FRotationKernel.hpp 66 KB
::Value; BRAMAS Berenger committed Aug 13, 2012 84 BRAMAS Berenger committed Aug 17, 2012 85 86 FReal DlmkCoefOTheta[8][SizeDlmkMatrix]; //< d_lmk for Multipole rotation FReal DlmkCoefOMinusTheta[8][SizeDlmkMatrix]; //< d_lmk for Multipole reverse rotation BRAMAS Berenger committed Aug 07, 2012 87 BRAMAS Berenger committed Aug 17, 2012 88 89 FReal DlmkCoefMTheta[8][SizeDlmkMatrix]; //< d_lmk for Local rotation FReal DlmkCoefMMinusTheta[8][SizeDlmkMatrix]; //< d_lmk for Local reverse rotation 90 BRAMAS Berenger committed Aug 17, 2012 91 92 93 94 95 96 FReal DlmkCoefM2LOTheta[343][SizeDlmkMatrix]; //< d_lmk for Multipole rotation FReal DlmkCoefM2LMMinusTheta[343][SizeDlmkMatrix]; //< d_lmk for Local reverse rotation /////////////////////////////////////////////////////// // Precomputation /////////////////////////////////////////////////////// 97 BRAMAS Berenger committed Aug 17, 2012 98 99 100 101 /** Compute the factorial from 0 to P*2 * Then the data is accessible in factorials array: * factorials[n] = n! with n <= 2*P */ BRAMAS Berenger committed Aug 13, 2012 102 void precomputeFactorials(){ BRAMAS Berenger committed Aug 28, 2015 103 104 105 106 107 factorials[0] = 1; FReal fidx = 1; for(int idx = 1 ; idx <= P2 ; ++idx, ++fidx){ factorials[idx] = fidx * factorials[idx-1]; } BRAMAS Berenger committed Aug 13, 2012 108 109 } BRAMAS Berenger committed Aug 17, 2012 110 111 112 113 114 115 116 117 118 119 /** This function precompute the translation coef. * Translation are independant of the angle between both cells. * So in the M2M/L2L the translation is the same for all children. * In the M2L the translation depend on the distance between the * source and the target (so a few number of possibilities exist) * * The number of possible translation depend of the tree height, * so the memory is allocated dynamically with a smart pointer to share * the data between threads. */ BRAMAS Berenger committed Aug 13, 2012 120 void precomputeTranslationCoef(){ BRAMAS Berenger committed Aug 28, 2015 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 {// M2M & L2L // Allocate M2MTranslationCoef = new FReal[treeHeight-1][P+1]; L2LTranslationCoef = new FReal[treeHeight-1][P+1]; // widthAtLevel represents half of the size of a box FReal widthAtLevel = boxWidth/4; // we go from the root to the leaf-1 for( int idxLevel = 0 ; idxLevel < treeHeight - 1 ; ++idxLevel){ // b is the parent-child distance = norm( vec(widthAtLevel,widthAtLevel,widthAtLevel)) const FReal b = FMath::Sqrt(widthAtLevel*widthAtLevel*3); // we compute b^idx iteratively FReal bPowIdx = 1.0; // we compute -1^idx iteratively FReal minus_1_pow_idx = 1.0; for(int idx = 0 ; idx <= P ; ++idx){ // coef m2m = (-b)^j/j! M2MTranslationCoef[idxLevel][idx] = minus_1_pow_idx * bPowIdx / factorials[idx]; // coef l2l = b^j/j! L2LTranslationCoef[idxLevel][idx] = bPowIdx / factorials[idx]; // increase bPowIdx *= b; minus_1_pow_idx = -minus_1_pow_idx; } // divide by two per level widthAtLevel /= 2; } } {// M2L // Allocate M2LTranslationCoef = new FReal[treeHeight][343][P+1]; // This is the width of a box at each level FReal boxWidthAtLevel = widthAtLeafLevel; // from leaf level to the root for(int idxLevel = treeHeight-1 ; idxLevel > 0 ; --idxLevel){ // we compute all possibilities for(int idxX = -3 ; idxX <= 3 ; ++idxX ){ for(int idxY = -3 ; idxY <= 3 ; ++idxY ){ for(int idxZ = -3 ; idxZ <= 3 ; ++idxZ ){ // if this is not a neighbour if( idxX < -1 || 1 < idxX || idxY < -1 || 1 < idxY || idxZ < -1 || 1 < idxZ ){ // compute the relative position const FPoint relativePosition( -FReal(idxX)*boxWidthAtLevel, -FReal(idxY)*boxWidthAtLevel, -FReal(idxZ)*boxWidthAtLevel); // this is the position in the index system from 0 to 343 const int position = ((( (idxX+3) * 7) + (idxY+3))) * 7 + idxZ + 3; // b is the distance between the two cells const FReal b = FMath::Sqrt( (relativePosition.getX() * relativePosition.getX()) + (relativePosition.getY() * relativePosition.getY()) + (relativePosition.getZ() * relativePosition.getZ())); // compute b^idx+1 iteratively FReal bPowIdx1 = b; for(int idx = 0 ; idx <= P ; ++idx){ // factorials[j+l] / FMath::pow(b,j+l+1) M2LTranslationCoef[idxLevel][position][idx] = factorials[idx] / bPowIdx1; bPowIdx1 *= b; } } } } } // multiply per two at each level boxWidthAtLevel *= FReal(2.0); } } BRAMAS Berenger committed Aug 13, 2012 186 187 } BRAMAS Berenger committed Aug 17, 2012 188 189 190 191 192 193 194 195 196 /////////////////////////////////////////////////////// // Precomputation rotation vector // This is a all in one function // First we compute the d_lmk needed, // then we compute vectors for M2M/L2L // finally we compute the vectors for M2L /////////////////////////////////////////////////////// COULAUD Olivier committed Apr 18, 2014 197 /** The following comments include formula taken from the original vectors BRAMAS Berenger committed Aug 17, 2012 198 199 200 * * * This function rotate a multipole vector by an angle azimuth phi BRAMAS Berenger committed Aug 13, 2012 201 202 203 204 205 206 207 208 * The formula used is present in several paper, but we refer to * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 * \f[ * O_{l,m}( \alpha, \beta + \phi ) = e^{-i \phi m} O_{l,m}( \alpha, \beta ) * \f] * The computation is simply a multiplication per a complex number \f$e^{-i \phi m} \f$ * Phi should be in [0,2pi] BRAMAS Berenger committed Aug 17, 2012 209 210 * * This function rotate a local vector by an angle azimuth phi BRAMAS Berenger committed Aug 13, 2012 211 212 213 214 215 216 217 218 * The formula used is present in several paper, but we refer to * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 * \f[ * M_{l,m}( \alpha, \beta + \phi ) = e^{i \phi m} M_{l,m}( \alpha, \beta ) * \f] * The computation is simply a multiplication per a complex number \f$e^{i \phi m} \f$ * Phi should be in [0,2pi] BRAMAS Berenger committed Aug 17, 2012 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 * * This function rotate a multipole vector by an angle inclination \theta * The formula used is present in several paper, but we refer to * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 * \f[ * O_{l,m}( \alpha + \theta, \beta ) = \sum_{k=-l}^l{ \sqrt{ \frac{(l-k)!(l+k)!}{(l-|m|)!(l+|m|)!} } * d^l_{km}( \theta ) O_{l,k}( \alpha, \beta ) } * \f] * Because we store only P_lm for l >= 0 and m >= 0 we use the relation of symetrie as: * \f$O_{l,-m} = \bar{ O_{l,m} } (-1)^m \f$ * Theta should be in [0,pi] * * This function rotate a local vector by an angle inclination \theta * The formula used is present in several paper, but we refer to * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 * \f[ * M_{l,m}( \alpha + \theta, \beta ) = \sum_{k=-l}^l{ \sqrt{ \frac{(l-|m|)!(l+|m|)!}{(l-k)!(l+k)!} } * d^l_{km}( \theta ) M_{l,k}( \alpha, \beta ) } * \f] * Because we store only P_lm for l >= 0 and m >= 0 we use the relation of symetrie as: * \f$M_{l,-m} = \bar{ M_{l,m} } (-1)^m \f$ * Theta should be in [0,pi] * * Remark about the structure of the structure of the matrixes DlmkCoef[O/M](Minus)Theta. * It is composed of "P" small matrix. COULAUD Olivier committed Jun 19, 2014 246 * The matrix M(l) (0 <= l <= P) has a size of (l*2+1) BRAMAS Berenger committed Aug 17, 2012 247 248 249 250 251 252 253 254 255 256 * It means indexes are going from -l to l for column and row. * l = 0: ( -0 <= m <= 0 ; -0 <= k <= 0) * [X] * l = 1: ( -1 <= m <= 1 ; -1 <= k <= 1) * [X X X] * [X X X] * [X X X] * etc. * The real size of such matrix is : * 1x1 + 3x3 + ... + (2P+1)x(2P+1) BRAMAS Berenger committed Aug 13, 2012 257 258 */ void precomputeRotationVectors(){ BRAMAS Berenger committed Aug 28, 2015 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 ///////////////////////////////////////////////////////////////// // We will need a Sqrt(factorial[x-y]*factorial[x+y]) // so we precompute it FReal sqrtDoubleFactorials[P+1][P+1]; for(int l = 0 ; l <= P ; ++l ){ for(int m = 0 ; m <= l ; ++m ){ sqrtDoubleFactorials[l][m] = FMath::Sqrt(factorials[l-m]*factorials[l+m]); } } ///////////////////////////////////////////////////////////////// // We compute the rotation matrix, we do not need 343 matrix // We will compute only a part of the since we compute the inclinaison // angle. inclinaison(+/-x,+/-y,z) = inclinaison(+/-y,+/-x,z) // we put the negative (-theta) with a negative x typedef FReal (*pMatrixDlmk) /*[P+1]*/[P2+1][P2+1]; pMatrixDlmk dlmkMatrix[7][4][7]; // Allocate matrix for(int idxX = 0 ; idxX < 7 ; ++idxX) for(int idxY = 0 ; idxY < 4 ; ++idxY) for(int idxZ = 0 ; idxZ < 7 ; ++idxZ) { dlmkMatrix[idxX][idxY][idxZ] = new FReal[P+1][P2+1][P2+1]; } // First we compute special vectors: DlmkBuild0(dlmkMatrix[0+3][0][1+3]); // theta = 0 DlmkBuildPi(dlmkMatrix[0+3][0][-1+3]); // theta = Pi DlmkBuild(dlmkMatrix[1+3][0][0+3],FMath::FPiDiv2()); // theta = Pi/2 DlmkInverse(dlmkMatrix[-1+3][0][0+3],dlmkMatrix[1+3][0][0+3]); // theta = -Pi/2 // Then other angle for(int x = 1 ; x <= 3 ; ++x){ for(int y = 0 ; y <= x ; ++y){ for(int z = 1 ; z <= 3 ; ++z){ const FReal inclinaison = FSpherical(FPoint(FReal(x),FReal(y),FReal(z))).getInclination(); DlmkBuild(dlmkMatrix[x+3][y][z+3],inclinaison); // For inclinaison between ]pi/2;pi[ DlmkZNegative(dlmkMatrix[x+3][y][(-z)+3],dlmkMatrix[x+3][y][z+3]); // For inclinaison between ]pi;3pi/2[ DlmkInverseZNegative(dlmkMatrix[(-x)+3][y][(-z)+3],dlmkMatrix[x+3][y][z+3]); // For inclinaison between ]3pi/2;2pi[ DlmkInverse(dlmkMatrix[(-x)+3][y][z+3],dlmkMatrix[x+3][y][z+3]); } } } ///////////////////////////////////////////////////////////////// // Manage angle for M2M/L2L const int index_P0 = atLm(P,0); // For all possible child (morton indexing from 0 to 7) for(int idxChild = 0 ; idxChild < 8 ; ++idxChild){ // Retrieve relative position of child to parent const FReal x = FReal((idxChild&4)? -boxWidth : boxWidth); const FReal y = FReal((idxChild&2)? -boxWidth : boxWidth); const FReal z = FReal((idxChild&1)? -boxWidth : boxWidth); const FPoint relativePosition( x , y , z ); // compute azimuth BRAMAS Berenger committed Mar 24, 2015 316 const FSpherical sph(relativePosition); PIACIBELLO Cyrille committed Sep 04, 2014 317 BRAMAS Berenger committed Aug 28, 2015 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 // First compute azimuth rotation // compute the last part with l == P { int index_lm = index_P0; for(int m = 0 ; m <= P ; ++m, ++index_lm ){ const FReal mphi = (sph.getPhiZero2Pi() + FMath::FPiDiv2()) * FReal(m); // O_{l,m}( \alpha, \beta + \phi ) = e^{-i \phi m} O_{l,m}( \alpha, \beta ) rotationExpMinusImPhi[idxChild][index_lm].setRealImag(FMath::Cos(-mphi), FMath::Sin(-mphi)); // M_{l,m}( \alpha, \beta + \phi ) = e^{i \phi m} M_{l,m}( \alpha, \beta ) rotationExpImPhi[idxChild][index_lm].setRealImag(FMath::Cos(mphi), FMath::Sin(mphi)); } } // Then for l < P it just a copy of the previous computed vector { int index_lm = 0; // for l < P for(int l = 0 ; l < P ; ++l){ // take the l + 1 numbers from the vector with l' = P FMemUtils::copyall(rotationExpMinusImPhi[idxChild] + index_lm, rotationExpMinusImPhi[idxChild] + index_P0, l + 1); FMemUtils::copyall(rotationExpImPhi[idxChild] + index_lm, rotationExpImPhi[idxChild] + index_P0, l + 1); // index(l+1,0) = index(l,0) + l + 1 index_lm += l + 1; } } { // Then compute the inclinaison rotation // For the child parent relation we always have a inclinaison // for (1,1,1) or (1,1,-1) const int dx = 1; const int dy = 1; const int dz = (idxChild&1)?-1:1; // int index_lmk = 0; for(int l = 0 ; l <= P ; ++l){ for(int m = 0 ; m <= l ; ++m ){ { // for k == 0 const FReal d_lmk_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][0+P]; const FReal d_lmk = dlmkMatrix[dx+3][dy][dz+3][l][m+P][0+P]; // \sqrt{ \frac{(l-k)!(l+k)!}{(l-|m|)!(l+|m|)!} } const FReal Ofactor = sqrtDoubleFactorials[l][0]/sqrtDoubleFactorials[l][m]; const FReal Mfactor = sqrtDoubleFactorials[l][m]/sqrtDoubleFactorials[l][0]; DlmkCoefOTheta[idxChild][index_lmk] = Ofactor * d_lmk; DlmkCoefMTheta[idxChild][index_lmk] = Mfactor * d_lmk; DlmkCoefOMinusTheta[idxChild][index_lmk] = Ofactor * d_lmk_minusTheta; DlmkCoefMMinusTheta[idxChild][index_lmk] = Mfactor * d_lmk_minusTheta; ++index_lmk; } // for 0 < k FReal minus_1_pow_k = -1.0; for(int k = 1 ; k <= l ; ++k){ const FReal d_lm_minus_k = dlmkMatrix[dx+3][dy][dz+3][l][m+P][-k+P]; const FReal d_lmk = dlmkMatrix[dx+3][dy][dz+3][l][m+P][k+P]; const FReal d_lm_minus_k_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][-k+P]; const FReal d_lmk_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][k+P]; const FReal Ofactor = sqrtDoubleFactorials[l][k]/sqrtDoubleFactorials[l][m]; const FReal Mfactor = sqrtDoubleFactorials[l][m]/sqrtDoubleFactorials[l][k]; // for k negatif DlmkCoefOTheta[idxChild][index_lmk] = Ofactor * (d_lmk + minus_1_pow_k * d_lm_minus_k); DlmkCoefMTheta[idxChild][index_lmk] = Mfactor * (d_lmk + minus_1_pow_k * d_lm_minus_k); DlmkCoefOMinusTheta[idxChild][index_lmk] = Ofactor * (d_lmk_minusTheta + minus_1_pow_k * d_lm_minus_k_minusTheta); DlmkCoefMMinusTheta[idxChild][index_lmk] = Mfactor * (d_lmk_minusTheta + minus_1_pow_k * d_lm_minus_k_minusTheta); ++index_lmk; // for k positif DlmkCoefOTheta[idxChild][index_lmk] = Ofactor * (d_lmk - minus_1_pow_k * d_lm_minus_k); DlmkCoefMTheta[idxChild][index_lmk] = Mfactor * (d_lmk - minus_1_pow_k * d_lm_minus_k); DlmkCoefOMinusTheta[idxChild][index_lmk] = Ofactor * (d_lmk_minusTheta - minus_1_pow_k * d_lm_minus_k_minusTheta); DlmkCoefMMinusTheta[idxChild][index_lmk] = Mfactor * (d_lmk_minusTheta - minus_1_pow_k * d_lm_minus_k_minusTheta); ++index_lmk; minus_1_pow_k = -minus_1_pow_k; } } } } } ///////////////////////////////////////////////////////////////// // Manage angle for M2L // For all possible cases for(int idxX = -3 ; idxX <= 3 ; ++idxX ){ for(int idxY = -3 ; idxY <= 3 ; ++idxY ){ for(int idxZ = -3 ; idxZ <= 3 ; ++idxZ ){ // Test if it is not a neighbors if( idxX < -1 || 1 < idxX || idxY < -1 || 1 < idxY || idxZ < -1 || 1 < idxZ ){ // Build relative position between target and source const FPoint relativePosition( -FReal(idxX)*boxWidth, -FReal(idxY)*boxWidth, -FReal(idxZ)*boxWidth); const int position = ((( (idxX+3) * 7) + (idxY+3))) * 7 + idxZ + 3; const FSpherical sph(relativePosition); // Compute azimuth rotation vector // first compute the last part with l == P { int index_lm = index_P0; for(int m = 0 ; m <= P ; ++m, ++index_lm ){ const FReal mphi = (sph.getPhiZero2Pi() + FMath::FPiDiv2()) * FReal(m); // O_{l,m}( \alpha, \beta + \phi ) = e^{-i \phi m} O_{l,m}( \alpha, \beta ) rotationM2LExpMinusImPhi[position][index_lm].setRealImag(FMath::Cos(-mphi), FMath::Sin(-mphi)); // M_{l,m}( \alpha, \beta + \phi ) = e^{i \phi m} M_{l,m}( \alpha, \beta ) rotationM2LExpImPhi[position][index_lm].setRealImag(FMath::Cos(mphi), FMath::Sin(mphi)); } } // Then for l < P copy the subpart of the previous vector { int index_lm = 0; for(int l = 0 ; l < P ; ++l){ FMemUtils::copyall(rotationM2LExpMinusImPhi[position] + index_lm, rotationM2LExpMinusImPhi[position] + index_P0, l + 1); FMemUtils::copyall(rotationM2LExpImPhi[position] + index_lm, rotationM2LExpImPhi[position] + index_P0, l + 1); index_lm += l + 1; } } // Compute inclination vector { // We have to find the right d_lmk matrix int dx = 0 , dy = 0, dz = 0; // if x == 0 && y == 0 it means we have an inclination of 0 or PI if(idxX == 0 && idxY == 0){ dx = 0; dy = 0; // no matter if z is big, we want [0][0][1] or [0][0][-1] if( idxZ < 0 ) dz = 1; else dz = -1; } // if z == 0 we have an inclination of Pi/2 else if ( idxZ == 0){ dx = 1; dy = 0; dz = 0; } // else we take the right indexes else { dx = FMath::Max(FMath::Abs(idxX),FMath::Abs(idxY)); dy = FMath::Min(FMath::Abs(idxX),FMath::Abs(idxY)); dz = -idxZ; } int index_lmk = 0; for(int l = 0 ; l <= P ; ++l){ for(int m = 0 ; m <= l ; ++m ){ { // k == 0 const FReal d_lmk = dlmkMatrix[dx+3][dy][dz+3][l][m+P][0+P]; const FReal d_lmk_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][0+P]; // \sqrt{ \frac{(l-k)!(l+k)!}{(l-|m|)!(l+|m|)!} } const FReal Ofactor = sqrtDoubleFactorials[l][0]/sqrtDoubleFactorials[l][m]; const FReal Mfactor = sqrtDoubleFactorials[l][m]/sqrtDoubleFactorials[l][0]; DlmkCoefM2LOTheta[position][index_lmk] = Ofactor * d_lmk; DlmkCoefM2LMMinusTheta[position][index_lmk] = Mfactor * d_lmk_minusTheta; ++index_lmk; } FReal minus_1_pow_k = -1.0; for(int k = 1 ; k <= l ; ++k){ const FReal d_lm_minus_k = dlmkMatrix[dx+3][dy][dz+3][l][m+P][-k+P]; const FReal d_lmk = dlmkMatrix[dx+3][dy][dz+3][l][m+P][k+P]; const FReal d_lm_minus_k_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][-k+P]; const FReal d_lmk_minusTheta = dlmkMatrix[-dx+3][dy][dz+3][l][m+P][k+P]; const FReal Ofactor = sqrtDoubleFactorials[l][k]/sqrtDoubleFactorials[l][m]; const FReal Mfactor = sqrtDoubleFactorials[l][m]/sqrtDoubleFactorials[l][k]; DlmkCoefM2LOTheta[position][index_lmk] = Ofactor * (d_lmk + minus_1_pow_k * d_lm_minus_k); DlmkCoefM2LMMinusTheta[position][index_lmk] = Mfactor * (d_lmk_minusTheta + minus_1_pow_k * d_lm_minus_k_minusTheta); ++index_lmk; DlmkCoefM2LOTheta[position][index_lmk] = Ofactor * (d_lmk - minus_1_pow_k * d_lm_minus_k); DlmkCoefM2LMMinusTheta[position][index_lmk] = Mfactor * (d_lmk_minusTheta - minus_1_pow_k * d_lm_minus_k_minusTheta); ++index_lmk; minus_1_pow_k = -minus_1_pow_k; } } } } } } } } // Deallocate matrix for(int idxX = 0 ; idxX < 7 ; ++idxX) for(int idxY = 0 ; idxY < 4 ; ++idxY) for(int idxZ = 0 ; idxZ < 7 ; ++idxZ) { delete[] dlmkMatrix[idxX][idxY][idxZ]; } BRAMAS Berenger committed Aug 13, 2012 516 517 } BRAMAS Berenger committed Aug 17, 2012 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 /////////////////////////////////////////////////////// // d_lmk computation // This part is constitued of 6 functions : // DlmkBuild computes the matrix from a angle ]0;pi/2] // DlmkBuild0 computes the matrix for angle 0 // DlmkBuildPi computes the matrix for angle pi // Then, others use the d_lmk to build a rotated matrix: // DlmkZNegative computes for angle \theta ]pi/2;pi[ using d_lmk(pi- \theta) // DlmkInverseZNegative computes for angle \theta ]pi;3pi/2[ using d_lmk(\theta-pi) // DlmkInverse computes for angle \theta ]3pi/2;2pi[ using d_lmk(2pi- \theta) /////////////////////////////////////////////////////// /** Compute d_mlk for \theta = 0 * \f[ COULAUD Olivier committed Aug 22, 2012 534 * d^l_{m,k}( \theta ) = \delta_{m,k,} \,\, \mbox{\textrm{ $\delta$ Kronecker symbol }} BRAMAS Berenger committed Aug 17, 2012 535 536 537 * \f] */ void DlmkBuild0(FReal dlmk[P+1][P2+1][P2+1]) const { BRAMAS Berenger committed Aug 28, 2015 538 539 540 541 542 543 544 545 546 547 for(int l = 0 ; l <= P ; ++l){ for(int m = -l ; m <= l ; ++m){ // first put 0 every where for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = FReal(0.0); } // then replace per 1 for m == k dlmk[l][P+m][P+m] = FReal(1.0); } } BRAMAS Berenger committed Aug 13, 2012 548 549 } BRAMAS Berenger committed Aug 17, 2012 550 551 /** Compute d_mlk for \theta = PI * \f[ COULAUD Olivier committed Aug 22, 2012 552 * d^l_{m,k}( \theta ) = (-1)^{l+k} \delta_{m,k},\,\, \mbox{\textrm{ $\delta$ Kronecker delta } } BRAMAS Berenger committed Aug 17, 2012 553 554 555 * \f] */ void DlmkBuildPi(FReal dlmk[P+1][P2+1][P2+1]) const { BRAMAS Berenger committed Aug 28, 2015 556 557 558 559 560 561 562 563 564 565 for(int l = 0 ; l <= P ; ++l){ for(int m = -l ; m <= l ; ++m){ // put 0 every where for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = FReal(0.0); } // -1^l+k * 1 where m == k dlmk[l][P+m][P-m] = ((l+m)&0x1 ? FReal(-1) : FReal(1)); } } BRAMAS Berenger committed Aug 13, 2012 566 567 } BRAMAS Berenger committed Aug 17, 2012 568 569 570 571 572 573 /** Compute d_mlk for \theta = ]PI/2;PI[ * \f[ * d^l_{m,k}( \theta ) = (-1)^{l+m} d^l_{m,-k}( \Pi - \theta ) * \f] */ void DlmkZNegative(FReal dlmk[P+1][P2+1][P2+1], const FReal dlmkZPositif[P+1][P2+1][P2+1]) const { BRAMAS Berenger committed Aug 28, 2015 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 for(int l = 0 ; l <= P ; ++l){ for(int m = -l ; m <= l ; ++m){ // if l+m is odd if( (l+m)&0x1 ){ // put -1 every where for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = -dlmkZPositif[l][P+m][P-k]; } } else{ // else just copy for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = dlmkZPositif[l][P+m][P-k]; } } } } BRAMAS Berenger committed Aug 17, 2012 591 } BRAMAS Berenger committed Aug 13, 2012 592 BRAMAS Berenger committed Aug 17, 2012 593 594 595 596 597 598 /** Compute d_mlk for \theta = ]PI;3PI/2[ * \f[ * d^l_{m,k}( \theta ) = (-1)^{l+m} d^l_{-m,k}( \theta - \Pi ) * \f] */ void DlmkInverseZNegative(FReal dlmk[P+1][P2+1][P2+1], const FReal dlmkZPositif[P+1][P2+1][P2+1]) const { BRAMAS Berenger committed Aug 28, 2015 599 600 601 602 603 604 605 606 607 608 609 610 611 612 for(int l = 0 ; l <= P ; ++l){ for(int m = -l ; m <= l ; ++m){ if( (l+m)&0x1 ){ for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = -dlmkZPositif[l][P-m][P+k]; } } else{ for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = dlmkZPositif[l][P-m][P+k]; } } } } 613 614 } BRAMAS Berenger committed Aug 17, 2012 615 616 617 /** Compute d_mlk for \theta = ]3PI/2;2PI[ * \f[ * d^l_{m,k}( \theta ) = (-1)^{m+k} d^l_{m,k}( 2 \Pi - \theta ) BRAMAS Berenger committed Aug 12, 2012 618 * \f] BRAMAS Berenger committed Aug 07, 2012 619 */ BRAMAS Berenger committed Aug 17, 2012 620 void DlmkInverse(FReal dlmk[P+1][P2+1][P2+1], const FReal dlmkZPositif[P+1][P2+1][P2+1]) const { BRAMAS Berenger committed Aug 28, 2015 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 for(int l = 0 ; l <= P ; ++l){ for(int m = -l ; m <= l ; ++m){ // we start with k == -l, so if (k+m) is odd if( (l+m)&0x1 ){ // then we start per (-1) for(int k = -l ; k < l ; k+=2){ dlmk[l][P+m][P+k] = -dlmkZPositif[l][P+m][P+k]; dlmk[l][P+m][P+k+1] = dlmkZPositif[l][P+m][P+k+1]; } // l is always odd dlmk[l][P+m][P+l] = -dlmkZPositif[l][P+m][P+l]; } else{ // else we start per (+1) for(int k = -l ; k < l ; k+=2){ dlmk[l][P+m][P+k] = dlmkZPositif[l][P+m][P+k]; dlmk[l][P+m][P+k+1] = -dlmkZPositif[l][P+m][P+k+1]; } // l is always odd dlmk[l][P+m][P+l] = dlmkZPositif[l][P+m][P+l]; } } } BRAMAS Berenger committed Aug 17, 2012 644 645 646 647 648 649 650 651 652 653 654 655 } /** Compute d_mlk for \theta = ]0;PI/2[ * This used the second formula from the paper: * Fast and accurate determination of the wigner rotation matrices in FMM * * We use formula 28,29 to compute "g" * Then 25,26,27 for the recurrence. */ // theta should be between [0;pi] as the inclinaison angle void DlmkBuild(FReal dlmk[P+1][P2+1][P2+1], const FReal inTheta) const { BRAMAS Berenger committed Aug 28, 2015 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 FAssertLF(0 <= inTheta && inTheta < FMath::FTwoPi()); // To have constants for very used values const FReal F0 = FReal(0.0); const FReal F1 = FReal(1.0); const FReal F2 = FReal(2.0); const FReal cosTheta = FMath::Cos(inTheta); const FReal sinTheta = FMath::Sin(inTheta); // First compute g FReal g[SizeArray]; {// Equ 29 // g{0,0} = 1 g[0] = F1; // g{l,0} = sqrt( (2l - 1) / 2l) g{l-1,0} for l > 0 { int index_l0 = 1; FReal fl = F1; for(int l = 1; l <= P ; ++l, ++fl ){ g[index_l0] = FMath::Sqrt((fl*F2-F1)/(fl*F2)) * g[index_l0-l]; index_l0 += l + 1; } } // g{l,m} = sqrt( (l - m + 1) / (l+m)) g{l,m-1} for l > 0, 0 < m <= l { int index_lm = 2; FReal fl = F1; for(int l = 1; l <= P ; ++l, ++fl ){ FReal fm = F1; for(int m = 1; m <= l ; ++m, ++index_lm, ++fm ){ g[index_lm] = FMath::Sqrt((fl-fm+F1)/(fl+fm)) * g[index_lm-1]; } ++index_lm; } } } { // initial condition // Equ 28 // d{l,m,l} = -1^(l+m) g{l,m} (1+cos(theta))^m sin(theta)^(l-m) , For l > 0, 0 <= m <= l int index_lm = 0; FReal sinTheta_pow_l = F1; for(int l = 0 ; l <= P ; ++l){ // build variable iteratively FReal minus_1_pow_lm = l&0x1 ? FReal(-1) : FReal(1); FReal cosTheta_1_pow_m = F1; FReal sinTheta_pow_l_minus_m = sinTheta_pow_l; for(int m = 0 ; m <= l ; ++m, ++index_lm){ dlmk[l][P+m][P+l] = minus_1_pow_lm * g[index_lm] * cosTheta_1_pow_m * sinTheta_pow_l_minus_m; // update minus_1_pow_lm = -minus_1_pow_lm; cosTheta_1_pow_m *= F1 + cosTheta; sinTheta_pow_l_minus_m /= sinTheta; } // update sinTheta_pow_l *= sinTheta; } } { // build the rest of the matrix FReal fl = F1; for(int l = 1 ; l <= P ; ++l, ++fl){ FReal fk = fl; for(int k = l ; k > -l ; --k, --fk){ // Equ 25 // For l > 0, 0 <= m < l, -l < k <= l, cos(theta) >= 0 // d{l,m,k-1} = sqrt( l(l+1) - m(m+1) / l(l+1) - k(k-1)) d{l,m+1,k} // + (m+k) sin(theta) d{l,m,k} / sqrt(l(l+1) - k(k-1)) (1+cos(theta)) FReal fm = F0; for(int m = 0 ; m < l ; ++m, ++fm){ dlmk[l][P+m][P+k-1] = (FMath::Sqrt((fl*(fl+F1)-fm*(fm+F1))/(fl*(fl+F1)-fk*(fk-F1))) * dlmk[l][P+m+1][P+k]) + ((fm+fk)*sinTheta*dlmk[l][P+m][P+k]/(FMath::Sqrt(fl*(fl+F1)-fk*(fk-F1))*(F1+cosTheta))); } // Equ 26 // For l > 0, -l < k <= l, cos(theta) >= 0 // d{l,l,k-1} = (l+k) sin(theta) d{l,l,k} // / sqrt(l(l+1)-k(k-1)) (1+cos(theta)) dlmk[l][P+l][P+k-1] = (fl+fk)*sinTheta*dlmk[l][P+l][P+k]/(FMath::Sqrt(fl*(fl+F1)-fk*(fk-F1))*(F1+cosTheta)); } // Equ 27 // d{l,m,k} = -1^(m+k) d{l,-m,-k} , For l > 0, -l <= m < 0, -l <= k <= l for(int m = -l ; m < 0 ; ++m){ FReal minus_1_pow_mk = (m-l)&0x1 ? FReal(-1) : FReal(1); for(int k = -l ; k <= l ; ++k){ dlmk[l][P+m][P+k] = minus_1_pow_mk * dlmk[l][P-m][P-k]; minus_1_pow_mk = -minus_1_pow_mk; } } } } 746 747 } BRAMAS Berenger committed Aug 07, 2012 748 749 750 751 752 753 /** Compute the legendre polynomial from {0,0} to {P,P} * the computation is made by recurence (P cannot be equal to 0) * * The formula has been taken from: * Fast and accurate determination of the wigner rotation matrices in the fast multipole method * Formula number (22) BRAMAS Berenger committed Aug 12, 2012 754 * \f[ BRAMAS Berenger committed Aug 07, 2012 755 756 757 758 * P_{0,0} = 1 * P_{l,l} = (2l-1) sin( \theta ) P_{l-1,l-1} ,l \ge 0 * P_{l,l-1} = (2l-1) cos( \theta ) P_{l-1,l-1} ,l \ge 0 * P_{l,m} = \frac{(2l-1) cos( \theta ) P_{l-1,m} - (l+m-1) P_{l-2,m}x}{(l-k)} ,l \ge 1, 0 \leq