\n", "\\begin{align}\n", "\\rm x_{i + 1} & = \\rm x_i + v_i\\Delta t + \\frac{1}{2}a_i\\Delta t^2 \\\\ \n", "\\rm v_{i + 1} & = \\rm v_i + \\frac{1}{2}\\left(a_i + a_{i + 1}\\right)\\Delta t\n", "\\end{align}\n", "

\n", "\n", "where \n", "\n", "

\n", "\\begin{align}\n", "\\rm a_i & = \\rm \\frac{-G \\cdot m_n}{r_n^3}\\cdot \\vec{r}\n", "\\end{align}\n", "

\n", "\n", "and $\\rm v_i$ and $\\rm x_i$ are velocity and position. \n", "\n", "It is 2nd order integrator, though higher order forms exist. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# Set the seed for reproducibility\n", "np.random.seed(5)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "def acceleration(obj, object_array):\n", " # The object array is constructed in the form of: [x, y, z, vx, vy, vz, r, m]\n", " # Where r is the radius of the particle. \n", " G = 6.67408*(10**-11) # Gravitational constant\n", " \n", " dx = obj[0] - object_array[:,0]\n", " dy = obj[1] - object_array[:,1]\n", " dz = obj[2] - object_array[:,2]\n", " \n", " dr = np.array([dx, dy, dz])\n", " dist = np.linalg.norm(dr, axis = 0)\n", " \n", " a = -G*np.sum(object_array[:,7]*dr/dist**3, axis = 1)\n", " \n", " return a\n", "\n", "def pos_check(object_array, max_dist):\n", " # Check how far away an object is\n", " good_dist = np.where(np.all(object_array[:,:3]**2 < max_dist**2, axis = 1))[0]\n", " \n", " object_array = object_array[good_dist]\n", " \n", " return object_array\n", " \n", "def inelastic(object_array):\n", " # Radius of our particle is in obj[6]\n", " \n", " agg_ind_list = []\n", " full_ind_set = set(range(len(object_array)))\n", " \n", " for ii in range(len(object_array)):\n", " dist = np.linalg.norm(object_array[ii, :3] - object_array[:,:3], axis = 1)\n", " \n", " close = np.where((dist < object_array[ii, 6]) & (dist != 0))[0]\n", " \n", " if len(close) > 0:\n", " agg_ind = [ii]\n", " agg_ind.extend(close)\n", " agg_ind_list.append(agg_ind)\n", " \n", " if len(agg_ind_list) > 0: \n", " agg_sorted = sorted(agg_ind_list, key = len, reverse = True)\n", " agg_done = []\n", " new_entries = []\n", " \n", " for agg in agg_sorted:\n", " current = [*agg]\n", " if not np.all(np.in1d(current, agg_done)):\n", " new_mass = np.sum(object_array[agg, 7]) #new mass\n", " new_rad = np.sum(object_array[agg, 6]**3)**(1/3) #new radius\n", " new_pos = list(np.sum(object_array[agg, :3] * object_array[agg, 7].reshape(len(agg), 1), axis = 0)/np.sum(object_array[agg, 7])) #new position\n", " \n", " new_vel = list(np.sum(object_array[agg, 3:6] * object_array[agg, 7].reshape(len(agg), 1), axis = 0)/(np.sum(object_array[agg, 7]))) #new velocity\n", " new_entry = np.array([*new_pos, *new_vel, new_rad, new_mass])\n", " new_entries.append(new_entry)\n", " agg_done.extend(agg)\n", " \n", " new_ind_list = list(full_ind_set - set(agg_done))\n", " new_array = np.vstack([object_array[new_ind_list], new_entries])\n", " \n", " return new_array\n", " \n", " else:\n", " return object_array\n", "\n", "def leapfrog(object_arr, del_t):\n", " \n", "# object_arr = pos_check(object_arr, 100000)\n", " \n", " object_array = inelastic(object_arr)\n", " \n", " for obj in range(len(object_array)):\n", " obj_list = list(object_array)\n", " obj_i = obj_list.pop(obj)\n", " obj_arr = np.array(obj_list)\n", " \n", " a_1 = acceleration(obj_i, obj_arr)\n", " \n", " obj_i[:3] += obj_i[3:6]*del_t + 0.5*a_1*del_t**2\n", " \n", " a_2 = acceleration(obj_i, obj_arr)\n", " \n", " obj_i[3:6] += 0.5*(a_1 + a_2)*del_t\n", " \n", " object_array[obj] = obj_i\n", " \n", " return object_array" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# Produce uniform sphere with 1000 points\n", "max_n = 1000\n", "radius = 250\n", "n_points = 0\n", "ang_rot = True\n", "x_list = []\n", "y_list = []\n", "z_list = []\n", "m_list = []\n", "r_list = []\n", "init_vx = np.zeros(max_n)\n", "init_vy = np.zeros(max_n)\n", "init_vz = np.zeros(max_n)\n", "while n_points < max_n:\n", " rand_x = np.random.randint(-radius, radius)\n", " rand_y = np.random.randint(-radius, radius)\n", " rand_z = np.random.randint(-radius, radius)\n", " rand_m = np.random.randint(1, high = 5)\n", " rand_r = 1 + np.random.rand()\n", " dist = np.linalg.norm([rand_x, rand_y, rand_z])\n", " if dist > radius:\n", " continue\n", " else:\n", " x_list.append(rand_x)\n", " y_list.append(rand_y)\n", " z_list.append(rand_z)\n", " m_list.append(rand_m)\n", " r_list.append(rand_r)\n", " n_points += 1 \n", " \n", "xyzm_array = np.transpose(np.vstack([x_list, y_list, z_list, init_vx, init_vy, init_vz, r_list, m_list]))\n", "\n", "if ang_rot: \n", " com = np.sum(xyzm_array[:,:3].transpose()*xyzm_array[:,7], axis = 1)/np.sum(xyzm_array[:,7])\n", " for obj in range(len(xyzm_array)):\n", " obj_list = list(xyzm_array)\n", " obj_i = obj_list.pop(obj)\n", " obj_arr = np.array(obj_list)\n", " \n", " a_1 = acceleration(obj_i, obj_arr)\n", " \n", " force = np.linalg.norm(a_1*obj_i[7])\n", " mu = np.sum(1/(1/xyzm_array[:,7]))\n", " r_cm = np.linalg.norm(com - xyzm_array[3,:3])\n", " \n", " tz = np.arctan2(obj_i[0], obj_i[1])\n", " tx = np.arctan2(obj_i[2], obj_i[1])\n", " \n", " cz = np.cos(-tz)\n", " sz = np.sin(-tz)\n", " cx = np.cos(-tx)\n", " sx = np.sin(-tx)\n", " \n", " rz = np.array([[cz, -sz, 0], [sz, cz, 0], [0, 0, 1]])\n", " rx = np.array([[1, 0, 0], [0, cx, -sx], [0, sx, cx]])\n", " \n", " v = np.array([np.sqrt(force*r_cm/mu), 0, 0])\n", " \n", " # Rotate v into particle\n", " v = rz@v\n", " v = rx@v\n", " \n", " xyzm_array[obj, 3] = v[0]\n", " xyzm_array[obj, 4] = v[1]\n", " xyzm_array[obj, 5] = v[2]" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Frame 6000 done in 1040.33 seconds, array length: 6...\r" ] } ], "source": [ "# Try to set up fading trails of each object\n", "trail_x = []\n", "trail_y = []\n", "trail_z = []\n", "trail_vx = []\n", "trail_vy = []\n", "trail_vz = []\n", "trail_m = []\n", "trail_r = []\n", "\n", "t1 = time.time()\n", "\n", "frames = 6000\n", "\n", "for iteration in range(frames):\n", " \n", " xyzm_array = leapfrog(xyzm_array, 2**12)\n", "\n", " trail_x.append([*xyzm_array[:,0]])\n", " trail_y.append([*xyzm_array[:,1]])\n", " trail_z.append([*xyzm_array[:,2]])\n", " trail_vx.append([*xyzm_array[:,3]])\n", " trail_vy.append([*xyzm_array[:,4]])\n", " trail_vz.append([*xyzm_array[:,5]])\n", " trail_r.append([*xyzm_array[:,6]])\n", " trail_m.append([*xyzm_array[:,7]])\n", " \n", " t2 = time.time()\n", " \n", " print('Frame {0} done in {1:.2f} seconds, array length: {2}.'.format(iteration + 1, t2 - t1, len(xyzm_array)), end = '\\r')" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ -465.71443016, 1404.57680483, 744.52356769])" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# trail_x, trail_y, trail_z, trail_m\n", "com = np.sum(np.array([trail_x[-1], trail_y[-1], trail_z[-1]])*np.array(trail_m[-1]), axis = 1)/np.sum(np.array(trail_m[-1]))\n", "com" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Frame 6000 of 6000 done in 798.33 seconds, array length: 6...\r" ] }, { "data": { "image/png": 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