Esta tese é dedicada ao estudo de uma classe de equações de ondas com condições de fronteira da acústica. Investigamos a dinâmica assintótica de tais equações no caso em que o sistema está sujeito à ação de uma força externa não autônoma. Nessa situação, adicionando uma dissipação fraca, provamos que o problema gera um processo de evolução dissipativo. O nosso objetivo é então o estudo da existência de atratores não autônomos. Num primeiro momento estabelecemos a existência de um atrator do tipo \\pullback\", minimal, dentro de um universo de conjuntos temperados. Também estudamos a semicontinuidade superior dos atratores quando a perturbação não autônoma tende para zero. Nosso resultado permite considerar forcas externas não limitadas e perturbações não lineares com crescimento crítico (de Sobolev). Num segundo momento, fazemos um estudo sobre a existência de atratores uniformes. Em vista de resultados recentes de Zelik (2015), consideramos forcas externas mais gerais do que a dita classe das forcas compactas por translação (translation-compact). Parte desta tese foi aceita para publicação na revista \\Differential and Integral Equations\" sob o ttulo \\Pullback dynamics of non-autonomous wave equations with acoustic boundary condition\". / This thesis is concerned with the study of a class of wave equations with acoustic boundary conditions. We investigate the long-time dynamics of such equations in the case where the system is subject to a non-autonomous external force. In this situation, by adding a weak dissipation, we prove that the problem generates a dissipative evolution process. Our goal is then the existence of non-autonomous attractors. In this direction, we first establishes the existence of a minimal pullback attractor within a universe of tempered sets. We also studied the upper semi-continuity of attractors when the non-autonomous perturbation tends to zero. Our result allows to consider unbounded external forces and nonlinear perturbation with critical (Sobolev) growth. Secondly, we establish the existence of uniform attractors, as well. In view of recent results Zelik (2015) we consider more general external forces than the so called class of translation-compact forces. Part of this thesis was accepted for publication in the journal \\Differential and Integral Equations\" under the title \\Pullback dynamics of non-autonomous wave equations with acoustic boundary condition\".

Darbą sudaro: įvadas, analitinė ir praktinės dalys. Analitinėje dalyje trumpai aptariama su nagrinėjamu uždaviniu susijusi teorija ir pats uždavinys. Analitinėje dalyje yra trys poskyriai, juose pateikiama teorija, reikalinga nagrinėjamo uždavinio tyrimui. Praktinėje dalyje nagrinėjamas Šturmo ir Liuvilio kraštinis uždavinys su viena klasikine ir kita nelokalia dvitaške kraštine sąlyga. Ištirti keturių nelokalių kraštinių sąlygų atvejai. Kiekvienu kraštinių sąlygų atveju ieškomos kompleksinės tikrinės reikšmės ir tiriama jų kokybinė priklausomybė nuo uždavinio nelokaliosios sąlygos parametrų ir . Darbas iliustruotas charakteristinių funkcijų grafikais. Nustatyta, kad dviejų pirmųjų sąlygų atveju yra pakankamai nesudėtinga charakterisitnių funkcijų priklausomybė nuo parametro . Kitiems dviems atvejams ta priklausomybė yra žymiai sudėtingesnė. Tokios situacijos kruopščiai ištirtos ketvirtame atvejyje. Surasta tikrinių reikšmių elgsena bifurkacinių taškų aplinkoje. / This master thesis consists of introduction, analitical and practical parts. In the analitical part are considered the master thesis problem and theoretical studies, that are correlative with it. This part is rubricated to tree sections. They are designed for theoretical studies that are used to solve all problem of the master thesis. In practical part are examinated four problems: Šturm Louville problem with classical and nonlocal boundary condition. Here are investigated types of four nonlocal boundary conditions. In case of every nonlocal boundary conditions locking for complex eigenvalues and investigating thier quality that dependsupon nonlocal boundary condition parametere and . The work is pictorial of charakterical fukcion graphics. Is determinated, that in case of two condition charakteristical funkcion dependen of parameter is simple. In case of other two condition the dependen is convulated. This situation is examinated in the fourth case. Besides here resolved behaviour of eigenvalues in their points environment.

In the thesis there is studied nonhomogenous boundary value problem for the stationary Navier-Stokes system in domains which may have two types of outlets to infinity: paraboloidal and layer type. The boundary is multiply connected. It consists of connected noncompact components, forming the outer boundary, and connected compact components, forming the inner boundary. We suppose that the fluxes over the components of the inner boundary are sufficiently small, while we do not impose any restrictions on fluxes over the infinite components of the outer boundary. We prove that the formulated problem admits at least one weak solution which, depending on the geometry of the domain, may have either finite or infinite Dirichlet integral. / Disertacijoje nagrinėjama stacionari Navjė-Stokso sistema su nehomogenine kraštine sąlyga srityse su išėjimais į begalybę. Bendru atveju išėjimai į begalybę gali būti tiek paraboloidiniai, tiek sluoksnio tipo. Srities kraštą sudaro baigtinis skaičius nekompaktiškų jungių komponenčių, kurios suformuoja išorininį kraštą, ir baigtinis skaičius kompaktiškų jungių komponenčių, kurios suformuoja vidinį srities kraštą. Darydami prielaidą, kad srautai per vidinio krašto komponentes yra pakankamai maži, o srautų dydžiui per išorinio krašto komponentes nedarant jokių apribojimų, įrodome suformuluoto uždavinio bent vieno sprendinio egzistavimą. Priklausomai nuo srities geometrijos, uždavinio sprendinys gali turėti tiek baigtinį, tiek begalinį Dirichlė integralą.

Disertacijoje nagrinėjama stacionari Navjė-Stokso sistema su nehomogenine kraštine sąlyga srityse su išėjimais į begalybę. Bendru atveju išėjimai į begalybę gali būti tiek paraboloidiniai, tiek sluoksnio tipo. Srities kraštą sudaro baigtinis skaičius nekompaktiškų jungių komponenčių, kurios suformuoja išorininį kraštą, ir baigtinis skaičius kompaktiškų jungių komponenčių, kurios suformuoja vidinį srities kraštą. Darydami prielaidą, kad srautai per vidinio krašto komponentes yra pakankamai maži, o srautų dydžiui per išorinio krašto komponentes nedarant jokių apribojimų, įrodome suformuluoto uždavinio bent vieno sprendinio egzistavimą. Priklausomai nuo srities geometrijos, uždavinio sprendinys gali turėti tiek baigtinį, tiek begalinį Dirichlė integralą. / In the thesis there is studied nonhomogenous boundary value problem for the stationary Navier-Stokes system in domains which may have two types of outlets to infinity: paraboloidal and layer type. The boundary is multiply connected. It consists of connected noncompact components, forming the outer boundary, and connected compact components, forming the inner boundary. We suppose that the fluxes over the components of the inner boundary are sufficiently small, while we do not impose any restrictions on fluxes over the infinite components of the outer boundary. We prove that the formulated problem admits at least one weak solution which, depending on the geometry of the domain, may have either finite or infinite Dirichlet integral.

