Finite Element Based Nonlinear Normalization of Human Lumbar Intervertebral Disc Stiffness to Account for Its Morphology

Disc degeneration, usually associated with low back pain and changes of intervertebral stiffness, represents a major health issue. As the intervertebral disc (IVD) morphology influences its stiffness, the link between mechanical properties and degenerative grade is partially lost without an efficient normalization of the stiffness with respect to the morphology. Moreover, although the behavior of soft tissues is highly nonlinear, only linear normalization protocols have been defined so far for the disc stiffness. Thus, the aim of this work is to propose a nonlinear normalization based on finite elements (FE) simulations and evaluate its impact on the stiffness of human anatomical specimens of lumbar IVD. First, a parameter study involving simulations of biomechanical tests (compression, flexion/extension, bilateral torsion and bending) on 20 FE models of IVDs with various dimensions was carried out to evaluate the effect of the disc's geometry on its compliance and establish stiffness/morphology relations necessary to the nonlinear normalization. The computed stiffness was then normalized by height (H), cross-sectional area (CSA), polar moment of inertia (J) or moments of inertia (Ixx, Iyy) to quantify the effect of both linear and nonlinear normalizations. In the second part of the study, T1-weighted MRI images were acquired to determine H, CSA, J, Ixx and Iyy of 14 human lumbar IVDs. Based on the measured morphology and pre-established relation with stiffness, linear and nonlinear normalization routines were then applied to the compliance of the specimens for each quasi-static biomechanical test. The variability of the stiffness prior to and after normalization was assessed via coefficient of variation (CV). The FE study confirmed that larger and thinner IVDs were stiffer while the normalization strongly attenuated the effect of the disc geometry on its stiffness. Yet, notwithstanding the results of the FE study, the experimental stiffness showed consistently higher CV after normalization. Assuming that geometry and material properties affect the mechanical response, they can also compensate for one another. Therefore, the larger CV after normalization can be interpreted as a strong variability of the material properties, previously hidden by the geometry's own influence. In conclusion, a new normalization protocol for the intervertebral disc stiffness in compression, flexion, extension, bilateral torsion and bending was proposed, with the possible use of MRI and FE to acquire the discs' anatomy and determine the nonlinear relations between stiffness and morphology. Such protocol may be useful to relate the disc's mechanical properties to its degree of degeneration.

Flujo de potencia entre estructuras complejas sometidas a excitaciones de alta frecuencia y aplicaciones a la detección de daño estructural

