#include <deal.II/base/symbolic_function.h>

Inheritance diagram for Functions::SymbolicFunction< dim, RangeNumberType >:

Public Member Functions
	SymbolicFunction (const std::vector< Differentiation::SD::Expression > &function, const Tensor< 1, dim, Differentiation::SD::Expression > &coordinate_symbols=get_default_coordinate_symbols(), const Differentiation::SD::Expression &time_symbol=Differentiation::SD::make_symbol("t"), const Differentiation::SD::types::substitution_map &user_substitution_map={})

	SymbolicFunction (const std::string &expressions)

void	update_user_substitution_map (const Differentiation::SD::types::substitution_map &substitutions)

void	set_additional_function_arguments (const Differentiation::SD::types::substitution_map &arguments)

const Tensor< 1, dim, Differentiation::SD::Expression > &	get_coordinate_symbols () const

const Differentiation::SD::Expression &	get_time_symbol () const

const std::vector< Differentiation::SD::Expression > &	get_symbolic_function_expressions () const

const Differentiation::SD::types::substitution_map &	get_user_substitution_map () const

SymbolicFunction< dim, RangeNumberType >	time_derivative () const

virtual RangeNumberType	value (const Point< dim > &p, const unsigned int component=0) const override

virtual Tensor< 1, dim, RangeNumberType >	gradient (const Point< dim > &p, const unsigned int component=0) const override

virtual RangeNumberType	laplacian (const Point< dim > &p, const unsigned int component=0) const override

virtual SymmetricTensor< 2, dim, RangeNumberType >	hessian (const Point< dim > &p, const unsigned int component=0) const override

template<typename StreamType >
StreamType &	print (StreamType &out) const

Public Member Functions inherited from Function< dim, double >
	Function (const unsigned int n_components=1, const time_type initial_time=0.0)

	Function (const Function &f)=default

virtual	~Function () override=0

Function &	operator= (const Function &f)

virtual void	vector_value (const Point< dim > &p, Vector< double > &values) const

virtual void	value_list (const std::vector< Point< dim >> &points, std::vector< double > &values, const unsigned int component=0) const

virtual void	vector_value_list (const std::vector< Point< dim >> &points, std::vector< Vector< double >> &values) const

virtual void	vector_values (const std::vector< Point< dim >> &points, std::vector< std::vector< double >> &values) const

virtual void	vector_gradient (const Point< dim > &p, std::vector< Tensor< 1, dim, double >> &gradients) const

virtual void	gradient_list (const std::vector< Point< dim >> &points, std::vector< Tensor< 1, dim, double >> &gradients, const unsigned int component=0) const

virtual void	vector_gradients (const std::vector< Point< dim >> &points, std::vector< std::vector< Tensor< 1, dim, double >>> &gradients) const

virtual void	vector_gradient_list (const std::vector< Point< dim >> &points, std::vector< std::vector< Tensor< 1, dim, double >>> &gradients) const

virtual void	vector_laplacian (const Point< dim > &p, Vector< double > &values) const

virtual void	laplacian_list (const std::vector< Point< dim >> &points, std::vector< double > &values, const unsigned int component=0) const

virtual void	vector_laplacian_list (const std::vector< Point< dim >> &points, std::vector< Vector< double >> &values) const

virtual void	vector_hessian (const Point< dim > &p, std::vector< SymmetricTensor< 2, dim, double >> &values) const

virtual void	hessian_list (const std::vector< Point< dim >> &points, std::vector< SymmetricTensor< 2, dim, double >> &values, const unsigned int component=0) const

virtual void	vector_hessian_list (const std::vector< Point< dim >> &points, std::vector< std::vector< SymmetricTensor< 2, dim, double >>> &values) const

std::size_t	memory_consumption () const

Public Member Functions inherited from FunctionTime< numbers::NumberTraits< double >::real_type >
	FunctionTime (const numbers::NumberTraits< double >::real_type initial_time=numbers::NumberTraits< double >::real_type(0.0))

virtual	~FunctionTime ()=default

numbers::NumberTraits< double >::real_type	get_time () const

virtual void	set_time (const numbers::NumberTraits< double >::real_type new_time)

virtual void	advance_time (const numbers::NumberTraits< double >::real_type delta_t)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Static Public Member Functions
static Tensor< 1, dim, Differentiation::SD::Expression >	get_default_coordinate_symbols ()

