Numerical integration errors in deep space orbit determination

Il progetto di tesi è stato sviluppato durante il periodo di tirocinio svolto all’interno del “Laboratorio di Radio Scienza ed Esplorazione Planetaria” da un'esperienza da cui prende il nome lo stesso elaborato: ”Numerical integration errors in deep space orbit determination”. Lo scopo del sopraccitato laboratorio è stato quello di studiare in modo approfondito il problema kepleriano dei due corpi, per poi passare ad un’analisi del problema dei tre corpi e successivamente a n corpi (con particolare attenzione alle orbite dei satelliti medicei di Giove). Lo studio è stato affiancato ad un costante utilizzo della piattaforma di programmazione Matlab per l’elaborazione e la stesura di codici per il calcolo di traiettorie orbitali ed errori numerici. Infatti, il fulcro del lavoro è stato proprio il confronto di vari integratori e degli errori numerici derivanti dall’integrazione. Nella tesi, dapprima, viene introdotto il sistema Gioviano, vengono presentati i satelliti medicei, delineate le caratteristiche fisiche fondamentali e i principali motivi che portano ad avere particolare interesse nel conoscere lo sviluppo orbitale di tale sistema. In seguito, l'elaborato, dopo una dettagliata descrizione teorica del problema dei due corpi, presenta un codice per la rappresentazione di orbite kepleriane e il calcolo dei relativi errori commessi dal metodo numerico rispetto a quello analitico. Nell'ultimo capitolo, invece, il problema è esteso a più corpi dotati di massa e a tal proposito viene proposto un codice per la rappresentazione delle orbite descritte nel tempo da n corpi, date le condizioni iniziali, e il calcolo dei rispettivi errori nel sistema di riferimento (r,t,n). In merito a ciò, vengono infine testati diversi integratori per cercare quello con le migliori performance e sono poi analizzati alcuni parametri in input al problema per verificare sotto quali condizioni l’integratore lavora meglio.

Mekanismien dynamiikan simuloinnissa sovellettavia numeerisia- ja mallinnusmenetelmiä

Koneet voidaan usein jakaa osajärjestelmiin, joita ovat ohjaus- ja säätöjärjestelmät, voimaa tuottavat toimilaitteet ja voiman välittävät mekanismit. Eri osajärjestelmiä on simuloitu tietokoneavusteisesti jo usean vuosikymmenen ajan. Osajärjestelmien yhdistäminen on kuitenkin uudempi ilmiö. Usein esimerkiksi mekanismien mallinnuksessa toimilaitteen tuottama voimaon kuvattu vakiona, tai ajan funktiona muuttuvana voimana. Vastaavasti toimilaitteiden analysoinnissa mekanismin toimilaitteeseen välittämä kuormitus on kuvattu vakiovoimana, tai ajan funktiona työkiertoa kuvaavana kuormituksena. Kun osajärjestelmät on erotettu toisistaan, on niiden välistenvuorovaikutuksien tarkastelu erittäin epätarkkaa. Samoin osajärjestelmän vaikutuksen huomioiminen koko järjestelmän käyttäytymissä on hankalaa. Mekanismien dynamiikan mallinnukseen on kehitetty erityisesti tietokoneille soveltuvia numeerisia mallinnusmenetelmiä. Useimmat menetelmistä perustuvat Lagrangen menetelmään, joka mahdollistaa vapaasti valittaviin koordinaattimuuttujiin perustuvan mallinnuksen. Numeerista ratkaisun mahdollistamiseksi menetelmän avulla muodostettua differentiaali-algebraaliyhtälöryhmää joudutaan muokkaamaan esim. derivoimalla rajoiteyhtälöitä kahteen kertaan. Menetelmän alkuperäisessä numeerisissa ratkaisuissa kaikki mekanismia kuvaavat yleistetyt koordinaatit integroidaan jokaisella aika-askeleella. Tästä perusmenetelmästä johdetuissa menetelmissä riippumattomat yleistetyt koordinaatit joko integroidaan ja riippuvat koordinaatit ratkaistaan rajoiteyhtälöiden perusteella tai yhtälöryhmän kokoa pienennetään esim. käyttämällä nopeus- ja kiihtyvyysanalyyseissä eri kiertymäkoordinaatteja kuin asema-analyysissä. Useimmat integrointimenetelmät on alun perin tarkoitettu differentiaaliyhtälöiden (ODE) ratkaisuunjolloin yhtälöryhmään liitetyt niveliä kuvaavat algebraaliset rajoiteyhtälöt saattavat aiheuttaa ongelmia. Nivelrajoitteiden virheiden korjaus, stabilointi, on erittäin tärkeää mekanismien dynamiikan simuloinnin onnistumisen ja tulosten oikeellisuuden kannalta. Mallinnusmenetelmien johtamisessa käytetyn virtuaalisen työn periaatteen oletuksena nimittäin on, etteivät rajoitevoimat tee työtä, eli rajoitteiden vastaista siirtymää ei tapahdu. Varsinkaan monimutkaisten järjestelmien pidemmissä analyyseissä nivelrajoitteet eivät toteudu tarkasti. Tällöin järjestelmän energiatasapainoei toteudu ja järjestelmään muodostuu virtuaalista energiaa, joka rikkoo virtuaalisen työn periaatetta, Tästä syystä tulokset eivät enää pidäpaikkaansa. Tässä raportissa tarkastellaan erityyppisiä mallinnus- ja ratkaisumenetelmiä, ja vertaillaan niiden toimivuutta yksinkertaisten mekanismien numeerisessa ratkaisussa. Menetelmien toimivuutta tarkastellaan ratkaisun tehokkuuden, nivelrajoitteiden toteutumisen ja energiatasapainon säilymisen kannalta.

