1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473
|
/*
* $Id: trans.c,v 1.1 2005-09-18 22:04:57 dhmunro Exp $
* Routines for transporting a ray through a cylindrical mesh.
*/
/* Copyright (c) 2005, The Regents of the University of California.
* All rights reserved.
* This file is part of yorick (http://yorick.sourceforge.net).
* Read the accompanying LICENSE file for details.
*/
#include "trans.h"
/* includes track.h and bound.h as well */
extern double exp(double);
extern double sqrt(double);
extern double fabs(double);
#define SQ(x) ((x)*(x))
#define FUZZ 1.0e-99
#ifdef NLTSS
/* Let Hybrid-C compiler find vector versions of exp, sqrt, etc. */
#include <lcc.h>
#endif
/* Also a CYCRED switch below, originally intended for Crays, but
optimized incorrectly there when last checked. */
/* 1-exp(-x) != x for small x. For x<ONE_MINUS_EXP, just use x. */
#define ONE_MINUS_EXP 1.0e-4
/* ---------------------------------------------------------------------- */
/* Solve the transport equation assuming that both the opacity and
* source function are constant within each zone. Return the overall
* absorption and self-emission along the ray.
*/
void
FlatSource(double *opac, double *source, /* opacity, emissivity */
long kxlm, long ngroup, /* opac, source dimens */
RayPath *path, /* ray path through mesh */
double *absorb, double *selfem, /* attenuation factor and
* self-emission for
* entire path */
double *work) /* scratch space */
{
/* Formulas:
* The transport equation is d(inu)/ds= (knu)(jnu - inu), or
* d(inu)/d(tau)= jnu - inu.
* In a region where knu and jnu are constant, the solution is
* inu= exp(-tau)*inu0 + (1-exp(-tau))*jnu.
*
* Thus, integrating inu along the path requires solving the
* first order recurrence:
* x[i+1]= a[i]*x[i] + b[i]
* where x is inu, a is exp(-tau), and b is (1-exp(-tau))*jnu.
*
* The solution to this recurrence is:
* inu= (absorb)*inu0 + (selfem)
* where absorb is the product of all a[i], and selfem is x[last]
* with x[0]=0. This total attenuation factor and self-emission
* specific intensity (in same units as input source function) are
* the outputs of FlatSource.
*/
long nzones= path->ncuts-1;
long *zone= path->zone;
double *ds= path->ds;
double *tau= work;
double *atten= tau+nzones;
double *jnu= atten+nzones;
long n,i;
register long j;
if (nzones<1) {
if (absorb && selfem) {
for (n=0 ; n<ngroup ; n++) {
*absorb++= 1.0;
*selfem++= 0.0;
}
}
return;
}
/* Eventually, may want to branch to a separate algorithm if the
* number of groups is very large, but the number of edge crossings
* is small...
* For now, the slightly less efficient inner loop on crossings
* is a better choice, since the number of groups is often quite
* small, while the number of crossings is rarely small.
*/
for (n=0 ; n<ngroup ; n++) {
/* Gather the opacity and source function along the ray path.
* Assume that opac[0]==0 and source[0]==0, so no special coding
* required for non-contiguous sections of the path.
*/
/* #pragma nohazard */
for (i=0 ; i<nzones ; i++) {
j= zone[i];
tau[i]= opac[j]*ds[i];
atten[i]= exp(-tau[i]);
jnu[i]= source[j];
}
/* Replace jnu by appropriate "b" function.
* Note that (1-exp(-tau))*jnu --> tau*jnu as tau-->0 may be
* important even for very small tau... */
/* #pragma nohazard */
for (i=0 ; i<nzones ; i++) {
if (fabs(tau[i]) > ONE_MINUS_EXP) jnu[i]*= (1.0-atten[i]);
else jnu[i]*= tau[i];
}
/* Solve the first order recurrence. */
Reduce(atten,jnu,nzones); /* clobbers atten, jnu arrays... */
*absorb= atten[0];
*selfem= jnu[0];
absorb++;
selfem++;
opac+= kxlm;
source+= kxlm;
}
}
/* ---------------------------------------------------------------------- */
/* Solve the transport equation assuming that the source function
* varies linearly with optical depth across a zone. The opacity is
* assumed to be constant within each zone. (Hence, opac must be zone
* centered, while source is point centered.) Return the overall
* absorption and self-emission along the ray.
