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YouTube-simulations/wave_billiard.c

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/*********************************************************************************/
/* */
/* Animation of wave equation in a planar domain */
/* */
/* N. Berglund, december 2012, may 2021 */
/* */
/* UPDATE 24/04: distinction between damping and "elasticity" parameters */
/* UPDATE 27/04: new billiard shapes, bug in color scheme fixed */
/* UPDATE 28/04: code made more efficient, with help of Marco Mancini */
/* */
/* Feel free to reuse, but if doing so it would be nice to drop a */
/* line to nils.berglund@univ-orleans.fr - Thanks! */
/* */
/* compile with */
/* gcc -o wave_billiard wave_billiard.c */
/* -L/usr/X11R6/lib -ltiff -lm -lGL -lGLU -lX11 -lXmu -lglut -O3 -fopenmp */
/* */
/* To make a video, set MOVIE to 1 and create subfolder tif_wave */
/* It may be possible to increase parameter PAUSE */
/* */
/* create movie using */
/* ffmpeg -i wave.%05d.tif -vcodec libx264 wave.mp4 */
/* */
/*********************************************************************************/
/*********************************************************************************/
/* */
/* NB: The algorithm used to simulate the wave equation is highly paralellizable */
/* One could make it much faster by using a GPU */
/* */
/*********************************************************************************/
#include <math.h>
#include <string.h>
#include <GL/glut.h>
#include <GL/glu.h>
#include <unistd.h>
#include <sys/types.h>
#include <tiffio.h> /* Sam Leffler's libtiff library. */
#include <omp.h>
#define MOVIE 0 /* set to 1 to generate movie */
/* General geometrical parameters */
#define WINWIDTH 1280 /* window width */
#define WINHEIGHT 720 /* window height */
#define NX 1280 /* number of grid points on x axis */
#define NY 720 /* number of grid points on y axis */
// #define NX 640 /* number of grid points on x axis */
// #define NY 360 /* number of grid points on y axis */
/* setting NX to WINWIDTH and NY to WINHEIGHT increases resolution */
/* but will multiply run time by 4 */
#define XMIN -2.0
#define XMAX 2.0 /* x interval */
#define YMIN -1.125
#define YMAX 1.125 /* y interval for 9/16 aspect ratio */
// #define XMIN -1.8
// #define XMAX 1.8 /* x interval */
// #define YMIN -1.0125
// #define YMAX 1.0125 /* y interval for 9/16 aspect ratio */
#define JULIA_SCALE 1.0 /* scaling for Julia sets */
/* Choice of the billiard table */
#define B_DOMAIN 3 /* choice of domain shape, see list in global_pdes.c */
#define CIRCLE_PATTERN 4 /* pattern of circles, see list in global_pdes.c */
#define P_PERCOL 0.25 /* probability of having a circle in C_RAND_PERCOL arrangement */
#define NPOISSON 300 /* number of points for Poisson C_RAND_POISSON arrangement */
#define LAMBDA 0.75 /* parameter controlling the dimensions of domain */
#define MU 0.025 /* parameter controlling the dimensions of domain */
#define NPOLY 3 /* number of sides of polygon */
#define APOLY 1.0 /* angle by which to turn polygon, in units of Pi/2 */
#define MDEPTH 4 /* depth of computation of Menger gasket */
#define MRATIO 3 /* ratio defining Menger gasket */
#define MANDELLEVEL 1000 /* iteration level for Mandelbrot set */
#define MANDELLIMIT 10.0 /* limit value for approximation of Mandelbrot set */
#define FOCI 1 /* set to 1 to draw focal points of ellipse */
#define NGRIDX 15 /* number of grid point for grid of disks */
