Dans cette thèse, on s'intéresse à la propagation d'ondes en milieu fortement hétérogène modélisée par l'équation d'Helmholtz. Les méthodes numériques permettant de résoudre ce problème souffrent de dispersion numérique, en particulier à haute fréquence. Ce phénomène, appelé "effet de pollution", est largement analysé dans la littérature quand le milieu de propagation est homogène et l'utilisation de "méthodes d'ordre élevé" est souvent proposée pour minimiser ce problème. Dans ce travail, on s'intéresse à un milieu de propagation hétérogène, cas pour lequel on dispose de moins de connaissances. On propose d'adapter des méthodes éléments finis d'ordre élevé pour résoudre l'équation d'Helmholtz en milieu hétérogène, afin de réduire l'effet de pollution. Les méthodes d'ordre élevé étant généralement basées sur des maillages "larges", une stratégie multi-échelle originale est développée afin de prendre en compte des hétérogénéités de petite échelle. La convergence de la méthode est démontrée. En particulier, on montre que la méthode est robuste vis-a-vis de l'effet de pollution. D'autre part, on applique la méthode a plusieurs cas-tests numériques. On s'intéresse d'abord à des problèmes académiques, qui permettent de valider la théorie de convergence développée. On considère ensuite des cas-tests "industriels" appliqués à la Géophysique. Ces derniers nous permettent de conclure que la méthode multi-échelle proposée est plus performante que les éléments finis "classiques" et que des problèmes 3D réalistes peuvent être considérés. / The main objective of this work is the design of an efficient numerical strategy to solve the Helmholtz equation in highly heterogeneous media. We propose a methodology based on coarse meshes and high order polynomials together with a special quadrature scheme to take into account fine scale heterogeneities. The idea behind this choice is that high order polynomials are known to be robust with respect to the pollution effect and therefore, efficient to solve wave problems in homogeneous media. In this work, we are able to extend so-called "asymptotic error-estimate" derived for problems homogeneous media to the case of heterogeneous media. These results are of particular interest because they show that high order polynomials bring more robustness with respect to the pollution effect even if the solution is not regular, because of the fine scale heterogeneities. We propose special quadrature schemes to take int account fine scale heterogeneities. These schemes can also be seen as an approximation of the medium parameters. If we denote by h the finite-element mesh step and by e the approximation level of the medium parameters, we are able to show a convergence theorem which is explicit in terms of h, e and f, where f is the frequency. The main theoretical results are further validated through numerical experiments. 2D and 3D geophysica benchmarks have been considered. First, these experiments confirm that high-order finite-elements are more efficient to approximate the solution if they are coupled with our multiscale strategy. This is in agreement with our results about the pollution effect. Furthermore, we have carried out benchmarks in terms of computational time and memory requirements for 3D problems. We conclude that our multiscale methodology is able to greatly reduce the computational burden compared to the standard finite-element method

This thesis presents how Tikhonov’s regularization can be used to solve an inverse problem of Helmholtz equation inside of a rectangle. The rectangle will be met with both Neumann and Dirichlet boundary conditions. A linear operator containing a Fourier series will be derived from the Helmholtz equation. Using this linear operator, an expression for the inverse operator can be formulated to solve the inverse problem. However, the inverse problem will be found to be ill-posed according to Hadamard’s definition. The regularization used to overcome this ill-posedness (in this thesis) is Tikhonov’s regularization. To compare the efficiency of this inverse operator with Tikhonov’s regularization, another inverse operator will be derived from Helmholtz equation in the partial frequency domain. The inverse operator from the frequency domain will also be regularized with Tikhonov’s regularization. Plots and error measurements will be given to understand how accurate the Tikhonov’s regularization is for both inverse operators. The main focus in this thesis is the inverse operator containing the Fourier series. A series of examples will also be given to strengthen the definitions, theorems and proofs that are made in this work.