*/
void
LinearSource(double *opac, double *source, /* opacity, emissivity */
long kxlm, long ngroup, /* opac, source dimens */
RayPath *path, /* ray path through mesh */
double *absorb, double *selfem, /* attenuation factor and
* self-emission for
* entire path */
double *work) /* scratch space */
{
/* Formulas:
* The transport equation is d(inu)/ds= (knu)(jnu - inu), or
* d(inu)/d(tau)= jnu - inu.
* In a region where knu is constant and jnu linear, the solution is
* inu1= exp(-tau)*inu0 + (1-exp(-tau))*jnu0 +
* (1-(1-exp(-tau))/tau)*(jnu1-jnu0).
*
* Thus, integrating inu along the path requires solving the
* first order recurrence:
* x[i+1]= a[i]*x[i] + b[i]
* where x is inu, a is exp(-tau), and b is (1-exp(-tau))*jnu0+
* (1-(1-exp(-tau))/tau)*(jnu1-jnu0).
*
* The solution to this recurrence is:
* inu= (absorb)*inu0 + (selfem)
* where absorb is the product of all a[i], and selfem is x[last]
* with x[0]=0. This total attenuation factor and self-emission
* specific intensity (in same units as input source function) are
* the outputs of LinearSource.
*/
long nzones= path->ncuts-1;
long *zone= path->zone;
double *ds= path->ds;
long *pt1= path->pt1;
long *pt2= path->pt2;
double *f= path->f;
double fi= path->fi;
double ff= path->ff;
double *tau= work;
double *atten= tau+nzones;
double *jnu= atten+nzones; /* jnu has nzones+1 elements! */
long n,i;
register long j,k;
register double f1,f2;
if (nzones<1) {
if (absorb && selfem) {
for (n=0 ; n<ngroup ; n++) {
*absorb++= 1.0;
*selfem++= 0.0;
}
}
return;
}
/* Eventually, may want to branch to a separate algorithm if the
* number of groups is very large, but the number of edge crossings
* is small...
* For now, the slightly less efficient inner loop on crossings
* is a better choice, since the number of groups is often quite
* small, while the number of crossings is rarely small.
*/
for (n=0 ; n<ngroup ; n++) {
/* Gather the opacity and source function along the ray path.
* Assume that opac[0]==0 and source[0]==0, so no special coding
* required for non-contiguous sections of the path.
*/
/* #pragma nohazard */
for (i=0 ; i<nzones ; i++) {
j= zone[i];
tau[i]= opac[j]*ds[i];
atten[i]= exp(-tau[i]);
}
/* Interpolate source to crossing point on each edge */
/* #pragma nohazard */
for (i=0 ; i<=nzones ; i++) {
j= pt1[i]; k= pt2[i];
jnu[i]= (0.5-f[i])*source[j] + (0.5+f[i])*source[k];
}
/* Interpolate source into interior of first and last zone */
f1= (1.0-fi)*jnu[0]+fi*jnu[1]; /* nzones=1 is possible... */
jnu[nzones]= ff*jnu[nzones-1]+(1.0-ff)*jnu[nzones];
jnu[0]= f1;
/* Replace jnu by appropriate "b" function.
* Note that (1-exp(-tau)-tau)*jnu --> (tau^2/2)*jnu as tau-->0
* may be important even for very small tau...
* Vectorizing algorithm requires small positive random number
* in order to make zero divide very unlikely. Magnitude is
* chosen in order to have sub-roundoff effect on
* tau > ONE_MINUS_EXP. */
#define RANDOM_SMALL_TAU (1.5261614e-20*ONE_MINUS_EXP)
/* Note: this vectorizes on Cray as of 3/jul/90 */
/* #pragma nohazard */
for (i=0 ; i<nzones ; i++) {
f1= -atten[i]+(1-atten[i])/(tau[i]+RANDOM_SMALL_TAU);
f2= 1.0 -(1-atten[i])/(tau[i]+RANDOM_SMALL_TAU);
if (fabs(tau[i]) > ONE_MINUS_EXP) {
jnu[i]= f1*jnu[i]+f2*jnu[i+1];
} else {
jnu[i]= 0.5*tau[i]*(jnu[i]+jnu[i+1]);
}
}
/* Solve the first order recurrence. */
Reduce(atten,jnu,nzones); /* clobbers atten, jnu arrays... */
*absorb= atten[0];
*selfem= jnu[0];
absorb++;
selfem++;
opac+= kxlm;
source+= kxlm;
}
}
/* ---------------------------------------------------------------------- */
/* Reduce x[i+1]= a[i]*x[i]+b[i] for i=0,...,n-1 to
* x[n]= a[0]*x[0]+b[0] -i.e.- return a[0], b[0]
* May use an algorithm which clobbers a[1:n-1], b[1:n-1] as well.
