function_lib.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 1999 - 2021 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.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------
 
#ifndef dealii_function_lib_h
#define dealii_function_lib_h
 
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/function.h>
#include <deal.II/base/point.h>
#include <deal.II/base/smartpointer.h>
#include <deal.II/base/table.h>
 
#include <array>
 
DEAL_II_NAMESPACE_OPEN
 
namespace Functions
{
  template <int dim>
  class SquareFunction : public Function<dim>
  {
  public:
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
    virtual void
    vector_value(const Point<dim> &p, Vector<double> &values) const override;
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
    virtual void
    vector_gradient(const Point<dim> &           p,
                    std::vector<Tensor<1, dim>> &gradient) const override;
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
  };
 
 
 
  template <int dim>
  class Q1WedgeFunction : public Function<dim>
  {
  public:
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<dim>> &,
      std::vector<std::vector<Tensor<1, dim>>> &) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
  };
 
 
 
  template <int dim>
  class PillowFunction : public Function<dim>
  {
  public:
    PillowFunction(const double offset = 0.);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
 
  private:
    const double offset;
  };
 
 
 
  template <int dim>
  class CosineFunction : public Function<dim>
  {
  public:
    CosineFunction(const unsigned int n_components = 1);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
 
    virtual SymmetricTensor<2, dim>
    hessian(const Point<dim> & p,
            const unsigned int component = 0) const override;
 
    virtual void
    hessian_list(const std::vector<Point<dim>> &       points,
                 std::vector<SymmetricTensor<2, dim>> &hessians,
                 const unsigned int component = 0) const override;
  };
 
 
 
  template <int dim>
  class CosineGradFunction : public Function<dim>
  {
  public:
    CosineGradFunction();
 
    virtual double
    value(const Point<dim> &p, const unsigned int component) const override;
    virtual void
    vector_value(const Point<dim> &p, Vector<double> &values) const override;
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> &p, const unsigned int component) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<dim>> &           points,
      std::vector<std::vector<Tensor<1, dim>>> &gradients) const override;
 
    virtual double
    laplacian(const Point<dim> &p, const unsigned int component) const override;
  };
 
 
 
  template <int dim>
  class ExpFunction : public Function<dim>
  {
  public:
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
  };
 
 
 
  class LSingularityFunction : public Function<2>
  {
  public:
    virtual double
    value(const Point<2> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<2>> &points,
               std::vector<double> &        values,
               const unsigned int           component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<2>> &points,
                      std::vector<Vector<double>> &values) const override;
 
    virtual Tensor<1, 2>
    gradient(const Point<2> &   p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<2>> &points,
                  std::vector<Tensor<1, 2>> &  gradients,
                  const unsigned int           component = 0) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<2>> &,
      std::vector<std::vector<Tensor<1, 2>>> &) const override;
 
    virtual double
    laplacian(const Point<2> &   p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<2>> &points,
                   std::vector<double> &        values,
                   const unsigned int           component = 0) const override;
  };
 
 
 
  class LSingularityGradFunction : public Function<2>
  {
  public:
    LSingularityGradFunction();
    virtual double
    value(const Point<2> &p, const unsigned int component) const override;
 
    virtual void
    value_list(const std::vector<Point<2>> &points,
               std::vector<double> &        values,
               const unsigned int           component) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<2>> &points,
                      std::vector<Vector<double>> &values) const override;
 
    virtual Tensor<1, 2>
    gradient(const Point<2> &p, const unsigned int component) const override;
 
    virtual void
    gradient_list(const std::vector<Point<2>> &points,
                  std::vector<Tensor<1, 2>> &  gradients,
                  const unsigned int           component) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<2>> &,
      std::vector<std::vector<Tensor<1, 2>>> &) const override;
 
    virtual double
    laplacian(const Point<2> &p, const unsigned int component) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<2>> &points,
                   std::vector<double> &        values,
                   const unsigned int           component) const override;
  };
 
 
 
  template <int dim>
  class SlitSingularityFunction : public Function<dim>
  {
  public:
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<dim>> &,
      std::vector<std::vector<Tensor<1, dim>>> &) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
  };
 
 
  class SlitHyperSingularityFunction : public Function<2>
  {
  public:
    virtual double
    value(const Point<2> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<2>> &points,
               std::vector<double> &        values,
               const unsigned int           component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<2>> &points,
                      std::vector<Vector<double>> &values) const override;
 
    virtual Tensor<1, 2>
    gradient(const Point<2> &   p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<2>> &points,
                  std::vector<Tensor<1, 2>> &  gradients,
                  const unsigned int           component = 0) const override;
 
    virtual void
    vector_gradient_list(
      const std::vector<Point<2>> &,
      std::vector<std::vector<Tensor<1, 2>>> &) const override;
 
    virtual double
    laplacian(const Point<2> &   p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<2>> &points,
                   std::vector<double> &        values,
                   const unsigned int           component = 0) const override;
  };
 
