Quantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the “best-fit” parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 10⁶ parameters.

Darbo metu apžvelgiami metodai pereinamiesiems procesams vamzdyne realizuoti. Aprašyta vamzdyno tėkmės pereinamųjų procesų dinamika matematinės lygtys ir sudarytas baigtinių elementų modelis skaitiniam vamzdynų pereinamųjų procesų modeliavimui. Panaudojant atgalinį slėgio bangos modeliavimą esant idealizuotam slėgio bangos sklidimo modeliui, sudaromas slėgio impulso vietos nustatymo algoritmas ir įvertinamos atsirandančios paklaidos. Realizuota programine įranga patikrinamas sudarytų modelių ir algoritmo teisingumas, bei pateikiami pasiūlymai, atsiradusioms paklaidoms mažinti. / During this research, the methods to realize the transient processes in piping systems are overviewed, also finite element method and idealized wave propagation of pressure mathematical models are set up. Using the backward pressure wave modelling at idealized pressure wave propagation model, piping system rupture location algorithm and the assessment of estimated errors are set up. With released software, the developed models and correctness of the algorithm are verified, and suggestions to reduce the resulting errors are presented.

In dieser Arbeit werden Konzepte für die Diagnostik der großskaligen Zirkulation in der Troposphäre und Stratosphäre entwickelt. Der Fokus liegt dabei auf dem Energiehaushalt, auf der Wellenausbreitung und auf der Interaktion der atmosphärischen Wellen mit dem Grundstrom. Die Konzepte werden hergeleitet, wobei eine neue Form des lokalen Eliassen-Palm-Flusses unter Einbeziehung der Feuchte eingeführt wird. Angewendet wird die Diagnostik dann auf den Reanalysedatensatz ERA-Interim und einen durch beobachtete Meerestemperatur- und Eisdaten angetriebenen Lauf des ECHAM6 Atmosphärenmodells. Die diagnostischen Werkzeuge zur Analyse der großskaligen Zirkulation sind einerseits nützlich, um das Verständnis der Dynamik des Klimasystems weiter zu fördern. Andererseits kann das gewonnene Verständnis des Zusammenhangs von Energiequellen und -senken sowie deren Verknüpfung mit synoptischen und planetaren Wellensystemen und dem resultierenden Antrieb des Grundstroms auch verwendet werden, um Klimamodelle auf die korrekte Wiedergabe dieser Beobachtungen zu prüfen. Hier zeigt sich, dass die Abweichungen im untersuchten ECHAM6-Modelllauf bezüglich des Energiehaushalts klein sind, jedoch teils starke Abweichungen bezüglich der Ausbreitung von atmosphärischen Wellen existieren. Planetare Wellen zeigen allgemein zu große Intensitäten in den Eliassen-Palm-Flüssen, während innerhalb der Strahlströme der oberen Troposphäre der Antrieb des Grundstroms durch synoptische Wellen verfälscht ist, da deren vertikale Ausbreitung gegenüber den Beobachtungen verschoben ist. Untersucht wird auch der Einfluss von arktischen Meereisänderungen ausgehend vom Bedeckungsminimum im August/September bis in den Winter. Es werden starke positive Temperaturanomalien festgestellt, welche an der Oberfläche am größten sind. Diese führen vor allem im Herbst zur Intensivierung von synoptischen Systemen in den arktischen Breiten, da die Stabilität der troposphärischen Schichtung verringert ist. Im darauffolgenden Winter stellen sich barotrope bis in die Stratosphäre reichende Änderungen der großskaligen Zirkulation ein, welche auf Meereisänderungen zurückzuführen sind. Der meridionale Druckgradient sinkt und führt so zu einem Muster ähnlich einer negativen Phase der arktischen Oszillation in der Troposphäre und einem geschwächten Polarwirbel in der Stratosphäre. Diese Zusammenhänge werden ebenfalls in einem ECHAM6-Modelllauf untersucht, wobei vor allem der Erwärmungstrend in der Arktis zu gering ist. Die großskaligen Veränderungen im Winter können zum Teil auch im Modelllauf festgestellt werden, jedoch zeigen sich insbesondere in der Stratosphäre Abweichungen für die Periode mit der geringsten Eisausdehnung. Die vertikale Ausbreitung planetarer Wellen von der Troposphäre in die Stratosphäre ist in ECHAM6 mit sehr großen Abweichungen wiedergegeben. Somit stellt die Wellenausbreitung insgesamt den größten in dieser Arbeit festgestellten Mangel in ECHAM6 dar. / In this study concepts for the diagnostics of the large-scale circulation in the troposphere and the stratosphere are developed. Therefore the energy budget, wave propagation and the interaction between waves and the mean flow are analyzed. A corresponding set of diagnostic methods is derived. Furthermore a new type of localized Eliassen Palm Fluxes including moisture fluxes is introduced. These diagnostic methods are then applied to the ERA-Interim reanalysis and to a run of the ECHAM6 atmospheric model forced with observed sea surface temperatures and sea ice data. The diagnostics of the large scale circulation are useful to enhance the understanding of the climate system dynamics. Furthermore the knowledge of the relation between energy sources and sinks, atmospheric waves on planetary and synoptic scales and their forcing of the mean flow is applicable to validate global climate models. The results presented here show small deviations in terms of the energy balance in ECHAM6 but large discrepancies in terms of wave propagation. On the one hand Eliassen Palm fluxes connected to planetary waves are generally too strong. On the other hand the mean flow forcing within upper tropospheric jet streams by synoptic scale waves does not agree with observations, since the vertical propagation is shifted. A second part of this study analyses the influence of Arctic sea ice anomalies at the sea ice minimum in August/September on atmospheric conditions. Strong positive temperature anomalies with their maximum at the surface are observed. In autumn they lead to intensified synoptic scale systems, because of a reduced atmospheric vertical stability. A large-scale barotropic circulation pattern up to the stratosphere appears in winter related to previous late summer sea ice changes. A reduced meridional pressure gradient leads to a pattern related to a negative phase of the Arctic Oscillation in the troposp here as well as related to a weaker stratospheric polar vortex. The same analysis performed with an ECHAM6 model run shows a too small warming of Arctic latitudes. While tropospheric changes in the Arctic are covered by the model to some degree, the stratosphere shows large discrepancies in reproducing the observed changes in the low ice period. The vertical propagation of planetary waves from the troposphere into the stratosphere is reproduced with large differences. Accordingly this study shows the largest errors in ECHAM6 related to atmospheric wave propagation.

The near field blast–wave propagation dynamics has been a subject of intense research in recent past. Since experiments on a large scale are difficult to carry out, focus has been directed towards recreating these blast waves inside the laboratory by expending minuscule amounts of energy(few joules),which have been termed here as micro–blast waves. In the present study, micro-blast waves are generated from the open end of a small diameter polymer tube (Inner Diameter of 1.3 mm)coated on its inner side with negligible amounts of HMX explosive (~18 mg/m), along with traces of aluminium powder. Experimental, numerical, and analytical approaches have been adopted in this investigation to understand the generation and subsequent propagation of these micro–blast waves in the open domain. Time–resolved schlieren flow visualization experiments, using a high speed digital camera, and dynamic pressure measurements (head–on and side–on pressures) have been carried out. Quasi one dimensional numerical modeling of the detonation process inside the tube, has been carried out by considering the reaction kinetics of a single(HMX) reaction to account for the reaction dynamics of HMX. The one dimensional numerical model is then coupled to a commercial Navier– Stokes equation solver to understand the propagation of the blast wave from the open end of the tube. A theory that is valid for large scale explosions of intermediate strength was then used for the first time to understand the propagation dynamics of these micro–blast waves. From the experiments, the trajectory of the blast wave was mapped, and its initial Mach number was found to be about 3.7. The side–on overpressure was found to be 5.5 psi at a distance of 20 mm from the tube, along an axis, offset by 30 mm from the tube axis. These values were found to compare quite well with the numerically obtained data in the open domain. From the numerical model of the tube, the energy in the blast wave was inferred to be 1.5 J. This value was then used in the analytical theory and excellent correlation was obtained, suggesting the exciting possibility of using such theories, validated for large-scale explosions, to describe these micro–blasts. Considering the uncertainties in the approximate model, a better estimate of energy was obtained by working back the energy(using the analytical model) from the trajectory data as 1.25 J. The average TNT equivalent, a measure of its strength relative to a TNT explosion, was found to be 0.3. A few benchmark experiments, demonstrating the capability of this novel blast device have also been done by comparing them against the extant large–scale explosion database, suggesting the possibility of using these micro–blast waves to study certain aspects of large–scale explosions.

The traffic engineering community currently faces the advent of a new generation of Intelligent Transportation Systems (ITS), known as cooperative systems. More specifically, the recent developments of connected and autonomous vehicles, i.e. cooperative vehicles, are expected to cause a societal shift, changing the way people commute on a daily basis and relate to transport in general. The research presented in this dissertation is motivated by the need for proper understanding of the possible inputs of cooperative vehicles in a traffic stream. Beyond legal aspects regarding the introduction of such vehicles and considerations on standardization and harmonization of the communication norms, the research focuses on the use of communication for highway traffic flow homogenization. In particular, the selected approach for the introduction of cooperation inherits from the theory of traffic flow and the recent developments of microscopic traffic models. Cooperation can first be introduced as a form of multi-anticipation, which can either come from drivers' behaviors or from communication. A mathematical framework for investigating the impact of perturbations into a steady-state traffic is proposed for the class of time continuous car-following models. Linear stability analyses are refined for forward and backward multi-anticipation, exploring the underlying importance of considering upstream information. The linear stability analyses for all wavelengths can be deepened by the mean of the graphical root locus analysis, which enables comparisons and design of strategies of cooperation. The positive influence of bilateral cooperation and of added linear control terms are highlighted. Weakly non-linear analyses are also performed, and the equations of solitary waves appearing at the frontier of the instability domain are obtained. A simple condition over the partial derivatives of the dynamical system is found to determine the acceleration regime of the leading edge of the travelling wave. Following these analytical results, one aim is to simulate a realistic traffic thereby reproducing the driving behavior variability. A Next Generation Simulation trajectory dataset is used to calibrate three continuous car-following models. A methodology involving data filtering, robust calibration, parameters estimation and sampling of realistic parameters is detailed, and allows realistic traffic with stop-and-go waves appearances to be replicated. Based on these simulated trajectories, previous analytical results are confirmed, and the growing perturbations are removed for various coverage rates of cooperative vehicles and adequately tuned cooperative strategies. Finally the issue of information reliability is assessed for a mixed fleet of cooperative and non-cooperative vehicles. The modeling choice consists in building a three layers multi-agent framework that enables the following properties to be defined: the physical behavior of vehicles, the communication possibilities, and the trust each vehicle -or agent- has in another vehicle information or in itself. The investigation of trust and communication rules allow the model to deal with high rates of disturbed cooperative vehicles sensors and to learn in real time the quality of the sent and received information. It is demonstrated that appropriate communication and trust rules sensibly increase the robustness of the network to perturbations coming from exchanges of unreliable information.

Two separate problems are to be investigated in this thesis. The first problem is the propagation of waves through a short rod (or slug) of viscoelastic material. The second problem is the study of impact of a short viscoelastic rod (or slug) on a stationary semi-infinite viscoelastic. rod. The viscoelastic materials are modelled as standard linear solids which involve 3 material parameters and the motion is treated. as one-dimensional. For the first study, a viscoelastic slug is placed between two semi-infinite elastic rods and a wave initiated in the first rod is transmitted through the slug into the second rod. The objective is to relate the transmitted signal to the material parameters of the slug. We solve the governing system of partial differential equations using the Laplace transform and we examine the propagating velocity discontinuity using discontinuity analysis and the limit theorem of the Laplace transform. We then approximate the solution of the propagating disturbance using the regular perturbation method. We invert the Laplace transformed solution numerically to obtain the transmitted signal for several viscosity time constants and ratios of acoustic impedances. We compare the results obtained using the above techniques. In the second problem, we first model the impact and solve the governing system of partial differential equations in the Laplace transform domain. Then we examine the propagating stress and velocity discontinuities using discontinuity analysis. We approximate the solutions of the propagating stress and velocity using the regular and multiple scales perturbation methods. In this problem, we first consider the slug is elastic and the rod is viscoelastic. Secondly, we consider the slug is viscoelastic and the rod is elastic and thirdly, we consider both materials are viscoelastic. Numerically we invert the Laplace transformed solutions for the interface stress and interface velocity for several viscosity time constants and ratios of acoustic impedances to determine whether the slug and the rod part company or remain in contact. Then we compare the results obtained using the discontinuity analysis, regular and multiple scales perturbation methods.

Boundary shape optimization is a technique to search for an optimal shape by modifying the boundary of a device with a pre-specified topology. We consider boundary shape optimization of acoustic horns in loudspeakers and brass wind instruments. A horn is an interfacial device, situated between a source, such as a waveguide or a transducer, and surrounding space. Horns are used to control both the transmission properties from the source and the spatial power distribution in the far-field (directivity patterns). Transmission and directivity properties of a horn are sensitive to the shape of the horn flare. By changing the horn flare we design transmission efficient horns. However, it is difficult to achieve both controllability of directivity patterns and high transmission efficiency by using only changes in the horn flare. Therefore we use simultaneous shape and so-called topology optimization to design a horn/acoustic-lens combination to achieve high transmission efficiency and even directivity. We also design transmission efficient interfacial devices without imposing an upper constraint on the mouth diameter. The results demonstrate that there appears to be a natural limit on the optimal mouth diameter. We optimize brasswind instruments with respect to its intonation properties. The instrument is modeled using a hybrid method between a one-dimensional transmission line analogy for the slowly flaring part of the instrument, and a finite element model for the rapidly flaring part. An experimental study is carried out to verify the transmission properties of optimized horn. We produce a prototype of an optimized horn and then measure the input impedance of the horn. The measured values agree reasonably well with the predicted optimal values. The finite element method and the boundary element method are used as discretization methods in the thesis. Gradient-based optimization methods are used for optimization, in which the gradients are supplied by the adjoint methods.

In ground based high frequency (HF) radio pumping experiments, absorption of ordinary (O) mode pump waves energises the ionospheric plasma, producing optical emissions and other effects. Pump-induced or natural kilometre-scale field-aligned density depletions are believed to play a role in self-focussing phenomena such as the magnetic zenith (MZ) effect, i.e., the increased plasma response observed in the direction of Earth's magnetic field. Using ray tracing, we study the propagation of ordinary (O) mode HF radio waves in an ionosphere modified by density depletions, with special attention to transmission through the radio window (RW), where O mode waves convert into the extraordinary (X, or Z) mode. The depletions are shown to shift the position of the RW, or to introduce RWs at new locations. In a simplified model neglecting absorption, we estimate the wave electric field strength perpendicular to the magnetic field at altitudes normally inaccessible. This field could excite upper hybrid waves on small scale density perturbations. We also show how transmission and focussing combine to give stronger fields in some directions, notably at angles close to the MZ, with possible implications for the MZ effect. In a separate study, we consider electromagnetic (e-m) beams with helical wavefronts (i.e., twisted beams), which are associated with orbital angular momentum (OAM). By applying geometrical optics to each plane wave component of a twisted nonparaxial e-m Bessel beam, we calculate analytically the shift of the beam's centre of gravity during propagation perpendicularly and obliquely to a weak refractive index gradient in an isotropic medium. In addition to the so-called Hall shifts expected from paraxial theory, the nonparaxial treatment reveals new shifts in both the transverse and lateral directions. In some situations, the new shifts should be significant also for nearly paraxial beams.

This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.   The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.   The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.

