Discretized integral hydrodynamics

Using an interpolant form for the gradient of a function of position, we write an integral version of the conservation equations for a fluid. In the appropriate limit, these become the usual conservation laws of mass, momentum, and energy. We also discuss the special cases of the Navier-Stokes equations for viscous flow and the Fourier law for thermal conduction in the presence of hydrodynamic fluctuations. By means of a discretization procedure, we show how the integral equations can give rise to the so-called particle dynamics of smoothed particle hydrodynamics and dissipative particle dynamics.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

An adaptive multiscale method for density-driven instabilities

Accurate modeling of flow instabilities requires computational tools able to deal with several interacting scales, from the scale at which fingers are triggered up to the scale at which their effects need to be described. The Multiscale Finite Volume (MsFV) method offers a framework to couple fine-and coarse-scale features by solving a set of localized problems which are used both to define a coarse-scale problem and to reconstruct the fine-scale details of the flow. The MsFV method can be seen as an upscaling-downscaling technique, which is computationally more efficient than standard discretization schemes and more accurate than traditional upscaling techniques. We show that, although the method has proven accurate in modeling density-driven flow under stable conditions, the accuracy of the MsFV method deteriorates in case of unstable flow and an iterative scheme is required to control the localization error. To avoid large computational overhead due to the iterative scheme, we suggest several adaptive strategies both for flow and transport. In particular, the concentration gradient is used to identify a front region where instabilities are triggered and an accurate (iteratively improved) solution is required. Outside the front region the problem is upscaled and both flow and transport are solved only at the coarse scale. This adaptive strategy leads to very accurate solutions at roughly the same computational cost as the non-iterative MsFV method. In many circumstances, however, an accurate description of flow instabilities requires a refinement of the computational grid rather than a coarsening. For these problems, we propose a modified iterative MsFV, which can be used as downscaling method (DMsFV). Compared to other grid refinement techniques the DMsFV clearly separates the computational domain into refined and non-refined regions, which can be treated separately and matched later. This gives great flexibility to employ different physical descriptions in different regions, where different equations could be solved, offering an excellent framework to construct hybrid methods.

A multilevel multiscale finite volume method

The Multiscale Finite Volume (MsFV) method has been developed to efficiently solve reservoir-scale problems while conserving fine-scale details. The method employs two grid levels: a fine grid and a coarse grid. The latter is used to calculate a coarse solution to the original problem, which is interpolated to the fine mesh. The coarse system is constructed from the fine-scale problem using restriction and prolongation operators that are obtained by introducing appropriate localization assumptions. Through a successive reconstruction step, the MsFV method is able to provide an approximate, but fully conservative fine-scale velocity field. For very large problems (e.g. one billion cell model), a two-level algorithm can remain computational expensive. Depending on the upscaling factor, the computational expense comes either from the costs associated with the solution of the coarse problem or from the construction of the local interpolators (basis functions). To ensure numerical efficiency in the former case, the MsFV concept can be reapplied to the coarse problem, leading to a new, coarser level of discretization. One challenge in the use of a multilevel MsFV technique is to find an efficient reconstruction step to obtain a conservative fine-scale velocity field. In this work, we introduce a three-level Multiscale Finite Volume method (MlMsFV) and give a detailed description of the reconstruction step. Complexity analyses of the original MsFV method and the new MlMsFV method are discussed, and their performances in terms of accuracy and efficiency are compared.

Using spatially distributed parameters and multi-response objective functions to solve parameterization of complex applications of semi-distributed hydrological models

Application of semi-distributed hydrological models to large, heterogeneous watersheds deals with several problems. On one hand, the spatial and temporal variability in catchment features should be adequately represented in the model parameterization, while maintaining the model complexity in an acceptable level to take advantage of state-of-the-art calibration techniques. On the other hand, model complexity enhances uncertainty in adjusted model parameter values, therefore increasing uncertainty in the water routing across the watershed. This is critical for water quality applications, where not only streamflow, but also a reliable estimation of the surface versus subsurface contributions to the runoff is needed. In this study, we show how a regularized inversion procedure combined with a multiobjective function calibration strategy successfully solves the parameterization of a complex application of a water quality-oriented hydrological model. The final value of several optimized parameters showed significant and consistentdifferences across geological and landscape features. Although the number of optimized parameters was significantly increased by the spatial and temporal discretization of adjustable parameters, the uncertainty in water routing results remained at reasonable values. In addition, a stepwise numerical analysis showed that the effects on calibration performance due to inclusion of different data types in the objective function could be inextricably linked. Thus caution should be taken when adding or removing data from an aggregated objective function.

