variational_smoother_constraint.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2025 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

// Local Includes
#include "libmesh/variational_smoother_constraint.h"
#include "libmesh/mesh_tools.h"
#include "libmesh/boundary_info.h"

namespace libMesh
{

// Helper function to orient a vector in the positive x/y/z direction
auto get_positive_vector = [](const Point & vec) -> Point {
  libmesh_error_msg_if(vec.norm() < TOLERANCE,
                       "Can't define a positively-oriented vector with a zero vector.");

  // Choose sign such that direction vector points in the positive x/y/z
  // direction This helps to eliminate duplicate lines/planes
  Point canonical{0, 0, 0};
  // Choose the canonical dimension to ensure the dot product below is nonzero
  for (const auto dim_id : make_range(3))
    if (!absolute_fuzzy_equals(vec(dim_id), 0.))
      {
        canonical(dim_id) = 1.;
        break;
      }

  const auto dot_prod = vec * canonical;
  libmesh_assert(!absolute_fuzzy_equals(dot_prod, 0.));

  return (dot_prod > 0) ? vec.unit() : -vec.unit();
};

PointConstraint::PointConstraint(const Point & point, const Real & tol) : _point(point), _tol(tol)
{
}

bool PointConstraint::operator<(const PointConstraint & other) const
{
  if (*this == other)
    return false;

  return _point < other.point();
}

bool PointConstraint::operator==(const PointConstraint & other) const
{
  return _point.absolute_fuzzy_equals(other.point(), _tol);
}

ConstraintVariant PointConstraint::intersect(const ConstraintVariant & other) const
{
  // using visit to resolve the variant to its actual type
  return std::visit(
      [&](auto && o) -> ConstraintVariant {
        if (!o.contains_point(*this))
          // Point is not on the constraint
          return InvalidConstraint();

        return *this;
      },
      other);
}

LineConstraint::LineConstraint(const Point & point, const Point & direction, const Real & tol)
  : _point(point), _direction(get_positive_vector(direction)), _tol(tol)
{
  libmesh_error_msg_if(_direction.norm() < _tol,
                       "Can't define a line with zero magnitude direction vector.");
}

bool LineConstraint::operator<(const LineConstraint & other) const
{
  if (*this == other)
    return false;

  if (!(_direction.absolute_fuzzy_equals(other.direction(), _tol)))
    return _direction < other.direction();
  return (_direction * _point) < (other.direction() * other.point());
}

bool LineConstraint::operator==(const LineConstraint & other) const
{
  if (!(_direction.absolute_fuzzy_equals(other.direction(), _tol)))
    return false;
  return this->contains_point(other.point());
}

bool LineConstraint::contains_point(const PointConstraint & p) const
{
  // If the point lies on the line, then the vector p - point is parallel to the
  // line In that case, the cross product of p - point with the line's direction
  // will be zero.
  return _direction.cross(p.point() - _point).norm() < _tol;
}

bool LineConstraint::is_parallel(const LineConstraint & l) const
{
  return _direction.absolute_fuzzy_equals(l.direction(), _tol);
}

bool LineConstraint::is_parallel(const PlaneConstraint & p) const
{
  return _direction * p.normal() < _tol;
}

ConstraintVariant LineConstraint::intersect(const ConstraintVariant & other) const
{
  // using visit to resolve the variant to its actual type
  return std::visit(
      [&](auto && o) -> ConstraintVariant {
        // Use std::decay_t to strip references/const/volatile from o's type,
        // so we can match the actual stored variant type in std::is_same_v.
        using T = std::decay_t<decltype(o)>;
        if constexpr (std::is_same_v<T, LineConstraint>)
          {
            if (*this == o)
              return *this;

            if (this->is_parallel(o))
              // Lines are parallel and do not intersect
              return InvalidConstraint();

