Reaction diffusion update

unsupported/Bembel/GrayScott

@@ -0,0 +1,28 @@
+// This file is part of Bembel, the higher order C++ boundary element library.
+//
+// Copyright (C) 2022 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_GRAYSCOTT_MODULE_
+#define BEMBEL_GRAYSCOTT_MODULE_
+
+/**
+ *  \ingroup Modules
+ *  \defgroup GrayScott GrayScott
+ *  \brief Everything which is needed to solve the Gray-Scott model using
+ * Bembel.
+ **/
+
+#include <Bembel/Identity>
+#include <Bembel/LaplaceBeltrami>
+#include <Bembel/LinearForm>
+
+#include "src/GrayScott/ReactionTerm.hpp"
+
+#endif  // BEMBEL_GRAYSCOTT_MODULE_

unsupported/Bembel/src/GrayScott/ReactionTerm.hpp

@@ -0,0 +1,72 @@
+// This file is part of Bembel, the higher order C++ boundary element library.
+//
+// Copyright (C) 2022 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_LINEARFORM_REACTIONTERM_HPP_
+#define BEMBEL_SRC_LINEARFORM_REACTIONTERM_HPP_
+
+namespace Bembel {
+/**
+ *  \ingroup LinearForm
+ *  \brief This class takes care of the
+ * assembly of the linear form of the reaction term, i.e., uv^2.
+ */
+template <typename Derived, typename Scalar, typename Functor>
+void reactionLinearForm(const AnsatzSpace<Derived> &ansatz_space,
+                        const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &u,
+                        const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> &v,
+                        const Functor &functor,
+                        Eigen::Matrix<Scalar, Eigen::Dynamic, 1> *reaction_lf,
+                        int deg = 4) {
+  GaussSquare<Constants::maximum_quadrature_degree> GS;
+  auto Q = GS[deg];
+  SurfacePoint qp;
+  const auto longu = (ansatz_space.get_transformation_matrix() * u).eval();
+  const auto longv = (ansatz_space.get_transformation_matrix() * v).eval();
+  const auto &super_space = ansatz_space.get_superspace();
+  const ElementTree &et = super_space.get_mesh().get_element_tree();
+  const unsigned int number_of_elements = et.get_number_of_elements();
+  const unsigned int polynomial_degree = super_space.get_polynomial_degree();
+  const unsigned n_shape_fun =
+      (polynomial_degree + 1) * (polynomial_degree + 1);
+  // compute linear form of reaction term
+  Eigen::Matrix<Scalar, Eigen::Dynamic, 1> reaction_lf_long =
+      Eigen::Matrix<Scalar, Eigen::Dynamic, 1>::Zero(
+          n_shape_fun * number_of_elements, 1);
+  for (auto i = 0; i < Q.w_.size(); ++i) {
+    const Eigen::Matrix<Scalar, Eigen::Dynamic, 1> basis_evaluated_at_pt =
+        super_space.basis(Q.xi_.col(i));
+    for (auto element = et.cpbegin(); element != et.cpend(); ++element) {
+      super_space.map2surface(*element, Q.xi_.col(i), Q.w_(i), &qp);
+      // get points on geometry and tangential derivatives
+      const auto &x_f_dx = qp.segment<3>(6);
+      const auto &x_f_dy = qp.segment<3>(9);
+      const auto &normal = x_f_dx.cross(x_f_dy);
+      // compute surface measures from tangential derivatives
+      Scalar x_kappa = normal.norm();
+      // integrand without basis functions
+      const Scalar h = element->get_h();
+      const Scalar u_val =
+          longu.segment(n_shape_fun * element->id_, n_shape_fun).transpose() *
+          basis_evaluated_at_pt;
+      const Scalar v_val =
+          longv.segment(n_shape_fun * element->id_, n_shape_fun).transpose() *
+          basis_evaluated_at_pt;
+      reaction_lf_long.block(n_shape_fun * element->id_, 0, n_shape_fun, 1) +=
+          x_kappa * Q.w_(i) * functor(u_val / h, v_val / h) * h *
+          basis_evaluated_at_pt;
+    }
+  }
+  reaction_lf->derived() =
+      ansatz_space.get_transformation_matrix().transpose() * reaction_lf_long;
+}
+
+}  // namespace Bembel
+#endif  // BEMBEL_SRC_LINEARFORM_REACTIONTERM_HPP_

unsupported/examples/CMakeLists.txt

@@ -7,6 +7,7 @@ set(OTHER_NOCIFILES
     AugmentedEFIE
     AugmentedEFIEStabilized
     AugmentedEFIEBalun
+    GrayScott
     )
 
 foreach(file IN LISTS OTHER_NOCIFILES)