Damage identification is an important topic in structural assessment and structural health monitoring (SHM). Vibration-based identification techniques use modal data to identify the existence, location and severity of possible damages in structures, often via a numerical model updating procedure. Among other factors influencing the practicality and reliability of a damage identification approach, two are of primary interest to this study. The first one concerns the amount and quality of modal data that can be used as ‘response’ data for the model updating. It is generally recognised that natural frequencies can be measured with relatively high accuracy; however, their number is limited. Mode shapes, on the other hand, are susceptible to larger measurement errors. Seeking additional modal frequency data is therefore of significant value. The second one concerns the errors at the numerical (finite element) model level, particularly in the representation of the effect of damage on the dynamic properties of the structure. An inadequate damage model can lead to inaccurate and even false damage identification. The first part of the thesis is devoted to enhancing the modal dataset by extracting the so called ‘artificial boundary condition’ (ABC) frequencies in a real measurement environment. The ABC frequencies correspond to the natural frequencies of the structure with a perturbed boundary condition, but can be generated without the need of actually altering the physical support condition. A comprehensive experimental study on the extraction of such frequencies has been conducted. The test specimens included steel beams of relatively flexible nature, as well as thick and stiffer beams made from metal material and reinforced concrete, to cover the typical variation of the dynamic characteristics of real-life structures in a laboratory condition. The extracted ABC frequencies are subsequently applied in the damage identification in beams. Results demonstrate that it is possible to extract the first few ABC frequencies from the modal testing in different beam settings for a variety of ABC incorporating one or two virtual pin supports. The inclusion of ABC frequencies enables the identification of structural damages satisfactorily without the necessity to involve the mode shape information. The second part of the thesis is devoted to developing a robust model updating and damage identification approach for beam cracks, with a special focus on thick beams which present a more challenging problem in terms of the effect of a crack than slender beams. The priority task has been to establish a crack model which comprehensively describes the effect of a crack to reduce the modelling errors. A cracked Timoshenko beam element model is introduced for explicit beam crack identification. The cracked beam element model is formulated by incorporating an additional flexibility due to a crack using the fracture mechanics principles. Complex effects in cracked thick beams, including shear deformation and coupling between transverse and longitudinal vibrations, are represented in the model. The accuracy of the cracked beam element model for predicting modal data of cracked thick beams is first verified against numerically simulated examples. The consistency of predictions across different modes is examined in comparison with the conventional stiffness reduction approach. Upon satisfactory verification, a tailored model updating procedure incorporating an adaptive discretisation approach is developed for the implementation of the cracked beam element model for crack identification. The updating procedure is robust in that it has no restriction on the location, severity and number of cracks to be identified. Example updating results demonstrate that satisfactory identification can be achieved for practically any configurations of cracks in a beam. Experimental study with five solid beam specimens is then carried out to further verify the developed cracked beam element model. Both forward verification and crack damage identification with the tested beams show similar level of accuracy to that with the numerically simulated examples. The cracked beam element model can be extended to crack identification of beams with complex cross sections. To do so the additional flexibility matrix for a specific cross-section type needs to be re-formulated. In the present study this is done for box sections. The stress intensity factors (SIF) for a box section as required for the establishment of the additional flexibility matrix are formulated with an empirical approach combining FE simulation, parametric analysis and regression analysis. The extended cracked beam element model is verified against both FE simulated and experimentally measured modal data. The model is subsequently incorporated in the crack identification for box beams. The successful extension of the cracked beam element model to the box beams paves the way for similar extension to the crack identification of other types of sections in real-life engineering applications.

Esta tese é dedicada ao estudo de uma classe de equações de ondas com condições de fronteira da acústica. Investigamos a dinâmica assintótica de tais equações no caso em que o sistema está sujeito à ação de uma força externa não autônoma. Nessa situação, adicionando uma dissipação fraca, provamos que o problema gera um processo de evolução dissipativo. O nosso objetivo é então o estudo da existência de atratores não autônomos. Num primeiro momento estabelecemos a existência de um atrator do tipo \\pullback\", minimal, dentro de um universo de conjuntos temperados. Também estudamos a semicontinuidade superior dos atratores quando a perturbação não autônoma tende para zero. Nosso resultado permite considerar forcas externas não limitadas e perturbações não lineares com crescimento crítico (de Sobolev). Num segundo momento, fazemos um estudo sobre a existência de atratores uniformes. Em vista de resultados recentes de Zelik (2015), consideramos forcas externas mais gerais do que a dita classe das forcas compactas por translação (translation-compact). Parte desta tese foi aceita para publicação na revista \\Differential and Integral Equations\" sob o ttulo \\Pullback dynamics of non-autonomous wave equations with acoustic boundary condition\". / This thesis is concerned with the study of a class of wave equations with acoustic boundary conditions. We investigate the long-time dynamics of such equations in the case where the system is subject to a non-autonomous external force. In this situation, by adding a weak dissipation, we prove that the problem generates a dissipative evolution process. Our goal is then the existence of non-autonomous attractors. In this direction, we first establishes the existence of a minimal pullback attractor within a universe of tempered sets. We also studied the upper semi-continuity of attractors when the non-autonomous perturbation tends to zero. Our result allows to consider unbounded external forces and nonlinear perturbation with critical (Sobolev) growth. Secondly, we establish the existence of uniform attractors, as well. In view of recent results Zelik (2015) we consider more general external forces than the so called class of translation-compact forces. Part of this thesis was accepted for publication in the journal \\Differential and Integral Equations\" under the title \\Pullback dynamics of non-autonomous wave equations with acoustic boundary condition\".

