# Shearing sheet (Akihiko Fujii) (C)¶

This example is identical to the shearing_sheet example but uses a different algorithm for resolving individual collisions. In some cases, this might give more realistic results. Particle properties resemble those found in Saturn’s rings.

In this collision resolve method, particles are displaced if they overlap. This example also shows how to implement your own collision routine. This is where one could add fragmentation, or merging of particles.

#include <stdio.h>
#include <stdlib.h>
#include <unistd.h>
#include <math.h>
#include "rebound.h"

int collision_resolve_hardsphere_pullaway(struct reb_simulation* r, struct reb_collision c);

double coefficient_of_restitution_bridges(const struct reb_simulation* const r, double v);
void heartbeat(struct reb_simulation* const r);

int main(int argc, char* argv[]) {
struct reb_simulation* r = reb_create_simulation();
// Setup constants
r->opening_angle2    = .5;                    // This determines the precission of the tree code gravity calculation.
r->integrator            = REB_INTEGRATOR_SEI;
r->boundary            = REB_BOUNDARY_SHEAR;
r->gravity            = REB_GRAVITY_TREE;
r->collision            = REB_COLLISION_TREE;
r->collision_resolve = collision_resolve_hardsphere_pullaway;
double OMEGA             = 0.00013143527;    // 1/s
r->ri_sei.OMEGA         = OMEGA;
r->G                 = 6.67428e-11;        // N / (1e-5 kg)^2 m^2
r->softening             = 0.1;            // m
r->dt                 = 1e-3*2.*M_PI/OMEGA;    // s
r->heartbeat            = heartbeat;    // function pointer for heartbeat
// This example uses two root boxes in the x and y direction.
// Although not necessary in this case, it allows for the parallelization using MPI.
// See Rein & Liu for a description of what a root box is in this context.
double surfacedensity         = 400;             // kg/m^2
double particle_density        = 400;            // kg/m^3
double particle_radius_min     = 1;            // m
double particle_radius_max     = 4;            // m
double boxsize             = 100;            // m
if (argc>1){                        // Try to read boxsize from command line
boxsize = atof(argv[1]);
}
reb_configure_box(r, boxsize, 2, 2, 1);
r->nghostx = 2;
r->nghosty = 2;
r->nghostz = 0;

// Initial conditions
printf("Toomre wavelength: %f\n",4.*M_PI*M_PI*surfacedensity/OMEGA/OMEGA*r->G);
// Use Bridges et al coefficient of restitution.
r->coefficient_of_restitution = coefficient_of_restitution_bridges;
// When two particles collide and the relative velocity is zero, the might sink into each other in the next time step.
// By adding a small repulsive velocity to each collision, we prevent this from happening.
r->minimum_collision_velocity = particle_radius_min*OMEGA*0.001;  // small fraction of the shear accross a particle

double total_mass = surfacedensity*r->boxsize.x*r->boxsize.y;
double mass = 0;
while(mass<total_mass){
struct reb_particle pt = {0};
pt.x         = reb_random_uniform(-r->boxsize.x/2.,r->boxsize.x/2.);
pt.y         = reb_random_uniform(-r->boxsize.y/2.,r->boxsize.y/2.);
pt.z         = reb_random_normal(1.);                    // m
pt.vy         = -1.5*pt.x*OMEGA;
pt.m         = particle_mass;     // kg
mass += particle_mass;
}
reb_integrate(r, INFINITY);
}

// This example is using a custom velocity dependend coefficient of restitution
double coefficient_of_restitution_bridges(const struct reb_simulation* const r, double v){
// assumes v in units of [m/s]
double eps = 0.32*pow(fabs(v)*100.,-0.234);
if (eps>1) eps=1;
if (eps<0) eps=0;
return eps;
}

void heartbeat(struct reb_simulation* const r){
if (reb_output_check(r, 1e-3*2.*M_PI/r->ri_sei.OMEGA)){
reb_output_timing(r, 0);
//reb_output_append_velocity_dispersion("veldisp.txt");
}
if (reb_output_check(r, 2.*M_PI/r->ri_sei.OMEGA)){
//reb_output_ascii("position.txt");
}
}