            // Solve for t in the equation p1 + t·d1 = p2 + s·d2
            // The shortest vector between skew lines lies along the normal vector
            // (d1 × d2). Projecting the vector (p2 - p1) onto this normal gives a
            // scalar proportional to the distance. This is equivalent to solving:
            //   ((p2 - p1) × d2) · (d1 × d2) = t · |d1 × d2|²
            //   ⇒ t = ((delta × d2) · (d1 × d2)) / |d1 × d2|²

            const Point delta = o.point() - _point;
            const Point cross_d1_d2 = _direction.cross(o.direction());
            const Real cross_dot = (delta.cross(o.direction())) * cross_d1_d2;
            const Real denom = cross_d1_d2.norm_sq();

            const Real t = cross_dot / denom;
            const Point intersection = _point + t * _direction;

            // Verify that intersection lies on both lines
            if (o.direction().cross(intersection - o.point()).norm() > _tol)
              // Lines do not intersect at a single point
              return InvalidConstraint();

            return PointConstraint{intersection};
          }

        else if constexpr (std::is_same_v<T, PlaneConstraint>)
          return o.intersect(*this);

        else if constexpr (std::is_same_v<T, PointConstraint>)
          {
            if (!this->contains_point(o))
              // Point is not on the line
              return InvalidConstraint();

            return o;
          }

        else
          libmesh_error_msg("Unsupported constraint type in Line::intersect.");
      },
      other);
}

PlaneConstraint::PlaneConstraint(const Point & point, const Point & normal, const Real & tol)
  : _point(point), _normal(get_positive_vector(normal)), _tol(tol)
{
  libmesh_error_msg_if(_normal.norm() < _tol,
                       "Can't define a plane with zero magnitude direction vector.");
}

bool PlaneConstraint::operator<(const PlaneConstraint & other) const
{
  if (*this == other)
    return false;

  if (!(_normal.absolute_fuzzy_equals(other.normal(), _tol)))
    return _normal < other.normal();
  return (_normal * _point) < (other.normal() * other.point());
}

bool PlaneConstraint::operator==(const PlaneConstraint & other) const
{
  if (!(_normal.absolute_fuzzy_equals(other.normal(), _tol)))
    return false;
  return this->contains_point(other.point());
}

bool PlaneConstraint::is_parallel(const PlaneConstraint & p) const
{
  return _normal.absolute_fuzzy_equals(p.normal(), _tol);
}

bool PlaneConstraint::is_parallel(const LineConstraint & l) const { return l.is_parallel(*this); }

bool PlaneConstraint::contains_point(const PointConstraint & p) const
{
  // distance between the point and the plane
  const Real dist = (p.point() - _point) * _normal;
  return std::abs(dist) < _tol;
}

bool PlaneConstraint::contains_line(const LineConstraint & l) const
{
  const bool base_on_plane = this->contains_point(PointConstraint(l.point()));
  const bool dir_orthogonal = std::abs(_normal * l.direction()) < _tol;
  return base_on_plane && dir_orthogonal;
}

ConstraintVariant PlaneConstraint::intersect(const ConstraintVariant & other) const
{
  // using visit to resolve the variant to its actual type
  return std::visit(
      [&](auto && o) -> ConstraintVariant {
        // Use std::decay_t to strip references/const/volatile from o's type,
        // so we can match the actual stored variant type in std::is_same_v.
        using T = std::decay_t<decltype(o)>;
        if constexpr (std::is_same_v<T, PlaneConstraint>)
          {
            // If planes are identical, return one of them
            if (*this == o)
              return *this;

            if (this->is_parallel(o))
              // Planes are parallel and do not intersect
              return InvalidConstraint();

            // Solve for a point on the intersection line of two planes.
            // Given planes:
            //   Plane 1: n1 · (x - p1) = 0
            //   Plane 2: n2 · (x - p2) = 0
            // The line of intersection has direction dir = n1 × n2.
            // To find a point on this line, we assume:
            //   x = p1 + s·n1 = p2 + t·n2
            //   ⇒ p1 - p2 = t·n2 - s·n1
            // Taking dot products with n1 and n2 leads to:
            //   [-n1·n1   n1·n2] [s] = [n1 · (p1 - p2)]
            //   [-n1·n2   n2·n2] [t]   [n2 · (p1 - p2)]

            const Point dir = this->_normal.cross(o.normal()); // direction of line of intersection
            libmesh_assert(dir.norm() > _tol);
            const Point w = _point - o.point();

