This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

No presente trabalho, estudamos e desenvolvemos algoritmos com análise de complexidade de avaliação de pior caso para problemas de programação não linear. Para minimização irrestrita, estabelecemos dois algoritmos semelhantes que exploram modelos de ordem superior com estratégia de regularização. Propusemos uma implementação computacional que preserva as boas propriedades teóricas de complexidade, e fizemos experimentos numéricas com problemas clássicos da literatura, a fim de atestar a implementação e avaliar a aplicabilidade de métodos que empreguem modelos de ordem superior. Para minimização com restrições, estabelecemos um algoritmo de duas fases que converge a pontos que satisfazem condições de otimalidade de primeira ordem não escaladas para o problema de programação não linear. / In the present work, we have studied and developed algorithms with worst-case evaluation complexity analysis for nonlinear programming problems. For the unconstrained optimization case, we have established two similar algorithms that explore high-order regularization models. We have proposed a computational implementation that preserves the good properties of the evaluation complexity theory, and we made numerical experiments with classical problems from the literature, in order to check the implementation and certify the practical applicability of methods that employ high-order models. For the constrained optimization case, we have established a two phases algorithm that converges to points that meet the unscaled first-order optimality condition for the nonlinear programming problem.

Estuarine ecosystems are under enormous stress due to rapid coastal developments and climate change. Proper management of these important ecosystems requires a good understanding of their key processes. In this thesis, riparian groundwater-surface water interaction is explored for an aquifer-estuary system primarily by a series of numerical experiments. The work focuses on riparian-scale groundwater flow and salinization. The overall aim of the study was to extend our understanding of aquifer-estuary exchange, which is currently centered on the lower marine estuarine reach, to middle estuaries (i.e., the estuary reach that has variable salinity). The numerical experiments were guided by previous studies and observations made from an exploratory field investigation conducted in and next to Sandy Creek, a macro-tidal estuary incised in the alluvial aquifer of the Pioneer Valley, North-eastern Australia (Longitude 49.11°, Latitude -21.27°). The following observations were made from the field investigation: Sandy Creek estuary experiences a variable salinity regime in its mid reaches that consists of periods of 1) freshwater flushing due to up catchment-derived flooding, 2) persistent freshwater conditions for at least 2 months following the flooding, 3) tidal salinity fluctuations and 4) constant near-seawater salinity; laterally extensive and disconnected aquitards were found to occur at the field site; Sandy Creek had an essentially ‘vertical’ bank slope. Numerical simulations were conducted using the finite element modeling code FEFLOW for saturated unsaturated, variable-density groundwater flow and solute transport, to examine the influence of the following factors on aquifer-estuary exchange: a tidally varying estuarine salinity and hydraulic head, a seasonal freshwater flush (i.e., estuary with freshwater and an elevated stage due to an up catchment sourced flood), near estuary aquitard