El tema central de investigación en esta Tesis es el estudio del comportamientodinámico de una estructura mediante modelos que describen la distribución deenergía entre los componentes de la misma y la aplicación de estos modelos parala detección de daños incipientes.Los ensayos dinámicos son un modo de extraer información sobre las propiedadesde una estructura. Si tenemos un modelo de la estructura se podría ajustar éstepara que, con determinado grado de precisión, tenga la misma respuesta que elsistema real ensayado. Después de que se produjese un daño en la estructura,la respuesta al mismo ensayo variará en cierta medida; actualizando el modelo alas nuevas condiciones podemos detectar cambios en la configuración del modeloestructural que nos condujeran a la conclusión de que en la estructura se haproducido un daño.De este modo, la detección de un daño incipiente es posible si somos capacesde distinguir una pequeña variación en los parámetros que definen el modelo. Unrégimen muy apropiado para realizar este tipo de detección es a altas frecuencias,ya que la respuesta es muy dependiente de los pequeños detalles geométricos,dado que el tamaño característico en la estructura asociado a la respuesta esdirectamente proporcional a la velocidad de propagación de las ondas acústicas enel sólido, que para una estructura dada es inalterable, e inversamente proporcionala la frecuencia de la excitación. Al mismo tiempo, esta característica de la respuestaa altas frecuencias hace que un modelo de Elementos Finitos no sea aplicable enla práctica, debido al alto coste computacional.Un modelo ampliamente utilizado en el cálculo de la respuesta de estructurasa altas frecuencias en ingeniería es el SEA (Statistical Energy Analysis). El SEAaplica el balance energético a cada componente estructural, relacionando la energíade vibración de estos con la potencia disipada por cada uno de ellos y la potenciatransmitida entre ellos, cuya suma debe ser igual a la potencia inyectada a cadacomponente estructural. Esta relación es lineal y viene caracterizada por los factoresde pérdidas. Las magnitudes que intervienen en la respuesta se consideranpromediadas en la geometría, la frecuencia y el tiempo.Actualizar el modelo SEA a datos de ensayo es, por lo tanto, calcular losfactores de pérdidas que reproduzcan la respuesta obtenida en éste. Esta actualización,si se hace de manera directa, supone la resolución de un problema inversoque tiene la característica de estar mal condicionado. En la Tesis se propone actualizarel modelo SEA, no en término de los factores de pérdidas, sino en términos deparámetros estructurales que tienen sentido físico cuando se trata de la respuestaa altas frecuencias, como son los factores de disipación de cada componente, susdensidades modales y las rigideces características de los elementos de acoplamiento.Los factores de pérdidas se calculan como función de estos parámetros. Estaformulación es desarrollada de manera original en esta Tesis y principalmente sefunda en la hipótesis de alta densidad modal, es decir, que en la respuesta participanun gran número de modos de cada componente estructural.La teoría general del método SEA, establece que el modelo es válido bajounas hipótesis sobre la naturaleza de las excitaciones externas muy restrictivas,como que éstas deben ser de tipo ruido blanco local. Este tipo de carga es difícil dereproducir en condiciones de ensayo. En la Tesis mostramos con casos prácticos queesta restricción se puede relajar y, en particular, los resultados son suficientementebuenos cuando la estructura se somete a una carga armónica en escalón.Bajo estas aproximaciones se desarrolla un algoritmo de optimización por pasosque permite actualizar un modelo SEA a un ensayo transitorio cuando la carga esde tipo armónica en escalón. Este algoritmo actualiza el modelo no solamente parauna banda de frecuencia en particular sino para diversas bandas de frecuencia demanera simultánea, con el objetivo de plantear un problema mejor condicionado.Por último, se define un índice de daño que mide el cambio en la matriz depérdidas cuando se produce un daño estructural en una localización concreta deun componente. Se simula numéricamente la respuesta de una estructura formadapor vigas donde producimos un daño en la sección de una de ellas; como se tratade un cálculo a altas frecuencias, la simulación se hace mediante el Método delos Elementos Espectrales para lo que ha sido necesario desarrollar dentro de laTesis un elemento espectral de tipo viga dañada en una sección determinada. Losresultados obtenidos permiten localizar el componente estructural en que se haproducido el daño y la sección en que éste se encuentra con determinado grado deconfianza.AbstractThe main subject under research in this Thesis is the study of the dynamic behaviourof a structure using models that describe the energy distribution betweenthe components of the structure and the applicability of these models to incipientdamage detection.Dynamic tests are a way to extract information about the properties of astructure. If we have a model of the structure, it can be updated in order toreproduce the same response as in experimental tests, within a certain degree ofaccuracy. After damage occurs, the response will change to some