            // Dot product terms used in 2x2 system
            const Real n1_dot_n1 = _normal * _normal;
            const Real n1_dot_n2 = _normal * o.normal();
            const Real n2_dot_n2 = o.normal() * o.normal();
            const Real n1_dot_w = _normal * w;
            const Real n2_dot_w = o.normal() * w;

            const Real denom = -(n1_dot_n1 * n2_dot_n2 - n1_dot_n2 * n1_dot_n2);
            libmesh_assert(std::abs(denom) > _tol);

            const Real s = -(n1_dot_n2 * n2_dot_w - n2_dot_n2 * n1_dot_w) / denom;
            const Point p0 = _point + s * _normal;

            return LineConstraint{p0, dir};
          }

        else if constexpr (std::is_same_v<T, LineConstraint>)
          {
            if (this->contains_line(o))
              return o;

            if (this->is_parallel(o))
              // Line is parallel and does not intersect the plane
              return InvalidConstraint();

            // Solve for t in the parametric equation:
            //   p(t) = point + t·d
            // such that this point also satisfies the plane equation:
            //   n · (p(t) - p0) = 0
            // which leads to:
            //   t = (n · (p0 - point)) / (n · d)

            const Real denom = _normal * o.direction();
            libmesh_assert(std::abs(denom) > _tol);
            const Real t = (_normal * (_point - o.point())) / denom;
            return PointConstraint{o.point() + t * o.direction()};
          }

        else if constexpr (std::is_same_v<T, PointConstraint>)
          {
            if (!this->contains_point(o))
              // Point is not on the plane
              return InvalidConstraint();

            return o;
          }

        else
          libmesh_error_msg("Unsupported constraint type in Plane::intersect.");
      },
      other);
}

VariationalSmootherConstraint::VariationalSmootherConstraint(
    System & sys, const bool & preserve_subdomain_boundaries)
  : Constraint(), _sys(sys), _preserve_subdomain_boundaries(preserve_subdomain_boundaries)
{
}

VariationalSmootherConstraint::~VariationalSmootherConstraint() = default;

void VariationalSmootherConstraint::constrain()
{
  const auto & mesh = _sys.get_mesh();
  const auto dim = mesh.mesh_dimension();

  // Only compute the node to elem map once
  std::unordered_map<dof_id_type, std::vector<const Elem *>> nodes_to_elem_map;
  MeshTools::build_nodes_to_elem_map(mesh, nodes_to_elem_map);

  const auto & boundary_info = mesh.get_boundary_info();

  const auto boundary_node_ids = MeshTools::find_boundary_nodes(mesh);

  // Identify/constrain subdomain boundary nodes, if requested
  std::unordered_map<dof_id_type, ConstraintVariant> subdomain_boundary_map;
  if (_preserve_subdomain_boundaries)
    {
      for (const auto * elem : mesh.active_element_ptr_range())
        {
          const auto & sub_id1 = elem->subdomain_id();
          for (const auto side : elem->side_index_range())
            {
              const auto * neighbor = elem->neighbor_ptr(side);
              if (neighbor == nullptr)
                continue;

              const auto & sub_id2 = neighbor->subdomain_id();
              if (sub_id1 == sub_id2)
                continue;

              // elem and neighbor are in different subdomains, and share nodes
              // that need to be constrained
              for (const auto local_node_id : elem->nodes_on_side(side))
                {
                  const auto & node = mesh.node_ref(elem->node_id(local_node_id));
                  // Make sure we haven't already processed this node
                  if (subdomain_boundary_map.count(node.id()))
                    continue;

                  // Get the relevant nodal neighbors for the subdomain constraint
                  const auto side_grouped_boundary_neighbors =
                      get_neighbors_for_subdomain_constraint(
                          mesh, node, sub_id1, nodes_to_elem_map);

                  // Determine which constraint should be imposed
                  const auto subdomain_constraint =
                      determine_constraint(node, dim, side_grouped_boundary_neighbors);

