Today is on products from the perspective of quality placed an increasing emphasis. This work is focused to the analyze of cross roll straghtening of long semifinished products with a circular cross-section. This process increased quality of semifinished products from the perspective of their curvature, because they do not reach the required limit values. Curvature of the semifinished product is due to residual stress, which is during straghtening using plastic deformation of the material redistributed and the semifinished product is leveled. This work analyzes the effect of rotation of straightening rolls on the distributiom of contact pressure between the straightening semifinished product and the roll of straightener. Author then analyzes effect of input parameters of program to straightening at the final curvature of the semifinished product. With the experience gained, then try to adjust some input parameters.

We study the dynamics of time-like geodesics in the field of black holes perturbed by a circular ring or disc, restricting to static and axisymmetric class of space-times. Two analytical methods are tested which do not require solving the equations of motion: (i) the so-called geometric criterion of chaos based on eigenvalues of the Riemann tensor, and (ii) the method of Melnikov which detects the chaotic layer arising by break-up of a homoclinic orbit. Predictions of both methods are compared with numerical results in order to learn how accurate and reliable they are.

This thesis is concerned with two problems in the Calculus of Variations touching on two central aspects of Materials Science: the structure of solid matter and its dynamic behavior. The problem pertaining to the first aspect is the analysis of the rigidity properties of possibly branched microstructures formed by shape memory alloys undergoing cubic-to-tetragonal transformations. On the basis of a variational model in the framework of linearized elasticity, we derive a non-convex and non-discrete valued differential inclusion describing the local volume fractions of such structures. Our main result shows the inclusion to be rigid without additional regularity assumptions and provides a list of all possible solutions. We give constructions ensuring that the various types of solutions indeed arise from the variational model and quantitatively describe their rigidity via H-measures. Our contribution to the second aspect is a conditional result on the convergence of the Allen-Cahn Equations to multi-phase mean curvature flow, which is a popular model for grain growth in polychrystalline metals. The proof relies on the gradient flow structure of both models and borrows ideas from certain convergence proofs for minimizing movement schemes.:1 Introduction 1.1 Shape memory alloys 1.2 Multi-phase mean curvature flow 2 Branching microstructures in shape memory alloys: Rigidity due to macroscopic compatibility 2.1 The main rigidity theorem 2.2 Outline of the proof 2.3 Proofs 3 Branching microstructures in shape memory alloys: Constructions 3.1 Outline and setup 3.2 Branching in two linearly independent directions 3.3 Combining all mechanisms for varying the volume fractions 4 Branching microstructures in shape memory alloys: Quantitative aspects via H-measures 4.1 Preliminary considerations 4.2 Structure of the H-measures 4.3 The transport property and accuracy of the approximation 4.4 Applications of the transport property 5 Convergence of the Allen-Cahn Equation to multi-phase mean curvature flow 5.1 Main results 5.2 Compactness 5.3 Convergence 5.4 Forces and volume constraints

In geometry, various tools have been developed to explore the topology and other features of a manifold from its geometrical structure. Among the two most powerful ones are the analysis of the critical points of a function, or more generally, the closed orbits of a dynamical system defined on the manifold, and the evaluation of curvature inequalities. When any (nondegenerate) function has to have many critical points and with different indices, then the topology must be rich, and when certain curvature inequalities hold throughout the manifold, that constrains the topology. It has been observed that these principles also hold for metric spaces more general than Riemannian manifolds, and for instance also for graphs. This thesis represents a contribution to this program. We study the relation between the closed orbits of a dynamical system and the topology of a manifold or a simplicial complex via the approach of Floer. And we develop notions of Ricci curvature not only for graphs, but more generally for, possibly directed, hypergraphs, and we draw structural consequences from curvature inequalities. It includes methods that besides their theoretical importance can be used as powerful tools for data analysis. This thesis has two main parts; in the first part we have developed topological methods based on the dynamic of vector fields defined on smooth as well as discrete structures. In the second part, we concentrate on some curvature notions which already proved themselves as powerful measures for determining the local (and global) structures of smooth objects. Our main motivation here is to develop methods that are helpful for the analysis of complex networks. Many empirical networks incorporate higher-order relations between elements and therefore are naturally modeled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraphs, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. We can then characterize various classes of hypergraphs by their curvature. In the last chapter, we show that our curvature notion is a powerful tool for determining complex local structures in a variety of real and random networks modeled as (directed) hypergraphs. Furthermore, it can nicely detect hyperloop structures; hyperloops are fundamental in some real networks such as chemical reactions as catalysts in such reactions are faithfully modeled as vertices of directed hyperloops. We see that the distribution of our curvature notion in real networks deviates from random models.

