# hub.darcs.net :: basilisk -> basilisk -> files

Basilisk source code

## root / src / heights.h.page

 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  /** # Height-Functions The "height-function" is a vector field which gives the distance, along each coordinate axis, from the center of the cell to the closest interface defined by a volume fraction field. This distance is estimated using the "column integral" of the volume fraction in the corresponding direction. This integral is not always defined (for example because the interface is too far i.e. farther than 5.5 cells in our implementation) in which case the value of the field is set to *nodata*. See e.g. [Popinet, 2009](references.bib#popinet2009) for more details on height functions. We also store the "orientation" of the height function together with its value by adding *HSHIFT* if the volume fraction is unity on the "top" end. The function below applied to the value will return the corresponding height and orientation. The distance is normalised with the cell size so that the coordinates of the interface are given by ~~~c (x, y + Delta*height(h.y[])) or (x + Delta*height(h.x[]), y) ~~~ */ #define HSHIFT 20. static inline double height (double H) { return H > HSHIFT/2. ? H - HSHIFT : H < -HSHIFT/2. ? H + HSHIFT : H; } static inline int orientation (double H) { return fabs(H) > HSHIFT/2.; } /** We make sure that two layers of ghost cells are defined on the boundaries (the default is one layer). */ #define BGHOSTS 2 /** ## Half-column integration This helper function performs the integration on half a column, either "downward" (*j = -1*) or "upward" (*j = 1*). */ static void half_column (Point point, scalar c, vector h, vector cs, int j) { /** The 'state' of the height function can be: *complete* if both ends were found, zero or one if one end was found and between zero and one if only the interface was found. */ const int complete = -1; foreach_dimension() { /** *S* is the state and *H* the (partial) value of the height function. If we are on the (first) downward integration (*j = -1*) we initialise *S* and *H* with the volume fraction in the current cell. */ double S = c[], H = S, ci, a; /** On the upward integration (*j = 1*), we recover the state of the downward integration. Both the state and the (possibly partial) height value are encoded in a single number using a base 100 shift for the state. */ typedef struct { int s; double h; } HState; HState state = {0, 0}; if (j == 1) { /** Check whether this is an inconsistent height. */ if (h.x[] == 300.) state.s = complete, state.h = nodata; /** Otherwise, this is either a complete or a partial height. */ else { int s = (h.x[] + HSHIFT/2.)/100.; state.h = h.x[] - 100.*s; state.s = s - 1; } /** If this is a complete height, we start a fresh upward integration. */ if (state.s != complete) S = state.s, H = state.h; } /** We consider the four neighboring cells of the half column, the corresponding volume fraction *ci* is recovered either from the standard volume fraction field *c* (first two cells) or from the shifted field *cs* (last two cells). The construction of *cs* is explained in the next section. */ for (int i = 1; i <= 4; i++) { ci = i <= 2 ? c[i*j] : cs.x[(i - 2)*j]; H += ci; /** We then check whether the partial height is complete or not. */ if (S > 0. && S < 1.) { S = ci; if (ci <= 0. || ci >= 1.) { /** We just left an interfacial cell (*S*) and found a full or empty cell (*ci*): this is a partial height and we can stop the integration. If the cell is full (*ci = 1*) we shift the origin of the height. */ H -= i*ci; break; } } /** If *S* is empty or full and *ci* is full or empty, we went right through he interface i.e. the height is complete and we can stop the integration. The origin is shifted appropriately and the orientation is encoded using the "HSHIFT trick". */ else if (S >= 1. && ci <= 0.) { H = (H - 0.5)*j + (j == -1)*HSHIFT; S = complete; break; } else if (S <= 0. && ci >= 1.) { H = (i + 0.5 - H)*j + (j == 1)*HSHIFT; S = complete; break; } /** If *ci* is identical to *S* (which is empty or full), we check that *H* is an integer (i.e. its fractional value is zero), otherwise we are in the case where we found an interface but didn't go through it: this is an inconsistent height and we stop the integration. */ else if (S == ci && modf(H, &a)) break; } /** We update the global state using the state *S* of the half-integration. */ if (j == -1) { /** For the downward integration, we check that the partial heights (*S != complete*) are consistent: if the first cell is full or empty or if the last cell is interfacial, the partial height is marked as inconsistent. */ if (S != complete && ((c[] <= 0. || c[] >= 1.) || (S > 0. && S < 1.))) h.x[] = 300.; // inconsistent else if (S == complete) h.x[] = H; else /** This is a partial height: we encode the state using a base 100 shift. */ h.x[] = H + 100.*(1. + (S >= 1.)); } else { // j = 1 /** For the upward integration, we update the current *state* using the state of the half-integration *S* only if the first downward integration returned a partial height, or if the upward integration returned a complete height with a smaller value than the (complete) height of the downward integration. */ if (state.s != complete || (S == complete && fabs(height(H)) < fabs(height(state.h)))) state.s = S, state.h = H; /** Finally, we set the vector field *h* using the state and height. */ if (state.s != complete) h.x[] = nodata; else h.x[] = (state.h > 1e10 ? nodata : state.h); } } } /** ## Column propagation Once columns are computed on a local 9-cells-high stencil, we will need to "propagate" these values upward or downward so that they are accessible at distances of up to 5.5 cells from the interface. This is important in 3D in particular where marginal (~45 degrees) cases may require such high stencils to compute consistent HF curvatures. We do this by selecting the smallest height in a 5-cells neighborhood along each direction. */ static void column_propagation (vector h) { foreach() for (int i = -2; i <= 2; i++) foreach_dimension() if (fabs(height(h.x[i])) <= 3.5 && fabs(height(h.x[i]) + i) < fabs(height(h.x[]))) h.x[] = h.x[i] + i; boundary ((scalar *){h}); } /** ## Multigrid implementation The *heights()* function takes a volume fraction field *c* and returns the height function vector field *h*. */ #if !TREE trace void heights (scalar c, vector h) { /** We need a 9-points-high stencil (rather than the default 5-points). To do this we store in *cs* the volume fraction field *c* shifted by 2 grid points in the respective directions. We make sure that this field uses the same boundary conditions as *c*. */ vector cs[]; foreach_dimension() for (int i = 0; i < nboundary; i++) cs.x.boundary[i] = c.boundary[i]; /** To compute the height function, we sum the volume fractions in a (half-)column starting at the current cell. We start by integrating downward (*j = -1*) and then integrate upward (*j = 1*). */ for (int j = -1; j <= 1; j += 2) { /** We first build the shifted (by $\pm 2$) volume fraction field in each direction. */ foreach() foreach_dimension() cs.x[] = c[2*j]; boundary ((scalar *){cs}); /** We sum the half-column, downward or upward. */ foreach() half_column (point, c, h, cs, j); } boundary ((scalar *){h}); /** Finally we "propagate" values along columns. */ column_propagation (h); } /** ## Tree implementation We first define the prolongation functions for heights. */ #else // TREE foreach_dimension() static void refine_h_x (Point point, scalar h) { /** We try to prolongate columns from nearby non-prolongation cells. */ bool complete = true; foreach_child() { for (int i = -2; i <= 2; i++) if (allocated(i) && !is_prolongation(neighbor(i)) && !is_boundary(neighbor(i)) && fabs(height(h[i])) <= 3.5 && fabs(height(h[i]) + i) < fabs(height(h[]))) h[] = h[i] + i; if (h[] == nodata) complete = false; } if (complete) return; /** If some children have not been initialised, we first check that the (three in 2D, nine in 3D) coarse heights are defined and have compatible orientations. If not, the children heights are undefined. Otherwise, a (bi)quadratic fit of the coarse heights is used to compute the children heights. */ int ori = orientation(h[]); #if dimension == 2 for (int i = -1; i <= 1; i++) if (h[0,i] == nodata || orientation(h[0,i]) != ori) return; double h0 = (30.*height(h[]) + height(h[0,1]) + height(h[0,-1]))/16. + HSHIFT*ori; double dh = (height(h[0,1]) - height(h[0,-1]))/4.; foreach_child() if (h[] == nodata) h[] = h0 + dh*child.y - child.x/2.; #else // dimension == 3 double H[3][3], H0 = height(h[]); for (int i = -1; i <= 1; i++) for (int j = -1; j <= 1; j++) if (h[0,i,j] == nodata || orientation(h[0,i,j]) != ori) return; else H[i+1][j+1] = height(h[0,i,j]) - H0; double h0 = 2.*H0 + (H[2][2] + H[2][0] + H[0][0] + H[0][2] + 30.*(H[2][1] + H[0][1] + H[1][0] + H[1][2]))/512. + HSHIFT*ori; double h1 = (H[2][2] + H[2][0] - H[0][0] - H[0][2] + 30.*(H[2][1] - H[0][1]))/128.; double h2 = (H[2][2] - H[2][0] - H[0][0] + H[0][2] + 30.*(H[1][2] - H[1][0]))/128.; double h3 = (H[0][0] + H[2][2] - H[0][2] - H[2][0])/32.; foreach_child() if (h[] == nodata) h[] = h0 + h1*child.y + h2*child.z + h3*child.y*child.z - child.x/2.; #endif // dimension == 3 } /** The *heights()* function implementation is similar to the multigrid case, but the construction of the shifted volume fraction field *cs* is more complex. */ trace void heights (scalar c, vector h) { vector cs[]; foreach_dimension() for (int i = 0; i < nboundary; i++) cs.x.boundary[i] = c.boundary[i]; /** To compute the shifted field, we first need to *restrict* the volume fraction on all levels. */ restriction ({c}); for (int j = -1; j <= 1; j += 2) { /** We traverse the tree level by level, from coarse to fine. On the root cell the height function is undefined. */ foreach_level(0) foreach_dimension() h.x[] = nodata; for (int l = 1; l <= depth(); l++) { /** We construct the ($\pm 2$) shifted field at this level. */ foreach_level (l) foreach_dimension() cs.x[] = c[2*j]; /** We then need to apply boundary conditions on the shifted field. This is more complex than for a constant resolution grid. We first construct the ($\pm 1$) shifted field for the immediately coarser level. This is done by copying the volume fraction field for pairs of adjacent cells. */ foreach_level (l - 1) foreach_dimension() { cs.x[] = c[j]; cs.x[j] = c[2*j]; } /** We can now use this shifted coarse field (which matches the shifted fine field) to apply boundary conditions on coarse/fine prolongation halos. */ foreach_halo (prolongation, l - 1) foreach_dimension() c.prolongation (point, cs.x); boundary_iterate (level, (scalar *){cs}, l); /** We can now sum the half-column at this level, downward or upward according to *j*. */ foreach_level (l) half_column (point, c, h, cs, j); } } /** We fill the prolongation cells with "nodata". The restriction function does nothing as we have already defined *h* on all levels. */ foreach_dimension() { h.x.prolongation = no_data; h.x.restriction = no_restriction; } boundary ((scalar *){h}); /** Final prolongation cells will be filled with values obtained either from neighboring columns or by interpolation from coarser levels (see *refine_h_x()* above). */ foreach_dimension() h.x.prolongation = refine_h_x; /** Finally, we "propagate" values along columns. */ column_propagation (h); } #endif // TREE