Some recent applications of non-linear models to the engineering practice

The Department of Structural Analysis of the University of Santander has been for a longtime involved in the solution of the country´s practical engineering problems. Some of these have required the use of non-conventional methods of analysis, in order to achieve adequate engineering answers. As an example of the increasing application of non-linear computer codes in the nowadays engineering practice, some cases will be briefly presented. In each case, only the main features of the problem involved and the solution used to solve it will be shown

Juvenile–adult tree associations in a continental Mediterranean ecosystem: no evidence for sustained and general facilitation at increased aridity

Question: How do tree species identity, microhabitat and water availability affect inter- and intra-specific interactions between juvenile and adult woody plants? Location: Continental Mediterranean forests in Alto Tajo Natural Park, Guadalajara, Spain. Methods: A total of 2066 juveniles and adults of four co-occurring tree species were mapped in 17 plots. The frequency of juveniles at different microhabitats and water availability levels was analysed using log-linear models. We used nearest-neighbour contingency table analysis of spatial segregation and J-functions to describe the spatial patterns. Results: We found a complex spatial pattern that varied according to species identity and microhabitat. Recruitment was more frequent in gaps for Quercus ilex, while the other three species recruited preferentially under shrubs or trees depending on the water availability level. Juveniles were not spatially associated to conspecific adults, experiencing segregation from them inmany cases. Spatial associations, both positive and negative, were more common at higher water availability levels. Conclusions: Our results do not agree with expectations from the stressgradient hypothesis, suggesting that positive interactions do not increase in importance with increasing aridity in the study ecosystem. Regeneration patterns are species-specific and depend on microhabitat characteristics and dispersal strategies. In general, juveniles do not look for conspecific adult protection. This work contributes to the understanding of species co-existence, proving the importance of considering a multispecies approach at several plots to overcome limitations of simple pair-wise comparisons in a limited number of sites.

Predictor paramétrico de glucosa subcutánea

La diabetes mellitus es una enfermedad que se caracteriza por la nula o insuficiente producción de insulina, o la resistencia del organismo a la misma. La insulina es una hormona que ayuda a que la glucosa (por ejemplo la obtenida a partir de los alimentos ingeridos) llegue a los tejidos periféricos y al sistema nervioso para suministrar energía. Hoy en día la tecnología actual permite abordar el desarrollo del llamado “páncreas endocrino artificial”, que consta de un sensor continuo de glucosa subcutánea, una bomba de infusión subcutánea de insulina y un algoritmo de control en lazo cerrado que calcule la dosis de insulina requerida por el paciente en cada momento, según la medida de glucosa obtenida por el sensor y según unos objetivos. El mayor problema que presentan los sistemas de control en lazo cerrado son los retardos, el sensor de glucosa subcutánea mide la glucosa del líquido intersticial, que representa la que hubo en la sangre un tiempo atrás, por tanto, un cambio en los niveles de glucosa en la sangre, debidos por ejemplo, a una ingesta, tardaría un tiempo en ser detectado por el sensor. Además, una dosis de insulina suministrada al paciente, tarda un tiempo aproximado de 20-30 minutos para la llegar a la sangre. Para evitar trabajar en la medida que sea posible con estos retardos, se intenta predecir cuál será el nivel de glucosa en un futuro próximo, para ello se utilizara un predictor de glucosa subcutánea, con la información disponible de glucosa e insulina. El objetivo del proyecto es diseñar una metodología para estimar el valor futuro de los niveles de glucosa obtenida a partir de un sensor subcutáneo, basada en la identificación recursiva del sistema glucorregulatorio a través de modelos lineales y determinando un horizonte de predicción óptimo de trabajo y analizando la influencia de la insulina en los resultados de la predicción. Se ha implementado un predictor paramétrico basado en un modelo autorregresivo ARX que predice con mejor precisión y con menor RMSE que un predictor ZOH a un horizonte de predicción de treinta minutos. Utilizar información relativa a la insulina no tiene efecto en la predicción. El preprocesado, postprocesado y el tratamiento de la estabilidad tienen un efecto muy beneficioso en la predicción. Diabetes mellitusis a group of metabolic diseases in which a person has high blood sugar, either because the body does not produce enough insulin, or because cells do not respond to the insulin produced. The insulin is a hormone that helps the glucose to reach to outlying tissues and the nervous system to supply energy. Nowadays, the actual technology allows raising the development of the “artificial endocrine pancreas”. It involves a continuous glucose sensor, an insulin bump, and a full closed loop algorithm that calculate the insulin units required by patient at any time, according to the glucose measure obtained by the sensor and any target. The main problem of the full closed loop systems is the delays, the glucose sensor measures the glucose in the interstitial fluid that represents the glucose was in the blood some time ago. Because of this, a change in the glucose in blood would take some time to be detected by the sensor. In addition, insulin units administered by a patient take about 20-30 minutes to reach the blood stream. In order to avoid this effect, it will try to predict the glucose level in the near future. To do that, a subcutaneous glucose predictor is used to predict the future glucose with the information about insulin and glucose. The goal of the proyect is to design a method in order to estimate the future valor of glucose obtained by a subcutaneous sensor. It is based on the recursive identification of the regulatory system through the linear models, determining optimal prediction horizon and analyzing the influence of insuline on the prediction results. A parametric predictor based in ARX autoregressive model predicts with better precision and with lesser RMSE than ZOH predictor in a thirty minutes prediction horizon. Using the relative insulin information has no effect in the prediction. The preprocessing, the postprocessing and the stability treatment have many advantages in the prediction.