m \le l-1 BRAMAS Berenger committed Aug 12, 2012 759 * \f] BRAMAS Berenger committed Aug 07, 2012 760 */ BRAMAS Berenger committed Aug 17, 2012 761 void computeLegendre(FReal legendre[], const FReal inCosTheta, const FReal inSinTheta) const { BRAMAS Berenger committed Aug 28, 2015 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 const FReal invSinTheta = -inSinTheta; legendre[0] = 1.0; // P_0,0(1) = 1 legendre[1] = inCosTheta; // P_1,0 = cos(theta) legendre[2] = invSinTheta; // P_1,1 = -sin(theta) // work with pointers FReal* FRestrict legendre_l1_m1 = legendre; // P{l-2,m} starts with P_{0,0} FReal* FRestrict legendre_l1_m = legendre + 1; // P{l-1,m} starts with P_{1,0} FReal* FRestrict legendre_lm = legendre + 3; // P{l,m} starts with P_{2,0} // Compute using recurrence FReal l2_minus_1 = 3; // 2 * l - 1 FReal fl = FReal(2.0);// To get 'l' as a float for(int l = 2; l <= P ; ++l, ++fl ){ FReal lm_minus_1 = fl - FReal(1.0); // l + m - 1 FReal l_minus_m = fl; // l - m for( int m = 0; m < l - 1 ; ++m ){ // P_{l,m} = \frac{(2l-1) cos( \theta ) P_{l-1,m} - (l+m-1) P_{l-2,m}x}{(l-m)} *(legendre_lm++) = (l2_minus_1 * inCosTheta * (*legendre_l1_m++) - (lm_minus_1++) * (*legendre_l1_m1++) ) / (l_minus_m--); } // P_{l,l-1} = (2l-1) cos( \theta ) P_{l-1,l-1} *(legendre_lm++) = l2_minus_1 * inCosTheta * (*legendre_l1_m); // P_{l,l} = (2l-1) sin( \theta ) P_{l-1,l-1} *(legendre_lm++) = l2_minus_1 * invSinTheta * (*legendre_l1_m); // goto P_{l-1,0} ++legendre_l1_m; l2_minus_1 += FReal(2.0); // 2 * l - 1 => progress by two } 793 794 } BRAMAS Berenger committed Aug 07, 2012 795 /////////////////////////////////////////////////////// BRAMAS Berenger committed Aug 17, 2012 796 797 798 // Multiplication for rotation // Here we have two function that are optimized // to compute the rotation fast! BRAMAS Berenger committed Aug 07, 2012 799 /////////////////////////////////////////////////////// 800 BRAMAS Berenger committed Aug 17, 2012 801 802 803 804 /** This function use a d_lmk vector to rotate the vec * multipole or local vector. * The result is copyed in vec. * Please see the structure of dlmk to understand this function. BRAMAS Berenger committed Mar 24, 2015 805 * Warning we cast the vec FComplex array into a FReal array BRAMAS Berenger committed Aug 07, 2012 806 */ BRAMAS Berenger committed Mar 24, 2015 807 static void RotationYWithDlmk(FComplex vec[], const FReal* dlmkCoef){ BRAMAS Berenger committed Aug 28, 2015 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 FReal originalVec[2*SizeArray]; FMemUtils::copyall((FComplex*)originalVec,vec,SizeArray); // index_lm == atLm(l,m) but progress iteratively to write the result int index_lm = 0; for(int l = 0 ; l <= P ; ++l){ const FReal*const FRestrict originalVecAtL0 = originalVec + (index_lm * 2); for(int m = 0 ; m <= l ; ++m, ++index_lm ){ FReal res_lkm_real = 0.0; FReal res_lkm_imag = 0.0; // To read all "m" value for current "l" const FReal* FRestrict iterOrignalVec = originalVecAtL0; { // for k == 0 // same coef for real and imaginary res_lkm_real += (*dlmkCoef) * (*iterOrignalVec++); res_lkm_imag += (*dlmkCoef++) * (*iterOrignalVec++); } for(int k = 1 ; k <= l ; ++k){ // coef contains first real value res_lkm_real += (*dlmkCoef++) * (*iterOrignalVec++); // then imaginary res_lkm_imag += (*dlmkCoef++) * (*iterOrignalVec++); } // save the result vec[index_lm].setRealImag(res_lkm_real, res_lkm_imag); } } 834 835 } BRAMAS Berenger committed Aug 17, 2012 836 /** This function is computing dest[:] *= src[:] BRAMAS Berenger committed Mar 24, 2015 837 * it computes inSize FComplex multiplication BRAMAS Berenger committed Aug 17, 2012 838 * to do so we first proceed per 4 and the the inSize%4 rest BRAMAS Berenger committed Aug 07, 2012 839 */ BRAMAS Berenger committed Mar 24, 2015 840 static void RotationZVectorsMul(FComplex* FRestrict dest, const FComplex* FRestrict src, const int inSize = SizeArray){ BRAMAS Berenger committed Aug 28, 2015 841 842 843 844 845 846 847 848 849 850 851 852 853 const FComplex*const FRestrict lastElement = dest + inSize; const FComplex*const FRestrict intermediateLastElement = dest + (inSize & ~0x3); // first the inSize - inSize%4 elements for(; dest != intermediateLastElement ;) { (*dest++) *= (*src++); (*dest++) *= (*src++); (*dest++) *= (*src++); (*dest++) *= (*src++); } // then the rest for(; dest != lastElement ;) { (*dest++) *= (*src++); } BRAMAS Berenger committed Jul 31, 2012 854 855 } BRAMAS Berenger committed Aug 17, 2012 856 857 858 /////////////////////////////////////////////////////// // Utils /////////////////////////////////////////////////////// 859 BRAMAS Berenger committed Aug 17, 2012 860 861 862 /** Return the position of a leaf from its tree coordinate * This is used only for the leaf BRAMAS Berenger committed Aug 07, 2012 863 */ BRAMAS Berenger committed Mar 24, 2015 864 FPoint getLeafCenter(const FTreeCoordinate coordinate) const { BRAMAS Berenger committed Aug 28, 2015 865 866 867 868 return FPoint( FReal(coordinate.getX()) * widthAtLeafLevel + widthAtLeafLevelDiv2 + boxCorner.getX(), FReal(coordinate.getY()) * widthAtLeafLevel + widthAtLeafLevelDiv2 + boxCorner.getY(), FReal(coordinate.getZ()) * widthAtLeafLevel + widthAtLeafLevelDiv2 + boxCorner.getZ()); 869 870 } BRAMAS Berenger committed Aug 17, 2012 871 872 873 874 875 876 877 878 /** Return position in the array of the l/m couple * P[atLm(l,m)] => P{l,m} * 0 * 1 2 * 3 4 5 * 6 7 8 9 ... */ int atLm(const int l, const int m) const { BRAMAS Berenger committed Aug 28, 2015 879 880 // summation series over l + m => (l*(l+1))/2 + m return ((l*(l+1))>>1) + m; BRAMAS Berenger committed Aug 07, 2012 881 } 882 BRAMAS Berenger committed Aug 07, 2012 883 public: 884 BRAMAS Berenger committed Aug 07, 2012 885 /** Constructor, needs system information */ BRAMAS Berenger committed Mar 24, 2015 886 FRotationKernel( const int inTreeHeight, const FReal inBoxWidth, const FPoint& inBoxCenter) : BRAMAS Berenger committed Aug 28, 2015 887 888 889 890 891 892 893 894 895 896 boxWidth(inBoxWidth), treeHeight(inTreeHeight), widthAtLeafLevel(inBoxWidth/FReal(1 << (inTreeHeight-1))), widthAtLeafLevelDiv2(widthAtLeafLevel/2), boxCorner(inBoxCenter.getX()-(inBoxWidth/2),inBoxCenter.getY()-(inBoxWidth/2),inBoxCenter.getZ()-(inBoxWidth/2)) { // simply does the precomputation precomputeFactorials(); precomputeTranslationCoef(); precomputeRotationVectors(); 897 898 899 900 901 902 903 } /** Default destructor */ virtual ~FRotationKernel(){ } /** P2M BRAMAS Berenger committed Aug 07, 2012 904 905 906 * The computation is based on the paper : * Parallelization of the fast multipole method * Formula number 10, page 3 BRAMAS Berenger committed Aug 12, 2012 907 * \f[ BRAMAS Berenger committed Aug 07, 2012 908 * \omega (q,a) = q \frac{a^{l}}{(l+|m|)!