extent; modelupdating to the new test conditions can help to detect changes in the structuralmodel leading to the conclusión that damage has occurred.In this way incipient damage detection is possible if we are able to detect srnallvariations in the model parameters. It turns out that the high frequency regimeis highly relevant for incipient damage detection, because the response is verysensitive to small structural geometric details. The characteristic length associatedwith the response is proportional to the propagation speed of acoustic waves insidethe solid, but inversely proportional to the excitation frequency. At the same time,this fact makes the application of a Finite Element Method impractical due to thehigh computational cost.A widely used model in engineering when dealing with the high frequencyresponse is SEA (Statistical Energy Analysis). SEA applies the energy balance toeach structural component, relating their vibrational energy with the dissipatedpower and the transmitted power between the different components; their summust be equal to the input power to each of them. This relationship is linear andcharacterized by loss factors. The magnitudes considered in the response shouldbe averaged in geometry, frequency and time.SEA model updating to test data is equivalent to calculating the loss factorsthat provide a better fit to the experimental response. This is formulated as an illconditionedinverse problem. In this Thesis a new updating algorithm is proposedfor the study of the high frequency response regime in terms of parameters withphysical meaning such as the internal dissipation factors, modal densities andcharacteristic coupling stiffness. The loss factors are then calculated from theseparameters. The approach is developed entirely in this Thesis and is mainlybased on a high modal density asumption, that is to say, a large number of modescontributes to the response.General SEA theory establishes the validity of the model under the asumptionof very restrictive external excitations. These should behave as a local white noise.This kind of excitation is difficult to reproduce in an experimental environment.In this Thesis we show that in practical cases this assumption can be relaxed, inparticular, results are good enough when the structure is excited with a harmonicstep function.Under these assumptions an optimization algorithm is developed for SEAmodel updating to a transient test when external loads are harmonic step functions.This algorithm considers the response not only in a single frequency band,but also for several of them simultaneously.A damage index is defined that measures the change in the loss factor matrixwhen a damage has occurred at a certain location in the structure. The structuresconsidered in this study are built with damaged beam elements; as we are dealingwith the high frequency response, the numerical simulation is implemented witha Spectral Element Method. It has therefore been necessary to develop a spectralbeam damaged element as well. The reported results show that damage detectionis possible with this algorithm, moreover, damage location is also possible withina certain degree of accuracy.

Impact And Explosive Loads On Concrete Buildings Using Shell And Beam Type Elements

The threat of impact or explosive loads is regrettably a scenario to be taken into account in the design of lifeline or critical civilian buildings. These are often made of concrete and not specifically designed for military threats. Numerical simulation of such cases may be undertaken with the aid of state of the art explicit dynamic codes, however several difficult challenges are inherent to such models: the material modeling for the concrete anisotropic failure, consideration of reinforcement bars and important structural details, adequate modeling of pressure waves from explosions in complex geometries, and efficient solution to models of complete buildings which can realistically assess failure modes. In this work we employ LS-Dyna for calculation, with Lagrangian finite elements and explicit time integration. Reinforced concrete may be represented in a fairly accurate fashion with recent models such as CSCM model [1] and segregated rebars constrained within the continuum mesh. However, such models cannot be realistically employed for complete models of large buildings, due to limitations of time and computer resources. The use of structural beam and shell elements for this purpose would be the obvious solution, with much lower computational cost. However, this modeling requires careful calibration in order to reproduce adequately the highly nonlinear response of structural concrete members, including bending with and without compression, cracking or plastic crushing, plastic deformation of reinforcement, erosion of vanished elements etc. The main objective of this work is to provide a strategy for modeling such scenarios based on structural elements, using available material models for structural elements [2] and techniques to include the reinforcement in a realistic way. These models are calibrated against fully three-dimensional models and shown to be accurate enough. At the same time they provide the basis for realistic simulation of impact and explosion on full-scale buildings