Experimental analysis of the breakage of a liquid bridge under microgravity conditions

The experimental results obtained in experiment “STACO” made on board the Spacelab D-2 are re-visited, with image-analysis tools not then available. The configuration consisted of a liquid bridge between two solid supporting discs. An expected breakage occurred during the experiment. The recorded images are analysed and the measured behaviour compared with the results of a three dimensional model of the liquid dynamics, obtaining a much better fit than with linear models

Frequency and Intensity of drought events over Ebro River basin.

Lately, several researchers have pointed out that climate change is expected to increase temperatures and lower rainfall in Mediterranean regions, simultaneously increasing the intensity of extreme rainfall events. These changes could have consequences regarding rainfall regime, erosion, sediment transport and water quality, soil management, and new designs in diversion ditches. Climate change is expected to result in increasingly unpredictable and variable rainfall, in amount and timing, changing seasonal patterns and increasing the frequency of extreme weather events. Consequently, the evolution of frequency and intensity of drought periods is of most important as in agro-ecosystems many processes will be affected by them. Realising the complex and important consequences of an increasing frequency of extreme droughts at the Ebro River basin, our aim is to study the evolution of drought events at this site statistically, with emphasis on the occurrence and intensity of them. For this purpose, fourteen meteorological stations were selected based on the length of the rainfall series and the climatic classification to obtain a representative untreated dataset from the river basin. Daily rainfall series from 1957 to 2002 were obtained from each meteorological station and no-rain period frequency as the consecutive numbers of days were extracted. Based on this data, we study changes in the probability distribution in several sub-periods. Moreover we used the Standardized Precipitation Index (SPI) for identification of drought events in a year scale and then we use this index to fit log-linear models to the contingency tables between the SPI index and the sub-periods, this adjusted is carried out with the help of ANOVA inference.

Risk assessment of the spanish national railway system

The principal risks in the railway industry are mainly associated with collisions, derailments and level crossing accidents. An understanding of the nature of previous accidents on the railway network is required to identify potential causes and develop safety systems and deploy safety procedures. Risk assessment is a process for determining the risk magnitude to assist with decision-making. We propose a three-step methodology to predict the mean number of fatalities in railway accidents. The first is to predict the mean number of accidents by analyzing generalized linear models and selecting the one that best fits to the available historical data on the basis of goodness-offit statistics. The second is to compute the mean number of fatalities per accident and the third is to estimate the mean number of fatalities. The methodology is illustrated on the Spanish railway system. Statistical models accounting for annual and grouped data for the 1992-2009 time period have been analyzed. After identifying the models for broad and narrow gauges, we predicted mean number of accidents and the number of fatalities for the 2010-18 time period.

Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate.

Crecimiento forestal en el bosque tropical de montaña: efectos de la diversidad florística y de la manipulación de nutrientes

El bosque tropical de montaña, es considerado zona de mega diversidad y de alto grado de endemismo, por las diferentes zonas ecológicas que presenta. Durante las últimas décadas estos bosques han recibido mayor atención por parte de investigadores, sin embargo, aún existe poca investigación en determinar cuáles son las respuestas de los bosques a los cambios ambientales a los que son sometidos. Estos bosques están sufriendo serias amenazas como pérdida de cobertura vegetal y cambios en los ciclos de nutrientes. El trabajo se dividió en cuatro objetivos específicos. i) Caracterización y análisis de patrones altitudinales de la riqueza de especies en el bosque tropical de montaña en el sur del Ecuador; con el fin de conocer cómo varía la diversidad de especies riqueza lo largo de un gradiente altitudinal. ii) Conocer los patrones espaciales del crecimiento en tres remanentes boscosos de un bosque tropical de montaña para determinar cómo la vecindad y la semejanza funcional de ésta influyen en el crecimiento forestal. iii) Conocer los efectos de la fertilización en el crecimiento diamétrico de especies arbóreas, en el bosque tropical de montaña; se analizó cómo reaccionan los árboles a la adición de nutrientes N y P en tres tipos de bosque. iv) Saber la respuesta de la comunidad de árboles a la adición de nutrientes en el bosque montano andino; este objetivo se basó con el supuesto de la deficiencia de tres tipos de nutrientes N, P y Ca, en esta formación boscosa y cómo reaccionan los árboles a la adición de nutrientes. El presente trabajo se llevó a cabo, en el bosque tropical de montaña que se encuentra localizada en la parte adyacente del Parque Nacional Podocarpus (PNP) en la cordillera del Consuelo, forma parte de la cadena oriental de los Andes del sur del Ecuador El trabajo de desarrollo entre los años 2008 y 2014. Para abordar el primer objetivo se establecieron 54 parcelas ubicadas aleatoriamente a lo largo de un gradiente altitudinal (3 niveles de altitud) y se e midieron e identificaron todos los individuos mayores a 5cm de DAP. Se construyó una filogenia con Phylocom y se calcularon diferentes componentes de diversidad para cada parcela ( riqueza taxonómica, diversidad filogenética y edad media de las especies). Ajustando modelos lineares se contrastó el efecto de la altitud sobre dichos componentes y se vio que la riqueza taxonómica y la edad media de las especies aumentaron con la altitud, en sentido contrario a las predicciones de la "hipótesis del conservadurismo tropical" (Tropical Conservatism Hypothesis). Para abordar el segundo objetivo se realizó una remedición de todos los árboles cartografiados en tres parcelas permanentes de alrededor de 5000 m2 cada una, representativas de tres estados diferentes de la sucesión del bosque montano. A partir de las coordenadas y de los datos de registrados, y empleando diferentes funciones de correlación de marca se analizó la distribución espacial del tamaño y del crecimiento relativo y del tamaño. Se constató que mientras que el tamaño de los árboles presentó una correlación espacial negativa, el crecimiento presentó correlación espacial positiva, en ambos casos a distancias cortas. El rango y la magnitud de ambas correlaciones aumentaron al avanzar la sucesión. La distribución espacial del crecimiento mostró una correlación negativa con la distribución espacial de tamaños. Por otro lado, la distribución espacial del crecimiento mostró una correlación negativa para árboles semejantes funcionalmente y positiva cuando se calculó entre árboles con diferente estrategia funcional. En conjunto, los resultados obtenidos señalan un aumento de la importancia de procesos competitivos y una mayor estructuración espacial del crecimiento y de la distribución de tamaños al avanzar la sucesión. Para el tercer y cuarto objetivo se instalaron 52 parcelas distribuidas en bloques donde se fertilizaron dos veces al año durante 6,4 años, se identificaron todos los individuos mayores a 10 cm de DAP, y se midió el crecimiento diamétrico durante estos años Con la adición de nutrientes realizada a los diferentes tipos de bosque en la gradiente altitudinal, encontramos que el efecto sobre el crecimiento diamétrico en la comunidad varia con el rango