                  // This subdomain boundary node does not lie on an external boundary,
                  // go ahead and impose constraint
                  if (boundary_node_ids.find(node.id()) == boundary_node_ids.end())
                    this->impose_constraint(node, subdomain_constraint);

                  // This subdomain boundary node could lie on an external boundary, save it
                  // for later to combine with the external boundary constraint.
                  // We also save constraints for non-boundary nodes so we don't try to
                  // re-constrain the node when accessed from the neighboring elem.
                  // See subdomain_boundary_map.count call above.
                  subdomain_boundary_map[node.id()] = subdomain_constraint;

                } // for local_node_id

            } // for side
        } // for elem
    }

  // Loop through boundary nodes and impose constraints
  for (const auto & bid : boundary_node_ids)
    {
      const auto & node = mesh.node_ref(bid);

      // Get the relevant nodal neighbors for the boundary constraint
      const auto side_grouped_boundary_neighbors = get_neighbors_for_boundary_constraint(
          mesh, node, boundary_node_ids, boundary_info, nodes_to_elem_map);

      // Determine which constraint should be imposed
      const auto boundary_constraint =
          determine_constraint(node, dim, side_grouped_boundary_neighbors);

      // Check for the case where this boundary node is also part of a subdomain id boundary
      if (const auto it = subdomain_boundary_map.find(bid); it != subdomain_boundary_map.end())
        {
          const auto & subdomain_constraint = it->second;
          // Combine current boundary constraint with previously determined
          // subdomain_constraint
          auto combined_constraint = intersect_constraints(subdomain_constraint, boundary_constraint);

          // This will catch cases where constraints have no intersection
          // Fall back to fixed node constraint
          if (std::holds_alternative<InvalidConstraint>(combined_constraint))
            combined_constraint = PointConstraint(node);

          this->impose_constraint(node, combined_constraint);
        }

      else
        this->impose_constraint(node, boundary_constraint);

    } // end bid
}

void VariationalSmootherConstraint::fix_node(const Node & node)
{
  for (const auto d : make_range(_sys.get_mesh().mesh_dimension()))
    {
      const auto constrained_dof_index = node.dof_number(_sys.number(), d, 0);
      DofConstraintRow constraint_row;
      // Leave the constraint row as all zeros so this dof is independent from other dofs
      const auto constrained_value = node(d);
      // Simply constrain this dof to retain it's current value
      _sys.get_dof_map().add_constraint_row(
          constrained_dof_index, constraint_row, constrained_value, true);
    }
}

void VariationalSmootherConstraint::constrain_node_to_plane(const Node & node,
                                                            const Point & ref_normal_vec)
{
  const auto dim = _sys.get_mesh().mesh_dimension();
  // determine equation of plane: c_x * x + c_y * y + c_z * z + c = 0
  std::vector<Real> xyz_coefs; // vector to hold c_x, c_y, c_z
  Real c = 0.;

  // We choose to constrain the dimension with the largest magnitude coefficient
  // This approach ensures the coefficients added to the constraint_row
  // (i.e., -c_xyz / c_max) have as small magnitude as possible

  // We initialize this to avoid maybe-uninitialized compiler error
  unsigned int constrained_dim = 0;

  // Let's assert that we have a nonzero normal to ensure that constrained_dim
  // is always set
  libmesh_assert(ref_normal_vec.norm() > TOLERANCE);

  Real max_abs_coef = 0.;
  for (const auto d : make_range(dim))
    {
      const auto coef = ref_normal_vec(d);
      xyz_coefs.push_back(coef);
      c -= coef * node(d);

      const auto coef_abs = std::abs(coef);
      if (coef_abs > max_abs_coef)
        {
          max_abs_coef = coef_abs;
          constrained_dim = d;
        }
    }

  DofConstraintRow constraint_row;
  for (const auto free_dim : make_range(dim))
    {
      if (free_dim == constrained_dim)
        continue;

      const auto free_dof_index = node.dof_number(_sys.number(), free_dim, 0);
      constraint_row[free_dof_index] = -xyz_coefs[free_dim] / xyz_coefs[constrained_dim];
    }

  const auto inhomogeneous_part = -c / xyz_coefs[constrained_dim];
  const auto constrained_dof_index = node.dof_number(_sys.number(), constrained_dim, 0);
  _sys.get_dof_map().add_constraint_row(
      constrained_dof_index, constraint_row, inhomogeneous_part, true);
}

void VariationalSmootherConstraint::constrain_node_to_line(const Node & node,
                                                           const Point & line_vec)
{
  const auto dim = _sys.get_mesh().mesh_dimension();

