weak_boundary_conditions.h

/*  ______________________________________________________________________
 *
 *  ExaDG - High-Order Discontinuous Galerkin for the Exa-Scale
 *
 *  Copyright (C) 2021 by the ExaDG authors
 *
 *  This program is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *
 *  This program 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 General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <https://www.gnu.org/licenses/>.
 *  ______________________________________________________________________
 */
 
#ifndef INCLUDE_EXADG_INCOMPRESSIBLE_NAVIER_STOKES_SPATIAL_DISCRETIZATION_OPERATORS_WEAK_BOUNDARY_CONDITIONS_H_
#define INCLUDE_EXADG_INCOMPRESSIBLE_NAVIER_STOKES_SPATIAL_DISCRETIZATION_OPERATORS_WEAK_BOUNDARY_CONDITIONS_H_
 
#include <exadg/functions_and_boundary_conditions/evaluate_functions.h>
#include <exadg/incompressible_navier_stokes/user_interface/boundary_descriptor.h>
#include <exadg/incompressible_navier_stokes/user_interface/enum_types.h>
#include <exadg/matrix_free/integrators.h>
#include <exadg/operators/operator_type.h>
 
namespace ExaDG
{
namespace IncNS
{
// clang-format off
/*
 * Velocity:
 *
 *  The following two functions calculate the interior/exterior value for boundary faces depending on the
 *  operator type, the type of the boundary face and the given boundary conditions.
 *
 *                            +-------------------------+--------------------+------------------------------+
 *                            | Dirichlet boundaries    | Neumann boundaries | symmetry boundaries          |
 *  +-------------------------+-------------------------+--------------------+------------------------------+
 *  | full operator           | u⁺ = -u⁻ + 2g           | u⁺ = u⁻            | u⁺ = u⁻ - 2 (u⁻*n)n          |
 *  +-------------------------+-------------------------+--------------------+------------------------------+
 *  | homogeneous operator    | u⁺ = -u⁻                | u⁺ = u⁻            | u⁺ = u⁻ - 2 (u⁻*n)n          |
 *  +-------------------------+-------------------------+--------------------+------------------------------+
 *  | inhomogeneous operator  | u⁺ = -u⁻ + 2g , u⁻ = 0  | u⁺ = u⁻ , u⁻ = 0   | u⁺ = u⁻ - 2 (u⁻*n)n , u⁻ = 0 |
 *  +-------------------------+-------------------------+--------------------+------------------------------+
 *
 */
// clang-format on
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>
  calculate_interior_value(unsigned int const                       q,
                           FaceIntegrator<dim, dim, Number> const & integrator,
                           OperatorType const &                     operator_type)
{
  // element e⁻
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> value_m;
 
  if(operator_type == OperatorType::full or operator_type == OperatorType::homogeneous)
  {
    value_m = integrator.get_value(q);
  }
  else if(operator_type == OperatorType::inhomogeneous)
  {
    // do nothing, value_m is already initialized with zeros
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
  }
 
  return value_m;
}
 
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>
  calculate_exterior_value(dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> const & value_m,
                           unsigned int const                                              q,
                           FaceIntegrator<dim, dim, Number> const &        integrator,
                           OperatorType const &                            operator_type,
                           BoundaryTypeU const &                           boundary_type,
                           dealii::types::boundary_id const                boundary_id,
                           std::shared_ptr<BoundaryDescriptorU<dim> const> boundary_descriptor,
                           double const &                                  time)
{
  // element e⁺
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> value_p;
 
  if(boundary_type == BoundaryTypeU::Dirichlet or boundary_type == BoundaryTypeU::DirichletCached)
  {
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> g;
 
      if(boundary_type == BoundaryTypeU::Dirichlet)
      {
        auto bc       = boundary_descriptor->dirichlet_bc.find(boundary_id)->second;
        auto q_points = integrator.quadrature_point(q);
 
        g = FunctionEvaluator<1, dim, Number>::value(*bc, q_points, time);
      }
      else if(boundary_type == BoundaryTypeU::DirichletCached)
      {
        auto bc = boundary_descriptor->get_dirichlet_cached_data();
        g       = FunctionEvaluator<1, dim, Number>::value(*bc,
                                                     integrator.get_current_cell_index(),
                                                     q,
                                                     integrator.get_quadrature_index());
      }
      else
      {
        AssertThrow(false, dealii::ExcMessage("Not implemented."));
      }
 
      value_p = -value_m + dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>(2.0 * g);
    }
    else if(operator_type == OperatorType::homogeneous)
    {
      value_p = -value_m;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
    }
  }
  else if(boundary_type == BoundaryTypeU::Neumann)
  {
    value_p = value_m;
  }
  else if(boundary_type == BoundaryTypeU::Symmetry)
  {
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> normal_m =
      integrator.get_normal_vector(q);
 
    value_p = value_m - 2.0 * (value_m * normal_m) * normal_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return value_p;
}
 
/*
 *  This function calculates the exterior velocity on boundary faces
 *  according to:
 *
 *  Dirichlet boundary: u⁺ = -u⁻ + 2g (mirror) or g (direct)
 *  Neumann boundary:   u⁺ = u⁻
 *  symmetry boundary:  u⁺ = u⁻ -(u⁻*n)n - (u⁻*n)n = u⁻ - 2 (u⁻*n)n
 *
 *  The name "convective" indicates that this function is used when
 *  evaluating the convective operator.
 *
 *  OperatorType:
 *  - To evaluate the nonlinear convective term (residual evaluation), use OperatorType::full.
 *  - To evaluate the linearization of the nonlinear convective term, use OperatorType::homogeneous.
 *  - To evaluate the linearly implicit convective term, use OperatorType::homogeneous for the
 * "matrix-vector" product and OperatorType::inhomogeneous for the "rhs" contribution.
 */
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>
  calculate_exterior_value_convective(
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> const & u_m,
    unsigned int const                                              q,
    FaceIntegrator<dim, dim, Number> &                              integrator,
    OperatorType const &                                            operator_type,
    BoundaryTypeU const &                                           boundary_type,
    TypeDirichletBCs const &                                        type_dirichlet_bc,
    dealii::types::boundary_id const                                boundary_id,
    std::shared_ptr<BoundaryDescriptorU<dim> const>                 boundary_descriptor,
    double const &                                                  time)
{
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> u_p;
 
  if(boundary_type == BoundaryTypeU::Dirichlet or boundary_type == BoundaryTypeU::DirichletCached)
  {
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> g;
 
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      if(boundary_type == BoundaryTypeU::Dirichlet)
      {
        auto bc       = boundary_descriptor->dirichlet_bc.find(boundary_id)->second;
        auto q_points = integrator.quadrature_point(q);
 
        g = FunctionEvaluator<1, dim, Number>::value(*bc, q_points, time);
      }
      else if(boundary_type == BoundaryTypeU::DirichletCached)
      {
        auto bc = boundary_descriptor->get_dirichlet_cached_data();
        g       = FunctionEvaluator<1, dim, Number>::value(*bc,
                                                     integrator.get_current_cell_index(),
                                                     q,
                                                     integrator.get_quadrature_index());
      }
      else
      {
        AssertThrow(false, dealii::ExcMessage("Not implemented."));
      }
    }
 
    if(type_dirichlet_bc == TypeDirichletBCs::Mirror)
    {
      u_p = -u_m + dealii::make_vectorized_array<Number>(2.0) * g;
    }
    else if(type_dirichlet_bc == TypeDirichletBCs::Direct)
    {
      u_p = g;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("TypeDirichletBCs is not implemented."));
    }
  }
  else if(boundary_type == BoundaryTypeU::Neumann)
  {
    u_p = u_m;
  }
  else if(boundary_type == BoundaryTypeU::Symmetry)
  {
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> normal_m =
      integrator.get_normal_vector(q);
 
    u_p = u_m - 2. * (u_m * normal_m) * normal_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return u_p;
}
 
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>
  calculate_exterior_value_from_dof_vector(
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> const & value_m,
    unsigned int const                                              q,
    FaceIntegrator<dim, dim, Number> const &                        integrator_bc,
    OperatorType const &                                            operator_type,
    BoundaryTypeU const &                                           boundary_type)
{
  // element e⁺
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> value_p;
 
  if(boundary_type == BoundaryTypeU::Dirichlet or boundary_type == BoundaryTypeU::DirichletCached)
  {
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> g = integrator_bc.get_value(q);
 
      value_p = -value_m + dealii::make_vectorized_array<Number>(2.0) * g;
    }
    else if(operator_type == OperatorType::homogeneous)
    {
      value_p = -value_m;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
    }
  }
  else if(boundary_type == BoundaryTypeU::Neumann)
  {
    value_p = value_m;
  }
  else if(boundary_type == BoundaryTypeU::Symmetry)
  {
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> normal_m =
      integrator_bc.get_normal_vector(q);
 
    value_p = value_m - 2.0 * (value_m * normal_m) * normal_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return value_p;
}
 
