O presente trabalho estuda o modelo de Blume-Capel na rede aleatória e também analisa a inclusão de um termo de campo cristalino aleatório e de um termo de campo local aleatório. Ao resolver o modelo na rede aleatória, uma técnica de conectividade finita foi utilizada, na qual cada spin é conectado a um número finito de outros spins. Os spins foram conectados de acordo com uma distribuição de Poisson, os termos de campo aleatório seguiram uma distribuição bimodal e as interações entre os spins foram consideradas uniformes. Desse modo, só há desordem nas conexões entre os spins. O foco desse trabalho foi determinar como a natureza da transição de fase é alterada com a conectividade e se há um comportamento reentrante das linhas de transição de fase. A técnica de réplicas é usada para obter equações de ponto de sela para a distribuição de campos locais. Um Ansatz de simetria de réplicas foi utilizado para a função de ordem e esse foi escrito em termos de uma distribuição bidimensional de campos efetivos, onde uma das componentes é associada com um termo linear dos spins e a outra com o termo de campo cristalino. Com isso, equações para as funções de ordem e a energia livre podem ser obtidas. Uma técnica de dinâmica populacional é usada para resolver numericamente a equação auto-consistente para a distribuição de campos locais e outros parâmetros, como a magnetização, a atividade da rede e a energia livre. Os resultados indicam que a natureza da transição ferromagnética-paramagnética, a posição do ponto tricrítico e a existência de reentrância dependem fortemente do valor da conectividade e, nos casos com um termo de campo aleatório, dependem da intensidade dos campos aleatórios. No caso em que o campo cristalino é aleatório, o ponto tricrítico é suprimido para valores acima de um certo valor de aleatoriedade. / The present work studies the Blume-Capel model in a random network and also analyses the inclusion of a random crystal-field term and a random field term. To solve the model in a random network a finite connectivity technique is used, in which each spin is connected to a finite number of other spins. The spins were connected according a Poisson distribution, the random field terms followed a bimodal distribution and the bonds between the spins were considered uniform. Thus, there is only a connection disorder. The focus of this work was on determining how the nature of the phase transition changes with the connectivity and the random fields and if there is a reentrant behavior of the phase boundaries. The replica technique is used to obtain saddle-point equations for the effective local-field distribution. The replica symmetric Ansatz for the order function is written in terms of a two-dimensional effective-field distribution, where one of the components is associated with a linear form in the spins and the other with the crystal-field term. This allows one to derive equations for the order function and for the free-energy. A population dynamics procedure is used to solve numerically a self-consistency equation for the distribution of the local field and with it some physical parameters, like magnetization and free-energy. The results obtained indicate that the nature of the F-P transition, the location of the tricritical point and the presence of a reentrant phase depend strongly on the connectivity. In the cases with a random field term, those are also dependent on the intensity of the fields. For the case with a random crystal-field term, the tricritical point is supressed above a certain value of randomness.