#define NGRIDY 20 /* number of grid point for grid of disks */
/* You can add more billiard tables by adapting the functions */
/* xy_in_billiard and draw_billiard below */
/* Physical parameters of wave equation */
#define TWOSPEEDS 1 /* set to 1 to replace hardcore boundary by medium with different speed */
#define OMEGA 0.9 /* frequency of periodic excitation */
#define COURANT 0.01 /* Courant number */
#define COURANTB 0.003 /* Courant number in medium B */
#define GAMMA 0.0 /* damping factor in wave equation */
#define GAMMA_BC 1.0e-4 /* damping factor on boundary */
// #define GAMMA 5.0e-10 /* damping factor in wave equation */
#define KAPPA 0.0 /* "elasticity" term enforcing oscillations */
#define KAPPA_BC 5.0e-4 /* "elasticity" term on absorbing boundary */
/* The Courant number is given by c*DT/DX, where DT is the time step and DX the lattice spacing */
/* The physical damping coefficient is given by GAMMA/(DT)^2 */
/* Increasing COURANT speeds up the simulation, but decreases accuracy */
/* For similar wave forms, COURANT^2*GAMMA should be kept constant */
/* Boundary conditions, see list in global_pdes.c */
#define B_COND 3
/* Parameters for length and speed of simulation */
#define NSTEPS 7000 /* number of frames of movie */
#define NVID 50 /* number of iterations between images displayed on screen */
#define NSEG 100 /* number of segments of boundary */
#define INITIAL_TIME 200 /* time after which to start saving frames */
#define PAUSE 1000 /* number of frames after which to pause */
#define PSLEEP 1 /* sleep time during pause */
#define SLEEP1 1 /* initial sleeping time */
#define SLEEP2 1 /* final sleeping time */
/* Plot type, see list in global_pdes.c */
#define PLOT 1
/* Color schemes */
#define BLACK 1 /* background */
#define COLOR_SCHEME 1 /* choice of color scheme, see list in global_pdes.c */
#define SCALE 0 /* set to 1 to adjust color scheme to variance of field */
#define SLOPE 1.0 /* sensitivity of color on wave amplitude */
// #define SLOPE 0.5 /* sensitivity of color on wave amplitude */
#define ATTENUATION 0.0 /* exponential attenuation coefficient of contrast with time */
#define E_SCALE 750.0 /* scaling factor for energy representation */
#define COLORHUE 260 /* initial hue of water color for scheme C_LUM */
#define COLORDRIFT 0.0 /* how much the color hue drifts during the whole simulation */
#define LUMMEAN 0.5 /* amplitude of luminosity variation for scheme C_LUM */
#define LUMAMP 0.3 /* amplitude of luminosity variation for scheme C_LUM */
#define HUEMEAN 220.0 /* mean value of hue for color scheme C_HUE */
#define HUEAMP -220.0 /* amplitude of variation of hue for color scheme C_HUE */
// #define HUEMEAN 320.0 /* mean value of hue for color scheme C_HUE */
// #define HUEAMP 100.0 /* amplitude of variation of hue for color scheme C_HUE */
#include "global_pdes.c"
#include "sub_wave.c"
double courant2, courantb2; /* Courant parameters squared */
void init_wave(x, y, phi, psi, xy_in)
/* initialise field with drop at (x,y) - phi is wave height, psi is phi at time t-1 */
double x, y, *phi[NX], *psi[NX]; short int * xy_in[NX];
{
int i, j;
double xy[2], dist2;
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
ij_to_xy(i, j, xy);
dist2 = (xy[0]-x)*(xy[0]-x) + (xy[1]-y)*(xy[1]-y);
xy_in[i][j] = xy_in_billiard(xy[0],xy[1]);
if ((xy_in[i][j])||(TWOSPEEDS)) phi[i][j] = 0.2*exp(-dist2/0.001)*cos(-sqrt(dist2)/0.01);
else phi[i][j] = 0.0;
psi[i][j] = 0.0;
}
}
void init_planar_wave(x, y, phi, psi, xy_in)
