Data from present day scientific simulations and observations of physical processes often consist of multiple scalar fields. It is important to study the interactions between the fields to understand the underlying phenomena. A visual representation of these interactions would assist the scientist by providing quick insights into complex relationships that exist between the fields. We describe new techniques for visual analysis of multifield scalar data where the relationships can be quantified by the gradients of the individual scalar fields and their mutual alignment. Empirically, gradients along with their mutual alignment have been shown to be a good indicator of the relationships between the different scalar variables. The Jacobi set, defined as the set of points where the gradients are linearly dependent, describes the relationship between the gradient fields. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a feature-rich dataset. For two dimensional domains, we pose the problem of simplification as the extraction of level sets and offset contours and describe a robust technique to simplify and create a multi-resolution representation of the Jacobi set. Existing isosurface-based techniques for scalar data exploration like Reeb graphs, contour spectra, isosurface statistics, etc., study a scalar field in isolation. We argue that the identification of interesting isovalues in a multifield data set should necessarily be based on the interaction between the different fields. We introduce a variation density function that profiles the relationship between multiple scalar fields over isosurfaces of a given scalar field. This profile serves as a valuable tool for multifield data exploration because it provides the user with cues to identify interesting isovalues of scalar fields. Finally, we introduce a new multifield comparison measure to capture relationships between scalar variables. We also show that our measure is insensitive to noise in the scalar fields and to noise in their gradients. Further, it can be computed robustly and efficiently. The comparison measure can be used to identify regions of interest in the domain where interactions between the scalar fields are significant. Subsequent visualization of the data focuses on these regions of interest leading to effective visual analysis. We demonstrate the effectiveness of our techniques by applying them to real world data from different domains like combustion studies, climate sciences and computer graphics.

La compréhension des écoulements à bulles dans les centrales nucléaires demeure encore un élément limitant dans l’analyse des opérations et de la sûreté des installations. Pour ne citer qu’un exemple, l’amélioration de la durée de vie etde la performance des générateurs de vapeur nécessite d’appréhender les régimes d’écoulement au sein des tubes qui sont responsables de leur vibration. Cependant, pour simuler avec précision ces écoulements, les codes de simulation numérique doivent relever de nombreux défis parmi lesquels la capacité à simuler des inclusions ayant des tailles très variées. Dans cette thèse, une nouvelle approche, appelée approche multi-champ, est implémentée dans le code NEPTUNE_CFD, basé sur un modèle bi-fluide. Cette approche inclut une méthode de suivi d’interface pour les grandes structures déformables et prend en compte les effets liés à la turbulence et aux changements de phase.Pour simuler de tels écoulements complexes en limitant le coût CPU, l’approche multi-champ considère séparément les petites inclusions sphériques des grandes inclusions déformables. Ainsi, les petites structures sphériques sont définies via un champ eulérien dispersé évoluant au sein d’un champ continu porteur, comme c’est habituellement le cas avec le modèle bi-fluide. Les grosses bulles déformables sont considérées comme des interfaces entre deux champs continus, un champ liquide et un champ gaz. Si on prend l’exemple d’un écoulement diphasique avec de l’eau et des bulles d’air de différentes tailles, trois champs sont alors définis pour cet écoulement: un champ continu liquide, un champ continu gaz et un champ dispersé gaz contenant les petites bulles sphériques. Cependant, simuler avec précision des interfaces entre deux champscontinus avec le modèle bi-fluide nécessite le développement de traitements spécifiques afin de coupler les deux champs à l’interface et de limiter la diffusion de cette interface.Après avoir amélioré la simulation des interfaces dans des écoulements laminaires, les effets liés à la turbulence sont étudiés. Une étude a priori de simulations aux grandes échelles est proposée pour identifier les termes sous-mailles et comparer différents modèles de turbulence disponibles dans la littérature. L’implémentation et la validation du modèle de turbulence retenu suite à l’étude sont détaillées. Les changements de phase sont ensuite explorés via le développement d’un modèle spécifique pour le terme de transfert de masse. Pour finir, des simulations trois champs sont présentées. De nouveauxcritères sont définis pour modéliser la fragmentation des grandes inclusions déformables en petites bulles sphériques ainsi que la coalescence de ces dernières pour former de grandes bulles déformables.A chaque étape de l’implémentation des différents modèles évoqués, des validations basées sur des données analytiques et issues d’expériences sont présentées afin de s’assurer que les phénomènes physiques sont bien prédits. Des cas tests dans des configurations industrielles sont également détaillés pour montrer la capacité de l’approche développée à simuler des écoulements complexes / Bubbly flows occurring in nuclear power plants remain a major limiting phenomenon for the analysis of operation and safety. As an example, the improvement of steam generator lifetime and performance relies on the comprehension of flow regimes inside the tubes responsible for tube vibrations. However, to ensure an accurate simulation of these flows, theComputational Multi-Fluid Dynamics (CMFD) codes have to take up many challenges, among others the ability of dealing with a variety of inclusion sizes. The classical two-fluid model allows simulating small spherical inclusions but is not able to compute large deformable inclusions. Thus, in this thesis, a new approach, called the multifield approach, is implementedin the CMFD code NEPTUNE_CFD, based on a two-fluid model. This approach includes an interface tracking method for large and deformable structures and takes into account turbulence and phase change effects.To simulate such complex flows with reasonable computational costs, the multifield approach considers separately the small spherical inclusions and the large deformable ones. Thus, the small spherical structures are defined as a dispersed field evolving in a continuous carrier field, as usually done in the two-fluid model. The large deformable bubbles are considered as interfaces between two continuous phases treated as two different fields in the two-fluid model. In the example of a two-phase flow with water and air bubbles of different sizes, three fields are defined: a continuous liquid field, a continuous gas field and a dispersed gas field containing the small spherical bubbles. However, the accurate simulation of interfaces between the two continuous fields within the two-fluid model requires specific treatments to couple the two fields at the interface and to limit the interface smearing.After improving the interface simulation in laminar flows, turbulence effects are investigated. An a priori Large Eddy Simulation (LES) study is performed to identify the predominant subgrid terms and to compare different availableturbulence models. The implementation and validation of the most suitable model is proposed. Phase change interfaces are then explored with the development of a specific model for the mass transfer term. Finally, three fields simulations are performed. New criteria are defined for the breakup of the large deformable inclusions into small spherical bubbles and for the coalescence of the latter forming large deformable bubbles.Validation at each step of the models implementations are presented using analytical and experimental data to ensure that the physical phenomena are well predicted. Test cases in industrial configurations are finally performed to show the ability of the developed approach to deal with complex flows

In CANDU reactors, under normal operating conditions, the inlet headers collect and distribute single-phase liquid flow (heavy water) to the fuel cooling channels via the feeders. However, under some postulated loss of coolant accidents, the inlet headers may receive two-phase fluid (steam/water) and the fluid forms a stratified region inside the header. To predict the thermalhydraulic behaviour of headers for the reactor safety analysis, the two-phase flow distribution within the headers and through the feeders must be modelled. In order to analyse the two-phase flow behaviour of a scaled CANDU inlet header / a transparent and instrumented version of a header with 5 feeders was previously built in the Mechanical Engineering Department of Middle East Technical University (METU-Two Phase Flow Test Facility / METU-TPFTF). The aim of this study is to investigate two-phase flow distribution through multiple outlets from such a horizontal drum both numerically and experimentally. For this purpose, three-dimensional incompressible finite difference equations in cylindrical coordinates were derived by using two-fluid model to simulate adiabatic two-phase flow (air/water) in the header numerically. The discretized equations were then programmed into a computer code which was developed specifically for modelling the header type geometry. A method based on the principles of Implicit Multi Field (IMF) technique has been utilised to solve those equations. The solution algorithm was tested by using some numerical benchmark problems. A number of experimental tests covering single and two-phase flow distribution through outlet pairs from the header were performed. Void fractions and flow rates obtained from these tests are in good agreement with the code results. The code also predicts the void fraction and pressure distribution in the header satisfactorily.

