Space calls: we need a specialist mathematician in the field of numerical solution of stochastic differential equations

    Alexander 4110 Shaenko (ex-engineer Dauria Aerospace, now the leader of the Mayak crowdsourcing satellite project) and Stepan Tezyunichev are writing open source software for modeling the thermal conditions of satellites.

    The repository is here .



    Prior to this, Sasha wrote a disser - “A method for solving the problem of radiant heat transfer without a matrix of angular coefficients” ( dissertation , abstract ). The code is here . (it’s on VB.NET, it’s braking, but it even works and there is documentation)

    Cosmoengineers use the Monte Carlo method and it suddenly dawned on them, apparently, the system of differential equations is stochastic, since some of them are obtained by random testing.

    The guys want to write an algorithm that would solve the system of diffours as quickly as possible, and therefore they need the advice of a specialist mathematician in the field of numerical solution of stochastic differential equations.

    The guys seemed to solve the main difficulty by constructing a massive-parallel algorithm for calculating the radiation course using the Monte Carlo method on CUDA. Now they want to use their system for integration, and it is of a large dimension, of the order of 100 thousand unknowns, and a tough, suitable method of integration over time. Conventional explicit methods require too small a time step, while implicit methods require many times to calculate the right side, which is resource-intensive.

    Therefore, it is necessary to find or build such an integration method that would be suitable for such a combination of requirements and would allow to calculate quickly.

    What is it for?


    Now in Russia and in the world as a whole there is no open software that would be sharpened for the calculation of satellite heat transfer. Those commercial ones that exist cost tens and hundreds of thousands of dollars and are not available to ordinary people. There are, of course, open codes just for calculating heat transfer, but in order to fully count on them any real satellite, you need to pretty dance.

    You will say that for ordinary people they are not needed, but this is only as long as you do not want to create your own small company to create satellites. Most domestic private space companies do not have money for such software, and many foreign companies do not.

    And if you are a student who designs his own satellite on a course or graduation project? The trial version for 30 days may not be enough.

    And if you want to understand the radiant heat transfer? Or use a good implementation of the ray tracing method for other applications?

    In a word, there will be many applications, and not only space ones.

    THORIUM


    The idea of ​​THORIUM was born at Shaenko back in 2007 at the IASS conference in Venice and developed in reflections on the automation of calculating the strength of interceptors for a 787 passenger aircraft in the Moscow branch of Boeing.


    " Radioastron "

    At that time, Alexander was a graduate student at the Astro Space Center of the Lebedev Physical Institute. Lebedev. There, at that time, they finished making the Radio Astron space telescope and thought about the Millimetron.

    image

    " Millimetron "

    In "Millimetron" had a few years to support a large 10-meter mirror at about 4 K (-269 C). To do this, it is planned to place the mirror in the shadow of a multilayer umbrella made of a polymer metallized film, which would protect delicate equipment from the thermal radiation of the Sun and the Earth. The umbrella was called the “heat shield”. Exactly the same is with the American competitor JWST, the Space Telescope James Webb.

    The real form of the heat shield is crumpled twenty-meter film panels with a small distance between them. This means that in a good way it is necessary to simulate each more or less large fold in order to understand how heat will be distributed between them. In addition, different layers have very different temperatures - from about +100 C in the layer illuminated by the Sun to -223 C in the layer facing the mirror. With this scatter, the properties of the materials changed very much.

    The existing settlement codes did not allow to take all this into account, and therefore the THORIUM - Thermooptic radiation iterative universal module was born. It is based on two ideas that arose at the end of 2007:

    1. To calculate the radiation path in three-dimensional graphics, a lot of software and hardware have been developed, and since outer heat exchange is radiant in space , you can use the operating time from the graphics in the calculation of heat transfer.

    2. If you want to calculate complex structures with a large number of elements in the model, then you should not remember which elements exchange radiation with which, before starting the calculation. It is more profitable to watch this at every step of integration over time.

    These two principles were implemented in both the old code on VB.NET and the new one. Millimetron was saved, and THORIUM began to live its own life, helping to design thermal management systems (COTS) for small devices. For example, COTR DX1 and MKA-N, small devices of Dauria were designed with its help.

    Vidos of the old THORIUM with Daurievsky MKA-N




    DX1 in THORIUM

    From the point of view of mathematics, THORIUM solves the system of differential equations of non-stationary radiation-conductive heat transfer taking into account the variability of thermophysical and thermo-optical characteristics of materials. These equations look like this:



    where
    c i is the specific heat of the i-th element; ρ i is its density; V i is the volume of the element; T i is the temperature of the element; t is the time; N i n is the number of elements having a common face with the i-th element; Qn k i is the power transmitted from the k-th neighbor of the i-th element through thermal conductivity; S i is the area of ​​the i-th element; Qe i- the density of the outgoing radiant flux of the i-th element; Qr ij is the power transmitted from the j-th element to the i-th element through radiation; N is the number of elements in the model; Qr j m is the power of radiation emitted by the m-th external source and absorbed in the i-th element; M is the number of radiation sources; Qi i p is the heat from the p-th internal heat source in the i-th element, P is the number of internal heat sources.

    The number N can be more than 10 5 , the properties of the elements and the boundary conditions can very sharply change with time, for example, when you turn on some device or exit the Sun from the shade, you need to integrate 10-20 turns, and this is about 120,000 seconds.

    So the search for a good method for integrating such a system would be very useful to the authors, and astronautics, and engineering as a whole.

    Who is interested - write in comments or mail to Alexander - ark4110@gmail.com.

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