layers, lateral asymmetry (about the estuary centerline) in hydraulic conductivity and regional hydraulic gradients. The simulations neglected seepage face development after numerical experiments showed that for a vertical bank estuary interacting with a sandy loam aquifer, seepage face effects on groundwater flow and associated salinity distribution were minimal. The following observations were drawn from the range of numerical experiments considered. Tidal salinity fluctuations in the estuary (varying between 0 and 1 - i.e., using a relative salinity scale where a salinity of 1 is seawater) produced flow paths and residence times that were distinctly different to the constant seawater salinity case. While the constant average 0.5 salinity case and the corresponding tidally-varying salinity case (i.e., salinity varying between 0 and 1) produced somewhat comparable results in terms of RUC and RLC (RUC represents groundwater discharge to the estuary that originated from recharge to the estuary bank and RLC groundwater discharge to the estuary that originated from recharge through the estuary bed), whereas flow paths and the total salt mass in the aquifer differed. Freshwater flushing simulations indicated that the near-estuary aquifer responds rapidly to a 2-day ‘wet season’ flushing event with a short-lived freshwater lens created through freshening of the hyporheic zone. Annual cycling of the seasonal flushing led to significant disruption of the estuary water circulation in the aquifer thereby impacting on residence times, transport pathways, and RUC and RLC, and acting to potentially remobilize groundwater and contaminants previously trapped in continuous and semi-continuous re-circulation cells. Although groundwater flow paths determined using tide-averaged velocity vectors were representative of flow paths from transient tidally driven flow vector field, residence times calculated from the two flow fields were markedly different. The influence of riparian scale aquitards and lateral asymmetry (about the estuary centreline) in hydraulic gradients and hydraulic conductivity on groundwater flow and associated salinity distribution was also found to be sensitive to estuarine salinity conditions. The results indicate that observations made about aquifer-estuary interaction in the lower estuary may not be directly applicable to the middle estuary. According to the simulations, tidal salinity variations in the estuary are important factors that affect hyporheic-riparian salt transport processes and that the use of a time averaged estuarine salinity as an approximation to variable salinity conditions is unsuitable for the accurate prediction of the near-estuary dynamics in middle estuaries. This study was based on a two dimensional representation of the riparian scale interaction and it is clear that future research needs to focus on the three-dimensionality of the aquifer-estuary system, incorporating spatially and temporally varying flow and transport characteristics. That is, many estuaries are tortuous and the aquifer geology spatially complex such that assumptions required for the two-dimensional section will most likely restrict application to the field. The tidal dynamics in the middle estuary is also expected to generate three dimensional aspects to the aquifer-estuary interaction. Thus further investigation that explicitly models the hydrodynamics and salt transport in the estuary and estuarine morphology is required to refine the insight provided by the simple conceptual model adopted in this study.

