[pt] Este trabalho apresenta a análise numérica de vasos de pressão cilíndricos modelados por cascas e placas axissimétricas submetidas a carregamento de pressão interna uniformemente distribuída, utilizando-se a técnica de elementos finitos. São consideradas análises de junções entre superfícies com diferentes espessuras, tais como paredes finas (razão entre o raio e a espessura superior a 10) e moderadamente espessas (razão entre o raio e a espessura inferior a 5). Os campos de deslocamento considerados são os referentes aos elementos planos axissimétricos. A partir deste modelo são avaliadas as tensões na transição entre as superfícies e os resultados comparados com soluções analíticas simplificadas. Conclui-se que a solução analítica aproximada é aceitável para uma grande faixa de valores envolvendo placas e cascas de espessuras moderadamente espessas, enquanto que, para paredes finas, a análise por elementos finitos é necessária para verificação do comportamento das tensões na junção. Testes numéricos utilizando o programa ANSYS são apresentados para demonstrar o desempenho de análises lineares axissimétricas, empregando elementos quadráticos em comparação com as soluções analíticas e avaliando também as limitações do modelo analítico na região da descontinuidade geométrica do modelo proposto. / [en] This work presents the numerical analysis of cylindrical pressure vessels, modeled using axisymmetric shells and plates elements under internal pressure loads. The numerical analysis considers surface joints for various surface thickness ratios, from thin (ratio between radius and thickness greater than 10) to thick (ratio between radius and thickness less than 5) shells. Element displacement fields of axisymmetric plane elements are used to evaluate the stress state at the surfaces junctions, and the obtained results are compared to simplified analytical solutions. It is concluded that analytical approximate results present an acceptable solution for a large range of plates to shells geometries up to moderately thick shells, whereas for thin shells the finite element solution is necessary to be considered in order to accurately verify the stresses at plate to shell junction. Numerical tests applying ANSYS program are presented to demonstrate the performance of linear axisymmetric analysis applying quadratic elements in comparison to the analytical solutions also evaluating the limitations of the analytical model in the region of the geometric discontinuity of the proposed model.

O presente trabalho está inserido no campo de estudo das cascas axissimétricas, tendo como objetivo a análise de seu comportamento estrutural mediante o desenvolvimento e aplicação de uma ferramenta numérica baseada no método dos elementos finitos generalizados. A utilização desse recurso é uma alternativa eficaz e difere do método dos elementos finitos convencional pela possibilidade de enriquecimento nodal das funções de aproximação. Como resultado pode-se dispensar o uso de redes muito refinadas. Com o intuito de evidenciar as vantagens do método adotado são apresentados exemplos comparando-se as soluções numéricas obtidas com soluções analíticas ou numéricas geradas com o do método dos elementos finitos convencional. Os resultados obtidos com um pequeno número de elementos finitos e com enriquecimento por funções polinomiais, mostraram-se convergentes já nos primeiros graus de enriquecimento. Desenvolve-se uma análise complementar de convergência baseada em estimativa de erro, mostrando que a metodologia adotada pode proporcionar melhores taxas de convergência em relação ao refino h quando predomina a regularidade da solução. A mesma análise aponta que a combinação dos refinos h e p pode levar a resultados mais precisos, com elevadas taxas de convergência, quando a solução (particularmente suas derivadas) apresentar regularidade menor. / The present dissertation is inserted in the field of study of the axisymmetric shells. The objective is to analyze the structural behavior by means of the development and application of a numerical tool based on the generalized finite element method. The use of this resource is an efficient alternative to the conventional finite element method for the possibility of nodal enrichment of the approach functions. Therefore one can avoid the use of very fine nets. In addition, in order to evidence the advantages of the adopted method, there are shown examples comparing the numerical solutions with analytical or numerical values generated with the conventional finite element method. The results obtained with a small number of elements, including enrichment by polynomial functions, had revealed convergence in the first degrees of enrichment. It is developed a complementary convergence analysis based on estimate of error, showing that the adopted methodology can provide better convergence ratios in relation to the h-refinement, in the cases where the regularity of the solution predominates. The same analysis shows that the combination of the refinement in its versions h and p can give more accurate results, by increasing convergence, when the solution (particularly its derivatives) presents lower regularity.

