linear_wave_elements.h

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// LIC// ====================================================================
// LIC// This file forms part of oomph-lib, the object-oriented,
// LIC// multi-physics finite-element library, available
// LIC// at http://www.oomph-lib.org.
// LIC//
// LIC// Copyright (C) 2006-2025 Matthias Heil and Andrew Hazel
// LIC//
// LIC// This library is free software; you can redistribute it and/or
// LIC// modify it under the terms of the GNU Lesser General Public
// LIC// License as published by the Free Software Foundation; either
// LIC// version 2.1 of the License, or (at your option) any later version.
// LIC//
// LIC// This library is distributed in the hope that it will be useful,
// LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
// LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// LIC// Lesser General Public License for more details.
// LIC//
// LIC// You should have received a copy of the GNU Lesser General Public
// LIC// License along with this library; if not, write to the Free Software
// LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
// LIC// 02110-1301  USA.
// LIC//
// LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
// LIC//
// LIC//====================================================================
// Header file for LinearWave elements
 
 
#ifndef OOMPH_WAVE_ELEMENTS_HEADER
#define OOMPH_WAVE_ELEMENTS_HEADER
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
#include "generic/nodes.h"
#include "generic/Qelements.h"
#include "generic/oomph_utilities.h"
 
namespace oomph
{
  //=============================================================
  /// A class for all isoparametric elements that solve the
  /// LinearWave equations.
  /// \f[ \frac{\partial^2 u}{\partial x_i^2}= \frac{\partial^2 u}{\partial t^2}+f(t,x_j) \f]
  /// This contains the generic maths. Shape functions, geometric
  /// mapping etc. must get implemented in derived class.
  //=============================================================
  template<unsigned DIM>
  class LinearWaveEquations : public virtual FiniteElement
  {
  public:
    /// Function pointer to source function fct(x,f(x)) --
    /// x is a Vector!
    typedef void (*LinearWaveSourceFctPt)(const double& time,
                                          const Vector<double>& x,
                                          double& u);
 
    /// Constructor (must initialise the Source_fct_pt to null)
    LinearWaveEquations() : Source_fct_pt(0) {}
 
    /// Broken copy constructor
    LinearWaveEquations(const LinearWaveEquations& dummy) = delete;
 
    /// Broken assignment operator
    void operator=(const LinearWaveEquations&) = delete;
 
    /// Return the index at which the unknown value
    /// is stored. The default value, 0, is appropriate for single-physics
    /// problems, when there is only one variable, the value that satisfies the
    /// linear wave equation.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknown is always stored at the same index at each node.
    virtual inline unsigned u_index_lin_wave() const
    {
      return 0;
    }
 
    /// du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_lin_wave(const unsigned& n) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = node_pt(n)->time_stepper_pt();
 
      // Initialise d^2u/dt^2
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Find the index at which the linear wave unknown is stored
        const unsigned u_nodal_index = u_index_lin_wave();
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        // Add the contributions to the derivative
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
      return dudt;
    }
 
    /// d^2u/dt^2 at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double d2u_dt2_lin_wave(const unsigned& n) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = node_pt(n)->time_stepper_pt();
 
      // Initialise d^2u/dt^2
      double ddudt = 0.0;
      // Loop over the timesteps, if there is a non steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Find the index at which the linear wave unknown is stored
        const unsigned u_nodal_index = u_index_lin_wave();
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        // Add the contributions to the derivative
        for (unsigned t = 0; t < n_time; t++)
        {
          ddudt +=
            time_stepper_pt->weight(2, t) * nodal_value(t, n, u_nodal_index);
        }
      }
      return ddudt;
    }
 
    /// Output with default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// Output with default number of plot points
    void output(FILE* file_pt)
    {
      unsigned nplot = 5;
      output(file_pt, nplot);
    }
 
    /// Output FE representation of soln: x,y,u or x,y,z,u at
    /// n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& nplot);
 
    /// Output exact soln: x,y,u_exact or x,y,z,u_exact at nplot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact soln: x,y,u_exact or x,y,z,u_exact at
    /// nplot^DIM plot points (time-dependent version)
    virtual void output_fct(
      std::ostream& outfile,
      const unsigned& nplot,
      const double& time,
      FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);
 
