young_laplace_elements.cc

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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) 2006-2023 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//====================================================================
 
// Non-inline functions for YoungLaplace elements
#include "young_laplace_elements.h"
 
namespace oomph
{
  //======================================================================
  // Set the data for the number of Variables at each node
  //======================================================================
  template<>
  const unsigned QYoungLaplaceElement<4>::Initial_Nvalue[16] = {
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
  template<>
  const unsigned QYoungLaplaceElement<3>::Initial_Nvalue[9] = {
    1, 1, 1, 1, 1, 1, 1, 1, 1};
  template<>
  const unsigned QYoungLaplaceElement<2>::Initial_Nvalue[4] = {1, 1, 1, 1};
 
 
  //======================================================================
  /// Get exact position vector to meniscus
  //======================================================================
  void YoungLaplaceEquations::exact_position(
    const Vector<double>& s,
    Vector<double>& r,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
  {
    // Get global coordinates
    Vector<double> x(2);
    interpolated_x(s, x);
 
    // Exact solution Vector (here a scalar)
    Vector<double> exact_soln(1);
 
    // Get exact solution at this point
    (*exact_soln_pt)(x, exact_soln);
 
    if (!use_spines())
    {
      r[0] = x[0];
      r[1] = x[1];
      r[2] = exact_soln[0];
    }
    else
    {
      /// Get spines values
      Vector<double> spine_base(3, 0.0);
      Vector<double> spine(3, 0.0);
      Vector<Vector<double>> dspine_base;
      allocate_vector_of_vectors(2, 3, dspine_base);
      Vector<Vector<double>> dspine;
      allocate_vector_of_vectors(2, 3, dspine);
 
      get_spine_base(x, spine_base, dspine_base);
      get_spine(x, spine, dspine);
 
      /// Global Eulerian cooordinates
      for (unsigned j = 0; j < 3; j++)
      {
        r[j] = spine_base[j] + exact_soln[0] * spine[j];
      }
    }
  }
 
  //======================================================================
  /// Get position vector to meniscus
  //======================================================================
  void YoungLaplaceEquations::position(const Vector<double>& s,
                                       Vector<double>& r) const
  {
    // Get global coordinates
    Vector<double> x(2);
    interpolated_x(s, x);
 
    // Displacement along spine (or cartesian displacement)
    double u = interpolated_u(s);
 
    // cartesian calculation case
    if (!use_spines())
    {
      r[0] = x[0];
      r[1] = x[1];
      r[2] = u;
    }
    // spine case
    else
    {
      /// Get spines values
      Vector<double> spine_base(3, 0.0);
      Vector<double> spine(3, 0.0);
      Vector<Vector<double>> dspine_base;
      allocate_vector_of_vectors(2, 3, dspine_base);
      Vector<Vector<double>> dspine;
      allocate_vector_of_vectors(2, 3, dspine);
 
      get_spine_base(x, spine_base, dspine_base);
      get_spine(x, spine, dspine);
 
      /// Global Eulerian cooordinates
      for (unsigned j = 0; j < 3; j++)
      {
        r[j] = spine_base[j] + u * spine[j];
      }
    }
  }
 
 
  //======================================================================
  /// Compute element residual vector. Pure version without hanging nodes
  //======================================================================
  void YoungLaplaceEquations::fill_in_contribution_to_residuals(
    Vector<double>& residuals)
  {
    // Find out how many nodes there are
    unsigned n_node = nnode();
 
    // Set up memory for the shape functions
    Shape psi(n_node);
    DShape dpsidzeta(n_node, 2);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Integers to store the local equation numbers
    int local_eqn = 0;
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Call the derivatives of the shape and test functions
      double J = dshape_eulerian_at_knot(ipt, psi, dpsidzeta);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      // Calculate local values of displacement along spine and its derivatives
      // Allocate and initialise to zero
      double interpolated_u = 0.0;
      Vector<double> interpolated_zeta(2, 0.0);
      Vector<double> interpolated_dudzeta(2, 0.0);
 
      // Calculate function value and derivatives:
      //-----------------------------------------
      // Loop over nodes
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += u(l) * psi(l);
        // Loop over directions
        for (unsigned j = 0; j < 2; j++)
        {
          interpolated_zeta[j] += nodal_position(l, j) * psi(l);
          interpolated_dudzeta[j] += u(l) * dpsidzeta(l, j);
        }
      }
 