/*
 * Pressure:
 *
 * These two function calculate the interior/exterior value on boundary faces depending on the
 * operator type, the type of the boundary face and the given boundary conditions.
 *
 *                            +--------------------+----------------------+
 *                            | Neumann boundaries | Dirichlet boundaries |
 *  +-------------------------+--------------------+----------------------+
 *  | full operator           | p⁺ = p⁻            | p⁺ = - p⁻ + 2g       |
 *  +-------------------------+--------------------+----------------------+
 *  | homogeneous operator    | p⁺ = p⁻            | p⁺ = - p⁻            |
 *  +-------------------------+--------------------+----------------------+
 *  | inhomogeneous operator  | p⁺ = 0 , p⁻ = 0    | p⁺ = 2g , p⁻ = 0     |
 *  +-------------------------+--------------------+----------------------+
 *
 */
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::VectorizedArray<Number>
  calculate_interior_value(unsigned int const                     q,
                           FaceIntegrator<dim, 1, Number> const & integrator,
                           OperatorType const &                   operator_type)
{
  // element e⁻
  dealii::VectorizedArray<Number> value_m = dealii::make_vectorized_array<Number>(0.0);
 
  if(operator_type == OperatorType::full or operator_type == OperatorType::homogeneous)
  {
    value_m = integrator.get_value(q);
  }
  else if(operator_type == OperatorType::inhomogeneous)
  {
    // do nothing, value_m is already initialized with zeros
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
  }
 
  return value_m;
}
 
 
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::VectorizedArray<Number>
  calculate_exterior_value(dealii::VectorizedArray<Number> const &         value_m,
                           unsigned int const                              q,
                           FaceIntegrator<dim, 1, Number> const &          integrator,
                           OperatorType const &                            operator_type,
                           BoundaryTypeP const &                           boundary_type,
                           dealii::types::boundary_id const                boundary_id,
                           std::shared_ptr<BoundaryDescriptorP<dim> const> boundary_descriptor,
                           double const &                                  time,
                           double const &                                  inverse_scaling_factor)
{
  dealii::VectorizedArray<Number> value_p = dealii::make_vectorized_array<Number>(0.0);
 
  if(boundary_type == BoundaryTypeP::Dirichlet)
  {
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      auto bc       = boundary_descriptor->dirichlet_bc.find(boundary_id)->second;
      auto q_points = integrator.quadrature_point(q);
 
      dealii::VectorizedArray<Number> g =
        FunctionEvaluator<0, dim, Number>::value(*bc, q_points, time);
 
      value_p = -value_m + 2.0 * inverse_scaling_factor * g;
    }
    else if(operator_type == OperatorType::homogeneous)
    {
      value_p = -value_m;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
    }
  }
  else if(boundary_type == BoundaryTypeP::Neumann)
  {
    value_p = value_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return value_p;
}
 
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::VectorizedArray<Number>
  calculate_exterior_value_from_dof_vector(dealii::VectorizedArray<Number> const & value_m,
                                           unsigned int const                      q,
                                           FaceIntegrator<dim, 1, Number> const &  integrator_bc,
                                           OperatorType const &                    operator_type,
                                           BoundaryTypeP const &                   boundary_type,
                                           double const & inverse_scaling_factor)
{
  dealii::VectorizedArray<Number> value_p = dealii::make_vectorized_array<Number>(0.0);
 
  if(boundary_type == BoundaryTypeP::Dirichlet)
  {
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      dealii::VectorizedArray<Number> g = integrator_bc.get_value(q);
 
      value_p = -value_m + 2.0 * inverse_scaling_factor * g;
    }
    else if(operator_type == OperatorType::homogeneous)
    {
      value_p = -value_m;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
    }
  }
  else if(boundary_type == BoundaryTypeP::Neumann)
  {
    value_p = value_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return value_p;
}
 
