miscellaneous_ex11.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
 
// This library is free software; you can 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.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
 
 
// <h1>Miscellaneous Example 11 - Using Loop Subdivision Shell Elements</h1>
// \author Roman Vetter
// \author Norbert Stoop
// \date 2014
//
// This example demonstrates how subdivision surface shell elements
// are used, and how boundary conditions can be applied to them.  To
// keep it simple, we solve the static deflection of a clamped, square,
// linearly elastic Kirchhoff-Love plate subject to a uniform
// load distribution.  Refer to Cirak et al., Int. J. Numer. Meth.
// Engng. 2000; 47: 2039-2072, for a detailed description of what's
// implemented.  In fact, this example follows that paper very closely.
 
 
// C++ include files that we need
#include <iostream>
 
// LibMesh includes
#include "libmesh/libmesh.h"
#include "libmesh/replicated_mesh.h"
#include "libmesh/mesh_refinement.h"
#include "libmesh/mesh_modification.h"
#include "libmesh/mesh_tools.h"
#include "libmesh/linear_implicit_system.h"
#include "libmesh/equation_systems.h"
#include "libmesh/fe.h"
#include "libmesh/quadrature.h"
#include "libmesh/node.h"
#include "libmesh/elem.h"
#include "libmesh/dof_map.h"
#include "libmesh/vector_value.h"
#include "libmesh/tensor_value.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/dense_submatrix.h"
#include "libmesh/dense_vector.h"
#include "libmesh/dense_subvector.h"
#include "libmesh/sparse_matrix.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/vtk_io.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/enum_solver_package.h"
#include "libmesh/parallel.h"
#include "libmesh/getpot.h"
 
// These are the include files typically needed for subdivision elements.
#include "libmesh/face_tri3_subdivision.h"
#include "libmesh/mesh_subdivision_support.h"
 
// Bring in everything from the libMesh namespace
using namespace libMesh;
 
#if defined(LIBMESH_ENABLE_SECOND_DERIVATIVES) && LIBMESH_DIM > 2
// Function prototype.  This is the function that will assemble
// the stiffness matrix and the right-hand-side vector ready
// for solution.
void assemble_shell (EquationSystems & es,
                     const std::string & system_name);
#endif
 
// Begin the main program.
int main (int argc, char ** argv)
{
  // Initialize libMesh.
  LibMeshInit init (argc, argv);
 
  // This example requires a linear solver package.
  libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
                           "--enable-petsc, --enable-trilinos, or --enable-eigen");
 
  // Skip this 3D example if libMesh was compiled as 1D/2D-only.
  libmesh_example_requires (3 == LIBMESH_DIM, "3D support");
 
  // Skip this example without --enable-node-valence
#ifndef LIBMESH_ENABLE_NODE_VALENCE
  libmesh_example_requires (false, "--enable-node-valence");
#endif
 
  // Skip this example without --enable-amr; requires MeshRefinement
#ifndef LIBMESH_ENABLE_AMR
  libmesh_example_requires(false, "--enable-amr");
#else
 
  // Skip this example without --enable-second; requires d2phi
#ifndef LIBMESH_ENABLE_SECOND_DERIVATIVES
  libmesh_example_requires(false, "--enable-second");
#elif LIBMESH_DIM > 2
 
  // Create a 2D mesh distributed across the default MPI communicator.
  // Subdivision surfaces do not appear to work with DistributedMesh yet.
  ReplicatedMesh mesh (init.comm(), 2);
 
  // Read the coarse square mesh.
  mesh.read ("square_mesh.off");
 
  // Resize the square plate to edge length L.
  const Real L = 100.;
  MeshTools::Modification::scale(mesh, L, L, L);
 
  // Get the number of mesh refinements from the command line
  const int n_refinements =
    libMesh::command_line_next("-n_refinements", 3);
 