  // We will free the dimension most parallel to line_vec to keep the
  // constraint coefficients small
  const std::vector<Real> line_vec_coefs{line_vec(0), line_vec(1), line_vec(2)};
  auto it = std::max_element(line_vec_coefs.begin(), line_vec_coefs.end(), [](double a, double b) {
    return std::abs(a) < std::abs(b);
  });
  const unsigned int free_dim = std::distance(line_vec_coefs.begin(), it);
  const auto free_dof_index = node.dof_number(_sys.number(), free_dim, 0);

  // A line is parameterized as r(t) = node + t * line_vec, so
  // x(t) = node(x) + t * line_vec(x)
  // y(t) = node(y) + t * line_vec(y)
  // z(t) = node(z) + t * line_vec(z)
  // Let's say we leave x free. Then t = (x(t) - node(x)) / line_vec(x)
  // Then y and z can be constrained as
  // y = node(y) + line_vec_y * (x(t) - node(x)) / line_vec(x)
  //   = x(t) * line_vec(y) / line_vec(x) + (node(y) - node(x) * line_vec(y) / line_vec(x))
  // z = x(t) * line_vec(z) / line_vec(x) + (node(z) - node(x) * line_vec(z) / line_vec(x))

  libmesh_assert(!relative_fuzzy_equals(line_vec(free_dim), 0.));
  for (const auto constrained_dim : make_range(dim))
    {
      if (constrained_dim == free_dim)
        continue;

      DofConstraintRow constraint_row;
      constraint_row[free_dof_index] = line_vec(constrained_dim) / line_vec(free_dim);
      const auto inhomogeneous_part =
          node(constrained_dim) - node(free_dim) * line_vec(constrained_dim) / line_vec(free_dim);
      const auto constrained_dof_index = node.dof_number(_sys.number(), constrained_dim, 0);
      _sys.get_dof_map().add_constraint_row(
          constrained_dof_index, constraint_row, inhomogeneous_part, true);
    }
}

void VariationalSmootherConstraint::find_nodal_or_face_neighbors(
    const MeshBase & mesh,
    const Node & node,
    const std::unordered_map<dof_id_type, std::vector<const Elem *>> & nodes_to_elem_map,
    std::vector<const Node *> & neighbors)
{
  // Find all the nodal neighbors... that is the nodes directly connected
  // to this node through one edge.
  MeshTools::find_nodal_neighbors(mesh, node, nodes_to_elem_map, neighbors);

  // If no neighbors are found, use all nodes on the containing side as
  // neighbors.
  if (!neighbors.size())
    {
      // Grab the element containing node
      const auto * elem = libmesh_map_find(nodes_to_elem_map, node.id()).front();
      // Find the element side containing node
      for (const auto &side : elem->side_index_range())
        {
          const auto &nodes_on_side = elem->nodes_on_side(side);
          const auto it =
              std::find_if(nodes_on_side.begin(), nodes_on_side.end(), [&](auto local_node_id) {
                return elem->node_id(local_node_id) == node.id();
              });

          if (it != nodes_on_side.end())
            {
              for (const auto &local_node_id : nodes_on_side)
                // No need to add node itself as a neighbor
                if (const auto *node_ptr = elem->node_ptr(local_node_id);
                    *node_ptr != node)
                  neighbors.push_back(node_ptr);
              break;
            }
        }
    }
  libmesh_assert(neighbors.size());
}