O presente trabalho estuda o modelo de Blume-Capel na rede aleatória e também analisa a inclusão de um termo de campo cristalino aleatório e de um termo de campo local aleatório. Ao resolver o modelo na rede aleatória, uma técnica de conectividade finita foi utilizada, na qual cada spin é conectado a um número finito de outros spins. Os spins foram conectados de acordo com uma distribuição de Poisson, os termos de campo aleatório seguiram uma distribuição bimodal e as interações entre os spins foram consideradas uniformes. Desse modo, só há desordem nas conexões entre os spins. O foco desse trabalho foi determinar como a natureza da transição de fase é alterada com a conectividade e se há um comportamento reentrante das linhas de transição de fase. A técnica de réplicas é usada para obter equações de ponto de sela para a distribuição de campos locais. Um Ansatz de simetria de réplicas foi utilizado para a função de ordem e esse foi escrito em termos de uma distribuição bidimensional de campos efetivos, onde uma das componentes é associada com um termo linear dos spins e a outra com o termo de campo cristalino. Com isso, equações para as funções de ordem e a energia livre podem ser obtidas. Uma técnica de dinâmica populacional é usada para resolver numericamente a equação auto-consistente para a distribuição de campos locais e outros parâmetros, como a magnetização, a atividade da rede e a energia livre. Os resultados indicam que a natureza da transição ferromagnética-paramagnética, a posição do ponto tricrítico e a existência de reentrância dependem fortemente do valor da conectividade e, nos casos com um termo de campo aleatório, dependem da intensidade dos campos aleatórios. No caso em que o campo cristalino é aleatório, o ponto tricrítico é suprimido para valores acima de um certo valor de aleatoriedade. / The present work studies the Blume-Capel model in a random network and also analyses the inclusion of a random crystal-field term and a random field term. To solve the model in a random network a finite connectivity technique is used, in which each spin is connected to a finite number of other spins. The spins were connected according a Poisson distribution, the random field terms followed a bimodal distribution and the bonds between the spins were considered uniform. Thus, there is only a connection disorder. The focus of this work was on determining how the nature of the phase transition changes with the connectivity and the random fields and if there is a reentrant behavior of the phase boundaries. The replica technique is used to obtain saddle-point equations for the effective local-field distribution. The replica symmetric Ansatz for the order function is written in terms of a two-dimensional effective-field distribution, where one of the components is associated with a linear form in the spins and the other with the crystal-field term. This allows one to derive equations for the order function and for the free-energy. A population dynamics procedure is used to solve numerically a self-consistency equation for the distribution of the local field and with it some physical parameters, like magnetization and free-energy. The results obtained indicate that the nature of the F-P transition, the location of the tricritical point and the presence of a reentrant phase depend strongly on the connectivity. In the cases with a random field term, those are also dependent on the intensity of the fields. For the case with a random crystal-field term, the tricritical point is supressed above a certain value of randomness.

In this manuscript we perform a rigorous mathematical investigation of the behavior opportunistic network models exhibit when two major real-world problems are taken into account. The first problem considered is obstruction. Here we model the network using an obstructed Gilbert graph which is a classical Gilbert graph but where there exist zones where no nodes are allowed to be placed. We take a look at percolation properties of this model, that is we investigate random graph configurations for which a component of infinite size has strictly positive probability to be created. The second problem considered in this thesis is mobility. Of course mobility in and of itself is not a problem but a feature in any network that follows the store-carry-forward paradigm. However it can be problematic to properly handle in a mathematical model. In the past this has been done by modelling movement by a series of static network configurations. However, with this technique it can be difficult to get a grasp on some of the time sensitive properties of the network. In this work we introduce the time bounded cylinder model which enables an analysis over a complete timeframe. We provide normal approximations for important properties of the model, like its covered volume and the number of isolated nodes. As we are using rigorous mathematics to tackle problems which computer scientists working in the field of distributed systems are faced with, we bring the two fields closer together.