Dans cette thèse, nous étudions les propriétés spectrales du Laplacien avec la condition de bord de Robin attractive sur des domaines du plan à coins. Notre but est de comprendre l’influence des coins convexes sur l’asymptotique des valeurs propres de cet opérateur lorsque le paramètre de Robin est grand. Nous montrons en particulier que l’asymptotique des premières valeurs propres de Robin sur des polygones curvilignes est déterminée par des opérateurs modèles : les Laplaciens agissant sur les secteurs tangents au domaine. Pour une certaine classe de polygones droits, nous montrons l’existence d’un opérateur effectif sur le bord du domaine qui détermine l’asymptotique des valeurs propres suivantes. Enfin, des asymptotiques de Weyl pour différents seuils dépendant du paramètre de Robin sont obtenues. / In this thesis, we are interested in the spectral properties of the Laplacian with the attractive Robin boundary condition on planar domains with corners. The aim is to understand the influence of the convex corners on the spectral properties of this operator when the Robin parameter is large. In particular, we show that the asymptotics of the first Robin eigenvalues on curvilinear polygons is determined by model operators: the Robin Laplacians acting on infinite sectors. For a particular class of polygons with straight edges, we prove the existence of an effective operator acting on the boundary of the domain and determining the asymptotics of the further eigenvalues. Finally, some Weyl-type asymptotics for different thresholds depending on the Robin parameter are obtained.

Doctoral thesis deals with the numerical simulations of two-phase flows, especially with prediction of movement of dispersed phase (particles) carried by fluid. The Euler-Lagrange approach was applied for description of the system fluid-particles. It means that the fluid is considered to be continuum and its movement is described using Euler approach. Particles are regarded as mass points and their movement is solved using Lagrangian approach. The Large Eddy Simulation method was adopted for solution of the fluid flow. The series of simulations of the backward-facing step flow laden with particles were performed. The concentration of the particles in the flow was high enough for consideration of the influence of particles on the turbulence of the carrier phase. The developed scheme for generation of turbulence on the inlet is applied. The influence of anisotropic decomposition of subgrid energy on movement of particles was studied in the frame of this work.

