Publications

The economical and ecological importance of tidal inlets has fostered the development of empirical tools forinlet management during the last century. This study aims at confronting these empirical theories with resultsobtained with a process-based numerical model, MORSYS2D. This 2DH morphodynamic modeling system isapplied to an idealized tidal inlet/lagoon system with different combinations of significant wave height,tidal range and tidal prism. The numerical model predictions are compared to the empirical models ofHayes, Bruun, O'Brien and FitzGerald and to morphologies observed at natural tidal inlets. The results presentgood accordance with observations as well as with some key behaviors predicted with the empirical theories.The predicted morphologies satisfy the relation of O'Brien between the tidal prism and the cross-sectionalarea, the model reproduces the conceptual model of sand by-passing by ebb-tidal delta breaching of FitzGeraldand the classifications of Hayes and Bruun are generally respected. However, some inconsistencies betweenmodel results andHayes classification highlight the limitations of applying this classification,which only considersthe yearly-averaged significant wave height and tidal range, to a single tidal inlet case.

The patterns of water quality and ecological dynamics in estuaries vary at different time scales. Characterizing thisvariability at short (diurnal and seasonal) time scales is a pre-requisite to investigate the effects of long-term climatechanges. Seasonal scales depend mostly on the natural climatic variability, while diurnal scales are generallyassociated with the tidal cycle and wind events. In order to study the role of these superimposed scales in the Aveirolagoon, an integrated approach, which combines data acquisition and numerical modelling, was followed. Theanalysis was conducted with the three-dimensional, fully coupled hydrodynamic and ecological numerical model,ECO-SELFE, extended to account for the oxygen cycle. For data acquisition three field campaigns were performedalong a salinity gradient (Mira channel), covering different seasonal situations: March 2009, September 2009 andJanuary 2010. These campaigns, each lasting 24 hours, included the measurement of physical, chemical andbiological parameters (river flow, water levels, currents velocities, water temperature, salinity, dissolved oxygen,nutrients and chlorophyll a). Results show the ability of the model to represent both the spatial and temporal patternsobserved for the different variables. Chlorophyll a and nutrients tend to present larger concentrations upstream thandownstream, with the largest values observed in spring. The distribution patterns are significantly influenced by thetidal dynamics.

ABSTRACT: This study investigates sediment transport at a very low-energy backbarrier beach in southern Portugal, from aspring-to-neap tide period, during fair-weather conditions. Rates and directions of transport were determined based on the applicationof fluorescent tracer techniques. Wind and currents were collected locally, whereas the dominant small and short-period wind waveswere characterized using a morphodynamic modelling system coupling a circulation model, a spectral wave model, and a bottomevolution model, well validated over the study area.For the recorded conditions sediment transport was small and ebb oriented, with daily transport rates below 0.02m3 day-1. Tidalcurrents (mainly ebb velocities) were found to be the main causative forcing controlling sediment displacements. Transport rateswere higher during spring tides, tending towards very small values at neap tides. Results herein reported points towards thedistinction between tracer advection and tracer dispersion in this type of environment. Transport by advection was low as a consequenceof the prevailing hydrodynamic conditions (Hs<0.1 m, and max. current velocity of 0.5ms-1) and the tracer adjustment tothe transport layer, whereas dispersion was relatively high (few metres per day). Tracer techniques allowed distinguishing the broadpicture of transport, but revealed the need for refinement in this type of environments (bi-directional forcing by ebb and flood cycles).

Tidal asymmetry is a key factor in the geomorphology of many coastal lagoons. Whilst tidal asymmetry insemi-diurnal tidal regimes has been extensively studied, its occurrence in mixed tidal regimes is less wellunderstood, and has a number of unique causes and characteristics. Tidal phase duration analysis andleast squares harmonic analysis have been used to study the temporal and spatial nature of asymmetryin both offshore and coastal lagoon tides at the Murray Mouth in south-eastern Australia. GIS analysis ofbathymetric survey data has been used to study the roles of lagoon bathymetry on tidal distortion.Tidal asymmetry in the Murray Mouth coastal lagoon results from frequency relationships between majorastronomical driving tides as well as bathymetry-induced tidal transformations. The frequency differencebetween the K1 and S2 tides produces an important 6-monthly ebb/flood-dominant asymmetry cycle in theincoming oceanic tides, whilst phase relationships between the K1 and K2 tides contribute to persistentasymmetry. The use of standard relationships between theM2 andM4 tides as asymmetry indicators is shown tobe invalid for this mixed tidal regime. In this shallow, microtidal system, the impacts of bathymetry on tidalasymmetry within the lagoon are profound, with inlet and channel configurations identified as the mostimportant controlling factors.The results of this study may inform the strategic management of tidal inlets in mixed tidal regimes, suchas the Murray Mouth, that are subject to intermittent or long-term constriction.

The generation of unstructured triangular grids for coastal applications requires extensive manual tunning, in order to improve the stability, accuracy and efficiency of the model simulations. As current grids reach 105-106 nodes, this manual tunning is no longer feasible. This paper presents a new post-processor that aims at optimizing the local distribution of nodes and elements by adding and deleting nodes and by changing the connections between nodes. The algorithms target a smooth transition between elements sizes by making the number of connections of each internal node as close to 6 as possible.An application of the post-processor to various grids shows that the grid roughness is reduced by 10-30%, while decreasing marginally the number of nodes and maintaining the global nodal distribution. To illustrate the benefits of the post-processor, a grid with 130.000 nodes of the maritime zone under Portuguese jurisdiction is used to simulate tides with a good accuracy (root mean square errors below 10 cm) and without numericaloscillations. The model results provide an extensive database of tidal elevations that can feed boundary conditions to local estuarine models.

The Portuguese southern coast national and international appeal turns it into a highly relevant area concerning the country's economy. Vale do Lobo is a resort located in the centre of this coast, where the beach has been suffering from erosion problems for several years now. Local interventions have been executed in order to prevent the erosion of the beach, yet the efficiency of such strategy has not been verified.This study's main purpose is to enable a better understanding of the evolutionary trends of the coastline between Quarteira and Ancão inlet and to analyse the efficiency of the beach nourishment interventions as a protection technique for a coastal environment with the wave energy exposure and geomorphologic characteristics of the present one.The study focussed on a 10-year period, during which, 700x103 and 370x103 m3 of sand with D50=0.76 mm (slightly larger than the D50 of the native beach) were extracted from offshore, and deposited in the foreshore. The wave climate in the study area was processed, analysed and used as input for modelling the nearshore processes. In addition, beach profiles measured alongshore have been analysed to obtain the representative beach profile. Coastlines of different dates have been derived from aerial photographs.The longshore transport model Litdrift and the coastline model Litline were applied to study the sediment dynamics. Litdrift was used to estimate the longshore transport capacity of the beach for the hydrodynamic series considering the representative beach profile previously obtained. These results were then used to calculate the evolution of the coastline due to the incident wave climate with Litline model. Aerial photographs were used to verify the model. The analysis of the two beach nourishments performed allowed to infer on the efficiency of this coastal protection technique to delay the erosion process.

This article addresses the problem of sedimentation at the entrance of a harbour by evaluating and understanding thesediment dynamics in the adjacent beaches. The results of the methodology applied to acknowledge the beaches

In the past decade, two-dimensional (2DH) morphodynamic modeling systems were developed to simulate the morphological changes due to the combined action of waves, tides and winds. In spite of some significantsuccesses, they are unable to predict complex morphological behaviors, such as inlet infillings in highly dynamicenvironments during winter energetic wave conditions. For these reasons, the state of the art in morphodynamic modeling tends to evolve towards three-dimensional (3D) modeling systems. In this context, the present study focuses on the development of an unstructured 3D coupled wave-current model taking into account the verticalstructure of radiation stresses based on recent theoretical developments. The model is applied to the common breakwater test case and succeeds efficiently to reproduce the wave-driven circulation behind the breakwater.The study investigates also the vertical structure of the flow and the wave-induced setup along a cross-shore profile. The undertow is well-predicted by the model in the surf zone and the wave-induced setups predicted by the 3D model are about 30-40% greater than those simulated by a 2DH approach. This important aspect could berelevant in storm surge and inundation studies for which the quality of the predictions could be strongly improved by using a 3D fully coupled wave-current model.

A extracção de sedimentos nas zonas costeiras é uma actividade de importância e prática crescentes há já vários anos e em diversos países. A evolução de uma zona de extracção depende essencialmente do tipo de sedimento e das características hidrodinâmicas do local. No presente trabalho pretendeu-se estudar a evolução morfológica e sedimentar de uma escavação ao largo de Vale do Lobo (Algarve, Portugal) com base em 3 campanhas batimétricas realizadas entre 2006 e 2008, antes e após a extracção, assim como numericamente, através da aplicação de um modelo morfodinâmico

Presently, conceptual and numerical sediment concentration modelling is largely based on the diffusionapproach. However, several works have shown that this approach is, frequently, unsuitable to model the profiledistribution of different grain size particles in the same flow. In order to understand the grain size effects onsuspended sediment concentration profiles under breaking wave conditions, a new data set was collected at threewest coast Portuguese beaches (Almagreira, Caparica and Comporta beaches). As expected, the relativeconcentration for the entire grain-size spectrum shows a systematic decrease with distance to the bed, thoughwith different profile shapes (exponential and power law). The analysis of the relative concentration profiles foreach sediment fraction show similar profile patterns; however, as expected, the decrease in the relativeconcentration with height strongly depends on particle size and assumes a different behaviour at each site. In thiscase, diffusion approach was unable to explain all the observed profile patterns, namely the overlapping ofcoarse sediment profiles and the different profile shapes in the same flow. These deficiencies were overcome bythe use of the convection-diffusion approach.

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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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