DoubleLayerOperator.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_DOUBLELAYEROPERATOR_HPP_
#define BEMBEL_SRC_LAPLACE_DOUBLELAYEROPERATOR_HPP_
 
namespace Bembel {
// forward declaration of class LaplaceDoubleLayerOperator in order to define
// traits
class LaplaceDoubleLayerOperator;
 
template <>
struct LinearOperatorTraits<LaplaceDoubleLayerOperator> {
  typedef Eigen::VectorXd EigenType;
  typedef Eigen::VectorXd::Scalar Scalar;
  enum {
    OperatorOrder = 0,
    Form = DifferentialForm::Discontinuous,
    NumberOfFMMComponents = 1
  };
};
 
class LaplaceDoubleLayerOperator
    : public LinearOperatorBase<LaplaceDoubleLayerOperator> {
  // implementation of the kernel evaluation, which may be based on the
  // information available from the superSpace
 public:
  LaplaceDoubleLayerOperator() {}
  template <class T>
  void evaluateIntegrand_impl(
      const T &super_space, const SurfacePoint &p1, const SurfacePoint &p2,
      Eigen::Matrix<
          typename LinearOperatorTraits<LaplaceDoubleLayerOperator>::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();
    auto y_n = y_f_dx.cross(y_f_dy).normalized();
 
    // integrand without basis functions
    auto integrand =
        evaluateKernelGrad(x_f, y_f, y_n) * y_kappa * x_kappa * ws * wt;
 
    // multiply basis functions with integrand and add to intval, this is an
    // efficient implementation of
    // (*intval) += super_space.BasisInteraction(s, t) * evaluateKernel(x_f,
    // y_f)
    // * x_kappa * y_kappa * ws * wt;
    super_space.addScaledBasisInteraction(intval, integrand, s, t);
 
    return;
  }
 
  Eigen::Matrix<double, 1, 1> 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);
 
    // compute surface measure in x and unnormalized normal in y from tangential
    // derivatives
    auto x_kappa = x_f_dx.cross(x_f_dy).norm();
    auto y_n = y_f_dx.cross(y_f_dy);
 
    // interpolation
    Eigen::Matrix<double, 1, 1> intval;
    intval(0) = evaluateKernelGrad(x_f, y_f, y_n) * x_kappa;
 
    return intval;
  }
 
  double evaluateKernelGrad(const Eigen::Vector3d &x, const Eigen::Vector3d &y,
                            const Eigen::Vector3d &y_n) const {
    auto c = x - y;
    auto r = c.norm();
    auto r3 = r * r * r;
    return c.dot(y_n) / 4. / BEMBEL_PI / r3;
  }
};
 
}  // namespace Bembel
#endif  // BEMBEL_SRC_LAPLACE_DOUBLELAYEROPERATOR_HPP_