Conformal geometry processing

This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. Conformal transformations are valuable in applications because they naturally preserve the integrity of geometric data. To date, however, there has been no clearly stated and consistent theory of conformal transformations that can be used to develop general-purpose geometry processing algorithms: previous methods for computing conformal maps have been restricted to the flat two-dimensional plane, or other spaces of constant curvature. In contrast, our formulation can be used to produce---for the first time---general surface deformations that are perfectly conformal in the limit of refinement. It is for this reason that we commandeer the title Conformal Geometry Processing.

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in our theory. Given an immersed surface, we wish to construct new immersions that (i) induce a conformally equivalent metric and (ii) exhibit a prescribed change in extrinsic curvature. Curvature determines the potential in the Dirac equation; the solution of this equation determines the geometry of the new surface. We derive the precise conditions under which curvature is allowed to evolve, and develop efficient numerical algorithms for solving the Dirac equation on triangulated surfaces.

From a practical perspective, this theory has a variety of benefits: conformal maps are desirable in geometry processing because they do not exhibit shear, and therefore preserve textures as well as the quality of the mesh itself. Our discretization yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications. We also present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces.

Interplay of martensitic phase transformation and plastic slip in polycrystals

Inspired by key experimental and analytical results regarding Shape Memory Alloys (SMAs), we propose a modelling framework to explore the interplay between martensitic phase transformations and plastic slip in polycrystalline materials, with an eye towards computational efficiency. The resulting framework uses a convexified potential for the internal energy density to capture the stored energy associated with transformation at the meso-scale, and introduces kinetic potentials to govern the evolution of transformation and plastic slip. The framework is novel in the way it treats plasticity on par with transformation.

We implement the framework in the setting of anti-plane shear, using a staggered implicit/explict update: we first use a Fast-Fourier Transform (FFT) solver based on an Augmented Lagrangian formulation to implicitly solve for the full-field displacements of a simulated polycrystal, then explicitly update the volume fraction of martensite and plastic slip using their respective stick-slip type kinetic laws. We observe that, even in this simple setting with an idealized material comprising four martensitic variants and four slip systems, the model recovers a rich variety of SMA type behaviors. We use this model to gain insight into the isothermal behavior of stress-stabilized martensite, looking at the effects of the relative plastic yield strength, the memory of deformation history under non-proportional loading, and several others.

We extend the framework to the generalized 3-D setting, for which the convexified potential is a lower bound on the actual internal energy, and show that the fully implicit discrete time formulation of the framework is governed by a variational principle for mechanical equilibrium. We further propose an extension of the method to finite deformations via an exponential mapping. We implement the generalized framework using an existing Optimal Transport Mesh-free (OTM) solver. We then model the $\alpha$--$\gamma$ and $\alpha$--$\varepsilon$ transformations in pure iron, with an initial attempt in the latter to account for twinning in the parent phase. We demonstrate the scalability of the framework to large scale computing by simulating Taylor impact experiments, observing nearly linear (ideal) speed-up through 256 MPI tasks. Finally, we present preliminary results of a simulated Split-Hopkinson Pressure Bar (SHPB) experiment using the $\alpha$--$\varepsilon$ model.

Assessment of inland waters fisheries in Nigeria with implications for improved freshwater fish production, poverty alleviation and food security

Rapid Appraisals of the current fisheries situations of some selected inland water bodies in Nigeria were carried out within the framework of Aquaculture and Inland Fisheries Project (AIFP) Annex II of the National Special Programme for Food Security (NSPFS). This paper presents the results of the fishery assessment of one of the selected inland water bodies (Sabke Lake) in Nigeria with a view to optimizing the fish yield through the adoption of improved culture based fishery techniques and community-based fisheries management. The lake is unmanaged and the fishing pressure was found to be very high. Also a number of fishermen were found using small mesh size nets to crop the juveniles of highly valued fish species for an optimum catch. About 14 fishermen ought to have been engaged in full time fishing activities if the fisheries of this lake is to be managed on a sustainable basis. However, a surplus of over 100 fishermen was recorded actively fishing during the period of the study. The results further revealed that Fisheries Rules and Regulations were not established for the national exploitation and proper management of the fisheries of many inland water bodies including Sabke Lake. All these have a depilatory effect on the abundance and sizes of fish harvested from the nigerian inland water bodies especially Sabke Lake. A community based management system that establishes a participatory involvement of fishermen in the conservation and national exploitation of fisheries resources for their own well being is recommended for Sabke Lake and other inland water bodies in Nigeria

