main_petsc.cpp 13.5 KB
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/*!
 * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
 *
 * # Stokes incompressible 2D # {#num_sk_inc_2D_ps}
 *
 * In this example we try to solve the 2D stokes equation for incompressible
 * fluid with Reynold number = 0
 *
 * Such equation require the inversion of a system. We will
 * show how to produce such system and solve it using Finite differences with
 * staggered grid. The system of equation to solve is the following
 *
 *
 * \f[ \eta \partial^{2}_{x} v_x + \eta \partial^{2}_{y} v_x - \partial_{x} P  = 0 \f]
 * \f[ \eta \partial^{2}_{x} v_y + \eta \partial^{2}_{y} v_y - \partial_{y} P  = 0 \f]
 * \f[ \partial_{x} v_x + \partial_{y} v_y  = 0 \f]
 *
 * \f$ v_x = 0 \quad v_y = 1 \f$ at x = L
 *
 * \f$ v_x = 0 \quad v_y = 0 \f$ otherwise
 *
 * ## General Model properties ##
 *
 * In order to do this we have to define a structure that contain the main information
 * about the system
 *
 * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_eigen.cpp def system
 *
 */

//! \cond [def system] \endcond

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#include "config.h"

#ifdef HAVE_PETSC

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#include "Grid/grid_dist_id.hpp"
#include "Matrix/SparseMatrix.hpp"
#include "Vector/Vector.hpp"
#include "FiniteDifference/FDScheme.hpp"
#include "FiniteDifference/util/common.hpp"
#include "FiniteDifference/eq.hpp"
#include "Solvers/petsc_solver.hpp"
#include "Solvers/petsc_solver.hpp"
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#include "FiniteDifference/operators.hpp"
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struct lid_nn
{
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  // dimensionaly of the equation (2D problem 3D problem ...)
  static const unsigned int dims = 2;
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  // number of fields in the system v_x, v_y, P so a total of 3
  static const unsigned int nvar = 3;
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  // boundary conditions PERIODIC OR NON_PERIODIC
  static const bool boundary[];
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  // type of space float, double, ...
  typedef float stype;
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  // type of base grid, it is the distributed grid that will store the result
  // Note the first property is a 2D vector (velocity), the second is a scalar (Pressure)
  typedef grid_dist_id<2,float,aggregate<float[2],float>,CartDecomposition<2,float>> b_grid;
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  // type of SparseMatrix, for the linear system, this parameter is bounded by the solver
  // that you are using, in case of umfpack using <double,int> it is the only possible choice
  // Here we choose PETSC implementation
  typedef SparseMatrix<double,int,PETSC_BASE> SparseMatrix_type;
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  // type of Vector for the linear system, this parameter is bounded by the solver
  // that you are using
  typedef Vector<double,PETSC_BASE> Vector_type;
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  // Define that the underline grid where we discretize the system of equation is staggered
  static const int grid_type = STAGGERED_GRID;
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};

const bool lid_nn::boundary[] = {NON_PERIODIC,NON_PERIODIC};

//! \cond [def system] \endcond


/*!
 *
 * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
 *
 * ## System of equation modeling ## {#num_sk_inc_2D_ps_sem}
 *
 * We model the equations. This part will change in the near future to use template expression
 * parsing and simplify the process of defining equations.
 *
 * \f$  \eta v_x \nabla v_x = eta\_lap\_vx \quad \nabla = \partial^{2}_{x} + \partial^{2}_{y} \f$ Step1
 *
 * \f$  \partial_{x} P = p\_x \f$ Step 2
 *
 * \f$  -p\_x = m\_ p\_ x \f$ Step 3
 *
 * \f$ eta\_lap\_vx - m\_p\_x \f$ Step4. This is the first equation in the system
 *
 * The second equation definition is similar to the first one
 *
 * \f$ \partial^{forward}_{x} v_x = dx\_vx \f$ Step5
 *
 * \f$ \partial^{forward}_{y} v_y = dy\_vy \f$ Step6
 *
 * \f$ dx\_vx + dy\_vy \f$ Step 7. This is the third equation in the system
 *
 *
 * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp def equation
 *
 */

//! \cond [def equation] \endcond

// Convenient constants
constexpr unsigned int v[] = {0,1};
constexpr unsigned int P = 2;
constexpr unsigned int ic = 2;
constexpr int x = 0;
constexpr int y = 1;

//! \cond [def equation] \endcond

/*!
 * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
 *
 * In case of boundary conditions and staggered grid we need a particular set of equations
 * at boundaries. Explain in detail is out of the scope of this example, but to qualitatively
 * have an idea consider the following staggered cell
 *
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 \verbatim
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 +--$--+
 |     |
 #  *  #
 |     |
 0--$--+
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 # = velocity(x)
 $ = velocity(y)
 * = pressure
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 \endverbatim
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 *
 * As we can see several properties has different position in the cell.
 * Consider we want to impose \f$v_y = 0\f$ on \f$x=0\f$. Because there are not
 * points at \f$x = 0\f$ we have to interpolate between $ of this cell
 * and $ of the previous cell on y Averaging using the Avg operator. The same apply for
 * \f$v_x\f$ on  \f$y=0\f$. Similar arguments can be done for other boundaries in
 *  order to finally reach a well defined system
 *
 * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp bond def eq
 *
 */

//! \cond [bond def eq] \endcond

// Usefull constants (as MACRO)
#define EQ_1 0
#define EQ_2 1
#define EQ_3 2

//! \cond [bond def eq] \endcond

#include "Vector/vector_dist.hpp"
#include "data_type/aggregate.hpp"

int main(int argc, char* argv[])
{
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  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * ## Initialization ## {#num_sk_inc_2D_ps_init}
   *
   * After model our equation we:
   * * Initialize the library
   * * Define some useful constants
   * * define Ghost size
   * * Non-periodic boundary conditions
   * * Padding domain expansion
   *
   * Padding and Ghost differ in the fact the padding extend the domain.
   * Ghost is an extension for each sub-domain
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp init
   *
   */

  //! \cond [init] \endcond

  // Initialize
  openfpm_init(&argc,&argv);

  // Names for the positions in cell
  std::initializer_list<char> cc = {0,0};
  std::initializer_list<char> ll = {0,-1};
  std::initializer_list<char> bl = {-1,-1};
  std::initializer_list<char> bb = {-1,0};

  //! \cond [def equation] \endcond
  
  // Create field that we have v_x, v_y, P
  Field<v[x],lid_nn> v_x{ll};                 // Definition 1 v_x
  Field<v[y],lid_nn> v_y{bb};                 // Definition 2 v_y
  Field<P,lid_nn> Prs{cc};		      // Definition 3 Prs
  coeff<double,lid_nn> eta{1.0,bl};           // Coefficient
  
  // Create the derivatives and Laplacians that are needed
  Laplacian<lid_nn,CENTRAL> lap;
  Der<x,lid_nn,CENTRAL> dx;
  Der<y,lid_nn,CENTRAL> dy;
  Der<x,lid_nn,FORWARD> dx_f;
  Der<y,lid_nn,FORWARD> dy_f;

  // Equations
  auto vx_eq = eta*lap(v_x) - dx(Prs);   // Eq1 V_x
  auto vy_eq = eta*lap(v_y) - dy(Prs);   // Eq2 V_y
  auto ic_eq = dx_f(v_x) + dy_f(v_y);    // Eq3 Incom

  Avg<x,decltype(v_y)> avg_vy{v_y};
  Avg<y,decltype(v_x)> avg_vx{v_x};
  
  Avg<x,decltype(v_y),FORWARD> avg_vy_f{v_y};
  Avg<y,decltype(v_x),FORWARD> avg_vx_f{v_x};
  
  //! \cond [def equation] \endcond

  // velocity in the grid is the property 0, pressure is the property 1
  constexpr int velocity = 0;
  constexpr int pressure = 1;

  // Domain, a rectangle
  Box<2,float> domain({0.0,0.0},{12.0,4.0});

  // Ghost (Not important in this case but required)
  Ghost<2,float> g(0.01);

  // Grid points on x=96 and y=32
  long int sz[] = {96,32};
  size_t szu[2];
  szu[0] = (size_t)sz[0];
  szu[1] = (size_t)sz[1];

  // We need one more point on the left and down part of the domain
  // This is given by the boundary conditions that we impose.
  //
  Padding<2> pd({1,1},{0,0});

  //! \cond [init] \endcond

  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * Distributed grid that store the solution
   *
   * \see \ref e0_s_grid_inst
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp grid inst
   *
   */

  //! \cond [grid inst] \endcond

  grid_dist_id<2,float,aggregate<float[2],float>> g_dist(szu,domain,g);

  //! \cond [grid inst] \endcond

  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * Solving the system above require the solution of a system like that
   *
   * \f$ Ax = b \quad x = A^{-1}b\f$
   *
   * where A is the system the discretize the left hand side of the equations + boundary conditions
   * and b discretize the right hand size + boundary conditions
   *
   * FDScheme is the object that we use to produce the Matrix A and the vector b.
   * Such object require the maximum extension of the stencil
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp fd scheme
   *
   */

  //! \cond [fd scheme] \endcond

  // It is the maximum extension of the stencil (order 2 laplacian stencil has extension 1)
  Ghost<2,long int> stencil_max(1);

  // Finite difference scheme
  FDScheme<lid_nn> fd(pd, stencil_max, domain, g_dist);

  //! \cond [fd scheme] \endcond

  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * ## Impose the equation on the domain ## {#num_sk_inc_2D_ps_ied}
   *
   * Here we impose the system of equation, we start from the incompressibility Eq imposed in the bulk with the
   * exception of the first point {0,0} and than we set P = 0 in {0,0}, why we are doing this is again
   * mathematical to have a well defined system, an intuitive explanation is that P and P + c are both
   * solution for the incompressibility equation, this produce an ill-posed problem to make it well posed
   * we set one point in this case {0,0} the pressure to a fixed constant for convenience P = 0
   *
   * The best way to understand what we are doing is to draw a smaller example like 8x8.
   * Considering that we have one additional point on the left for padding we have a grid
   * 9x9. If on each point we have v_x v_y and P unknown we have
   * 9x9x3 = 243 unknown. In order to fully determine and unique solution we have to
   * impose 243 condition. The code under impose (in the case of 9x9) between domain
   * and bulk 243 conditions.
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp impose eq dom
   *
   *
   */

  //! \cond [impose eq dom] \endcond

  fd.impose<EQ_3>(ic_eq,0.0,{0,0},{sz[0]-2,sz[1]-2},cc,true);
  fd.impose<EQ_3>(Prs,0.0,{0,0},{0,0},cc);

  // Here we impose the Eq1 and Eq2
  fd.impose<EQ_1>(vx_eq,0.0,{1,0},{sz[0]-2,sz[1]-2},ll);
  fd.impose<EQ_2>(vy_eq,0.0,{0,1},{sz[0]-2,sz[1]-2},bb);

  // v_x and v_y
  // Imposing B1
  fd.impose<EQ_1>(v_x,0.0,{0,0},{0,sz[1]-2},ll);
  fd.impose<EQ_2>(avg_vy_f,0.0,{-1,0},{-1,sz[1]-1},bb);
  // Imposing B2
  fd.impose<EQ_1>(v_x,0.0,{sz[0]-1,0},{sz[0]-1,sz[1]-2},ll);
  fd.impose<EQ_2>(avg_vy,1.0,{sz[0]-1,0},{sz[0]-1,sz[1]-1},bb);

  // Imposing B3
  fd.impose<EQ_1>(avg_vx_f,0.0,{0,-1},{sz[0]-1,-1},ll);
  fd.impose<EQ_2>(v_y,0.0,{0,0},{sz[0]-2,0},bb);
  // Imposing B4
  fd.impose<EQ_1>(avg_vx,0.0,{0,sz[1]-1},{sz[0]-1,sz[1]-1},ll);
  fd.impose<EQ_2>(v_y,0.0,{0,sz[1]-1},{sz[0]-2,sz[1]-1},bb);

  // When we pad the grid, there are points of the grid that are not
  // touched by the previous condition. Mathematically this lead
  // to have too many variables for the conditions that we are imposing.
  // Here we are imposing variables that we do not touch to zero
  //

  // Padding pressure
  fd.impose<EQ_3>(Prs,0.0,{-1,-1},{sz[0]-1,-1},cc);
  fd.impose<EQ_3>(Prs,0.0,{-1,sz[1]-1},{sz[0]-1,sz[1]-1},cc);
  fd.impose<EQ_3>(Prs,0.0,{-1,0},{-1,sz[1]-2},cc);
  fd.impose<EQ_3>(Prs,0.0,{sz[0]-1,0},{sz[0]-1,sz[1]-2},cc);

  // Impose v_x Padding Impose v_y padding
  fd.impose<EQ_1>(v_x,0.0,{-1,-1},{-1,sz[1]-1},cc);
  fd.impose<EQ_2>(v_y,0.0,{-1,-1},{sz[0]-1,-1},cc);

  //! \cond [impose eq dom] \endcond

  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * ## Solve the system of equation ## {#num_sk_inc_2D_sse}
   *
   * Once we imposed all the equations we can retrieve the Matrix A and the vector b
   * and pass these two element to the solver. In this example we are using  PETSC solvers
   *  direct/Iterative solvers. While Umfpack
   * has only one solver, PETSC wrap several solvers. The function best_solve set the solver in
   * the modality to try multiple solvers to solve your system. The subsequent call to solve produce a report
   * of all the solvers tried comparing them in error-convergence and speed. If you do not use
   * best_solve try to solve your system with the default solver GMRES (That is the most robust iterative solver
   *  method)
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp solver
   *
   */

  //! \cond [solver] \endcond

  // Create a PETSC solver
  petsc_solver<double> solver;

  // Set the maxumum nunber if iterations
  solver.setMaxIter(1000);

  solver.setRestart(200);

  // Give to the solver A and b, return x, the solution
  auto x = solver.try_solve(fd.getA(),fd.getB());

  //! \cond [solver] \endcond

  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * ## Copy the solution on the grid and write on VTK ## {#num_sk_inc_2D_ps_csg}
   *
   * Once we have the solution we copy it on the grid
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp copy write
   *
   */

  //! \cond [copy write] \endcond

  fd.template copy<velocity,pressure>(x,{0,0},{sz[0]-1,sz[1]-1},g_dist);

  g_dist.write("lid_driven_cavity_p_petsc");

  //! \cond [cony write] \endcond


  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * ## Finalize ## {#num_sk_inc_2D_ps_fin}
   *
   *  At the very end of the program we have always to de-initialize the library
   *
   * \snippet Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp fin lib
   *
   */

  //! \cond [fin lib] \endcond

  openfpm_finalize();

  //! \cond [fin lib] \endcond


  /*!
   * \page Stokes_0_2D_petsc Stokes incompressible 2D petsc
   *
   * # Full code # {#num_sk_inc_2D_ps_code}
   *
   * \include Numerics/Stoke_flow/0_2D_incompressible/main_petsc.cpp
   *
   */
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}

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#else

int main(int argc, char* argv[])
{
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  return 0;
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}

#endif