altitudinal, y el tipo de nutriente, analizando a nivel de las especies, en la mayoría de los casos las especies comunes no tuvieron cambios significativos a la adición de nutrientes. Los resultados de este estudio aportan nuevas evidencias para el entendimiento de la diversidad, estructura y dinámica de los bosques tropicales de montaña. ABSTRACT The montane tropical forest is considered a megadiverse habitat that harbor an enormous degree of endemism. This is mainly due to the high degree of environmental heterogeneity found and the presence of different well defined ecological areas. These forests have received more attention during the last decades, however, the information regarding the responses of these forests to environmental change, is still scarce. These forests are seriously endangered and are suffering serious threats, such as loss of vegetative cover, changes in the nutrient cycles. The work was divided in four specific objectives: i) Characterization and analysis of the species richness altitudinal patterns in the montane tropical forest of south Ecuador. Specifically, how species diversity changes along altitudinal gradients. ii) Exploring the spatial patterns of tree growth in three remnants of a montane tropical forest, and analyze how tree neighborhood and functional similarity among trees influence tree growth. Tropical Conservatism Hypothesis iii) Understanding the effects of fertilization in arboreal species growth (increase in diameter) of the montane tropical forest. Specifically we studied the effects of P and N addition on three different forests across an altitudinal gradient. iv) Know the response of the community of trees to the addition of nutrients in the Andean montane forest; this objective was based on the supposition of deficiency of three types of nutrients: P, N and Ca in this forest all formation and how the trees react to the addition of these nutrients. The present work was carried out in the montane tropical forest located in Bombuscaro, San Francisco and Cajanuma close to Podocarpus National Park (PNP) on Consuelo mountain range (Andean oriental range) at South of Ecuador. Field work was carried out during 2008 and 2014. To address the first objective, we randomly placed 54 plots along an altitudinal gradient. In these plots, every individual larger than 5 cm of DBH was measured and identified. A phylogeny was build with Phylocom and different diversity components (taxonomic richness, phylogenetic diversity and average species age) were computed for each plot. Linear models were used to test the effects of altitude on the diversity components. Our results showed that, contrary to the Tropical Conservatism Hypothesis, both taxonomic richness and average species age increased with altitude. To address our second objective, all mapped trees in three successional permanent plots (around ~5000 m2 each) were re-measured. Using different mark correlation functions, we analyzed the spatial distribution of tree-size and tree relative growth rate. Whereas tree size showed negative spatial correlation at fine spatial scales, relative growth rate showed positive correlation at the same scales. The range and magnitude of those correlations increased along successional stage. The spatial distribution of the relative growth rate was negatively correlated with the spatial distribution of tree sizes. Additionally, we found that the spatial correlation of the relative growth rate was negative for functionally similar trees and positive when computed for functionally different trees. In synthesis, our results point to an increase of competitive processes and strong spatial structure of relative growth rate and tree size along succession. For the third and fourth objectives, 52 plots were placed in a block design and were fertilized twice a year for 6,4 years. In these plots all the individuals with DBH > 10 cm were identified, and the diametrical growth was measured during these years. The nutrient addition at the three different altitude forests, revealed that the effect on the diametrical growth in the community varied with the altitudinal range. When analyzed at species level, the addition of nutrients was no significant in most cases. These results represent new evidences that will improved our understanding of diversity patterns and structure, and the dynamics of tropical montane forests.