unsupported/examples/GrayScott.cpp

@@ -0,0 +1,141 @@
+// This file is part of Bembel, the higher order C++ boundary element library.
+//
+// Copyright (C) 2022 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.
+
+#include <Bembel/IO>
+#include <Bembel/Identity>
+#include <Bembel/LaplaceBeltrami>
+#include <Bembel/LinearForm>
+#include <Eigen/Dense>
+#include <iostream>
+#include <unsupported/Bembel/GrayScott>
+
+int main() {
+  using namespace Bembel;
+  using namespace Eigen;
+  IO::Stopwatch sw;
+  std::function<double(const Vector3d&)> init_u = [](const Vector3d& in) {
+    // small perturbation
+    return 0.6581 + 0.01 * (double)rand() / RAND_MAX - 0.005;
+    // return 0.6581 + 0.005;
+    // return 1.0;
+  };
+
+  std::function<double(const Vector3d&)> init_v = [](const Vector3d& in) {
+    // small perturbation
+    return 0.2279 + 0.01 * (double)rand() / RAND_MAX - 0.005;
+    // return 0.2279 - 0.005;
+  };
+
+  std::function<double(const Vector3d&)> init_cf = [](const Vector3d& in) {
+    return 1.0;
+  };
+
+  std::function<double(const double& u_val, const double& v_val)> reaction =
+      [](const double& u_val, const double& v_val) {
+        return u_val * v_val * v_val;
+      };
+
+  Eigen::VectorXd u_init;
+  Eigen::VectorXd v_init;
+
+  Geometry geometry("sphere.dat");
+  std::cout << "\n" << std::string(60, '=') << "\n";
+  // Iterate over polynomial degree.
+  unsigned int polynomial_degree = 2;
+  // Iterate over refinement levels
+  unsigned int refinement_level = 4;
+  std::cout << "Degree " << polynomial_degree << " Level " << refinement_level
+            << "\n";
+  sw.tic();
+  // Build ansatz space
+  AnsatzSpace<LaplaceBeltramiOperator> ansatz_space_lb(
+      geometry, refinement_level, polynomial_degree);
+
+  AnsatzSpace<MassMatrixScalarCont> ansatz_space_mass(
+      geometry, refinement_level, polynomial_degree);
+  // Gray Scott Model
+  // u_t = r_u * \laplacian u -uv^2 + f(1-u)
+  // v_t = r_v * \laplacian v +uv^2 - (f+k)v
+  double delta_t = 0.1;
+  double diffusion_rate_u = 0.01;
+  double diffusion_rate_v =
+      0.001;  // for sake of spots on the surface, please set the
+              // diffusion_rate_v to 0.0005 or less
+  double f = 0.1;
+  double k = 0.05;
+  // Build discrete operator for Laplace Beltrami and Identity
+  DiscreteLocalOperator<LaplaceBeltramiOperator> disc_op_lb(ansatz_space_lb);
+  disc_op_lb.compute();
+  DiscreteLocalOperator<MassMatrixScalarCont> disc_op_mass(ansatz_space_mass);
+  disc_op_mass.compute();
+  // Initial solutions of u and v
+  SparseLU<SparseMatrix<double>, COLAMDOrdering<int>> solver;
+  solver.compute(disc_op_mass.get_discrete_operator());
+  DiscreteLinearForm<DirichletTrace<double>, MassMatrixScalarCont> disc_lf(
+      ansatz_space_mass);
+  disc_lf.get_linear_form().set_function(init_u);
+  disc_lf.compute();
+  u_init = solver.solve(disc_lf.get_discrete_linear_form());
+  disc_lf.get_linear_form().set_function(init_v);
+  disc_lf.compute();
+  v_init = solver.solve(disc_lf.get_discrete_linear_form());
+  disc_lf.get_linear_form().set_function(init_cf);
+  disc_lf.compute();
+  // Left hand side
+  const Eigen::SparseMatrix<double>& stiff_mat =
+      disc_op_lb.get_discrete_operator();
+  const Eigen::SparseMatrix<double>& mass_mat =
+      disc_op_mass.get_discrete_operator();
+
+  Eigen::SimplicialLLT<Eigen::SparseMatrix<double>> solver_u;
+  Eigen::SimplicialLLT<Eigen::SparseMatrix<double>> solver_v;
+  solver_u.compute((1 + delta_t * f) * mass_mat +
+                   delta_t * diffusion_rate_u * stiff_mat);
+  solver_v.compute((1 + (f + k) * delta_t) * mass_mat +
+                   delta_t * diffusion_rate_v * stiff_mat);
+  sw.tic();
+  Eigen::VectorXd u = u_init;
+  Eigen::VectorXd v = v_init;
+  // Solve the solutions of u and v at the end of the time
+  for (int i = 0; i < 20000; ++i) {
+    // Right hand side
+    Eigen::VectorXd reaction_term;
+    reactionLinearForm(ansatz_space_lb, u, v, reaction, &reaction_term);
+    Eigen::Matrix<double, Eigen::Dynamic, 1> rhs_u =
+        (delta_t * f) * disc_lf.get_discrete_linear_form() +
+        disc_op_mass.get_discrete_operator() * u - delta_t * reaction_term;
+
+    Eigen::Matrix<double, Eigen::Dynamic, 1> rhs_v =
+        disc_op_mass.get_discrete_operator() * v + delta_t * reaction_term;
+    u = solver_u.solve(rhs_u);
+    v = solver_v.solve(rhs_v);
+  }
+  auto difft = sw.toc();
+  std::cout << "RD is solved in " << difft << " sec.\n";
+
+  // Visualization
+  FunctionEvaluator<MassMatrixScalarCont> evaluator(ansatz_space_mass);
+  evaluator.set_function(u);
+  VTKSurfaceExport writer(geometry, 6);
+  std::function<double(int, const Eigen::Vector2d&)> fun1 =
+      [&](int patch_number, const Eigen::Vector2d& reference_domain_point) {
+        auto retval =
+            evaluator.evaluateOnPatch(patch_number, reference_domain_point);
+        return double(retval(0, 0));
+      };
+  writer.addDataSet("u", fun1);
+  evaluator.set_function(v);
+  writer.addDataSet("v", fun1);
+  writer.writeToFile("sol.vtp");
+  std::cout << "\n" << std::string(60, '=') << "\n";
+
+  return 0;
+}