  // Quadrisect the mesh triangles a few times to obtain a
  // finer mesh.  Subdivision surface elements require the
  // refinement data to be removed afterward.
  MeshRefinement mesh_refinement (mesh);
  mesh_refinement.uniformly_refine (n_refinements);
  MeshTools::Modification::flatten (mesh);
 
  // Write the mesh before the ghost elements are added.
#if defined(LIBMESH_HAVE_VTK)
  VTKIO(mesh).write ("without_ghosts.pvtu");
#endif
#if defined(LIBMESH_HAVE_EXODUS_API)
  ExodusII_IO(mesh).write ("without_ghosts.e");
#endif
 
  // Print information about the triangulated mesh to the screen.
  mesh.print_info();
 
  // Turn the triangulated mesh into a subdivision mesh
  // and add an additional row of "ghost" elements around
  // it in order to complete the extended local support of
  // the triangles at the boundaries.  If the second
  // argument is set to true, the outermost existing
  // elements are converted into ghost elements, and the
  // actual physical mesh is thus getting smaller.
  MeshTools::Subdivision::prepare_subdivision_mesh (mesh, false);
 
  // Print information about the subdivision mesh to the screen.
  mesh.print_info();
 
  // Write the mesh with the ghost elements added.
  // Compare this to the original mesh to see the difference.
#if defined(LIBMESH_HAVE_VTK)
  VTKIO(mesh).write ("with_ghosts.pvtu");
#endif
#if defined(LIBMESH_HAVE_EXODUS_API)
  ExodusII_IO(mesh).write ("with_ghosts.e");
#endif
 
  // Create an equation systems object.
  EquationSystems equation_systems (mesh);
 
  // Declare the system and its variables.
  // Create a linear implicit system named "Shell".
  LinearImplicitSystem & system = equation_systems.add_system<LinearImplicitSystem> ("Shell");
 
  // Add the three translational deformation variables
  // "u", "v", "w" to "Shell".  Since subdivision shell
  // elements meet the C1-continuity requirement, no
  // rotational or other auxiliary variables are needed.
  // Loop Subdivision Elements are always interpolated
  // by quartic box splines, hence the order must always
  // be FOURTH.
  system.add_variable ("u", FOURTH, SUBDIVISION);
  system.add_variable ("v", FOURTH, SUBDIVISION);
  system.add_variable ("w", FOURTH, SUBDIVISION);
 
  // Give the system a pointer to the matrix and rhs assembly
  // function.
  system.attach_assemble_function (assemble_shell);
 
  // Use the parameters of the equation systems object to
  // tell the shell system about the material properties, the
  // shell thickness, and the external load.
  const Real h  = 1.;
  const Real E  = 1.e7;
  const Real nu = 0.;
  const Real q  = 1.;
  equation_systems.parameters.set<Real> ("thickness")       = h;
  equation_systems.parameters.set<Real> ("young's modulus") = E;
  equation_systems.parameters.set<Real> ("poisson ratio")   = nu;
  equation_systems.parameters.set<Real> ("uniform load")    = q;
 
  // Initialize the data structures for the equation system.
  equation_systems.init();
 
  // Print information about the system to the screen.
  equation_systems.print_info();
 
  // Solve the linear system.
  system.solve();
 
  // After solving the system, write the solution to a VTK
  // or ExodusII output file ready for import in, e.g.,
  // Paraview.
#if defined(LIBMESH_HAVE_VTK)
  VTKIO(mesh).write_equation_systems ("out.pvtu", equation_systems);
#endif
#if defined(LIBMESH_HAVE_EXODUS_API)
  ExodusII_IO(mesh).write_equation_systems ("out.e", equation_systems);
#endif
 
  // Find the center node to measure the maximum deformation of the plate.
  Node * center_node = 0;
  Real nearest_dist_sq = mesh.point(0).norm_sq();
  for (unsigned int nid=1; nid<mesh.n_nodes(); ++nid)
    {
      const Real dist_sq = mesh.point(nid).norm_sq();
      if (dist_sq < nearest_dist_sq)
        {
          nearest_dist_sq = dist_sq;
          center_node = mesh.node_ptr(nid);
        }
    }
 
  // Finally, we evaluate the z-displacement "w" at the center node.
  const unsigned int w_var = system.variable_number ("w");
  dof_id_type w_dof = center_node->dof_number (system.number(), w_var, 0);
  Number w = 0;
  if (w_dof >= system.get_dof_map().first_dof() &&
      w_dof <  system.get_dof_map().end_dof())
    w = system.current_solution(w_dof);
  system.comm().sum(w);
 
 
  // The analytic solution for the maximum displacement of
  // a clamped square plate in pure bending, from Taylor,
  // Govindjee, Commun. Numer. Meth. Eng. 20, 757-765, 2004.
  const Real D = E * h*h*h / (12*(1-nu*nu));
  const Real w_analytic = 0.001265319 * L*L*L*L * q / D;
 
  // Print the finite element solution and the analytic
  // prediction of the maximum displacement of the clamped
  // square plate to the screen.
  libMesh::out << "z-displacement of the center point: " << w << std::endl;
  libMesh::out << "Analytic solution for pure bending: " << w_analytic << std::endl;
 
#endif // LIBMESH_ENABLE_SECOND_DERIVATIVES, LIBMESH_DIM > 2
 
#endif // #ifdef LIBMESH_ENABLE_AMR
 
  // All done.
  return 0;
}
 
#if defined(LIBMESH_ENABLE_SECOND_DERIVATIVES) && LIBMESH_DIM > 2
 
// We now define the matrix and rhs vector assembly function
// for the shell system.  This function implements the
// linear Kirchhoff-Love theory for thin shells.  At the
// end we also take into account the boundary conditions
// here, using the penalty method.
void assemble_shell (EquationSystems & es,
                     const std::string & libmesh_dbg_var(system_name))
{
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Shell");
 
  // Get a constant reference to the mesh object.
  const MeshBase & mesh = es.get_mesh();
 
  // Get a reference to the shell system object.
  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem> ("Shell");
 
  // Get the shell parameters that we need during assembly.
  const Real h  = es.parameters.get<Real> ("thickness");
  const Real E  = es.parameters.get<Real> ("young's modulus");
  const Real nu = es.parameters.get<Real> ("poisson ratio");
  const Real q  = es.parameters.get<Real> ("uniform load");
 
  // Compute the membrane stiffness K and the bending
  // rigidity D from these parameters.
  const Real K = E * h     /     (1-nu*nu);
  const Real D = E * h*h*h / (12*(1-nu*nu));
 
  // Numeric ids corresponding to each variable in the system.
  const unsigned int u_var = system.variable_number ("u");
  const unsigned int v_var = system.variable_number ("v");
  const unsigned int w_var = system.variable_number ("w");
 
  // Get the Finite Element type for "u".  Note this will be
  // the same as the type for "v" and "w".
  FEType fe_type = system.variable_type (u_var);
 
  // Build a Finite Element object of the specified type.
  std::unique_ptr<FEBase> fe (FEBase::build(2, fe_type));
 
  // A Gauss quadrature rule for numerical integration.
  // For subdivision shell elements, a single Gauss point per
  // element is sufficient, hence we use extraorder = 0.
  const int extraorder = 0;
  std::unique_ptr<QBase> qrule (fe_type.default_quadrature_rule (2, extraorder));
 
  // Tell the finite element object to use our quadrature rule.
  fe->attach_quadrature_rule (qrule.get());
 
  // The element Jacobian * quadrature weight at each integration point.
  const std::vector<Real> & JxW = fe->get_JxW();
 
  // The surface tangents in both directions at the quadrature points.
  const std::vector<RealGradient> & dxyzdxi  = fe->get_dxyzdxi();
  const std::vector<RealGradient> & dxyzdeta = fe->get_dxyzdeta();
 