As many of the world’s nuclear power plants are near the end of their supposed life span a need arise to assess the components crucial to the safety of these plants. One of these crucial components is the concrete reactor confinement; to assess its condition, non-destructive testing (NDT) is an attractive method. Traditional testing of concrete structures has comprised of drilling out a sample and performing stress tests on it, but because of the radioactive environment inside the containment this method is far from ideal. NDT is of course possible to use at any structure but at reactor containments the benefits from not creating holes in the structure are prominent; NDT is also an attractive option from an esthetical point of view because it leaves the structure intact. The NDT method pertaining to this study is the impact echo method which comprise of applying a force on the structure, usually a hammer blow, and measuring the response with a receiver. The impact will excite waves propagating in the structure which gives rise to Lamb modes. Lamb modes are structural oscillations of the wall and it is the frequency of these modes that are used to determine the thickness of the wall. The elastic properties of the structure can in turn be obtained by measuring the velocities of the waves propagation. It is also possible to use the impact echo method to detect irregularities in the structure such as cracks or delamination. To simulate the dynamics of a system using NDT numerical methods such as finite element modeling (FEM) is often used. The purpose of this study is to assess the possibility to utilize absorbing layers using increasing damping (ALID) in models to reduce the computational time of FEM analyses. ALIDs are used at the edges to simulate an infinite system and are thus supposed to cancel out incoming waves to prevent unwanted reflection from the edges. The models in this study have all pertained to two dimensional plates utilizing infinitesimal strain theory; the decrease in computational time is significant when using ALIDs and for three dimensional models it would be even more so. The ALIDs are specified by length and maximum mass proportional Rayleigh damping (CMmax), in this study three different lengths are tested, 0.5, 1.5 and 4.5 m for CMmax ranging from 103 to 2*105 Ns/m. The damping is increased with increasing distance into the ALID with specified maximum value at the back edge. However, it should be noted that the increase in damping causes difference in impedance between elements and if this difference is too large it will cause reflections of waves at the boundary between the elements. The ALID must thus be defined so that it sufficiently cancels out the wave without causing unwanted reflections due to impedance differences. The conclusion is that the 0.5 m long ALID does not provide good results for any choice of maximum mass proportional Rayleigh damping. Both the 1.5 and 4.5 m long ALIDs are, however, concluded to be applicable; the 1.5 m ALID having 2*104 < CMmax <5*104 Ns/m and the 4.5 m ALID having 5*103 < CMmax < 104 Ns/m are choices that have shown promise in the performed simulations. The hope is that the results obtained in this study will aid in the development of numerical analysis techniques for NDT methods that can be used in the construction of new reactor confinements and/or maintenance of existing reactor confinements and other thick concrete structures. / Många av världens kärnkraftverk närmar sig slutet på sin beräknade livslängd och ett behov uppstår då att kunna utvärdera de komponenter som är väsentliga för säkerheten på dessa verk. Reaktoromslutningen i betong är en av dessa komponenter och oförstörande provning (NDT) är en attraktiv metod för att bedöma dess tillstånd. Traditionellt har utvärdering av betongkonstruktioner bestått av stresstester på borrprover men p.g.a. den radioaktiva miljön på insidan av omslutningen är denna metod ej att föredra. NDT är självklart möjligt att använda på allsköns betongkonstruktioner då det ger både konstruktionsmässiga och estetiska fördelar. NDT metoden som rör denna studie kallas impact echo och går ut på att man med en hammare slår till en punkt på väggen och mäter responsen en bit därifrån. Lasten ger upphov till vågor i form av deformation som propagerar i väggen och dessa ger i sin tur upphov till Lamb moder. Lamb moderna är strukturella oscillationer av väggen och genom att studera dess frekvenser kan väggens tjocklek bestämmas. Elastiska egenskaper i väggen erhålls utifrån de olika vågornas propageringshastigheter. Impact echo metoden kan även användas för att finna strukturella oegentligheter inuti väggen så som sprickor och delaminering. För att utföra numeriska simuleringar av dynamiska system med NDT-metoder är finita elementmetoden (FEM) användbar. Syftet med denna studie är att bedöma vilka möjligheter som finns för att implementera absorberande ränder med ökande dämpning (ALID) i datamodeller för att minska beräkningstiden av FEM-analyser. ALID används vid kanterna för att simulera ett oändligt system, dess uppgift är att dämpa bort inkommande vågor så att dessa ej reflekteras tillbaka och stör mätningarna. Samtliga modeller i denna studie är två-dimensionella med antagen oändligt liten spänning i normalriktningen. Vinsten i beräkningstid av att använda ALID är stor och ökar ytterligare om modellen utökas till tre dimensioner. Ett ALID definieras genom dess längd och maximala massproportionerlig Rayleigh-dämpning (CMmax). I denna rapport har längderna 0.5, 1.5 and 4.5 m använts med CMmax i intervallet från 103 till 2*105 Ns/m. Dämpningen ökar med ökat avstånd in i ALID med det specificerade maxvärdet vid den bakre kanten. Det bör noteras att skillnad i dämpning mellan element leder till skillnad i impedans; reflektioner av vågorna uppstår vid övergång från ett element med lägre impedans till ett med högre impedans. Ett ALID måste således vara definierat så att det dämpar bort tillräckligt av de inkommande vågorna utan att oönskade reflektioner i ALID uppstår. Studien pekar på att ett 0.5 m långt ALID inte åstadkommer önskvärda resultat för något av valen för CMmax som använts i denna rapport. Både det 1.5 och 4.5 m långa ALID har däremot get bra resultat; ett 1.5 m långt ALID bör ha 2*104 < CMmax <5*104 Ns/m och ett 4.5 m långt ALID 5*103 < CMmax < 104 Ns/m. Förhoppningen med studien är att resultaten skall underlätta utvecklingen av NDT-metoder som kan användas vid konstruktion och underhåll av reaktoromslutningar och andra tjocka betongkonstruktioner.