symfem.functionals

Functionals used to define the dual sets.

Module Contents

Classes

BaseFunctional	A functional.
PointEvaluation	A point evaluation.
WeightedPointEvaluation	A point evaluation.
DerivativePointEvaluation	A point evaluation of a given derivative.
PointDirectionalDerivativeEvaluation	A point evaluation of a derivative in a fixed direction.
PointNormalDerivativeEvaluation	A point evaluation of a normal derivative.
PointComponentSecondDerivativeEvaluation	A point evaluation of a component of a second derivative.
PointInnerProduct	An evaluation of an inner product at a point.
DotPointEvaluation	A point evaluation in a given direction.
PointDivergenceEvaluation	A point evaluation of the divergence.
IntegralAgainst	An integral against a function.
IntegralOfDivergenceAgainst	An integral of the divergence against a function.
IntegralOfDirectionalMultiderivative	An integral of a directional derivative of a scalar function.
IntegralMoment	An integral moment.
DerivativeIntegralMoment	An integral moment of the derivative of a scalar function.
DivergenceIntegralMoment	An integral moment of the divergence of a vector function.
TangentIntegralMoment	An integral moment in the tangential direction.
NormalIntegralMoment	An integral moment in the normal direction.
NormalDerivativeIntegralMoment	An integral moment in the normal direction.
InnerProductIntegralMoment	An integral moment of the inner product with a vector.
NormalInnerProductIntegralMoment	An integral moment of the inner product with the normal direction.

Functions

_to_tex(→ str)	Convert an expresson to TeX.
_nth(→ str)	Add th, st or nd to a number.

Attributes

ScalarValueOrFloat
ListOfFunctionals

symfem.functionals.ScalarValueOrFloat

symfem.functionals._to_tex(f: symfem.functions.FunctionInput, tfrac: bool = False) → str

Convert an expresson to TeX.

Parameters:

• f – the function

• tfrac – Should \tfrac be used in the place of \frac?

Returns:

The function as TeX

symfem.functionals._nth(n: int) → str

Add th, st or nd to a number.

Parameters:

n – The number

Returns:

The string (n)th, (n)st, (n)rd or (n)nd

class symfem.functionals.BaseFunctional(reference: symfem.references.Reference, entity: Tuple[int, int], mapping: str | None)

Bases: abc.ABC

A functional.

name = 'Base functional'

entity_dim() → int

Get the dimension of the entity this DOF is associated with.

Returns:

The dimension of the entity this DOF is associated with

entity_number() → int

Get the number of the entity this DOF is associated with.

Returns:

The number of the entity this DOF is associated with

perform_mapping(fs: List[symfem.functions.AnyFunction], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → List[symfem.functions.AnyFunction]

Map functions to a cell.

Parameters:

• fs – functions

• map – Map from the reference cell to a physical cell

• inverse_map – Map to the reference cell from a physical cell

Returns:

Mapped functions

entity_tex() → str

Get the entity the DOF is associated with in TeX format.

Returns:

TeX representation of entity

entity_definition() → str

Get the definition of the entity the DOF is associated with.

Returns:

The definition

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

eval(function: symfem.functions.AnyFunction, symbolic: bool = True) → symfem.functions.ScalarFunction | float

Apply to the functional to a function.

Parameters:

• function – The function

• symbolic – Should it be applied symbolically?

Returns:

The value of the functional for the function

abstract dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.ScalarFunction

Symbolically apply the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

abstract _eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

abstract get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation.

name = 'Point evaluation'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.WeightedPointEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, weight: sympy.core.expr.Expr, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation.

name = 'Weighted point evaluation'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.DerivativePointEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, derivative: Tuple[int, Ellipsis], entity: Tuple[int, int], mapping: str | None = None)

Bases: BaseFunctional

A point evaluation of a given derivative.

name = 'Point derivative evaluation'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

perform_mapping(fs: List[symfem.functions.AnyFunction], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → List[symfem.functions.AnyFunction]

Map functions to a cell.

