symfem.functionals

Functionals used to define the dual sets.

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.
TraceIntegralMoment	An integral moment of the trace of a matrix.
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.

Module Contents

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

Bases: abc.ABC

A functional.

reference

entity

mapping

property nderivs: int

Abstractmethod:

Number of derivatives.

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.Function], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → list[symfem.functions.Function]

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.Function, 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

abstractmethod 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.Function) → symfem.functions.ScalarFunction

Symbolically apply the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

abstractmethod _eval_symbolic(function: symfem.functions.Function) → symfem.functions.Function

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

abstractmethod 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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Base functional'

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.

point

property nderivs: int

Number of derivatives.

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

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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point evaluation'

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.

point

weight

property nderivs: int

Number of derivatives.

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

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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Weighted point evaluation'

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.

point

derivative

property nderivs: int

Number of derivatives.

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

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.Function], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → list[symfem.functions.Function]

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point derivative evaluation'

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.

point

dir

property nderivs: int

Number of derivatives.

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

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point evaluation of directional derivative'

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.

reference

property nderivs: int

Number of derivatives.

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point evaluation of normal derivative'

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.

point

component

property nderivs: int

Number of derivatives.

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

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point evaluation of Hessian component'

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.

point

lvec

rvec

property nderivs: int

Number of derivatives.

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

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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point inner product'

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.

point

vector

property nderivs: int

Number of derivatives.

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

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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Dot point evaluation'

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.

point

property nderivs: int

Number of derivatives.

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

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Point evaluation of divergence'

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.Function

integral_domain

property nderivs: int

Number of derivatives.

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

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

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dot(function: symfem.functions.Function) → 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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Integral against'

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.

integral_domain

f

property nderivs: int

Number of derivatives.

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

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

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Integral of divergence against'

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.

integral_domain

directions

orders

scale = 1

property nderivs: int

Number of derivatives.

dof_point() → symfem.geometry.PointType

Get the location of the DOF in the cell.

Returns:

The point

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

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.Function], map: symfem.geometry.PointType, inverse_map: symfem.geometry.PointType) → list[symfem.functions.Function]

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Integral of a directional derivative'

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.Function

integral_domain

dof

property nderivs: int

Number of derivatives.

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

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

dot(function: symfem.functions.Function) → 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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Integral moment'

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.

dot_with

property nderivs: int

Number of derivatives.

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

Apply to the functional to a function.

Parameters:

function – The function

Returns:

The value of the functional for the function

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Derivative integral moment'

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.

property nderivs: int

Number of derivatives.

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

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Integral moment of divergence'

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.

_scalar_f

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

name = 'Tangential integral moment'

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.

_scalar_f

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

name = 'Normal integral moment'

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

Bases: IntegralMoment

An integral moment of the trace of a matrix.

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

Take the product of a function with the trace of the matrix.

Parameters:

function – The function

Returns:

The inner product of the function and the matrix’s trace

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

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Trace integral moment'

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.

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

property nderivs: int

Number of derivatives.

abstractmethod discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Normal derivative integral moment'

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.

inner_with_left

inner_with_right

dot(function: symfem.functions.Function) → 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

discrete(poly_degree: int) → DiscreteDof

Get points and weights that define this DOF discretely.

Parameters:

poly_degree – The polynomial degree of the element. This may be used to decide which degree quadrature rule to use

Returns:

Points (a list of lists whose indices are [point_index][dimension]) and weights (a list of list of lists whose indices are [dimension][point_index][derivative])

name = 'Inner product integral moment'

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.

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

name = 'Normal inner product integral moment'