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The definition of geostatistical domains is a stage in the estimation of mineral resources, in which a sample resulting from a mining exploration process is divided into zones that show homogeneity or minimal variation in the main element of interest or mineral grade, having geological and spatial meaning. Its importance lies in the fact that the quality of the estimation techniques, and therefore, the correct quantification of the mineral resource, will improve in geostatistically stationary areas. The present study seeks to define geostatistical domains of estimation for a mineral grade, using a non-traditional approach based on the k-prototype clustering algorithm. This algorithm is based on the k-means paradigm of unsupervised machine learning, but it is exempt from the one-time restriction on numeric data. The latter is especially convenient, as it allows the incorporation of categorical variables such as geological attributes in the grouping. The case study corresponds to a hydrothermal gold deposit of high sulfidation, located in the southern zone of Peru, where estimation domains are defined from a historical record of data recovered from 131 diamond drill holes and 37 trenches. The characteristics directly involved were the gold grade (Au), silver grade (Ag), type of hydrothermal alteration, and type of mineralization. The results obtained showed that clustering with k-prototypes is an efficient approach and can be used as an alternative or complement to the traditional methodology.

143 p., En este trabajo, desarrollamos algoritmos automáticos orientados a un objetivo hp diseñados para problemas no elípticos. Estos algoritmos se destacan en términos de robustez y simplicidad en su implementación, atributos que los hacen especialmente adecuados para aplicaciones industriales. Una ventaja clave de nuestras metodologías es que no requieren calcular soluciones de referencia en mallas globalmente refinadas. Sin embargo, nuestro enfoque se limita a refinamientos anisotrópicos p eisotrópicos h.Los resultados numéricos 1D muestran la convergencia de nuestros algoritmos orientados a un objetivo,tanto h como p, usando las ecuaciones de Helmholtz y convección-difusión. Además, los resultados numéricos en 2D muestran la convergencia de los algoritmos hp usando las ecuaciones de Poisson,Helmholtz y convección-difusión. También, probamos estos algoritmos hp en casos 3D con la ecuación de Helmholtz para demostrar la versatilidad de nuestros algoritmos.Finalmente, extendemos nuestros algoritmos orientados a un objetivo hp para generar grandes bases dedatos confiables e ideales para entrenar redes neuronales. Como resultado, mostramos la generación eficiente de grandes bases de datos potencialmente con cientos de miles de datos sintéticos.

139 p., El subsuelo terrestre está formado por diferentes materiales, principalmente por rocas porosas que posiblemente contienen minerales y están rellenas de agua salada y/o hidrocarburos. Por lo general, las formaciones que crean estos materiales son irregulares y con materiales de diferentes propiedades mezclados en el mismo estrato.Uno de los principales objetivos en geofísica es determinar las propiedades petrofísicas del subsuelo de la Tierra. De este modo, las compañías pueden determinar la localización de las reservas de hidrocarburos para maximizar su producción o descubrir localizaciones óptimas para el almacenamiento de hidrógeno o el depósito de CO$_2$. Para este propósito, las compañías registran mediciones electromagnéticas utilizando herramientas de Medición Durante Perforación (LWD por sus siglas en inglés -- Logging While Drilling), las cuales son capaces de recabar datos mientras se lleva a cabo el proceso de prospección. Los datos obtenidos se procesan para producir un mapa del subsuelo de la Tierra. Basándose en el mapa generado, el operador ajusta en tiempo real la trayectoria de la herramienta de prospección para seguir explorando objetivos de explotación, incluidos los yacimientos de petróleo y gas, y maximizar la posterior productividad de las reservas disponibles. Esta técnica de ajuste en tiempo real se denomina geo-navegación.Hoy en día, la geo-navegación desempeña un papel esencial en geofísica. Sin embargo, requiere la resolución de problemas inversos en tiempo real. Esto supone un reto, ya que los problemas inversos suelen estar mal planteados.Existen múltiples métodos tradicionales para resolver los problemas inversos, principalmente, los métodos basados en el gradiente o en la estadística. Sin embargo, estos métodos tienen graves limitaciones. En particular, a menudo necesitan calcular el problema inverso cientos de veces para cada conjunto de mediciones, lo que es computacionalmente caro en problemas tridimensionales (3D).Para superar estas

Euskera: Leioako UPV/EHU-ko Campus-era joateko bidegorri baten edo sare baten proposamena egitea da lan honen helburua. Proposamena jasangarritasunaren ikuspuntutik emango egingo da. Hau lortzeko jada eraikita dauden errepideen trazatuak erabiliko dira eta ahalik eta diru gutxienarekin eraiki daitekeen bidegorri bat edo sare bat proposatuko da. Bidegorriaren trazatu egokiena aukeratzeko Geoestatikaren teknika erabiliko dahainbat informazio iturri erabili dira. Datu hauenHonen bidez, Leioako UPV/EHU-ko Campus inguruko garraio azpiegituren eta mugikortasunaren azterketa egingo da. Lan honen bidez, Unibertsitateko Campus-eko erabiltzaileen mugikortasun-beharrak asetzea lortu nahi da, Campusa-ren iraunkortasunari laguntzeko., Castellano: Este trabajo tiene como objetivo proponer un vía o red de vías bici al Campus de la UPV / EHU en Leioa. La propuesta se presentará desde el punto de vista de la sostenibilidad. Para lograrlo se utilizarán los trazados de carreteras ya construidas y se propondrá una vía o red de vías que se pueda construir con la menor cantidad de dinero posible. Para seleccionar la ruta más adecuada para el carril bici, se han utilizado diferentes tipos de datos. Por medio de estos datos, se ha realizado el análisis de la infraestructura de transporte y de la movilidad alrededor del Campus UPV / EHU en Leioa. Por medio de este trabajo se pretende satisfacer las necesidades de movilidad del Campus Universitario y contribuir así a la sostenibilidad del mismo., English: The objective of this work is to propose a cycle path or network of paths to the UPV/EHU Campus in Leioa. The proposal will be presented from the point of view of sustainability. In order to achieve this, the roads that are already built will be used and the road or network will be proposed to be built with the least amount of money as possible. To select the most suitable route for the bike lane, various sources of information has been used. Through this data, we analyse the transport infrastructure and the mobility around the UPV/EHU Campus in Leioa. This work aims to satisfy the mobility needs of users on the University Campus to contribute to the sustainability of the Campus.

The growth of the renewable energy industry is happening at a swift pace pushed, by the emergence of Industry 4.0. Smart technologies like artificial intelligence (AI), Big Data, the Internet of Things (IoT), Digital Twin (DT), etc. enable companies within the sector of renewable energies to drastically improve their operations. In this sectoral context, where upgraded sustainability standards also play a vital role, it is necessary to fulfil the human capital requirements of the imminent technological advances. This article aims to determine the current skills of the renewable energy industry workforce and to predict the upcoming skill requirements linked to a digital transition by creating a unified database that contains both types of skills. This will serve as a tool for renewable energy businesses, education centers, and policymakers to plan the training itinerary necessary to close the skills gap, as part of the sectoral strategy to achieve a competent future workforce.

Due to the importance of road transport an adequate identification of the various road network levels is necessary for an efficient and sustainable management of the road infrastructure. Additionally, traffic values are key data for any pavement management system. In this work traffic volume data of 2019 in the Basque Autonomous Community (Spain) were analyzed and modeled. Having a multidimensional sample, the average annual daily traffic (AADT) was considered as the main variable of interest, which is used in many areas of the road network management. First, an exploratory analysis was performed, from which descriptive statistical information was obtained continuing with the clustering by various variables in order to standardize its behavior by translation. In a second stage, the variable of interest was estimated in the entire road network of the studied country using linear-based radial basis functions (RBFs). The estimated model was compared with the sample statistically, evaluating the estimation using cross-validation and highest-traffic sectors are defined. From the analysis, it was observed that the clustering analysis is useful for identifying the real importance of each road segment, as a function of the real traffic volume and not based on other criteria. It was also observed that interpolation methods based on linear-type radial basis functions (RBF) can be used as a preliminary method to estimate the AADT.

The aim of this paper is to characterize the mechanical behavior of corrugated cardboard boxes using simple models that allow an approach to the load capacity and the deformation of the boxes. This is very interesting during a box design stage, in which the box does not exist yet. On the one hand, a mathematical model of strength and deformation of boxes with different geometry is obtained from experiments according to the Box Compression Test and Edge Crush Test standards. On the second hand, a finite element simulation is proposed in which only the material elastic modulus in the compression direction is needed. For that, corrugated cardboard sheets are glued to build billets for testing, and an equivalent elastic modulus is obtained. This idea arises from the fact that the collapse of the box is given by the local bucking of the corrugated cardboard panels, due to the slenderness itself, and the properties in the compression direction are predominant. As a result, the numerical models show satisfactory agreement with experiments, concluding that it is an adequate methodology to simulate in a simple and efficient way this type of boxes built with corrugated cardboard.

In recent times, new learning methodologies known as student-based methodologies have been introduced to simplify the learning process for the students and facilitate the acquisition of skills for them. Among them, problem based learning (PBL) and project-based learning (PjBL) are widely used methods in the world of education. Real case solving (RCS) is a variant of the PBL where students solve real cases through the application of the PBL methodology. RCS seems to be a relevant approach for educators, but it has an apparently limited implementation degree at the academic level. This article presents the successful implementation of four different RCS approaches in the lecturing process in five different classes in the engineering degree of University of Deusto. The initiative has been analyzed both quantitative and qualitatively; the overall performance and success rate of the students were compared with the ones acquired from conventional teaching methods. The results were found to be promising, demonstrating a significantly better performance than the traditional teaching methodologies. The successful results encouraged the university to continue working further in this direction.

The machine tool industry, which is the starting point of all the metal producing activities, is presently undergoing rapid and continuous changes as a result of the fourth industrial revolution Industry 4.0. Manufacturing models are profoundly transforming with emerging digitalization. Smart technologies like artificial intelligence (AI), big data, the Internet of Things (IoT), digital twin, allow the machine tool companies to optimize processes, increase efficiency and reduce waste through a new phase of automation. These technologies, as well, enable the machine tool producers to reach the aim of creating products with improved performance, extended life, high reliability that are eco-efficient. Therefore, Industry 4.0 could be perceived as an invaluable opportunity for the machine tool sector, only if the sector has a competent workforce capable of handling the implementation of new business models and technological developments. The main condition to create this highly qualified workforce is reskilling and upskilling of the current workforce. Once we define the expected evolution of skills requirements, we can clarify the skills mismatch between the workers and job profiles. Only then, we can reduce them by delivering well-developed trainings. For this purpose, this article identifies the current and foreseen skills requirements demanded by the machine tool industry workforce. To this end, we generated an integrated database for the sector with the present and prospective skills needs of the metal processing sector professionals. The presented sectoral database is a fundamental structure that will make the sector acquire targeted industrial reforms. It can also be an essential instrument for machine tool companies, policymakers, academics and education or training centers to build well-designed and effective training programs to enhance the skills of the labor force

This work presents the results obtained from a spatial modeling and analysis process on pollutants measured in the air through forty-three monitoring stations located in the three provinces of the Basque Autonomous Community (Spain). The pollutants measured correspond to the set of nitrogen oxides (nitric oxide, NO; nitrogen dioxide, NO 2 ; and nitrogen oxides, NO x ) and atmospheric particulate matter with a diameter less than or equal to 10 micrometers (PM 10 ). The objective of this work was to generate a map of the pollutants that exhaustively covers the entire area of the Basque Autonomous Community using geostatistical techniques, in such a way that it serves as a basis for short and midterm environmental studies.

In the present study, the influence of the sampling density on the coestimation error of a regionalized, locally stationary and geo-mining nature variable is analyzed. The case study is two-dimensional (2D) and synthetic-type, and it has been generated using a non-conditional Sequential Gaussian Simulation (SGS), with subsequent transformation to Gaussian distribution, seeking to emulate the structural behavior of the aforementioned variable. A primary and an auxiliary variable with different spatial and statistical properties are constructed using the same methodology. The collocated ordinary cokriging method has been applied, in which the auxiliary variable is spatially correlated with the primary one and it is known exhaustively. Fifteen sampling densities are extracted from the target population of the primary variable, which are compared with the simulated values after performing coestimation. The obtained results follow a potential function that indicates the mean global error (MGE) based on the sampling density percentage (SDP) (MGE=1.2366⋅SDP−0.224).

The food industry has recently faced rapid and constant changes due to the current industrial revolution, Industry 4.0, which has also profoundly altered the dynamics of the industry overall. Due to the emerging digitalisation, manufacturing models are changing through the use of smart technologies, such as robotics, Artificial Intelligence (AI), Internet of Things (IoT), machine learning, etc. They are experiencing a new phase of automation that enables innovative and more efficient processes, products and services. The introduction of these novel business models demands new professional skills requirements in the workforce of the food industry. In this work, we introduce an industry-driven proactive strategy to achieve a successful digital transformation in the food sector. For that purpose, we focus on defining the current and near-future key skills and competencies demanded by each of the professional profiles related to the food industry. To achieve this, we generated an automated database of current and future professions and competencies and skills. This database can be used as a fundamental roadmap guiding the sector through future changes caused by Industry 4.0. The interest shown by the local sectorial cluster and related entities reinforce the idea. This research will be a key tool for both academics and policy-makers to provide well-developed and better-oriented continuous training programs in order to reduce the skill mismatch between the workforce and the jobs.

Calculating adequate vehicle routes for collecting municipal waste is still an unsolved issue, even though many solutions for this process can be found in the literature. A gap still exists between academics and practitioners in the field. One of the apparent reasons why this rift exists is that academic tools often are not easy to handle and maintain by actual users. In this work, the problem of municipal waste collection is modeled using a simple but efficient and especially easy to maintain solution. Real data have been used, and it has been solved using a Genetic Algorithm (GA). Computations have been done in two different ways: using a complete random initial population, and including a seed in this initial population. In order to guarantee that the solution is efficient, the performance of the genetic algorithm has been compared with another well-performing algorithm, the Variable Neighborhood Search (VNS). Three problems of different sizes have been solved and, in all cases, a significant improvement has been obtained. A total reduction of 40% of itineraries is attained with the subsequent reduction of emissions and costs.

The construction sector has always occupied a strategic place in the European economy. The European construction industry suffered during the 2007–2008 global financial crisis, and today the sector is undergoing a recovery process. Among all the construction subsectors, civil engineering has the highest growth rate. Currently, the sector has to face profound industrial changes emerging with digital transformations (Industry 4.0), sustainability, climate change and energy efficiency. To promote the growth of the civil engineering sector and accelerate the recovery, we need to create a highly qualified and competent workforce that can handle the challenges coming up with the technological progress and global competitiveness. The main condition to achieve this capable workforce is to define the expected evolution of skills requirements. For that purpose, our work focuses on identifying current and near-future key skills required by the civil engineering occupations. To achieve this, we developed an automated sectoral database for the current and near-future skills requirements of the selected professional profiles. It is our belief that this sectoral database is a fundamental framework that will guide the sector through the future changes. We also believe that our research can be used as a key tool for construction companies, policy-makers, academics and training centers to develop well-designed and efficient training programs for upskilling and reskilling the workforce.

Thanks to the digitalization of industry, maintenance is a trending topic. The amount of data available for analyses and optimizations in this field has increased considerably. In addition, there are more and more complex systems to maintain, and to keep all these devices in proper conditions, which requires maintenance management to gain efficiency and effectiveness. Within maintenance, Condition-Based Maintenance (CBM) programs can provide significant advantages, but often these programs are complex to manage and understand. The problem becomes more complex when equipment is analyzed in the context of a plant, where equipment can be more or less saturated, critical regarding quality, etc. Thus, this paper focuses on CBM optimization of a full industrial chain, with the objective of determining its optimal values of preventive intervention limits for equipment under economic criteria. It develops a mathematical plus discrete-event-simulation based model that takes the evolution in quality and production speed into consideration as well as condition based, corrective and preventive maintenance. The optimization process is performed using a Multi-Objective Evolutionary Algorithm. Both the model and the optimization approach are applied to an industrial case, where the data gathered by the IoT (Internet of Things) devices at edge level can detect when some premises of the CBM model are no longer valid and request a new simulation. The simulation performed in a centralized way can thus obtain new optimal values who fit better to the actual system than the existing ones. Finally, these new optimal values can be transferred to the model whenever it is necessary. The approach developed has raised the interest of a partner of the Deusto Digital Industry Chair.

Maintenance has always been a key activity in the manufacturing industry because of its economic consequences. Nowadays, its importance is increasing thanks to the Industry 4.0 or fourth industrial revolution. There are more and more complex systems to maintain, and maintenance management must gain efficiency and effectiveness in order to keep all these devices in proper conditions. Within maintenance, Condition-Based Maintenance (CBM) programs can provide significant advantages, even though often these programs are complex to manage and understand. For this reason, several research papers propose approaches that are as simple as possible and can be understood by users and modified by experts. In this context, this paper focuses on CBM optimization in an industrial environment, with the objective of determining the optimal values of preventive intervention limits for equipment under corrective and preventive maintenance cost criteria. In this work, a cost-benefit mathematical model is developed. It considers the evolution in quality and production speed, along with condition based, corrective and preventive maintenance. The cost-benefit optimization is performed using a Multi-Objective Evolutionary Algorithm. Both the model and the optimization approach are applied to an industrial case.

143 p., Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevantfeatures of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solvechallenging engineering problems. In time dependent problems, GOA requires to represent the error inthe Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it viaadaptive refinements. A full space-time variational formulation of the problem allows theaforementioned error representation. Thus, variational space-time formulations for PDEs have been ofgreat interest in the last decades, among other things, because they allow to develop mesh-adaptivealgorithms. Since it is known that implicit time marching schemes have variational structure, they areoften employed for GOA in time-domain problems. When coming to explicit-in-time methods, thesewere introduced for Ordinary Differential Equations (ODEs) employing specific inexact quadrature rules.In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed asdiscontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. Wesystematically build trial and test functions that, after exact integration in time, lead to one, two, andgeneral stage explicit RK methods. This approach enables us to reproduce the existing time-domain goalorientedadaptive algorithms using explicit methods in time. Here, we employ the lowest order dPGformulation that we propose to recover the Forward Euler method and we derive an appropriate errorrepresentation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs localrefinements in space. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy (CFL)condition to ensure the stability of the method. We provide some numerical results in one-dimensional(1D)+time for the diffusion and advection-diffusion equations to show the performance of the proposedalgorithm.O

Labaurpena: lan honetan problema baten eredu matematikoa sortu da eta konputazio teknikak erabilita ebatzi da. Zehazki, Sopelako hondakinen bilketa egiteko jarraitzen diren ibilbideak aztertu dira eta horietako batzuen optimizazioa ikertu da. Problema definitu ondoren algoritmo genetikoa erabilita ebatzi da algoritmoaren eraginkortasuna neurtuz. Horretarako, lehenik eta behin, dimentsio txikiko matrizearekin algoritmo genetikoa eta indar basatian oinarritutako kalkuluak egin dira. Emaitzei esker algoritmo genetikoaren eraginkortasuna egiaztatuta dagoenean, dimentsio handiagoko matrizean algoritmo genetikoa aplikatu da, honen ibilbide optimoaren lortzen saiatuz., Resumen: en este proyecto se ha modelado matemáticamente un problema y se ha resuelto mediante técnicas computacionales. Concretamente, se ha ajustado el problema de planificación de gestión de residuos, por poder ser moldeable matemáticamente y solucionable mediante diferentes estrategias. Una vez que el problema está definido, se ha resuelto usando el algoritmo genético y se ha medido la eficiencia del algoritmo. Para ello, primero se han hecho los cálculos en una matriz de pequeña dimensión aplicando la fuerza bruta y el algoritmo genético. Cuando la eficiencia del algoritmo genético está comprobada, se ha aplicado el algoritmo genético a una matriz de mayor dimensión, intentando conseguir un resultado optimo de la ruta., in this work a problem is modeled mathematically and it is solved using mathematical software. Concretely, the problem of garbage pickup route planning is tackled, as it can be modeled mathematically and solved computationally using different strategies. We define the problem, we solve it using a genetic algorithm and the efficiency of the algorithm has been measured. To do so, the first step is to apply both brute force and genetic algorithm to the short dimension matrix. When the efficiency of the genetic algorithm is proven, the method is applied on another matrix with greater dimension, trying to get an optimum result for the route.

Laburpena: ekuazio diferentzialak erabiltzen dira gure inguruneko fenomeno asko deskribatzeko. Ekuazio diferentzialetan funtzio ezezagun baten aldaketak (hau da, deribatuak) agertzen dira, eta helburua ekuazioa betetzen duen funtzioa aurkitzea izaten da. Beraien konplexutasuna dela eta, gehienetan, ez da posible izaten emaitza zehatza edo analitikoa kalkulatzea. Ondorioz, metodo matematiko ugari sortu izan dira ekuazio diferentzialen emaitza hurbilduak kalkulatzeko. Metodo horiei zenbakizko metodo deritze, eta lan honetan zehar horietako batzuk aztertu eta landu dira. Ekuazio diferentzialak ingeniaritza zibileko alorrean ere erabiliak dira. Lan honetan ekuazio diferentzialen bidez deskribatutako ingeniaritza zibileko problema batzuk aurkezten eta garatzen dira. Helburuak honako hauek izan dira: Ekuazio diferentzialak ebazteko zenbakizko metodoak ulertzea eta aplikatzea. Ekuazio diferentzialen bidez deskribatutako ingeniaritza zibileko problema bat ebaztea. Zehazki, presa baten apurketaren ondorioz uraren fluxuak duen portaera aztertu da. Aipatutako helburuak lortzeko zenbait pauso jarraitu dira. Lehendabizi, aldagai bakarraren menpekotasuna duen problema lineala aztertu eta ebatzi da. Zehazki, espazioan dimentsio bakarrekoa den problema ebatzi da. Jarraian, espazioan dimentsio bakarrekoa izanik denborarekiko menpekotasuna duen problema lineala ulertu eta ebatzi da. Bukatzeko, presa baten apurketaren osteko fluxuaren jarrera aztertu da. Fenomeno hori Saint-Venant-en ekuazioek definitzen dute. Bi ekuazio diferentzialek osatzen dute problema. Espazioan dimentsio bakarrekoak dira, denborarekiko menpekotasuna dute eta problema ez-lineala da., Resumen: se utilizan ecuaciones diferenciales para describir muchos fenómenos de nuestro entorno. En las ecuaciones diferenciales, aparecen derivadas de la función incógnita y el objetivo suele ser hallar la función que satisface dicha ecuación. Dada su complejidad, en la mayor parte de los casos no suele ser posible encontrar el resultado exacto o analítico de éstas. Por este motivo, se han desarrollado diversos métodos matemáticos para obtener los resultados aproximados de las ecuaciones diferenciales. A estos métodos se los conoce como métodos numéricos y algunos de ellos han sido estudiados y tratados en este trabajo. Las ecuaciones diferenciales son utilizadas también en el campo de la ingeniería civil. En este trabajo se presentan y desarrollan varios problemas de ingeniería civil descritos mediante ecuaciones diferenciales. Los objetivos han sido los siguientes: Entender y aplicar métodos numéricos para la resolución de ecuaciones diferenciales. Resolver un problema de la ingeniería civil descrito mediante ecuaciones diferenciales. Específicamente, se ha estudiado el comportamiento del flujo del agua tras la rotura de una presa. Para conseguir estos objetivos se han seguido varios pasos. En primer lugar, se ha analizado y resuelto un problema lineal dependiente de una sola variable. Más específicamente, se ha resuelto un problema unidimensional en espacio. A continuación, se ha comprendido y resuelto un problema lineal unidimensional que depende del tiempo. Finalmente, se ha analizado el comportamiento del flujo del agua a causa de la rotura de una presa. Este fenómeno viene definido por las ecuaciones de Saint-Venant. El problema lo completan dos ecuaciones diferenciales, siendo estas unidimensionales y dependientes del tiempo, que constituyen un problema no-lineal., differential equations are used to describe several phenomena of our environment. Differential equations consist in the derivatives of an unknown function, and the objective is to find the function that satisfies such equation. Due to their complexity, it is not in general possible to find an exact or analytic solution for these equations. For this reason, several mathematical methods have been developed in order to obtain an approximation of the solutions of differential equations. Differential equations are also used in the field of civil engineering. In this project, some civil engineering problems described by differential equations are shown and developed. The objectives have been the following ones: To understand and apply numerical methods on the resolution of differential equations. To solve a problem related to civil engineering described by differential equations. Specifically, the flux of the water after a dam-break has been analysed. In order to achieve these objectives, the following steps have been followed. On the first place, a linear problem that depends on a single variable has been analysed and solved. Specifically, an unidimensional problem has been solved. Secondly, an unidimensional time-dependent linear problem has been understood and solved. Finally, the behaviour of the flux after a dam-break has been studied. This phenomenon is described by Saint-Venant equations, which are two unidimensional and time-dependent equations that define a non-linear problem.

We focus on the inversion of borehole resistivity measurements in real time. To perform this task, we propose the use of Deep Learning methods. One critical task on this en- deavor is to produce a large database that can be used to train Deep Neural Networks. In this work, we explore different venues to obtain such database conforming the ground truth data via massive finite element computer simulations of the so-called forward prob- lem. This consists of solving multiple Boundary Value Problems governed by a Partial Differential Equation with different material coefficients. After describing the Finite Ele- ment Method, we investigate a venue to achieve high performance for performing a large amount of simulations using a Fourier approximation based Finite Element Method. The idea is to exploit the orthogonality of Fourier basis functions under reasonable assumptions often satisfied in our geophysics applications to reduce the computational cost of building the corresponding systems of linear equations. Solving such systems requires the use of advanced iterative solvers, which will be analyzed during the Ph.D. studies of Carlos Uri- arte.

Konposizio metodoek, Ekuazio Diferentzial Arruntak (EDAak) ebazteko oinarrizko zenbakizko integrazio-metodo bat modu egokian konposatuz emaitzak hobetzeko aukera ematen dute. Lan honetan erreparatuko diegu bigarren ordenako zehaztasuna duen oinarrizko integratzaile simetriko bat erabiliz lortzen den konposizio metodo simetrikoei. Simetrien erabilerak, ordena-baldintzak sinplifikatzeaz gain, ezezagunen kopurua gutxitzea eragiten du. Asko dira honelako metodoen koefizienteak bilatzen jardun duten autoreak. Honela, ezaguna da aipaturiko eran lortutako 8 ordenako metodorik onenak 17 atal dituela, eta 10 ordenako eta 31, 33 eta 35 ataleko metodo oso onak ere lortu direla. Lan honetan 10 ordenako eta 31 ataleko konposizio metodo simetrikoak lortzeko sortu ditugun bi teknika aurkezten dira.; Composition methods are useful when solving Ordinary Differential Equa-tions (ODEs) as they can be used to increase the order of accuracy of a given basic nu-merical integration scheme. We will focus on symmetric composition methods involv-ing some basic second order symmetric integrator with different step sizes. The introduction of symmetries into these methods simplifies the order conditions and re-duces the number of unknowns. Several authors have worked in the search of the coef-ficients of these type of methods: the best method of order 8 has 17 stages, 10-order methods of 31, 33 and 35 stages have been also found. In this work two techniques that we have built to obtain 10-order symmetric composition methods of symmetric inte-grators of s = 31 stages are explored.

Composition methods are useful when solving Ordinary Differential Equations (ODEs) as they increase the order of accuracy of a given basic numerical integration scheme. We will focus on sy-mmetric composition methods involving some basic second order symmetric integrator with different step sizes [17]. The introduction of symmetries into these methods simplifies the order conditions and reduces the number of unknowns. Several authors have worked in the search of the coefficients of these type of methods: the best method of order 8 has 17 stages [24], methods of order 8 and 15 stages were given in [29, 39, 40], 10-order methods of 31, 33 and 35 stages have been also found [24, 34]. In this work some techniques that we have built to obtain 10-order symmetric composition methods of symmetric integrators of s = 31 stages (16 order conditions) are explored. Given some starting coefficients that satisfy the simplest five order conditions, the process followed to obtain the coefficients that satisfy the sixteen order conditions is provided.

Uh in erako Deribatu Partz ia leko Eku azioa (DPE) askatzeko Elementu Finituen Metodoa (EFM) erabilita, 2. orde nako Ekuazio Dife re ntzial Arrunteko (EDA) sistema zurruna lortzen da . Sistema honetako maiztasunik handi eneko uhinak ez dira deribatu partzialeko ekuazioaren portaeraren adi erazgarri eta beraien erag ina moteltzea komeni da. HHT-a metodoak EFM erabili ostean sortzen diren maiztasun handiko uhinen moteltze algoritmikoari eragiten dio . 2 . ordenako BDFa (BDF2a) oinarritzat hartuz, BDF-a izeneko metodo berri bat sortu dugu; metodo honek, HHT-a metodoaren antzeko moteltze algoritmikoa du. Sortu dugun metodo berria 2. ordenakoa da, eta a parametroaren balio batzuetarako A-egonkorra izateaz gain mote ltze algoritmiko kontro latua erakusten du.; After applying the Fini te Element Method (FEM) to the wave-type Partial Differential Equ ation (PDE), a second order stiff Ordinary Differe nti al Equati on system (ODE) is obtained . The high freq uency modes associated to this system are not representative of the PDE and they must be numeri ca lly damped . Simil arly to the HHT-a method , whi ch allows the numerical damping of these undes irable hi gh frequency modes, we have constructed a modi f icati on of the 2-order BDF method (the BDF2 method), which we have called BDF-a . This new method is seco nd-order accurate, and for some values of a it is unconditi onally stable and it permits a parametric control of numerical dissipation.

295 p., La semidiscretización del Método de Elementos Finitos (MEF) de los problemas de difusión y onda conduce a sistemas de Ecuaciones Diferenciales Ordinarias (EDOs) fuertemente rígidos que pueden integrarse con las funciones de la odesuite de Matlab. Sin embargo, como la función ode15s que ofrece Matlab para sistemas rígidos se muestra poco eficiente en los problemas vibratorios, hemos profundizado en los métodos BDF que la sustentan hasta comprender los motivos y proponer alternativas. Ello nos ha llevado a trabajar en dos direcciones. Por una parte, en la ampliación de las regiones de estabilidad utilizando puntos super-futuros, lo que nos ha conducido a métodos de orden 4 incondicionalmente estables. Y por otra, en la ponderación de los métodos clásicos de la mecánica computacional que abordan directamente la resolución de EDOs de orden 2 y conducen a métodos incondicionalmente estables de orden de precisión 2, con control paramétrico del amortiguamiento algorítmico y que permiten la disipación de los modos de alta frecuencia que están mal definidos por la semidiscretización MEF y sólo aportan ruido a la solución. En este sentido, hemos podido incorporar al BDF de orden 2, el control paramétrico del amortiguamiento de las frecuencias de manera similar a como lo hace el método HHT-alfa. Estos desarrollos los hemos hecho en una metodología orientada a objeto en Matlab con el fin de disponer de una herramienta de laboratorio que facilite la experimentación y la incorporación de nuevos desarrollos sin necesidad de alterar los anteriores.