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Numerical simulation of large atmospheric multiphase mechanisms and detailed combustion kinetics

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but streaming OK and around a morning my talk is about a numerical simulation of large chemical kinetics systems using the atmospheric chemistry so which is an experimental enviroment which I have developed over the last couple of years under the supervision of also acknowledged during my studies
and this is not a finished product so yeah he was 1st there was somebody
online At 1st I will talk about the introduction where chemical kinetics systems far applied to real world problems and what motivate my talk then I wanna talk about the model description and also the math behind it in the 3rd part we come to the numerical solver which are used to integrate numerical schemes with Rosenbrock methods is a special part of on American soil which areas here and then I will come to a summary and some further work which I want to future
so the 1st time the introduction modulating the next couple of slides I'll give you a little overview over practical applications
of reaction kinetics or applied these is the 1st of all is mulling of atmospheric chemical processes where did it troubles fear the main actor for cleaning the atmosphere and the vigorous for new particle formations there are a couple of mechanisms used forward is kind of studies like the master chemical mechanism which is developed at the University of Leeds and the UK Met Office and also would this is saying yeah explicit gas phase them mechanism and we have also a liquid-phase mechanism which is developed the troubles both of these mechanisms can be cobble together to form a real multiphase chemistry problem also found these kind of schemes can be incorporated into a high-dimensional sea of these simulations which makes its highly computational yeah of the added costs a lot of of numerical computations here because you have to evaluate a chemical called in every so that you use and the finer the mesh to more you have to it so
next practical application is small ling of vision and combustion this is used mainly for modeling power stations turbines engines and so on here as well as the forward tropospheric mechanisms the size of the mechanism depends on to you number of used hydrocarbons so the larger the molecules get some the larger mechanism gets and this is an exponential growth also for combustion systems we wanna minimize or investigate that the systems to minimize emission rates and all that stuff so a 3rd of an of
allocation as from systems biology where for example metabolic networks are investigated for of the material a truck to concision for example or modeling the cell cycle and offer stuff so the next I will come to the
model and the math behind all model
so we start with the box model framework or Apple's and this is filled with with gas and the control trouble it's so we start with cloud condensation nuclei of some oratory size and where income these particles can condense water on to form such clout problems and year for a simplification purposes the monitors persistent is assumed so all the droplets scroll with the same rate then we can add to the emission values and deposition Darrell values which are usually time-dependent and now we can zoom in into
a such water droplets to see what's going on there we have guessed festive user materials transport of ultimate droplets and away from it and we have also gas-phase reactions then we model the phase transfer according to the sports approach so material masses transferred into the droplets and all of the problem and then of course we have also the the country's reactions and liquid-phase diffusion here use more locates it denotes the concentration off but species a in the liquid phase and the actual Caballe denotes the concentration in the gas phase so next succumb to or
mathematical formulation you can see here we have the reaction system which consists of a linear combination of the units but is transformed into a linear combination of products the matrices here these new and UP or stored efficiently as sparse matrices and will be used later on for the calculation next we have some equations 1st of all the rate of progression which is here the
reaction right this is usually it depends usually on the concentration itself and the temperature where decay is the reaction constants and also for simplification purposes we assume that it's only depends on temperature and yeah not 1 time for example so these erected considers multiplied with the concentration values of that's a race to the power of its the stereometric coefficient then we have
formats conservation equation which describes the evolution of the species concentrations over time and we can also add an emission values or the position that use or some dilution rates and also for simulating combustion systems we all salute to track the change in temperature which is described by this equation here where you must conservation equation multiplied with some molar until energy for each species and here is a constant volume enviroment assumed there or another kind of environment like constant pressure but we chose from the constant voice apartment so