} P_{lm}(cos( \alpha ) )e^{-im \beta} BRAMAS Berenger committed Aug 12, 2012 909 * \f] 910 */ BRAMAS Berenger committed Aug 28, 2015 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 void P2M(CellClass* const inPole, const ContainerClass* const inParticles ) override { const FReal i_pow_m[4] = {0, FMath::FPiDiv2(), FMath::FPi(), -FMath::FPiDiv2()}; // w is the multipole moment FComplex* FRestrict const w = inPole->getMultipole(); // Copying the position is faster than using cell position const FPoint cellPosition = getLeafCenter(inPole->getCoordinate()); // We need a legendre array FReal legendre[SizeArray]; FReal angles[P+1][2]; // For all particles in the leaf box const FReal*const physicalValues = inParticles->getPhysicalValues(); const FReal*const positionsX = inParticles->getPositions()[0]; const FReal*const positionsY = inParticles->getPositions()[1]; const FReal*const positionsZ = inParticles->getPositions()[2]; for(FSize idxPart = 0 ; idxPart < inParticles->getNbParticles() ; ++ idxPart){ // P2M const FPoint position(positionsX[idxPart],positionsY[idxPart],positionsZ[idxPart]); const FSpherical sph(position - cellPosition); // The physical value (charge, mass) const FReal q = physicalValues[idxPart]; // The distance between the SH and the particle const FReal a = sph.getR(); // Compute the legendre polynomial computeLegendre(legendre, sph.getCosTheta(), sph.getSinTheta()); // w{l,m}(q,a) = q a^l/(l+|m|)! P{l,m}(cos(alpha)) exp(-i m Beta) FReal q_aPowL = q; // To consutrct q*a^l continously int index_l_m = 0; // To construct the index of (l,m) continously FReal fl = 0.0; for(int l = 0 ; l <= P ; ++l, ++fl ){ { // We need to compute the angles to use in the "m" loop // So we can compute only the one needed after "l" inc const FReal angle = fl * sph.getPhi() + i_pow_m[l & 0x3]; angles[l][0] = FMath::Cos(angle); angles[l][1] = FMath::Sin(angle); } for(int m = 0 ; m <= l ; ++m, ++index_l_m){ const FReal magnitude = q_aPowL * legendre[index_l_m] / factorials[l+m]; w[index_l_m].incReal(magnitude * angles[m][0]); w[index_l_m].incImag(magnitude * angles[m][1]); } q_aPowL *= a; } } 961 962 963 } /** M2M BRAMAS Berenger committed Aug 07, 2012 964 965 966 * The operator A has been taken from : * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 as the operator A BRAMAS Berenger committed Aug 12, 2012 967 * \f[ BRAMAS Berenger committed Aug 07, 2012 968 * O_{l,m}(a+b') = \sum_{j=|m|}^l{ \frac{ b^{l-j} }{ (l-j)! } O_{j,m}(a) } BRAMAS Berenger committed Aug 12, 2012 969 * \f] BRAMAS Berenger committed Aug 07, 2012 970 971 972 * As describe in the paper, when need first to rotate the SH * then transfer using the formula * and finaly rotate back. 973 */ BRAMAS Berenger committed Aug 28, 2015 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 void M2M(CellClass* const FRestrict inPole, const CellClass*const FRestrict *const FRestrict inChildren, const int inLevel) override { // Get the translation coef for this level (same for all child) const FReal*const coef = M2MTranslationCoef[inLevel]; // A buffer to copy the source w allocated once FComplex source_w[SizeArray]; // For all children for(int idxChild = 0 ; idxChild < 8 ; ++idxChild){ // if child exists if(inChildren[idxChild]){ // Copy the source FMemUtils::copyall(source_w, inChildren[idxChild]->getMultipole(), SizeArray); // rotate it forward RotationZVectorsMul(source_w,rotationExpMinusImPhi[idxChild]); RotationYWithDlmk(source_w,DlmkCoefOTheta[idxChild]); // Translate it FComplex target_w[SizeArray]; int index_lm = 0; for(int l = 0 ; l <= P ; ++l ){ for(int m = 0 ; m <= l ; ++m, ++index_lm ){ // w{l,m}(a+b) = sum(j=m:l, b^(l-j)/(l-j)! w{j,m}(a) FReal w_lm_real = 0.0; FReal w_lm_imag = 0.0; int index_jm = atLm(m,m); // get atLm(l,m) int index_l_minus_j = l-m; // get l-j continuously for(int j = m ; j <= l ; ++j, --index_l_minus_j, index_jm += j ){ //const coef = (b^l-j) / (l-j)!; w_lm_real += coef[index_l_minus_j] * source_w[index_jm].getReal(); w_lm_imag += coef[index_l_minus_j] * source_w[index_jm].getImag(); } target_w[index_lm].setRealImag(w_lm_real,w_lm_imag); } } // Rotate it back RotationYWithDlmk(target_w,DlmkCoefOMinusTheta[idxChild]); RotationZVectorsMul(target_w,rotationExpImPhi[idxChild]); // Sum the result FMemUtils::addall( inPole->getMultipole(), target_w, SizeArray); } } 1017 1018 1019 } /** M2L BRAMAS Berenger committed Aug 07, 2012 1020 1021 1022 * The operator B has been taken from : * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 as the operator B BRAMAS Berenger committed Aug 12, 2012 1023 * \f[ COULAUD Olivier committed Aug 22, 2012 1024 * M_{l,m}(a-b') = \sum_{j=|m|}^{\infty}{ \frac{ (j+l)! } { b^{j+l+1} } O_{j,-m}(a) } , \mbox{\textrm{ j bounded by P-l } } BRAMAS Berenger committed Aug 12, 2012 1025 * \f] BRAMAS Berenger committed Aug 07, 2012 1026 1027 1028 * As describe in the paper, when need first to rotate the SH * then transfer using the formula * and finaly rotate back. 1029 1030 */ void M2L(CellClass* const FRestrict inLocal, const CellClass* inInteractions[], const int /*inSize*/, const int inLevel) { BRAMAS Berenger committed Aug 28, 2015 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 // To copy the multipole data allocated once FComplex source_w[SizeArray]; // For all children for(int idxNeigh = 0 ; idxNeigh < 343 ; ++idxNeigh){ // if interaction exits if(inInteractions[idxNeigh]){ const FReal*const coef = M2LTranslationCoef[inLevel][idxNeigh]; // Copy multipole data into buffer FMemUtils::copyall(source_w, inInteractions[idxNeigh]->getMultipole(), SizeArray); // Rotate RotationZVectorsMul(source_w,rotationM2LExpMinusImPhi[idxNeigh]); RotationYWithDlmk(source_w,DlmkCoefM2LOTheta[idxNeigh]); // Transfer to u FComplex target_u[SizeArray]; int index_lm = 0; for(int l = 0 ; l <= P ; ++l ){ FReal minus_1_pow_m = 1.0; for(int m = 0 ; m <= l ; ++m, ++index_lm ){ // u{l,m}(a-b) = sum(j=|m|:P-l, (j+l)!/b^(j+l+1) w{j,-m}(a) FReal u_lm_real = 0.0; FReal u_lm_imag = 0.0; int index_jl = m + l; // get j+l int index_jm = atLm(m,m); // get atLm(l,m) for(int j = m ; j <= P-l ; ++j, ++index_jl, index_jm += j ){ // coef = (j+l)!/b^(j+l+1) // because {l,-m} => {l,m} conjugate -1^m with -i u_lm_real += minus_1_pow_m * coef[index_jl] * source_w[index_jm].getReal(); u_lm_imag -= minus_1_pow_m * coef[index_jl] * source_w[index_jm].getImag(); } target_u[index_lm].setRealImag(u_lm_real,u_lm_imag); minus_1_pow_m = -minus_1_pow_m; } } // Rotate it back RotationYWithDlmk(target_u,DlmkCoefM2LMMinusTheta[idxNeigh]); RotationZVectorsMul(target_u,rotationM2LExpMinusImPhi[idxNeigh]); // Sum FMemUtils::addall(inLocal->getLocal(), target_u, SizeArray); } } } void M2L(CellClass* const FRestrict inLocal, const CellClass* inInteractions[], const int neighborPositions[], const int inSize, const int inLevel) override { // To copy the multipole data allocated once FComplex source_w[SizeArray]; // For all children for(int idxExistingNeigh = 0 ; idxExistingNeigh < inSize ; ++idxExistingNeigh){ const int idxNeigh = neighborPositions[idxExistingNeigh]; // if interaction exits const FReal*const coef = M2LTranslationCoef[inLevel][idxNeigh]; // Copy multipole data into buffer FMemUtils::copyall(source_w, inInteractions[idxExistingNeigh]->getMultipole(), SizeArray); // Rotate RotationZVectorsMul(source_w,rotationM2LExpMinusImPhi[idxNeigh]); RotationYWithDlmk(source_w,DlmkCoefM2LOTheta[idxNeigh]); // Transfer to u FComplex target_u[SizeArray]; int index_lm = 0; for(int l = 0 ; l <= P ; ++l ){ FReal minus_1_pow_m = 1.0; for(int m = 0 ; m <= l ; ++m, ++index_lm ){ // u{l,m}(a-b) = sum(j=|m|:P-l, (j+l)!/b^(j+l+1) w{j,-m}(a) FReal u_lm_real = 0.0; FReal u_lm_imag = 0.0; int index_jl = m + l; // get j+l int index_jm = atLm(m,m); // get atLm(l,m) for(int j = m ; j <= P-l ; ++j, ++index_jl, index_jm += j ){ // coef = (j+l)!/b^(j+l+1) // because {l,-m} => {l,m} conjugate -1^m with -i u_lm_real += minus_1_pow_m * coef[index_jl] * source_w[index_jm].getReal(); u_lm_imag -= minus_1_pow_m * coef[index_jl] * source_w[index_jm].getImag(); } target_u[index_lm].setRealImag(u_lm_real,u_lm_imag); minus_1_pow_m = -minus_1_pow_m; } } // Rotate it back RotationYWithDlmk(target_u,DlmkCoefM2LMMinusTheta[idxNeigh]); RotationZVectorsMul(target_u,rotationM2LExpMinusImPhi[idxNeigh]); // Sum FMemUtils::addall(inLocal->getLocal(), target_u, SizeArray); } 1122 1123 1124 } /** L2L BRAMAS Berenger committed Aug 07, 2012 1125 1126 1127 * The operator C has been taken from : * Implementation of rotation-based operators for Fast Multipole Method in X10 * At page 5 .1 as the operator C BRAMAS Berenger committed Aug 12, 2012 1128 * \f[ BRAMAS Berenger committed Aug 07, 2012 1129 * M_{l,m}(a-b') = \sum_{j=l}^{\infty}{ \frac{ b^{j-l} }{ (j-l)! } M_{j,m}(a) } , \textrm{j bounded by P} BRAMAS Berenger committed Aug 12, 2012 1130 * \f] BRAMAS Berenger committed Aug 07, 2012 1131 1132 1133 * As describe in the paper, when need first to rotate the SH * then transfer using the formula * and finaly rotate back. 1134 */ BRAMAS Berenger committed Aug 28, 2015 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 void L2L(const CellClass* const FRestrict inLocal, CellClass* FRestrict *const FRestrict inChildren, const int inLevel) override { // Get the translation coef for this level (same for all chidl) const FReal*const coef = L2LTranslationCoef[inLevel]; // To copy the source local allocated once FComplex source_u[SizeArray]; // For all children for(int idxChild = 0 ; idxChild < 8 ; ++idxChild){ // if child exists if(inChildren[idxChild]){ // Copy the local data into the buffer FMemUtils::copyall(source_u, inLocal->getLocal(), SizeArray); // Rotate RotationZVectorsMul(source_u,rotationExpImPhi[idxChild]); RotationYWithDlmk(source_u,DlmkCoefMTheta[idxChild]); // Translate FComplex target_u[SizeArray]; for(int l = 0 ; l <= P ; ++l ){ for(int m = 0 ; m <= l ; ++m ){ // u{l,m}(r-b) = sum(j=0:P, b^(j-l)/(j-l)! u{j,m}(r); FReal u_lm_real = 0.0; FReal u_lm_imag = 0.0; int index_jm = atLm(l,m); // get atLm(j,m) int index_j_minus_l = 0; // get l-j continously for(int j = l ; j <= P ; ++j, ++index_j_minus_l, index_jm += j){ // coef = b^j-l/j-l! u_lm_real += coef[index_j_minus_l] * source_u[index_jm].getReal(); u_lm_imag += coef[index_j_minus_l] * source_u[index_jm].getImag(); } target_u[atLm(l,m)].setRealImag(u_lm_real,u_lm_imag); } } // Rotate RotationYWithDlmk(target_u,DlmkCoefMMinusTheta[idxChild]); RotationZVectorsMul(target_u,rotationExpMinusImPhi[idxChild]); // Sum in child FMemUtils::addall(inChildren[idxChild]->getLocal(), target_u, SizeArray); } } 1177 1178 1179 } /** L2P BRAMAS Berenger committed Aug 12, 2012 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 * Equation are coming from the PhD report of Pierre Fortin. * We have two different computations, one for the potential (end of function) * the other for the forces. * * The potential use the fallowing formula, page 36, formula 2.14 + 1: * \f[ * \Phi = \sum_{j=0}^P{\left( u_{j,0} I_{j,0}(r, \theta, \phi) + \sum_{k=1}^j{2 Re(u_{j,k} I_{j,k}(r, \theta, \phi))} \right)}, * \textrm{since } u_{l,-m} = (-1)^m \overline{ u_{l,m} } * \f] * * The forces are coming form the formulas, page 37, formulas 2.14 + 3: * \f[ * F_r = -\frac{1}{r} \left( \sum_{j=1}^P{j u_{j,0} I_{j,0}(r, \theta, \phi) } + \sum_{k=1}^j{2 j Re(u_{j,k} I_{j,k}(r, \theta, \phi))} \right) * F_{ \theta } = -\frac{1}{r} \left( \sum_{j=0}^P{j u_{j,0} \frac{ \partial I_{j,0}(r, \theta, \phi) }{ \partial \theta } } + \sum_{k=1}^j{2 Re(u_{j,k} \frac{ \partial I_{j,k}(r, \theta, \phi) }{ \partial \theta })} \right) BRAMAS Berenger committed Nov 12, 2012 1194 * F_{ \phi } = -\frac{1}{r sin \phi} \sum_{j=0}^P \sum_{k=1}^j{(-2k) Im(u_{j,k} I_{j,k}(r, \theta, \phi)) } BRAMAS Berenger committed Aug 12, 2012 1195 * \f] 1196 */ BRAMAS Berenger committed Aug 28, 2015 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 void L2P(const CellClass* const inLocal, ContainerClass* const inParticles) override { const FReal i_pow_m[4] = {0, FMath::FPiDiv2(), FMath::FPi(), -FMath::FPiDiv2()}; // Take the local value from the cell const FComplex* FRestrict const u = inLocal->getLocal(); // Copying the position is faster than using cell position const FPoint cellPosition = getLeafCenter(inLocal->getCoordinate()); // For all particles in the leaf box const FReal*const physicalValues = inParticles->getPhysicalValues(); const FReal*const positionsX = inParticles->getPositions()[0]; const FReal*const positionsY = inParticles->getPositions()[1]; const FReal*const positionsZ = inParticles->getPositions()[2]; FReal*const forcesX = inParticles->getForcesX(); FReal*const forcesY = inParticles->getForcesY(); FReal*const forcesZ = inParticles->getForcesZ(); FReal*const potentials = inParticles->getPotentials(); for(FSize idxPart = 0 ; idxPart < inParticles->getNbParticles() ; ++ idxPart){ // L2P const FPoint position(positionsX[idxPart],positionsY[idxPart],positionsZ[idxPart]); const FSpherical sph(position - cellPosition); // The distance between the SH and the particle const FReal r = sph.getR(); // Compute the legendre polynomial FReal legendre[SizeArray]; computeLegendre(legendre, sph.getCosTheta(), sph.getSinTheta()); // pre compute what is used more than once FReal minus_r_pow_l_div_fact_lm[SizeArray]; FReal minus_r_pow_l_legendre_div_fact_lm[SizeArray]; { int index_lm = 0; FReal minus_r_pow_l = 1.0; // To get (-1*r)^l for(int l = 0 ; l <= P ; ++l){ for(int m = 0 ; m <= l ; ++m, ++index_lm){ minus_r_pow_l_div_fact_lm[index_lm] = minus_r_pow_l / factorials[l+m]; minus_r_pow_l_legendre_div_fact_lm[index_lm] = minus_r_pow_l_div_fact_lm[index_lm] * legendre[index_lm]; } minus_r_pow_l *= -r; } } // pre compute what is use more than once FReal cos_m_phi_i_pow_m[P+1]; FReal sin_m_phi_i_pow_m[P+1]; { for(int m = 0 ; m <= P ; ++m){ const FReal m_phi_i_pow_m = FReal(m) * sph.getPhi() + i_pow_m[m & 0x3]; cos_m_phi_i_pow_m[m] = FMath::Cos(m_phi_i_pow_m); sin_m_phi_i_pow_m[m] = FMath::Sin(m_phi_i_pow_m); } } // compute the forces { FReal Fr = 0; FReal FO = 0; FReal Fp = 0; int index_lm = 1; // To get atLm(l,m), warning starts with l = 1 FReal fl = 1.0; // To get "l" as a float for(int l = 1 ; l <= P ; ++l, ++fl){ // first m == 0 { Fr += fl * u[index_lm].getReal() * minus_r_pow_l_legendre_div_fact_lm[index_lm]; } { const FReal coef = minus_r_pow_l_div_fact_lm[index_lm] * (fl * (sph.getCosTheta()*legendre[index_lm] - legendre[index_lm-l]) / sph.getSinTheta()); const FReal dI_real = coef; // F(O) += 2 * Real(L dI/dO) FO += u[index_lm].getReal() * dI_real; } ++index_lm; // then 0 < m for(int m = 1 ; m <= l ; ++m, ++index_lm){ { const FReal coef = minus_r_pow_l_legendre_div_fact_lm[index_lm]; const FReal I_real = coef * cos_m_phi_i_pow_m[m]; const FReal I_imag = coef * sin_m_phi_i_pow_m[m]; // F(r) += 2 x l x Real(LI) Fr += 2 * fl * (u[index_lm].getReal() * I_real - u[index_lm].getImag() * I_imag); // F(p) += -2 x m x Imag(LI) Fp -= 2 * FReal(m) * (u[index_lm].getReal() * I_imag + u[index_lm].getImag() * I_real); } { const FReal legendre_l_minus_1 = (m == l) ? FReal(0.0) : FReal(l+m)*legendre[index_lm-l]; const FReal coef = minus_r_pow_l_div_fact_lm[index_lm] * ((fl * sph.getCosTheta()*legendre[index_lm] - legendre_l_minus_1) / sph.getSinTheta()); const FReal dI_real = coef * cos_m_phi_i_pow_m[m]; const FReal dI_imag = coef * sin_m_phi_i_pow_m[m]; // F(O) += 2 * Real(L dI/dO) FO += FReal(2.0) * (u[index_lm].getReal() * dI_real - u[index_lm].getImag() * dI_imag); } } } // div by r Fr /= sph.getR(); FO /= sph.getR(); Fp /= sph.getR() * sph.getSinTheta(); // copy variable from spherical position const FReal cosPhi = FMath::Cos(sph.getPhi()); const FReal sinPhi = FMath::Sin(sph.getPhi()); const FReal physicalValue = physicalValues[idxPart]; // compute forces const FReal forceX = ( cosPhi * sph.getSinTheta() * Fr + cosPhi * sph.getCosTheta() * FO + (-sinPhi) * Fp) * physicalValue; const FReal forceY = ( sinPhi * sph.getSinTheta() * Fr + sinPhi * sph.getCosTheta() * FO + cosPhi * Fp) * physicalValue; const FReal forceZ = ( sph.getCosTheta() * Fr + (-sph.getSinTheta()) * FO) * physicalValue; // inc particles forces forcesX[idxPart] += forceX; forcesY[idxPart] += forceY; forcesZ[idxPart] += forceZ; } // compute the potential { FReal magnitude = 0; // E = sum( l = 0:P, sum(m = -l:l, u{l,m} )) int index_lm = 0; for(int l = 0 ; l <= P ; ++l ){ {//for m == 0 // (l-|m|)! * P{l,0} / r^(l+1) magnitude += u[index_lm].getReal() * minus_r_pow_l_legendre_div_fact_lm[index_lm]; ++index_lm; } for(int m = 1 ; m <= l ; ++m, ++index_lm ){ const FReal coef = minus_r_pow_l_legendre_div_fact_lm[index_lm]; const FReal I_real = coef * cos_m_phi_i_pow_m[m]; const FReal I_imag = coef * sin_m_phi_i_pow_m[m]; magnitude += FReal(2.0) * ( u[index_lm].getReal() * I_real - u[index_lm].getImag() * I_imag ); } } // inc potential potentials[idxPart] += magnitude; } } 1348 1349 1350 } BRAMAS Berenger committed Aug 07, 2012 1351 /** P2P BRAMAS Berenger committed Aug 12, 2012 1352 1353 1354 1355 * This function proceed the P2P using particlesMutualInteraction * The computation is done for interactions with an index <= 13. * (13 means current leaf (x;y;z) = (0;0;0)). * Calling this method in multi thread should be done carrefully. BRAMAS Berenger committed Aug 07, 2012 1356 */ BRAMAS Berenger committed Aug 31, 2015 1357 void P2P(const FTreeCoordinate& inPosition, BRAMAS Berenger committed Nov 06, 2015 1358 ContainerClass* const FRestrict inTargets, const ContainerClass* const FRestrict inSources, BRAMAS Berenger committed Aug 28, 2015 1359 1360 ContainerClass* const inNeighbors[], const int neighborPositions[], const int inSize) override { BRAMAS Berenger committed Nov 06, 2015 1361 1362 1363 1364 1365 1366 1367 1368 1369 if(inTargets == inSources){ FP2PRT::template Inner(inTargets); P2POuter(inPosition, inTargets, inNeighbors, neighborPositions, inSize); } else{ const ContainerClass* const srcPtr[1] = {inSources}; FP2PRT::template FullRemote(inTargets,srcPtr,1); FP2PRT::template FullRemote(inTargets,inNeighbors,inSize); } BRAMAS Berenger committed Aug 31, 2015 1370 1371 1372 1373 1374 1375 } void P2POuter(const FTreeCoordinate& /*inLeafPosition*/, ContainerClass* const FRestrict inTargets, ContainerClass* const inNeighbors[], const int neighborPositions[], const int inSize) override { BRAMAS Berenger committed Aug 28, 2015 1376 1377 1378 1379 1380 1381 int nbNeighborsToCompute = 0; while(nbNeighborsToCompute < inSize && neighborPositions[nbNeighborsToCompute] < 14){ nbNeighborsToCompute += 1; } FP2PRT::template FullMutual(inTargets,inNeighbors,nbNeighborsToCompute); 1382 1383 1384 } BRAMAS Berenger committed Aug 07, 2012 1385 /** Use mutual even if it not useful and call particlesMutualInteraction */ 1386 void P2PRemote(const FTreeCoordinate& /*inPosition*/, BRAMAS Berenger committed Aug 28, 2015 1387 ContainerClass* const FRestrict inTargets, const ContainerClass* const FRestrict /*inSources*/, BRAMAS Berenger committed Nov 06, 2015 1388 const ContainerClass* const inNeighbors[], const int neighborPositions[], BRAMAS Berenger committed Aug 28, 2015 1389 1390 const int inSize) override { FP2PRT::template FullRemote(inTargets,inNeighbors,inSize); 1391 } 1392 1393 1394 1395 }; #endif // FROTATIONKERNEL_HPP