Do ponto de vista da hidrodinâmica clássica, o problema de impacto hidrodinâmico configura-se como um problema de contorno com fronteiras móveis cuja posição deve ser determinada simultaneamente à solução da equação de campo. Essa característica traz dificuldades para obtenção de soluções analíticas e numéricas. Nesse sentido, o presente trabalho propõe o desenvolvimento de um método numérico específico para analisar o problema de impacto hidrodinâmico de corpos sólidos rígidos contra a superfície livre da água. A solução da equação dinâmica não linear do problema de impacto depende da determinação do tensor de massa adicional a cada instante de tempo, o qual depende da posição e atitude do corpo no instante considerado. Um método variacional específico é empregado, através do qual os coeficientes de massa adicional são determinados com erro de segunda ordem, na posição considerada. Tal método é exemplo de técnicas numéricas dessingularizadas, através das quais o potencial de velocidade é aproximado em um espaço finito-dimensional formado por funções-teste derivadas de soluções potenciais elementares, tais como pólos, dipolos, anéis de dipolos, de vórtices, etc. O problema potencial de impacto hidrodinâmico, que se caracteriza pela dominância das forças de inércia, é formulado admitindo-se a superfície líquida como equipotencial, o que permite a analogia com o limite assintótico de frequência infinita do problema de radiação de ondas causada pelo movimento de corpos flutuantes. O método desenvolvido é então aplicado ao caso de impacto vertical de corpos axissimétricos, formulando o problema sob o chamado modelo de von Kármán generalizado (GvKM). Nesse modelo as condições de contorno na geometria exata do corpo são satisfeitas, porém os efeitos do empilhamento de água junto às raízes do jato, que se forma ao longo da intersecção com a superfície livre, não são considerados no caso geral. Resultados numéricos do coeficiente de massa adicional para uma família de esferoides são apresentados e tabulados para o pronto uso em análise e projeto. Além disso, considerações acerca da inclusão do efeito de empilhamento de água junto às raízes do jato, ou seja, da elevação da superfície livre são também feitas para o caso de esferas, fazendo uso de abordagens analíticas encontradas na literatura especializada. / In terms of classical hydrodynamics, the hydrodynamic impact problem is characterized as a boundary problem with moving boundary which position must be determined simultaneously with the solution of the field equation. This feature brings difficulties to get analytical and numerical solutions. In this sense, the purpose of this work is to present a variational method technique specifically designed for the hydrodynamic impact problem of axisymmetric rigid bodies on the free surface. The solution of the nonlinear dynamic equation of the impacting motion depends on the determination of the added mass tensor and its derivative with respect to time at each integration time step. This is done through a variational method technique that leads to a second-order error approximation for the added mass if a first-order error approximation is sought for the velocity potential. This method is an example of desingularized numerical techniques, through which the velocity potential is approximated in a sub-space of finite dimension, formed by trial functions derived from elementary potential solutions, such as poles, dipoles, and vortex rings, which are placed inside the body. The potential problem of hydrodynamic impact, characterized by the dominance of inertial forces, is here formulated by assuming the liquid surface as equipotential, what allows the analogy with the infinity frequency limit in the usual free surface oscillating floating body problem. The method is applied to the vertical hydrodynamic impact of axisymmetric bodies within the so-called Generalized von Kármán Model (GvKM). In such approach, the exact body boundary condition is full-filled and the wet correction is not taken into account. Numerical results for the added mass coefficient for a family of spheroids are presented. Moreover, considerations are made on the effects of the free surface elevation for the specific case of an impacting sphere, through analytical approaches.