/* initialise field with drop at (x,y) - phi is wave height, psi is phi at time t-1 */
/* beta version, works for vertical planar wave only so far */
double x, y, *phi[NX], *psi[NX]; short int * xy_in[NX];
{
int i, j;
double xy[2], dist2;
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
ij_to_xy(i, j, xy);
dist2 = (xy[0]-x)*(xy[0]-x);
xy_in[i][j] = xy_in_billiard(xy[0],xy[1]);
if ((xy_in[i][j])||(TWOSPEEDS)) phi[i][j] = 0.01*exp(-dist2/0.0005)*cos(-sqrt(dist2)/0.01);
else phi[i][j] = 0.0;
psi[i][j] = 0.0;
}
}
void init_wave_flat(phi, psi, xy_in)
/* initialise flat field - phi is wave height, psi is phi at time t-1 */
double *phi[NX], *psi[NX]; short int * xy_in[NX];
{
int i, j;
double xy[2], dist2;
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
ij_to_xy(i, j, xy);
xy_in[i][j] = xy_in_billiard(xy[0],xy[1]);
phi[i][j] = 0.0;
psi[i][j] = 0.0;
}
}
void add_drop_to_wave(factor, x, y, phi, psi)
/* add drop at (x,y) to the field with given prefactor */
double factor, x, y, *phi[NX], *psi[NX];
{
int i, j;
double xy[2], dist2;
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
ij_to_xy(i, j, xy);
dist2 = (xy[0]-x)*(xy[0]-x) + (xy[1]-y)*(xy[1]-y);
phi[i][j] += 0.2*factor*exp(-dist2/0.001)*cos(-sqrt(dist2)/0.01);
}
}
void oscillate_linear_wave(amplitude, t, x1, y1, x2, y2, phi, psi)
/* oscillating boundary condition at (x,y), beta version */
double amplitude, t, x1, y1, x2, y2, *phi[NX], *psi[NX];
{
int i, j, ij1[2], ij2[2], imin, imax, jmin, jmax, d = 5;
double xy[2], dist2;
xy_to_ij(x1, y1, ij1);
xy_to_ij(x2, y2, ij2);
imin = ij1[0] - d; if (imin < 0) imin = 0;
imax = ij2[0] + d; if (imax >= NX) imax = NX-1;
jmin = ij1[1] - d; if (jmin < 0) jmin = 0;
jmax = ij2[1] + d; if (jmax >= NY) jmax = NY-1;
for (i = imin; i < imax; i++)
for (j = jmin; j < jmax; j++)
{
ij_to_xy(i, j, xy);
dist2 = (xy[0]-x1)*(xy[0]-x1); /* to be improved */
// dist2 = (xy[0]-x)*(xy[0]-x) + (xy[1]-y)*(xy[1]-y);
// if (dist2 < 0.01)
if (dist2 < 0.001)
phi[i][j] = amplitude*exp(-dist2/0.001)*cos(-sqrt(dist2)/0.01)*cos(t*OMEGA);
// phi[i][j] += 0.2*exp(-dist2/0.001)*cos(-sqrt(dist2)/0.01)*cos(t*OMEGA);
}
}
/*********************/
/* animation part */
/*********************/
double compute_energy(phi, psi, xy_in, i, j)
double *phi[NX], *psi[NX];
short int *xy_in[NX];
int i, j;
{
double velocity, energy, gradientx2, gradienty2;
int iplus, iminus, jplus, jminus;
velocity = (phi[i][j] - psi[i][j]);
iplus = (i+1); if (iplus == NX) iplus = NX-1;
iminus = (i-1); if (iminus == -1) iminus = 0;
jplus = (j+1); if (jplus == NY) jplus = NY-1;
jminus = (j-1); if (jminus == -1) jminus = 0;
gradientx2 = (phi[iplus][j]-phi[i][j])*(phi[iplus][j]-phi[i][j])
+ (phi[i][j] - phi[iminus][j])*(phi[i][j] - phi[iminus][j]);
gradienty2 = (phi[i][jplus]-phi[i][j])*(phi[i][jplus]-phi[i][j])
+ (phi[i][j] - phi[i][jminus])*(phi[i][j] - phi[i][jminus]);
if (xy_in[i][j]) return(E_SCALE*E_SCALE*(velocity*velocity + 0.5*COURANT*COURANT*(gradientx2+gradienty2)));
else if (TWOSPEEDS) return(E_SCALE*E_SCALE*(velocity*velocity + 0.5*COURANTB*COURANTB*(gradientx2+gradienty2)));
else return(0);
}
void draw_wave(phi, psi, xy_in, scale, time)
/* draw the field */
double *phi[NX], *psi[NX], scale;
short int *xy_in[NX];
int time;
{
int i, j, iplus, iminus, jplus, jminus;
double rgb[3], xy[2], x1, y1, x2, y2, velocity, energy, gradientx2, gradienty2;
static double dtinverse = ((double)NX)/(COURANT*(XMAX-XMIN)), dx = (XMAX-XMIN)/((double)NX);
glBegin(GL_QUADS);
// printf("dtinverse = %.5lg\n", dtinverse);
for (i=0; i<NX; i++)
for (j=0; j<NY; j++)
{
if ((TWOSPEEDS)||(xy_in[i][j]))
{
if (PLOT == P_AMPLITUDE)
color_scheme(COLOR_SCHEME, phi[i][j], scale, time, rgb);
else if (PLOT == P_ENERGY)
color_scheme(COLOR_SCHEME, compute_energy(phi, psi, xy_in, i, j), scale, time, rgb);
else if (PLOT == P_MIXED)
{
if (j > NY/2) color_scheme(COLOR_SCHEME, phi[i][j], scale, time, rgb);
else color_scheme(COLOR_SCHEME, compute_energy(phi, psi, xy_in, i, j), scale, time, rgb);
}
glColor3f(rgb[0], rgb[1], rgb[2]);
glVertex2i(i, j);
glVertex2i(i+1, j);
glVertex2i(i+1, j+1);
glVertex2i(i, j+1);
}
}
glEnd ();
}
void evolve_wave_half(phi_in, psi_in, phi_out, psi_out, xy_in)
/* time step of field evolution */
/* phi is value of field at time t, psi at time t-1 */
double *phi_in[NX], *psi_in[NX], *phi_out[NX], *psi_out[NX]; short int *xy_in[NX];
{
int i, j, iplus, iminus, jplus, jminus;
double delta, x, y, c, cc;
// c = COURANT;
// cc = courant2;
#pragma omp parallel for private(i,j,iplus,iminus,jplus,jminus,delta,x,y)
for (i=0; i<NX; i++){
for (j=0; j<NY; j++){
if (xy_in[i][j])
{
c = COURANT;
cc = courant2;
}
else if (TWOSPEEDS)
{
c = COURANTB;
cc = courantb2;
}
if ((TWOSPEEDS)||(xy_in[i][j])){
/* discretized Laplacian for various boundary conditions */
if ((B_COND == BC_DIRICHLET)||(B_COND == BC_ABSORBING))
{
iplus = (i+1); if (iplus == NX) iplus = NX-1;
iminus = (i-1); if (iminus == -1) iminus = 0;
jplus = (j+1); if (jplus == NY) jplus = NY-1;
jminus = (j-1); if (jminus == -1) jminus = 0;
}
else if (B_COND == BC_PERIODIC)
{
iplus = (i+1) % NX;
iminus = (i-1) % NX;
if (iminus < 0) iminus += NX;
jplus = (j+1) % NY;
jminus = (j-1) % NY;
if (jminus < 0) jminus += NY;
}
else if (B_COND == BC_VPER_HABS)
{
iplus = (i+1); if (iplus == NX) iplus = NX-1;
iminus = (i-1); if (iminus == -1) iminus = 0;
jplus = (j+1) % NY;
jminus = (j-1) % NY;
if (jminus < 0) jminus += NY;
}
delta = phi_in[iplus][j] + phi_in[iminus][j] + phi_in[i][jplus] + phi_in[i][jminus] - 4.0*phi_in[i][j];
x = phi_in[i][j];
y = psi_in[i][j];
/* evolve phi */
if ((B_COND == BC_PERIODIC)||(B_COND == BC_DIRICHLET))
phi_out[i][j] = -y + 2*x + cc*delta - KAPPA*x - GAMMA*(x-y);
else if (B_COND == BC_ABSORBING)
{
if ((i>0)&&(i<NX-1)&&(j>0)&&(j<NY-1))
phi_out[i][j] = -y + 2*x + cc*delta - KAPPA*x - GAMMA_BC*(x-y);
/* upper border */
else if (j==NY-1)
phi_out[i][j] = x - c*(x - phi_in[i][NY-2]) - KAPPA_BC*x - GAMMA_BC*(x-y);
/* lower border */
else if (j==0)
phi_out[i][j] = x - c*(x - phi_in[i][1]) - KAPPA_BC*x - GAMMA_BC*(x-y);
/* right border */
if (i==NX-1)
phi_out[i][j] = x - c*(x - phi_in[NX-2][j]) - KAPPA_BC*x - GAMMA_BC*(x-y);
/* left border */
else if (i==0)
phi_out[i][j] = x - c*(x - phi_in[1][j]) - KAPPA_BC*x - GAMMA_BC*(x-y);
}
else if (B_COND == BC_VPER_HABS)
{
if ((i>0)&&(i<NX-1))
phi_out[i][j] = -y + 2*x + cc*delta - KAPPA*x - GAMMA*(x-y);
/* right border */
else if (i==NX-1)
phi_out[i][j] = x - c*(x - phi_in[NX-2][j]) - KAPPA_BC*x - GAMMA_BC*(x-y);
/* left border */
else if (i==0)
phi_out[i][j] = x - c*(x - phi_in[1][j]) - KAPPA_BC*x - GAMMA_BC*(x-y);
}
psi_out[i][j] = x;
if (FLOOR)
{
if (phi_out[i][j] > VMAX) phi_out[i][j] = VMAX;
if (phi_out[i][j] < -VMAX) phi_out[i][j] = -VMAX;
if (psi_out[i][j] > VMAX) psi_out[i][j] = VMAX;
if (psi_out[i][j] < -VMAX) psi_out[i][j] = -VMAX;
}
}
}
}
// printf("phi(0,0) = %.3lg, psi(0,0) = %.3lg\n", phi[NX/2][NY/2], psi[NX/2][NY/2]);
}
void evolve_wave(phi, psi, phi_tmp, psi_tmp, xy_in)
/* time step of field evolution */
/* phi is value of field at time t, psi at time t-1 */
double *phi[NX], *psi[NX], *phi_tmp[NX], *psi_tmp[NX]; short int *xy_in[NX];
{
evolve_wave_half(phi, psi, phi_tmp, psi_tmp, xy_in);
evolve_wave_half(phi_tmp, psi_tmp, phi, psi, xy_in);
}
double compute_variance(phi, psi, xy_in)
/* compute the variance of the field, to adjust color scheme */
double *phi[NX], *psi[NX]; short int * xy_in[NX];
{
int i, j, n = 0;
double variance = 0.0;
for (i=1; i<NX; i++)
for (j=1; j<NY; j++)
{
if (xy_in[i][j])
{
n++;
variance += phi[i][j]*phi[i][j];
}
}
if (n==0) n=1;
return(variance/(double)n);
}
void animation()
{
double time, scale;
double *phi[NX], *psi[NX], *phi_tmp[NX], *psi_tmp[NX];
short int *xy_in[NX];
int i, j, s;
/* Since NX and NY are big, it seemed wiser to use some memory allocation here */
for (i=0; i<NX; i++)
{
phi[i] = (double *)malloc(NY*sizeof(double));
psi[i] = (double *)malloc(NY*sizeof(double));
phi_tmp[i] = (double *)malloc(NY*sizeof(double));
psi_tmp[i] = (double *)malloc(NY*sizeof(double));
xy_in[i] = (short int *)malloc(NY*sizeof(short int));
}
/* initialise positions and radii of circles */
if (B_DOMAIN == D_CIRCLES) init_circle_config();
courant2 = COURANT*COURANT;
courantb2 = COURANTB*COURANTB;
/* initialize wave with a drop at one point, zero elsewhere */
init_planar_wave(XMIN + 0.01, 0.0, phi, psi, xy_in);
// init_planar_wave(XMIN + 1.0, 0.0, phi, psi, xy_in);
// init_wave(-1.5, 0.0, phi, psi, xy_in);
// init_wave(0.0, 0.0, phi, psi, xy_in);
/* add a drop at another point */
// add_drop_to_wave(1.0, 0.7, 0.0, phi, psi);
// add_drop_to_wave(1.0, -0.7, 0.0, phi, psi);
// add_drop_to_wave(1.0, 0.0, -0.7, phi, psi);
blank();
glColor3f(0.0, 0.0, 0.0);
draw_wave(phi, psi, xy_in, 1.0, 0);
draw_billiard();
glutSwapBuffers();
sleep(SLEEP1);
for (i=0; i<=INITIAL_TIME + NSTEPS; i++)
{
//printf("%d\n",i);
/* compute the variance of the field to adjust color scheme */
/* the color depends on the field divided by sqrt(1 + variance) */
if (SCALE)
{
scale = sqrt(1.0 + compute_variance(phi,psi, xy_in));
// printf("Scaling factor: %5lg\n", scale);
}
else scale = 1.0;
draw_wave(phi, psi, xy_in, scale, i);
for (j=0; j<NVID; j++)
{
evolve_wave(phi, psi, phi_tmp, psi_tmp, xy_in);
// if (i % 10 == 9) oscillate_linear_wave(0.2*scale, 0.15*(double)(i*NVID + j), -1.5, YMIN, -1.5, YMAX, phi, psi);
}
draw_billiard();
glutSwapBuffers();
if (MOVIE)
{
if (i >= INITIAL_TIME) save_frame();
else printf("Initial phase time %i of %i\n", i, INITIAL_TIME);
/* it seems that saving too many files too fast can cause trouble with the file system */
/* so this is to make a pause from time to time - parameter PAUSE may need adjusting */
if (i % PAUSE == PAUSE - 1)
{
printf("Making a short pause\n");
sleep(PSLEEP);
s = system("mv wave*.tif tif_wave/");
}
}
}
if (MOVIE)
{
for (i=0; i<20; i++) save_frame();
s = system("mv wave*.tif tif_wave/");
}
for (i=0; i<NX; i++)
{
free(phi[i]);
free(psi[i]);
free(phi_tmp[i]);
free(psi_tmp[i]);
free(xy_in[i]);
}
}
void display(void)
{
glPushMatrix();
blank();
glutSwapBuffers();
blank();
glutSwapBuffers();
animation();
sleep(SLEEP2);
glPopMatrix();
glutDestroyWindow(glutGetWindow());
}
int main(int argc, char** argv)
{
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_RGB | GLUT_DOUBLE | GLUT_DEPTH);
glutInitWindowSize(WINWIDTH,WINHEIGHT);
glutCreateWindow("Wave equation in a planar domain");
init();
glutDisplayFunc(display);
glutMainLoop();
return 0;
}
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