Problems proposed by Robert Bosch Company at BEEs01/ESGI123
Two different interesting problems were proposed by Robert Bosch Company. Both of them were aimed at obtaining analytical solutions. According to the results that teams showed at their final presentations Robert Bosch Company made contracts with the members of the groups to continue solving the problems.
Overlap of spherical harmonics
Attending at BEEs01/ESGI123 workshop that held at Peter the Great St. Petersburg Polytechnic University in Russia, we faced to the problem “Overlap of spherical harmonics” and now we would like to share our impressions. The main goal of the problem was to identify whether spherical harmonics (as in the picture below) intersect or not and if they do intersect to find the value of the intersection.
Such problem arises while modeling powder-based mediums or particles in fluids and has been posed us by Robert BOSH company. The most common approach is to consider particles as spheres or spherecylinders. There is analytical expression for such approximations that gives the answer for the main question: do the particles intersect, or not? However the accuracy of these approaches is low when particles have complex shape. That’s why BOSH company started thinking about decomposition of particles shapes in spherical functions and ask us to find an analytical expression for its intersection or other possibilities to increase accuracy of shapes approximations.
Our German-Russian team attacked the problem and very fast we managed to find out that today there is no analytical expression of intersection even for more simple convex shapes such as ellipsoids. So, we focused on the other variants of problem solution such as numerical algorithms and approximations of particles by several intersecting spheres.
This short cooperation with industry was interesting experience for all of us. Even though one week is quite narrow time frame to receive complete solution of the problem, this is great opportunity to apply academic knowledge to real life problem as well as to communicate with specialists from allied sciences.
Potential of a vortex
My name is Artem Eliseev, I’m a student of the Peter the Great Saint-Petersburg Polytechnic University. I was one of the participants of BEEs01/ESGI123 conference, which was held in our university in October 2016. I was invited to participate in this conference along with some of my groupmates because all of us had been interested in solving the mathematical problems connected with industry.
At the first day of the conference several problems proposed by different industrial companies were presented. One of them was connected with hydrodynamics, which is a subject of my field of interest, so I decided to take part in solving this exact problem. I was not alone in my decision because my groupmates Danila and Vasily also found this project suitable for them. The last but obviously not least member of our team was Prof. Thomas Götz from the University of Koblenz, Germany.
Our task came from the Robert Bosch Company and consisted in finding analytically scalar potential of a specified function. It was formulated as ‘In order to calculate pressure in vortex-based CFD methods using Cauchy-Lagrange integral, one must know velocity potential, created by vortices at the surface of a body. For a triangular vortex frame, an analytical expression of the potential is known and has relatively simple form. Use of frame-based potentials only severely limits accessible vortex configurations. The problem can be solved by using potential of a single vortex. According to claims in literature, it can be expressed through elementary functions, but it is very complicated and could not be found in existing publications.’
This task rises while using the discrete vortex method in order to solve Navier-Stokes equations. While solving this problem we found very good contact with Prof. Götz and that helped us to unite efforts from all members of our team. Below you can see how many equations and formulas we wrote during the conference.
At the last day we presented results of our work to the other members of the conference and to the company representative, Timofey Kruglov. After the conversation Timofey found our results fine and offered me, Danila and Vasily to continue work on this problem after the conference and we accepted his offer.
To sum up, I want to thank the organizers of this conference for the given opportunity to establish contacts with specialists of applied mathematics from all over the Europe and solve tasks connected with real industrial problems.