*/
void Reduce(double *a, double *b, long n)
{
long i;
#ifndef CYCRED
/* on a non-vector machine, use the straightforward algorithm */
register double tmpa,tmpb;
tmpa= a[0]; tmpb= b[0]; /* miscoded tmpa*b[0] originally?? */
for (i=1 ; i<n ; i++) {
tmpa*= a[i];
tmpb= tmpb*a[i]+b[i];
}
a[0]= tmpa; b[0]= tmpb;
#else
/* on vector machine, a cyclic reduction algorithm is faster */
while (n > 2) { /* log2(n) reduction passes */
/* #pragma nohazard */
for (i=0 ; i<n/2 ; i++) {
a[i]= a[2*i]*a[2*i+1];
b[i]= b[2*i]*a[2*i+1]+b[2*i+1];
}
if (n & 01) { /* keep final odd point if present */
a[i]= a[2*i];
b[i]= b[2*i];
n= i+1;
} else n= i;
}
if (n == 2) {
a[0]= a[0]*a[1];
b[0]= b[0]*a[1]+b[1];
}
#endif
}
/* ---------------------------------------------------------------------- */
/* The LinearSource transport solver requires a point centered source
* function; the given source function is zone centered.
* This point centering scheme is local to the four zones surrounding
* the point, with weights which are large for zones whose optical
* depth (along either k or l) is near 1.
* The algorithm requires 4*(kmax*(lmax+1)+1) doubles as working space.
*
* An unrelated Milne condition is applied for points which lie on
* the vacuum boundary of the problem.
*/
void PtCenterSource(double *opac, double *source, long kxlm, long ngroup,
Mesh *mesh, Boundary *boundary, double *work)
/* Historical note: This algorithm should be algebraically equivalent
* to the algorithm used in DIRT or TDG, excepting a case in the
* boundary correction involving a boundary between non-void zones,
* which can, in fact, never occur.
*/
{
long klmax= mesh->klmax;
long kmax= mesh->kmax;
double *z= mesh->z;
double *r= mesh->r;
int *ireg= mesh->ireg;
/* partition the working space */
double *wp= work; /* wp only needs kmax*lmax... */
double *wk= wp+klmax+kmax+1;
double *wl= wk+klmax+kmax+1;
double *b= wl+klmax+kmax+1;
long i,n;
register double area,dddk,dddl;
register double w00,w01,w10,w11;
/* applying of Milne condition at boundaries logically complex... */
long npoints= boundary->npoints;
long *zone= boundary->zone;
int *side= boundary->side;
/* zone+pt1[side] --> zone+pt2[side] is boundary edge */
long pt1index[4]= { -1L, -1L /* -kmax */, 0L /* -kmax */, 0L };
long pt2index[4]= { 0L, -1L, -1L /* -kmax */, 0L /* -kmax */ };
/* zone+zon[side] is zone interior to boundary by one more than zone */
long zonindex[4]= { 0L /* -kmax */, +1L, 0L /* +kmax */, -1L };
double b1,w1,milne1,b2,w2,milne2,dx,dtau,bx,dtaux,b1b,w1b,milne1b;
long j1,j2,jz,jx,j1b;
int beginSection;
/* without this, not obvious that these are initialized */
b1= w1= milne1= b2= w2= milne2= b1b= w1b= milne1b= 0.0;
j2= j1b= 0;
pt1index[1]-= kmax;
pt1index[2]-= kmax;
pt2index[2]-= kmax;
pt2index[3]-= kmax;
zonindex[0]-= kmax;
zonindex[2]+= kmax;
/* zero workspace in guard zones */
for (i=0 ; i<=kmax ; i++) wk[i]= wl[i]= 0.0;
for (i=klmax ; i<klmax+kmax+1 ; i++) wk[i]= wl[i]= b[i]= 0.0;
for (n=0 ; n<ngroup ; n++) {
/* Compute weighting factors -- zone centered, but separate
* values wk, wl for the delta-k and delta-l directions. */
/* #pragma nohazard */
for (i=kmax+1 ; i<klmax ; i++) {
/* area is actually twice zonal area */
area= (z[i]-z[i-kmax-1])*(r[i-1]-r[i-kmax]) -
(z[i-1]-z[i-kmax])*(r[i]-r[i-kmax-1]);