Distributed static linear Gaussian models using consensus

Algorithms for distributed agreement are a powerful means for formulating distributed versions of existing centralized algorithms. We present a toolkit for this task and show how it can be used systematically to design fully distributed algorithms for static linear Gaussian models, including principal component analysis, factor analysis, and probabilistic principal component analysis. These algorithms do not rely on a fusion center, require only low-volume local (1-hop neighborhood) communications, and are thus efficient, scalable, and robust. We show how they are also guaranteed to asymptotically converge to the same solution as the corresponding existing centralized algorithms. Finally, we illustrate the functioning of our algorithms on two examples, and examine the inherent cost-performance tradeoff.

Comparing linear and non linear wind flow models

Assessing wind conditions on complex terrain has become a hard task as terrain complexity increases. That is why there is a need to extrapolate in a reliable manner some wind parameters that determine wind farms viability such as annual average wind speed at all hub heights as well as turbulence intensities. The development of these tasks began in the early 90´s with the widely used linear model WAsP and WAsP Engineering especially designed for simple terrain with remarkable results on them but not so good on complex orographies. Simultaneously non-linearized Navier Stokes solvers have been rapidly developed in the last decade through CFD (Computational Fluid Dynamics) codes allowing simulating atmospheric boundary layer flows over steep complex terrain more accurately reducing uncertainties. This paper describes the features of these models by validating them through meteorological masts installed in a highly complex terrain. The study compares the results of the mentioned models in terms of wind speed and turbulence intensity.

Linear latent force models using Gaussian Processes

Purely data-driven approaches for machine learning present difficulties when data are scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data-driven modeling with a physical model of the system. We show how different, physically inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology, and geostatistics.

A study of the mechanical behaviour of the human aortic artery by means of non-linear finite element models

En la presente tesis desarrollamos una estrategia para la simulación numérica del comportamiento mecánico de la aorta humana usando modelos de elementos finitos no lineales. Prestamos especial atención a tres aspectos claves relacionados con la biomecánica de los tejidos blandos. Primero, el análisis del comportamiento anisótropo característico de los tejidos blandos debido a las familias de fibras de colágeno. Segundo, el análisis del ablandamiento presentado por los vasos sanguíneos cuando estos soportan cargas fuera del rango de funcionamiento fisiológico. Y finalmente, la inclusión de las tensiones residuales en las simulaciones en concordancia con el experimento de apertura de ángulo. El análisis del daño se aborda mediante dos aproximaciones diferentes. En la primera aproximación se presenta una formulación de daño local con regularización. Esta formulación tiene dos ingredientes principales. Por una parte, usa los principios de la teoría de la fisura difusa para garantizar la objetividad de los resultados con diferentes mallas. Por otra parte, usa el modelo bidimensional de Hodge-Petruska para describir el comportamiento mesoscópico de los fibriles. Partiendo de este modelo mesoscópico, las propiedades macroscópicas de las fibras de colágeno son obtenidas a través de un proceso de homogenización. En la segunda aproximación se presenta un modelo de daño no-local enriquecido con el gradiente de la variable de daño. El modelo se construye a partir del enriquecimiento de la función de energía con un término que contiene el gradiente material de la variable de daño no-local. La inclusión de este término asegura una regularización implícita de la implementación por elementos finitos, dando lugar a resultados de las simulaciones que no dependen de la malla. La aplicabilidad de este último modelo a problemas de biomecánica se estudia por medio de una simulación de un procedimiento quirúrgico típico conocido como angioplastia de balón. In the present thesis we develop a framework for the numerical simulation of the mechanical behaviour of the human aorta using non-linear finite element models. Special attention is paid to three key aspects related to the biomechanics of soft tissues. First, the modelling of the characteristic anisotropic behaviour of the softue due to the collagen fibre families. Secondly, the modelling of damage-related softening that blood vessels exhibit when subjected to loads beyond their physiological range. And finally, the inclusion of the residual stresses in the simulations in accordance with the opening-angle experiment The modelling of damage is addressed with two major and different approaches. In the first approach a continuum local damage formulation with regularisation is presented. This formulation has two principal ingredients. On the one hand, it makes use of the principles of the smeared crack theory to avoid the mesh size dependence of the structural response in softening. On the other hand, it uses a Hodge-Petruska bidimensional model to describe the fibrils as staggered arrays of tropocollagen molecules, and from this mesoscopic model the macroscopic material properties of the collagen fibres are obtained using an homogenisation process. In the second approach a non-local gradient-enhanced damage formulation is introduced. The model is built around the enhancement of the free energy function by means of a term that contains the referential gradient of the non-local damage variable. The inclusion of this term ensures an implicit regularisation of the finite element implementation, yielding mesh-objective results of the simulations. The applicability of the later model to biomechanically-related problems is studied by means of the simulation of a typical surgical procedure, namely, the balloon angioplasty.