An Adaptive Remeshing Procedure for Proximity Maneuvering Spacecraft Formations

We consider a methodology to optimally obtain reconfigurations of spacecraft formations. It is based on the discretization of the time interval in subintervals (called the mesh) and the obtainment of local solutions on them as a result of a variational method. Applied to a libration point orbit scenario, in this work we focus on how to find optimal meshes using an adaptive remeshing procedure and on the determination of the parameter that governs it

General electromagnetic simulation tool to predict the microwave nonlinear response of planar, arbitrarily-shaped HTS structures

This work describes a simulation tool being developed at UPC to predict the microwave nonlinear behavior of planar superconducting structures with very few restrictions on the geometry of the planar layout. The software is intended to be applicable to most structures used in planar HTS circuits, including line, patch, and quasi-lumped microstrip resonators. The tool combines Method of Moments (MoM) algorithms for general electromagnetic simulation with Harmonic Balance algorithms to take into account the nonlinearities in the HTS material. The Method of Moments code is based on discretization of the Electric Field Integral Equation in Rao, Wilton and Glisson Basis Functions. The multilayer dyadic Green's function is used with Sommerfeld integral formulation. The Harmonic Balance algorithm has been adapted to this application where the nonlinearity is distributed and where compatibility with the MoM algorithm is required. Tests of the algorithm in TM010 disk resonators agree with closed-form equations for both the fundamental and third-order intermodulation currents. Simulations of hairpin resonators show good qualitative agreement with previously published results, but it is found that a finer meshing would be necessary to get correct quantitative results. Possible improvements are suggested.

Verkkovaihtosuuntaajan vektorisäädön toteutus FPGA-piirillä

Verkkovaihtosuuntaajalla pystytään muuntamaan tasajännite vaihtojännitteeksi ja päinvastoin. Verkkovaihtosuuntaajan toiminta perustuu tehokytkinten ohjaukseen ja sopivan modulointimenetelmän käyttöön. Vektorisäädössä vaihtosuuntaajanvirrat ja jännitteet esitetään kompleksitasossa, jolloin virta- ja jännitekomponentit voidaan esittää vektoreina. Vektorisäädössä verkkovaihtosuuntaajan ohjaustoteutetaan laskemalla kompleksitasossa vektoreille arvot, jotka tuottavat vaihtosuuntaajan lähtöön halutun vektorin. Koska FPGA-piirit mahdollistavat nopean rinnakkaisen laskennan, soveltuvat ne hyvin vektorisäädön toteuttamiseen. FPGA-piirien rakenteesta johtuen on säätöjärjestelmän suunnittelussa huomioitava kiinteän pilkun lukujen riittävä bittileveys ja järjestelmän diskretointiaika. Työssä suunnitellaan verkkovaihtosuuntaajan vektorisäätö ja tutkitaan bittileveyden vaikutusta säädön toteuttamiseen FPGA-piirillä. Bittileveyden tarkasteluun esitetään käytettäväksi tilastollisia menetelmiä. Työssä tarkastellaan kiinteän pilkun järjestelmän ja liukulukujärjestelmän erosuureen tilastollisia tunnusmerkkejä sekä histogrammia. Tarkasteluissa huomattiin, että maksimivirhe itsessään ei tarjoa riittävästi tietoa erosuureen jakautumisesta. Näin ollen maksimivirhe ei ole kaikissa tilanteissa sovelias menetelmä riittävän bittitarkkuuden määrittämiseen. Työssä esitetään riittävän bittitarkkuuden määrittelemiseen käytettäväksi otossuureista otosvarianssia, keskipoikkeamaa ja vaihteluväliä.

Laskentamalli PKL-koelaitteiston pienen vuodon kokeen simuloimiseksi Apros-ohjelmistolla

Työn tavoitteena oli tehdä Apros-laskentamalli PKL-koelaitteistosta ja testata kuinka hyvin Apros pystyy laskemaan PKL-koelaitteistolla suoritetun E2.2 pienen vuodon kokeen. Tavoitteena oli myös tarkastella boorittoman veden tulpan etenemistä pienen vuodon kokeen aikana. PKL-koelaitteisto vastaa saksalaista sähköteholtaan 1300 MW olevaa Philippsburg 2 painevesilaitosta. Koelaitteiston tilavuudet ja teho on skaalattu kertoimella 145. Työssä tehdyllä laskentamallilla tarkasteltiin boorittoman veden tulpan liikkumista pienen vuodon kokeen aikana. Kun malli oli valmis, laskenta suoritettiin Apros 5.05 versiolla. Boorittoman veden tulpan etenemisen laskennassa käytettiin toisen kertaluvun diskretointia, jolla booripitoisuuden muutokset säilyvät teräväreunaisina. Laskentamalli pystyi kuvaamaan koelaitteistolla suoritetussa pienen vuodon kokeessa tapahtuneet ilmiöt varsin hyvin. Eroa koetuloksiintuli pääkiertopiirien luonnonkiertojen alkamishetkistä ja primääripaineen käyttäytymisessä. Kokeen alkutilanne ei ollut stationääritila, joten alkutilanteen asettamisessa oli hankaluuksia. Varsinkin pääkiertopiirien veden pinnankorkeuksienasettamisessa oli vaikeuksia, koska veden pinnankorkeuksien erot pyrkivät tasoittumaan nopeasti kokeen aikana. Apros pystyi laskemaan PKL-koelaitteistolla suoritetun pienen vuodon kokeen hyvin. Mallilla tulisi kuitenkin laskea vielä toisentyyppisiäkin kokeita, ennen kuin voidaan varmuudella tietää mallin toimivuus. PKL-koelaitteisto vastaa pääpiirteiltään Suomeen rakennettavaa Olkiluoto 3 ydinvoimalaitosta. Tehty työ antaa lisävarmuutta, kun Olkiluoto 3 laitoksen turvallisuustarkasteluita tehdään.

Probabilistic electrical resistivity tomography of a CO2 sequestration analog

Electrical resistivity tomography (ERT) is a well-established method for geophysical characterization and has shown potential for monitoring geologic CO2 sequestration, due to its sensitivity to electrical resistivity contrasts generated by liquid/gas saturation variability. In contrast to deterministic inversion approaches, probabilistic inversion provides the full posterior probability density function of the saturation field and accounts for the uncertainties inherent in the petrophysical parameters relating the resistivity to saturation. In this study, the data are from benchtop ERT experiments conducted during gas injection into a quasi-2D brine-saturated sand chamber with a packing that mimics a simple anticlinal geological reservoir. The saturation fields are estimated by Markov chain Monte Carlo inversion of the measured data and compared to independent saturation measurements from light transmission through the chamber. Different model parameterizations are evaluated in terms of the recovered saturation and petrophysical parameter values. The saturation field is parameterized (1) in Cartesian coordinates, (2) by means of its discrete cosine transform coefficients, and (3) by fixed saturation values in structural elements whose shape and location is assumed known or represented by an arbitrary Gaussian Bell structure. Results show that the estimated saturation fields are in overall agreement with saturations measured by light transmission, but differ strongly in terms of parameter estimates, parameter uncertainties and computational intensity. Discretization in the frequency domain (as in the discrete cosine transform parameterization) provides more accurate models at a lower computational cost compared to spatially discretized (Cartesian) models. A priori knowledge about the expected geologic structures allows for non-discretized model descriptions with markedly reduced degrees of freedom. Constraining the solutions to the known injected gas volume improved estimates of saturation and parameter values of the petrophysical relationship. (C) 2014 Elsevier B.V. All rights reserved.

Radiaalikompressorin spiraalivirtauksen numeerinen mallinnus

Diplomityössä mallinnetaan numeerisesti radiaalikompressorin spiraalin virtaus. Spiraalin tarkoituksena radiaalikompressorissa on kerätä tasaisesti virtaus diffuusorin kehältä. Spiraaliin on viime aikoina kiinnitetty enemmän huomiota, koska on havaittu, että kompressorin hyötysuhdetta voidaan parantaa spiraalia optimoimalla. Spiraalin toimintaa tarkastellaan kolmella eri massavirralla. Työn alussa käsitellään spiraalin toimintaperiaatteita. Numeerisena ratkaisijana käytetään Teknillisessä korkeakoulussa kehitettyä FIN-FLO -koodia. FINFLO -laskentaohjelmassa ratkaistaan Navier-Stokes yhtälöt kolmeulotteiselle laskenta-alueelle. Diskretointi perustuu kontrollitilavuus menetelmään. Työssä käsitellään laskentakoodin toimintaperiaatteitta. Turbulenssia mallinnetaan algebrallisella Baldwin-Lomaxin ja kahden yhtälön Chienin k-e turbulenssimalleilla. Laskentatuloksia verrataan Lappeenrannan teknillisessä korkeakoulussa tehtyihin mittauksiin kyseessä olevasta kompressorin spiraalista. Myös eri turbulenssimalleilla ja hilatasoilla saatuja tuloksia verrataan keskenään. Laskentatuloksien jälkikäsittelyä varten ohjelmoitiin neljä eri tietokoneohjelmaa. Laskennalla pyritään saamaan lisäselvyyttä virtauksen käyttäytymiseen spiraalissa ja erityisesti ns. kielen alueella. Myös kahden eri turbulenssimallin toimivuutta kompressorin numeerisessa mallinnuksessa tutkitaan.

