main.cpp

/*
 * ### WIKI 1 ###
 *
 * ## Simple example
 *
 * In this example we show 1D PSE derivative function approximation
 *
 * ### WIKI END ###
 *
 */
 
#include "Vector/vector_dist.hpp"
#include "Decomposition/CartDecomposition.hpp"
#include "PSE/Kernels.hpp"
#include "data_type/aggregate.hpp"
#include <cmath>
 
 
/*
 * ### WIKI 2 ###
 *
 * Here we define the function x*e^(-x*x) and its
 * second derivative in analytic form
 *
 * 2x*(2*x-3)*e^(-x^2)
 *
 */
 
double f_xex2(double x)
{
    return x*exp(-x*x);
}
 
double f_xex2(Point<1,double> & x)
{
    return x.get(0)*exp(-x.get(0)*x.get(0));
}
 
double Lapf_xex2(Point<1,double> & x)
{
    return 2.0*x.get(0)*(2.0*x.get(0)*x.get(0) - 3.0)*exp(-x.get(0)*x.get(0));
}
 
/*
 *
 * ### WIKI END ###
 *
 */
 
int main(int argc, char* argv[])
{
    //
    // ### WIKI 3 ###
    //
    // Some useful parameters. Like
    //
    // * Number of particles
    // * Minimum number of padding particles
    // * The computational domain
    // * The spacing
    // * The mollification length
    // * Second order Laplacian kernel in 1D
    //
 
    // Number of particles
    const size_t Npart = 125;
 
    // Number of padding particles (At least)
    const size_t Npad = 40;
 
    // The domain
    Box<1,double> box({0.0},{4.0});
 
    // Calculated spacing
    double spacing = box.getHigh(0) / Npart;
 
    // Epsilon of the particle kernel
    const double eps = 2*spacing;
 
    // Laplacian PSE kernel 1 dimension, on double, second order
    Lap_PSE<1,double,2> lker(eps);
 
    //
    // ### WIKI 2 ###
    //
    // Here we Initialize the library and we define Ghost size
    // and non-periodic boundary conditions
    //
    openfpm_init(&argc,&argv);
    Vcluster<> & v_cl = create_vcluster();
 
    size_t bc[1]={NON_PERIODIC};
    Ghost<1,double> g(12*eps);
 
    //
    // ### WIKI 3 ###
    //
    // Here we are creating a distributed vector defined by the following parameters
    //
    // we create a set of N+1 particles to have a fully covered domain of particles between 0.0 and 4.0
    // Suppose we have a spacing given by 1.0 you need 4 +1 particles to cover your domain
    //
    vector_dist<1,double, aggregate<double> > vd(Npart+1,box,bc,g);
 
    //
    // ### WIKI 4 ###
    //
    // We assign the position to the particles, the scalar property is set to
    // the function x*e^(-x*x) value.
    // Each processor has parts of the particles and fill part of the space, the
    // position is assigned independently from the decomposition.
    //
    // In this case, if there are 1001 particles and 3 processors the in the
    // domain from 0.0 to 4.0
    //
    // * processor 0 place particles from 0.0 to 1.332 (334 particles)
    // * processor 1 place particles from 1.336 to 2.668 (334 particles)
    // * processor 2 place particles from 2.672 to 4.0 (333 particles)
    //
 
    // It return how many particles the processors with id < rank has in total
    size_t base = vd.init_size_accum(Npart+1);
    auto it2 = vd.getIterator();
 
    while (it2.isNext())
    {
        auto key = it2.get();
 
        // set the position of the particles
        vd.getPos(key)[0] = (key.getKey() + base) * spacing;
        //set the property of the particles
        vd.template getProp<0>(key) = f_xex2((key.getKey() + base) * spacing);
 
        ++it2;
    }
 
    //
    // ### WIKI 5 ###
    //
    // Once defined the position, we distribute them across the processors
    // following the decomposition, finally we get the ghost part
    //
    vd.map();
    vd.ghost_get<0>();
 
