Gaspar J. Machado

Irrigation planning: an optimal control approach

We aim to plan the water usage in the irrigation of a given farmland keeping, at same time, the field cultivation in a good state of preservation. This problem is modeled and tackled as an optimal control problem: minimize the water flow (control) so that the extent water amount in the soil (trajectory) fulfills the cultivation water requirements. To estimate rainfall, we consider two models: one based on the average monthly rainfall of the last 10 years in Lisbon area and another which considers the best linear combination of average monthly rainfall from the last 10 years and the amount of rainfall in the previous month. We study the behavior of the solutions under different weather scenarios and we compare the solutions obtained using our model of rainfall with solutions obtained having a prior knowledge of the rainfall.

Finite Volume Scheme Based on Cell-Vertex Reconstructions for Anisotropic Diffusion Problems with Discontinuous Coefficients

We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusion problems based on cell to vertex reconstructions involving minimization of functionals to provide the coefficients of the cell to vertex mapping. The method handles complex situations such as large preconditioning number diffusion matrices and very distorted meshes. Numerical examples are provided to show the effectiveness of the method.

A very high-order finite volume method for the time-dependent convection–diffusion problem with Butcher Tableau extension

The time discretization of a very high-order nite volume method may give rise to new numerical diculties resulting into accuracy degradations. Indeed, for the simple onedimensional unstationary convection-diusi on equation for instance, a conicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specic diculty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new nite volume method is designed based on specic polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.

New cell–vertex reconstruction for finite volume scheme: Application to the convection–diffusion–reaction equation

Abstract The design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink’s approach based on a cell-to-vertex interpolation to compute vertex values with neighbour cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two- and three-dimensional geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method.

Numerical simulation of breast reduction with a new knitting condition

Breast reduction is one of the most common procedures in breast surgery. The aim of this work is to develop a computational model allowing one to forecast the final breast geometry according to the incision marking parameters. This model can be used in surgery simulators that provide preoperative planning and training, allowing the study of the origin of the errors in breast reduction. From the mathematical point of view, this is a problem of calculus of variations with unusual boundary conditions, known as knitting conditions. The breast tissue is considered as a hyperelastic material, discretized with three-dimensional finite elements for the body, whereas the skin is modelled with two-dimensional finite elements on the curved surface. Although the model is of low precision, we show that it is sufficient for a satisfactory prediction of breast reduction surgery results, allowing an analysis of errors frequently performed during the surgery and giving an understanding of how to avoid or correct them. Copyright

A Sixth-Order Finite Volumemethod for the 1D Biharmonic Operator

Abstract A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.

Irrigation planning: replanning and numerical solution

A model to optimize the water use in the irrigation of a farm field via optimal control (water flow) that take into account the evapotranspiration, rainfall, and losses by filtration and runoff was developed in [1]. Here we improve the previous model to take into account real data of rainfall. Model predictive control is applied to replan. We test and compare different nonlinear constrained optimization techniques for solving the nonlinear constrained optimization problem that arises from the discretization of the proposed optimal control problem. Furthermore, we test different time discretization steps.

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.

Analysis of effective efficiency in decision making for irrigation interventions

Multiple stresses are putting great pressure on water resources systems. Population growth, climate change, prosperity, energy production, food crisis, and water governance are among the factors straining water resources. Decision makers from rich to poor countries and from commercial to non-governmental organisations are struggling to devise schemes to adapt to these stressed water conditions. Better efficiency for water resources systems, and particularly irrigation systems, is recommended as one of the most important responses to climate change, unsustainable development, and water shortage. However, using certain efficiencies such as Classical Efficiency caused systems not to perform according to decision makers’ objectives. Effective Efficiency is a robust composite indicator that includes in its formulation both a flow weight, taking into account the leaching fraction, and reuse of return flows. Classical Efficiency is defined as the percentage of the diversion consumed beneficially, such as by crop evapotranspiration. Effective Efficiency, on the other hand, is defined as the ratio of beneficial consumptive use to total consumption, expressed as a percentage. In this paper, a normalised and non-dimensional form of Effective Efficiency is developed and necessary constraints for its successful application are explained. These constraints express water balance, flow weights and their thresholds, water reuse, and total consumptive use. Basic guidelines are proposed for better decision making in determining possible interventions for improving Effective Efficiency. This is done by analysing its domain through analytical and graphical methods. Three real cases are considered, namely, Imperial Irrigation District and Grand Valley irrigation systems in the United States and Nile Valley in upper Egypt. Three-dimensional sensitivity analysis is performed on Effective Efficiency and its variables using the three cases. This leads to an examination of the validity of the analysis and to suggestions for better intervention options. Meanwhile, it is also shown why Classical Efficiency should be used with care.

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier---Stokes and Euler Equations on Unstructured Meshes

We propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier–Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order convergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme.