HypersingularOperator.hpp

// This file is part of Bembel, the higher order C++ boundary element library.
//
// Copyright (C) 2024 see <http://www.bembel.eu>
//
// It was written as part of a cooperation of J. Doelz, H. Harbrecht, S. Kurz,
// M. Multerer, S. Schoeps, and F. Wolf at Technische Universitaet Darmstadt,
// Universitaet Basel, and Universita della Svizzera italiana, Lugano. This
// source code is subject to the GNU General Public License version 3 and
// provided WITHOUT ANY WARRANTY, see <http://www.bembel.eu> for further
// information.
#ifndef BEMBEL_SRC_LAPLACE_HYPERSINGULAROPERATOR_HPP_
#define BEMBEL_SRC_LAPLACE_HYPERSINGULAROPERATOR_HPP_
 
namespace Bembel {
// forward declaration of class LaplaceHypersingularOperator in order to define
// traits
class LaplaceHypersingularOperator;
 
template <>
struct LinearOperatorTraits<LaplaceHypersingularOperator> {
  typedef Eigen::VectorXd EigenType;
  typedef Eigen::VectorXd::Scalar Scalar;
  enum {
    OperatorOrder = 1,
    Form = DifferentialForm::Continuous,
    NumberOfFMMComponents = 2
  };
};
 
class LaplaceHypersingularOperator
    : public LinearOperatorBase<LaplaceHypersingularOperator> {
  // implementation of the kernel evaluation, which may be based on the
  // information available from the superSpace
 public:
  LaplaceHypersingularOperator() {}
  template <class T>
  void evaluateIntegrand_impl(
      const T &super_space, const SurfacePoint &p1, const SurfacePoint &p2,
      Eigen::Matrix<
          typename LinearOperatorTraits<LaplaceHypersingularOperator>::Scalar,
          Eigen::Dynamic, Eigen::Dynamic> *intval) const {
    auto polynomial_degree = super_space.get_polynomial_degree();
    auto polynomial_degree_plus_one_squared =
        (polynomial_degree + 1) * (polynomial_degree + 1);
 
    // get evaluation points on unit square
    auto s = p1.segment<2>(0);
    auto t = p2.segment<2>(0);
 
    // get quadrature weights
    auto ws = p1(2);
    auto wt = p2(2);
 
    // get points on geometry and tangential derivatives
    auto x_f = p1.segment<3>(3);
    auto x_f_dx = p1.segment<3>(6);
    auto x_f_dy = p1.segment<3>(9);
    auto y_f = p2.segment<3>(3);
    auto y_f_dx = p2.segment<3>(6);
    auto y_f_dy = p2.segment<3>(9);
 
    // compute surface measures from tangential derivatives
    auto x_kappa = x_f_dx.cross(x_f_dy).norm();
    auto y_kappa = y_f_dx.cross(y_f_dy).norm();
 
    // compute h
    auto h = 1. / (1 << super_space.get_refinement_level());  // h = 1 ./ (2^M)
 
    // integrand without basis functions
    auto integrand =
        evaluateKernel(x_f, y_f) * x_kappa * y_kappa * ws * wt / h / h;
 
    // multiply basis functions with integrand and add to intval, this is an
    // efficient implementation of
    super_space.addScaledSurfaceCurlInteraction(intval, integrand, p1, p2);
 
    return;
  }
 
  Eigen::Matrix<double, 2, 2> evaluateFMMInterpolation_impl(
      const SurfacePoint &p1, const SurfacePoint &p2) const {
    // get evaluation points on unit square
    auto s = p1.segment<2>(0);
    auto t = p2.segment<2>(0);
 
    // get points on geometry and tangential derivatives
    auto x_f = p1.segment<3>(3);
    auto x_f_dx = p1.segment<3>(6);
    auto x_f_dy = p1.segment<3>(9);
    auto y_f = p2.segment<3>(3);
    auto y_f_dx = p2.segment<3>(6);
    auto y_f_dy = p2.segment<3>(9);
 
    // evaluate kernel
    auto kernel = evaluateKernel(x_f, y_f);
 
    // interpolation
    Eigen::Matrix<double, 2, 2> intval;
    intval.setZero();
    intval(0, 0) = kernel * x_f_dy.dot(y_f_dy);
    intval(0, 1) = -kernel * x_f_dy.dot(y_f_dx);
    intval(1, 0) = -kernel * x_f_dx.dot(y_f_dy);
    intval(1, 1) = kernel * x_f_dx.dot(y_f_dx);
 
    return intval;
  }
 
  double evaluateKernel(const Eigen::Vector3d &x,
                        const Eigen::Vector3d &y) const {
    return 1. / 4. / BEMBEL_PI / (x - y).norm();
  }
};
 
template <typename InterpolationPoints>
struct H2Multipole::Moment2D<InterpolationPoints,
                             LaplaceHypersingularOperator> {
  static std::vector<Eigen::MatrixXd> compute2DMoment(
      const SuperSpace<LaplaceHypersingularOperator> &super_space,
      const int cluster_level, const int cluster_refinements,
      const int number_of_points) {
    Eigen::MatrixXd moment_dx = moment2DComputer<
        Moment1DDerivative<InterpolationPoints, LaplaceHypersingularOperator>,
        Moment1D<InterpolationPoints, LaplaceHypersingularOperator>>(
        super_space, cluster_level, cluster_refinements, number_of_points);
    Eigen::MatrixXd moment_dy = moment2DComputer<
        Moment1D<InterpolationPoints, LaplaceHypersingularOperator>,
        Moment1DDerivative<InterpolationPoints, LaplaceHypersingularOperator>>(
        super_space, cluster_level, cluster_refinements, number_of_points);
 
    Eigen::MatrixXd moment(moment_dx.rows() + moment_dy.rows(),
                           moment_dx.cols());
    moment << moment_dx, moment_dy;
 
    std::vector<Eigen::MatrixXd> vector_of_moments;
    vector_of_moments.push_back(moment);
 
    return vector_of_moments;
  }
};
 
}  // namespace Bembel
#endif  // BEMBEL_SRC_LAPLACE_HYPERSINGULAROPERATOR_HPP_