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Numerical modelling methodologies are important by their application to engineering and scientific problems, because there are processes where analytical mathematical expressions cannot be obtained to model them. When the only available information is a set of experimental values for the variables that determine the state of the system, the modelling problem is equivalent to determining the hyper-surface that best fits the data. This paper presents a methodology based on the Galerkin formulation of the finite elements method to obtain representations of relationships that are defined a priori, between a set of variables: y = z(x 1 , x 2 ,...., x d ). These representations are generated from the values of the variables in the experimental data. The approximation, piecewise, is an element of a Sobolev space and has derivatives defined in a general sense into this space. The using of this approach results in the need of inverting a linear system with a structure that allows a fast solver algorithm. The algorithm can be used in a variety of fields, being a multidisciplinary tool. The validity of the methodology is studied considering two real applications: a problem in hydrodynamics and a problem of engineering related to fluids, heat and transport in an energy generation plant. Also a test of the predictive capacity of the methodology is performed using a cross-validation method.

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When studying and modelling Complex Systems where there can be a considerable number of variables and relations, it is very important to obtain methodologies that help us to model natural phenomena, these being phenomena present in the physical world and real life.Hypothetically, if we consider a relation defined by a set of variables, ( ), ,......, n y f x x x = , symbolic and numerical methodologies can be obtained. This article presents a model of n-dimensional finite elements that provides the basis for defining a numerical methodology for studying and modelling complex systems. The use of n-dimensional elements then allows us to represent the relation using the values of the same for a finite number of points, this being carried out by resolving an optimization problem.To obtain the geometric model it was necessary to use the correct data structure design and programming to allow effective management of the acquisition and storage of the elements, the nodes considered for each, as well as the functions and procedures needed to approach the problem of optimisation.Finally, we applied the methodology to determine the geometric model and the problem of optimization to study and model an environmental problem

Monitoring of the quality of bathing water in line with the European Commission bathing water directive (Directive 2006/7/EC) is a significant economic expense for those countries with great lengths of coastline. In this study a numerical model based on finite elements is generated whose objective is partially substituting the microbiological analysis of the quality of coastal bathing waters. According to a study of the concentration of Escherichia coli in 299 Spanish Mediterranean beaches, it was established that the most important variables that influence the concentration are: monthly sunshine hours, mean monthly precipitation, number of goat cattle heads, population density, presence of Posidonia oceanica, UV, urbanization level, type of sediment, wastewater treatment ratio, salinity, distance to the nearest discharge, and wave height perpendicular to the coast. Using these variables, a model with an absolute error of 10.6±1.5CFU/100ml is achieved. With this model, if there are no significant changes in the beach environment and the variables remain more or less stable, the concentration of E. coli in bathing water can be determined, performing only specific microbiological analyses to verify the water quality.

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System modeling is a main task in several research fields. The development of numerical models is of crucial importance at the present because of its wide use in the applications of the generically named machine learning technology, including different kinds of neural networks, random field models, and kernel-based methodologies. However, some problems involving the reliability of their predictions are common to their use in the real world. Octahedric regression is a kernel averaged methodology developed by the authors that tries to simplify the entire process from raw data acquisition to model generation. A discussion about the treatment and prevention of overfitting is presented and, as a result, models are obtained that allow for the measurement of this effect. In this paper, this methodology is applied to the problem of estimating the energetic needs of different buildings according to their principal characteristics, a problem that has importance in architecture and civil and environmental engineering due to increasing concerns about energetic efficiency and ecological footprint.

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