Simulation of dynamic pinion course using Runge-Kutta's method and impact modeling

Once defined the relationship between the Starter Motor components and their functions, it is possible to develop a mathematical model capable to predict the Starter behavior during operation. One important aspect is the engagement system behavior. The development of a mathematical tool capable of predicting it is a valuable step in order to reduce the design time, cost and engineering efforts. A mathematical model, represented by differential equations, can be developed using physics laws, evaluating force balance and energy flow through the systems degrees of freedom. Another important physical aspect to be considered in this modeling is the impact conditions (particularly on the pinion and ring-gear contact). This work is a report of those equations application on available mathematical software and the resolution of those equations by Runge-Kutta's numerical integration method, in order to build an accessible engineering tool. Copyright © 2011 SAE International.

Caracterização e redução das oscilações espúrias resultantes da representação de linhas de transmissão por meio de elementos discretos de circuitos

Pós-graduação em Engenharia Elétrica - FEIS

Representation of transmission lines: comparison of models and parameters distributed discrete parameters

A transmission line is characterized by the fact that its parameters are distributed along its length. This fact makes the voltages and currents along the line to behave like waves and these are described by differential equations. In general, the differential equations mentioned are difficult to solve in the time domain, due to the convolution integral, but in the frequency domain these equations become simpler and their solutions are known. The transmission line can be represented by a cascade of π circuits. This model has the advantage of being developed directly in the time domain, but there is a need to apply numerical integration methods. In this work a comparison of the model that considers the fact that the parameters are distributed (Universal Line Model) and the fact that the parameters considered concentrated along the line (π circuit model) using the trapezoidal integration method, and Simpson's rule Runge-Kutta in a single-phase transmission line length of 100 km subjected to an operation power. © 2003-2012 IEEE.

A Poisson mixed model with nonnormal random effect distribution

In this paper, we propose a random intercept Poisson model in which the random effect is assumed to follow a generalized log-gamma (GLG) distribution. This random effect accommodates (or captures) the overdispersion in the counts and induces within-cluster correlation. We derive the first two moments for the marginal distribution as well as the intraclass correlation. Even though numerical integration methods are, in general, required for deriving the marginal models, we obtain the multivariate negative binomial model from a particular parameter setting of the hierarchical model. An iterative process is derived for obtaining the maximum likelihood estimates for the parameters in the multivariate negative binomial model. Residual analysis is proposed and two applications with real data are given for illustration. (C) 2011 Elsevier B.V. All rights reserved.