Friedrich August von Hayek hat meines Erachtens die elaborierteste Neufassung der liberalen Theorie im 20. Jahrhundert formuliert. Hayek selbst sah die Gefahr für eine liberale Großgesellschaft in der Überschätzung der menschlichen Erkenntnisfähigkeit, die Menschen dazu veranlasst, Gesellschaften zu planen. Die Konsequenz dieses falschen Anspruchs sah er in der Zerstörung der sozialen Strukturen, welche die Menschen kognitiv zur Planung erst befähigen. In dieser Arbeit zeige ich, dass es davon abweichend eine zweite Form des Scheiterns des liberalen Projekts gibt, welche aber selbst als Resultat der konsequenten Umsetzung desselben verstanden werden muss. Die liberale Großgesellschaft beruht auf einer bestimmten Geltung von Werten, welche die Individuen anerkennen. Meine These ist, dass einige dieser starken Werte selbst durch Marktkräfte selektiert werden und die Individuen im falschen Bewusstsein, dass diese Werte für die Marktgesellschaft konstituierend sind, das politische System zu einer antiliberalen Gesetzgebung animieren. Um dies zu plausibilisieren, soll der methodologische Individualismus konsequent angewendet werden. Dann muss aber der Darstellung der sozialen Ordnung ein Diskurs vorgelagert werden, der sich mit der Erkenntnisfähigkeit des Individuums beschäftigt. Die Art und Weise, wie es erkennt und was es über die Welt überhaupt wissen kann, ist grundlegend für die Art sozialer Ordnungen, die es realisiert. Die Soziologie von Hayek kann meines Erachtens nur aus der Perspektive seiner theoretischen Psychologie und Erkenntnistheorie angemessen interpretiert werden. Hier kommt der Referenz auf Hermann von Helmholtz Theorie des unbewussten Schlusses eine überragende Bedeutung zu. Mit diesem Ansatz gelingt der Anschluss der Überlegungen Hayeks an die neueste Interpretation des Gehirns als prediction machine durch Karl Friston. Dies führt zu der Ansicht, dass eine Herrschaft der in der Vergangenheit aufgebauten neuronalen Strukturen über den neu einlaufenden sensorischen Input behauptet werden kann. Das Gehirn ist eine statistische Maschine, die auf Basis eines registrierten Ereignisses die Folgeereignisse prognostiziert. Dies hat zufolge, dass die Wahrnehmung der Gegenwart immer eine Simulation einer möglichen Zukunft darstellt. Wir nehmen die Welt wahr, bevor sie sich gezeigt hat. Dadurch, dass Hayek zugleich eine Variante des Funktionalismus eine Dekade vor Hilary Putnam entworfen hat, und die epistemische Funktion des Marktes hervorhebt, kann der Markt im Sinne der von Clark und Chalmers formulierten These als aktiver Externalismus verstanden werden. Er ist somit als soziales System rekonstruierbar, dass die Wissensgrenze jedes Individuums dadurch erweitert, dass die ihn konstituierenden Prozesse der Außenwelt als notwendiger Teil der intern laufenden kognitiven Prozesse der Individuen verstanden werden. Möglich werden Marktgesellschaften dann, wenn Individuen über ganz bestimmte neuronale Repräsentationen verfügen, die sie zu einem ganz bestimmten Handeln und Urteilen bewegen. / In my opinion, Friedrich August von Hayek formulated the most elaborate rewriting of liberal theory in the 20th century. Hayek himself saw the danger to a liberal grand society in the overestimation of human cognition that prompts people to plan societies. He saw the consequence of this false claim in the destruction of the social structures that cognitively empower People to Plan. In this work I show that there is a different form of failure of the liberal project, which itself has to be regarded as a result of the consistent implementation of it. The liberal great society is based on a certain validity of values that individuals recognize. My thesis is that some of these strong Values are self-selected by market forces, and individuals, in the mistaken awareness that these Values are constituent for market society, animate the political system into anti-liberal legislation. In my opinion, Hayek's sociology can only be interpreted appropriately from the perspective of his theoretical psychology and epistemology. In this context, the reference to Hermann von Helmholtz's theory of unconscious inference is of paramount importance. This approach allows the connection of Hayek's reflections to the latest interpretation of the brain as a prediction machine by Karl Friston. This leads to the view that a domination of neural structures built up in the past can be predicated over the newly incoming sensory input. The brain is a statistical Machine that predicts follow-up events based on a registered event. According to this, the perception of the presence is always a simulation of a possible Future. We perceive the world before it has shown itself. By designing a variant of functionalism a decade before Hilary Putnam, highlighting the epistemic function of the market, according to Hayek, the market can be understood as active externalism in the Sense of the thesis formulated by Clark and Chalmers. It can therefore be reconstructed as a social system that expands the knowledge limit of each individual by understanding the processes of the outside world that constitutes it as a necessary part of the internal cognitive processes of Individuals. Market societies become possible when individuals have very specific neural representations that induce them to act and judge in a very specific way. These representations are located on such a high level within Hayek's neural architecture that they decide upon perceptual content. Traditionally, they can be described as norms or values. One of their essential characteristics is their consistency in time. In my opinion, therefore, there is a stage of development within the evolution of the market society, in which the values and normative attitudes that have historically led to the application of the market as a social system must now be abandoned in order to ensure the evolution of the market society. If the adaption time is too small, an interruption of the evolution of the market society may occur. This leads to the paradoxical situation that individuals nevertheless believe that they are realizing a liberal society, which, by the standards of the position developed by Hayek, can only be described as socialist. I will describe this as the self-annulment process of liberal society. In this scenario, the market society does not fail economically, but morally.

Wind power is a rapidly increasing resource of electrical power world-wide. With the increasing number of wind turbines installed one major concern is the noise they generate. Sometimes already built wind turbines have to be put down or down-regulated, when certain noise levels are exceeded, resulting in economical and environmental losses. Therefore, accurate sound propagation calculations would be beneficial already in a planning stage of a wind farm. A model that can account for varying wind speeds and complex terrains could therefore be of great importance when future wind farms are planned. In this report an extended version of the classical wave equation that allows for variations in wind speed and terrain is derived which can be used to solve complex terrain and wind settings. The equation are solved with the use of Fourier transforms and Chebyshev polynomials and a numerical code is developed. The numerical code is evaluated against test cases where analytical and simple solutions exist. Tests with no wind for both totally free propagation and with a ground surface is evaluated in both 2D and 3D settings. For these simple cases the developed code shows good agreement to analytical solutions if the computational domain is sufficiently large. More advanced test cases with wind and terrain is not evaluated in this report and needs further validation. If the sound pressure needs to be calculated for a large area, and if the frequency is high, the developed model has problems regarding computational time and memory. These problems could be solved by further development of the numerical code or by using other solution methods.

