We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

In this thesis we look at acoustic wave propagation under water. We look in particular at waves generated by a point source and what happens with the propagation when we model the bottom as flat or as curvilinear. We assume the source to be working at a certain frequency and therefore we model this problem by solving the Helmholtz equation. Since Helmholtz equation has some unwanted numerical properties we are interested in finding new numerical methods that could accelerate the solver. In this thesis we use the Waveholtz iteration, which solves Helmholtz equation by connecting it to the time-dependent wave equation. We use finite differences and the SBP-SAT method to approximate the spatial problem numerically and for modelling the sea bottom we use curvilinear coordinates.  To compare the Waveholtz iteration we also solve Helmholtz equation with a naive solver. The naive solver consists of approximating the equation with finite differences and then solving the linear system of equation by some iterative solver, which for our tests will be GMRES. The results show that the Waveholtz iteration converges in less iterations than our naive solver. It also shows that the number of iterations stays unchanged when changing our discretization, which otherwise is a big problem for our naive solver. This allows us to increase the accuracy of our numerical solution without changing the computation time too much.  We show that the number of iterations increases according to theory for an increasing frequency, and that for open problems we even see a smaller increase. For certain resonant frequencies in Helmholtz equation we do not expect the Waveholtz iteration to converge. In the neighbourhood of these frequencies the convergence becomes slow and we need many iterations for a solution of a certain accuracy. By reformulating the Waveholtz iteration as a Krylov solution we can see that resonances in Helmholtz equation have a smaller impact of the convergence. / I detta examensarbete undersöker vi akustisk vågutbredning i vatten. Vi kollar specifikt på vågor som genereras av en punktkälla och vad som sker när vi modellerar botten som plan eller som kurvlinjär. Då vi antar att punktkällan arbetar vid en bestämd frekvens, kommer vi modellera det fysikaliska problemet genom att lösa Helmholtz ekvation. Helmholtz ekvation har dock några numeriska egenskaper som är oönskade, och därför finns ett intresse av att hitta nya numeriska metoder som löser ekvationen. I detta examensarbete undersöker vi Waveholtz iteration, som löser Helmholtz ekvation genom att koppla den till den tidsberoende vågekvationen. Vi använder finita differenser och SBP-SAT metoden för att approximera det rumsliga problemet numeriskt. För att ge en detaljerad beskrivning av botten använder vi kurvlinjära koordinater. För att jämföra Waveholtz iterationen med något löser vi även Helmholtz med hjälp av en naiv lösare. Den naiva lösaren består av att approximera problemet med finita differenser och sedan lösa det linjära systemet rakt av med en iterativ lösare (vilket för våra fall kommer vara GMRES). Resultatet visar att Waveholtz iteration konvergerar på ett lägre antal iterationer än vår naiva lösare. Det visar även att antalet iterationer inte förändras när vi ändrar diskretisering, vilket annars är ett problem för vår naiva lösare. Detta innebär att vi kan få en högre noggrannhet utan att förlänga beräkningstiden alltför mycket.  Vi visar även att antalet iterationer ökar som förväntat med en ökad frekvens, samt att för öppna problem så ökar antalet iteration mindre än enligt teorin. Vid vissa resonanta frekvenser i Helmholtz ekvation förväntar vi oss att Waveholtz iteration inte kommer konvergerar. I närheten av dessa frekvenser blir konvergensen långsam och vi behöver många iterationer för att lösa problemet. Genom att formulera Waveholtz iteration som en Krylov lösning kommer resonanser i Helmholtz ekvation ge en mindre negativ effekt på konvergensen än om den är formulerad som en fixpunkts iteration.