// Utility function to determine whether two nodes share a boundary ID.
// The motivation for this is that a sliding boundary node on a triangular
// element can have a neighbor boundary node in the same element that is not
// part of the same boundary
// Consider the below example with nodes A, C, D, F, G that comprise elements
// E1, E2, E3, with boundaries B1, B2, B3, B4. To determine the constraint
// equations for the sliding node C, the neighboring nodes A and D need to
// be identified to construct the line that C is allowed to slide along.
// Note that neighbors A and D both share the boundary B1 with C.
// Without ensuring that neighbors share the same boundary as the current
// node, a neighboring node that does not lie on the same boundary
// (i.e. F and G) might be selected to define the constraining line,
// resulting in an incorrect constraint.
// Note that if, for example, boundaries B1 and B2 were to be combined
// into boundary B12, the node F would share a boundary id with node C
// and result in an incorrect constraint. It would be useful to design
// additional checks to detect cases like this.

//         B3
//    G-----------F
//    | \       / |
// B4 |  \  E2 /  | B2
//    |   \   /   |
//    | E1 \ / E3 |
//    A-----C-----D
//         B1

bool VariationalSmootherConstraint::nodes_share_boundary_id(const Node & boundary_node,
                                                            const Node & neighbor_node,
                                                            const Elem & containing_elem,
                                                            const BoundaryInfo & boundary_info)
{
  bool nodes_share_bid = false;

  // Node ids local to containing_elem
  const auto node_id = containing_elem.get_node_index(&boundary_node);
  const auto neighbor_id = containing_elem.get_node_index(&neighbor_node);

  for (const auto side_id : containing_elem.side_index_range())
    {
      // We don't care about this side if it doesn't contain our boundary and neighbor nodes
      if (!(containing_elem.is_node_on_side(node_id, side_id) &&
            containing_elem.is_node_on_side(neighbor_id, side_id)))
        continue;

      // If the current side, containing boundary_node and neighbor_node, lies on a boundary,
      // we can say that boundary_node and neighbor_node have a common boundary id.
      std::vector<boundary_id_type> boundary_ids;
      boundary_info.boundary_ids(&containing_elem, side_id, boundary_ids);
      if (boundary_ids.size())
        {
          nodes_share_bid = true;
          break;
        }
    }
  return nodes_share_bid;
}

std::set<std::set<const Node *>>
VariationalSmootherConstraint::get_neighbors_for_subdomain_constraint(
    const MeshBase & mesh,
    const Node & node,
    const subdomain_id_type sub_id,
    const std::unordered_map<dof_id_type, std::vector<const Elem *>> & nodes_to_elem_map)
{

  // Find all the nodal neighbors... that is the nodes directly connected
  // to this node through one edge, or if none exists, use other nodes on the
  // containing face
  std::vector<const Node *> neighbors;
  VariationalSmootherConstraint::find_nodal_or_face_neighbors(
      mesh, node, nodes_to_elem_map, neighbors);

  // Each constituent set corresponds to neighbors sharing a face on the
  // subdomain boundary
  std::set<std::set<const Node *>> side_grouped_boundary_neighbors;

  for (const auto * neigh : neighbors)
    {
      // Determine whether the neighbor is on the subdomain boundary
      // First, find the common elements that both node and neigh belong to
      const auto & elems_containing_node = libmesh_map_find(nodes_to_elem_map, node.id());
      const auto & elems_containing_neigh = libmesh_map_find(nodes_to_elem_map, neigh->id());
      const Elem * common_elem = nullptr;
      for (const auto * neigh_elem : elems_containing_neigh)
        {
          if ((std::find(elems_containing_node.begin(), elems_containing_node.end(), neigh_elem) !=
               elems_containing_node.end())
              // We should be able to find a common element on the sub_id boundary
              && (neigh_elem->subdomain_id() == sub_id))
            common_elem = neigh_elem;
          else
            continue;

