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// $Id: function.h 30036 2013-07-18 16:55:32Z maier $
//
// Copyright (C) 1998 - 2013 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef __deal2__function_h
#define __deal2__function_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/function_time.h>
#include <deal.II/base/subscriptor.h>
#include <deal.II/base/tensor.h>
#include <deal.II/base/point.h>
#include <deal.II/base/std_cxx1x/function.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
template <typename number> class Vector;
template <int rank, int dim> class TensorFunction;
/**
* This class is a model for a general function that, given a point at which
* to evaluate the function, returns a vector of values with one or more
* components.
*
* The class serves the purpose
* of representing both scalar and vector valued functions. To this end, we
* consider scalar functions as a special case of vector valued
* functions, in the former case only having a single component return
* vector. Since handling vectors is comparatively expensive, the interface
* of this class has functions which only ask for a single component of the
* vector-valued results (this is what you will usually need in case you know that
* your function is scalar-valued) as well as functions you can ask for an
* entire vector of results with as many components as the function object
* represents. Access to function objects therefore is through the following
* methods:
* @code
* // access to one component at one point
* double value (const Point<dim> &p,
* const unsigned int component = 0) const;
*
* // return all components at one point
* void vector_value (const Point<dim> &p,
* Vector<double> &value) const;
* @endcode
*
* For more efficiency, there are other functions returning one or all
* components at a list of points at once:
* @code
* // access to one component at several points
* void value_list (const std::vector<Point<dim> > &point_list,
* std::vector<double> &value_list,
* const unsigned int component = 0) const;
*
* // return all components at several points
* void vector_value_list (const std::vector<Point<dim> > &point_list,
* std::vector<Vector<double> > &value_list) const;
* @endcode
*
* Furthermore, there are functions returning the gradient of the
* function or even higher derivatives at one or several points.
*
* You will usually only overload those functions you need; the
* functions returning several values at a time (value_list(),
* vector_value_list(), and gradient analogs) will call those
* returning only one value (value(), vector_value(), and gradient
* analogs), while those ones will throw an exception when called but
* not overloaded.
*
* Note however, that conversely the functions returning all components of the
* function at one or several points (i.e. vector_value(),
* vector_value_list()), will not call the function returning one
* component at one point repeatedly, once for each point and
* component. The reason is efficiency: this would amount to too many
* virtual function calls. If you have vector-valued functions, you
* should therefore also provide overloads of the virtual functions
* for all components at a time.
*
* Also note, that unless only called a very small number of times,
* you should overload all sets of functions (returning only one
* value, as well as those returning a whole array), since the cost of
* evaluation of a point value is often less than the virtual function
* call itself.
*
*
* Support for time dependent functions can be found in the base
* class FunctionTime.
*
* @note If the functions you are dealing with have sizes which
* are a priori known (for example, <tt>dim</tt> elements), you might
* consider using the TensorFunction class instead. On the other hand,
* functions like VectorTools::interpolate or
* VectorTools::interpolate_boundary_values definitely only want objects
* of the current type. You can use the VectorFunctionFromTensorFunction
* class to convert the former to the latter.
*
* @ingroup functions
* @author Wolfgang Bangerth, 1998, 1999
*/
template <int dim>
class Function : public FunctionTime,
public Subscriptor
{
public:
/**
* Export the value of the template parameter as a static member
* constant. Sometimes useful for some expression template programming.
*/
static const unsigned int dimension = dim;
/**
* Number of vector components.
*/
const unsigned int n_components;
/**
* Constructor. May take an initial value for the number of components
* (which defaults to one, i.e. a scalar function), and the time
* variable, which defaults to zero.
*/
Function (const unsigned int n_components = 1,
const double initial_time = 0.0);
/**
* Virtual destructor; absolutely necessary in this case.
*
* This destructor is declared pure virtual, such that objects of this class
* cannot be created. Since all the other virtual functions have a
* pseudo-implementation to avoid overhead in derived classes, they can not
* be abstract. As a consequence, we could generate an object of this class
* because none of this class's functions are abstract.
*
* We circumvent this problem by making the destructor of this class
* abstract virtual. This ensures that at least one member function is
* abstract, and consequently, no objects of type Function can be
* created. However, there is no need for derived classes to explicitly
* implement a destructor: every class has a destructor, either explicitly
* implemented or implicitly generated by the compiler, and this resolves
* the abstractness of any derived class even if they do not have an
* explicitly declared destructor.
*
* Nonetheless, since derived classes want to call the destructor of a
* base class, this destructor is implemented (despite it being pure
* virtual).
*/
virtual ~Function () = 0;
/**
* Assignment operator. This is here only so that you can have objects of
* derived classes in containers, or assign them otherwise. It will raise
* an exception if the object from which you assign has a different
* number of components than the one being assigned to.
*/
Function &operator= (const Function &f);
/**
* 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.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Return all components of a vector-valued function at a given point.
*
* <tt>values</tt> shall have the right size beforehand, i.e.
* #n_components.
*/
virtual void vector_value (const Point<dim> &p,
Vector<double> &values) const;
/**
* Set <tt>values</tt> to the point values of the specified component of
* the function at the <tt>points</tt>. It is assumed that
* <tt>values</tt> already has the right size, i.e. the same size as the
* <tt>points</tt> array.
*
* Be default, this function repeatedly calls value() for each point
* separately, to fill the output array.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Set <tt>values</tt> to the point values of the function at the
* <tt>points</tt>. It is assumed that <tt>values</tt> already has the
* right size, i.e. the same size as the <tt>points</tt> array, and that
* all elements be vectors with the same number of components as this
* function has.
*
* Be default, this function repeatedly calls vector_value() for each
* point separately, to fill the output array.
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
/**
* For each component of the function, fill a vector of values, one for
* each point.
*
* The default implementation of this function in Function calls
* value_list() for each component. In order to improve performance, this
* can be reimplemented in derived classes to speed up performance.
*/
virtual void vector_values (const std::vector<Point<dim> > &points,
std::vector<std::vector<double> > &values) const;
/**
* Return the gradient of the specified component of the function at the
* given point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Return the gradient of all components of the function at the given
* point.
*/
virtual void vector_gradient (const Point<dim> &p,
std::vector<Tensor<1,dim> > &gradients) const;
/**
* Set <tt>gradients</tt> to the gradients of the specified component of
* the function at the <tt>points</tt>. It is assumed that
* <tt>gradients</tt> already has the right size, i.e. the same size as
* the <tt>points</tt> array.
*/
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
/**
* For each component of the function, fill a vector of gradient values,
* one for each point.
*
* The default implementation of this function in Function calls
* value_list() for each component. In order to improve performance, this
* can be reimplemented in derived classes to speed up performance.
*/
virtual void vector_gradients (const std::vector<Point<dim> > &points,
std::vector<std::vector<Tensor<1,dim> > > &gradients) const;
/**
* Set <tt>gradients</tt> to the gradients of the function at the
* <tt>points</tt>, for all components. It is assumed that
* <tt>gradients</tt> already has the right size, i.e. the same size as
* the <tt>points</tt> array.
*
* The outer loop over <tt>gradients</tt> is over the points in the list,
* the inner loop over the different components of the function.
*/
virtual void vector_gradient_list (const std::vector<Point<dim> > &points,
std::vector<std::vector<Tensor<1,dim> > > &gradients) const;
/**
* Compute the Laplacian of a given component at point <tt>p</tt>.
*/
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Compute the Laplacian of all components at point <tt>p</tt> and store
* them in <tt>values</tt>.
*/
virtual void vector_laplacian (const Point<dim> &p,
Vector<double> &values) const;
/**
* Compute the Laplacian of one component at a set of points.
*/
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Compute the Laplacians of all components at a set of points.
*/
virtual void vector_laplacian_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
/**
* Determine an estimate for the memory consumption (in bytes) of this
* object. Since sometimes the size of objects can not be determined
* exactly (for example: what is the memory consumption of an STL
* <tt>std::map</tt> type with a certain number of elements?), this is
* only an estimate. however often quite close to the true value.
*/
std::size_t memory_consumption () const;
};
/**
* Provide a function which always returns zero. Obviously, also the
* derivates of this function are zero. Also, it returns zero on all
* components in case the function is not a scalar one, which can be
* obtained by passing the constructor the appropriate number of
* components.
*
* This function is of use when you want to implement homogeneous boundary
* conditions, or zero initial conditions.
*
* @ingroup functions
* @author Wolfgang Bangerth, 1998, 1999
*/
template <int dim>
class ZeroFunction : public Function<dim>
{
public:
/**
* Constructor. The number of components is preset to one.
*/
ZeroFunction (const unsigned int n_components = 1);
/**
* Virtual destructor; absolutely necessary in this case.
*
*/
virtual ~ZeroFunction ();
virtual double value (const Point<dim> &p,
const unsigned int component) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &return_value) 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 Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void vector_gradient (const Point<dim> &p,
std::vector<Tensor<1,dim> > &gradients) const;
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
virtual void vector_gradient_list (const std::vector<Point<dim> > &points,
std::vector<std::vector<Tensor<1,dim> > > &gradients) const;
};
/**
* Provide a function which always returns the constant value
* handed to the constructor.
*
* Obviously, the derivates of this function are zero, which is why we
* derive this class from <tt>ZeroFunction</tt>: we then only have to
* overload the value functions, not all the derivatives. In some way, it
* would be more obvious to do the derivation in the opposite direction,
* i.e. let <tt>ZeroFunction</tt> be a more specialized version of
* <tt>ConstantFunction</tt>; however, this would be less efficient, since
* we could not make use of the fact that the function value of the
* <tt>ZeroFunction</tt> is known at compile time and need not be looked up
* somewhere in memory.
*
* You can pass to the constructor an integer denoting the number of
* components this function shall have. It defaults to one. If it is
* greater than one, then the function will return the constant value in
* all its components, which might not be overly useful a feature in most
* cases, however.
*
* @ingroup functions
* @author Wolfgang Bangerth, 1998, 1999
*/
template <int dim>
class ConstantFunction : public ZeroFunction<dim>
{
public:
/**
* Constructor; takes the constant function value as an argument. The
* number of components is preset to one.
*/
ConstantFunction (const double value,
const unsigned int n_components = 1);
/**
* Virtual destructor; absolutely necessary in this case.
*/
virtual ~ConstantFunction ();
virtual double value (const Point<dim> &p,
const unsigned int component) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &return_value) 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;
std::size_t memory_consumption () const;
protected:
/**
* Store the constant function value.
*/
const double function_value;
};
/**
* This is a constant vector-valued function, in which one or more
* components of the vector have a constant value and all other components
* are zero. It is especially useful as a weight function for
* VectorTools::integrate_difference, where it allows to integrate only one
* or a few vector components, rather than the entire vector-valued
* solution. In other words, it acts as a component mask with a single
* component selected (see the @ref GlossComponentMask "the glossary entry
* on component masks"). See the step-20 tutorial program for a detailed
* explanation and a use case.
*
* @ingroup functions
* @author Guido Kanschat, 2000, Wolfgang Bangerth 2006
*/
template <int dim>
class ComponentSelectFunction : public ConstantFunction<dim>
{
public:
/**
* Constructor if only a single component shall be non-zero. Arguments
* denote the component selected, the value for that component and the
* total number of vector components.
*/
ComponentSelectFunction (const unsigned int selected,
const double value,
const unsigned int n_components);
/**
* Constructor. As before, but the value for the selected component is
* assumed to be one. In essence, this function then works as a mask.
*/
ComponentSelectFunction (const unsigned int selected,
const unsigned int n_components);
/**
* Constructor if multiple components shall have non-zero, unit values
* (i.e. this should be a mask for multiple components). The first
* argument denotes a half-open interval of components (for example
* std::pair(0,dim) for the first dim components), and the second
* argument is the total number of vector components.
*/
ComponentSelectFunction (const std::pair<unsigned int, unsigned int> &selected,
const unsigned int n_components);
/**
* Return the value of the function at the given point for all
* components.
*/
virtual void vector_value (const Point<dim> &p,
Vector<double> &return_value) const;
/**
* Set <tt>values</tt> to the point values of the function at the
* <tt>points</tt>, for all components. It is assumed that
* <tt>values</tt> already has the right size, i.e. the same size as the
* <tt>points</tt> array.
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
/**
* Determine an estimate for the memory consumption (in bytes) of this
* object. Since sometimes the size of objects can not be determined
* exactly (for example: what is the memory consumption of an STL
* <tt>std::map</tt> type with a certain number of elements?), this is
* only an estimate. however often quite close to the true value.
*/
std::size_t memory_consumption () const;
protected:
/**
* Half-open interval of the indices of selected components.
*/
const std::pair<unsigned int,unsigned int> selected_components;
};
/**
* This class provides a way to convert a scalar function of the kind
* @code
* double foo (const Point<dim> &);
* @endcode
* into an object of type Function@<dim@>. Since the argument returns a
* scalar, the result is clearly a Function object for which
* <code>function.n_components==1</code>. The class works by storing a
* pointer to the given function and every time
* <code>function.value(p,component)</code> is called, calls
* <code>foo(p)</code> and returns the corresponding value. It also makes
* sure that <code>component</code> is in fact zero, as needs be for scalar
* functions.
*
* The class provides an easy way to turn a simple global function into
* something that has the required Function@<dim@> interface for operations
* like VectorTools::interpolate_boundary_values() etc., and thereby allows
* for simpler experimenting without having to write all the boiler plate
* code of declaring a class that is derived from Function and implementing
* the Function::value() function.
*
* The class gains additional expressvive power because the argument it
* takes does not have to be a pointer to an actual function. Rather, it is
* a function object, i.e., it can also be the result of call to std::bind
* (or boost::bind) or some other object that can be called with a single
* argument. For example, if you need a Function object that returns the
* norm of a point, you could write it like so:
* @code
* template <int dim>
* class Norm : public Function<dim> {
* public:
* virtual double value (const Point<dim> &p,
* const unsigned int component) const {
* Assert (component == 0, ExcMessage ("This object is scalar!"));
* return p.norm();
* }
* };
*
* Norm<2> my_norm_object;
* @endcode
* and then pass the <code>my_norm_object</code> around, or you could write
* it like so:
* @code
* ScalarFunctionFromFunctionObject<dim> my_norm_object (&Point<dim>::norm);
* @endcode
*
* Similarly, to generate an object that computes the distance to a point
* <code>q</code>, we could do this:
* @code
* template <int dim>
* class DistanceTo : public Function<dim> {
* public:
* DistanceTo (const Point<dim> &q) : q(q) {}
* virtual double value (const Point<dim> &p,
* const unsigned int component) const {
* Assert (component == 0, ExcMessage ("This object is scalar!"));
* return q.distance(p);
* }
* private:
* const Point<dim> q;
* };
*
* Point<2> q (2,3);
* DistanceTo<2> my_distance_object;
* @endcode
* or we could write it like so:
* @code
* ScalarFunctionFromFunctionObject<dim>
* my_distance_object (std_cxx1x::bind (&Point<dim>::distance,
* q,
* std_cxx1x::_1));
* @endcode
* The savings in work to write this are apparent.
*
* @author Wolfgang Bangerth, 2011
*/
template <int dim>
class ScalarFunctionFromFunctionObject : public Function<dim>
{
public:
/**
* Given a function object that takes a Point and returns a double value,
* convert this into an object that matches the Function<dim> interface.
*/
ScalarFunctionFromFunctionObject (const std_cxx1x::function<double (const Point<dim> &)> &function_object);
/**
* Return the value of the function at the given point. Returns the value
* the function given to the constructor produces for this point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
private:
/**
* The function object which we call when this class's value() or
* value_list() functions are called.
**/
const std_cxx1x::function<double (const Point<dim> &)> function_object;
};
/**
* This class is similar to the ScalarFunctionFromFunctionObject class in
* that it allows for the easy conversion of a function object to something
* that satisfies the interface of the Function base class. The difference
* is that here, the given function object is still a scalar function (i.e.
* it has a single value at each space point) but that the Function object
* generated is vector valued. The number of vector components is specified
* in the constructor, where one also selectes a single one of these vector
* components that should be filled by the passed object. The result is a
* vector Function object that returns zero in each component except the
* single selected one where it returns the value returned by the given as
* the first argument to the constructor.
*
* @note In the above discussion, note the difference between the (scalar)
* "function object" (i.e., a C++ object <code>x</code> that can be called
* as in <code>x(p)</code>) and the capitalized (vector valued) "Function
* object" (i.e., an object of a class that is derived from the Function
* base class).
*
* To be more concrete, let us consider the following example:
* @code
* double one (const Point<2> &p) { return 1; }
* VectorFunctionFromScalarFunctionObject<2>
* component_mask (&one, 1, 3);
* @endcode
* Here, <code>component_mask</code> then represents a Function object that
* for every point returns the vector $(0, 1, 0)^T$, i.e. a mask function
* that could, for example, be passed to
* VectorTools::integrate_difference(). This effect can also be achieved
* using the ComponentSelectFunction class but is obviously easily extended
* to functions that are non-constant in their one component.
*
* @author Wolfgang Bangerth, 2011
*/
template <int dim>
class VectorFunctionFromScalarFunctionObject : public Function<dim>
{
public:
/**
* Given a function object that takes a Point and returns a double
* value, convert this into an object that matches the Function@<dim@>
* interface.
*
* @param function_object The scalar function that will form one component
* of the resulting Function object.
* @param n_components The total number of vector components of the
* resulting Function object.
* @param selected_component The single component that should be
* filled by the first argument.
**/
VectorFunctionFromScalarFunctionObject (const std_cxx1x::function<double (const Point<dim> &)> &function_object,
const unsigned int selected_component,
const unsigned int n_components);
/**
* Return the value of the function at the given point. Returns the value
* the function given to the constructor produces for this point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Return all components of a vector-valued function at a given point.
*
* <tt>values</tt> shall have the right size beforehand, i.e.
* #n_components.
*/
virtual void vector_value (const Point<dim> &p,
Vector<double> &values) const;
private:
/**
* The function object which we call when this class's value() or
* value_list() functions are called.
**/
const std_cxx1x::function<double (const Point<dim> &)> function_object;
/**
* The vector component whose value is to be filled by the given scalar
* function.
*/
const unsigned int selected_component;
};
/**
* This class is built as a means of translating the <code>Tensor<1,dim>
* </code> values produced by objects of type TensorFunction and returning
* them as a multiple component version of the same thing as a Vector for
* use in, for example, the VectorTools::interpolate or the many other
* functions taking Function objects. It allows the user to place the
* desired components into an <tt>n_components</tt> long vector starting at
* the <tt>selected_component</tt> location in that vector and have all
* other components be 0.
*
* For example: Say you created a class called
* @code
* class RightHandSide : public TensorFunction<rank,dim>
* @endcode
* which extends the TensorFunction class and you have an object
* @code
* RightHandSide<1,dim> rhs;
* @endcode
* of that class which you want to interpolate onto your mesh using the
* VectorTools::interpolate function, but the finite element you use for
* the DoFHandler object has 3 copies of a finite element with
* <tt>dim</tt> components, for a total of 3*dim components. To
* interpolate onto that DoFHandler, you need an object of type Function
* that has 3*dim vector components. Creating such an object from the
* existing <code>rhs</code> object is done using this piece of code:
* @code
* RighHandSide<1,dim> rhs;
* VectorFunctionFromTensorFunction<dim> rhs_vector_function (rhs, 0, 3*dim);
* @endcode
* where the <code>dim</code> components of the tensor function are placed into
* the first <code>dim</code> components of the function object.
*
* @author Spencer Patty, 2013
*/
template <int dim>
class VectorFunctionFromTensorFunction : public Function<dim>
{
public:
/**
* Given a TensorFunction object that takes a <tt>Point</tt> and returns
* a <tt>Tensor<1,dim></tt> value, convert this into an object that
* matches the Function@<dim@> interface.
*
* By default, create a Vector object of the same size as
* <tt>tensor_function</tt> returns, i.e., with <tt>dim</tt> components.
*
* @param tensor_function The TensorFunction that will form one component
* of the resulting Vector Function object.
* @param n_components The total number of vector components of the
* resulting TensorFunction object.
* @param selected_component The first component that should be
* filled by the first argument. This should be such that the entire
* tensor_function fits inside the <tt>n_component</tt> length return
* vector.
*/
VectorFunctionFromTensorFunction (const TensorFunction<1,dim> &tensor_function,
const unsigned int selected_component=0,
const unsigned int n_components=dim);
/**
* This destructor is defined as virtual so as to coincide with all other
* aspects of class.
*/
virtual ~VectorFunctionFromTensorFunction();
/**
* Return a single component of a vector-valued function at a given
* point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Return all components of a vector-valued function at a given point.
*
* <tt>values</tt> shall have the right size beforehand, i.e.
* #n_components.
*/
virtual void vector_value (const Point<dim> &p,
Vector<double> &values) const;
/**
* Return all components of a vector-valued function at a list of points.
*
* <tt>value_list</tt> shall be the same size as <tt>points</tt> and each
* element of the vector will be passed to vector_value() to evaluate the
* function
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &value_list) const;
private:
/**
* The TensorFunction object which we call when this class's
* vector_value() or vector_value_list() functions are called.
**/
const TensorFunction<1,dim> &tensor_function;
/**
* The first vector component whose value is to be filled by the given
* TensorFunction. The values will be placed in components
* selected_component to selected_component+dim-1 for a
* <tt>TensorFunction<1,dim></tt> object.
*/
const unsigned int selected_component;
};
DEAL_II_NAMESPACE_CLOSE
#endif
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