Cette thèse adresse le problème du consensus dans les réseaux. On étudie des réseaux composés d'agents identiques capables de communiquer entre eux, qui ont une mémoire et des capacités de calcul. Le réseau ne possède pas de nœud central de fusion. Chaque agent stocke une valeur qui n'est pas initialement connue par les autres agents. L'objectif est d'atteindre le consensus, i.e. tous les agents ont la même valeur, d'une manière distribuée. De plus, seul les agents voisins peuvent communiquer entre eux. Ce problème a une longue et riche histoire. Si toutes les valeurs appartiennent à un espace vectoriel, il existe plusieurs protocoles pour résoudre le problème. Une des solutions connues est l'algorithme du gossip qui atteint le consensus de manière asymptotique. C'est un protocole itératif qui consiste à choisir deux nœuds adjacents à chaque itération et de les moyenner. La spécificité de cette thèse est dans le fait que les données stockées par les agents n'appartiennent pas nécessairement à un espace vectoriel, mais à un espace métrique. Par exemple, chaque agent stocke une direction (l'espace métrique est l'espace projectif) ou une position dans un graphe métrique (l'espace métrique est le graphe sous-jacent). Là, les protocoles de gossip mentionnés plus haut n'ont plus de sens car l'addition qui n'est plus disponibles dans les espaces métriques. Cependant, dans les espaces métriques les points milieu ont du sens dans certains cas. Et là ils peuvent se substituer aux moyennes arithmétiques. Dans ce travail, on a compris que la convergence du gossip avec les points milieu dépend de la courbure. On s'est focalisés sur le cas où l'espace des données appartient à une classe d'espaces métriques appelés les espaces CAT(k). Et on a pu démontrer que si les données initiales sont suffisamment "proches" dans un sens bien précis, alors le gossip avec les points milieu - qu'on a appelé le Random Parwise Midpoints- converge asymptotiquement vers un consensus / This thesis deals with the problem of consensus on networks. Networks under study consists of identical agents that can communicate with each other, have memory and computational capacity. The network has no central node. Each agent stores a value that, initially, is not known by other agents. The goal is to achieve consensus, i.e. all agents having the same value, in a fully distributed way. Hence, only neighboring agents can have direct communication. This problem has a long and fruitful history. If all values belong to some vector space, several protocols are known to solve this problem. A well-known solution is the pairwise gossip protocol that achieves consensus asymptotically. It is an iterative protocol that consists in choosing two adjacent nodes at each iteration and average them. The specificity of this Ph.D. thesis lies in the fact that the data stored by the agents does not necessarily belong to a vector space, but some metric space. For instance, each agent stores a direction (the metric space is the projective space) or position on a sphere (the metric space is a sphere) or even a position on a metric graph (the metric space is the underlying graph). Then the mentioned pairwise gossip protocols makes no sense since averaging implies additions and multiplications that are not available in metric spaces: what is the average of two directions, for instance? However, in metric spaces midpoints sometimes make sense and when they do, they can advantageously replace averages. In this work, we realized that, if one wants midpoints to converge, curvature matters. We focused on the case where the data space belongs to some special class of metric spaces called CAT(k) spaces. And we were able to show that, provided initial data is "close enough" is some precise meaning, midpoints-based gossip algorithm – that we refer to as Random Pairwise Midpoints - does converge to consensus asymptotically. Our generalization allows to treat new cases of data spaces such as positive definite matrices, the rotations group and metamorphic systems

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19468号 / 理博第4128号 / 新制||理||1594(附属図書館) / 32504 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 上田 哲生, 教授 重川 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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The femur is the longest limb bone found in humans. Almost all the long limb bones found in terrestrial mammals, including the femur studied herein, have been observed to be loaded in bending and are curved longitudinally. The curvature in these long bones increases the bending stress developed in the bone, potentially reducing the bone’s load carrying capacity, i.e. its mechanical strength. Therefore, bone curvature poses a paradox in terms of the mechanical function of long limb bones. The aim of this study is to investigate and explain the role of longitudinal bone curvature in the design of long bones. In particular, it has been hypothesized that curvature of long bones results in a trade-off between the bone’s mechanical strength and its bending predictability. This thesis employs finite element analysis of human femora to address this issue. Simplified human femora with different curvatures were modeled and analyzed using ANSYS Workbench finite element analysis software. The results obtained are compared between different curvatures including a straight bone. We examined how the bone curvature affects the bending predictability and load carrying capacity of bones. Results were post processed to yield probability density functions (PDFs) for circumferential location of maximum equivalent stress for various bone curvatures to assess the bending predictability of bones. To validate our findings on the geometrically simplified ANSYS Workbench femur models, a digitally reconstructed femur model from a CT scan of a real human femur was employed. For this model we performed finite element analysis in the FEA tool, Strand7, executing multiple simulations for different load cases. The results from the CT scanned femur model and those from the CAD femur model were then compared. We found general agreement in trends but some quantitative differences most likely due to the geometric differences between the digitally reconstructed femur model and the simplified CAD models. As postulated by others, our results support the hypothesis that the bone curvature is a trade-off between the bone strength and its bending predictability. Bone curvature increases bending predictability at the expense of load carrying capacity.

This thesis quantifies the flexural behavior (strength, stiffness and failure) of IsoTruss®-reinforced concrete beam-columns for use in deep foundation pile applications. Four-point bending tests were performed in the laboratory on two instrumented carbon/epoxy IsoTruss® reinforced concrete piles (IRC piles) and two instrumented steel reinforced concrete piles (SRC piles). The piles were approximately 14 ft (4.3 m) in length and 14 in (36 cm) in diameter and were loaded to failure while monitoring load, deflection, and strain data. The steel and IsoTruss®® reinforcement cages were designed to have equal flexural stiffness to permit a relative strength comparison. Moment curvature diagrams reveal that the stiffness values were indeed close, verifying the design objective. At failure, the IsoTruss®-reinforced concrete beams held nearly twice the bending moment as the steel-reinforced concrete beams [1,719 kip-in vs. 895 kip-in (194 kN-m vs.101 kN-m)], although the failure modes were quite different. The SRC piles exhibited the traditional ductile failure behavior, as expected, while the IRC piles lacked ductility. The IRC pile deflections remained linear to failure, while the SRC piles yielded significantly. At 35 kips (165 kN), the maximum load on the SRC piles, the ductility of the SRC piles was twice that of the IRC piles (0.0084 and 0.0042, respectively). Toughness measurements reveal that due to the lack of ductility in the IRC piles, the SRC piles absorbed approximately twice as much energy as the IRC piles. Further investigations are required to explain the absence of ductility in the IRC piles, since ductility has been observed in other IsoTruss®-reinforced concrete structures in flexure. Even with this low level of ductility, the IRC piles are substantially stronger than the SRC piles and provide an alternative for use in deep foundation environments. Not only is the IRC pile strong enough for the job, but the IsoTruss® reinforcement is approximately 62% lighter, more rigid, and more corrosion resistant than the steel reinforcement.