In aircraft with jet propulsion engine intakes at supersonic speed, strong pressure waves referred to as shockwaves occur, which may interact with any present boundary layers along the intake surface. The adverse pressure gradients associated with Shock Wave-Boundary Layer Interaction (SWBLI) may cause boundary layer flow separation, which can result in disturbances of the flow that can be harmful to the device or decrease engine performance. A common way in dealing with the adverse effects of SWBLI is through removal of low-momentum flow in the boundary layer, a process referred to as boundary layer bleed. In the process of bleed, the boundary layer is subjected to a pressure difference promoting flow out of the system, through a porous surface, and into a plenum. The porous surfaces used in the mass flow removal process contain orifices in small scales. Thus, in Computational Fluid Dynamics (CFD), creating a mesh resolving both the orifice scales and the bulk flow is a cumbersome task, and the computational cost becomes substantially increased. To this end, several boundary conditions which effectively model the large-scale effects of bleed have been developed. The aim of this study is to implement the Boundary Condition (BC) developed by John W. Slater into M-EDGE, the in-house compressible CFD-solver of SAAB Aeronautics. The bleed boundary condition model is based on a dimensionless surface sonic flow coefficient, which is derived from empirical wind-tunnel measurements of the bleed mass flow. In previous work, the Slater bleed BC has been shown to correlate well with wind-tunnel data. Furthermore, a simple transpiration law formulated by Reynald Bur was implemented in order get familiarized with the M-EDGE Fortran source code. However, this model is expected to yield unsatisfactory results, as reported in previous work in the field. The implemented Slater BC is tested on two different two-dimensional flow cases; flow over a flat plate without SWBLI, and flow including a shock wave generator creating SWBLI. In the flat plate case, simulations were run at Mach numbers 1.27, 1.58, 1.98 and 2.46 over a 6.85cm plate of 19% porosity. In the SWBLI-case, only flow at Mach 2.46 was considered, with a 9.53cm plate of 21% porosity. The Reynolds number range used throughout was 1.39−1.76·10^7/m. Simulations were run at different bleed rates over a structured grid using steady state RANS with the Spalart-Allmaras one-equation turbulence model. The boundary condition performance was assessed by its ability to recreate the sonic flow coefficients on which it is based. Further, the shape of downstream pitot pressure profiles are compared with experimental data. Results from the studies indicate that the implementation manages to recreate the data for the sonic flow coefficient with small error margins. The implementation can be used to simulate porous plates of different dimensions and porosities, even though the bleed model is based on empirical mass flow measurements of a 6.85cmplate of 19% porosity. The implementation is able to predict global bleed effects in the flow field, as indicated by comparisons of pitot pressure profiles at various downstream reference planes, despite differences in reference boundary layer intake profiles. Further, the overall flow field was compared visually with other simulation-studies, indicating that the global Mach distributions of the geometries were in accordance with the reference data. However, pitot profiles should be further studied with better matched intake boundary layer profiles. The main limitation of the boundary condition is that it relies on the wind-tunnel data of the surface sonic flow coefficients for specific bleed plate configurations. Furthermore, the implementation has only been verified to work within specific Mach number range of the underlying empirical measurements. In future work, the generality of the model could be increased by extending the data to other configurations and Mach numbers by conducting new experiments or using other published empirical data.

The thermal boundary condition to a fire exposed surface was quantified with a hybrid heat flux gage. Methods were developed to determine the net heat flux through the gage, incident heat flux, cold surface heat flux, convective heat transfer coefficient, adiabatic surface temperature, and the separated components of radiative and convective heat flux. Experiments were performed in a cone calorimeter with the hybrid gage flush mounted into UNIFRAX Duraboard LD ceramic board. The results were then compared to results obtained with a Schmidt-Boelter gage and a plate thermometer. The hybrid heat flux gage predicted a cold surface heat flux within 5% of cold surface heat fluxes measured with a Schmidt-Boelter gage. Adiabatic surface temperature measurements compared well with the plate thermometer measurements at steady state. Hybrid gage measurements were performed on flat plate samples of Aluminum 5083, Marinite P, and UNIFRAX Duraboard LD ceramic board. The gage and sample assemblies were exposed to mixed-mode heat transfer conditions in a cone calorimeter. Temperature measurements were performed at the top, center, bottom surfaces of the marinite and ceramic board samples. A single midpoint temperature was performed on the aluminum. Boundary condition details obtained with the hybrid gage were then input to the commercial finite element analysis package Abaqus. Abaqus was used to create the flat plate geometries of the sample and variable temperature dependent material properties were used for each material. Measured temperatures were then compared to the model predicted temperatures with good results. Hybrid gage measurements were verified using a new experimental apparatus. The apparatus consisted of an impinging jet assembly, a tungsten lamp, and a gage holster assembly. The impinging jet was used to expose the gage to isolated convection and the lamp was used to expose the gage to isolated radiation. The gage holster assembly was used to water cool the gage when desired. Measurements performed with the gage water cooled in isolated convection allowed for the convective heat transfer coefficient to be determined. Two methods were developed to determine the convective heat transfer coefficient in mixed-mode heat transfer conditions. These methods were then verified by comparison to the isolated heat transfer coefficient. Similarly, the incident radiation was isolated by water cooling the gage while only the lamp was on. The components of heat flux were then separated for mixed-mode comparisons and were verified against this isolated radiation. The hybrid gage predicted convective heat transfer coefficients within 10% of the isolated heat transfer coefficient and incident heat fluxes within 11% of the isolated radiation. / Master of Science