1. Barnes-Hut Algorithm (TREECODE) The BarnesHut simulation (Josh Barnes and Piet Hut) is an algorithm for performing an n-body simulation. It is notable for having order O(n log n) compared to a direct-sum algorithm which would be O(n2). Berkembang dan sudah banyak digunakan dikasus N-body dan SPH karena bisa di paralelisasi. NOTE

2. The crucial idea in speeding up the brute force algorithm is to group nearby bodies and approximate them as a single body. If the group is sufficiently far away, we can approximate its gravitational effects by using its center of mass. If two bodies have positions (x1 , y1) and (x2, y2), and masses m1 and m2, center of mass (x, y) are given by: m = m1 + m2 x = (x1.m1 + x2.m2) / m y = (y1.m1 + y2.m2) / m It recursively divides the set of bodies into groups by storing them in a quad-tree (2D) or Octree (3D), recursively subdivided into quadrants until each subdivision contains 0 or 1 bodies 3. To calculate the net force on a particular body, traverse the nodes of the tree, starting from the root. If the center-of-mass of an internal node is sufficiently far from the body, approximate the bodies contained in that part of the tree as a single body, whose position is the groups center of mass and whose mass is the groups total mass. The algorithm is fast because we dont need to individually examine any of the bodies in the group. If the internal node is not sufficiently far from the body, recursively traverse each of its subtrees. To determine if a node is sufficiently far away, compute the quotient s / d, s = width of the region represented by the internal node, d = the distance between the body and the nodes center-of-mass. Then, compare this ratio against a threshold value . If s / d < , then the internal node is sufficiently far away. By adjusting the parameter, we can change the speed and accuracy of the simulation. We will always use = 0.5, a value commonly used in practice. Note that if = 0, then no internal node is treated as a single body, and the algorithm degenerates to brute force. 4. CONSTRUCTING THE BARNES-HUT TREE To construct the Barnes-Hut tree, insert the bodies one after another. To insert a body b into the tree rooted at node x, use the following recursive procedure: 1. If node x does not contain a body, put the new body b here. 2. If node x is an internal node, update the center-of-mass and total mass of x. Recursively insert the body b in the appropriate quadrant. 3. If node x is an external node, say containing a body named c, then there are two bodies b and c in the same region. Subdivide the region further by creating four children. Then, recursively insert both b and c into the appropriate quadrant(s). Since b and c may still end up in the same quadrant, there may be several subdivisions during a single insertion. Finally, update the center-of-mass and total mass of x. 5. Example: 6. Every internal node contain the center of mass and total mass of their bodies. 7. CALCULATING THE FORCE ACTING ON A BODY To calculate the net force acting on body b, use the following recursive procedure, starting with the root of the quad-tree: 1. If the current node is an external node (and it is not body b), calculate the force exerted by the current node on b, and add this amount to bs net force. 2. Otherwise, calculate the ratio s/d. If s/d < , treat this internal node as a single body, and calculate the force it exerts on body b, and add this amount to bs net force. 3. Otherwise, run the procedure recursively on each of the current nodes children. 8. As an example, to calculate the net force acting on body a, we start at the root node, which is an internal node. It represents the center-of-mass of the six bodies a, b, c, d, e, and f, which have masses 1, 2, 3, 4, 5, and 6 kg, respectively. 9. Jika A terlalu jauh dari pusat massa sistem total (kriteria opening angle terpenuhi) A akan di anggap terkena gaya dari sistem itu sendiri (tidak perlu lagi menghitung per node internal). 10. Calculate Quadrapole moment (for b, c and d) if it is included. 11. Referensi: http://www.cs.princeton.edu/courses/archive/fall03/cs126/assignments/barnes-hut.html http://arborjs.org/docs/barnes-hut http://www.ifa.hawaii.edu/~barnes/treecode/treeguide.html http://en.wikipedia.org/wiki/Barnes%E2%80%93Hut_simulation