  // The second partial derivatives at the quadrature points.
  const std::vector<RealGradient> & d2xyzdxi2    = fe->get_d2xyzdxi2();
  const std::vector<RealGradient> & d2xyzdeta2   = fe->get_d2xyzdeta2();
  const std::vector<RealGradient> & d2xyzdxideta = fe->get_d2xyzdxideta();
 
  // The element shape function and its derivatives evaluated at the
  // quadrature points.
  const std::vector<std::vector<Real>> &          phi = fe->get_phi();
  const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();
  const std::vector<std::vector<RealTensor>> &  d2phi = fe->get_d2phi();
 
  // A reference to the DofMap object for this system.  The DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.
  const DofMap & dof_map = system.get_dof_map();
 
  // The global system matrix
  SparseMatrix<Number> & matrix = system.get_system_matrix();
 
  // Define data structures to contain the element stiffness matrix
  // and right-hand-side vector contribution.  Following
  // basic finite element terminology we will denote these
  // "Ke" and "Fe".
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;
 
  DenseSubMatrix<Number>
    Kuu(Ke), Kuv(Ke), Kuw(Ke),
    Kvu(Ke), Kvv(Ke), Kvw(Ke),
    Kwu(Ke), Kwv(Ke), Kww(Ke);
 
  DenseSubVector<Number>
    Fu(Fe),
    Fv(Fe),
    Fw(Fe);
 
  // This vector will hold the degree of freedom indices for
  // the element.  These define where in the global system
  // the element degrees of freedom get mapped.
  std::vector<dof_id_type> dof_indices;
  std::vector<dof_id_type> dof_indices_u;
  std::vector<dof_id_type> dof_indices_v;
  std::vector<dof_id_type> dof_indices_w;
 
  // Now we will loop over all the elements in the mesh.  We will
  // compute the element matrix and right-hand-side contribution.
  for (const auto & elem : mesh.active_local_element_ptr_range())
    {
      // The ghost elements at the boundaries need to be excluded
      // here, as they don't belong to the physical shell,
      // but serve for a proper boundary treatment only.
      libmesh_assert_equal_to (elem->type(), TRI3SUBDIVISION);
      const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *> (elem);
      if (sd_elem->is_ghost())
        continue;
 
      // Get the degree of freedom indices for the
      // current element.  These define where in the global
      // matrix and right-hand-side this element will
      // contribute to.
      dof_map.dof_indices (elem, dof_indices);
      dof_map.dof_indices (elem, dof_indices_u, u_var);
      dof_map.dof_indices (elem, dof_indices_v, v_var);
      dof_map.dof_indices (elem, dof_indices_w, w_var);
 
      const std::size_t n_dofs   = dof_indices.size();
      const std::size_t n_u_dofs = dof_indices_u.size();
      const std::size_t n_v_dofs = dof_indices_v.size();
      const std::size_t n_w_dofs = dof_indices_w.size();
 
      // Compute the element-specific data for the current
      // element.  This involves computing the location of the
      // quadrature points and the shape functions
      // (phi, dphi, d2phi) for the current element.
      fe->reinit (elem);
 
      // Zero the element matrix and right-hand side before
      // summing them.  We use the resize member here because
      // the number of degrees of freedom might have changed from
      // the last element.
      Ke.resize (n_dofs, n_dofs);
      Fe.resize (n_dofs);
 
      // Reposition the submatrices...  The idea is this:
      //
      //         -           -          -  -
      //        | Kuu Kuv Kuw |        | Fu |
      //   Ke = | Kvu Kvv Kvw |;  Fe = | Fv |
      //        | Kwu Kwv Kww |        | Fw |
      //         -           -          -  -
      //
      // The DenseSubMatrix.reposition () member takes the
      // (row_offset, column_offset, row_size, column_size).
      //
      // Similarly, the DenseSubVector.reposition () member
      // takes the (row_offset, row_size)
      Kuu.reposition (u_var*n_u_dofs, u_var*n_u_dofs, n_u_dofs, n_u_dofs);
      Kuv.reposition (u_var*n_u_dofs, v_var*n_u_dofs, n_u_dofs, n_v_dofs);
      Kuw.reposition (u_var*n_u_dofs, w_var*n_u_dofs, n_u_dofs, n_w_dofs);
 