          // Now, determine whether node and neigh are on a side coincident
          // with the subdomain boundary
          for (const auto common_side : common_elem->side_index_range())
            {
              bool node_found_on_side = false;
              bool neigh_found_on_side = false;
              for (const auto local_node_id : common_elem->nodes_on_side(common_side))
                {
                  if (common_elem->node_id(local_node_id) == node.id())
                    node_found_on_side = true;
                  else if (common_elem->node_id(local_node_id) == neigh->id())
                    neigh_found_on_side = true;
                }

              if (!(node_found_on_side && neigh_found_on_side &&
                    common_elem->neighbor_ptr(common_side)))
                continue;

              const auto matched_side = common_side;
              // There could be multiple matched sides, so keep this next part
              // inside the common_side loop
              //
              // Does matched_side, containing both node and neigh, lie on the
              // sub_id subdomain boundary?
              const auto matched_neighbor_sub_id =
                  common_elem->neighbor_ptr(matched_side)->subdomain_id();
              const bool is_matched_side_on_subdomain_boundary = matched_neighbor_sub_id != sub_id;

              if (is_matched_side_on_subdomain_boundary)
                {
                  // Store all nodes that live on this side
                  const auto nodes_on_side = common_elem->nodes_on_side(common_side);
                  std::set<const Node *> node_ptrs_on_side;
                  for (const auto local_node_id : nodes_on_side)
                    node_ptrs_on_side.insert(common_elem->node_ptr(local_node_id));
                  node_ptrs_on_side.erase(node_ptrs_on_side.find(&node));
                  side_grouped_boundary_neighbors.insert(node_ptrs_on_side);

                  continue;
                }

            } // for common_side

        } // for neigh_elem
    }

  return side_grouped_boundary_neighbors;
}

std::set<std::set<const Node *>>
VariationalSmootherConstraint::get_neighbors_for_boundary_constraint(
    const MeshBase & mesh,
    const Node & node,
    const std::unordered_set<dof_id_type> & boundary_node_ids,
    const BoundaryInfo & boundary_info,
    const std::unordered_map<dof_id_type, std::vector<const Elem *>> & nodes_to_elem_map)
{

  // Find all the nodal neighbors... that is the nodes directly connected
  // to this node through one edge, or if none exists, use other nodes on the
  // containing face
  std::vector<const Node *> neighbors;
  VariationalSmootherConstraint::find_nodal_or_face_neighbors(
      mesh, node, nodes_to_elem_map, neighbors);

  // Each constituent set corresponds to neighbors sharing a face on the
  // boundary
  std::set<std::set<const Node *>> side_grouped_boundary_neighbors;

  for (const auto * neigh : neighbors)
    {
      const bool is_neighbor_boundary_node =
          boundary_node_ids.find(neigh->id()) != boundary_node_ids.end();
      if (!is_neighbor_boundary_node)
        continue;

      // Determine whether nodes share a common boundary id
      // First, find the common element that both node and neigh belong to
      const auto & elems_containing_node = libmesh_map_find(nodes_to_elem_map, node.id());
      const auto & elems_containing_neigh = libmesh_map_find(nodes_to_elem_map, neigh->id());
      const Elem * common_elem = nullptr;
      for (const auto * neigh_elem : elems_containing_neigh)
        {
          const bool is_neigh_common =
              std::find(elems_containing_node.begin(), elems_containing_node.end(), neigh_elem) !=
              elems_containing_node.end();
          if (!is_neigh_common)
            continue;
          common_elem = neigh_elem;
          // Keep this in the neigh_elem loop because there can be multiple common
          // elements Now, determine whether node and neigh share a common boundary
          // id
          const bool nodes_have_common_bid = VariationalSmootherConstraint::nodes_share_boundary_id(
              node, *neigh, *common_elem, boundary_info);
          if (nodes_have_common_bid)
            {
              // Find the side coinciding with the shared boundary
              for (const auto side : common_elem->side_index_range())
                {
                  // We only care about external boundaries here, make sure side doesn't
                  // have a neighbor
                  if (common_elem->neighbor_ptr(side))
                    continue;

                  bool node_found_on_side = false;
                  bool neigh_found_on_side = false;
                  const auto nodes_on_side = common_elem->nodes_on_side(side);
                  for (const auto local_node_id : nodes_on_side)
                    {
                      if (common_elem->node_id(local_node_id) == node.id())
                        node_found_on_side = true;
                      else if (common_elem->node_id(local_node_id) == neigh->id())
                        neigh_found_on_side = true;
                    }
                  if (!(node_found_on_side && neigh_found_on_side))
                    continue;

                  std::set<const Node *> node_ptrs_on_side;
                  for (const auto local_node_id : nodes_on_side)
                    node_ptrs_on_side.insert(common_elem->node_ptr(local_node_id));
                  node_ptrs_on_side.erase(node_ptrs_on_side.find(&node));
                  side_grouped_boundary_neighbors.insert(node_ptrs_on_side);
                }
              continue;
            }
        }
    }

  return side_grouped_boundary_neighbors;
}

ConstraintVariant VariationalSmootherConstraint::determine_constraint(
    const Node & node,
    const unsigned int dim,
    const std::set<std::set<const Node *>> & side_grouped_boundary_neighbors)
{
  // Determines the appropriate geometric constraint for a node based on its
  // neighbors.

  // Extract neighbors in flat vector
  std::vector<const Node *> neighbors;
  for (const auto & side : side_grouped_boundary_neighbors)
    neighbors.insert(neighbors.end(), side.begin(), side.end());

  // Constrain the node to it's current location
  if (dim == 1 || neighbors.size() == 1)
    return PointConstraint{node};

  if (dim == 2 || neighbors.size() == 2)
    {
      // Determine whether node + all neighbors are colinear
      bool all_colinear = true;
      const Point ref_dir = (*neighbors[0] - node).unit();
      for (const auto i : make_range(std::size_t(1), neighbors.size()))
        {
          const Point delta = *(neighbors[i]) - node;
          libmesh_assert(delta.norm() >= TOLERANCE);
          const Point dir = delta.unit();
          if (!dir.relative_fuzzy_equals(ref_dir) && !dir.relative_fuzzy_equals(-ref_dir))
          {
            all_colinear = false;
            break;
          }
        }

      if (all_colinear)
        return LineConstraint{node, ref_dir};

      return PointConstraint{node};
    }

  // dim == 3, neighbors.size() >= 3
  std::set<PlaneConstraint> valid_planes;
  for (const auto & side_nodes : side_grouped_boundary_neighbors)
    {
      std::vector<const Node *> side_nodes_vec(side_nodes.begin(), side_nodes.end());
      for (const auto i : index_range(side_nodes_vec))
        {
          const Point vec_i = (*side_nodes_vec[i] - node);
          for (const auto j : make_range(i))
            {
              const Point vec_j = (*side_nodes_vec[j] - node);
              Point candidate_normal = vec_i.cross(vec_j);
              if (candidate_normal.norm() <= TOLERANCE)
                continue;

              const PlaneConstraint candidate_plane{node, candidate_normal};
              valid_planes.emplace(candidate_plane);
            }
        }
    }

  // Fall back to point constraint
  if (valid_planes.empty())
    return PointConstraint(node);

  // Combine all the planes together to get a common constraint
  auto it = valid_planes.begin();
  ConstraintVariant current = *it++;
  for (; it != valid_planes.end(); ++it)
    {
      current = intersect_constraints(current, *it);

      // This will catch cases where constraints have no intersection
      // (i.e., the element surface is non-planar)
      // Fall back to fixed node constraint
      if (std::holds_alternative<InvalidConstraint>(current))
        {
          current = PointConstraint(node);
          break;
        }
    }

  return current;
}

// Applies the computed constraint (PointConstraint, LineConstraint, or
// PlaneConstraint) to a node.
void VariationalSmootherConstraint::impose_constraint(const Node & node,
                                                      const ConstraintVariant & constraint)
{

  libmesh_assert_msg(!std::holds_alternative<InvalidConstraint>(constraint),
                     "Cannot impose constraint using InvalidConstraint.");

  if (std::holds_alternative<PointConstraint>(constraint))
    fix_node(node);
  else if (std::holds_alternative<LineConstraint>(constraint))
    constrain_node_to_line(node, std::get<LineConstraint>(constraint).direction());
  else if (std::holds_alternative<PlaneConstraint>(constraint))
    constrain_node_to_plane(node, std::get<PlaneConstraint>(constraint).normal());

  else
    libmesh_assert_msg(false, "Unknown constraint type.");
}

} // namespace libMesh