Evaluation of a new fishing pot trap (Lege) in River Rima, north western Nigeria

Based on the findings and diagnostic survey, a new fishing trap, christened Lege trap was designed and fabricated, and the performance evaluated. The 8-valve Lege trap was assessed concurrently with Malian and Ndurutu traps by the fishermen in the study area. The experiment was conducted in a randomized complete block design with one factor each replicated three times. Data collected on fish diversity number, biomass and size were subjected to descriptive statistics and analysis of variance. The results showed that 22 fish species belonging to thirteen families were caught. The prototype (Lege) trap recorded higher species diversity index (0.90) than the Malian (0.50) and Ndurutu (0.50) traps. The Lege trap also accounted for the largest number (55%) and biomass (63%) of fish caught which was significantly (P<0.05) higher than those of the Malian and Nduutu traps were. The mean length (15.03~c5.70cm), weight (60.43~c48.61g) and girth (4.77~c1.65cm) of fishes caught in the Lege trap were also significantly (P0.05) higher than those caught in the other two traps. These results demonstrated better performance of the new trap than the two conventional traps, even though the sizes of some of the fish species caught in all the traps were below those allowed by the Sokoto State Fisheries Edict where the study was conducted. Therefore, since it is desirable to develop conservation-oriented trap at a least cost, it is necessary to research further on the number of valves and mesh size of the new trap

Influence of number of valves on catch composition of a fishing pot trap (Lege) in River Rima, north western Nigeria

Three types of prototype (Lege) traps with different numbers of entrance valves were evaluated in River Rima, north western Nigeria. The traps contained 4,6 and 8 valves, tagged 4-V, 6-V and 8-V respectively. The experiment was carried out in a randomized complete block design with one factor each replicated three times. Data collected on fish diversity, number biomass and sizes were subjected to descriptive statistics and analysis of variance. The results of the catch composition showed close diversity index of 0.86 for 6-V, 0.80 for 8-V and 0.60 for 4-V Lege traps. However, the number (41%) and biomass 48%) of fish caught in the 6-V Lege trap were significantly (P0.05) higher than those caught in the other traps. There was no definite trend in the sizes (length and girth) of fish caught in the traps. On the basis of species diversity, and the number and biomass of fish caught, the 6-V Lege showed preference for adoption than the other two traps. However, further studies are recommended on the appropriate mesh size net for the trap in line with the provisions of fisheries edicts

Impact assessment of Niger State fisheries legislation on fisheries conservation in Edozhigi local government area (L.G.A.) of Niger State

A preliminary survey was conducted among the fishermen in five selected villages in Edozhigi L.G.A. of Niger State. One hundred and fifty fishermen were randomly selected and interviewed to find out the impact of Niger State fisheries legislation on fisheries conservation resources in the area. The analysis of data collected using descriptive statistics indicated that undersized mesh of gill nets, beach seines and traps are being used unabated. Also, fenced barriers across the entrance of flood plain ponds and Ex-bow lakes from the main stream are in the area. The fisheries rules and regulations implementers are rarely seen or not seen at all in the area. The decreasing nature of fish catches was detected. It is observed that government policy on fish conversation is neglected due to inadequate or lack of funding for meaningful extension and implementation of the fisheries rules and regulations

Study on the effects of three fishing baits on the catch composition of malian traps in Lake Kainji

The studies on the effects of three fishing baits on the catch composition of Malian traps in Lake Kainji were investigated. The traps were set between Monai and Taafa fishing villages in the Southern basin of the lake, baited with their respective treatment and were inspected daily for twelve days. A total of 218 fish were caught, of which the highest (54.59%) was caught by corn bran, while the lowest (11.01%) was caught by stomach content and rice bran caught 34.4%. The fish caught comprised of 15 species belonging to 8 families. There was no significant different (P>0.05) in the catch of the various baits. The weight also followed the same trend as the number of fish caught. However, both baits showed better efficiency for Alestes baremose. Tilapia zilli, S. galilaeus, Oreochromis niloticus, Labeo coubie and Distichodus rostratus than other species caught. There was a wide range between the inimum and maximum size of species caught, which showed the efficiency of the traps in capturing small size, juveniles and the adult of large fish species due to small mesh size (1") net-cover of the trap. Recommendations were made on the use of corn and rice bran as baits enhancing catch efficiency for fishes such as O.niloticus, T. zilli, T. galilaeus and D. rostratus

Responsible fisheries enhancing poverty alleviation of fishing communities of Lake Kainji

The attributes of responsible fisheries in enhancing Poverty Alleviation of fishing Communities of Lake Kainji were appraised. After impoundment in 1968, the catch per unit effort of fish continued on a downward trend, which suggested a decline in the fish population as a result of improper fishing methods such as the use of small mesh size nets, fish poisoning, destructive fishing gears like beach seine nets. The Nigerian-German (GTZ) Kainji Lake Fisheries Promotion Project intervention brought an increase in yields for the lake in 1996. The estimated annual yield was put at 38.244 metric tones in 1996, an increase of 18 % from the yield recorded in 1995 due to improvement in fishing and conservation methods. The result of catch assessment study carried out in 2003 as a follow up to GTZ intervention is presented. This reveals 15 family and 40 species of fish recorded. Which shows that Lake Kainji still has very high diversity and abundance of fish species. The paper suggests ways to encourage fishing communities to adopt new innovation or strategies by the government and non-governmental organization, which will go a long way in alleviation poverty among the fisher folks