      Kvu.reposition (v_var*n_v_dofs, u_var*n_v_dofs, n_v_dofs, n_u_dofs);
      Kvv.reposition (v_var*n_v_dofs, v_var*n_v_dofs, n_v_dofs, n_v_dofs);
      Kvw.reposition (v_var*n_v_dofs, w_var*n_v_dofs, n_v_dofs, n_w_dofs);
 
      Kwu.reposition (w_var*n_w_dofs, u_var*n_w_dofs, n_w_dofs, n_u_dofs);
      Kwv.reposition (w_var*n_w_dofs, v_var*n_w_dofs, n_w_dofs, n_v_dofs);
      Kww.reposition (w_var*n_w_dofs, w_var*n_w_dofs, n_w_dofs, n_w_dofs);
 
      Fu.reposition (u_var*n_u_dofs, n_u_dofs);
      Fv.reposition (v_var*n_u_dofs, n_v_dofs);
      Fw.reposition (w_var*n_u_dofs, n_w_dofs);
 
      // Now we will build the element matrix and right-hand-side.
      for (unsigned int qp=0; qp<qrule->n_points(); ++qp)
        {
          // First, we compute the external force resulting
          // from a load q distributed uniformly across the plate.
          // Since the load is supposed to be transverse to the plate,
          // it affects the z-direction, i.e. the "w" variable.
          for (unsigned int i=0; i<n_u_dofs; ++i)
            Fw(i) += JxW[qp] * phi[i][qp] * q;
 
          // Next, we assemble the stiffness matrix.  This is only valid
          // for the linear theory, i.e., for small deformations, where
          // reference and deformed surface metrics are indistinguishable.
 
          // Get the three surface basis vectors.
          const RealVectorValue & a1 = dxyzdxi[qp];
          const RealVectorValue & a2 = dxyzdeta[qp];
          RealVectorValue   a3 = a1.cross(a2);
          const Real jac = a3.norm(); // the surface Jacobian
          libmesh_assert_greater (jac, 0);
          a3 /= jac; // the shell director a3 is normalized to unit length
 
          // Get the derivatives of the surface tangents.
          const RealVectorValue & a11 = d2xyzdxi2[qp];
          const RealVectorValue & a22 = d2xyzdeta2[qp];
          const RealVectorValue & a12 = d2xyzdxideta[qp];
 
          // Compute the three covariant components of the first
          // fundamental form of the surface.
          const RealVectorValue a(a1*a1, a2*a2, a1*a2);
 
          // The elastic H matrix in Voigt's notation, computed from the
          // covariant components of the first fundamental form rather
          // than the contravariant components, exploiting that the
          // contravariant first fundamental form is the inverse of the
          // covariant first fundamental form (hence the determinant etc.).
          RealTensorValue H;
          H(0,0) = a(1) * a(1);
          H(0,1) = H(1,0) = nu * a(1) * a(0) + (1-nu) * a(2) * a(2);
          H(0,2) = H(2,0) = -a(1) * a(2);
          H(1,1) = a(0) * a(0);
          H(1,2) = H(2,1) = -a(0) * a(2);
          H(2,2) = 0.5 * ((1-nu) * a(1) * a(0) + (1+nu) * a(2) * a(2));
          const Real det = a(0) * a(1) - a(2) * a(2);
          libmesh_assert_not_equal_to (det * det, 0);
          H /= det * det;
 
          // Precompute come cross products for the bending part below.
          const RealVectorValue a11xa2 = a11.cross(a2);
          const RealVectorValue a22xa2 = a22.cross(a2);
          const RealVectorValue a12xa2 = a12.cross(a2);
          const RealVectorValue a1xa11 =  a1.cross(a11);
          const RealVectorValue a1xa22 =  a1.cross(a22);
          const RealVectorValue a1xa12 =  a1.cross(a12);
          const RealVectorValue a2xa3  =  a2.cross(a3);
          const RealVectorValue a3xa1  =  a3.cross(a1);
 
          // Loop over all pairs of nodes I,J.
          for (unsigned int i=0; i<n_u_dofs; ++i)
            {
              for (unsigned int j=0; j<n_u_dofs; ++j)
                {
                  // The membrane strain matrices in Voigt's notation.
                  RealTensorValue MI, MJ;
                  for (unsigned int k=0; k<3; ++k)
                    {
                      MI(0,k) = dphi[i][qp](0) * a1(k);
                      MI(1,k) = dphi[i][qp](1) * a2(k);
                      MI(2,k) = dphi[i][qp](1) * a1(k)
                        + dphi[i][qp](0) * a2(k);
 
                      MJ(0,k) = dphi[j][qp](0) * a1(k);
                      MJ(1,k) = dphi[j][qp](1) * a2(k);
                      MJ(2,k) = dphi[j][qp](1) * a1(k)
                        + dphi[j][qp](0) * a2(k);
                    }
 
                  // The bending strain matrices in Voigt's notation.
                  RealTensorValue BI, BJ;
                  for (unsigned int k=0; k<3; ++k)
                    {
                      const Real term_ik = dphi[i][qp](0) * a2xa3(k)
                        + dphi[i][qp](1) * a3xa1(k);
                      BI(0,k) = -d2phi[i][qp](0,0) * a3(k)
                        +(dphi[i][qp](0) * a11xa2(k)
                          + dphi[i][qp](1) * a1xa11(k)
                          + (a3*a11) * term_ik) / jac;
                      BI(1,k) = -d2phi[i][qp](1,1) * a3(k)
                        +(dphi[i][qp](0) * a22xa2(k)
                          + dphi[i][qp](1) * a1xa22(k)
                          + (a3*a22) * term_ik) / jac;
                      BI(2,k) = 2 * (-d2phi[i][qp](0,1) * a3(k)
                                     +(dphi[i][qp](0) * a12xa2(k)
                                       + dphi[i][qp](1) * a1xa12(k)
                                       + (a3*a12) * term_ik) / jac);
 
                      const Real term_jk = dphi[j][qp](0) * a2xa3(k)
                        + dphi[j][qp](1) * a3xa1(k);
                      BJ(0,k) = -d2phi[j][qp](0,0) * a3(k)
                        +(dphi[j][qp](0) * a11xa2(k)
                          + dphi[j][qp](1) * a1xa11(k)
                          + (a3*a11) * term_jk) / jac;
                      BJ(1,k) = -d2phi[j][qp](1,1) * a3(k)
                        +(dphi[j][qp](0) * a22xa2(k)
                          + dphi[j][qp](1) * a1xa22(k)
                          + (a3*a22) * term_jk) / jac;
                      BJ(2,k) = 2 * (-d2phi[j][qp](0,1) * a3(k)
                                     +(dphi[j][qp](0) * a12xa2(k)
                                       + dphi[j][qp](1) * a1xa12(k)
                                       + (a3*a12) * term_jk) / jac);
                    }
 
                  // The total stiffness matrix coupling the nodes
                  // I and J is a sum of membrane and bending
                  // contributions according to the following formula.
                  const RealTensorValue KIJ = JxW[qp] * K * MI.transpose() * H * MJ
                    + JxW[qp] * D * BI.transpose() * H * BJ;
 
                  // Insert the components of the coupling stiffness
                  // matrix KIJ into the corresponding directional
                  // submatrices.
                  Kuu(i,j) += KIJ(0,0);
                  Kuv(i,j) += KIJ(0,1);
                  Kuw(i,j) += KIJ(0,2);
 
                  Kvu(i,j) += KIJ(1,0);
                  Kvv(i,j) += KIJ(1,1);
                  Kvw(i,j) += KIJ(1,2);
 
                  Kwu(i,j) += KIJ(2,0);
                  Kwv(i,j) += KIJ(2,1);
                  Kww(i,j) += KIJ(2,2);
                }
            }
 
        } // end of the quadrature point qp-loop
 
      // The element matrix and right-hand-side are now built
      // for this element.  Add them to the global matrix and
      // right-hand-side vector.  The NumericMatrix::add_matrix()
      // and NumericVector::add_vector() members do this for us.
      matrix.add_matrix         (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);
    } // end of non-ghost element loop
 
  // Next, we apply the boundary conditions.  In this case,
  // all boundaries are clamped by the penalty method, using
  // the special "ghost" nodes along the boundaries.  Note
  // that there are better ways to implement boundary conditions
  // for subdivision shells.  We use the simplest way here,
  // which is known to be overly restrictive and will lead to
  // a slightly too small deformation of the plate.
  for (const auto & elem : mesh.active_local_element_ptr_range())
    {
      // For the boundary conditions, we only need to loop over
      // the ghost elements.
      libmesh_assert_equal_to (elem->type(), TRI3SUBDIVISION);
      const Tri3Subdivision * gh_elem = static_cast<const Tri3Subdivision *> (elem);
      if (!gh_elem->is_ghost())
        continue;
 
      // Find the side which is part of the physical plate boundary,
      // that is, the boundary of the original mesh without ghosts.
      for (auto s : elem->side_index_range())
        {
          const Tri3Subdivision * nb_elem = static_cast<const Tri3Subdivision *> (elem->neighbor_ptr(s));
          if (nb_elem == nullptr || nb_elem->is_ghost())
            continue;
 
          /*
           * Determine the four nodes involved in the boundary
           * condition treatment of this side.  The MeshTools::Subdiv
           * namespace provides lookup tables next and prev
           * for an efficient determination of the next and previous
           * nodes of an element, respectively.
           *
           *      n4
           *     /  \
           *    / gh \
           *  n2 ---- n3
           *    \ nb /
           *     \  /
           *      n1
           */
          const Node * nodes [4]; // n1, n2, n3, n4
          nodes[1] = gh_elem->node_ptr(s); // n2
          nodes[2] = gh_elem->node_ptr(MeshTools::Subdivision::next[s]); // n3
          nodes[3] = gh_elem->node_ptr(MeshTools::Subdivision::prev[s]); // n4
 
          // The node in the interior of the domain, n1, is the
          // hardest to find.  Walk along the edges of element nb until
          // we have identified it.
          unsigned int n_int = 0;
          nodes[0] = nb_elem->node_ptr(0);
          while (nodes[0]->id() == nodes[1]->id() || nodes[0]->id() == nodes[2]->id())
            nodes[0] = nb_elem->node_ptr(++n_int);
 
          // The penalty value.  \f$ \frac{1}{\epsilon} \f$
          const Real penalty = 1.e10;
 
          // With this simple method, clamped boundary conditions are
          // obtained by penalizing the displacements of all four nodes.
          // This ensures that the displacement field vanishes on the
          // boundary side s.
          for (unsigned int n=0; n<4; ++n)
            {
              const dof_id_type u_dof = nodes[n]->dof_number (system.number(), u_var, 0);
              const dof_id_type v_dof = nodes[n]->dof_number (system.number(), v_var, 0);
              const dof_id_type w_dof = nodes[n]->dof_number (system.number(), w_var, 0);
              matrix.add (u_dof, u_dof, penalty);
              matrix.add (v_dof, v_dof, penalty);
              matrix.add (w_dof, w_dof, penalty);
            }
        }
    } // end of ghost element loop
}
 
#endif // defined(LIBMESH_ENABLE_SECOND_DERIVATIVES) && LIBMESH_DIM > 2