Unmanned aerial vehicle (UAV) networks, a flying platform, are a promising wireless communications infrastructure with wide-ranging applications in both commercial and military domain. Owing to the appealing characteristics, such as high mobility, high feasibility, and low cost, UAV networks can be applied in various scenarios, such as emergency communications, cellular networks, device-to-device (D2D) networks, and sensor networks, regardless of infrastructure and spatial constraints. To handle complicated missions, provide wireless coverage for a large range, and have a long lifetime, a UAV network may consist of a large amount of UAVs, working cooperatively as a swarm, also referred to as swarm UAV networks. Although high mobility and numerous UAVs offer high flexibility, high scalability, and performance enhancement for swarm UAV networks, they also incur some technical challenges. One of the major challenges is the routing protocol design. With high mobility, a dynamic network topology may be encountered. As a result, traditional routing protocols based on routing path discovery are not applicable in swarm UAV networks, as the discovered routing path may be outdated especially when the amount of UAVs is large causing considerable routing path discovery delay. Multicast is an essential and key technology in the scenarios, where swarm UAV networks are employed as aerial small base station (BSs), like relay or micro BS. Swarm UAV networks consisting of a large amount of UAVs will encounter severe multicast delay with existing multicast methods using acknowledgement (ACK) feedback and retransmissions. This issue will be deteriorated when a swarm UAV network is deployed far away from BSs, causing high packet loss. Data exchange is another major technical challenge in swarm UAV networks, where UAVs exchange data packets with each other, such as requesting and retrieving lost packets. Due to numerous UAVs, data exchange between UAVs can cause message and signaling storm, resulting in a long data exchange delay and severe ovehead. In this dissertation, I focus on developing novel routing protocols, multicast schemes, and data exchange schemes, enabling efficient, robust, and high-performance routing, multicast, and data exchange in swarm UAV networks. To be specific, two novel flooding-based routing protocols are designed in this dissertation, where random network coding (RNC) is utilized to improve the efficiency of the flooding-based routing in swarm UAV networks without relying on network topology information and routing path discovery. Using the property of RNC that as long as sufficient different versions of encoded packets/generations are accumulated, original packets could be decoded, RNC is naturally able to accelerate the routing process. This is because the use of RNC can reduce the number of encoded packets that are required to be delivered in some hop. In a hop, the receiver UAV may have already overheard some generations in previous hops, so that it only needs to receive fewer generations from the transmitter UAV in the current hop. To further expedite the flooding-based routing, the second flooding-based routing protocol is designed, where each forwarding UAV creates a new version of generation by linearly combining received generations rather than by decode original packets. Despite the flooding-based routing significantly hastened by RNC, the inherent drawback of the flooding-based routing is still unsolved, namely numerous hops. Aiming at reducing the amount of hops, a novel enhanced flooding-based routing protocol leveraging clustering is designed, where the whole UAV network will be partitioned into multiple clusters and in each cluster only one UAV will be selected as the representative of this cluster, participating in the flooding-based routing process. By this way, the number of hops is restricted by the number of representatives, since packets are only flooded between limited representatives rather than numerous UAVs. To address the multicast issue in swarm UAV networks, a novel multicast scheme is proposed based on clustering, where a UAV experiencing packet loss will retrieve the lost packets by requesting other UAVs in the same cluster without depending on retransmissions of BSs. In this way, the lost packet retrieval is carried out through short-distance data exchange between UAVs with reliable transmissions and a short delay. Tractable stochastic geometry tools are used to model swarm UAV networks with a dynamic network topology, based on which comprehensive analytical performance analysis is given. To enable efficient data exchange between UAVs in swarm UAV networks, a data exchange scheme is proposed utilizing unsupervised learning. With the proposed scheme, all UAVs are assigned to multiple clusters and a UAV can only carry out data exchange within its cluster. By this way, UAVs in different clusters perform data exchange in a parallel fashion to expedite data exchange. The agglomerative hierarchical clustering, a type of unsupervised learning, is used to conduct clustering in order to guarantee that UAVs in the same cluster are able to supply and supplement each other's lost packets. Additionally, a data exchange mechanism, including a novel random backoff procedure, is designed, where the priorities of UAVs in data exchange determined by the number of their lost packets or requested packets that they can provide. As a result, each request-reply process would be taken fully advantage, maximally supplying lost packets not only to the UAV sending request, but also to other UAVs in the same cluster. For all the developed technologies in this dissertation, their technical details and the corresponding system procedures are designed based on low-complexity and well-developed technologies, such as the carrier sense multiple access/collision avoidance (CSMA/CA), for practicability in practice and without loss of generality. Moreover, extensive simulation studies are conducted to demonstrate the effectiveness and superiority of the proposed and developed technologies. Additionally, system