We present a method for anisotropic mesh refinement to high-order numerical solutions. We accomplish this by assigning metrics to vertices that approximate the error in that region. To choose values for each metric, we first reconstruct an error equation from the leading order terms of the Taylor expansion. Then, we use a Fourier approximation to choose the metric associated with that vertex. After assigning a metric to each vertex, we refine the mesh anisotropically using three mesh operations. The three mesh operations we use are swapping to maximize quality, inserting at approximate circumcenters to decrease cell size, and vertex removal to eliminate small edges. Because there are no guarantees on the results of these modification tools, we use them iteratively to produce a quasi-optimal mesh. We present examples demonstrating that our anisotropic refinement algorithm improves solution accuracy for both second and third order solutions compared with uniform refinement and isotropic refinement. We also analyze the effect of using second derivatives for refining third order solutions.

We present a method for anisotropic mesh refinement to high-order numerical solutions. We accomplish this by assigning metrics to vertices that approximate the error in that region. To choose values for each metric, we first reconstruct an error equation from the leading order terms of the Taylor expansion. Then, we use a Fourier approximation to choose the metric associated with that vertex. After assigning a metric to each vertex, we refine the mesh anisotropically using three mesh operations. The three mesh operations we use are swapping to maximize quality, inserting at approximate circumcenters to decrease cell size, and vertex removal to eliminate small edges. Because there are no guarantees on the results of these modification tools, we use them iteratively to produce a quasi-optimal mesh. We present examples demonstrating that our anisotropic refinement algorithm improves solution accuracy for both second and third order solutions compared with uniform refinement and isotropic refinement. We also analyze the effect of using second derivatives for refining third order solutions.

We present a method for anisotropic mesh refinement to high-order numerical solutions. We accomplish this by assigning metrics to vertices that approximate the error in that region. To choose values for each metric, we first reconstruct an error equation from the leading order terms of the Taylor expansion. Then, we use a Fourier approximation to choose the metric associated with that vertex. After assigning a metric to each vertex, we refine the mesh anisotropically using three mesh operations. The three mesh operations we use are swapping to maximize quality, inserting at approximate circumcenters to decrease cell size, and vertex removal to eliminate small edges. Because there are no guarantees on the results of these modification tools, we use them iteratively to produce a quasi-optimal mesh. We present examples demonstrating that our anisotropic refinement algorithm improves solution accuracy for both second and third order solutions compared with uniform refinement and isotropic refinement. We also analyze the effect of using second derivatives for refining third order solutions. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

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Pour les écoulements turbulents d’intérêt pratique, la turbulence interagit avec le confinement et les forces externes, ce qui cause inhomogénéité et anisotropie statistiques. Isoler leur contribution à des statistiques ciblées est indispensable pour comprendre les différents phénomènes physiques. Le but de cette thèse a donc été d’acquérir une meilleure connaissance de l’anisotropie en fonction de la direction et de l’échelle dans un ensemble de contextes idéalisés et réalistes. On a utilisé une caractérisation statistique dans l’espace spectral ainsi que dans l’espace de séparation. La caractérisation dans l’espace spectral concerne les statistiques anisotropes de turbulence sous forme de spectres directionnels d’énergie, polarisation et hélicité. La caractérisation dans l’espace de séparation s’appuie sur les moments des incréments de vitesse à deux points du deuxième et troisième ordre, et sur les corrélations de vitesse à deux points. Tout d’abord, on a étudié l’effet du forçage spectral de grandes échelles. Les schémas de forçage considérés sont le schéma de forçage de type Euler, non hélicitaire et hélicitaire, et le schéma ABC. On a montré que les deux forçages ont un inconvénient, dans le sens que, si le nombre de modes suffisamment excités est petit, de l’anisotropie se produit même aux petites échelles. Dans le cas du forçage Euler, cela dépend de la gamme de nombres d’onde forcés ainsi que de leur hélicité. Le forçage ABC, pour lequel le niveau d’hélicité injectée ne peut pas être contrôlé, n’excite que six modes et donc il produit toujours de l’anisotropie et à toutes les échelles résolues. Ensuite, on a analysé l’anisotropie en fonction de l’échelle et de la direction pour la turbulence homogène en rotation. Chose étonnante, l’anisotropie se produit à toutes les échelles même si la rotation est faible. En particulier, on a identifié deux gammes d’échelles anisotropes qualitativement différentes. Aux grandes échelles, l’anisotropie directionnelle est plus grande et décroît avec le nombre d’onde. Aux petites échelles, elle est beaucoup plus faible—mais encore significative—et croit lentement avec le nombre d’onde jusqu’aux échelles dissipatives. Une autre conclusion intéressante et originale de cette partie du travail concerne le rôle de l’échelle de Zeman et son lien avec l’anisotropie aux différentes échelles de l’écoulement. D’après des travaux précédents, l’échelle de Zeman devrait être l’échelle de longueur caractéristique qui sépare les échelles affectées par la rotation par les échelles isotropes. Après une plus ample investigation, en utilisant simulations à différents paramètres, on a découvert que l’échelle de séparation entre grande et faible anisotropie est plutôt l’échelle de longueur caractéristique pour laquelle les effets de rotation et de dissipation s’équilibrent. Ce résultat, toutefois, n’est pas en contradiction avec l’argument de Zeman sur le rétablissement de l’isotropie dans la limite asymptotique de viscosité nulle, comme l’échelle de séparation s’annule à nombre de Reynolds infini, et donc seulement la gamme d’anisotropie décroissante devrait persister et les échelles beaucoup plus petite que celle de Zeman pourraient récupérer l’isotropie. Enfin, on a considéré l’écoulement de von Kármán entre deux disques équipés de pales en contre-rotation dans une cavité cylindrique. On a répété l’analyse dans l’espace de séparation dans plusieurs petites sous-régions, afin d’enquêter les analogies possibles entre la dynamique de l’écoulement et celle de la turbulence homogène en rotation. On a découvert que, dans les régions du domaine où l’écoulement a un taux de rotation moyen plus grand, les distributions des statistiques dans l’espace de séparation montrent certaines des caractéristiques typiques de la turbulence en rotation. / In turbulent flows of practical interest, turbulence interacts with confinement and external forces, leading to statistical inhomogeneity and anisotropy. Isolating their contributions to some targeted statistics is indispensable for understanding the underlying physical phenomena. The aim of this thesis has therefore been to gain further insight into direction- and scale-dependent anisotropy in a set of idealized and realistic contexts. Both spectral space and separation space statistical characterizations have been employed. The spectral characterization concerns the anisotropic statistics of turbulence under the form of directional energy, polarization and helicity spectra. The separation space characterization is built on two-point second- and third-order velocity increment moments, and two-point velocity correlations. First, we studied the effect of large-scale spectral forcing. The considered forcing methods are the non-helical and the helical Euler scheme, and the ABC-scheme. We showed that both forcings have a drawback in that, if the number of sufficiently excited modes is too low, anisotropy is bound to arise even at small scales. In the case of Euler forcing, this depends on both the range of forcing wavenumbers and its helicity contents. The ABC forcing, for which the amount of injected helicity cannot be controlled, excites only six modes and therefore always generates anisotropy at all resolved scales. Our second step was to analyze the scale- and direction-dependent anisotropy of homogeneous rotating turbulence. Surprisingly, anisotropy arises at all scales even at low rotation rate. In particular, we identified two anisotropic ranges with different features. In the large scales, directional anisotropy is larger and decreases with wavenumber. At smaller scales, it is much weaker—although still significant—and slowly increases with wavenumber all the way to the dissipative scales. Another interesting and original conclusion of this part of the work concerns the role of the Zeman scale and its link with the flow scale-dependent anisotropy. The Zeman scale was previously argued to be the characteristic lengthscale separating rotation-affected scales 2 from isotropic ones. Upon closer investigation using several simulations at different parameters, we found that the separating scale between large and weak anisotropy is rather the characteristic lengthscale at which rotation and dissipation effects balance. This result, however, does not contradict Zeman’s argument about isotropy recovery in the asymptotic limit of vanishing viscosity, since the separating scale vanishes at infinite Reynolds number, and therefore only the decreasing anisotropy range should persist and scales much smaller than the Zeman one may recover isotropy. Finally, we considered the von Kármán flow between two counter-rotating bladed disks in a cylindrical cavity. We repeated the separation space analysis in different small sub-regions, in order to question the possible analogies in the flow dynamics with that of homogeneous rotating turbulence. We found that, in the regions of the domain where the mean flow has a larger average rotation rate, the distributions of the statistics in separation space display some of the features typical of rotating turbulence.

The purpose of this thesis is to give a classification of anisotropic algebraic groups over number fields of higher real rank. This will complete the classification of algebraic groups over number fields of higher real rank, which was begun by V. Chernousov, L. Lifschitz and D.W. Morris in their paper "Almost-Minimal Non-Uniform Lattices of Higher Rank''. The classification of anisotropic groups of higher real rank is also used to provide a classification of uniform lattices of higher rank contained in semisimple Lie groups with no compact factors. In particular, it is shown that all such lattices sit inside Lie groups of type An. This thesis proceeds as follows: The first chapter provides motivation for the classification and introduces all the main results of the thesis. The second chapter provides relevant definitions and background material for the proof. The next chapters provide a proof of the classification theorem, with chapters 3-5 examining the absolutely simple groups and the final chapter examining the simple groups which are not absolutely simple. / Mathematics

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This study is to investigate that the polishing behavior will be independent of or dependent on the direction of particle motion by the anisotropic polishing phenomenon of hydrodynamic polishing process under the semi-contact lubricating condition. There are two types of experiments to be examined to get to the objectives. First, taking polishing on the work surface which possesses the isotropic surface roughness, to discuss the variation of the smoothing efficiency of the surface irregularities in the five different directions on the work surface. Second, taking three kinds of polishing, ¡§longitudinal, transverse, and oblique roughness polishing¡¨, on the work surface which possesses the anisotropic surface roughness. Then to discuss the variation of the smoothing efficiency of the surface irregularities on the work surface. Both the results of two types experiments should be take to distinguish the difference between one smoothing efficiency and the others from using the hypothesis testing. All hypothesis tests about the experiment results of the work piece which possesses the isotropic surface roughness are accepting . But, most hypothesis tests about the experiment results of the work piece which possesses the anisotropic surface roughness are rejecting . The theory analysis about the smoothing efficiency is discussing. The discussion about the smoothing efficiency can explain the phenomenon due to taking polishing on the work surface which possesses the anisotropic surface roughness. The reason why the phenomenon happened is possible the effects of different lubrication condition. Last, from the lubrication theory, the effects of different lubrication condition due to different surface texture can be employed to verify the explanation about the phenomenon is suitable. The conclusion from the experiment results and the theory analysis is: the polishing behavior is possible independent of the direction of particle motion by the anisotropic polishing phenomenon of hydro- dynamic polishing process under the semi-contact lubricating condition.

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