Introduction : L’altération des systèmes de récompense et de douleur serait impliquée dans le développement et le maintien de la douleur chronique. En contexte expérimental, l’arrêt d’une stimulation douloureuse et désagréable peut déclencher une sensation de plaisir (pleasant pain relief, PPR), mais contrairement aux mécanismes endogène d’inhibition de douleur (ICPM) la variabilité interindividuelle du PPR semble ne jamais avoir été étudiée. Objectifs : Étude 1) Investiguer la variabilité interindividuelle du PPR et de l’ICPM chez des sous-groupes de sujets sains ayant des réponses de douleur dynamiques similaires (analyses de trajectoires) durant un test à l’eau froide (CPT) Étude 2) Mesurer la connectivité au repos de régions du système de récompense en IRMf en lien avec le PPR. Méthode : 1) Une thermode Peltier et un CPT (10 ℃) ont été administrés séquentiellement (N=122). Le PPR a été mesuré pendant quatre minutes après l’arrêt du CPT et, 2) suite à l’application d’un gel froid en imagerie fonctionnelle (N=38). Résultats : 1) Quatre trajectoires ont été identifiées selon les réponses de douleur durant le CPT. Le PPR était corrélé aux trajectoires de douleur, mais pas l’efficacité d’ICPM. 2) Les connexions au repos, significativement corrélées au PPR, étaient les suivantes : noyau accumbens (NAcc) gauche-cortex cingulaire postérieur, NAcc gauche-cortex cingulaire antérieur dorsal, amygdale gauche-cortex préfrontal dorsolatéral, NAcc droit-cervelet crus II gauche et droit et cortex préfrontal ventromédian-cervelet crus II droit. Discussion : La sensibilisation et l’aspect désagréable moyen de douleur durant une stimulation nociceptive tonique ainsi que la connectivité au repos entre des régions modulant le plaisir (récompense) et la cognition affectent le degré de PPR. / Introduction: Alteration of reward and pain systems is related to the development of chronic pain. In experimental settings, the cessation of a painful and unpleasant stimulation has been shown to elicit a pleasant pain relief (PPR), but contrary to inhibitory conditioned pain modulation mechanisms (ICPM) inter-individual variability of PPR appears to have never been studied. Objectives: Study 1) Investigate inter-individual variability of PPR and ICPM in subgroups of healthy subjects with similar dynamic pain responses (trajectory analyses) during a cold pressor test (CPT) Study 2) Measure resting state functional connectivity with regions of the reward system related to PPR. Method: 1) Peltier thermode and CPT (10℃) were administered sequentially (N=122). The PPR was measured for four minutes after CPT offset 2) PPR was measured (N=38) following the application of a cold gel during functional imaging. Results: 1) Four trajectories were identified based on pain responses during CPT. PPR was correlated with pain trajectories, but not the efficacy of ICPM. 2) Resting state connections, significantly correlated with PPR, were the following: left nucleus accumbens (NAcc)-posterior cingulate cortex, left NAcc-dorsal anterior cingulate cortex, left amygdala-dorsolateral prefrontal cortex, right NAcc-left and right cerebellum crus II and ventromedial prefrontal cortex-right cerebellum crus II. Discussion: Sensitization and average pain unpleasantness during tonic nociceptive stimulation, as well as connectivity between regions modulating pleasure (reward) and cognition, affect the degree of PPR.

No description available.

The moment-to-moment fluctuations in pain-evoked sensory and emotional qualities, and how the pain experience differs between sexes are not well understood. Therefore, this thesis sought to 1) characterise the temporal profiles of the most prominent noxious heat-evoked sensations, 2) characterise sex differences in these sensations, 3) evaluate the magnitude of sharp pain quality evoked in hairy and glabrous skin, and 4) determine the role of absolute stimulus temperatures on sex differences in pain adaptation and habituation. A broad-based heat pain model was developed for this study that incorporates a temporally-continuous assessment of multiple sensory and affective pain dimensions, including pain, burning, sharp, stinging, cutting, and annoyance evoked by two types (static, dynamic) of repeated prolonged noxious heat stimuli. The salient hypotheses were: 1) Burning sensations have a different temporal profile compared with sharp and other related qualities, 2) The temporal dynamics of heat pain intensity and annoyance differ between males and females, 3) Sex differences in heat pain are associated with specific pain qualities and specific types of skin, and 4) Moderate-high temperatures induce pain adaptation and habituation in females but not in males. The most prominent findings were 1) sharp, stinging and cutting sensations adapted when stimulus intensity was static, but burning sensations were evoked during static and dynamic stimulus phases, 2) pain and annoyance in women were greater than men during the dynamic phases of the first stimulus but less than men during static stimulus phases and on stimulus repetition, 3) the sex difference in pain adaptation occurred with percept-fixed stimulus intensities and with absolute stimulus temperatures, 4) the sex effects associated with dynamic stimuli occurred in hairy but not glabrous skin. These findings give new insights into the relationships between pain intensity, quality and affect and have strong implications for views on sex differences in pain sensitivity.