Future developments in stiff integration techniques : stability of methods that do not use an exact Jacobian /

"COO-2383-0035."

Methods and Models for Accelerating Dynamic Simulation of Fluid Power Circuits

The objective of this dissertation is to improve the dynamic simulation of fluid power circuits. A fluid power circuit is a typical way to implement power transmission in mobile working machines, e.g. cranes, excavators etc. Dynamic simulation is an essential tool in developing controllability and energy-efficient solutions for mobile machines. Efficient dynamic simulation is the basic requirement for the real-time simulation. In the real-time simulation of fluid power circuits there exist numerical problems due to the software and methods used for modelling and integration. A simulation model of a fluid power circuit is typically created using differential and algebraic equations. Efficient numerical methods are required since differential equations must be solved in real time. Unfortunately, simulation software packages offer only a limited selection of numerical solvers. Numerical problems cause noise to the results, which in many cases leads the simulation run to fail. Mathematically the fluid power circuit models are stiff systems of ordinary differential equations. Numerical solution of the stiff systems can be improved by two alternative approaches. The first is to develop numerical solvers suitable for solving stiff systems. The second is to decrease the model stiffness itself by introducing models and algorithms that either decrease the highest eigenvalues or neglect them by introducing steady-state solutions of the stiff parts of the models. The thesis proposes novel methods using the latter approach. The study aims to develop practical methods usable in dynamic simulation of fluid power circuits using explicit fixed-step integration algorithms. In this thesis, twomechanisms whichmake the systemstiff are studied. These are the pressure drop approaching zero in the turbulent orifice model and the volume approaching zero in the equation of pressure build-up. These are the critical areas to which alternative methods for modelling and numerical simulation are proposed. Generally, in hydraulic power transmission systems the orifice flow is clearly in the turbulent area. The flow becomes laminar as the pressure drop over the orifice approaches zero only in rare situations. These are e.g. when a valve is closed, or an actuator is driven against an end stopper, or external force makes actuator to switch its direction during operation. This means that in terms of accuracy, the description of laminar flow is not necessary. But, unfortunately, when a purely turbulent description of the orifice is used, numerical problems occur when the pressure drop comes close to zero since the first derivative of flow with respect to the pressure drop approaches infinity when the pressure drop approaches zero. Furthermore, the second derivative becomes discontinuous, which causes numerical noise and an infinitely small integration step when a variable step integrator is used. A numerically efficient model for the orifice flow is proposed using a cubic spline function to describe the flow in the laminar and transition areas. Parameters for the cubic spline function are selected such that its first derivative is equal to the first derivative of the pure turbulent orifice flow model in the boundary condition. In the dynamic simulation of fluid power circuits, a tradeoff exists between accuracy and calculation speed. This investigation is made for the two-regime flow orifice model. Especially inside of many types of valves, as well as between them, there exist very small volumes. The integration of pressures in small fluid volumes causes numerical problems in fluid power circuit simulation. Particularly in realtime simulation, these numerical problems are a great weakness. The system stiffness approaches infinity as the fluid volume approaches zero. If fixed step explicit algorithms for solving ordinary differential equations (ODE) are used, the system stability would easily be lost when integrating pressures in small volumes. To solve the problem caused by small fluid volumes, a pseudo-dynamic solver is proposed. Instead of integration of the pressure in a small volume, the pressure is solved as a steady-state pressure created in a separate cascade loop by numerical integration. The hydraulic capacitance V/Be of the parts of the circuit whose pressures are solved by the pseudo-dynamic method should be orders of magnitude smaller than that of those partswhose pressures are integrated. The key advantage of this novel method is that the numerical problems caused by the small volumes are completely avoided. Also, the method is freely applicable regardless of the integration routine applied. The superiority of both above-mentioned methods is that they are suited for use together with the semi-empirical modelling method which necessarily does not require any geometrical data of the valves and actuators to be modelled. In this modelling method, most of the needed component information can be taken from the manufacturer’s nominal graphs. This thesis introduces the methods and shows several numerical examples to demonstrate how the proposed methods improve the dynamic simulation of various hydraulic circuits.

The Kineticist's Workbench: Combining Symbolic and Numerical Methods in the Simulation of Chemical Reaction Mechanisms

The Kineticist's Workbench is a program that simulates chemical reaction mechanisms by predicting, generating, and interpreting numerical data. Prior to simulation, it analyzes a given mechanism to predict that mechanism's behavior; it then simulates the mechanism numerically; and afterward, it interprets and summarizes the data it has generated. In performing these tasks, the Workbench uses a variety of techniques: graph- theoretic algorithms (for analyzing mechanisms), traditional numerical simulation methods, and algorithms that examine simulation results and reinterpret them in qualitative terms. The Workbench thus serves as a prototype for a new class of scientific computational tools---tools that provide symbiotic collaborations between qualitative and quantitative methods.

Flux comparison of Eulerian and Lagrangian estimates of Agulhas leakage: A case study using a numerical model

Estimating the magnitude of Agulhas leakage, the volume flux of water from the Indian to the Atlantic Ocean, is difficult because of the presence of other circulation systems in the Agulhas region. Indian Ocean water in the Atlantic Ocean is vigorously mixed and diluted in the Cape Basin. Eulerian integration methods, where the velocity field perpendicular to a section is integrated to yield a flux, have to be calibrated so that only the flux by Agulhas leakage is sampled. Two Eulerian methods for estimating the magnitude of Agulhas leakage are tested within a high-resolution two-way nested model with the goal to devise a mooring-based measurement strategy. At the GoodHope line, a section halfway through the Cape Basin, the integrated velocity perpendicular to that line is compared to the magnitude of Agulhas leakage as determined from the transport carried by numerical Lagrangian floats. In the first method, integration is limited to the flux of water warmer and more saline than specific threshold values. These threshold values are determined by maximizing the correlation with the float-determined time series. By using the threshold values, approximately half of the leakage can directly be measured. The total amount of Agulhas leakage can be estimated using a linear regression, within a 90% confidence band of 12 Sv. In the second method, a subregion of the GoodHope line is sought so that integration over that subregion yields an Eulerian flux as close to the float-determined leakage as possible. It appears that when integration is limited within the model to the upper 300 m of the water column within 900 km of the African coast the time series have the smallest root-mean-square difference. This method yields a root-mean-square error of only 5.2 Sv but the 90% confidence band of the estimate is 20 Sv. It is concluded that the optimum thermohaline threshold method leads to more accurate estimates even though the directly measured transport is a factor of two lower than the actual magnitude of Agulhas leakage in this model.

A temperature-based thermodynamically consistent integration scheme for discrete thermo-elastodynamics

The formulation of thermodynamically consistent (TC) time integration methods was introduced by a general procedure based on the GENERIC form of the evolution equations for thermo-mechanical problems. The use of the entropy was reported to be the best choice for the thermodynamical variable to easily provide TC integrators. Also the employment of the internal energy was proved to not involve excessive complications. However, attempts towards the use of the temperature in the design of GENERIC-based TC schemes have so far been unfruitful. This paper complements the said procedure to attain TC integrators by presenting a TC scheme based on the temperature as thermodynamical state variable. As a result, the problems which arise due to the use of the entropy are overcome, mainly the definition of boundary conditions. What is more, the newly proposed method exhibits the general enhanced numerical stability and robustness properties of the entropy formulation.

Methods for automatic error analysis of numerical algorithms /

Vita.

Automatic multirate methods for ordinary differential equations /

"UILU-ENG 80 1701."

Optimal stiffly stable methods for ordinary differential equations,

"COO-1469-0164."

Effective cell-centred time-domain Maxwell's equations numerical solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed. © 2004 Elsevier Inc. All rights reserved.