Simulácia šírenia ultrazvuku prostredníctvom mäkkých biologických tkanív má širokú škálu praktických aplikácií. Patria sem dizajn prevodníkov pre diagnostický a terapeutický ultrazvuk, vývoj nových metód spracovania signálov a zobrazovacích techník, štúdium anomálií ultrazvukových lúčov v heterogénnych médiách, ultrazvuková klasifikácia tkanív, učenie rádiológov používať ultrazvukové zariadenia a interpretáciu ultrazvukových obrazov, modelové vrstvenie medicínskeho obrazu a plánovanie liečby pre ultrazvuk s vysokou intenzitou. Ultrazvuková simulácia však predstavuje výpočtovo zložitý problém, pretože simulačné domény sú veľmi veľké v porovnaní s akustickými vlnovými dĺžkami, ktoré sú predmetom záujmu. Ale ak je problém osovo symetrický, problém môže byť riešený v 2D.To umožňuje spúšťanie simulácií na mriežke s väčším počtom bodov, s menším využitím výpoč- tových zdrojov za kratšiu dobu. Táto práca modeluje a implementuje zrýchlenie vlnovej nelineárnej ultrazvukovej simulácie v axisymetrickom súradnicovom systéme realizovanom v Matlabe pomocou Mex súborov pre diskrétne sínové a kosínové transformácie. Axisymetrická simulácia bola implementovaná v C++ ako open source rozšírenie K-WAVE toolboxu. Kód je optimalizovaný na beh na jednom uzle superpočítaču Salomon (IT4Innovations, Ostrava, Česká republika) s dvoma dvanásť-jadrovými procesormi Intel Xeon E5-2680v3. Na maximalizáciu výpočtovej efektívnosti boli vykonané viaceré optimalizácie kódu. Po prvé, fourierové tramsformácie boli vypočítané pomocou real-to-complex FFT z knižnice FFTW. V porovnaní s complex-to-complex FFT to znížilo čas výpočtu a pamäť spojenú s výpočtom FFT o takmer 50%. Taktiež diskrétne sínové a kosínové transformácie sa počítali pomocou knižnice FFTW, ktoré v Matlab verzii museli byť vyvolané z dynamicky načítaných MEX súborov. Po druhé, aby sa znížilo zaťaženie priepustnosti pamäte, boli všetky operácie počítané jednoduchej presnosti pohyblivej rádovej čiarky. Po tretie, elementárne operá- cie boli paralelizované pomocou OpenMP a potom vektorizované pomocou rozšírení SIMD (SSE). Celkový výpočet C++ verzie je až do 34-násobne rýchlejší a využíva menej ako tretinu pamäte ako Matlab verzia simulácie. Simulácia ktorá by trvala takmer dva dni tak môže byť vypočítaná za jeden a pol hodinu. Toto všetko umožňuje počítať simuláciu na výpočetnej mriežke s veľkosťou 16384 × 8192 bodov v primeranom čase.

In this paper, we study the partial Fourier method for treating the Lamé equations in three-­dimensional axisymmetric domains subjected to nonaxisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement u, the body force f \in (L_2)^3 and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three­dimensional boundary value problem to an infinite sequence of two­dimensional boundary value problems, whose solutions u_n (n = 0,1,2,...) are the Fourier coefficients of u. This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients u_n are proved, which are important for further numerical treatment, e.g. by the finite-element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two­dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients u_n and of the error of the partial Fourier approximation are given.

Ce sujet de thèse s’inscrit dans le cadre des études actuellement menées au CEA/DAM concernant des schémas numériques ALE (Arbitrary-Lagrangian-Eulerian)de type « Lagrange + Projection », dans le contexte des simulations hydrodynamiques mutli-matériaux en grandes déformations. Ces schémas doivent respecter les équations de conservation de la masse, de la quantité de mouvement et de l’énergie totale.Les schémas décalés en temps et en espace sont très utilisés dans les codes industriels. Ils sont robustes et permettent une bonne approximation des comportements complexes, mais sont connus pour ne pas conserver exactement l’énergie totale. Cela pose un problème dans le traitement des chocs, sur maillages raffinés ou dans la simulation des milieux réactifs.En 2016, des travaux originaux on été proposés par A. Llor et. al. pour rendre conservatif ce type de schéma dans un contexte lagrangien (sans projection), notamment en proposant une correction pour retrouver la conservation de l’énergie totale.Le travail de cette thèse a été d’étendre ce schéma lagrangien dans un contexte ALE multi-matériaux (avec interface), en garantissant la conservation de toutes les quantités, le respect du second principe de la thermodynamique et la robustesse. De nombreux cas tests ont été menés (en 2D plan et en 2D axisymétrique) et comparés aux méthodes existantes afin de montrer la pertinence de cette approche. / This PhD subject comes within actual studies managed by CEA/DAM about ALE (Arbitrary-Lagrangian-Eulerian) schemes (with a splitting of Lagrangian and Remapping steps) in the context of hydrodynamic simulations. These numerical schemes have to respect mass, momentum and total energy conservation, which are the fundamental equations of the studied systems.Space- and Time-Staggered are widely used in industrial codes for their simplicity androbustness despite their known lack of exact energy conservation. This is a major drawbackin presence of strong shocks. Among all existing schemes, none of them meet the expectations of robustness, conservation,thermodynamic consistency (both shocks and relaxations capture), accuracy andadaptibility to complex behaviors. Recently, some novel works have been proposed by A.Llor et. al. in order to make conservative this type of scheme in a Lagrangian context (without remapping step). Current remap methods, necessary in large deformations, donot guarantee simultaneously total energy conservation and thermodynamic consistency.This work aims at extending this conservative Lagrangian space- and time-staggeredscheme to a multi-material ALE methodology, keeping its good properties (conservation,accuracy, thermodynamic consistency, robustness) intact. Classical, but demanding, test cases have been performed (both in plane and axisymmetric 2D geometries) and have been compared to existing numerical methods in order to assess the relevance of our approach.

No description available.

The dissertation explores a framework for numerically simulating the deformation of the liquid metal wall's free surface in Z-pinch fusion devices. This research is conducted in the context of utilizing liquid metals as plasma-facing components in fusion reactors. In the Z-pinch fusion process, electric current travels through a plasma column and enters into a pool of liquid metal. The current flowing through the liquid metal generates Lorentz force, which deforms the free surface of the liquid metal. Modeling this phenomenon is essential as it offers insights into the feasibility of using liquid metal as an electrode wall in such fusion devices. The conventional magneto-hydrodynamic (MHD) formulation aims at modeling the situation where an external magnetic field is applied to flows involving electrically conducting liquids, with the initial magnetic field is known and then evolved over time through magnetic induction equation. However, in Z-pinch fusion devices, the electric current is directly injected into a conducting liquid. In these situations, an analytical expression for the magnetic field generated by the applied current is not readily available, necessitating numerical calculations. Moreover, the deformation of the liquid metal surface changes the geometry of the current path over time and the resulting magnetic field. By directly solving the Maxwell equations in combination with Navier-Stokes equations, it becomes possible to predict the magnetic field even when the fluid is in motion. In this dissertation, a numerical framework utilizing the Maxwell-Navier-Stokes system is explored to successfully capture the deformation of the liquid metal's free surface due to applied electric current. / Doctor of Philosophy / In this dissertation, a method is described that uses a computer to simulate how the initially stable, flat surface of liquid metal deforms when subjected to electrical currents in Z-pinch fusion devices, a specific type of nuclear fusion technology. Z-pinch fusion devices generate plasma, a hot fluid-like substance, through the nuclear fusion process, triggered and maintained by strong pulsated current. There's a growing interest in using liquid metal as the first layer of material to isolate the hot plasma from the rest of the nuclear fusion reactor body, rather than solid materials, due to its unique benefits. However, the Z-pinch fusion process, by introducing electric currents through the liquid metal layer, induces a Lorentz force that consequently deforms the surface of the liquid metal. Developing a tool to predict this deformation is vital as it aids in evaluating the potential of using liquid metal as a plasma-facing layer over solid materials in these fusion devices. The simulation tools presented in this dissertation are able to successfully captures the dynamics of how the liquid metal surface deforms under the impact of electrical currents.