A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balance laws, variational formulation, and linearization

A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

Experimental and numerical analysis of reinforced concrete elements subjected to blast loading

El hormigón es uno de los materiales de construcción más empleados en la actualidad debido a sus buenas prestaciones mecánicas, moldeabilidad y economía de obtención, entre otras ventajas. Es bien sabido que tiene una buena resistencia a compresión y una baja resistencia a tracción, por lo que se arma con barras de acero para formar el hormigón armado, material que se ha convertido por méritos propios en la solución constructiva más importante de nuestra época. A pesar de ser un material profusamente utilizado, hay aspectos del comportamiento del hormigón que todavía no son completamente conocidos, como es el caso de su respuesta ante los efectos de una explosión. Este es un campo de especial relevancia, debido a que los eventos, tanto intencionados como accidentales, en los que una estructura se ve sometida a una explosión son, por desgracia, relativamente frecuentes. La solicitación de una estructura ante una explosión se produce por el impacto sobre la misma de la onda de presión generada en la detonación. La aplicación de esta carga sobre la estructura es muy rápida y de muy corta duración. Este tipo de acciones se denominan cargas impulsivas, y pueden ser hasta cuatro órdenes de magnitud más rápidas que las cargas dinámicas impuestas por un terremoto. En consecuencia, no es de extrañar que sus efectos sobre las estructuras y sus materiales sean muy distintos que las que producen las cargas habitualmente consideradas en ingeniería. En la presente tesis doctoral se profundiza en el conocimiento del comportamiento material del hormigón sometido a explosiones. Para ello, es crucial contar con resultados experimentales de estructuras de hormigón sometidas a explosiones. Este tipo de resultados es difícil de encontrar en la literatura científica, ya que estos ensayos han sido tradicionalmente llevados a cabo en el ámbito militar y los resultados obtenidos no son de dominio público. Por otra parte, en las campañas experimentales con explosiones llevadas a cabo por instituciones civiles el elevado coste de acceso a explosivos y a campos de prueba adecuados no permite la realización de ensayos con un elevado número de muestras. Por este motivo, la dispersión experimental no es habitualmente controlada. Sin embargo, en elementos de hormigón armado sometidos a explosiones, la dispersión experimental es muy acusada, en primer lugar, por la propia heterogeneidad del hormigón, y en segundo, por la dificultad inherente a la realización de ensayos con explosiones, por motivos tales como dificultades en las condiciones de contorno, variabilidad del explosivo, o incluso cambios en las condiciones atmosféricas. Para paliar estos inconvenientes, en esta tesis doctoral se ha diseñado un novedoso dispositivo que permite ensayar hasta cuatro losas de hormigón bajo la misma detonación, lo que además de proporcionar un número de muestras estadísticamente representativo, supone un importante ahorro de costes. Con este dispositivo se han ensayado 28 losas de hormigón, tanto armadas como en masa, de dos dosificaciones distintas. Pero además de contar con datos experimentales, también es importante disponer de herramientas de cálculo para el análisis y diseño de estructuras sometidas a explosiones. Aunque existen diversos métodos analíticos, hoy por hoy las técnicas de simulación numérica suponen la alternativa más avanzada y versátil para el cálculo de elementos estructurales sometidos a cargas impulsivas. Sin embargo, para obtener resultados fiables es crucial contar con modelos constitutivos de material que tengan en cuenta los parámetros que gobiernan el comportamiento para el caso de carga en estudio. En este sentido, cabe destacar que la mayoría de los modelos constitutivos desarrollados para el hormigón a altas velocidades de deformación proceden del ámbito balístico, donde dominan las grandes tensiones de compresión en el entorno local de la zona afectada por el impacto. En el caso de los elementos de hormigón sometidos a explosiones, las tensiones de compresión son mucho más moderadas, siendo las tensiones de tracción generalmente las causantes de la rotura del material. En esta tesis doctoral se analiza la validez de algunos de los modelos disponibles, confirmando que los parámetros que gobiernan el fallo de las losas de hormigón armado ante explosiones son la resistencia a tracción y su ablandamiento tras rotura. En base a los resultados anteriores se ha desarrollado un modelo constitutivo para el hormigón ante altas velocidades de deformación, que sólo tiene en cuenta la rotura por tracción. Este modelo parte del de fisura cohesiva embebida con discontinuidad fuerte, desarrollado por Planas y Sancho, que ha demostrado su capacidad en la predicción de la rotura a tracción de elementos de hormigón en masa. El modelo ha sido modificado para su implementación en el programa comercial de integración explícita LS-DYNA, utilizando elementos finitos hexaédricos e incorporando la dependencia de la velocidad de deformación para permitir su utilización en el ámbito dinámico. El modelo es estrictamente local y no requiere de remallado ni conocer previamente la trayectoria de la fisura. Este modelo constitutivo ha sido utilizado para simular dos campañas experimentales, probando la hipótesis de que el fallo de elementos de hormigón ante explosiones está gobernado por el comportamiento a tracción, siendo de especial relevancia el ablandamiento del hormigón. Concrete is nowadays one of the most widely used building materials because of its good mechanical properties, moldability and production economy, among other advantages. As it is known, it has high compressive and low tensile strengths and for this reason it is reinforced with steel bars to form reinforced concrete, a material that has become the most important constructive solution of our time. Despite being such a widely used material, there are some aspects of concrete performance that are not yet fully understood, as it is the case of its response to the effects of an explosion. This is a topic of particular relevance because the events, both intentional and accidental, in which a structure is subjected to an explosion are, unfortunately, relatively common. The loading of a structure due to an explosive event occurs due to the impact of the pressure shock wave generated in the detonation. The application of this load on the structure is very fast and of very short duration. Such actions are called impulsive loads, and can be up to four orders of magnitude faster than the dynamic loads imposed by an earthquake. Consequently, it is not surprising that their effects on structures and materials are very different than those that cause the loads usually considered in engineering. This thesis broadens the knowledge about the material behavior of concrete subjected to explosions. To that end, it is crucial to have experimental results of concrete structures subjected to explosions. These types of results are difficult to find in the scientific literature, as these tests have traditionally been carried out by armies of different countries and the results obtained are classified. Moreover, in experimental campaigns with explosives conducted by civil institutions the high cost of accessing explosives and the lack of proper test fields does not allow for the testing of a large number of samples. For this reason, the experimental scatter is usually not controlled. However, in reinforced concrete elements subjected to explosions the experimental dispersion is very pronounced. First, due to the heterogeneity of concrete, and secondly, because of the difficulty inherent to testing with explosions, for reasons such as difficulties in the boundary conditions, variability of the explosive, or even atmospheric changes. To overcome these drawbacks, in this thesis we have designed a novel device that allows for testing up to four concrete slabs under the same detonation, which apart from providing a statistically representative number of samples, represents a significant saving in costs. A number of 28 slabs were tested using this device. The slabs were both reinforced and plain concrete, and two different concrete mixes were used. Besides having experimental data, it is also important to have computational tools for the analysis and design of structures subjected to explosions. Despite the existence of several analytical methods, numerical simulation techniques nowadays represent the most advanced and versatile alternative for the assessment of structural elements subjected to impulsive loading. However, to obtain reliable results it is crucial to have material constitutive models that take into account the parameters that govern the behavior for the load case under study. In this regard it is noteworthy that most of the developed constitutive models for concrete at high strain rates arise from the ballistic field, dominated by large compressive stresses in the local environment of the area affected by the impact. In the case of concrete elements subjected to an explosion, the compressive stresses are much more moderate, while tensile stresses usually cause material failure. This thesis discusses the validity of some of the available models, confirming that the parameters governing the failure of reinforced concrete slabs subjected to blast are the tensile strength and softening behaviour after failure. Based on these results we have developed a constitutive model for concrete at high strain rates, which only takes into account the ultimate tensile strength. This model is based on the embedded Cohesive Crack Model with Strong Discontinuity Approach developed by Planas and Sancho, which has proved its ability in predicting the tensile fracture of plain concrete elements. The model has been modified for its implementation in the commercial explicit integration program LS-DYNA, using hexahedral finite elements and incorporating the dependence of the strain rate, to allow for its use in dynamic domain. The model is strictly local and does not require remeshing nor prior knowledge of the crack path. This constitutive model has been used to simulate two experimental campaigns, confirming the hypothesis that the failure of concrete elements subjected to explosions is governed by their tensile response, being of particular relevance the softening behavior of concrete.

3D hp-adaptive finite element simulations of a magic-T electromagnetic waveguide structure

This paper employs a 3D hp self-adaptive grid-refinement finite element strategy for the solution of a particular electromagnetic waveguide structure known as Magic-T. This structure is utilized as a power divider/combiner in communication systems as well as in other applications. It often incorporates dielectrics, metallic screws, round corners, and so on, which may facilitate its construction or improve its design, but significantly difficult its modeling when employing semi-analytical techniques. The hp-adaptive finite element method enables accurate modeling of a Magic-T structure even in the presence of these undesired materials/geometries. Numerical results demonstrate the suitability of the hp-adaptive method for modeling a Magic-T rectangular waveguide structure, delivering errors below 0.5% with a limited number of unknowns. Solutions of waveguide problems delivered by the self-adaptive hp-FEM are comparable to those obtained with semi-analytical techniques such as the Mode Matching method, for problems where the latest methods can be applied. At the same time, the hp-adaptive FEM enables accurate modeling of more complex waveguide structures.

Simulación en tiempo real de vehículos industriales con modelos multicuerpo de gran complejidad

El objetivo de esta Tesis ha sido la consecución de simulaciones en tiempo real de vehículos industriales modelizados como sistemas multicuerpo complejos formados por sólidos rígidos. Para el desarrollo de un programa de simulación deben considerarse cuatro aspectos fundamentales: la modelización del sistema multicuerpo (tipos de coordenadas, pares ideales o impuestos mediante fuerzas), la formulación a utilizar para plantear las ecuaciones diferenciales del movimiento (coordenadas dependientes o independientes, métodos globales o topológicos, forma de imponer las ecuaciones de restricción), el método de integración numérica para resolver estas ecuaciones en el tiempo (integradores explícitos o implícitos) y finalmente los detalles de la implementación realizada (lenguaje de programación, librerías matemáticas, técnicas de paralelización). Estas cuatro etapas están interrelacionadas entre sí y todas han formado parte de este trabajo. Desde la generación de modelos de una furgoneta y de camión con semirremolque, el uso de tres formulaciones dinámicas diferentes, la integración de las ecuaciones diferenciales del movimiento mediante métodos explícitos e implícitos, hasta el uso de funciones BLAS, de técnicas de matrices sparse y la introducción de paralelización para utilizar los distintos núcleos del procesador. El trabajo presentado en esta Tesis ha sido organizado en 8 capítulos, dedicándose el primero de ellos a la Introducción. En el Capítulo 2 se presentan dos formulaciones semirrecursivas diferentes, de las cuales la primera está basada en una doble transformación de velocidades, obteniéndose las ecuaciones diferenciales del movimiento en función de las aceleraciones relativas independientes. La integración numérica de estas ecuaciones se ha realizado con el método de Runge-Kutta explícito de cuarto orden. La segunda formulación está basada en coordenadas relativas dependientes, imponiendo las restricciones por medio de penalizadores en posición y corrigiendo las velocidades y aceleraciones mediante métodos de proyección. En este segundo caso la integración de las ecuaciones del movimiento se ha llevado a cabo mediante el integrador implícito HHT (Hilber, Hughes and Taylor), perteneciente a la familia de integradores estructurales de Newmark. En el Capítulo 3 se introduce la tercera formulación utilizada en esta Tesis. En este caso las uniones entre los sólidos del sistema se ha realizado mediante uniones flexibles, lo que obliga a imponer los pares por medio de fuerzas. Este tipo de uniones impide trabajar con coordenadas relativas, por lo que la posición del sistema y el planteamiento de las ecuaciones del movimiento se ha realizado utilizando coordenadas Cartesianas y parámetros de Euler. En esta formulación global se introducen las restricciones mediante fuerzas (con un planteamiento similar al de los penalizadores) y la estabilización del proceso de integración numérica se realiza también mediante proyecciones de velocidades y aceleraciones. En el Capítulo 4 se presenta una revisión de las principales herramientas y estrategias utilizadas para aumentar la eficiencia de las implementaciones de los distintos algoritmos. En primer lugar se incluye una serie de consideraciones básicas para aumentar la eficiencia numérica de las implementaciones. A continuación se mencionan las principales características de los analizadores de códigos utilizados y también las librerías matemáticas utilizadas para resolver los problemas de álgebra lineal tanto con matrices densas como sparse. Por último se desarrolla con un cierto detalle el tema de la paralelización en los actuales procesadores de varios núcleos, describiendo para ello el patrón empleado y las características más importantes de las dos herramientas propuestas, OpenMP y las TBB de Intel. Hay que señalar que las características de los sistemas multicuerpo problemas de pequeño tamaño, frecuente uso de la recursividad, y repetición intensiva en el tiempo de los cálculos con fuerte dependencia de los resultados anteriores dificultan extraordinariamente el uso de técnicas de paralelización frente a otras áreas de la mecánica computacional, tales como por ejemplo el cálculo por elementos finitos. Basándose en los conceptos mencionados en el Capítulo 4, el Capítulo 5 está dividido en tres secciones, una para cada formulación propuesta en esta Tesis. En cada una de estas secciones se describen los detalles de cómo se han realizado las distintas implementaciones propuestas para cada algoritmo y qué herramientas se han utilizado para ello. En la primera sección se muestra el uso de librerías numéricas para matrices densas y sparse en la formulación topológica semirrecursiva basada en la doble transformación de velocidades. En la segunda se describe la utilización de paralelización mediante OpenMP y TBB en la formulación semirrecursiva con penalizadores y proyecciones. Por último, se describe el uso de técnicas de matrices sparse y paralelización en la formulación global con uniones flexibles y parámetros de Euler. El Capítulo 6 describe los resultados alcanzados mediante las formulaciones e implementaciones descritas previamente. Este capítulo comienza con una descripción de la modelización y topología de los dos vehículos estudiados. El primer modelo es un vehículo de dos ejes del tipo chasis-cabina o furgoneta, perteneciente a la gama de vehículos de carga medianos. El segundo es un vehículo de cinco ejes que responde al modelo de un camión o cabina con semirremolque, perteneciente a la categoría de vehículos industriales pesados. En este capítulo además se realiza un estudio comparativo entre las simulaciones de estos vehículos con cada una de las formulaciones utilizadas y se presentan de modo cuantitativo los efectos de las mejoras alcanzadas con las distintas estrategias propuestas en esta Tesis. Con objeto de extraer conclusiones más fácilmente y para evaluar de un modo más objetivo las mejoras introducidas en la Tesis, todos los resultados de este capítulo se han obtenido con el mismo computador, que era el top de la gama Intel Xeon en 2007, pero que hoy día está ya algo obsoleto. Por último los Capítulos 7 y 8 están dedicados a las conclusiones finales y las futuras líneas de investigación que pueden derivar del trabajo realizado en esta Tesis. Los objetivos de realizar simulaciones en tiempo real de vehículos industriales de gran complejidad han sido alcanzados con varias de las formulaciones e implementaciones desarrolladas. ABSTRACT The objective of this Dissertation has been the achievement of real time simulations of industrial vehicles modeled as complex multibody systems made up by rigid bodies. For the development of a simulation program, four main aspects must be considered: the modeling of the multibody system (types of coordinates, ideal joints or imposed by means of forces), the formulation to be used to set the differential equations of motion (dependent or independent coordinates, global or topological methods, ways to impose constraints equations), the method of numerical integration to solve these equations in time (explicit or implicit integrators) and the details of the implementation carried out (programming language, mathematical libraries, parallelization techniques). These four stages are interrelated and all of them are part of this work. They involve the generation of models for a van and a semitrailer truck, the use of three different dynamic formulations, the integration of differential equations of motion through explicit and implicit methods, the use of BLAS functions and sparse matrix techniques, and the introduction of parallelization to use the different processor cores. The work presented in this Dissertation has been structured in eight chapters, the first of them being the Introduction. In Chapter 2, two different semi-recursive formulations are shown, of which the first one is based on a double velocity transformation, thus getting the differential equations of motion as a function of the independent relative accelerations. The numerical integration of these equations has been made with the Runge-Kutta explicit method of fourth order. The second formulation is based on dependent relative coordinates, imposing the constraints by means of position penalty coefficients and correcting the velocities and accelerations by projection methods. In this second case, the integration of the motion equations has been carried out by means of the HHT implicit integrator (Hilber, Hughes and Taylor), which belongs to the Newmark structural integrators family. In Chapter 3, the third formulation used in this Dissertation is presented. In this case, the joints between the bodies of the system have been considered as flexible joints, with forces used to impose the joint conditions. This kind of union hinders to work with relative coordinates, so the position of the system bodies and the setting of the equations of motion have been carried out using Cartesian coordinates and Euler parameters. In this global formulation, constraints are introduced through forces (with a similar approach to the penalty coefficients) are presented. The stabilization of the numerical integration is carried out also by velocity and accelerations projections. In Chapter 4, a revision of the main computer tools and strategies used to increase the efficiency of the implementations of the algorithms is presented. First of all, some basic considerations to increase the numerical efficiency of the implementations are included. Then the main characteristics of the code’ analyzers used and also the mathematical libraries used to solve linear algebra problems (both with dense and sparse matrices) are mentioned. Finally, the topic of parallelization in current multicore processors is developed thoroughly. For that, the pattern used and the most important characteristics of the tools proposed, OpenMP and Intel TBB, are described. It needs to be highlighted that the characteristics of multibody systems small size problems, frequent recursion use and intensive repetition along the time of the calculation with high dependencies of the previous results complicate extraordinarily the use of parallelization techniques against other computational mechanics areas, as the finite elements computation. Based on the concepts mentioned in Chapter 4, Chapter 5 is divided into three sections, one for each formulation proposed in this Dissertation. In each one of these sections, the details of how these different proposed implementations have been made for each algorithm and which tools have been used are described. In the first section, it is shown the use of numerical libraries for dense and sparse matrices in the semirecursive topological formulation based in the double velocity transformation. In the second one, the use of parallelization by means OpenMP and TBB is depicted in the semi-recursive formulation with penalization and projections. Lastly, the use of sparse matrices and parallelization techniques is described in the global formulation with flexible joints and Euler parameters. Chapter 6 depicts the achieved results through the formulations and implementations previously described. This chapter starts with a description of the modeling and topology of the two vehicles studied. The first model is a two-axle chassis-cabin or van like vehicle, which belongs to the range of medium charge vehicles. The second one is a five-axle vehicle belonging to the truck or cabin semi-trailer model, belonging to the heavy industrial vehicles category. In this chapter, a comparative study is done between the simulations of these vehicles with each one of the formulations used and the improvements achieved are presented in a quantitative way with the different strategies proposed in this Dissertation. With the aim of deducing the conclusions more easily and to evaluate in a more objective way the improvements introduced in the Dissertation, all the results of this chapter have been obtained with the same computer, which was the top one among the Intel Xeon range in 2007, but which is rather obsolete today. Finally, Chapters 7 and 8 are dedicated to the final conclusions and the future research projects that can be derived from the work presented in this Dissertation. The objectives of doing real time simulations in high complex industrial vehicles have been achieved with the formulations and implementations developed.

