The modeling week is a project week in which high school students work together with teachers and scientific support on real problems from the research of companies or university institutes.
Some examples are the following problems:
Each group is given an individual task to work on during the five days. They are always supervised by a scientist. At the end of the week, the student teams present their results to company representatives at a representative closing event in Aachen or Karlsruhe. The families of the participants and representatives of the schools are cordially invited to attend. The week also includes a supporting program with a sports competition, a barbecue and other leisure activities, as well as a study information event.
For the purpose of in-situ exploration of thick ice layers both on Earth and on the icy moons of our solar system, e.g. Europa, fusion robots are being developed. One advantage of this technology is that it requires extremely low maintenance and can also be operated well autonomously. The fusion robots, also called cryo-robots, can be individually loaded with scientific payloads (i.e., equipped with certain measuring instruments) melt through the ice. However, work is also underway to send an entire smart grid of smaller melting probes into the ice. Each of the elements of this smart grid, also called pingers or probes, can melt into the ice in a directionally stable manner, i.e., along a straight trajectory that is not necessarily vertically oriented. Furthermore, each pinger is equipped with an acoustic emitter and an acoustic receiver, so that acoustic so that acoustic waves can be transmitted and received. There are large pingers (cylinders of length 1m with a diameter of 20cm) which can emit an acoustic signal up to a distance of 3km and small pingers (cylinders of length 1m with a diameter of 15cm) which can emit an acoustic signal up to a distance of 2km. The individual 'wave beams' of the signal cover a circular cross-section with approximate diameter 2*pinger diameter. Only one signal packet can be transmitted at a time.
The pingers are synchronized in time, so that the travel times of the waves between the pingers can be measured and stored (we assume here that the challenge of data communication is solved ...). By means of a so-called time-of-flight tomography, it is thus possible to infer local heterogeneities in the ice, for example due to density variations, or salt, or water deposition. Here, however, we are not concerned with the time-of-flight tomography itself, but with the ice volume covered by the pinger-to-pinger beams. The pinger is in constant sinking motion. In terrestrial applications, the necessary melting power comes from a generator that sets a nominal sinking rate and can be turned off. For extraterrestrial applications, RTGs (so-called Radio Thermal Generators) are planned, in which radioactive material decays according to a certain decay curve and produces heat (continuously). Candidate materials are curium, uranium or plutonium (see data sheet). The goal is to control and send the pingers (or melting probes) of the RTGS, arranged in a smart grid, into the ice in such a way that the investigated ice volume can be maximized.
Your Task
In order to calculate the scanned volume with the now derived formulas, I was mainly responsible for developing an algorithm for this in Python. With my previous knowledge in Java and some help from the supervisor in the beginning, I was able to develop it.
import numpy as np import scipy.integrate as integrate from matplotlib import pyplot as plt
s = 20000 # thickness of ice in m T = 200 # difference of temperature r = 0.1 # radius of big pinger t = 10*365*86400 # 10 years in seconds l = 18.1*365*86400 # 18.1 years in seconds A = np.pi*r*r # Surface of pinger at front a=3
