subentities in back-to-front plotting order when using an extra dimension.

Returns:

List of lists of subentity dimensions and numbers