The B.I.E.M. applied to flow through porous media

This paper refers to the numerical solution of the classical Darcy's problem of plane fluid through isotropic media. Regarding the numerical procedure,the Laplace equation, is a classical one in mathematical physics and several procedures have been devised in order to solve it. So as to show the capability of the method, the paper presents some exemples.

A FEM-BEM coupling procedure through the Steklov-Poincarè operator

Many advantages can be got in combining finite and boundary elements.It is the case, for example, of unbounded field problems where boundary elements can provide the appropriate conditions to represent the infinite domain while finite elements are suitable for more complex properties in the near domain. However, in spite of it, other disadvantages can appear. It would be, for instance, the loss of symmetry in the finite elements stiffness matrix, when the combination is made. On the other hand, in our days, with the strong irruption of the parallel proccessing the techniques of decomposition of domains are getting the interest of numerous scientists. With their application it is possible to separate the resolution of a problem into several subproblems. That would be beneficial in the combinations BEM-FEM as the loss of symmetry would be avoided and every technique would be applicated separately. Evidently for the correct application of these techniques it is necessary to establish the suitable transmission conditions in the interface between BEM domain and FEM domain. In this paper, one parallel method is presented which is based in the interface operator of Steklov Poincarè.

Efecto del Cross-talk en la respuesta de un Transductor Ultrasónico tipo PZT

El presente trabajo de tesis investiga el efecto del fenómeno conocido como “Cross-talk” generado por el modo lateral de vibración, en la respuesta de un transductor ultrasónico formado por un arreglo de elementos piezoeléctricos tipo PZT (Zircanato Titanato de Plomo), la investigación se lleva a cabo desde el punto de vista de la naturaleza física de este efecto, así como de los parámetros asociados al mismo, así como un análisis del efecto del “Cross-talk” en la respuesta del transductor, formado por arreglos de elementos piezoeléctricos. Diversas investigaciones han abordado el fenómeno del “Cross-talk” y de sus efectos en la respuesta de los transductores, estos se han enfocado principalmente al modo espesor (thickness) de vibración. Sin embargo no ha habido un estudio a fondo para el estudio de este fenómeno en el modo lateral de vibración tema de interés de este trabajo de tesis. Este trabajo incluye simulaciones del fenómeno del “Cross-talk” mediante el método de los elementos finitos (MEF), así como la construcción de un transductor tipo matricial (arrray) de 2x3 elementos, en el que fueron realizadas las mediciones físicas del fenómeno. El trabajo abarca un estudio comparativo entre las simulaciones y las mediciones realizadas en el transductor, considerando que las cerámicas del transductor están montadas sobre diferentes materiales (backing) en donde la propagación de la energía emitida por las cerámicas piezoeléctricas provoca un mayor o menor grado de “Cross-talk” dependiendo de la velocidad en que se propaga dicha energía. Esta investigación también llevó a cabo el estudio del efecto del “Cross-talk” en el patrón de radiación que emite el arreglo de elementos piezoeléctricos, siendo este patrón de radiación un factor importante en la respuesta del transductor, motivo por el cual se realizó un análisis de cómo se ve afectado este patrón bajo la influencia del fenómeno del “Cross-talk”. Como ya se mencionó debido a la falta de un estudio a profundidad del fenómeno del “Cross-talk” en el modo lateral, la contribución del presente trabajo es importante ya que se enfoca al modo lateral de vibración de los elementos piezoeléctricos del arreglo. En particular se desarrollo una ecuación que permite cuantificar el fenómeno del “Cross-talk” y visualizar sus efectos en el arreglo. Derivando de este estudio se concluye que el fenómeno del “Cross-talk” generado por el modo lateral de vibración tiene un efecto significativo en la respuesta de los diferentes transductores matriciales considerados. ABSTRACT This thesis investigates the effect of the phenomenon known as crosstalk from the point of view of its physical nature and the elements that lead to the formation of this phenomenon to an analysis of how it may affect the performance of the ultrasonic transducer. This phenomenon occurs primarily in matrix arrays and this phenomenon is magnified by certain factors causing serious problems in the performance of a transducer. Researchers have addressed the phenomenon of crosstalk and their effects on the response of these transducers. They have mainly focused in the thickness vibration mode, and there has been no comprehensive study of this phenomenon in the lateral vibration mode, issue of interest of this thesis. This work includes simulations of the crosstalk phenomenon using the finite element method (FEM), and the construction of a matrix type transducer (array) of 2x3 elements, in which physical measurements were made. The work includes a comparative study between simulations and measurements in the transducer, whereas the ceramic transducer are mounted on different materials (backing) where the spread of the energy emitted by the piezoelectric ceramic causes a greater or lesser degree of crosstalk depending on the speed at which this energy spreads. This research also carried out the study of the effect of the crosstalk in the radiation pattern emitted by the piezoelectric array. The radiation pattern is an important factor in the response of the transducer that is why we conducted an analysis of how this pattern is affected under the influence of the crosstalk phenomenon. As mentioned before because of the lack of an in-depth study of the crosstalk phenomenon in the lateral vibration mode, the contribution of this work is important because it focuses in this vibration mode of the piezoelectric elements in the array. In particular, an equation was developed to quantify the crosstalk phenomenon and to see its effects in the array. Deriving from this study it is possible to conclude that the crosstalk phenomenon generated by the lateral vibration mode has a significant effect on the response of the different matrix transducers considered in this work.