design insights are also explored and revealed through simulations. / Doctor of Philosophy / Compared to fixed infrastructures in wireless communications, unmanned aerial vehicle (UAV) networks possess some significant advantages, such as low cost, high mobility, and high feasibility, making UAV networks have a wide range of applications in both military and commercial fields. However, some characteristics of UAV networks, including dynamic network topology and numerous UAVs, may become technical barriers for wireless communications. One of the major challenges is the routing protocol design. Routing is the process of selecting a routing path, enabling data delivered from a node (source) to another desired node (destination). Traditionally, routing is performed based on routing path discovery, where control packets are broadcasted and the path, on which a control packet first reaches the destination, will be selected as routing path. However, in UAV networks, routing path discovery may experience a long delay, as control packets go through many UAVs. Besides, the discovered routing path may be outdated, as the topology of UAV networks change over time. Another key technology in wireless communications that may not work well in UAV networks is multicast, where a transmitter, like a base station (BS), broadcasts data to UAVs and all UAVs are required to receive this data. With numerous UAVs, multicast delay may be severe, since the transmitter will keep retransmitting a data packet to UAVs until all UAVs successfully receive the packet. This issue will be deteriorated when a UAV network is deployed far away from BSs, causing high packet loss. Data exchange between UAVs is a fundamental and important system procedure in UAV networks. A large amount of UAV in a UAV network will cause serious data exchange delay, as many UAVs have to compete for limited wireless resources to request or send data. In this dissertation, I focus on developing novel technologies and schemes for swarm UAV networks, where a large amount of UAVs exist to make UAV networks powerful and handle complicated missions, enable efficient, robust, and high-performance routing, multicast, and data exchange system procedures. To be specific, two novel flooding-based routing protocols are designed, where random network coding (RNC) is utilized to improve the efficiency of flooding-based routing without relying on any network topology information or routing path discovery. The use of RNC could naturally expedite flooding-based routing process. With RNC, a receiver can decode original packets as long as it accumulates sufficient encoded packets, which may be sent by different transmitters in different hops. As a result, in some hops, fewer generations may be required to be transmitted, as receivers have already received and accumulated some encoded in previous hops. To further improve the efficiency of flooding-based routing, another routing protocol using RNC is designed, where UAVs create new encoded packets by linearly combining received encoded packets rather than linearly combing original packets. Apparently, this method would be more efficient. UAVs do not need to collect sufficient encoded packets and decode original packets, while only linearly combining all received encoded packets. Although RNC could effectively improve the efficiency of flooding-based routing, the inherent drawback is still unsolved, which is a large amount of hops caused by numerous UAVs. Thus, an enhanced flooding-based routing protocol using clustering is designed, where the whole UAV network will be partitioned into multiple clusters. In each cluster only one UAV will be selected as the representative of this cluster, participating in the flooding-based routing process. By this way, the number of hops could be greatly reduced, as packets are only flooded between limited representatives rather than numerous UAVs. To address the multicast issue in swarm UAV networks, a novel multicast scheme is proposed, where a UAV experiencing packet loss will retrieve its lost packets by requesting other UAVs in the same cluster without depending on retransmissions of BSs. In this way, the lost packet retrieval is carried out through short-distance data exchange between UAVs with reliable transmissions and a short delay. Then, the optimal number of clusters and the performance of the proposed multicast scheme are investigated by tractable stochastic geometry tools. If all UAVs closely stay together in a swarm UAV network, long data exchange delay would be significant technical issue, since UAVs will cause considerable interference to each other and all UAVs will compete for spectrum access. To cope with that, a data exchange scheme is proposed leveraging unsupervised learning. To avoid interference between UAVs and a long-time waiting for spectrum access, all UAVs are assigned to multiple clusters and different clusters use different frequency bands to carry out data exchange simultaneously. The agglomerative hierarchical clustering, a type of unsupervised learning, is used to conduct clustering, guaranteeing that UAVs in the same cluster are able to supply and supplement each other's lost packets. Additionally, a data exchange mechanism is designed, facilitating that a UAV with more lost packets or more requested packets has a higher priority to carry out data exchange. In this way, each request-reply process would be taken fully advantage, maximally supplying lost packets not only to the UAV sending request, but also to other UAVs in the same cluster. For all the developed technologies in this dissertation, their technical details and the corresponding system procedures are designed based on low-complexity and well-developed technologies, such as the carrier sense multiple access/collision avoidance (CSMA/CA), for practicability in reality and without loss of generality. Moreover, extensive simulation studies are conducted to demonstrate the effectiveness and superiority of the developed technologies. Additionally, system design insights are also explored and revealed through simulations.