Influence of number of valves on catch composition of a fishing pot trap (Lege) in River Rima, north western Nigeria

Three types of prototype (Lege) traps with different numbers of entrance valves were evaluated in River Rima, north western Nigeria. The traps contained 4,6 and 8 valves, tagged 4-V, 6-V and 8-V respectively. The experiment was carried out in a randomized complete block design with one factor each replicated three times. Data collected on fish diversity, number, biomass and sizes were subjected to descriptive statistics and analysis of variance. The results of the catch composition showed close diversity index of 0.86 for 6-V, 0.80 for 8-V and 0.60 for 4-V Lege traps. However, the number (41%) and biomass (48%) of fish caught in the 6-V Lege trap were significantly (P<0.05) higher than those caught in the other traps. There was no definite trend in the sizes (length and girth) of fish caught in the traps. On the basis of species diversity, and the number and biomass of fish caught, the 6-V Lege showed preference for adoption than the other two traps. However, further studies are recommended on the appropriate mesh size net for the trap in line with the provisions of fisheries edicts

Neue Chance für Scholle und Kabeljau in der Nordsee

Two surveys in the German Bight revealed a relatively good recruitment for plaice and cod. However, with the current high effort and uneconomic way of fishing, most of these fish will end as discards. To protect young plaice during the winter months an increase of the mesh opening in beam trawls within the plaice box is suggested. Therefore, the fishermen are called upon to make a more considerated selection of their nets.

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Effects of a flash-flood on channel morphology and some insects of a small stony stream in the Peak District, Derbyshire

In this paper, some observations are made following a flash-flood that occurred in Stake Clough, a small tributary of the River Goyt, during the evening of 6 August 1996. The site was visited eight times between 8 August - 30 October 1996 to take samples and make observations on the stream. The flood scoured the bed of Stake Clough but more significantly, caused it to change course along the middle part of the floodplain. Initially after the flood, the numbers of insects in all stretches of the stream channel were low (100-200 m super(2)), but then gradually rose to population densities approaching ten times this figure. The benthos was dominated by the Chironomidae and also leuctrid stoneflies (Leuctra nigra, L. hippopus and L. inermis). On 8th August 1996, 12 mesh bags, each containing oak leaves, were placed in the stream and collected after 24 hours. These were also dominated by chironomids, and contained relatively high numbers of the caddis, Potamophylax cingulatus.

Feeding selectivity of brown trout (Salmo trutta) in Loch Ness, Scotland

The aim of this study was to compare statistically the zooplankton assemblage ingested by brown trout (Salmo trutta) in Loch Ness with that of the zooplankton in the water column. This would allow the examination of the apparent paradox that very few copepods appear to be consumed by trout at a time of year when they are numerous and readily available as food. The investigation was limited to the crustacean zooplankters, since the Rotifera are generally so small that they are only of interest to fish in the first few days of life. 25 trout were obtained from anglers, and the stomach contents of non-"ferox" animals analysed. Samples of pelagic zooplankton were obtained approximately monthly from 30-m vertical net-hauls (mesh size 100 km). It is concluded that the variation in dietary composition with trout wet weight indicates an ontogenetic habitat shift producing spatial separation of young and older individuals.

Análise de fadiga em pontes rodoviárias de concreto armado

As pontes rodoviárias de concreto armado estão sujeitas às ações dinâmicas variáveis devido ao tráfego de veículos sobre o tabuleiro. Estas ações dinâmicas podem gerar o surgimento das fraturas ou mesmo a sua propagação na estrutura. A correta consideração destes aspectos objetivou o desenvolvimento de um estudo, de forma a avaliar os esforços do tráfego de veículos pesados sobre o tabuleiro. As técnicas para a contagem de ciclos de esforços e a aplicação das regras de dano acumulado foram analisadas através das curvas S-N de diversas normas estudadas. A ponte rodoviária investigada é constituída por quatro vigas longitudinais, três transversinas e por um tabuleiro de concreto armado. O modelo computacional, desenvolvido para a análise dinâmica da ponte, foi concebido com base no emprego de técnicas usuais de discretização através do método dos elementos finitos. O modelo estrutural da obra de arte rodoviária estudada foi simulado com base no emprego de elementos finitos sólidos tridimensionais. Os veículos são representados a partir de sistemas massa-mola-amortecedor. O tráfego dessas viaturas é considerado mediante a simulação de comboios semi-infinitos, deslocando-se com velocidade constante sobre o tabuleiro da ponte. As conclusões deste trabalho versam acerca da vida útil de serviço dos elementos estruturais de pontes rodoviárias de concreto armado submetidas às ações dinâmicas provenientes do tráfego de veículos pesados sobre o tabuleiro.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.