Reference documentation for deal.II version 9.2.0
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Public Member Functions | Static Public Member Functions | Private Member Functions | Private Attributes | List of all members
Functions::SymbolicFunction< dim, RangeNumberType > Class Template Reference

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

Inheritance diagram for Functions::SymbolicFunction< dim, RangeNumberType >:
[legend]

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::Expressionget_time_symbol () const
 
const std::vector< Differentiation::SD::Expression > & get_symbolic_function_expressions () const
 
const Differentiation::SD::types::substitution_mapget_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, RangeNumberType >
 Function (const unsigned int n_components=1, const time_type initial_time=0.0)
 
 Function (const Function &f)=default
 
virtual ~Function () override=0
 
Functionoperator= (const Function &f)
 
virtual void vector_value (const Point< dim > &p, Vector< RangeNumberType > &values) const
 
virtual void value_list (const std::vector< Point< dim >> &points, std::vector< RangeNumberType > &values, const unsigned int component=0) const
 
virtual void vector_value_list (const std::vector< Point< dim >> &points, std::vector< Vector< RangeNumberType >> &values) const
 
virtual void vector_values (const std::vector< Point< dim >> &points, std::vector< std::vector< RangeNumberType >> &values) const
 
virtual void vector_gradient (const Point< dim > &p, std::vector< Tensor< 1, dim, RangeNumberType >> &gradients) const
 
virtual void gradient_list (const std::vector< Point< dim >> &points, std::vector< Tensor< 1, dim, RangeNumberType >> &gradients, const unsigned int component=0) const
 
virtual void vector_gradients (const std::vector< Point< dim >> &points, std::vector< std::vector< Tensor< 1, dim, RangeNumberType >>> &gradients) const
 
virtual void vector_gradient_list (const std::vector< Point< dim >> &points, std::vector< std::vector< Tensor< 1, dim, RangeNumberType >>> &gradients) const
 
virtual void vector_laplacian (const Point< dim > &p, Vector< RangeNumberType > &values) const
 
virtual void laplacian_list (const std::vector< Point< dim >> &points, std::vector< RangeNumberType > &values, const unsigned int component=0) const
 
virtual void vector_laplacian_list (const std::vector< Point< dim >> &points, std::vector< Vector< RangeNumberType >> &values) const
 
virtual void vector_hessian (const Point< dim > &p, std::vector< SymmetricTensor< 2, dim, RangeNumberType >> &values) const
 
virtual void hessian_list (const std::vector< Point< dim >> &points, std::vector< SymmetricTensor< 2, dim, RangeNumberType >> &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, RangeNumberType >>> &values) const
 
std::size_t memory_consumption () const
 
- Public Member Functions inherited from FunctionTime< numbers::NumberTraits< RangeNumberType >::real_type >
 FunctionTime (const numbers::NumberTraits< RangeNumberType >::real_type initial_time=numbers::NumberTraits< RangeNumberType >::real_type(0.0))
 
virtual ~FunctionTime ()=default
 
numbers::NumberTraits< RangeNumberType >::real_type get_time () const
 
virtual void set_time (const numbers::NumberTraits< RangeNumberType >::real_type new_time)
 
virtual void advance_time (const numbers::NumberTraits< RangeNumberType >::real_type delta_t)
 
- Public Member Functions inherited from Subscriptor
 Subscriptor ()
 
 Subscriptor (const Subscriptor &)
 
 Subscriptor (Subscriptor &&) noexcept
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (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::Expressionget_default_coordinate_symbols ()
 
- Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (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::Expressionuser_function
 
Differentiation::SD::types::substitution_map user_substitution_map
 
Differentiation::SD::types::substitution_map additional_function_arguments
 
std::vector< Differentiation::SD::Expressionfunction
 
std::vector< Tensor< 1, dim, Differentiation::SD::Expression > > function_gradient
 
std::vector< Tensor< 2, dim, Differentiation::SD::Expression > > function_hessian
 
std::vector< Differentiation::SD::Expressionfunction_laplacian
 
Tensor< 1, dim, Differentiation::SD::Expressioncoordinate_symbols
 
Differentiation::SD::Expression time_symbol
 

Additional Inherited Members

- Public Types inherited from Function< dim, RangeNumberType >
using time_type = typename FunctionTime< typename numbers::NumberTraits< RangeNumberType >::real_type >::time_type
 
- Public Types inherited from FunctionTime< numbers::NumberTraits< RangeNumberType >::real_type >
using time_type = numbers::NumberTraits< RangeNumberType >::real_type
 
- Public Attributes inherited from Function< dim, RangeNumberType >
const unsigned int n_components
 
- Static Public Attributes inherited from Function< dim, RangeNumberType >
static const unsigned int dimension = dim
 

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 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
functionA vector of symbolic expressions of type Differentiation::SD::Expression, representing the components of this Function.
coordinate_symbolsA 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_symbolA symbolic variable representing time. It defaults to a symbolic variable named "t".
user_substitution_mapAny 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, RangeNumberType >.

◆ 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, RangeNumberType >.

◆ 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, RangeNumberType >.

◆ 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, RangeNumberType >.

◆ 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: