Publicações

A first characterization of greenhouse gases had been carried out to study their role and impact in a productive transitional coastal system of the southern Portugal

Near estuaries and harbours, submerged shoals and defence structures impact the exposure to overtopping. These features may be accounted for in two-dimensional horizontal (2DH) numerical models, either based on phase resolving or phase-average solvers for wave propagation. In between, surfbeat solvers, such as in XBeach, combine an affordable computational cost with the ability to generate and propagate the longer infragravity (IG) waves. However, surfzone wave characteristics and the overtopping exposure modelled with XBeach are sensi tive to settings such as the shape of the forcing wave spectra and the numerical scheme for wave propagation. The present paper explores this sensitivity and assesses the performance of different inundation models built with the 2DH surfbeat solver of XBeach. These models were forced with downscaled water levels and directional wave spectra and the results fuelled a discussion bounded by data collected downdrift of the entrance to the harbour of Figueira da Foz (Portugal). The original second-order upwind scheme, which propagates short-waves with a lower numerical diffusion improved the model performance in terms of long-wave height, and an unconventional breaking criterion better represented the cross-shore distribution of short-wave height near the shore. A cali brated model was validated through the hindcast of an overtopping event observed under moderate swell forcing, and was used to map the overtopping exposure during a hypothetical combination of an energetic swell with a water level having a return period of ~70 years. Compared to the default wave spectra shape and model settings, using an appropriate representation of the short-wave directional spectrum at the open boundary was necessary to reproduce the observed overtopping extent. Refining the cross-shore resolution of the model helped to better represent the observed inundation extents, as also did the reduction of the friction coefficient. Addi tional phase-resolving simulations in 1DH overestimated IG wave energy and produced higher and more frequent overtopping discharges. The differences with the calibrated 2DH surfbeat model increased with the proximity of the inlet and with short-wave height and angle of incidence. Overall, required calibration steps were provided. They aim at making 2DH XBeach surfbeat a credible tool for 1) predicting short and IG wave characteristics up to the shoreline, as well as for 2) providing first estimates of exposure to overtopping in areas with shallow and alongshore-irregular morphologies.

The efects of climate change on the estuarine environment are not fully understood. In temperate regions, droughts are expected to increase in frequency and severity, due to lower precipitation, and the mean sea level is expected to rise. This study aimed to assess how the estuarine environment will be afected by river fow

Mainland Portugal presents an extensive, diversified and complex coastal zone. This diversity gives rise to diverse geomorphologic features which support different densities of human occupancy and a range of land uses and activities. In this sense, the coastal zone becomes an area of great national strategic value, but has also into a multi-hazard zone. To understand this complexity, two statistical techniques, namely Cluster Analysis and Principal Component Analysis, were used to identify and differentiate, between the different coastal typologies, based on a set of variables that express the territorial complexity and the occurrences and impacts of coastal flooding. The chosen methodology enabled a diverse set of coastal typologies to be identified, which contributes towards differentiating local specific characteristics. The holistic and differentiating nature of the applied methodology and the results obtained allows it to contribute to the definition of mitigation strategies and the implementation of adaptation measures. The results enable us to define a set of typologies associated with Coastal Disaster Risk Reduction and Management which are an important contribution to comprehensive coastal flood risk assessment and management.

Ministry of the Sea promoted an expert conference to discuss the currentstate of the Tagus river estuary and adjacent oceanic areas and to promotefuture action. The Tagus river estuary shapes the Lisbon MetropolitanArea, bordering 11 water-facing municipalities across 320 km2(among the largest in Europe and critically positioned along the North-South bird migration corridor of the East Atlantic [1]). It communicateswith the deep North Atlantic through the Tagus submarine prodelta andthe Lisbon/Cascais/Setúbal canyon system (Fig. 1). This proximity allowsinteraction of water masses and biological communities betweenthe estuary and the deep ocean and deserves scientific attention.The conference integrated perspectives on current threats, aftermillennia of human occupation and decades of local efforts to improveenvironmental status and contributed towards an action plan to maintainor improve ecosystem services for this densely populated area,following an ecosystem stewardship framework [2].

The characterization of estuarine hydrodynamics primarily depends on knowledge of the bathymetryand topography. Here, we present the first comprehensive, high-resolution dataset of the topography andbathymetry of the Amazon River estuary, the world

Analisa-se a evolução batimétrica naenvolvente do esporão EV2,na Cova do Vapor, ao longo dos últimos 50 anos,com base em levantamentoshidrográficos.Aocorrênciade erosões localizadas está associadaao prolongamento inicial do esporão, em 1969-1971. A profundidade máxima(-14,1mZH, 200m a WNW da cabeça)foi atingida em 2006, na sequência de uma intervençãodereparaçãoda estrutura.Verifica-se que,em certos períodos (e.g.1990-1995), aposição dos pontos de profundidade máxima sofrea influênciada evolução do sector nascenteda Golada, confundindo-secom essa mesma evolução.

A baía da cidade da Praia, na ilha de Santiago, Cabo Verde, acolhe um dos principais portos comerciais e de cruzeiros de Cabo Verde, tendo também uma relação importante, do ponto de vista ambiental e paisagístico, com a frente ribeirinha da cidade. Procedeu-se a uma análise da evolução morfológica do sistema, com recurso a levantamentos hidrográficos (de 1882 a 2016) e imagens aéreas (de 1968 a 2018), para determinar tendências evolutivas históricas. Realizou-se também a aplicação de um sistema de modelos numéricos, envolvendo a simulação da hidrodinâmica, do transporte sedimentar e da evolução do fundo, para identificar os processos litorais dominantes. Calculou-se uma tendência média de avanço da praia Grande ou da Gamboa de 0,6 a 1,0 m/ano, correspondendo este último valor à média dos últimos 50 anos. As condições de maior hidrodinamismo no interior da baía estão associadas aos períodos de agitação marítima de maior altura. As intervenções mais recentes, nomeadamente na zona portuária, tendem a diminuir esse hidrodinamismo.

This study investigates the roleof infragravity waves (IGW) on the hydrodynamics of a wave-dominated inlet (Albufeira lagoon, Portugal) during storm Leslie (October 2018) through the analysis of field observations. Measured significant IGW heights reached up to 0.9 m in the surf zone, at adjacent ocean beaches, and 0.4 m at the flood-delta. At the flood-delta, IGW were more energetic than wind-generated short-waves by up to a factor of 2. These large IGW are expected to contribute significantly to sediment transport during storm conditions.

Existing knowledge on the evolution of the Tagus estuary inlet morphology derives mostlyfrom analyses of bathymetric surveys and aerial images. The limitations of this approachprevent a detailed understanding of the processes responsible for this evolution, as well as theprediction of the future evolution. A 2DH morhodynamic model was therefore applied to shednew light into the sediment dynamics of this tidal inlet. Description of the method

Due to their exposure to various forcings such as tides and waves, estuary mouths constitute dynamic coastal environments where sediment transport can result in abrupt and significant morphological changes. They concentrate relevant socio-economic and ecological values, which will become even more challenging in the climate change context. Despite their importance worldwide, the dynamics of these environments remains partly understood only and fosters many studies from the scientific community. This work is part of a citizen-science project (named Nouveaux Commanditaires Sciences) which aims at studying the hydro-sedimentary dynamics of the Tagus estuary mouth (bordered by a metropolitan area of 2 million inhabitants), especially during storms, by combining in-situ field observations, remote sensing data and numerical modelling. This research program has been set up following the request for a better understanding of the coastal erosion and marine flooding during storms expressed by severely affected inhabitants from an informal neighbourhood, Segundo Torrão (Southern margin of the inlet). In addition, local populations provide support for instrumental deployments and data acquisitions. In return, new knowledge derived from scientific studies will be used to improve coastal management over these threatened areas. In this presentation, we will present the main actions that are being carried out in the scope of this project.

To increase the response capability to coastal flooding events in an accelerating sea level rise context, the ability to predict floods has to improve and be integrated with other components of risk, such as vulnerability and exposure. The MOSAIC.pt project aims at developing an advanced flood risk management framework for coastal zones, integrating robust predictive tools and multi-source data with the different dimensions of the risk, to adequately contribute to emergency planning and response

Estuarine uses and activities may be negatively affected by climatic variability andclimate change

Os observatórios costeiros podem apoiar a gestão diária e a longo prazo dos sistemascosteiros, permitindo antecipar eventos de contaminação, contribuir para a vigilânciacontínua destas zonas e implementar e adaptar planos de gestão. Ao integrarem previsões,análises de cenários e indicadores, para além de observações históricas e em tempo real, estasferramentas têm aumentado de complexidade ao longo dos anos.

Coastal forecast systems have become reliable tools to support daily coastal management and strategic long-term planning. However, the availability of these systems for most coastal areas remains scarce, in particular for small estuaries and coastal bays, which limits the efficiency of management for these areas from both economic and environmental perspectives. Building a new forecast system or maintaining in operation an existing one requires considerable effort, from both financial and personnel points of view. Accurate forecasts require a team comprising experts in coastal processes and computer science and the availability of robust and large computational resources to maintain the predictions available on a daily basis. Herein, a new concept in forecast tools is presented, building the foundations for forecast systems to be built and made available for all coastal systems worldwide. Denoted OPENCoastS, this freely available web service builds forecasts systems for the coastal area selected by the user and maintains them in operation using the European Open Science Cloud infrastructure. Forecasts are based on the SCHISM community model

The influence of freshwater discharge and sea level rise in the salinity in the upper Tagusestuary is assessed using a numerical model. Results suggest that the river flow is the main driver of the salinity in the upper estuary and that salinity intrusion increases with the duration of the droughts. For the analyzed scenarios, salinity reaches concentrations that are inadequate for irrigation during some periods, which can harm the agricultural activities.

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The hydrodynamics and sediment dynamics of a low energy estuarine beach was analysed. The beach is subjected to local wind waves generated in an area of restricted fetch, wake waves generated by catamarans and a semi diurnal meso-tidal regime. Only storm events of short duration modify the beach profile, which, once the normal hydrodynamic conditions are restored, naturally recovers its initial shape. The beach is characterised by having a steep upper slope until mean sea level, followed, seaward, by a low gradient terrace, and a bimodal sediment distribution. To investigate the hydro sedimentologic beach behaviour, wind, topo-hydrographic and sedimentologic data were used as input and validation of process based numerical modelling. Wave generation and transformation in an area of restricted fetch, nearshore circulation, sediment transport and morphological evolution were simulated for average annual conditions and storm conditions. The statistical analysis of a six-year wind data series allowed to derive the average wind regime, based on which, the average annual wave regime and the average annual longshore sediment budget, 14.5x103 m3year-1, were calculated. The contribution of the individual components of the representative wave regime, discretised by directional sector and height class of incidence, was evaluated as well as the spatial distribution of the longshore sediment transport in the active part of the beach profile. The characteristics of the wave groups generated by the passage of catamarans at different speeds were estimated and their action on the beach morphology was simulated. Although when the catamarans travel at 20 knots speed the average annual wake wave energy dissipated at the beach is 2.5 times higher than the average annual wind wave energy, the erosion effect on the beach profile is still not relevant for the present traffic. In opposition, short duration storm events generate the formation of an erosion scarp at the upper part of the beach face. The numerical modelling of this phenomenon allowed to acknowledge on the protective effect that the low gradient terrace has on the beach face.

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A two-dimensional model of wind-driven circulation in a closed basin was developed using a finite-volume technique for generalized curvilinear grids and applying some of the recent developments in hydrodynamic modeling. The terms in the Navier¬Stokes equations were treated separately according to the fractional step method and the propagation step, including the continuity equation and the pressure and stress terms in the momentum equations, was solved using a conjugate gradient method.The model was then applied to a number of test cases to examine the feasibility of the approach used by comparing with results obtained with the two-dimensional version of the three-dimensional model CH3D. These included a square basin with constant slope and with a V-shaped bottom and Lake Okeechobee, in South Florida. To evaluate the long-term numerical stability of the model, a ten-day model simu¬lation with varying wind was also run for Lake Okeechobee.

An examination of the causes for generation and dynamics of turbidity maxima in estuaries reveals the critical role of sediment tidal pumping phenomenon and, to a lesser extent, of the well-known effect of residual gravitational circulation due to salt water penetration. Both phenomena depend on the vertical sediment concentration profile and, consequently, on the magnitude of the vertical mass transport fluxes. Where high concentration suspensions occur regularly, the erosion/deposition fluxes can be drastically modified by sediment stratification, consequently influencing sus¬pended sediment response to currents and wave action. This influence is inherent in flow-sediment hysteresis, which therefore reflects the role of vertical mass transport in the estuarine and coastal suspended fine sediment regime.A vertical transport numerical model was used to investigate the influence of several key parameters describing sediment settling, bed properties and stabilized diffusion on the concentration profile. The model was also applied to simulate the influence of the same parameters on the time-lagged sediment response to flow variations, reflected in the characteristics of flow-sediment hysteresis loops.Field data obtained in Hangzhou Bay (People´s Republic of China), a high concentration environment, showed typical features of flow-sediment hysteresis and confirmed the importance of the vertical mass fluxes in contributing to sediment transport in the bay. A qualitative simulation provided by the numerical model, using settling parameters corresponding to local sediment, while confirming the importance of the hysteresis phenomenon, also revealed the critical need to use algorithms describing adequately stabilized diffusion and bed fluxes.Additional evidence of hysteresis was obtained through analysis of microscale variables, such as the Reynolds stresses and the variances of the velocity components resulting from combined effects of wave action and turbulence. Spectral analysis of the measured random variations did not support the commonly accepted hypothesis of similarity between the responses to turbulent flow of sediment concentration and temperature. The normalized turbulent intensities for all the measured veloc¬ity components showed their highest values during the period of lowest sediment concentration; this result is consistent with the hypothesis of turbulent intensity damping by suspended sediment.

A contribution to increased use and development of the method of artificial nourishment of beaches is the main objective of this thesis. It also aims at being a guide for the design of the reconsctrution or outright creation of beaches.In the first chapter, the subject is introduced and the objectives of the thesis are outlined. The second chapter cover´s some general considerations on littoral dynamics, beach equilibrium and sediment budget.Since the theoretical foundations of the artificial nourishment of beaches still leave much to be desired, a description of this method and its principles are presented in the third chapter. Important technical and economic aspects are included, such as: sampling, analysis and characterization of native and borrow material; selection Of nourishment sources; extraction and transport of borrow sand; and characteristics of the theoretical final beach profile. Such problems as the integration of ecological aspects, and cost-benefit analysis are also treated.in the fourth chapter, the Portuguese experience in this domain is described. The experience of such countries as France and the U.S., where this method has frequently been used is also mentioned. In this way a considerable amount of hitherto scattered information is presented in a unified way.The fifth chapter, treats briefly the problem of coastal model similitude, both as a support for the experimental work: which was carried out and to help engineers in the design of a coastal physical model or in merely assessing the possibilities of such a model.In the sixth chapter, an experimental study is described of the influence of both the grain size of borrow material and the nourishment method on the redistribution of sediments after the nourishment. The obtained results are compared with existing theoretical methods and some prototype results.Finally, in the last chapter the main conclusions and recomendations of the thesis are presented.

We provide a systematic analysis of the consistency, stability, convergence and accuracy of the numerical solution of the transport equation by a general Eulerian-Lagrangian Method (ELM). The method involvesthree basic steps: the backwards tracking of characteristic lines following the flow, the interpolation of concentrations at the feet of these lines, and the solution of dispersion taking such concentrations as initial conditions. The first two steps constitute the Backwards Method of Characteristics (BMC); the third step involves a time-discretization along the characteristic lines, and a spatial discretization of the dispersion operator, both based on conventional techniques (e.g.. Euler or Crank¬Nicholson for time; finite-elements or finite-differences for space).The choice of the spatial interpolator is shown to impact the consistency, stability and convergence, as well as the accuracy of the BMC. Most interpolators ensure consistency, but only a few ensure stability, hence convergence; stability criteria are derived from a newly developed generalized Fourier analysis, which can account for non-linearities introduced by quadratic grids. The comparison of formally derived propagation and truncation errors, complemented by numerical experimentation, provides a reference for the choice of the interpolator, given a specific transport problem characterized by prevailing concentration gradients.The BMC potentiates the use of large time-steps, well above Courant number of order one. In the limiting case of pure advection, optimal accuracy would be obtained for a At close to the total time of interest; the presence of dispersion constrains, however, the size of At, especially in the case of non-uniform flows. The comparison of the truncation errors for the three basic steps of ELM provides a reference to select At. as a function of Ax, of the spatial interpolators and time-discretization schemes, and of the gradients of flow and concentrations.

Apresenta-se um estudo sistemático, baseado simultaneamente em análises formais e em experimentação numérica, da precisão e estabilidade da solução da equação de transporte por um método Euleriana-Lagrangeana (MEL). 0 método decompõe a equação de transporte em equações separadas de advecção e de difusão, resolvendo a primeira pelo método das características regressivas no tempo (MCR), e a segundo por um método de elementos finitos, do tipo Galerkin.Mostra-se que as interpolações no espaço requeridas pelo MCR são um factor limitativo da precisão global do MEL, e comparam-se diversas técnicas alternativas de interpolação, algumas das quais originais. A combinação de esquemas compactos e não-compactos de interpolação, baseados em polinómios de Lagrange, é apontada como uma potencial solução óptima.Demonstra-se que, para uma adequada escolha do esquema de interpolação, o MCR é consistente, estável e convergente e tem boas características de precisão. Escolhas inadequadas do esquema de interpolação podem, no entanto, causar instabilidade e inconsistência.Estabelece-se a dependência da precisão do MCR no passo de cálculo, mostrando-se que essa precisão aumenta, em geral (para um tempo total fixo), quando se reduz o número de passos de cálculo, isto é, quando se aumenta o passo de cálculo (uma propriedade simultaneamente pouco habitual e muito conveniente). No entanto, na gama de valores muito pequenos do Número de Courant, a precisão é praticamente independente do passo de cálculo, o que evita que o método se torne divergente.Analisa-se brevemente o efeito de malhas irregulares e pluridimensionais sobre a aplicabilidade e precisão do MCR. Apesar de sensível à irregularidade da malha, o método mantém boas características de precisão desde que as distorções geométricas não sejam excessivas. Malhas pluri-dimensionais, quando simultaneamente irregulares, levantam problemas específicos de aplicabilidade de esquemas não-compactos de interpolação; esses problemas poderão se resolvidos através da utilização conjugada de esquemas compactos e não-compactos, para um mesmo problema. Investigação adicional a ainda requerida nestas áreas.Mostra-se ainda que a presença de mecanismos físicos de difusão beneficia tanta a precisão da solução da equação de advecção como a precisão global do MEL.Finalmente, demonstra-se a eficácia da aplicação do MEL (numa forma particular, restricta a esquemas de interpolação compactos) à simulação do transporte de poluentes em águas costeiras. 0 método permite, em particular, realizar simulações longas (várias marés) a custos moderados, mesma para malhas irregulares com elevado número de nós, e, também, evitar a necessidade do uso de difusividades artificiais como garante de estabilidade; em geral, e nestes dois aspectos em particular, o método revela-se superior a métodos Eulerianos, mais convencionais (por exemplo, métodos de elementos finitos, do tipo Galrkin ou Petrov-Galerkin, aplicados à equação de transporte indivisa).

No âmbito de um projecto de investigação intitulado caracterizarão de Formações Estuárias e do Meio Marítimo Litoral Por Técnicas de Detecção Remota , insere-se o estu¬do da avaliação de suspensões na água do mar com recurso à aplicação de técnicas de detecção remota como meio de caracterização de parâmetros de qualidade de água. 0 trabalho que se apresenta foi desenvolvido com os seguintes objectivos:1- Desenvolvimento de metodologias de análise e quantificação de suspensões na água do mar a partir de dados de satélite;2- Implementação sob forma de programas para computador de modelos de quantificação de suspensões orgânicas e inorgânicas na água do mar, a aplicar aos dados do radiómetro CZCS do satélite NIMBUS-7;3- Aplicação e verificação do modelo ao litoral português.Corn vista à consecussão destes objectivos, começa-se por apresentar uma extensa análise das formas de abordagem do problema, enumerando-se os diversos métodos actualmente seguidos, nomeadamente o Empírico, o Teórico e, neste, os métodos microscópico e macroscópico.Com base nesta análise, propõe-se um modelo numérico de quantificção das suspensões a partir dos dados do radiómetro CZCS do satélite NIMBUS-7, o qual, embora empírico na sua formulação como a maioria dos actualmente existentes, se apoia no comportamento radiomótrico da água do mar e dos seus constituintes típicos, cujas características se apresentam também.A implementação prática do modelo exige a filtragem da componente atmosférica presente nos dados rediométricos. Com este objectivo, apresenta-se um modelo baseado na metodo¬logia correntemente utilizada. Consiste na consideração da difusão simples, causada pelos constituintes básicos da atmosfera (moléculas gasosas, aerosóis e ozono) na gama de comprimentos de onda considerada (440, 520, 550 e 670nm). Esta metodologia conduz, porém, a um sistema de equações indeterminado e que é resolvido pela consideração de uma equação ´exterior´ ao processo de transferência radiativa que tem lugar na atmasfera. Para este efeito, utilizou-se a equação de Smith e Wilson por ser a mais divulgada, ressalvando-se, contudo, a necessidade de verificação da sua efectiva aplicação às águas do litoral português, o oqe não foi possível concretizar dada a inexistência de dados.

A numerical method for the solution of the two-dimensional, unsteady, transport equation is formulated, and its accuracy is tested.The method uses a Eulerian-Lagrangian approach, in which the transport equation is divided into a diffusion equation (solved by a finite element method) and a convection equation (solved by the method of characteristics). This approach leads to results that are free of spurious oscillations and excessive numerical damping, even in the case where advection strongly dominates diffusion. For pure diffusion problems, optimal accuracy is approached as the time-step, At, goes to zero; conversely, for pure-convection problems, accuracy improves with increasing At; for convection-diffusion problems the At leading to optimal accuracy depends on the characteristics of the spatial discretization and on the relative importance of convection and diffusion.The method is cost-effective in modeling pollutant transport in coastal waters, as demonstrated by an illustrative application to a case study (sludge dumping in Massachusetts Bay). Numerical diffusion is eliminated or greatly reduced, raising the need for realistic estimation of dispersion coefficients. Costs (based on CPU time) should not exceed those of conventional Eulerian methods and, in some cases (e.g., problems involving predictions over several tidal cycles), considerable savings may even be achieved.

0 principal objectivo desta tese é mostrar a habilidade do métodos dos elementos finitos e do método dos elementos de fronteira na modelação das ondas gravíticas de superfície.Na análise linear, as equações reduzidas do movimento ondulatório são resolvidas pelo método dos elementos finitos, levando em linha de conta a reflexão parcial, a radiação e o atrito de fundo. A modelação da radiação é feita com elementos infinitos ou com elementos de radiação, sendo estes encarados como um caso particular dos elementos de reflexão. 0 atrito de fundo é introduzido após a discretização e é levado a efeito através de um coeficiente empírico que se pode relacionar com o coeficiente de Chezy.Quer o modelo determinístico quer o modelo estocástico, obtido do primeiro com base na noção do estimador natural, permitem a realização de ensaios de ressonância ou de agitação. Nos ensaios de ressonância de uma bacia portuária tem particular interesse o conhecimento do espectro de resposta ao ruído bran¬co de banda limitada.A título de exemplo de aplicação mostram-se alguns aspectos do estudo das condições de ressonância do porto de Leixões.Logo que não sejam válidas as hipóteses simplificativas que permitem uma abordagem linear das ondas gravíticas de superfície, são resolvidas as equações não lineares do movimento ondulatório com superfície livre, pelo método dos elementos de fronteira. As suas peculiaridades permitem, sem qualquer dúvida, considerar o método dos elementos de fronteira como o mais vocacionado para este tipo de análise.Os programas de cálculo automático para o estudo da análise não linear estão ainda em fase de teste, pelo que não se apresentam exemplos de aplicação.

Este trabalho é dirigido principalmente para o uso da simulação numérica dos escoamentos em estuários como uma ferramenta pa¬ra estudos de engenharia. Fazem-se algumas considerações gerais sobre a definição e classificação de estuários seguidas por uma descrição sumária da hidrodinâmica dos estuários. Descrevem-se e comparam-se as várias técnicas utilizadas na solução de problemas de hidradinâmica de stuários.Faz-se a dedução formal de um modelo matemático do escoamento em estuários homogéneos e integram-se verticalmente as suas e¬quações, obtendo-se as conhecidas equações do escoamento em águas pouco profundas ("shallow water equations"). Estas equações são discretizadas utilizando os métodos dos elementos finitos e das diferenças finitas, respectivamente no espaço e no tempo.Faz-se a descrição de um sistema de cálculo para determinar a solução do modelo matemático discretizado e tecem-se várias considerações acerca do seu desenvolvimento e uso.Finalmente apresentam-se os resultados de alguns testes utilizados para verificar o funcionamento do sistema de cálculo.

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