 
      // Allocation and definition of variables necessary for
      // further calculations
 
      /// "Simple" case
      /// --------------
      double nonlinearterm = 1.0;
      double sqnorm = 0.0;
 
      /// Spine case
      /// -----------
 
      // Derivs of position vector w.r.t. global intrinsic coords
      Vector<Vector<double>> dRdzeta;
      allocate_vector_of_vectors(2, 3, dRdzeta);
 
      // Unnormalised normal
      Vector<double> N_unnormalised(3, 0.0);
 
      // Spine and spine basis vectors, entries initialised to zero
      Vector<double> spine_base(3, 0.0), spine(3, 0.0);
 
      // Derivative of spine basis vector w.r.t to the intrinsic
      // coordinates: dspine_base[i,j] = j-th component of the deriv.
      // of the spine basis vector w.r.t. to the i-th global intrinsic
      // coordinate
      Vector<Vector<double>> dspine_base;
      allocate_vector_of_vectors(2, 3, dspine_base);
 
      // Derivative of spine vector w.r.t to the intrinsic
      // coordinates: dspine[i,j] = j-th component of the deriv.
      // of the spine vector w.r.t. to the i-th global intrinsic
      // coordinate
      Vector<Vector<double>> dspine;
      allocate_vector_of_vectors(2, 3, dspine);
 
      // Vector v_\alpha contains the numerator of the variations of the
      // area element {\cal A}^{1/2} w.r.t. the components of dR/d\zeta_\alpha
      Vector<double> area_variation_numerator_0(3, 0.0);
      Vector<double> area_variation_numerator_1(3, 0.0);
 
      // Vector position
      Vector<double> r(3, 0.0);
 
      // No spines
      //---------
      if (!use_spines())
      {
        for (unsigned j = 0; j < 2; j++)
        {
          sqnorm += interpolated_dudzeta[j] * interpolated_dudzeta[j];
        }
        nonlinearterm = 1.0 / sqrt(1.0 + sqnorm);
      }
 
      // Spines
      //------
      else
      {
        // Get the spines
        get_spine_base(interpolated_zeta, spine_base, dspine_base);
        get_spine(interpolated_zeta, spine, dspine);
 
        // calculation of dR/d\zeta_\alpha
        for (unsigned alpha = 0; alpha < 2; alpha++)
        {
          // Product rule for d(u {\bf S} ) / d \zeta_\alpha
          Vector<double> dudzeta_times_spine(3, 0.0);
          scalar_times_vector(
            interpolated_dudzeta[alpha], spine, dudzeta_times_spine);
 
          Vector<double> u_times_dspinedzeta(3, 0.0);
          scalar_times_vector(
            interpolated_u, dspine[alpha], u_times_dspinedzeta);
 
          Vector<double> d_u_times_spine_dzeta(3, 0.0);
          vector_sum(
            dudzeta_times_spine, u_times_dspinedzeta, d_u_times_spine_dzeta);
 
          // Add derivative of spine base
          vector_sum(d_u_times_spine_dzeta, dspine_base[alpha], dRdzeta[alpha]);
        }
 
        /// Get the unnormalized normal
        cross_product(dRdzeta[0], dRdzeta[1], N_unnormalised);
 
        // Tmp storage
        Vector<double> v_tmp_1(3, 0.0);
        Vector<double> v_tmp_2(3, 0.0);
 
        // Calculation of
        // |dR/d\zeta_1|^2 dR/d\zeta_0 - <dR/d\zeta_0,dR/d\zeta_1>dR/d\zeta_1
        scalar_times_vector(pow(two_norm(dRdzeta[1]), 2), dRdzeta[0], v_tmp_1);
        scalar_times_vector(
          -1 * scalar_product(dRdzeta[0], dRdzeta[1]), dRdzeta[1], v_tmp_2);
        vector_sum(v_tmp_1, v_tmp_2, area_variation_numerator_0);
 
        // Calculation of
        // |dR/d\zeta_0|^2 dR/d\zeta_1 - <dR/d\zeta_0,dR/d\zeta_1>dR/d\zeta_0
        scalar_times_vector(pow(two_norm(dRdzeta[0]), 2), dRdzeta[1], v_tmp_1);
        scalar_times_vector(
          -1 * scalar_product(dRdzeta[0], dRdzeta[1]), dRdzeta[0], v_tmp_2);
        vector_sum(v_tmp_1, v_tmp_2, area_variation_numerator_1);
 
        // Global Eulerian cooordinates
        for (unsigned j = 0; j < 3; j++)
        {
          r[j] = spine_base[j] + interpolated_u * spine[j];
        }
      }
 
 
      // Assemble residuals
      //-------------------
 
      // Loop over the test (shape) functions
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the local equation
        local_eqn = u_local_eqn(l);
 
        /*IF it's not a boundary condition*/
        if (local_eqn >= 0)
        {
          // "simple" calculation case
          if (!use_spines())
          {
            // Add source term: The curvature
            residuals[local_eqn] += get_kappa() * psi(l) * W;
 
            // The YoungLaplace bit itself
            for (unsigned k = 0; k < 2; k++)
            {
              residuals[local_eqn] +=
                nonlinearterm * interpolated_dudzeta[k] * dpsidzeta(l, k) * W;
            }
          }
 
          // Spine calculation case
          else
          {
            // Calculation of d(u S)/d\zeta_0
            //-------------------------------
            Vector<double> v_tmp_1(3, 0.0);
            scalar_times_vector(dpsidzeta(l, 0), spine, v_tmp_1);
 
            Vector<double> v_tmp_2(3, 0.0);
            scalar_times_vector(psi(l), dspine[0], v_tmp_2);
 
            Vector<double> d_uS_dzeta0(3, 0.0);
            vector_sum(v_tmp_1, v_tmp_2, d_uS_dzeta0);
 
            // Add contribution to residual
            residuals[local_eqn] +=
              W * scalar_product(area_variation_numerator_0, d_uS_dzeta0) /
              two_norm(N_unnormalised);
 
            // Calculation of d(u S)/d\zeta_1
            scalar_times_vector(dpsidzeta(l, 1), spine, v_tmp_1);
            scalar_times_vector(psi(l), dspine[1], v_tmp_2);
            Vector<double> d_uS_dzeta1(3, 0.0);
            vector_sum(v_tmp_1, v_tmp_2, d_uS_dzeta1);
 
            // Add contribution to residual
            residuals[local_eqn] +=
              W * scalar_product(area_variation_numerator_1, d_uS_dzeta1) /
              two_norm(N_unnormalised);
 
            // Curvature contribution to the residual : kappa N S test
            residuals[local_eqn] += W * (get_kappa()) *
                                    scalar_product(N_unnormalised, spine) *
                                    psi(l);
          }
        }
      }
 
    } // End of loop over integration points
  }
 
 
  //======================================================================
  /// Self-test:  Return 0 for OK
  //======================================================================
  unsigned YoungLaplaceEquations::self_test()
  {
    bool passed = true;
 
    // Check lower-level stuff
    if (FiniteElement::self_test() != 0)
    {
      passed = false;
    }
 
    // Return verdict
    if (passed)
    {
      return 0;
    }
    else
    {
      return 1;
    }
  }
 
 
  //======================================================================
  /// Output solution at nplot points in each coordinate direction
  //======================================================================
  void YoungLaplaceEquations::output(std::ostream& outfile,
                                     const unsigned& nplot)
  {
    // Vector of local coordinates
    Vector<double> s(2);
 
    // Tecplot header info
    outfile << tecplot_zone_string(nplot);
 
    // Loop over plot points
    unsigned num_plot_points = nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      get_s_plot(iplot, nplot, s);
 
      // Compute intrinsic coordinates
      Vector<double> xx(2, 0.0);
      for (unsigned i = 0; i < 2; i++)
      {
        xx[i] = interpolated_x(s, i);
      }
 
      // Calculate the cartesian coordinates of point on meniscus
      Vector<double> r(3, 0.0);
 
      // Position
      if (use_spines())
      {
        position(s, r);
      }
      else
      {
        r[0] = xx[0];
        r[1] = xx[1];
        r[2] = interpolated_u(s);
      }
 
      // Output positon on meniscus
      for (unsigned i = 0; i < 3; i++)
      {
        outfile << r[i] << " ";
      }
 
      // Get spine stuff
      Vector<double> spine_base(3, 0.0), spine(3, 0.0);
      Vector<Vector<double>> dspine_base;
      allocate_vector_of_vectors(2, 3, dspine_base);
      Vector<Vector<double>> dspine;
      allocate_vector_of_vectors(2, 3, dspine);
 