    //
    // ### WIKI 6 ###
    //
    // near the boundary we have two options, or we use one sided kernels,
    // or we add additional particles, it is required that such particles
    // produces a 2 time differentiable function. In order to obtain such
    // result we extend for x < 0.0 and x > 4.0 with the test function xe^(-x*x).
    //
    // Note that for x < 0.0 such extension is equivalent to mirror the
    // particles changing the sign of the strength
    //
    // \verbatim
    //
    // 0.6  -0.5      0.5  0.6   Strength
    //  +----+----*----*----*-
    //          0.0              Position
    //
    //  with * = real particle
    //       + = mirror particle
    //
    // \endverbatim
    //
    //
    Box<1,double> m_pad({0.0},{0.1});
    Box<1,double> m_pad2({3.9},{4.0});
    double enlarge = 0.1;
 
    // Create a box
    if (Npad * spacing > 0.1)
    {
        m_pad.setHigh(0,Npad * spacing);
        m_pad2.setLow(0,4.0 - Npad*spacing);
        enlarge = Npad * spacing;
    }
 
    auto it = vd.getDomainIterator();
 
    while (it.isNext())
    {
        auto key = it.get();
 
        // set the position of the particles
        if (m_pad.isInsideNB(vd.getPos(key)) == true)
        {
            vd.add();
            vd.getLastPos()[0] = - vd.getPos(key)[0];
            vd.template getLastProp<0>() = - vd.template getProp<0>(key);
        }
 
        // set the position of the particles
        if (m_pad2.isInsideNB(vd.getPos(key)) == true)
        {
            vd.add();
            vd.getLastPos()[0] = 2.0 * box.getHigh(0) - vd.getPos(key)[0];
            vd.template getLastProp<0>() = f_xex2(vd.getLastPos()[0]);
        }
 
        ++it;
    }
 
    //
    // ### WIKI 7 ###
    //
    // We create a CellList with cell spacing 12 sigma
    //
 
    // get and construct the Cell list
    auto cl = vd.getCellList(12*eps, CL_NON_SYMMETRIC, false, enlarge);
 
    // Maximum infinity norm
    double linf = 0.0;
 
    //
    // ### WIKI 8 ###
    //
    // For each particle get the neighborhood of each particle
    //
    // This cycle is literally the formula from PSE operator approximation
    //
    // \$ \frac{1}{\epsilon^{2}} h (u_{q} - u_{p}) \eta_{\epsilon}(x_q - x_p) \$
    //
 
    auto it_p = vd.getDomainIterator();
    while (it_p.isNext())
    {
        // double PSE integration accumulator
        double pse = 0;
 
        // key
        vect_dist_key_dx key = it_p.get();
 
        // Get the position of the particles
        Point<1,double> p = vd.getPos(key);
 
        // We are not interested in calculating out the domain
        // note added padding particle are considered domain particles
        if (p.get(0) < 0.0 || p.get(0) >= 4.0)
        {
            ++it_p;
            continue;
        }
 
        // Get f(x) at the position of the particle
        double prp_x = vd.template getProp<0>(key);
 
        // Get the neighborhood of the particles
    auto NN = cl.getNNIteratorBox(cl.getCell(p));
        while(NN.isNext())
        {
            auto nnp = NN.get();
 
            // Calculate contribution given by the kernel value at position p,
            // given by the Near particle
            if (nnp != key.getKey())
            {
                // W(x-y)
                double ker = lker.value(p,vd.getPos(nnp));
 
                // f(y)
                double prp_y = vd.template getProp<0>(nnp);
 
                // 1.0/(eps)^2 [f(y)-f(x)]*W(x,y)*V_q
                double prp = 1.0/eps/eps * (prp_y - prp_x) * spacing;
                pse += prp * ker;
            }
 
            // Next particle
            ++NN;
        }
 
        // Here we calculate the L_infinity norm or the maximum difference
        // of the analytic solution from the PSE calculated
 
        double sol = Lapf_xex2(p);
 
        if (fabs(pse - sol) > linf)
            linf = fabs(pse - sol);
 
        ++it_p;
    }
 
    //
    // ### WIKI 9 ###
    //
    // Calculate the maximum infinity norm across processors and
    // print it
    //
 
    v_cl.max(linf);
    v_cl.execute();
 
    if (v_cl.getProcessUnitID() == 0)
        std::cout << "Norm infinity: " << linf << "\n";
 
    //
    // ### WIKI 10 ###
    //
    // Deinitialize the library
    //
    openfpm_finalize();
}