The moment-to-moment fluctuations in pain-evoked sensory and emotional qualities, and how the pain experience differs between sexes are not well understood. Therefore, this thesis sought to 1) characterise the temporal profiles of the most prominent noxious heat-evoked sensations, 2) characterise sex differences in these sensations, 3) evaluate the magnitude of sharp pain quality evoked in hairy and glabrous skin, and 4) determine the role of absolute stimulus temperatures on sex differences in pain adaptation and habituation. A broad-based heat pain model was developed for this study that incorporates a temporally-continuous assessment of multiple sensory and affective pain dimensions, including pain, burning, sharp, stinging, cutting, and annoyance evoked by two types (static, dynamic) of repeated prolonged noxious heat stimuli. The salient hypotheses were: 1) Burning sensations have a different temporal profile compared with sharp and other related qualities, 2) The temporal dynamics of heat pain intensity and annoyance differ between males and females, 3) Sex differences in heat pain are associated with specific pain qualities and specific types of skin, and 4) Moderate-high temperatures induce pain adaptation and habituation in females but not in males. The most prominent findings were 1) sharp, stinging and cutting sensations adapted when stimulus intensity was static, but burning sensations were evoked during static and dynamic stimulus phases, 2) pain and annoyance in women were greater than men during the dynamic phases of the first stimulus but less than men during static stimulus phases and on stimulus repetition, 3) the sex difference in pain adaptation occurred with percept-fixed stimulus intensities and with absolute stimulus temperatures, 4) the sex effects associated with dynamic stimuli occurred in hairy but not glabrous skin. These findings give new insights into the relationships between pain intensity, quality and affect and have strong implications for views on sex differences in pain sensitivity.

One method to increase efficiency, robustness and accuracy of automatic control, is to introduce mathematical models of the system in question to increase performance. With these models, it is possible to predict the behavior of the system, which enables control according to the predictions. The problem here is that if these models do not describe the dynamics of the system well enough, this method could fail to increase performance. To address this problem, one idea is to estimate the dynamics of the system during operation, using methods for system identification, signal processing and sensor fusion. In this thesis, the possibilities of estimating a ship's dynamics during operation have been investigated. The mathematical model describing the dynamics of the ship is a graybox model, which is based on the physical and mechanical relations. This model's properties are therefore described by physical quantities such as mass and moment of inertia, all of which are unknown. This means that, when estimating the model, these physical properties will be estimated. For a systematic approach, first a simulation environment with a 4-degrees-of-freedom ship model has been developed. This environment has been used for validation of system identification methods. A model of a podded propulsion system has also been derived and validated. The methods for estimating the properties of the ship have been analyzed using the data collected from the simulations. For system identification and estimation of ship properties, the influence of measurement noise and potential of detecting a change in dynamics has been analyzed. This has been done through Monte Carlo simulations of the estimation method with different noise realizations in the simulations, to analyze how the measurement noise affects the variance and bias for the estimates. The results show that variance and bias vary a lot between the parameters and that even a small change in dynamics is visible in some parameter estimates when only ten minutes of data have been used. A method based on cumulative summation (CUSUM) has been proposed and validated to analyze if such a method could yield fast and effective detection of system deviations. The results show that the method is rather effective a with robust detection of changes in the dynamics after about four minutes of data collection. Finally, the methods have been validated on data collected on a real ship to analyze the potential of the methods under actual circumstances. The results show that the particular data is not appropriate for this kind of application along with some additional problems that can yield impaired results. / Genom att inkludera matematiska modeller som beskriver ett systems dynamik i styrningsalgoritmer, kan man åstadkomma en automatisk styrning med förbättrad effektivitet, robusthet och noggrannhet. Med dessa modeller går det att förutsäga beteendet hos systemet och därmed öppnas också möjligheten att använda sig av detta i styrningen. Problemet är att om dessa modeller inte beskriver systemets dynamik tillräckligt bra kan prestandan istället sänkas genom dessa metoder. Den här sortens problem kan man lösa genom att aktivt skatta systemets dynamik under körning, med hjälp av metoder för systemidentifiering, signalbehandling och sensorfusion. I denna exjobbsrapport har möjligheterna att skatta ett skepps girdynamik undersökts. Den matematiska modell som beskriver skeppets dynamik är en grålådemodell som baserar sig på fysikaliska och mekaniska samband. Denna modells egenskaper beskrivs därför av fysikaliska storheter så som massa, tröghetsmoment och tyngdpunkt, vilka alla är okända. Detta innebär att vid modellskattning skattas dessa fysikaliska storheter, vilka kan vara av stort intresse. En simuleringsmiljö med en skeppsmodell med fyra frihetsgrader har skapats och använts för att validera metoder för systemidentifiering. En modell av ett roterbart framdrivningssystem har också härletts och inkluderats i simuleringsmodellen. Vid systemidentifiering och skattning av skeppets egenskaper har dels inverkan av mätbrus analyserats samt även möjligheter till att detektera skillnader i dynamik. Detta har gjorts med Monte Carlo-simuleringar av skattningsmetoden med olika brusrealiseringar för att analysera hur mätbrus påverkar variansen och metodfelet hos skattningarna. Resultaten visar att vissa parametrar skattas med större noggrannhet och hos dessa kan därmed en förändring i dynamik identifieras när endast tio minuter av data har använts. En metod baserad på kumulativ summering av residualer har formulerats och validerats, detta för att undersöka om en sådan metod kan ge snabb och effektiv detektion av systemförändringar. Resultat visar på robusthet i att detektera skillnader i dynamik efter ungefär fyra minuter av datainsamling. Slutligen har metoderna validerats på data insamlad på ett riktigt skepp för att undersöka potentialen under verkliga omständigheter. Resultaten visar att just denna data inte är lämplig för denna applikation samt några problem som kan leda till försämrade resultat.

La thèse est composée d’un chapitre de préliminaires et de deux articles sur le sujet du déploiement de singularités d’équations différentielles ordinaires analytiques dans le plan complexe. L’article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity traite le problème de l’équivalence analytique de familles paramétriques de systèmes linéaires en dimension 2 qui déploient une singularité résonante générique de rang de Poincaré 1 dont la matrice principale est composée d’un seul bloc de Jordan. La question: quand deux telles familles sontelles équivalentes au moyen d’un changement analytique de coordonnées au voisinage d’une singularité? est complètement résolue et l’espace des modules des classes d’équivalence analytiques est décrit en termes d’un ensemble d’invariants formels et d’un invariant analytique, obtenu à partir de la trace de la monodromie. Des déploiements universels sont donnés pour toutes ces singularités. Dans l’article Confluence of singularities of non-linear differential equations via Borel–Laplace transformations on cherche des solutions bornées de systèmes paramétriques des équations non-linéaires de la variété centre de dimension 1 d’une singularité col-noeud déployée dans une famille de champs vectoriels complexes. En général, un système d’ÉDO analytiques avec une singularité double possède une unique solution formelle divergente au voisinage de la singularité, à laquelle on peut associer des vraies solutions sur certains secteurs dans le plan complexe en utilisant les transformations de Borel–Laplace. L’article montre comment généraliser cette méthode et déployer les solutions sectorielles. On construit des solutions de systèmes paramétriques, avec deux singularités régulières déployant une singularité irrégulière double, qui sont bornées sur des domaines «spirals» attachés aux deux points singuliers, et qui, à la limite, convergent vers une paire de solutions sectorielles couvrant un voisinage de la singularité confluente. La méthode apporte une description unifiée pour toutes les valeurs du paramètre. / The thesis is composed of a chapter of preliminaries and two articles on the theme of unfolding of singularities of analytic differential equations in a complex domain. They are both related to the problem of local analytic classification of parametric families of linear systems: When two parametric families of linear systems are equivalent by means of an analytic change of coordinates in a neighborhood of the singularity? The article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity deals with the question of analytic equivalence of parametric families of systems of linear differential equations in dimension 2 unfolding a generic resonant singularity of Poincaré rank 1 whose leading matrix is a Jordan bloc. The problem is completely solved and the moduli space of analytic equivalence classes is described in terms of a set of formal invariants and a single analytic invariant obtained from the trace of the monodromy. Universal unfoldings are provided for all such singularities. The article Confluence of singularities of non-linear differential equations via Borel-Laplace transformations investigates bounded solutions of systems of differential equations describing a 1-dimensional center manifold of an unfolded saddle-node singularity in a family of complex vector fields. Generally, a system of analytic ODE at a double singular point possesses a unique formal solution in terms of a divergent power series. The classical Borel summation method associates to it true solutions that are asymptotic to the series on certain sectors in the complex plane. The article shows how to unfold the Borel and Laplace integral transformations of the summation procedure. A new kind of solutions of parameter dependent systems of ODE with two simple (regular) singular points unfolding a double (irregular) singularity are constructed, which are bounded on certain “spiraling” domains attached to both singular points, and which at the limit converge uniformly to a pair of the classical sectorial solutions. The method provides a unified treatment for all values of parameter.