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Private Member Functions
Differentiation::SD::types::substitution_map	create_evaluation_substitution_map (const Point< dim > &point) const

void	update_values () const

void	update_first_derivatives () const

void	update_second_derivatives () const

Private Attributes
const std::vector< Differentiation::SD::Expression >	user_function

Differentiation::SD::types::substitution_map	user_substitution_map

Differentiation::SD::types::substitution_map	additional_function_arguments

std::vector< Differentiation::SD::Expression >	function

std::vector< Tensor< 1, dim, Differentiation::SD::Expression > >	function_gradient

std::vector< Tensor< 2, dim, Differentiation::SD::Expression > >	function_hessian

std::vector< Differentiation::SD::Expression >	function_laplacian

Tensor< 1, dim, Differentiation::SD::Expression >	coordinate_symbols

Differentiation::SD::Expression	time_symbol

Additional Inherited Members
Public Types inherited from Function< dim, double >
using	time_type = typename FunctionTime< typename numbers::NumberTraits< double >::real_type >::time_type

Public Types inherited from FunctionTime< numbers::NumberTraits< double >::real_type >
using	time_type = numbers::NumberTraits< double >::real_type

Public Attributes inherited from Function< dim, double >
const unsigned int	n_components

Static Public Attributes inherited from Function< dim, double >
static const unsigned int	dimension

Detailed Description

template<int dim, typename RangeNumberType = double>
class Functions::SymbolicFunction< dim, RangeNumberType >

A Function class that leverages symbolic differentiation to compute gradients, Laplacians, Hessians, and time derivatives.

This class can be used to define functions using methods provided by the Differentiation::SD namespace. In particular, one can define a symbolic evaluation point (the argument of the function), as well as a symbolic expression.

The symbolic gradients and the symbolic Hessians are computed at construction time, and when a substitution in the symbolic functions is requested by the user using the method update_user_substitution_map().

Whenever one of the evaluation methods is called, a substitution is attempted with the coordinate symbols argument replaced by the evaluation point and the symbolic time replaced by the current time, as returned by the get_time() method. The user has to make sure that at evaluation time argument substitution provides a fully evaluated expression (i.e., no other symbols are contained in the function expression, except numerical values), or an exception will be thrown. Additional symbols can be partially evaluated or substituted by storing them in a user supplied substitution maps, that can be updated by calling update_user_substitution_map() or the set_additional_function_arguments() methods.

The simplest use case of this class is given in the following example:

SymbolicFunction<2> fun("x^2+y; t*x*y");
fun.set_time(3.0);
Point<2> p(1.0, 2.0);
 
auto a = fun.value(p, / * component * / 0); // a = 3.0
auto b = fun.value(p, / * component * / 1); // b = 6.0
 
auto df_dt = fun.time_derivative();
 
auto c = df_dt.value(p, / * component * / 0); // c = 0.0
auto d = df_dt.value(p, / * component * / 1); // d = 2.0

where a Function with two components is defined using a string containing their expressions separated by semicolons.

A more involved example, that explicitly uses Differentiation::SD::Expression objects, is given by

using namespace Differentiation::SD;
// Create a position Tensor<1,2,Differentiation::SD::Expression>
// with symbols "x" and "y", and the symbol "t"
const auto x = SymbolicFunction<2>::get_default_coordinate_symbols();
const auto t = make_symbol("t");
 
// Use directly x[0] (the symbol "x"), x[1] (the symbol "y"), and t
// (the symbol "t").
Expression f = std::sin(x[0])*std::cos(x[1])*std::sin(t);
// Alternatively, you can achieve the same result parsing a string:
// Expression f("sin(x)*cos(y)*sin(t)", true);
SymbolicFunction<2> function({f}, x);
 
// Evaluate the function, its gradient, and its Laplacian
Point<2> p(1.0, 2.0);
auto fp = function.value(p);
auto gradfp = function.gradient(p);
auto lapfp = function.laplacian(p);
 
// Evaluate the time derivative of the function, its gradient, and its
// Laplacian
auto time_derivative = function.time_derivative();
auto dt_fp = time_derivative.value(p);
auto dt_gradfp = time_derivative.gradient(p);
auto dt_lapfp = time_derivative.laplacian(p);

Partial substitution is possible (i.e., you can define the function using additional symbols). However, as soon as you evaluate the function, you have to make sure that all extraneous symbols (i.e., those not referring to the spacial coordinate_symbols or to the time_symbol variable) have been substituted with numerical values, or expressions of the spatial or temporal argument, by calling the update_user_substitution_map() or the set_additional_function_arguments() methods.

If your function requires additional arguments to be evaluated, you can specify them by calling the set_additional_function_arguments() method.

If you call update_user_substitution_map() and set_additional_function_arguments() with the same argument, the effect on the function evaluation will be the same, however, the internal behaviour and function derivatives will be different. The method update_user_substitution_map() performs the substitution once (the first time it is required), and then stores internally a copy of the resulting expression, together with its derivatives (if required). These are then used in all subsequent evaluations. Calling set_additional_function_arguments() will evaluate the passed substitution map on the fly during evaluation time, after all derivatives have been computed.

Note: The difference between this class and the FunctionParser class is that this class allows to compute first and second order derivatives (in a symbolic way), while the FunctionParser class computes first order derivatives only, using finite differences. For complicated expressions, this class may be slower than the FunctionParser class.

Author: Luca Heltai 2019

Definition at line 154 of file symbolic_function.h.

Constructor & Destructor Documentation

◆ SymbolicFunction() [1/2]

template<int dim, typename RangeNumberType = double>

Functions::SymbolicFunction< dim, RangeNumberType >::SymbolicFunction	(	const std::vector< Differentiation::SD::Expression > &	function,
		const Tensor< 1, dim, Differentiation::SD::Expression > &	coordinate_symbols = `get_default_coordinate_symbols()`,
		const Differentiation::SD::Expression &	time_symbol = `Differentiation::SD::make_symbol("t")`,
		const Differentiation::SD::types::substitution_map &	user_substitution_map = `{}`
	)

Constructor.

The resulting Function object will have as many components as there are entries in the vector of symbolic expressions function.

The vector function should contain a list of symbolic expression involving the coordinate symbols argument coordinate_symbols and possibly the symbolic time argument time_symbol. It is possible to define it in terms of other symbols, as long as the optional parameter user_substitution_map replaces all symbols except coordinate_symbols and time_symbol. This is useful if, for example, you want to express formulas in terms of material parameters that you want to name symbolically, rather than through their numeric values when defining the formula, or when you want to express your formula in terms of polar coordinates rather than cartesian ones, and you want the symbolic engine to compute the derivatives for you. You may later update the symbol map contained in user_substitution_map by calling update_user_substitution_map().

Parameters

function	A vector of symbolic expressions of type Differentiation::SD::Expression, representing the components of this Function.
coordinate_symbols	A tensor of symbols representing coordinates, used as input argument in the symbolic expressions contained in the `function` vector. The default `coordinate_symbols` is a Tensor<1,dim,Differentiation::SD::Expression> containing the symbols "x" for `dim` equal to one, "x", "y" for `dim` equal to two, and "x", "y", "z" for `dim` equal to three.
time_symbol	A symbolic variable representing time. It defaults to a symbolic variable named "t".
user_substitution_map	Any other symbol that may be contained in the symbolic function needs to be specified in this map. The map may be empty, and the functions may still contain unevaluated symbols, provided that you call update_user_substitution_map() and provide a replacement of all symbols except `coordinate_symbols` and `time_symbol` before any evaluation occurs.

◆ SymbolicFunction() [2/2]

template<int dim, typename RangeNumberType = double>

Functions::SymbolicFunction< dim, RangeNumberType >::SymbolicFunction ( const std::string & expressions )

Constructor that takes a single string that describes the function expression as a semicolon separated list of expressions.

The symbolic expression can use the default argument and the default symbolic time variable, plus any additional symbols that you may need, provided that you update the user substitution map that substitutes all of them before you try to evaluate the function or its derivatives, by calling update_user_substitution_map(), and that you provide all the additional function arguments of your function using the method set_additional_function_arguments().

Member Function Documentation

◆ update_user_substitution_map()

template<int dim, typename RangeNumberType = double>

void Functions::SymbolicFunction< dim, RangeNumberType >::update_user_substitution_map ( const Differentiation::SD::types::substitution_map & substitutions )

Store and apply the substitution map substitutions to each symbolic component of this Function object.

Notice that this method will trigger a recomputation of the gradients, Hessians, and Laplacians of each component.

◆ set_additional_function_arguments()

template<int dim, typename RangeNumberType = double>

void Functions::SymbolicFunction< dim, RangeNumberType >::set_additional_function_arguments ( const Differentiation::SD::types::substitution_map & arguments )

Set the additional arguments to be substituted in next evaluation step.

Notice that the arguments are substituted after evaluating the permanent_user_substitution_map, and after all derivatives are computed. If the additional arguments you pass still depend on the coordinate or time symbols, then evaluation of derivatives will result in a partial derivative evaluation.

This method provides a way to evaluate functions that depend on more arguments than simply the coordinates and time. If you want to compute the total derivative w.r.t. to complicated symbolic expressions, you should call update_user_substitution_map() instead.

◆ get_default_coordinate_symbols()

template<int dim, typename RangeNumberType = double>

static Tensor<1, dim, Differentiation::SD::Expression> Functions::SymbolicFunction< dim, RangeNumberType >::get_default_coordinate_symbols ( )

static

Return a tensor of coordinate symbols that can be used to define the expressions of this symbolic function object.

The default argument is a Tensor<1,dim,Differentiation::SD::Expression> containing the symbols "x" for dim equal to one, "x", "y" for dim equal to two, and "x", "y", "z" for dim equal to three.

◆ get_coordinate_symbols()

template<int dim, typename RangeNumberType = double>

const Tensor<1, dim, Differentiation::SD::Expression>& Functions::SymbolicFunction< dim, RangeNumberType >::get_coordinate_symbols ( ) const

Get the actual arguments used for the coordinates in the symbolic function. This object does not include any user-defined arguments.

◆ get_time_symbol()

template<int dim, typename RangeNumberType = double>

const Differentiation::SD::Expression& Functions::SymbolicFunction< dim, RangeNumberType >::get_time_symbol ( ) const

Get the actual symbolic time in use in this symbolic function.

◆ get_symbolic_function_expressions()

template<int dim, typename RangeNumberType = double>

const std::vector<Differentiation::SD::Expression>& Functions::SymbolicFunction< dim, RangeNumberType >::get_symbolic_function_expressions ( ) const

Get the actual symbolic expressions used in this symbolic function.

◆ get_user_substitution_map()

template<int dim, typename RangeNumberType = double>

const Differentiation::SD::types::substitution_map& Functions::SymbolicFunction< dim, RangeNumberType >::get_user_substitution_map ( ) const

Get the currently stored user_substitution_map.

◆ time_derivative()

template<int dim, typename RangeNumberType = double>

SymbolicFunction<dim, RangeNumberType> Functions::SymbolicFunction< dim, RangeNumberType >::time_derivative ( ) const

Return a SymbolicFunction object that represents the time derivative of this function. The spatial argument, the symbolic time, and the currently stored user substitution map are forwarded to the new function.

◆ value()

template<int dim, typename RangeNumberType = double>

virtual RangeNumberType Functions::SymbolicFunction< dim, RangeNumberType >::value	(	const Point< dim > &	p,
		const unsigned int	component = `0`
	)		const

overridevirtual

Return the value of the function at the given point. Unless there is only one component (i.e. the function is scalar), you should state the component you want to have evaluated; it defaults to zero, i.e. the first component.

Reimplemented from Function< dim, double >.

◆ gradient()

template<int dim, typename RangeNumberType = double>

virtual Tensor<1, dim, RangeNumberType> Functions::SymbolicFunction< dim, RangeNumberType >::gradient	(	const Point< dim > &	p,
		const unsigned int	component = `0`
	)		const

overridevirtual

Return the gradient of the specified component of the function at the given point.

Reimplemented from Function< dim, double >.

◆ laplacian()

template<int dim, typename RangeNumberType = double>

virtual RangeNumberType Functions::SymbolicFunction< dim, RangeNumberType >::laplacian	(	const Point< dim > &	p,
		const unsigned int	component = `0`
	)		const

overridevirtual

Compute the Laplacian of a given component at point p.

Reimplemented from Function< dim, double >.

◆ hessian()

template<int dim, typename RangeNumberType = double>

virtual SymmetricTensor<2, dim, RangeNumberType> Functions::SymbolicFunction< dim, RangeNumberType >::hessian	(	const Point< dim > &	p,
		const unsigned int	component = `0`
	)		const

overridevirtual

Compute the Hessian of a given component at point p, that is the gradient of the gradient of the function.

Reimplemented from Function< dim, double >.

◆ print()

template<int dim, typename RangeNumberType >

template<typename StreamType >

StreamType & Functions::SymbolicFunction< dim, RangeNumberType >::print ( StreamType & out ) const

Print the stored arguments and function expression, as it would be evaluated when calling the method value().

Definition at line 442 of file symbolic_function.h.

◆ create_evaluation_substitution_map()

template<int dim, typename RangeNumberType = double>

Differentiation::SD::types::substitution_map Functions::SymbolicFunction< dim, RangeNumberType >::create_evaluation_substitution_map ( const Point< dim > & point ) const

private

Return a substitution map that replaces the argument with the values of point, the symbolic time with the value of this->get_time(), and any additional arguments with the substitution map given by additional_function_arguments.

◆ update_values()

template<int dim, typename RangeNumberType = double>

void Functions::SymbolicFunction< dim, RangeNumberType >::update_values ( ) const

private

Recompute the symbolic value of the function, applying the user substitution map. This may be an expensive computation, and it is called only if necessary.

◆ update_first_derivatives()

template<int dim, typename RangeNumberType = double>

void Functions::SymbolicFunction< dim, RangeNumberType >::update_first_derivatives ( ) const

private

Recompute the symbolic gradient of the function, applying the user substitution map. This may be an expensive computation, and it is called only if necessary.

◆ update_second_derivatives()

template<int dim, typename RangeNumberType = double>

void Functions::SymbolicFunction< dim, RangeNumberType >::update_second_derivatives ( ) const

private

Recompute the symbolic Hessian and the symbolic Lapalacian of the function. This may be an expensive computation, and it is called only if necessary.

Member Data Documentation

◆ user_function

template<int dim, typename RangeNumberType = double>

const std::vector<Differentiation::SD::Expression> Functions::SymbolicFunction< dim, RangeNumberType >::user_function

private

The components of this symbolic function, before any subustitution took place. This is immutable, and generated at construction time.

Before any evaluation takes place, the user_substitution_map is applied to this object, and the result is stored in the internal variable function.

During evaluation, the symbolic_coordinate, the symbolic_time, and any remaining symbols are substituted with the input evaluation point, the current time, and the content of additional_function_arguments.

Definition at line 369 of file symbolic_function.h.

◆ user_substitution_map

template<int dim, typename RangeNumberType = double>

Differentiation::SD::types::substitution_map Functions::SymbolicFunction< dim, RangeNumberType >::user_substitution_map

private

Store the user substitution map used for expression substitutions. This may be updated with a call to update_user_substitution_map(). Notice that the function may still have unresolved symbols, provided that they are resolved by a call to set_additional_function_arguments().

Definition at line 377 of file symbolic_function.h.

◆ additional_function_arguments

template<int dim, typename RangeNumberType = double>

Differentiation::SD::types::substitution_map Functions::SymbolicFunction< dim, RangeNumberType >::additional_function_arguments

private

Store a user substitution map used for additional argument substitutions. This will be updated by a call to set_additional_function_arguments().

Definition at line 384 of file symbolic_function.h.

◆ function

template<int dim, typename RangeNumberType = double>

std::vector<Differentiation::SD::Expression> Functions::SymbolicFunction< dim, RangeNumberType >::function

mutableprivate

The actual components of this symbolic function. This is obtained from the user_function, after applying the user_substitution_map.

Definition at line 390 of file symbolic_function.h.

◆ function_gradient

template<int dim, typename RangeNumberType = double>

std::vector<Tensor<1, dim, Differentiation::SD::Expression> > Functions::SymbolicFunction< dim, RangeNumberType >::function_gradient

mutableprivate

The gradients of each component of this symbolic function. This is obtained by computing the symbolic gradient of the object function, that is, after applying the user_substitution_map to user_function.

Definition at line 398 of file symbolic_function.h.

◆ function_hessian

template<int dim, typename RangeNumberType = double>

std::vector<Tensor<2, dim, Differentiation::SD::Expression> > Functions::SymbolicFunction< dim, RangeNumberType >::function_hessian

mutableprivate

The Hessians of each component of this symbolic function. This is obtained by computing the symbolic Hessian of the object function, that is, after applying the user_substitution_map to user_function.

Definition at line 406 of file symbolic_function.h.

◆ function_laplacian

template<int dim, typename RangeNumberType = double>

std::vector<Differentiation::SD::Expression> Functions::SymbolicFunction< dim, RangeNumberType >::function_laplacian

mutableprivate

The Laplacians of each component of this symbolic function. This is obtained by computing the symbolic Laplacian of the object function, that is, after applying the user_substitution_map to user_function.

Definition at line 413 of file symbolic_function.h.

◆ coordinate_symbols

template<int dim, typename RangeNumberType = double>

Tensor<1, dim, Differentiation::SD::Expression> Functions::SymbolicFunction< dim, RangeNumberType >::coordinate_symbols

private

The coordinate symbols argument of the function.

Definition at line 418 of file symbolic_function.h.

◆ time_symbol

template<int dim, typename RangeNumberType = double>

Differentiation::SD::Expression Functions::SymbolicFunction< dim, RangeNumberType >::time_symbol

mutableprivate

The symbolic time argument of the function.

Definition at line 423 of file symbolic_function.h.

The documentation for this class was generated from the following file:

include/deal.II/base/symbolic_function.h

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<int dim, typename RangeNumberType = double> class Functions::SymbolicFunction< dim, RangeNumberType >

Constructor & Destructor Documentation

◆ SymbolicFunction() [1/2]

◆ SymbolicFunction() [2/2]

Member Function Documentation

◆ update_user_substitution_map()

◆ set_additional_function_arguments()

◆ get_default_coordinate_symbols()

◆ get_coordinate_symbols()

◆ get_time_symbol()

◆ get_symbolic_function_expressions()

◆ get_user_substitution_map()

◆ time_derivative()

◆ value()

◆ gradient()

◆ laplacian()

◆ hessian()

◆ print()

◆ create_evaluation_substitution_map()

◆ update_values()

◆ update_first_derivatives()

◆ update_second_derivatives()

Member Data Documentation

◆ user_function

◆ user_substitution_map

◆ additional_function_arguments

◆ function

◆ function_gradient

◆ function_hessian

◆ function_laplacian

◆ coordinate_symbols

◆ time_symbol

template<int dim, typename RangeNumberType = double>
class Functions::SymbolicFunction< dim, RangeNumberType >