This paper is the last part of a three-fold article aimed at some efficient numerical methods for solving the Poisson problem in three-dimensional prismatic and axisymmetric domains. In the first and second parts, the Fourier singular complement method (FSCM) was introduced and analysed for prismatic and axisymmetric domains with reentrant edges, as well as for the axisymmetric domains with sharp conical vertices. In this paper we shall mainly conduct numerical experiments to check and compare the accuracies and efficiencies of FSCM and some other related numerical methods for solving the Poisson problem in the aforementioned domains. In the case of prismatic domains with a reentrant edge, we shall compare the convergence rates of three numerical methods: 3D finite element method using prismatic elements, FSCM, and the 3D finite element method combined with the FSCM. For axisymmetric domains with a non-convex edge or a sharp conical vertex we investigate the convergence rates of the Fourier finite element method (FFEM) and the FSCM, where the FFEM will be implemented on both quasi-uniform meshes and locally graded meshes. The complexities of the considered algorithms are also analysed.

Estuarine ecosystems are under enormous stress due to rapid coastal developments and climate change. Proper management of these important ecosystems requires a good understanding of their key processes. In this thesis, riparian groundwater-surface water interaction is explored for an aquifer-estuary system primarily by a series of numerical experiments. The work focuses on riparian-scale groundwater flow and salinization. The overall aim of the study was to extend our understanding of aquifer-estuary exchange, which is currently centered on the lower marine estuarine reach, to middle estuaries (i.e., the estuary reach that has variable salinity). The numerical experiments were guided by previous studies and observations made from an exploratory field investigation conducted in and next to Sandy Creek, a macro-tidal estuary incised in the alluvial aquifer of the Pioneer Valley, North-eastern Australia (Longitude 49.11°, Latitude -21.27°). The following observations were made from the field investigation: Sandy Creek estuary experiences a variable salinity regime in its mid reaches that consists of periods of 1) freshwater flushing due to up catchment-derived flooding, 2) persistent freshwater conditions for at least 2 months following the flooding, 3) tidal salinity fluctuations and 4) constant near-seawater salinity; laterally extensive and disconnected aquitards were found to occur at the field site; Sandy Creek had an essentially ‘vertical’ bank slope. Numerical simulations were conducted using the finite element modeling code FEFLOW for saturated unsaturated, variable-density groundwater flow and solute transport, to examine the influence of the following factors on aquifer-estuary exchange: a tidally varying estuarine salinity and hydraulic head, a seasonal freshwater flush (i.e., estuary with freshwater and an elevated stage due to an up catchment sourced flood), near estuary aquitard layers, lateral asymmetry (about the estuary centerline) in hydraulic conductivity and regional hydraulic gradients. The simulations neglected seepage face development after numerical experiments showed that for a vertical bank estuary interacting with a sandy loam aquifer, seepage face effects on groundwater flow and associated salinity distribution were minimal. The following observations were drawn from the range of numerical experiments considered. Tidal salinity fluctuations in the estuary (varying between 0 and 1 - i.e., using a relative salinity scale where a salinity of 1 is seawater) produced flow paths and residence times that were distinctly different to the constant seawater salinity case. While the constant average 0.5 salinity case and the corresponding tidally-varying salinity case (i.e., salinity varying between 0 and 1) produced somewhat comparable results in terms of RUC and RLC (RUC represents groundwater discharge to the estuary that originated from recharge to the estuary bank and RLC groundwater discharge to the estuary that originated from recharge through the estuary bed), whereas flow paths and the total salt mass in the aquifer differed. Freshwater flushing simulations indicated that the near-estuary aquifer responds rapidly to a 2-day ‘wet season’ flushing event with a short-lived freshwater lens created through freshening of the hyporheic zone. Annual cycling of the seasonal flushing led to significant disruption of the estuary water circulation in the aquifer thereby impacting on residence times, transport pathways, and RUC and RLC, and acting to potentially remobilize groundwater and contaminants previously trapped in continuous and semi-continuous re-circulation cells. Although groundwater flow paths determined using tide-averaged velocity vectors were representative of flow paths from transient tidally driven flow vector field, residence times calculated from the two flow fields were markedly different. The influence of riparian scale aquitards and lateral asymmetry (about the estuary centreline) in hydraulic gradients and hydraulic conductivity on groundwater flow and associated salinity distribution was also found to be sensitive to estuarine salinity conditions. The results indicate that observations made about aquifer-estuary interaction in the lower estuary may not be directly applicable to the middle estuary. According to the simulations, tidal salinity variations in the estuary are important factors that affect hyporheic-riparian salt transport processes and that the use of a time averaged estuarine salinity as an approximation to variable salinity conditions is unsuitable for the accurate prediction of the near-estuary dynamics in middle estuaries. This study was based on a two dimensional representation of the riparian scale interaction and it is clear that future research needs to focus on the three-dimensionality of the aquifer-estuary system, incorporating spatially and temporally varying flow and transport characteristics. That is, many estuaries are tortuous and the aquifer geology spatially complex such that assumptions required for the two-dimensional section will most likely restrict application to the field. The tidal dynamics in the middle estuary is also expected to generate three dimensional aspects to the aquifer-estuary interaction. Thus further investigation that explicitly models the hydrodynamics and salt transport in the estuary and estuarine morphology is required to refine the insight provided by the simple conceptual model adopted in this study.

