Class 21. N -body Techniques, Part 4

Transcription

1 Class. N -body Techniques, Pat Tee Codes Efficiency can be inceased by gouping paticles togethe: Neaest paticles exet geatest foces diect summation. Distant paticles exet smallest foces teat in goups. Teat distant paticles as one lage paticle of equivalent mass Evaluate foces diectly fo neaby paticles. But how do we oganize paticles into goups? Will sketch one method (Banes & Hut 986, Natue, 6; also see Henquist 987, ApJS 6, 7), then go into moe detail. Banes & Hut method: Oveview The BH method is a hieachical foce-calculation algoithm: Place paticles on mesh one at a time. Divide mesh into equal volume subdomains at each placement so that each paticle occupies a single subdomain. E.g., in -D: paticle paticles paticles paticles paticles 6 paticles 7 paticles 8 paticles

2 Now, oganize paticles based on nesting of subdomains: ROOT Level node banch Level 8 Level 6 7 leaf Level How does this speed up foce evaluation? Conside evaluation of foce on paticle : If any subdomain subtends an angle θ = l/d < θ cit as seen fom paticle (l is size of subdomain, d is distance fom paticle ), then teat all paticles in that subdomain as one. E.g., Paticle, 8: teat diectly. Top-left subdomain: teat as goup. = just summations, instead of 7. Banes & Hut method: Details Cost of tee build depends on equied depth (numbe of levels). Fo homogeneous paticle distibution (i.e., no cells empty), tee depth +log k N. Fok =,depth +logn. time equied to constuct tee O(N log N). Must also compute total mass and cente-of-mass position = one moe O(N log N) pass though tee. Finally, foce evaluation ( puning ) = k sums pe paticle at each level = O(log N) sums pe paticle (depends on θ cit )= O(N log N) scaling N fo N. How bad an appoximation is it? Conside expanding potential of cell (e.g., Maion & Heald 980, pp. 8 0; this comes fom Taylo seies expansion of potential nea oigin): Φ = Gm ( ) + Gm x,i Gm ( ) x x i,ix,j + x i x j i i,j

3 Mass point m (x ),i Field point P (x i ) Oigin of expansion (e.g. cente of mass) so whee Φ= Φ =Φ () +Φ () +Φ () + +Φ (l) + Φ () Gm = GM Φ () Gm x,i x i i Φ () Gm Φ (l ) ( )(l+) l! i,j is the monopole, ( ) x,ix,j i,j,...,l is the dipole, x i x j ( ) is the quadupole, Gm x,ix,j x l,l x i x j x l ( ) is the l -pole. If we choose expansion cente to be cente of mass of goup, then m =0. But then notice that Φ () = Gm (/) = 0, so dipole vanishes. eo tem dominated by quadupole. (Can also wite whee Φ= GM G (Q), Q ij = m k (x k,i x k,j kδ ij ) k is the taceless quadupole tenso, k is ove the mass components, and k is elative to the cell cente of mass. With this notation, and invoking the paallel axis theoem, the quadupole of a paent cell can be constucted via the quadupoles of its daughte cells: Q = i Q i + i m i( i i i ), whee i is ove the daughte cells and i is elative to the paent cente of mass.) Often, quadupole not needed (monopole is good enough ). With quadupole, fo θ cit =,focestypicallyaccuateto % (in pactice, keep θ cit < / =0.7 fo-dtee,< / =0.6 fo-dtee). Thisisaveage eo; cetain pathological configuations can give much lage eos. Also, tees in geneal beak F ij = F ji...

4 Fo high pecision, might conside octopole. Tuns out the octopole does not help convegence much need to go to next highe ode, the hexadecapole! Obviously this means many moe computations to compute foce (still scales as O(N log N)), but can use lage θ cit. On balance, pobably neve need bette than hexadecapole. Banes & Hut method: Pseudocode Define a node stuct: contains size, cente, mass, position, Q, etc. of cell, plus info on subcells (may be nodes). Following example stoes only monopole (i.e., total mass). Tee build stat with special cell ( oot ) stat oot = new node [includes initialization] loop ove paticles i put_in_tee(i,oot) calc_moments(oot) function put_in_tee(paticle,node) to which (sub)cell does paticle belong? is cell......empty? : make paticle a leaf in cell beak...a leaf? : make cell a node cell = new node put_in_tee(leaf,cell)...a node? : put_in_tee(paticle,cell) function calc_moments(node) [loop ove non-empty (sub)cells is cell......a leaf? : node->mass += cell.leaf->mass node->pos += (cell.leaf->mass)*(cell.leaf->pos) beak...a node? : calc_moments(cell.node) node->mass += cell.node->mass node->pos += (cell.node->mass)*(cell.node->pos) ] node->pos /= node->mass

5 Tee walk stat at oot function add_to_foce(pos,node,foce) theta = (node->size)/(distance to node) theta < theta_cit? : multipole_expansion(pos,node,foce) ["pune"] else : [loop ove non-empty (sub)cells is cell......a leaf? : diect_foce(pos,cell.leaf,foce) beak...a node? : add_to_foce(pos,cell.node,foce) ] Othe Types of Tees Diffe pimaily in oganization of paticle infomation. Mutually neaest neighbou E.g., Appel 98, Jenigan 98, Pote 98. Given N paticles, two neaest joined togethe node, leaving N entities(n paticles plus node) in list. Node contains total mass and cente-of-mass position of cluste. Repeat until only cluste emains. O(log N) levels (binay tee), O(N log N) updatetime. Advantage: Peseves physical poximity of paticles (binaies). Can also let paticles dift a while befoe update. Disadvantage: Abitay node shapes, had to estimate eo when expanding potentials. k-d tee (ecusively bisect longest dimension) E.g., Olson & Packe 996. Fist detemine dimension (x, y, o z) that spans lagest spatial ange of paticle distibution.

6 Sot data on this dimension and divide into halves containing equal numbes of paticles. Repeat with sublists until each contains only paticle. Often used fo domain decomposition to balance wok between multiple pocessos. Advantage: No empty cells, moe efficient shape. Disadvantage: Exteme oblong shapes lage eo. Fast Multipole Method Impoved tee walking/puning. In pinciple can achieve O(N) scaling, and momentum consevation (!), but complex implementation. Idea is that local infomation is passed up the tee so it can be swapped with distant nodes: mutual multipole expansion (postal sevice analogy). Cutting edge of tee code development, much of it done hee at U Mayland (Compute Science)! Summay PP method (diect summation) most accuate, but O(N ). PM method O(N g log N g ), but esolution limited. Tee codes O(N log N), but sometimes difficult to implement. Also: PP-PM = P M diectsummationoveneabypaticles,usegidfodistant inteactions. 6