Parameters:

• fs – functions

• map – Map from the reference cell to a physical cell

• inverse_map – Map to the reference cell from a physical cell

Returns:

Mapped functions

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointDirectionalDerivativeEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, direction_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation of a derivative in a fixed direction.

name = 'Point evaluation of directional derivative'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointNormalDerivativeEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, edge: symfem.references.Reference, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: PointDirectionalDerivativeEvaluation

A point evaluation of a normal derivative.

name = 'Point evaluation of normal derivative'

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointComponentSecondDerivativeEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, component: Tuple[int, int], entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation of a component of a second derivative.

name = 'Point evaluation of Jacobian component'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointInnerProduct(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, lvec: symfem.functions.FunctionInput, rvec: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

An evaluation of an inner product at a point.

name = 'Point inner product'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.DotPointEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, vector_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation in a given direction.

name = 'Dot point evaluation'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.PointDivergenceEvaluation(reference: symfem.references.Reference, point_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

A point evaluation of the divergence.

name = 'Point evaluation of divergence'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

adjusted_dof_point() → symfem.geometry.PointType

Get the adjusted position of the DOF in the cell for plotting.

Returns:

The point

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.IntegralAgainst(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

An integral against a function.

f: symfem.functions.AnyFunction

name = 'Integral against'

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dot(function: symfem.functions.AnyFunction) → symfem.functions.ScalarFunction

Dot a function with the moment function.

Parameters:

function – The function

Returns:

The product of the function and the moment function

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.IntegralOfDivergenceAgainst(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: BaseFunctional

An integral of the divergence against a function.

name = 'Integral of divergence against'

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dot(function: symfem.functions.ScalarFunction) → symfem.functions.ScalarFunction

Dot a function with the moment function.

Parameters:

function – The function

Returns:

The product of the function and the moment function

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.IntegralOfDirectionalMultiderivative(reference: symfem.references.Reference, directions: symfem.geometry.SetOfPoints, orders: Tuple[int, Ellipsis], entity: Tuple[int, int], scale: int = 1, mapping: str | None = 'identity')

Bases: BaseFunctional

An integral of a directional derivative of a scalar function.

name = 'Integral of a directional derivative'

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

perform_mapping(fs: List[symfem.functions.AnyFunction], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → List[symfem.functions.AnyFunction]

Map functions to a cell.

Parameters:

• fs – functions

• map – Map from the reference cell to a physical cell

• inverse_map – Map to the reference cell from a physical cell

Returns:

Mapped functions

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.IntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'identity', map_function: bool = True)

Bases: BaseFunctional

An integral moment.

f: symfem.functions.AnyFunction

name = 'Integral moment'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dot(function: symfem.functions.AnyFunction) → symfem.functions.ScalarFunction

Dot a function with the moment function.

Parameters:

function – The function

Returns:

The product of the function and the moment function

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.DerivativeIntegralMoment(reference: symfem.references.Reference, f: symfem.functions.FunctionInput, dot_with_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: IntegralMoment

An integral moment of the derivative of a scalar function.

name = 'Derivative integral moment'

dot(function: symfem.functions.AnyFunction) → symfem.functions.ScalarFunction

Dot a function with the moment function.

Parameters:

function – The function

Returns:

The product of the function and the moment function

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

class symfem.functionals.DivergenceIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: IntegralMoment

An integral moment of the divergence of a vector function.

name = 'Integral moment of divergence'

_eval_symbolic(function: symfem.functions.AnyFunction) → symfem.functions.AnyFunction

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.TangentIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'covariant')

Bases: IntegralMoment

An integral moment in the tangential direction.

name = 'Tangential integral moment'

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.NormalIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'contravariant')

Bases: IntegralMoment

An integral moment in the normal direction.

name = 'Normal integral moment'

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.NormalDerivativeIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: DerivativeIntegralMoment

An integral moment in the normal direction.

name = 'Normal derivative integral moment'

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.InnerProductIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, inner_with_left_in: symfem.functions.FunctionInput, inner_with_right_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'identity')

Bases: IntegralMoment

An integral moment of the inner product with a vector.

name = 'Inner product integral moment'

dot(function: symfem.functions.AnyFunction) → symfem.functions.ScalarFunction

Take the inner product of a function with the moment direction.

Parameters:

function – The function

Returns:

The inner product of the function and the moment direction

dof_direction() → symfem.geometry.PointType | None

Get the direction of the DOF.

Returns:

The direction

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

class symfem.functionals.NormalInnerProductIntegralMoment(reference: symfem.references.Reference, f_in: symfem.functions.FunctionInput, dof: BaseFunctional, entity: Tuple[int, int], mapping: str | None = 'double_contravariant')

Bases: InnerProductIntegralMoment

An integral moment of the inner product with the normal direction.

name = 'Normal inner product integral moment'

get_tex() → Tuple[str, List[str]]

Get a representation of the functional as TeX, and list of terms involved.

Returns:

Representation of the functional as TeX, and list of terms involved

symfem.functionals.ListOfFunctionals