Wireless digital broadcasting applications such as digital audio broadcast (DAB) and digital video broadcast (DVB) are becoming increasingly popular since the digital format allows for quality improvements as compared to traditional analogue broadcast. The broadcasting is commonly based on packet transmission. In this thesis, we consider broadcasting over packet erasure channels. To achieve reliable transmission, error-control schemes are needed. By carefully designing the error-control schemes, transmission efficiency can be improved compared to traditiona lautomatic repeat-request (ARQ) schemes and rateless codes. Here, we first study the application of a novel binary deterministic rateless (BDR) code. Then, we focus on the design of network coding for the wireless broadcasting system, which can significantly improve the system performance compared to traditional ARQ. Both the one-hop broadcasting system and a relay-aided broadcasting system areconsidered. In the one-hop broadcasting system, we investigate the application of systematic BDR (SBDR) codes and instantaneously decodable network coding (IDNC). For the SBDR codes, we determine the number of encoded redundancy packets that guarantees high broadcast transmission efficiencies and simultaneous lowcomplexity. Moreover, with limited feedback the efficiency performance can be further improved. Then, we propose an improved network coding scheme that can asymptotically achieve the theoretical lower bound on transmission overhead for a sufficiently large number of information packets. In the relay-aided system, we consider a scenario where the relay node operates in half duplex mode, and transmissions from the BS and the relay, respectively, are over orthogonal channels. Based on random network coding, a scheduling problem for the transmissions of redundancy packets from the BS and the relay is formulated. Two scenarios; namely instantaneous feedback after each redundancy packet, and feedback after multiple redundancy packets are investigated. We further extend the algorithms to multi-cell networks. Besides random network coding, IDNC based schemes are proposed as well. We show that significant improvements in transmission efficiency are obtained as compared to previously proposed ARQ and network-coding-based schemes. / QC 20110907

<p>Moving boundary problems such as solidification, crack propagation, multi-body contact or shape optimal design represent an important class of engineering problems. Common to these problems are one or more moving interfaces or boundaries. One of the main challenges associated with boundary evolution is the difficulty that arises when the topology of the geometry changes. Other geometric issues such as distance to the boundary, projected point on the boundary and intersection between surfaces are also important and need to be efficiently solved. In general, the present thesis is concerned with the geometric arrangement and behavioral analysis of evolving parametric boundaries immersed in a domain. </p> <p>The first problem addressed in this thesis is the packing of ellipsoidal fillers in a regular domain and to estimate their effective physical behavior. Particle packing problem arises when one generates simulated microstructures of particulate composites. Such particulate composites used as thermal interface materials (TIMs) motivates this work. The collision detection and distance calculation between ellipsoids is much more difficult than other regular shapes such as spheres or polyhedra.  While many existing methods address the spherical packing problems, few appear to achieve volume loading exceeding 60%. The packing of ellipsoidal particles is even more difficult than that of spherical particles due to the need to detect contact between the particles. In this thesis, an efficient and robust ultra-packing algorithm termed Modified Drop-Fall-Shake is developed. The algorithm is used to simulate the real mixing process when manufacturing TIMs with hundreds of thousands ellipsoidal particles. The effective thermal conductivity of the particulate system is evaluated using an algorithm based on Random Network Model. </p> <p><br></p> <p>In problems where general free-form parametric surfaces (as opposed to the ellipsoidal fillers) need to be evolved inside a regular domain, the geometric distance from a point in the domain to the boundary is necessary to determine the influence of the moving boundary on the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge,  an algebraic procedure is presented here to find the level sets of rational parametric surfaces commonly utilized by commercial CAD systems. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bezier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain can also be used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G₀ continuity---problems where the Newton--Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively. </p> <p><br></p> <p>Next, ALS is also extended for boundary representation (B-rep) models that are popularly used in CAD systems for modeling solids. B-rep model generally contains multiple NURBS patches due to the trimming feature used to construct such models, and as a result are not ``watertight" or mathematically compatible at patch edges. A time consuming geometry clean-up procedure is needed to preprocess geometry prior to finite element mesh generation using a B-rep model, which can take up to 70% of total analysis time according to literature. To avoid the need to clean up geometry and directly provide link between CAD and CAE integration,  signed algebraic level sets using novel inner/outer bounding box strategy is proposed for point classification of B-rep model. Several geometric examples are demonstrated, showing that this technique naturally models single patch NURBS geometry as well, and can deal with multiple patches involving planar trimming feature and Boolean operation. During the investigation of algebraic level sets, a complex self-intersection problem is also reported, especially for three-dimensional surface. The self-intersection may occur within an interval of interest during implicitization of a curve or surface since the implicitized curve or surface is not trimmed and extends to infinity. Although there is no robust and universal solution the problem, two potential solutions are provided and discussed in this thesis.</p> <p><br></p> <p>In order to improve the computational efficiency of analysis in immersed boundary problems, an efficient local refinement technique for both mesh and quadrature  using the kd-tree data structure is further proposed. The kd-tree sub-division is theoretically proved to be more efficient against traditional quad-/oct-tree subdivision methods. In addition, an efficient local refinement strategy based on signed algebraic level sets is proposed to divide the cells. The efficiency of kd-tree based mesh refinement and adaptive quadrature is later shown through numerical examples comparing with oct-tree subdivision, revealing significant reduction of degrees of freedom and quadrature points.</p> <p><br></p> <p>Towards analysis of moving boundaries problems, an explicit interface tracking method termed enriched isogeometric analysis (EIGA) is adopted in this thesis. EIGA utilizes NURBS shape function for both geometry representation and field approximation. The behavior field is modeled by a weighted blending of the underlying domain approximation and enriching field, allowing high order continuity naturally. Since interface is explicitly represented, EIGA provides direct geometric information such as normals and curvatures. In addition, the blending procedure ensures strong enforced boundary conditions. An important moving boundary problem, namely, reaction-diffusion problem, is investigated using EIGA. In reaction-diffusion problems, the phase interfaces evolve due to chemical reaction and diffusion under multi-physics driven forces, such as mechanical, electrical, thermal, etc. Typical failure phenomenon due to reaction-diffusion problems include void formation and intermetallic compound (IMC) growth. EIGA is applied to study factors and behavior patterns in these failure phenomenon, including void size, current direction, current density, etc. A full joint simulation is also conducted to study the degradation of solder joint under thermal aging and electromigration. </p>

This dissertation is related to mathematical modelling of the spread of respiratory syncytial virus (RSV) in Valencia and is still causing a large number of hospitalizations of children in this community. A mathematical model based on a system of nonlinear differential equations of first order has been built. This model considers the population divided into two age groups to pay particular attention to children under one year who are the most affected by this disease. This model has been fitted with hospitalizations data of Valencia and has been used to perform a cost analysis of a potential vaccination strategy. We also propose a complete network model to study the seasonal evolution of RSV epidemics in which seasonal parameters were fitted with the previous continuous model. Both developments are contrasted. On the complete network model we propose a strategy for vaccination of children based on the administration of three doses, and develop a cost-effectiveness study for different vaccination rates. Finally we have defined a SIRS model for RSV epidemics on a random social network of contacts among individuals. In this model has not forced the seasonality. The seasonality arises naturally for certain values of the duration of immunity of a patient recovered, the number of contacts and the likelihood of infection from a contact in a day. In this social network model only a narrow range of parameters can support RSV epidemic seasons. / Moraño Fernández, JA. (2010). Mathematical modelling of Respiratory Syncytial Virus spread in the Spanish region of Valencia. Preventive applications [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8638

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal