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// $Id: function_lib.h 30036 2013-07-18 16:55:32Z maier $
//
// Copyright (C) 1999 - 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_lib_h
#define __deal2__function_lib_h
#include <deal.II/base/config.h>
#include <deal.II/base/function.h>
#include <deal.II/base/point.h>
DEAL_II_NAMESPACE_OPEN
/**
* Namespace implementing some concrete classes derived from the
* Function class that describe actual functions. This is rather
* a collection of classes that we have needed for our own programs
* once and thought might be useful to others as well at some
* point.
*
* @ingroup functions
*/
namespace Functions
{
/**
* The distance to the origin squared.
*
* This function returns the square norm of the radius vector of a point.
*
* Together with the function, its derivatives and Laplacian are defined.
*
* @ingroup functions
* @author: Guido Kanschat, 1999
*/
template<int dim>
class SquareFunction : public Function<dim>
{
public:
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
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 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> > &gradient) const;
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* The function <tt>xy</tt> in 2d and 3d, not implemented in 1d.
* This function serves as an example for
* a vanishing Laplacian.
*
* @ingroup functions
* @author: Guido Kanschat, 2000
*/
template<int dim>
class Q1WedgeFunction : public Function<dim>
{
public:
virtual double value (const Point<dim> &p,
const unsigned int component = 0) 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 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> > &,
std::vector<std::vector<Tensor<1,dim> > > &) const;
/**
* Laplacian of the function at one point.
*/
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Laplacian of the function at multiple points.
*/
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* d-quadratic pillow on the unit hypercube.
*
* This is a function for testing the implementation. It has zero Dirichlet
* boundary values on the domain $(-1,1)^d$. In the inside, it is the
* product of $1-x_i^2$ over all space dimensions.
*
* Providing a non-zero argument to the constructor, the whole function
* can be offset by a constant.
*
* Together with the function, its derivatives and Laplacian are defined.
*
* @ingroup functions
* @author: Guido Kanschat, 1999
*/
template<int dim>
class PillowFunction : public Function<dim>
{
public:
/**
* Constructor. Provide a
* constant that will be added to
* each function value.
*/
PillowFunction (const double offset=0.);
/**
* The value at a single point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Gradient at a single point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Gradients at multiple points.
*/
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
/**
* Laplacian at a single point.
*/
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Laplacian at multiple points.
*/
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
private:
const double offset;
};
/**
* Cosine-shaped pillow function.
* This is another function with zero boundary values on $[-1,1]^d$. In the interior
* it is the product of $\cos(\pi/2 x_i)$.
*
* @ingroup functions
* @author Guido Kanschat, 1999
*/
template<int dim>
class CosineFunction : public Function<dim>
{
public:
/**
* Constructor which allows to
* optionally generate a vector
* valued cosine function with
* the same value in each
* component.
*/
CosineFunction (const unsigned int n_components = 1);
virtual double value (const Point<dim> &p,
const unsigned int component = 0) 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 gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Second derivatives at a
* single point.
*/
virtual Tensor<2,dim> hessian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Second derivatives at
* multiple points.
*/
virtual void hessian_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<2,dim> > &hessians,
const unsigned int component = 0) const;
};
/**
* Gradient of the cosine-shaped pillow function.
*
* This is a vector-valued function with @p dim components, the
* gradient of CosineFunction. On the square [-1,1], it has tangential
* boundary conditions zero. Thus, it can be used to test
* implementations of Maxwell operators without bothering about
* boundary terms.
*
* @ingroup functions
* @author Guido Kanschat, 2010
*/
template<int dim>
class CosineGradFunction : public Function<dim>
{
public:
/**
* Constructor, creating a
* function with @p dim components.
*/
CosineGradFunction ();
virtual double value (const Point<dim> &p,
const unsigned int component) const;
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) 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) const;
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component) const;
virtual void vector_gradient_list (const std::vector<Point<dim> > &points,
std::vector<std::vector<Tensor<1,dim> > > &gradients) const;
virtual double laplacian (const Point<dim> &p,
const unsigned int component) const;
};
/**
* Product of exponential functions in each coordinate direction.
*
* @ingroup functions
* @author Guido Kanschat, 1999
*/
template<int dim>
class ExpFunction : public Function<dim>
{
public:
/**
* The value at a single point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Gradient at a single point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Gradients at multiple points.
*/
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
/**
* Laplacian at a single point.
*/
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Laplacian at multiple points.
*/
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* Harmonic singularity on the L-shaped domain in 2D.
*
* @ingroup functions
* @author Guido Kanschat
* @date 1999
*/
class LSingularityFunction : public Function<2>
{
public:
virtual double value (const Point<2> &p,
const unsigned int component = 0) const;
virtual void value_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
virtual void vector_value_list (const std::vector<Point<2> > &points,
std::vector<Vector<double> > &values) const;
virtual Tensor<1,2> gradient (const Point<2> &p,
const unsigned int component = 0) const;
virtual void gradient_list (const std::vector<Point<2> > &points,
std::vector<Tensor<1,2> > &gradients,
const unsigned int component = 0) const;
virtual void vector_gradient_list (const std::vector<Point<2> > &,
std::vector<std::vector<Tensor<1,2> > > &) const;
virtual double laplacian (const Point<2> &p,
const unsigned int component = 0) const;
virtual void laplacian_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* Gradient of the harmonic singularity on the L-shaped domain in 2D.
*
* The gradient of LSingularityFunction, which is a vector valued
* function with vanishing curl and divergence.
*
* @ingroup functions
* @author Guido Kanschat, 2010
*/
class LSingularityGradFunction : public Function<2>
{
public:
/**
* Default constructor setting
* the dimension to 2.
*/
LSingularityGradFunction ();
virtual double value (const Point<2> &p,
const unsigned int component) const;
virtual void value_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component) const;
virtual void vector_value_list (const std::vector<Point<2> > &points,
std::vector<Vector<double> > &values) const;
virtual Tensor<1,2> gradient (const Point<2> &p,
const unsigned int component) const;
virtual void gradient_list (const std::vector<Point<2> > &points,
std::vector<Tensor<1,2> > &gradients,
const unsigned int component) const;
virtual void vector_gradient_list (const std::vector<Point<2> > &,
std::vector<std::vector<Tensor<1,2> > > &) const;
virtual double laplacian (const Point<2> &p,
const unsigned int component) const;
virtual void laplacian_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component) const;
};
/**
* Singularity on the slit domain in 2D and 3D.
*
* @ingroup functions
* @author Guido Kanschat, 1999, 2006
*/
template <int dim>
class SlitSingularityFunction : public Function<dim>
{
public:
virtual double value (const Point<dim> &p,
const unsigned int component = 0) 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 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> > &,
std::vector<std::vector<Tensor<1,dim> > > &) const;
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* Singularity on the slit domain with one Neumann boundary in 2D.
*
* @ingroup functions
* @author Guido Kanschat, 2002
*/
class SlitHyperSingularityFunction : public Function<2>
{
public:
virtual double value (const Point<2> &p,
const unsigned int component = 0) const;
virtual void value_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
virtual void vector_value_list (const std::vector<Point<2> > &points,
std::vector<Vector<double> > &values) const;
virtual Tensor<1,2> gradient (const Point<2> &p,
const unsigned int component = 0) const;
virtual void gradient_list (const std::vector<Point<2> > &points,
std::vector<Tensor<1,2> > &gradients,
const unsigned int component = 0) const;
virtual void vector_gradient_list (const std::vector<Point<2> > &,
std::vector<std::vector<Tensor<1,2> > > &) const;
virtual double laplacian (const Point<2> &p,
const unsigned int component = 0) const;
virtual void laplacian_list (const std::vector<Point<2> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
};
/**
* A jump in x-direction transported into some direction.
*
* If the advection is parallel to the y-axis, the function is
* <tt>-atan(sx)</tt>, where <tt>s</tt> is the steepness parameter provided in
* the constructor.
*
* For different advection directions, this function will be turned in
* the parameter space.
*
* Together with the function, its derivatives and Laplacian are defined.
*
* @ingroup functions
* @author: Guido Kanschat, 2000
*/
template<int dim>
class JumpFunction : public Function<dim>
{
public:
/**
* Constructor. Provide the
* advection direction here and
* the steepness of the slope.
*/
JumpFunction (const Point<dim> &direction,
const double steepness);
/**
* Function value at one point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Function values at multiple
* points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Gradient at one point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
/**
Gradients at multiple points.
*/
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
/**
* Laplacian of the function at one point.
*/
virtual double laplacian (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Laplacian of the function at multiple points.
*/
virtual void laplacian_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) 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:
/**
* Advection vector.
*/
const Point<dim> direction;
/**
* Steepness (maximal derivative)
* of the slope.
*/
const double steepness;
/**
* Advection angle.
*/
double angle;
/**
* Sine of <tt>angle</tt>.
*/
double sine;
/**
* Cosine of <tt>angle</tt>.
*/
double cosine;
};
/**
* Given a wavenumber vector generate a cosine function. The
* wavenumber coefficient is given as a $d$-dimensional point $k$
* in Fourier space, and the function is then recovered as $f(x) =
* \cos(\sum_i k_i x_i) = Re(\exp(i k.x))$.
*
* The class has its name from the fact that it resembles one
* component of a Fourier cosine decomposition.
*
* @ingroup functions
* @author Wolfgang Bangerth, 2001
*/
template <int dim>
class FourierCosineFunction : public Function<dim>
{
public:
/**
* Constructor. Take the Fourier
* coefficients in each space
* direction as argument.
*/
FourierCosineFunction (const Point<dim> &fourier_coefficients);
/**
* 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 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;
/**
* 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;
private:
/**
* Stored Fourier coefficients.
*/
const Point<dim> fourier_coefficients;
};
/**
* Given a wavenumber vector generate a sine function. The
* wavenumber coefficient is given as a $d$-dimensional point $k$
* in Fourier space, and the function is then recovered as $f(x) =
* \sin(\sum_i k_i x_i) = Im(\exp(i k.x))$.
*
* The class has its name from the fact that it resembles one
* component of a Fourier sine decomposition.
*
* @ingroup functions
* @author Wolfgang Bangerth, 2001
*/
template <int dim>
class FourierSineFunction : public Function<dim>
{
public:
/**
* Constructor. Take the Fourier
* coefficients in each space
* direction as argument.
*/
FourierSineFunction (const Point<dim> &fourier_coefficients);
/**
* 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 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;
/**
* 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;
private:
/**
* Stored Fourier coefficients.
*/
const Point<dim> fourier_coefficients;
};
/**
* Given a sequence of wavenumber vectors and weights generate a sum
* of sine functions. Each wavenumber coefficient is given as a
* $d$-dimensional point $k$ in Fourier space, and the entire
* function is then recovered as
* $f(x) = \sum_j w_j sin(\sum_i k_i x_i) = Im(\sum_j w_j \exp(i k.x))$.
*
* @ingroup functions
* @author Wolfgang Bangerth, 2001
*/
template <int dim>
class FourierSineSum : public Function<dim>
{
public:
/**
* Constructor. Take the Fourier
* coefficients in each space
* direction as argument.
*/
FourierSineSum (const std::vector<Point<dim> > &fourier_coefficients,
const std::vector<double> &weights);
/**
* 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 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;
/**
* 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;
private:
/**
* Stored Fourier coefficients
* and weights.
*/
const std::vector<Point<dim> > fourier_coefficients;
const std::vector<double> weights;
};
/**
* Given a sequence of wavenumber vectors and weights generate a sum
* of cosine functions. Each wavenumber coefficient is given as a
* $d$-dimensional point $k$ in Fourier space, and the entire
* function is then recovered as
* $f(x) = \sum_j w_j cos(\sum_i k_i x_i) = Re(\sum_j w_j \exp(i k.x))$.
*
* @ingroup functions
* @author Wolfgang Bangerth, 2001
*/
template <int dim>
class FourierCosineSum : public Function<dim>
{
public:
/**
* Constructor. Take the Fourier
* coefficients in each space
* direction as argument.
*/
FourierCosineSum (const std::vector<Point<dim> > &fourier_coefficients,
const std::vector<double> &weights);
/**
* 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 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;
/**
* 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;
private:
/**
* Stored Fourier coefficients
* and weights.
*/
const std::vector<Point<dim> > fourier_coefficients;
const std::vector<double> weights;
};
/**
* Base function for cut-off function. This class stores the center
* and the radius of the supporting ball of a cut-off function. It
* also stores the number of the non-zero component, if the function
* is vector-valued.
*
* @ingroup functions
* @author Guido Kanschat, 2002
*/
template <int dim>
class CutOffFunctionBase : public Function<dim>
{
public:
/**
* Value used in the
* constructor of this and
* derived classes to denote
* that no component is
* selected.
*/
static const unsigned int no_component = numbers::invalid_unsigned_int;
/**
* Constructor. Arguments are the
* center of the ball and its
* radius.
*
* If an argument <tt>select</tt> is
* given and not -1, the
* cut-off function will be
* non-zero for this component
* only.
*/
CutOffFunctionBase (const double radius = 1.,
const Point<dim> = Point<dim>(),
const unsigned int n_components = 1,
const unsigned int select = CutOffFunctionBase<dim>::no_component);
/**
* Move the center of the ball
* to new point <tt>p</tt>.
*/
void new_center (const Point<dim> &p);
/**
* Set the radius of the ball to <tt>r</tt>.
*/
void new_radius (const double r);
protected:
/**
* Center of the integration ball.
*/
Point<dim> center;
/**
* Radius of the ball.
*/
double radius;
/**
* Component selected. If
* <tt>no_component</tt>, the function is
* the same in all components.
*/
const unsigned int selected;
};
/**
* Cut-off function in L-infinity for an arbitrary ball. This
* function is the characteristic function of a ball around <tt>center</tt>
* with a specified <tt>radius</tt>, that is,
* \f[
* f = \chi(B_r(c)).
* \f]
* If vector valued, it can be restricted
* to a single component.
*
* @ingroup functions
* @author Guido Kanschat, 2001, 2002
*/
template<int dim>
class CutOffFunctionLinfty : public CutOffFunctionBase<dim>
{
public:
/**
* Constructor. Arguments are the
* center of the ball and its
* radius.
*
* If an argument <tt>select</tt> is
* given and not -1, the
* cut-off function will be
* non-zero for this component
* only.
*/
CutOffFunctionLinfty (const double radius = 1.,
const Point<dim> = Point<dim>(),
const unsigned int n_components = 1,
const unsigned int select = CutOffFunctionBase<dim>::no_component);
/**
* Function value at one point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
};
/**
* Cut-off function for an arbitrary ball. This function is a cone
* with support in a ball of certain <tt>radius</tt> around <tt>center</tt>. The
* maximum value is 1. If vector valued, it can be restricted
* to a single component.
*
* @ingroup functions
* @author Guido Kanschat, 2001, 2002
*/
template<int dim>
class CutOffFunctionW1 : public CutOffFunctionBase<dim>
{
public:
/**
* Constructor. Arguments are the
* center of the ball and its
* radius.
* radius.
*
* If an argument <tt>select</tt> is
* given, the cut-off function
* will be non-zero for this
* component only.
*/
CutOffFunctionW1 (const double radius = 1.,
const Point<dim> = Point<dim>(),
const unsigned int n_components = 1,
const unsigned int select = CutOffFunctionBase<dim>::no_component);
/**
* Function value at one point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
};
/**
* Cut-off function for an arbitrary ball. This is the traditional
* cut-off function in C-infinity for a ball of certain <tt>radius</tt>
* around <tt>center</tt>, $f(r)=exp(1-1/(1-r**2/s**2))$, where $r$ is the
* distance to the center, and $s$ is the radius of the sphere. If
* vector valued, it can be restricted to a single component.
*
* @ingroup functions
* @author Guido Kanschat, 2001, 2002
*/
template<int dim>
class CutOffFunctionCinfty : public CutOffFunctionBase<dim>
{
public:
/**
* Constructor. Arguments are the
* center of the ball and its
* radius.
* radius.
*
* If an argument <tt>select</tt> is
* given, the cut-off function
* will be non-zero for this
* component only.
*/
CutOffFunctionCinfty (const double radius = 1.,
const Point<dim> = Point<dim>(),
const unsigned int n_components = 1,
const unsigned int select = CutOffFunctionBase<dim>::no_component);
/**
* Function value at one point.
*/
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Function values at multiple points.
*/
virtual void vector_value_list (const std::vector<Point<dim> > &points,
std::vector<Vector<double> > &values) const;
/**
* Function gradient at one point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
};
/**
* A class that represents a function object for a monomial. Monomials are
* polynomials with only a single term, i.e. in 1-d they have the form
* $x^\alpha$, in 2-d the form $x_1^{\alpha_1}x_2^{\alpha_2}$, and in 3-d
* $x_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3}$. Monomials are therefore
* described by a $dim$-tuple of exponents. Consequently, the class's
* constructor takes a Tensor<1,dim> to describe the set of exponents. Most of
* the time these exponents will of course be integers, but real exponents are
* of course equally valid.
*
* @author Wolfgang Bangerth, 2006
*/
template <int dim>
class Monomial : public Function<dim>
{
public:
/**
* Constructor. The first argument is
* explained in the general description
* of the class. The second argument
* denotes the number of vector
* components this object shall
* represent. All vector components
* will have the same value.
*/
Monomial (const Tensor<1,dim> &exponents,
const unsigned int n_components = 1);
/**
* Function value at one 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;
/**
* Function values at multiple points.
*/
virtual void value_list (const std::vector<Point<dim> > &points,
std::vector<double> &values,
const unsigned int component = 0) const;
/**
* Function gradient at one point.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
private:
/**
* The set of exponents.
*/
const Tensor<1,dim> exponents;
};
}
DEAL_II_NAMESPACE_CLOSE
#endif
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