On the development of an unstructured grid solver for inert and reactive high speed flow simulations

An unstructured grid Euler solver for reactive compressible flow applications is presented. The method is implemented in a cell centered, finite volume context for unstructured triangular grids. Three different schemes for spatial discretization are implemented and analyzed. Time march is implemented in a time-split fashion with independent integrators for the flow and chemistry equations. The capability implemented is tested for inert flows in a hypersonic inlet and for inert and reactive supersonic flows over a 2-D wedge. The results of the different schemes are compared with each other and with independent calculations using a structured grid code. The strengths and the possible weaknesses of the proposed methods are discussed.

A temperature predicting model for manufacturing processes requiring coiling

A model for predicting temperature evolution for automatic controling systems in manufacturing processes requiring the coiling of bars in the transfer table is presented. Although the method is of a general nature, the presentation in this work refers to the manufacturing of steel plates in hot rolling mills. The predicting strategy is based on a mathematical model of the evolution of temperature in a coiling and uncoiling bar and is presented in the form of a parabolic partial differential equation for a shape changing domain. The mathematical model is solved numerically by a space discretization via geometrically adaptive finite elements which accomodate the change in shape of the domain, using a computationally novel treatment of the resulting thermal contact problem due to coiling. Time is discretized according to a Crank-Nicolson scheme. Since the actual physical process takes less time than the time required by the process controlling computer to solve the full mathematical model, a special predictive device was developed, in the form of a set of least squares polynomials, based on the off-line numerical solution of the mathematical model.

Solution of aerospace problems using structured and unstructured strategies

Products developed at industries, institutes and research centers are expected to have high level of quality and performance, having a minimum waste, which require efficient and robust tools to numerically simulate stringent project conditions with great reliability. In this context, Computational Fluid Dynamics (CFD) plays an important role and the present work shows two numerical algorithms that are used in the CFD community to solve the Euler and Navier-Stokes equations applied to typical aerospace and aeronautical problems. Particularly, unstructured discretization of the spatial domain has gained special attention by the international community due to its ease in discretizing complex spatial domains. This work has the main objective of illustrating some advantages and disadvantages of numerical algorithms using structured and unstructured spatial discretization of the flow governing equations. Numerical methods include a finite volume formulation and the Euler and Navier-Stokes equations are applied to solve a transonic nozzle problem, a low supersonic airfoil problem and a hypersonic inlet problem. In a structured context, these problems are solved using MacCormack’s implicit algorithm with Steger and Warming’s flux vector splitting technique, while, in an unstructured context, Jameson and Mavriplis’ explicit algorithm is used. Convergence acceleration is obtained using a spatially variable time stepping procedure.

Two-dimensional drift-diffusion simulation of organic solar cells

The aim of this master's thesis is to develop a two-dimensional drift-di usion model, which describes charge transport in organic solar cells. The main bene t of a two-dimensional model compared to a one-dimensional one is the inclusion of the nanoscale morphology of the active layer of a bulk heterojunction solar cell. The developed model was used to study recombination dynamics at the donor-acceptor interface. In some cases, it was possible to determine e ective parameters, which reproduce the results of the two-dimensional model in the one-dimensional case. A summary of the theory of charge transport in semiconductors was presented and discussed in the context of organic materials. Additionally, the normalization and discretization procedures required to nd a numerical solution to the charge transport problem were outlined. The charge transport problem was solved by implementing an iterative scheme called successive over-relaxation. The obtained solution is given as position-dependent electric potential, free charge carrier concentrations and current densities in the active layer. An interfacial layer, separating the pure phases, was introduced in order to describe charge dynamics occurring at the interface between the donor and acceptor. For simplicity, an e ective generation of free charge carriers in the interfacial layer was implemented. The pure phases simply act as transport layers for the photogenerated charges. Langevin recombination was assumed in the two-dimensional model and an analysis of the apparent recombination rate in the one-dimensional case is presented. The recombination rate in a two-dimensional model is seen to e ectively look like reduced Langevin recombination at open circuit. Replicating the J-U curves obtained in the two-dimensional model is, however, not possible by introducing a constant reduction factor in the Langevin recombination rate. The impact of an acceptor domain in the pure donor phase was investigated. Two cases were considered, one where the acceptor domain is isolated and another where it is connected to the bulk of the acceptor. A comparison to the case where no isolated domains exist was done in order to quantify the observed reduction in the photocurrent. The results show that all charges generated at the isolated domain are lost to recombination, but the domain does not have a major impact on charge transport. Trap-assisted recombination at interfacial trap states was investigated, as well as the surface dipole caused by the trapped charges. A theoretical expression for the ideality factor n_id as a function of generation was derived and shown to agree with simulation data. When the theoretical expression was fitted to simulation data, no interface dipole was observed.