      // Get the spines
      if (use_spines())
      {
        get_spine_base(xx, spine_base, dspine_base);
        get_spine(xx, spine, dspine);
      }
 
 
      // Output spine base
      for (unsigned i = 0; i < 3; i++)
      {
        outfile << spine_base[i] << " ";
      }
 
      // Output spines
      for (unsigned i = 0; i < 3; i++)
      {
        outfile << spine[i] << " ";
      }
 
 
      // Output intrinsic coordinates
      for (unsigned i = 0; i < 2; i++)
      {
        outfile << xx[i] << " ";
      }
 
      // Output unknown
      outfile << interpolated_u(s) << " ";
 
 
      // Done
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //======================================================================
  /// Output exact solution
  ///
  /// Solution is provided via function pointer.
  /// Plot at a given number of plot points.
  //======================================================================
  void YoungLaplaceEquations::output_fct(
    std::ostream& outfile,
    const unsigned& nplot,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt)
  {
    // Vector of local coordinates
    Vector<double> s(2);
 
    // Vector for coordinates
    Vector<double> x(2);
 
    // Tecplot header info
    outfile << tecplot_zone_string(nplot);
 
    // Exact solution Vector (here a scalar)
    Vector<double> exact_soln(1);
 
    // Loop over plot points
    unsigned num_plot_points = nplot_points(nplot);
    for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
    {
      // Get local coordinates of plot point
      get_s_plot(iplot, nplot, s);
 
      // Get x position as Vector
      interpolated_x(s, x);
 
      /// Calculate the cartesian coordinates of point on meniscus
      Vector<double> r_exact(3, 0.0);
      exact_position(s, r_exact, exact_soln_pt);
 
      // Output x_exact,y_exact,z_exact
      for (unsigned i = 0; i < 3; i++)
      {
        outfile << r_exact[i] << " ";
      }
 
      // Done
      outfile << std::endl;
    }
 
    // Write tecplot footer (e.g. FE connectivity lists)
    write_tecplot_zone_footer(outfile, nplot);
  }
 
 
  //======================================================================
  /// Validate against exact solution
  ///
  /// Solution is provided via function pointer.
  /// Plot error at a given number of plot points.
  ///
  //======================================================================
  void YoungLaplaceEquations::compute_error(
    std::ostream& outfile,
    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
    double& error,
    double& norm)
  {
    // Initialise
    error = 0.0;
    norm = 0.0;
 
    // Vector of local coordinates
    Vector<double> s(2);
 
    // Vector for coordinates
    Vector<double> x(2);
 
    // Set the value of n_intpt
    unsigned n_intpt = integral_pt()->nweight();
 
    // Tecplot
    outfile << "ZONE" << std::endl;
 
    // Exact solution Vector (here a scalar)
    Vector<double> exact_soln(1);
 
    // Loop over the integration points
    for (unsigned ipt = 0; ipt < n_intpt; ipt++)
    {
      // Assign values of s
      for (unsigned i = 0; i < 2; i++)
      {
        s[i] = integral_pt()->knot(ipt, i);
      }
 
      // Get the integral weight
      double w = integral_pt()->weight(ipt);
 
      // Get jacobian of mapping
      double J = J_eulerian(s);
 
      // Premultiply the weights and the Jacobian
      double W = w * J;
 
      /// Calculate the cartesian coordinates of point on meniscus
      Vector<double> r(3, 0.0);
      position(s, r);
 
      /// Calculate the exact position
      Vector<double> r_exact(3, 0.0);
      exact_position(s, r_exact, exact_soln_pt);
 
      // Output x,y,...,error
      for (unsigned i = 0; i < 2; i++)
      {
        outfile << r[i] << " ";
      }
 
      for (unsigned i = 0; i < 2; i++)
      {
        outfile << r_exact[i] << " ";
      }
 
      outfile << std::endl;
 
      // Add to error and norm
      norm += 0.0;
      for (unsigned i = 0; i < 2; i++)
      {
        error += (r[i] - r_exact[i]) * (r[i] - r_exact[i]) * W;
      }
    }
  }
 
 
  //====================================================================
  // Force build of templates
  //====================================================================
  template class QYoungLaplaceElement<2>;
  template class QYoungLaplaceElement<3>;
  template class QYoungLaplaceElement<4>;
 
} // namespace oomph