Estuarine ecosystems are under enormous stress due to rapid coastal developments and climate change. Proper management of these important ecosystems requires a good understanding of their key processes. In this thesis, riparian groundwater-surface water interaction is explored for an aquifer-estuary system primarily by a series of numerical experiments. The work focuses on riparian-scale groundwater flow and salinization. The overall aim of the study was to extend our understanding of aquifer-estuary exchange, which is currently centered on the lower marine estuarine reach, to middle estuaries (i.e., the estuary reach that has variable salinity). The numerical experiments were guided by previous studies and observations made from an exploratory field investigation conducted in and next to Sandy Creek, a macro-tidal estuary incised in the alluvial aquifer of the Pioneer Valley, North-eastern Australia (Longitude 49.11°, Latitude -21.27°). The following observations were made from the field investigation: Sandy Creek estuary experiences a variable salinity regime in its mid reaches that consists of periods of 1) freshwater flushing due to up catchment-derived flooding, 2) persistent freshwater conditions for at least 2 months following the flooding, 3) tidal salinity fluctuations and 4) constant near-seawater salinity; laterally extensive and disconnected aquitards were found to occur at the field site; Sandy Creek had an essentially ‘vertical’ bank slope. Numerical simulations were conducted using the finite element modeling code FEFLOW for saturated unsaturated, variable-density groundwater flow and solute transport, to examine the influence of the following factors on aquifer-estuary exchange: a tidally varying estuarine salinity and hydraulic head, a seasonal freshwater flush (i.e., estuary with freshwater and an elevated stage due to an up catchment sourced flood), near estuary aquitard layers, lateral asymmetry (about the estuary centerline) in hydraulic conductivity and regional hydraulic gradients. The simulations neglected seepage face development after numerical experiments showed that for a vertical bank estuary interacting with a sandy loam aquifer, seepage face effects on groundwater flow and associated salinity distribution were minimal. The following observations were drawn from the range of numerical experiments considered. Tidal salinity fluctuations in the estuary (varying between 0 and 1 - i.e., using a relative salinity scale where a salinity of 1 is seawater) produced flow paths and residence times that were distinctly different to the constant seawater salinity case. While the constant average 0.5 salinity case and the corresponding tidally-varying salinity case (i.e., salinity varying between 0 and 1) produced somewhat comparable results in terms of RUC and RLC (RUC represents groundwater discharge to the estuary that originated from recharge to the estuary bank and RLC groundwater discharge to the estuary that originated from recharge through the estuary bed), whereas flow paths and the total salt mass in the aquifer differed. Freshwater flushing simulations indicated that the near-estuary aquifer responds rapidly to a 2-day ‘wet season’ flushing event with a short-lived freshwater lens created through freshening of the hyporheic zone. Annual cycling of the seasonal flushing led to significant disruption of the estuary water circulation in the aquifer thereby impacting on residence times, transport pathways, and RUC and RLC, and acting to potentially remobilize groundwater and contaminants previously trapped in continuous and semi-continuous re-circulation cells. Although groundwater flow paths determined using tide-averaged velocity vectors were representative of flow paths from transient tidally driven flow vector field, residence times calculated from the two flow fields were markedly different. The influence of riparian scale aquitards and lateral asymmetry (about the estuary centreline) in hydraulic gradients and hydraulic conductivity on groundwater flow and associated salinity distribution was also found to be sensitive to estuarine salinity conditions. The results indicate that observations made about aquifer-estuary interaction in the lower estuary may not be directly applicable to the middle estuary. According to the simulations, tidal salinity variations in the estuary are important factors that affect hyporheic-riparian salt transport processes and that the use of a time averaged estuarine salinity as an approximation to variable salinity conditions is unsuitable for the accurate prediction of the near-estuary dynamics in middle estuaries. This study was based on a two dimensional representation of the riparian scale interaction and it is clear that future research needs to focus on the three-dimensionality of the aquifer-estuary system, incorporating spatially and temporally varying flow and transport characteristics. That is, many estuaries are tortuous and the aquifer geology spatially complex such that assumptions required for the two-dimensional section will most likely restrict application to the field. The tidal dynamics in the middle estuary is also expected to generate three dimensional aspects to the aquifer-estuary interaction. Thus further investigation that explicitly models the hydrodynamics and salt transport in the estuary and estuarine morphology is required to refine the insight provided by the simple conceptual model adopted in this study.

No presente trabalho, estudamos e desenvolvemos algoritmos com análise de complexidade de avaliação de pior caso para problemas de programação não linear. Para minimização irrestrita, estabelecemos dois algoritmos semelhantes que exploram modelos de ordem superior com estratégia de regularização. Propusemos uma implementação computacional que preserva as boas propriedades teóricas de complexidade, e fizemos experimentos numéricas com problemas clássicos da literatura, a fim de atestar a implementação e avaliar a aplicabilidade de métodos que empreguem modelos de ordem superior. Para minimização com restrições, estabelecemos um algoritmo de duas fases que converge a pontos que satisfazem condições de otimalidade de primeira ordem não escaladas para o problema de programação não linear. / In the present work, we have studied and developed algorithms with worst-case evaluation complexity analysis for nonlinear programming problems. For the unconstrained optimization case, we have established two similar algorithms that explore high-order regularization models. We have proposed a computational implementation that preserves the good properties of the evaluation complexity theory, and we made numerical experiments with classical problems from the literature, in order to check the implementation and certify the practical applicability of methods that employ high-order models. For the constrained optimization case, we have established a two phases algorithm that converges to points that meet the unscaled first-order optimality condition for the nonlinear programming problem.

This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

<p>In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. </p><p>The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. </p><p>First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. </p><p>Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. </p><p>Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks.</p>

The problem of packing ellipsoids consists in arranging a given collection of ellipsoids within a particular set. The ellipsoids can be freely rotated and translated, and must not overlap each other. A particular case of this problem arises when the ellipsoids are balls. The problem of packing balls has been the subject of intense theoretical and empirical research. In particular, many works have tackled the problem with optimization tools. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years. This problem appears in a large number of practical applications, such as the design of high-density ceramic materials, the formation and growth of crystals, the structure of liquids, crystals and glasses, the flow and compression of granular materials, the thermodynamics of liquid to crystal transition, and, in biological sciences, in the chromosome organization in human cell nuclei. In this work, we deal with the problem of packing ellipsoids within compact sets from an optimization perspective. We introduce continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space. We present two different models for the non-overlapping of ellipsoids. As these models have quadratic numbers of variables and constraints, we also propose an implicit variables models that has a linear number of variables and constraints. We also present models for the inclusion of ellipsoids within half-spaces and ellipsoids. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, we present illustrative numerical experiments that show the capabilities of the proposed models. / O problema de empacotamento de elipsoides consiste em arranjar uma dada coleção de elipsoides dentro de um determinado conjunto. Os elipsoides podem ser rotacionados e transladados e não podem se sobrepor. Um caso particular desse problema surge quando os elipsoides são bolas. O problema de empacotamento de bolas tem sido alvo de intensa pesquisa teórica e experimental. Em particular, muitos trabalhos têm abordado esse problema com ferramentas de otimização. O problema de empacotamento de elipsoides, por outro lado, começou a receber mais atenção apenas recentemente. Esse problema aparece em um grande número de aplicações práticas, como o projeto de materiais cerâmicos de alta densidade, na formação e crescimento de cristais, na estrutura de líquidos, cristais e vidros, no fluxo e compressão de materiais granulares e vidros, na termodinâmica e cinética da transição de líquido para cristal e em ciências biológicas, na organização de cromossomos no núcleo de células humanas. Neste trabalho, tratamos do problema de empacotamento de elipsoides dentro de conjuntos compactos do ponto de vista de otimização. Introduzimos modelos de programação não-linear contínuos e diferenciáveis e algoritmos para o empacotamento de elipsoides no espaço n-dimensional. Apresentamos dois modelos diferentes para a não-sobreposição de elipsoides. Como esses modelos têm números quadráticos de variáveis e restrições em função do número de elipsoides a serem empacotados, também propomos um modelo com variáveis implícitas que possui uma quantidade linear de variáveis e restrições. Também apresentamos modelos para a inclusão de elipsoides em semi-espaços e dentro de elipsoides. Através da aplicação de uma estratégia multi-start simples combinada com uma escolha inteligente de pontos iniciais e um resolvedor para otimização local de programas não-lineares, apresentamos experimentos numéricos que mostram as capacidades dos modelos propostos.