// Function written by Akihiko Fujii
int collision_resolve_hardsphere_pullaway(struct reb_simulation* r, struct reb_collision c){
struct reb_particle* particles = r->particles;
struct reb_particle p1 = particles[c.p1];
struct reb_particle p2 = particles[c.p2];
struct reb_ghostbox gb = c.gb;
double x21  = p1.x + gb.shiftx  - p2.x;
double y21  = p1.y + gb.shifty  - p2.y;
double z21  = p1.z + gb.shiftz  - p2.z;
double _r = sqrt(x21*x21 + y21*y21 + z21*z21);
/* double r21 = sqrt(x21*x21 + y21*y21 + z21*z21); */
double rp   = p1.r+p2.r;
double oldvyouter;

if (x21>0){
oldvyouter = p1.vy;
}else{
oldvyouter = p2.vy;
}

if (rp*rp < x21*x21 + y21*y21 + z21*z21) return 0;

double vx21 = p1.vx + gb.shiftvx - p2.vx;
double vy21 = p1.vy + gb.shiftvy - p2.vy;
double vz21 = p1.vz + gb.shiftvz - p2.vz;

if (vx21*x21 + vy21*y21 + vz21*z21 >0) return 0; // not approaching

// Bring the to balls in the xy plane.
// NOTE: this could probabely be an atan (which is faster than atan2)
double theta = atan2(z21,y21);
double stheta = sin(theta);
double ctheta = cos(theta);
double vy21n = ctheta * vy21 + stheta * vz21;
double y21n = ctheta * y21 + stheta * z21;

// Bring the two balls onto the positive x axis.
double phi = atan2(y21n,x21);
double cphi = cos(phi);
double sphi = sin(phi);
double vx21nn = cphi * vx21  + sphi * vy21n;
double vy21nn = -sphi* vx21  + cphi * vy21n;

// Coefficient of restitution
double eps= r->coefficient_of_restitution(r, vx21nn);
double dvx2 = -(1.0+eps)*vx21nn;
double dvy2 = (_r/rp-1.)*vy21nn;

double minr = (p1.r>p2.r)?p2.r:p1.r;
double maxr = (p1.r<p2.r)?p2.r:p1.r;
double mindv= minr*r->minimum_collision_velocity;
mindv *= 1.-(_r - maxr)/minr;
if (mindv>maxr*r->minimum_collision_velocity)mindv = maxr*r->minimum_collision_velocity;
if (dvx2<mindv) dvx2 = mindv;

double dxx2 = rp-_r;
double dxx2n = cphi * dxx2;
double dxy2n = sphi * dxx2;
double dxy2nn = ctheta * dxy2n;
double dxz2nn = stheta * dxy2n;

double dvx2n = cphi * dvx2 - sphi * dvy2;
double dvy2n = sphi * dvx2 + cphi * dvy2;

double dvy2nn = ctheta * dvy2n;
double dvz2nn = stheta * dvy2n;

// Applying the changes to the particles.
const double p1pf = p1.m/(p1.m+p2.m);
const double p2pf = p2.m/(p1.m+p2.m);
particles[c.p2].vx -=    p1pf*dvx2n;
particles[c.p2].vy -=    p1pf*dvy2nn;
particles[c.p2].vz -=    p1pf*dvz2nn;
particles[c.p2].lastcollision = r->t;
particles[c.p2].x -=    p1pf*dxx2n;
particles[c.p2].y -=    p1pf*dxy2nn;
particles[c.p2].z -=    p1pf*dxz2nn;

particles[c.p1].vx +=    p2pf*dvx2n;
particles[c.p1].vy +=    p2pf*dvy2nn;
particles[c.p1].vz +=    p2pf*dvz2nn;
particles[c.p1].x +=    p2pf*dxx2n;
particles[c.p1].y +=    p2pf*dxy2nn;
particles[c.p1].z +=    p2pf*dxz2nn;

particles[c.p1].lastcollision = r->t;

return 0; // Do not remove any particle
}


This example is located in the directory examples/shearing_sheet_2