/* dddk, dddl actually twice median lengths */
dddk= sqrt( SQ(z[i]-z[i-1] + z[i-kmax]-z[i-kmax-1]) +
SQ(r[i]-r[i-1] + r[i-kmax]-r[i-kmax-1]));
dddl= sqrt( SQ(z[i]-z[i-kmax] + z[i-1]-z[i-kmax-1]) +
SQ(r[i]-r[i-kmax] + r[i-1]-r[i-kmax-1]));
if (ireg[i]) {
wp[i]= 1.0/(opac[i]*area+FUZZ);
wk[i]= wp[i]*SQ(1.0-exp(-0.5*opac[i]*dddk));
wl[i]= wp[i]*SQ(1.0-exp(-0.5*opac[i]*dddl));
} else {
wp[i]= wk[i]= wl[i]= 0.0;
}
}
/* Copy source to temporary to add guard zones */
/* #pragma nohazard */
for (i=0 ; i<klmax ; i++) b[i]= source[i];
/* Point centered source is weighted average over
* the four surrounding edges, each of which is centered
* between its two bounding zones. */
/* #pragma nohazard */
for (i=0 ; i<klmax ; i++) {
w00= wk[i]+ wl[i];
w10= wk[i+1]+ wl[i+1];
w01= wk[i+kmax]+ wl[i+kmax];
w11= wk[i+kmax+1]+wl[i+kmax+1];
b[i]= (b[i]*w00+b[i+1]*w10+b[i+kmax]*w01+b[i+kmax+1]*w11) /
(w00+w01+w10+w11+FUZZ);
}
/* Instead of interior weighting, use Milne condition at
* (vacuum) boundaries.
* Note that r=0 or z=0 symmetry boundaries are not in the list,
* because FindBoundaryPoints will not find them. TrimBoundary
* must be used to eliminate khold, lhold, and norad boundaries. */
/* Note that this loop is usually much shorter than the others. */
beginSection= 1;
for (i=0 ; i<npoints ; i++) {
/* Get point indices j1, j2, zone index jz */
jz= zone[i];
if (jz) {
if (beginSection) {
j1= jz + pt1index[side[i]];
} else {
j1= j2;
b1= b2; w1= w2; milne1= milne2;
}
j2= jz + pt2index[side[i]];
b2= source[jz];
if (side[i]&01) w2= wk[jz];
else w2= wl[jz];
/* dtau is half the optical depth of the zone perpendicular
* to the boundary edge -- i.e.- the optical depth from
* zone center to boundary. The Milne factor reduces the
* source so a linear extrapolation reaches 0.0 at
* optical depth 2/3 beyond the zone boundary. If dtau>1
* and the source function is decreasing toward the boundary,
* then extrapolate the source function to optical depth 1
* before applying the Milne reduction. */
dx= sqrt(SQ(z[j2]-z[j1])+SQ(r[j2]-r[j1]));
dtau= 0.25/(dx*wp[jz]+FUZZ);
if (dtau>1.0 && ireg[jx= jz+zonindex[side[i]]]) {
bx= source[jx];
dtaux= 0.25/(dx*wp[jx]+FUZZ);
if (bx > b2) { /* flux (gradient) out of boundary */
if (b2*(dtau+dtaux) > (bx-b2)*(dtau+(2./3.)))
/* effectively extrapolate to optical depth 1 first */
dtau= 1.0+(5./3.)*(bx-b2)*(dtau-1.0)/
(b2*(dtau+dtaux)-(bx-b2)*(dtau-1.0));
} else dtau= 1.0; /* Huh? At least this is continuous... */
}
milne2= 1.0+1.5*dtau;
/* Source at boundary point is weighted mean between the
* two adjacent boundary edges. */
if (beginSection) {
beginSection= 0;
j1b= j1;
b1b= b2; w1b= w2; milne1b= milne2;
b[j1]= b2 / (milne2+FUZZ); /* if closed boundary, changed later */
} else {
b[j1]= (b1*w1+b2*w2) / (w1*milne1+w2*milne2+FUZZ);
}
} else if (!beginSection) { /* final point of boundary section */
beginSection= 1;
if (j2 == j1b) { /* handle closed boundary */
b[j2]= (b1b*w1b+b2*w2) / (w1b*milne1b+w2*milne2+FUZZ);
} else { /* open boundary */
b[j2]= b2 / (milne2+FUZZ);
}
}
}
/* Copy point centered source over zone centered input */
/* #pragma nohazard */
for (i=0 ; i<klmax ; i++) source[i]= b[i];
source+= kxlm;
opac+= kxlm;
}
}
/* ---------------------------------------------------------------------- */
|