With the increased frequency of accidental and deliberate explosions, the response of civil infrastructure systems to blast loading has become a research topic of great interest. However, with the high cost and complex safety and logistical issues associated with live explosives testing, North American blast resistant construction standards (e.g. ASCE 59-11 & CSA S850-12) recommend the use of shock tubes to simulate blast loads and evaluate relevant structural response. This study aims first at developing a 2D axisymmetric shock tube model, implemented in ANSYS Fluent, a computational fluid dynamics (CFD) software, and then validating the model using the classical Sod’s shock tube problem solution, as well as available shock tube experimental test results. Subsequently, the developed model is compared to a more complex 3D model in terms of the pressure, velocity and gas density. The analysis results show that there is negligible difference between the two models for axisymmetric shock tube performance simulation. However, the 3D model is necessary to simulate non-axisymmetric shock tubes. The design of a shock tube depends on the intended application. As such, extensive analyses are performed in this study, using the developed 2D axisymmetric model, to evaluate the relationships between the blast wave characteristics and the shock tube design parameters. More specifically, the blast wave characteristics (e.g. peak reflected pressure, positive phase duration and the reflected impulse), were compared to the shock tube design parameters (e.g. the driver section pressure and length, the driven v section length, and perforation diameter and their locations). The results show that the peak reflected pressure increases as the driver pressure increases, while a decrease of the driven length increases the peak reflected pressure. In addition, the positive phase duration increases as both the driver length and driven length are increased. Finally, although shock tubes generally generate long positive phase durations, perforations located along the expansion section showed promising results in this study to generate short positive durations. Finally, the developed 2D axisymmetric model is used to optimize the dimensions of a proposed large-scale conical shock tube system developed for civil infrastructure blast response evaluation applications. The capabilities of this proposed shock tube system are further investigated by correlating its design parameters to a range of explosion threats identified by different hemispherical TNT charge weight and distance scenarios. / Thesis / Master of Applied Science (MASc)

We set out to analyze 5D stationary and bi-axisymmetric solutions to the vacuum Einstein equations. These are in the cohomogeneity 2 setting where the orbit space is a right half plane. They can have a wide range of behaviour at the boundary of the orbit space. The goal is to understand in detail the soliton example in Khuri, Weinstein and Yamada's paper ``5-dimensional space-periodic solutions of the static vacuum Einstein equations". This example is periodic and has alternating axis rods as its boundary data. We start by deriving the harmonic equations which determines the behaviour of the metric in the interior of the orbit space. Then we analyze what conditions the boundary data imposes on the metric. These are called the smoothness conditions which we derive for solely the alternating axis rod case. We show that with an ellipticity assumption they predict that the twist potentials are constant and that the metric is of the form which appears in Khuri, Weinstein and Yamada's paper. We then analyze the Schwarzschild metric in its standard form which is cohomogeneity 1 and its Weyl form which is cohomogeneity 2. This Weyl form can be made periodic and this serves as an inspiration for the examples in Khuri, Weinstein and Yamada's paper. Finally we analyze the soliton example in detail and show that it satisfies the smoothness conditions. We then provide a new example which has a single axis rod on the boundary with non-constant twist potentials but that is missing a point on the boundary. / Thesis / Master of Science (MSc) / We study the geometry of 5D blackholes. These blackholes are idealized by certain spatial symmetries and time invariance. They are solutions to the vacuum Einstein equations. The unique characteristic of these blackholes is the range of behaviour they may exhibit at the boundary of the domain of outer communication. There could be a standard event horizon called a horizon rod or an axis rod where a certain part of the spatial symmetry becomes trivial. In this thesis we start by deriving the harmonic map equations which are satisfied in the interior of the domain of communication. Then we show how this boundary data affects the metric through the smoothness conditions. We then analyze the soliton example in a paper by Khuri, Weinstein and Yamada and show that it respects the smoothness conditions. We then provide a new example which is interesting in the fact it has non-constant twist potentials.