next we come to the numerical so as I mentioned before we chose the specific kind of America soul 40 systems which are
committed to the 1st of all we have all ordinary differential equation system this this consists of and as plus 1 equations were and this is the number of species of or mechanism and the plus 1 comes from the temperature equation for tropospheric mechanisms we can drop the temperature equation and just take this because it in the system of equations that into account also we have given the initial value the vector of initial values so we start with some concentration and some temperature and forward to you Speech is profiles that up also we need for we need to Jacobian matrix of the system for all integration scheme which I will come later so you're are some properties of our ODE system or ODE system contains only 1st order derivatives if you can see and it's all 1st which are usually non-linear highly nonlinear functions and also several concentrations influenced the production of each species so we have a couple of ODE system and there is not an issue with these kind of systems we have the Jedi reaction rates so this different several rows of magnitude so we have recht it's which are very slow or we have reactions which goes very fast so we have a stiff ODE system also for a simplification purposes as image before we use say that rection constant is just depending on at a temperature and not on time like like some of the of the solar radiation or pressure or lot of stuff so next we come to our integration scheme we choose Rosenbrock methods which are the diagonal implicit and or put on methods and we have to solve as linear systems for each time step here can see we need to the Jacobian matrix to evaluate the formula and all right side we have to make a function call for each set and yeah and then waited freedom calculated all of our slopes we can form a new concentration of diese Rosenbrock method has already transformed them to avoid a matrix vector product you on the right side this is a hint done by substituting them that some variables to pretty easy so also these rows and prepare events can be used to evaluate the local for new a concentration values while and that a formula which is usually 1 portal less than the actual method so next come duty undergoing formulation we calculated Jacobian actually analytically by doing so matrix modifications 0 we multiply today these titrimetric metrics them with a diagonal matrix where d irector rents reaction rates are contained on the main diagonal these is multiplied with the the symmetric coefficients of the left side of the reaction times are diagonal matrix which contains the inverse values of the concentrations of all the main diagonal here we have to ensure that all the concentrations on the because is 0 which is done by adding a small value if it is actually 0 so this is the 1st part of the Jacobian formulation the 2nd port on the top right is a kilometer and it can be expressed as the symmetric spectrum of allocating so if your matrix A diagonal matrix and vector and on this part we have to ensure that or action constants are not equal to 0 FIL same as before it is 0 we have a really small attitude the lower left part of the Jacobi matrix this HiColor by this formula and the same as same as the top right part we have fewer over time and we see here Jesus Jesus Reuter of multiplied by a scalar and also that is this part it's actually the part of the list top left and this is multiplied with the internal energy is to get a row vector and and to topic right plot perspective you there is the derivative of tension temperature was prospective temperature which looks like this so does the sperm this is pretty computational intensive because we have to look like a lot of sparse matrices together and lot of a sparse matrix vector multiplications have to be done so now we have Jacobian formulation and we can plug it into our Rosemont formula to get the the system of linear equations which we have to solve for each term for each stage in the methods and yet they can see here all a lot of matrix matrix of
modifications metrics like would occasions which are ID computational intensive when the systems get really large we have also implemented they another approach for use kind of linear systems which might matrix over years where by plot this whole these coefficient matrix looks like you can see it it's really sparse I believe in this example this this point 0 8 per cent of nonzero so contains just to hear this amount of nonzero so so what it's it's yeah we need to exploit the use of nonzero structures for during the solving phase here so the next approach is to expound too might the matrix to avoid the actual computation of the Jacobian matrix which is then delayed to the of factorization so we just have to update all men Diokno here on the metrics so you're here is no matrix vector multiplied securities diagonal matrix times pectoris componentwise vector vector modification which not that costly in terms of and these can save up to 20 % of floating-point operations so here are just a few words about those that have control it can be implemented and with these embedded rules in Proc methods we calculate the local of the new concentration values then normalize them so we can accept a time step if the arrow is less than 1 and if this is not the case we have to review see and lower all step-size according to this formula and yet do we stop the stance of again also we have implemented not only 1 error measurement so we can yeah can invade investigated different such kind of norms to we have implemented the maximum on-board Leucadia norm where do the Euclidian norm gives us a much more smooth behavior for choosing new time steps so we can make a larger time steps very small enough anyone's interested is dental is actually 1 of people where p is the or of the method plus 1 OK next we have to solve all linear systems and we chose the sparse direct-sold for this kind of problem where we have given a sparse matrix a and which is assumed to be regular and unsymmetric and peace metrics has now to be factorized into lower and upper triangular matrix and then at that but substitution so Award du during his factorization their accuse a problem which is called which transforms in 0 element the nonzero element so 2 matrix gets more filled and so we have more or more calculations has to be done to avoid a massive field and we urge users metric imitation of given matrix symmetric because or diagonal values should be placed all diagonal also even after to them penetration so did knowing as your elements and go on the diagonal so approaches defining such kind of permutations or because of all the minimum degree of wood and developed by Markowitz in the late fifties I believe some variants of the use of as an approximate minimum degree approximate minimum filled or we have also another kind of method for finding such appeal which Hopcroft theory other items like Scotch and meet is using this here is a little graphic to show you that the minimum degree is superior to all that the ordering methods which we investigated this is done by applying the use of warring strategies studio large systems that us a chemical mechanism the mission of 3 comma decimal 2 when combined with an uncoupled 4 comma decimal 0 so this some output scenarios for this is a small mechanism the Regional Atmospheric Chemistry mall Wilson come from the was developed here as the earlier version 2 comma decimal 4 which is roughly 800 directions and 1 of 50 species I will not go into detail hold you whitey's Professor looks as they look a lot this is applaud output can look like this also our said or opera prudence was done with it so you have he calls them that I used in a search lines and it's it's also pretty fast so I'm not system we can simulate or open combustion systems like like this 1 here it's a become mechanism with roughly 17 thousand reactions and roughly free solvents from species yet again no detail on the profiles but yes this is how the output can look like so I wanna
sunrise at the end and give some for the work of my years we do hold the whole
idea behind the use of the simulation package was to improve the efficiency of the common cold where I have tried to increase the speed by using that dress calculations of directorate constants so we build index sets and can calculate a whole bunch of the same equations at once so to use that can save a lot of time then also we use them sparse matrix techniques to calculated linear of rock systems the solution of linear out our systems also we chose them stable and robust integration seems like that was what methods or W methods where T except Jacobian or did become Jacobian matrix don't have to be exact which gives us more tolerance and they also it's there is a need to reduce that Aquarius in the simulation software so the Delessert if-else statements or a select CASE statements that the better also for exploiting compiler optimizations he should choose and berets and not objects was tracts because of Fortran where I really prefer annotations instead of logic objects also you should awards from function calls of procedure calls with in loops which have a high iteration count and also 1st I always is necessary because we have lodged it how to write and read which can done by I'm writing some formative made about of metadata fires and then according to these metadata fires you right on for metadata which is sperm much more facile than right informative data so with some ideas for further work parallelization of some of our computational intensive returns like directorates and linear arc about using the Message Passing Interface of an anti all correlated colorectal trying then we also look at for improved ordering methods to get less filling during the factorization and also we want to incorporate iterative methods and compared him to direct efforts so a day and to further developed a model itself we wanna try to similar put it is past systems and also implement a real microphysical scheme of no it's no we do give of it suit a function for the liquid water content and we can then calculate all droplet radio out of these given functions also other features which implemented in the software these are not the fish but I began with that is they commit apostle which converts them some Texas into each other like an format or KBP or to use format which is used to propose and also I implemented today automated production software to reduce the number of species and reactions of the detailed mechanism to minimum to make a reality is the system small and get faster results for a chemical code so that's for me thanks for your attention and
the question
Beobachtungsstudie
Kinetik
Näherungsverfahren
Physikalisches System
Kinematik
Kinetik
Mathematik
Physikalisches System
Kardinalzahl
Biprodukt
Gesetz <Physik>
Integral
Deskriptive Statistik
Flächeninhalt
Reelle Zahl
Mereologie
Modelltheorie
Modelltheorie
Kinetik
Beobachtungsstudie
Prozess <Physik>
Prozess <Physik>
Flüssiger Zustand
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Mathematik
Rechenschieber
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Modelltheorie
Numerische Strömungssimulation
Multiplikation
Extremwert
Gerichteter Graph
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Sterbeziffer
Transformation <Mathematik>
Zahlenbereich
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Mathematik
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Modelltheorie
Modelltheorie
Quader
Sterbeziffer
Mathematik
Wasserdampftafel
Phasenumwandlung
Mathematik
Bilinearform
Modelltheorie
Modelltheorie
Tropfen
Sterbeziffer
Matrizenrechnung
Wärmeübergang
Gleichungssystem
Mathematik
Schwach besetzte Matrix
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Koeffizient
Arithmetische Folge
Konstante
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Tropfen
Phasenumwandlung
Gammafunktion
Streuungsdiagramm
Sterbeziffer
Matrizenring
Materialisation <Physik>
Ruhmasse
Kombinator
Schwach besetzte Matrix
Physikalisches System
Flüssiger Zustand
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Rechnen
Linearisierung
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Mittelwert
Ortsoperator
Sterbeziffer
Dichte <Physik>
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Physikalisches System
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Isochore
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Koeffizient
Erhaltungssatz
Evolute
Ruhmasse
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Modelltheorie
Differential
Matrizenrechnung
Partielle Differentiation
Mathematik
Gleichungssystem
Euler-Winkel
Skalarfeld
Gewöhnliche Differentialgleichung
Hausdorff-Dimension
Jacobi-Verfahren
Standardabweichung
Ausdruck <Logik>
Kompakter Raum
Steifes Anfangswertproblem
Gasdruck
Ordnung <Mathematik>
Sterbeziffer
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Innere Energie
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Kategorie <Mathematik>
Profil <Aerodynamik>
Temperaturstrahlung
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Biprodukt
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Lineares Gleichungssystem
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Verschlingung
Rechter Winkel
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Term
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Unendlichkeit
Physikalisches System
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Physikalisches System
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Integral
Gewöhnliche Differentialgleichung
Diagonalform
Mereologie
Analytische Menge
Größenordnung
Modelltheorie
Matrizenrechnung
Stellenring
Punkt
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Mathematik
Element <Mathematik>
Extrempunkt
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Approximation
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Physikalisches System
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Menge
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Linearisierung
Integral
Objekt <Kategorie>
Konstante
Loop
Modelltheorie
Algebraische Struktur

Metadaten

Formale Metadaten

Titel Numerical simulation of large atmospheric multiphase mechanisms and detailed combustion kinetics
Serientitel The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018
Autor Schimmel, Willi
Mitwirkende Leibniz-Institut für Oberflächenmodifizierung e.V. (IOP)
Leibniz-Institut für Troposphärenforschung (TROPOS)
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/35355
Herausgeber Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB)
Erscheinungsjahr 2018
Sprache Englisch
Produktionsort Leipzig

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik
Abstract A simulation tool for the numerical solution of large kinetic systems with special emphasis on examples from atmospheric chemistry and combustion is presented. The chemical mechanism has to be provided in readable ASCII format, whereas the program is able to read the TROPOS syntax for chemical systems and also the widely used ChemKin format. From this, a system of ordinary differential equations is generated internally and solved numerically by Rosenbrock-type methods. Efficiency is obtained by carefully exploiting the sparsity structures of the Jacobian. Additionally, another approach is implemented, where the direct evaluation of the Jacobian is no longer required. The effects of both strategies are investigated by simulating detailed atmospheric multiphase mechanisms and gas-phase combustion mechanisms.

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