In this thesis, we study the phenomenon of thermally induced polarisation using a combination of theory and computer simulation. Molecules of sufficiently low symmetry exhibit thermo-molecular orientation when subjected to a temperature gradient, leading to considerable electrostatic fields in polar liquids. Here, we first use non-equilibrium molecular dynamics simulations to study this interesting effect numerically. To this end, we propose an integration algorithm to impose a constant heat flux in simulations and show that it greatly improves energy conservation compared to a previous algorithm. We next investigate the thermal polarisation of water and find that truncation of electrostatic interactions can lead to severe artefacts, such as the wrong sign of polarisation and an overestimation of the electric field. We further show that the quadrupole-moment contribution to the electric field is significant and responsible for an inversion of its sign. To facilitate the theoretical description of electrostatic interactions, we propose a new dipolar model fluid as a perturbation of a Stockmayer fluid. Using this modified Stockmayer model, we provide numerical evidence for the recently proposed phenomenon of thermally induced monopoles. We show that the electrostatic field generated by a pair of heated/cooled colloidal particles immersed in such a solvent can be trivially described by two Coulomb charges. Finally, we propose a mean-field theory to predict the thermo-polarisation effect exhibited by our model fluid theoretically, and demonstrate near quantitative agreement with simulation results.

Magister Scientiae - MSc / Pressure drop within a field can be attributed to several factors. Pressure drop occurs when fractional forces cause resistance to flowing fluid through a porous medium. In this thesis, the sciences of petrophysics and rock physics were employed to develop understanding of the physical processes that occurs in reservoirs. This study focussed on the physical properties of rock and fluid in order to provide understanding of the system and the mechanism controlling its behaviour. The change in production capacity of wells E-M 1, 2, 3, 4&5 prompted further research to find out why the there will be pressure drop from the suits of wells and which well was contributing to the drop in production pressure. The E-M wells are located in the Bredasdorp Basin and the reservoirs have trapping mechanisms of stratigraphical and structural systems in a moderate to good quality turbidite channel sandstone. The basin is predominantly an elongated north-west and south-east inherited channel from the synrift sub basin and was open to relatively free marine circulation. By the southwest the basin is enclose by southern Outeniqua basin and the Indian oceans. Sedimentation into the Bredasdorp basin thus occurred predominantly down the axis of the basin with main input direction from the west. Five wells were studied E-M1, E-M2, E-M3, E-M4, and E-M5 to identify which well is susceptible to flow within this group. Setting criteria for discriminator the result generated four well as meeting the criteria except for E-M1. The failure of E-M1 reservoir well interval was in consonant with result showed by evaluation from the log, pressure and rock physics analyses for E-M1.iv Various methods in rock physics were used to identify sediments and their conditions and by applying inverse modelling (elastic impedance) the interval properties were better reflected. Also elastic impedance proved to be an economical and quicker method in describing the lithology and depositional environment in the absence of seismic trace.