    /// Get error against and norm of exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Get error against and norm of exact solution
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
    /// Access function: Pointer to source function
    LinearWaveSourceFctPt& source_fct_pt()
    {
      return Source_fct_pt;
    }
 
    /// Access function: Pointer to source function. Const version
    LinearWaveSourceFctPt source_fct_pt() const
    {
      return Source_fct_pt;
    }
 
 
    /// Get source term at continous time t and (Eulerian) position x.
    /// Virtual so it can be overloaded in derived multiphysics elements.
    inline void get_source_lin_wave(const double& t,
                                    const unsigned& ipt,
                                    const Vector<double>& x,
                                    double& source) const
    {
      // If no source function has been set, return zero
      if (Source_fct_pt == 0)
      {
        source = 0.0;
      }
      else
      {
        // Get source strength
        (*Source_fct_pt)(t, x, source);
      }
    }
 
 
    /// Get flux: flux[i] = du/dx_i
    void get_flux(const Vector<double>& s, Vector<double>& flux) const
    {
      // Find out how many nodes there are in the element
      unsigned n_node = nnode();
 
      // Find the index at which the linear wave unknown is stored
      unsigned u_nodal_index = u_index_lin_wave();
 
      // Set up memory for the shape and test functions
      Shape psi(n_node);
      DShape dpsidx(n_node, DIM);
 
      // Call the derivatives of the shape and test functions
      dshape_eulerian(s, psi, dpsidx);
 
      // Initialise to zero
      for (unsigned j = 0; j < DIM; j++)
      {
        flux[j] = 0.0;
      }
 
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        // Loop over derivative directions
        for (unsigned j = 0; j < DIM; j++)
        {
          flux[j] += nodal_value(l, u_nodal_index) * dpsidx(l, j);
        }
      }
    }
 
 
    /// Compute element residual Vector (wrapper)
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // using the dummy matrix argument
      fill_in_generic_residual_contribution_lin_wave(
        residuals, GeneralisedElement::Dummy_matrix, 0);
    }
 
 
    /// Compute element residual Vector and element Jacobian matrix (wrapper)
    virtual void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                                  DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_residual_contribution_lin_wave(residuals, jacobian, 1);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_u_lin_wave(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Find the index at which the linear wave unknown is stored
      unsigned u_nodal_index = u_index_lin_wave();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_u = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_du_dt_lin_wave(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_du_dt = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_du_dt += du_dt_lin_wave(l) * psi[l];
      }
 
      return (interpolated_du_dt);
    }
 
    /// Return FE representation of function value u(s) at local coordinate s
    inline double interpolated_d2u_dt2_lin_wave(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
 
      // Local shape function
      Shape psi(n_node);
 
      // Find values of shape function
      shape(s, psi);
 
      // Initialise value of u
      double interpolated_d2u_dt2 = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_d2u_dt2 += d2u_dt2_lin_wave(l) * psi[l];
      }
 
      return (interpolated_d2u_dt2);
    }
 
 
    /// Self-test: Return 0 for OK
    unsigned self_test();
 
 
  protected:
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// local coord. s; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_lin_wave(
      const Vector<double>& s,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return  Jacobian of mapping
    virtual double dshape_and_dtest_eulerian_at_knot_lin_wave(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Compute element residual Vector only (if flag=and/or element
    /// Jacobian matrix
    virtual void fill_in_generic_residual_contribution_lin_wave(
      Vector<double>& residuals, DenseMatrix<double>& jacobian, unsigned flag);
 
 
    /// Pointer to source function:
    LinearWaveSourceFctPt Source_fct_pt;
  };
 
 
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////
 
 
  //======================================================================
  /// QLinearWaveElement elements are linear/quadrilateral/brick-shaped
  /// LinearWave elements with isoparametric interpolation for the function.
  ///
  /// Empty, just establishes the template parameters
  ///
  ///
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class QLinearWaveElement : public virtual QElement<DIM, NNODE_1D>,
                             public virtual LinearWaveEquations<DIM>
  {
  private:
    /// Static array of ints to hold number of variables at
    /// nodes: Initial_Nvalue[n]
    static const unsigned Initial_Nvalue[];
 
 
  public:
    /// Constructor: Call constructors for QElement and
    /// LinearWave equations
    QLinearWaveElement() : QElement<DIM, NNODE_1D>(), LinearWaveEquations<DIM>()
    {
    }
 
    /// Broken copy constructor
    QLinearWaveElement(const QLinearWaveElement<DIM, NNODE_1D>& dummy) = delete;
 
 
    /// Broken assignment operator
    void operator=(const QLinearWaveElement<DIM, NNODE_1D>&) = delete;
 
    ///  Required  # of `values' (pinned or dofs)
    /// at node n
    inline unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue[n];
    }
 
    /// Output function:
    ///  x,y,u   or    x,y,z,u
    void output(std::ostream& outfile)
    {
      LinearWaveEquations<DIM>::output(outfile);
    }
 
    ///  Output function:
    ///   x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      LinearWaveEquations<DIM>::output(outfile, n_plot);
    }
 
    /// Output function:
    ///  x,y,u   or    x,y,z,u
    void output(FILE* file_pt)
    {
      LinearWaveEquations<DIM>::output(file_pt);
    }
 
    ///  Output function:
    ///   x,y,u   or    x,y,z,u at n_plot^DIM plot points
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      LinearWaveEquations<DIM>::output(file_pt, n_plot);
    }
 
 
    /// Output function for an exact solution:
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
    {
      LinearWaveEquations<DIM>::output_fct(outfile, n_plot, exact_soln_pt);
    }
 
 
    /// Output function for a time-dependent exact solution.
    ///  x,y,u_exact   or    x,y,z,u_exact at n_plot^DIM plot points
    /// (Calls the steady version)
    void output_fct(std::ostream& outfile,
                    const unsigned& n_plot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt)
    {
      LinearWaveEquations<DIM>::output_fct(
        outfile, n_plot, time, exact_soln_pt);
    }
 
 
  protected:
    /// Shape, test functions & derivs. w.r.t. to global coords. Return
    /// Jacobian.
    inline double dshape_and_dtest_eulerian_lin_wave(const Vector<double>& s,
                                                     Shape& psi,
                                                     DShape& dpsidx,
                                                     Shape& test,
                                                     DShape& dtestdx) const;
 
 
    /// Shape, test functions & derivs. w.r.t. to global coords. at
    /// integration point ipt. Return Jacobian.
    inline double dshape_and_dtest_eulerian_at_knot_lin_wave(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
  };
 
  // Inline functions:
 
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  double QLinearWaveElement<DIM, NNODE_1D>::dshape_and_dtest_eulerian_lin_wave(
    const Vector<double>& s,
    Shape& psi,
    DShape& dpsidx,
    Shape& test,
    DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < NNODE_1D; i++)
    {
      test[i] = psi[i];
      for (unsigned j = 0; j < DIM; j++)
      {
        dtestdx(i, j) = dpsidx(i, j);
      }
    }
 
    // Return the jacobian
    return J;
  }
 
  //======================================================================
  /// Define the shape functions and test functions and derivatives
  /// w.r.t. global coordinates and return Jacobian of mapping.
  ///
  /// Galerkin: Test functions = shape functions
  //======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  double QLinearWaveElement<DIM, NNODE_1D>::
    dshape_and_dtest_eulerian_at_knot_lin_wave(const unsigned& ipt,
                                               Shape& psi,
                                               DShape& dpsidx,
                                               Shape& test,
                                               DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
 
    // Set the test functions equal to the shape functions
    //(sets internal pointers)
    test = psi;
    dtestdx = dpsidx;
 
    // Return the jacobian
    return J;
  }
 
 
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the QLinearWaveElement elements: The spatial
  /// dimension of the face elements is one lower than that of the
  /// bulk element but they have the same number of points
  /// along their 1D edges.
  //=======================================================================
  template<unsigned DIM, unsigned NNODE_1D>
  class FaceGeometry<QLinearWaveElement<DIM, NNODE_1D>>
    : public virtual QElement<DIM - 1, NNODE_1D>
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : QElement<DIM - 1, NNODE_1D>() {}
  };
 
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////
 
 
  //=======================================================================
  /// Face geometry for the 1D QLinearWaveElement elements: Point elements
  //=======================================================================
  template<unsigned NNODE_1D>
  class FaceGeometry<QLinearWaveElement<1, NNODE_1D>>
    : public virtual PointElement
  {
  public:
    /// Constructor: Call the constructor for the
    /// appropriate lower-dimensional QElement
    FaceGeometry() : PointElement() {}
  };
 
} // namespace oomph
 
#endif