La thèse est décomposée en trois parties. Dans la première partie, les modèles de la CUF sont présentés. La CUF permet d’obtenir, dans un formalisme générale, de nombreux modèles qui diffèrent 1) selon l’ordre d’expansion dans l'épaisseur choisie pour les variables primaires; 2) selon le type de modèle: modèles couche équivalente (ESL) ou couche discrète (LW); 3) selon le principe variationnel : “Principle of Virtual Displacements” (raffinée) ou “Reissner’s Mixed Variational Theorem” (avancée). Des géométries cylindrique et à double courbure sont traités. La deuxième partie de la thèse est consacrée à l'obtention des équations fondamentales en utilisant différentes méthodes: la méthode analytique de Navier et deux méthodes numériques approchées; la “Finite Element Method” (FEM) et la “Radial Basis Functions” (RBF). La méthode RBF est une méthode sans maillage “meshless” et peut être considérée comme une méthode alternative à la FEM. La FEM est la plus utilisée dans la littérature et le sujet principal de cette thèse.Dans la dernière partie, différents problèmes sont proposés. Navier est utilisé pour l’analyse thermomécanique de coques FGM, l’analyse de coques piézo-électrique et l’analyse dynamique de nanotubes de carbone. Un élément fini coque, présenté dans cette thèse, est utilisé pour l’analyse de coques composites et FGM. Les résultats obtenus démontrent la supériorité de cet élément par rapport aux éléments finis basés sur les théories classiques pour l’analyse des matériaux avancés. Enfin, la méthode RBF est utilisée pour l’analyse de coques composites, permettant d'illustrer l'avantage des méthodes sans maillage. / The dissertation is organized in three main parts. In the first part, the shell models contained in the CUF are presented. The CUF permits to obtain, in a general and unified manner, several models that can differ by 1) the chosen order of expansion in the thickness direction, 2) the equivalent single layer or layer wise approach and 3) the variational statement used: “Principle of Virtual Displacements” (refined models) or “Reissner’s Mixed Variational Theorem” (advanced models). Both the cylindrical and the double-curvature geometries are considered. The second part is devoted to the derivation of the governing equations by means of different methods: an analytical method, that is the Navier method, and two approximated numerical methods, that are the Finite Element Method (FEM) and the Radial Basis Functions (RBF) method. The RBF method is based on a meshless approach and it can be considered a good alternative to the FEM. The finite element method is the most common method used in literature and it is the main topic of this thesis. In the last part, different problems are analyzed. The thermo-mechanical analysis of FGM shells, the electromechanical analysis of piezoelectric shells and the dynamic analysis of carbon nanotubes are performed by means of the Navier method. Then, the CUF shell finite element, presented in this thesis, is tested and used for the analysis of composite and FGM shells. The superiority of this element in respect to finite elements based on classical theories is shown. Finally, the RBF method is combined with the CUF for the analysis of composite and FGM shells in order to overcome the numerical problems relative to the mesh that usually affect the finite elements.

Modern unsteady (multi-)field visualizations require an effective reduction of the data to be displayed. From a huge amount of information the most informative parts have to be extracted. Instead of the fuzzy application dependent notion of feature, a new approach based on information theoretic concepts is introduced in this paper to detect important regions. This is accomplished by extending the concept of local statistical complexity from finite state cellular automata to discretized (multi-)fields. Thus, informative parts of the data can be highlighted in an application-independent, purely mathematical sense. The new measure can be applied to unsteady multifields on regular grids in any application domain. The ability to detect and visualize important parts is demonstrated using diffusion, flow, and weather simulations.

Stimuli-sensitive hydrogels are polymeric materials, which are able to reversibly swell in water in response to evironmental changes. Relevant stimuli include variations of pH, temperature, concentration of specific ions etc. Stacked layers composed of multiple thin hydrogels - also referred to as hydrogel-layer composites - combine the distinct sensing properties of different hydrogels. This approach enables the development of sophisticated micro uidic devices such as bisensitive valves or uid-sensitive de ectors. In order to numerically simulate the swelling of a polyelectrolyte hydrogel in response to an ion concentration change the multifield theory is adopted. The set of partial differential equations - including the description of the chemical, the electrical and the mechanical field - are solved using the Finite Element Method. Simulations are carried out on a twodimensional domain in order to capture interactions between the different fields. In the present work, the ion transport is governed by diffusive and migrative uxes. The distribution of ions in the gel and the solution bath result in an osmotic pressure difference, which is responsible for the mechanical deformation of the hydrogel-layer composite. The realized numerical investigation gives an insight into the evolution of the displacement field, the distribution of ions and the electric potential within the bulk material and the interface between gel and solution bath. The predicted behavior of the relevant field variables is in excellent agreement with results available in the literature.

Polyelectrolyte hydrogels are viscoelastic electroactive polymers which respond to external physical or chemical stimuli by a reversible volume phase transition. Novel fabrication methods allow the creation of hydrogel layer composites in which each layer shows a different sensitivity (e.g. to a different stimulus). This offers new opportunities, for example, in the design of new microsensors, microactuators and microfluidic devices as well as for high-selective membranes and target-specific drug delivery systems. Since only few research groups numerically investigated the transport mechanisms in hydrogel layer composites, a gap remains to describe the movement and transient distribution of ions inside the layer system. In this article, the multifield formulation is adopted to describe the transient distribution of ions in salt-sensitive hydrogel layer composites on the basis of a numerical simulation. For this, the Nernst-Planck and the Poisson equation are solved using one-dimensional finite elements for both anionic-anionic and anionic-cationic gel layer composites under chemical stimulation. Between adjacent gels, an additional interlayer is introduced to account for the physical and chemical bonding region between the gels. Adaptive mesh refinement provides a good resolution close to the interface between the adjacent gel layers. The obtained results are used to predict the osmotic pressure inside the gels and the dependent swelling of the gel layer composite. The excellent agreement of the obtained results with the Donnan equilibrium demonstrates the high potential of the method applied to predict the behavior of hydrogel layer composites.