A parallel domain decomposition Laplace transform Boundary Element Method, BEM, algorithm for the solution of large-scale transient heat conduction problems will be developed. This is accomplished by building on previous work by the author and including several new additions (most note-worthy is the extension to 3-D) aimed at extending the scope and improving the efficiency of this technique for large-scale problems. A Laplace transform method is utilized to avoid time marching and a Proper Orthogonal Decomposition, POD, interpolation scheme is used to improve the efficiency of the numerical Laplace inversion process. A detailed analysis of the Stehfest Transform (numerical Laplace inversion) is performed to help optimize the procedure for heat transfer problems. A domain decomposition process is described in detail and is used to significantly reduce the size of any single problem for the BEM, which greatly reduces the storage and computational burden of the BEM. The procedure is readily implemented in parallel and renders the BEM applicable to large-scale transient conduction problems on even modest computational platforms. A major benefit of the Laplace space approach described herein, is that it readily allows adaptation and integration of traditional BEM codes, as the resulting governing equations are time independent. This work includes the adaptation of two such traditional BEM codes for steady-state heat conduction, in both two and three dimensions. Verification and validation example problems are presented which show the accuracy and efficiency of the techniques. Additionally, comparisons to commercial Finite Volume Method results are shown to further prove the effectiveness.

Acoustic resonators, such as the Helmholtz and quarter-wave resonator, can be used to attenuate unwanted noise in an enclosed space. Classical formulations can be used to approximate resonator performance for a given resonator configuration, but may lack sufficient accuracy for some applications. This research aims to improve the analytical characterization of resonators to provide better correlation to experimental results. Using higher-order approximations and proper end corrections, more accuracy can be obtained in calculating the impedance and resonance frequency of a single resonator, which will then carry over into the overall configuration of the model. The impedance of a system of resonators in parallel is also considered, where the effects of acoustic coupling can be observed. Resonators with complex, non-ideal geometries are explored for applications where space is limited. The effects of tapers and toroidal curves are considered using impedance translation methods. These theoretical predictions are found to compare favorably with empirical data. Coupling between an enclosure and resonator system is explored experimentally. The effects of resonator placement, damping, and relative cavity and enclosure volume are considered. These data are used to design and test a resonator system with 10 dB of attenuation over a bandwidth of 10 Hz.

The most common equipment used for attitude determination in small satellites are magnetometers. However, using magnetometers gives rise to many challenges. One of these challenges is the calibration of the magnetometer. Magnetometer calibration takes many factors into account. There are external and internal factors. External factors can be the satellite itself. Satellites are built of many complex subsystems. These subsystems can produce magnetic disturbances and affect the measurements taken by the magnetometer ,which also affects the attitude determination of the satellite. Internal factors are nonorthogonalityand scale factors. In this project, we aim to test different calibration methods and compare the results. Another objective is to provide a complete procedure for a calibration of the magnetometer using the Helmholtz coils. The comparison of the results with other methods can help with the decision of which should be used to calibrate the magnetometer onboard the satellite for future calibrations for MIST satellites. / En av de mest vanliga verktyg för attitydbestämning i små satteliter är magnetometer, men att använda magnetometer kan leda till många utmaningar, en av de kalibrering av magnetometern. Magnetometer kalibrering är beroende av många faktorer. Det finns inre och yttre faktorer. Yttre faktorer kan vara själva satelliten. Del system som bildar satelliten kan påverka mätningar i magnetometern och då påverkar attitydbestämning av hela satelliten. Inre faktorer är icke ortogonalitet och skalära faktorer. I det här projektet vi ska testa olika kalibrerings metoder och jämföra resultaten. Ett annat mål är att bygga en komplett procedur för att kalibrera magnetometer med hjälp av Helmholtz spolar. Jämförelsen och resultaten från kalibreringen visar hur det är möjligt att kalibrnetometer som är integrerad i satelliten för kommande kalibreringar i MIST. / Kandidatexjobb i elektroteknik 2022, KTH, Stockholm

A bass-reflex system is a type of loudspeaker design that uses a port or a vent to improve low-frequency performance. The port acts as a Helmholtz resonator which extends the bass response of the system. However, at high drive levels, the air inside the port can become turbulent and cause distortion, noise, and compression. From previous works, it is known that the geometry of the port plays a crucial role in reducing these unwanted effects. This paper serves to provide more insight into optimal port shape by performing several objective tests on a group of 5 different prototype port shapes based on findings from previous literature. Total Harmonic Distortion (THD) and port compression tests were conducted to determine which port presented the highest performance.

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