 
 
  template <int dim>
  class JumpFunction : public Function<dim>
  {
  public:
    JumpFunction(const Point<dim> &direction, const double steepness);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual void
    gradient_list(const std::vector<Point<dim>> &points,
                  std::vector<Tensor<1, dim>> &  gradients,
                  const unsigned int             component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
    virtual void
    laplacian_list(const std::vector<Point<dim>> &points,
                   std::vector<double> &          values,
                   const unsigned int             component = 0) const override;
 
    virtual std::size_t
    memory_consumption() const override;
 
  protected:
    const Point<dim> direction;
 
    const double steepness;
 
    double angle;
 
    double sine;
 
    double cosine;
  };
 
 
 
  template <int dim>
  class FourierCosineFunction : public Function<dim>
  {
  public:
    FourierCosineFunction(const Tensor<1, dim> &fourier_coefficients);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
  private:
    const Tensor<1, dim> fourier_coefficients;
  };
 
 
 
  template <int dim>
  class FourierSineFunction : public Function<dim>
  {
  public:
    FourierSineFunction(const Tensor<1, dim> &fourier_coefficients);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
  private:
    const Tensor<1, dim> fourier_coefficients;
  };
 
 
  template <int dim>
  class FourierSineSum : public Function<dim>
  {
  public:
    FourierSineSum(const std::vector<Point<dim>> &fourier_coefficients,
                   const std::vector<double> &    weights);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
  private:
    const std::vector<Point<dim>> fourier_coefficients;
    const std::vector<double>     weights;
  };
 
 
 
  template <int dim>
  class FourierCosineSum : public Function<dim>
  {
  public:
    FourierCosineSum(const std::vector<Point<dim>> &fourier_coefficients,
                     const std::vector<double> &    weights);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual double
    laplacian(const Point<dim> & p,
              const unsigned int component = 0) const override;
 
  private:
    const std::vector<Point<dim>> fourier_coefficients;
    const std::vector<double>     weights;
  };
 
 
  template <int dim>
  class CutOffFunctionBase : public Function<dim>
  {
  public:
    static const unsigned int no_component = numbers::invalid_unsigned_int;
 
    CutOffFunctionBase(
      const double       radius       = 1.,
      const Point<dim>   center       = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      const bool         integrate_to_one       = false,
      const double       unitary_integral_value = 1.0);
 
    virtual ~CutOffFunctionBase() = default;
 
    virtual void
    set_center(const Point<dim> &p);
 
    virtual void
    set_radius(const double r);
 
    const Point<dim> &
    get_center() const;
 
    double
    get_radius() const;
 
    bool
    integrates_to_one() const;
 
  protected:
    Point<dim> center;
 
    double radius;
 
    const unsigned int selected;
 
    bool integrate_to_one;
 
    const double unitary_integral_value;
 
    double rescaling;
  };
 
 
  template <int dim>
  class CutOffFunctionTensorProduct : public CutOffFunctionBase<dim>
  {
  public:
    CutOffFunctionTensorProduct(
      double             radius       = 1.0,
      const Point<dim> & center       = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      const bool         integrate_to_one = false);
 
    template <template <int> class CutOffFunctionBaseType>
    void
    set_base();
 
    virtual void
    set_center(const Point<dim> &center) override;
 
    virtual void
    set_radius(const double radius) override;
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
  private:
    std::array<std::unique_ptr<CutOffFunctionBase<1>>, dim> base;
 
    bool initialized;
  };
 
 
 
  template <int dim>
  class CutOffFunctionLinfty : public CutOffFunctionBase<dim>
  {
  public:
    CutOffFunctionLinfty(
      const double radius             = 1.,
      const Point<dim>                = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      const bool         integrate_to_one = false);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
  };
 
 
  template <int dim>
  class CutOffFunctionW1 : public CutOffFunctionBase<dim>
  {
  public:
    CutOffFunctionW1(
      const double radius             = 1.,
      const Point<dim>                = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      const bool         integrate_to_one = false);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
  };
 
 
  template <int dim>
  class CutOffFunctionC1 : public CutOffFunctionBase<dim>
  {
  public:
    CutOffFunctionC1(
      const double radius             = 1.,
      const Point<dim>                = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      bool               integrate_to_one = false);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
  };
 
 
  template <int dim>
  class CutOffFunctionCinfty : public CutOffFunctionBase<dim>
  {
  public:
    CutOffFunctionCinfty(
      const double radius             = 1.,
      const Point<dim>                = Point<dim>(),
      const unsigned int n_components = 1,
      const unsigned int select       = CutOffFunctionBase<dim>::no_component,
      bool               integrate_to_one = false);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual void
    vector_value_list(const std::vector<Point<dim>> &points,
                      std::vector<Vector<double>> &  values) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
  };
 
 
 
  template <int dim, typename Number = double>
  class Monomial : public Function<dim, Number>
  {
  public:
    Monomial(const Tensor<1, dim, Number> &exponents,
             const unsigned int            n_components = 1);
 
    virtual Number
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual void
    vector_value(const Point<dim> &p, Vector<Number> &values) const override;
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<Number> &          values,
               const unsigned int             component = 0) const override;
 
    virtual Tensor<1, dim, Number>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
  private:
    const Tensor<1, dim, Number> exponents;
  };
 
 
 
  template <int dim>
  class InterpolatedTensorProductGridData : public Function<dim>
  {
  public:
    InterpolatedTensorProductGridData(
      const std::array<std::vector<double>, dim> &coordinate_values,
      const Table<dim, double> &                  data_values);
 
    InterpolatedTensorProductGridData(
      std::array<std::vector<double>, dim> &&coordinate_values,
      Table<dim, double> &&                  data_values);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual std::size_t
    memory_consumption() const override;
 
    const Table<dim, double> &
    get_data() const;
 
  protected:
    TableIndices<dim>
    table_index_of_point(const Point<dim> &p) const;
 
    const std::array<std::vector<double>, dim> coordinate_values;
 
    const Table<dim, double> data_values;
  };
 
 
  template <int dim>
  class InterpolatedUniformGridData : public Function<dim>
  {
  public:
    InterpolatedUniformGridData(
      const std::array<std::pair<double, double>, dim> &interval_endpoints,
      const std::array<unsigned int, dim> &             n_subintervals,
      const Table<dim, double> &                        data_values);
 
    InterpolatedUniformGridData(
      std::array<std::pair<double, double>, dim> &&interval_endpoints,
      std::array<unsigned int, dim> &&             n_subintervals,
      Table<dim, double> &&                        data_values);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual std::size_t
    memory_consumption() const override;
 
    const Table<dim, double> &
    get_data() const;
 
  private:
    const std::array<std::pair<double, double>, dim> interval_endpoints;
 
    const std::array<unsigned int, dim> n_subintervals;
 
    const Table<dim, double> data_values;
  };
 
 
  template <int dim>
  class Polynomial : public Function<dim>
  {
  public:
    Polynomial(const Table<2, double> &   exponents,
               const std::vector<double> &coefficients);
 
    virtual double
    value(const Point<dim> &p, const unsigned int component = 0) const override;
 
 
    virtual void
    value_list(const std::vector<Point<dim>> &points,
               std::vector<double> &          values,
               const unsigned int             component = 0) const override;
 
    virtual Tensor<1, dim>
    gradient(const Point<dim> & p,
             const unsigned int component = 0) const override;
 
    virtual std::size_t
    memory_consumption() const override;
 
  private:
    const Table<2, double> exponents;
 
    const std::vector<double> coefficients;
  };
 
#ifndef DOXYGEN
 
 
 
  // Template definitions
  template <int dim>
  template <template <int> class CutOffFunctionBaseType>
  void
  CutOffFunctionTensorProduct<dim>::set_base()
  {
    initialized = true;
    static_assert(
      std::is_base_of<CutOffFunctionBase<1>, CutOffFunctionBaseType<1>>::value,
      "You can only construct a CutOffFunctionTensorProduct function from "
      "a class derived from CutOffFunctionBase.");
 
    for (unsigned int i = 0; i < dim; ++i)
      base[i].reset(new CutOffFunctionBaseType<1>(this->radius,
                                                  Point<1>(this->center[i]),
                                                  this->n_components,
                                                  this->selected,
                                                  this->integrate_to_one));
  }
 
 
 
#endif
 
} // namespace Functions
DEAL_II_NAMESPACE_CLOSE
 
#endif