class Pinger: x = [] # X values for plotting xCord = 0 # x-axis positon down = 0 # how far the pinger goes into the ice (in km) def __init__(self, px, pdown): self.x.append(px) self.xCord = px self.down = pdown # Each functions are explained in the report @staticmethod def calcNeededPower(x): l = 18.1*365*86400 #18.1 years in seconds return (((2-x/s)*A*x*920*2100*T/2+A*x*920*335000)*np.power(np.e,(np.log(2)/l)*t)) /(t) @staticmethod def length(p0): p = s*(1+335000/(T*2100)) length = -p + np.sqrt((p*p)-(p0*np.power(np.e,(-np.log(2)/l)*t)*t*2*s)/(A*920*T*2100)) return length @staticmethod def volumen(a1, x1, x2): d = np.sqrt((x1-x2)*(x1-x2)+a1*a1) return a1*((x1+x2)/2)*4*r+(np.pi*2*r*r*d)
def totalVolume(pingers): tempVolume = 0 # will be returned length = len(pingers)-1 # array size of pingers for i in range(length): # iterate through all pingers currentPinger = pingers[i]; if i == length: break # No nextPinger possible -> no volume else: nextPinger = pingers[i+1]; # next pinger in array (index+1) m1 = currentPinger.calcNeededPower(currentPinger.down) # power for pinger1 m2 = nextPinger.calcNeededPower(nextPinger.down) # power for pinger2 l1 = currentPinger.length(m1) # depth for pinger1 l2 = nextPinger.length(m2) # depth for pinger1 x1 = currentPinger.xCord # x coordinate for pinger1 x2 = nextPinger.xCord # x coordinate for pinger1 a1 = 0 # calc delta x of pingers if x1 < x2: a1 = x2-x1 else: a1 = x1-x2 v = volumenWithLimitation(l1, l2, x1, x2) # volume between current and next pinger tempVolume = tempVolume+v # added to the other volumes plt.plot([currentPinger.xCord, currentPinger.xCord], [0, l1], 'o:r') # plot without realspace # Real area between vx = [currentPinger.xCord, nextPinger.xCord, nextPinger.xCord, currentPinger.xCord] vy = [-currentPinger.down, -nextPinger.down, 0, 0] plt.fill(vx, vy, 'o:r', alpha=0.1) if i == length-1: # draw last red area plt.plot([nextPinger.xCord, nextPinger.xCord], [0, l2], 'o:r') return tempVolume*1000 # added volume
def volumenWithLimitation(y1, y2, x1, x2): # days until > 3km m1 = calcNeededPower(y1) m2 = calcNeededPower(y2) # day = int(t/86400) # t = 15years in sec -> diveded by sec a day volume = 0 count = t/86400 for i in range(int(count)): y1 = lengthT(m1, i*86400) y2 = lengthT(m2, i*86400) delta = abs(y2-y1) space = abs(x2-x1) realspace = np.sqrt(space*space + delta*delta) if(realspace > 3): return volumen(x1, x2, space, abs(y1), abs(y2)) #abs because crazy issue of wed. return volumen(x1, x2, space, abs(y1), abs(y2))
# Config of figure figure, axes = plt.subplots() axes.set_ylabel('y in km') axes.set_xlabel('x in km') figure.set_size_inches(10, 10) figure.savefig('test.png', dpi=100) plt.xlim(-10, 10) plt.ylim(-30, 20) # Circle for Europa circle1 = plt.Circle((0, -1500), 1500, color='#A5E1F7', fill=True, label = "Europa (Moon)") plt.gca().add_patch(circle1) # Circle for Ice circle2 = plt.Circle((-0, -1500), 1480, color = '#1978CB', fill=True, label = "Ocean") plt.gca().add_patch(circle2) pingers = [] # Adding of the pingers pingers.append(Pinger(-6, 10)) pingers.append(Pinger(-4.8, 5)) pingers.append(Pinger(-2.8, 17)) pingers.append(Pinger(0, 18)) pingers.append(Pinger(3.8, 11.5)) pingers.append(Pinger(4.8, 12)) pingers.append(Pinger(6.8, 13)) pingers.append(Pinger(8, 16)) vGes = totalVolume(pingers) # Figure for data table fig, ax = plt.subplots(1,1) fig.set_size_inches(10, 10) fig.savefig('table.png', dpi=200) data=[] # Adding values in array c=0 for p in pingers: c+=1 ppow = p.calcNeededPower(p.down) plen = p.length(ppow) data.append([c, abs(plen), round(ppow/2.5*1000, 2)]) # Configure first row data.append(["", "", ""]) totalV = str(round(vGes,2)) + "m3" data.append(["Total Volume:", totalV, ""]) column_labels=["Pinger n", "Depth (km)", "Curium (g)"] ax.axis('off') table = ax.table(cellText=data,colLabels=column_labels,loc="center") table.set_fontsize(25) cellDict = table.get_celld() # Changing size of each cell for i in range(0,len(column_labels)): cellDict[(0,i)].set_height(.1) for j in range(1,len(pingers)+3): cellDict[(j,i)].set_height(.05) print(vGes) plt.show()
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