// clang-format off
/*
 *  This function calculates the exterior velocity gradient in normal direction depending
 *  on the operator type, the type of the boundary face and the given boundary conditions.
 *
 *  The function argument normal_gradient_m has the meaning of F(u⁻)*n with
 *
 *  Divergence formulation: F(u) = F_nu(u) / nu = ( grad(u) + grad(u)^T )
 *  Laplace formulation:    F(u) = F_nu(u) / nu =  grad(u)
 *
 *                            +---------------------------------+---------------------------------------+----------------------------------------------------+
 *                            | Dirichlet boundaries            | Neumann boundaries                    | symmetry boundaries                                |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | full operator           | F(u⁺)*n = F(u⁻)*n               | F(u⁺)*n = -F(u⁻)*n + 2h               | F(u⁺)*n = -F(u⁻)*n + 2*[(F(u⁻)*n)*n]n              |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | homogeneous operator    | F(u⁺)*n = F(u⁻)*n               | F(u⁺)*n = -F(u⁻)*n                    | F(u⁺)*n = -F(u⁻)*n + 2*[(F(u⁻)*n)*n]n              |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | inhomogeneous operator  | F(u⁺)*n = F(u⁻)*n, F(u⁻)*n = 0  | F(u⁺)*n = -F(u⁻)*n + 2h , F(u⁻)*n = 0 | F(u⁺)*n = -F(u⁻)*n + 2*[(F(u⁻)*n)*n]n, F(u⁻)*n = 0 |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *
 *                            +---------------------------------+---------------------------------------+----------------------------------------------------+
 *                            | Dirichlet boundaries            | Neumann boundaries                    | symmetry boundaries                                |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | full operator           | {{F(u)}}*n = F(u⁻)*n            | {{F(u)}}*n = h                        | {{F(u)}}*n = [(F(u⁻)*n)*n]n                        |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | homogeneous operator    | {{F(u)}}*n = F(u⁻)*n            | {{F(u)}}*n = 0                        | {{F(u)}}*n = [(F(u⁻)*n)*n]n                        |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 *  | inhomogeneous operator  | {{F(u)}}*n = 0                  | {{F(u)}}*n = h                        | {{F(u)}}*n = 0                                     |
 *  +-------------------------+---------------------------------+---------------------------------------+----------------------------------------------------+
 */
// clang-format on
 
template<int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE //
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>>
  calculate_exterior_normal_gradient(
    dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> const & normal_gradient_m,
    unsigned int const                                              q,
    FaceIntegrator<dim, dim, Number> const &                        integrator,
    OperatorType const &                                            operator_type,
    BoundaryTypeU const &                                           boundary_type,
    dealii::types::boundary_id const                                boundary_id,
    std::shared_ptr<BoundaryDescriptorU<dim> const>                 boundary_descriptor,
    double const &                                                  time,
    bool const                                                      variable_normal_vector)
{
  dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> normal_gradient_p;
 
  if(boundary_type == BoundaryTypeU::Dirichlet or boundary_type == BoundaryTypeU::DirichletCached)
  {
    normal_gradient_p = normal_gradient_m;
  }
  else if(boundary_type == BoundaryTypeU::Neumann)
  {
    if(operator_type == OperatorType::full or operator_type == OperatorType::inhomogeneous)
    {
      auto bc       = boundary_descriptor->neumann_bc.find(boundary_id)->second;
      auto q_points = integrator.quadrature_point(q);
 
      dealii::Tensor<1, dim, dealii::VectorizedArray<Number>> h;
      if(variable_normal_vector == false)
      {
        h = FunctionEvaluator<1, dim, Number>::value(*bc, q_points, time);
      }
      else
      {
        auto normals_m = integrator.get_normal_vector(q);
        h              = FunctionEvaluator<1, dim, Number>::value(
          *(std::dynamic_pointer_cast<FunctionWithNormal<dim>>(bc)), q_points, normals_m, time);
      }
 
      normal_gradient_p = -normal_gradient_m + 2.0 * h;
    }
    else if(operator_type == OperatorType::homogeneous)
    {
      normal_gradient_p = -normal_gradient_m;
    }
    else
    {
      AssertThrow(false, dealii::ExcMessage("Specified OperatorType is not implemented!"));
    }
  }
  else if(boundary_type == BoundaryTypeU::Symmetry)
  {
    auto normal_m     = integrator.get_normal_vector(q);
    normal_gradient_p = -normal_gradient_m + 2.0 * (normal_gradient_m * normal_m) * normal_m;
  }
  else
  {
    AssertThrow(false, dealii::ExcMessage("Boundary type of face is invalid or not implemented."));
  }
 
  return normal_gradient_p;
}
 
} // namespace IncNS
} // namespace ExaDG
 
#endif /* INCLUDE_EXADG_INCOMPRESSIBLE_NAVIER_STOKES_SPATIAL_DISCRETIZATION_OPERATORS_WEAK_BOUNDARY_CONDITIONS_H_ \
        */