Elastostatics

In this chapter, we are going to describe the main features as well as the basic steps of the Boundary Element Method (BEM) as applied to elastostatic problems and to compare them with other numerical procedures. As we shall show, it is easy to appreciate the adventages of the BEM, but it is also advisable to refrain from a possible unrestrained enthusiasm, as there are also limitations to its usefulness in certain types of problems. The number of these problems, nevertheless, is sufficient to justify the interest and activity that the new procedure has aroused among researchers all over the world. Briefly speaking, the most frequently used version of the BEM as applied to elastostatics works with the fundamental solution, i.e. the singular solution of the governing equations, as an influence function and tries to satisfy the boundary conditions of the problem with the aid of a discretization scheme which consists exclusively of boundary elements. As in other numerical methods, the BEM was developed thanks to the computational possibilities offered by modern computers on totally "classical" basis. That is, the theoretical grounds are based on linear elasticity theory, incorporated long ago into the curricula of most engineering schools. Its delay in gaining popularity is probably due to the enormous momentum with which Finite Element Method (FEM) penetrated the professional and academic media. Nevertheless, the fact that these methods were developed before the BEM has been beneficial because de BEM successfully uses those results and techniques studied in past decades. Some authors even consider the BEM as a particular case of the FEM while others view both methods as special cases of the general weighted residual technique. The first paper usually cited in connection with the BEM as applied to elastostatics is that of Rizzo, even though the works of Jaswon et al., Massonet and Oliveira were published at about the same time, the reason probably being the attractiveness of the "direct" approach over the "indirect" one. The work of Tizzo and the subssequent work of Cruse initiated a fruitful period with applicatons of the direct BEM to problems of elastostacs, elastodynamics, fracture, etc. The next key contribution was that of Lachat and Watson incorporating all the FEM discretization philosophy in what is sometimes called the "second BEM generation". This has no doubt, led directly to the current developments. Among the various researchers who worked on elastostatics by employing the direct BEM, one can additionallly mention Rizzo and Shippy, Cruse et al., Lachat and Watson, Alarcón et al., Brebbia el al, Howell and Doyle, Kuhn and Möhrmann and Patterson and Sheikh, and among those who used the indirect BEM, one can additionally mention Benjumea and Sikarskie, Butterfield, Banerjee et al., Niwa et al., and Altiero and Gavazza. An interesting version of the indirct method, called the Displacement Discontinuity Method (DDM) has been developed by Crounh. A comprehensive study on various special aspects of the elastostatic BEM has been done by Heisse, while review-type articles on the subject have been reported by Watson and Hartmann. At the present time, the method is well established and is being used for the solution of variety of problems in engineering mechanics. Numerous introductory and advanced books have been published as well as research-orientated ones. In this sense, it is worth noting the series of conferences promoted by Brebbia since 1978, wich have provoked a continuous research effort all over the world in relation to the BEM. In the following sections, we shall concentrate on developing the direct BEM as applied to elastostatics.

An embedded cohesive crack model for finite element analysis of quasi-brittle materials

This paper presents a numerical implementation of the cohesive crack model for the anal-ysis of quasibrittle materials based on the strong discontinuity approach in the framework of the finite element method. A simple central force model is used for the stress versus crack opening curve. The additional degrees of freedom defining the crack opening are determined at the crack level, thus avoiding the need for performing a static condensation at the element level. The need for a tracking algorithm is avoided by using a consistent pro-cedure for the selection of the separated nodes. Such a model is then implemented into a commercial program by means of a user subroutine, consequently being contrasted with the experimental results. The model takes into account the anisotropy of the material. Numerical simulations of well-known experiments are presented to show the ability of the proposed model to simulate the fracture of quasibrittle materials such as mortar, concrete and masonry.

Guía de introducción al Método de los Elementos de Contorno

Entre la impresionante floración de procedimientos de cálculo, provocada por la aplicación intensiva del ordenador, el llamado Método de los Elementos de Contorno (Boundary Element Method o Boundary Integral Equation Method) parece afianzarse como una alternativa útil al omnipresente Método de los Elementos Finitos que ya ha sido incorporado, como una herramienta de trabajo más, al cotidiano quehacer de la ingeniería. En España, tras unos intentos precursores que se señalan en el texto, la actividad más acusada en su desarrollo y mejora se ha centrado alrededor del Departamento que dirige uno de los autores. Después de la tesis doctoral de J. Domínguez en 1977 que introdujo en España la técnica del llamado "método directo", se han producido numerosas aportaciones en forma de artículos o tesis de investigación que han permitido alcanzar un nivel de conocimientos notable. En esta obrita se pretende transmitir parte de la experiencia adquirida, siquiera sea a nivel elemental y en un campo limitado de aplicación. La filosofía es semejante a la del pequeño libro de Hinton y Owen "A simple guide to finite elements" (Pineridge Press, 1980) que tanta aceptación ha tenido entre los principiantes. El libro se articula alrededor de un sólo tema, la solución del problema de Laplace, y se limitan los desarrollos matemáticos al mínimo imprescindible para el fácil seguimiento de áquel. Tras unos capítulos iniciales de motivación y centrado se desarrolla la técnica para problemas planos, tridimensionales y axisimétricos, limitando los razonamientos a los elementos más sencillos de variación constante o lineal. Finalmente, se incluye un capítulo descriptivo donde se avizoran temas que pueden provocar un futuro interés del estudioso. Para completar la información se ha añadido un apéndice en el que se recoge un pequeño programa para microordenador, con el objetivo de que se contemple la sencillez de programación para el caso plano. El programa es mejorable en muchos aspectos pero creemos que, con ello, mantiene un nivel de legibilidad adecuado para que el lector ensaye sobre él las modificaciones que se indican en los ejercicios al final del capítulo y justamente la provocación de ese aprendizaje es nuestro objetivo final.

The Boundary Element Method in elastodynamics

As is well known B.E.M. is obtained as a mixture of the integral representation formula of classical elasticity and the discretization philosophy of the finite element method (F.E.M.). The paper presents the application of B.E.M. to elastodynamic problems. Both the transient and steady state solutions are presented as well as some techniques to simplify problems with a free-stress boundary.

High fidelity simulations of failure in fiber-reinforced composites

Los ensayos virtuales de materiales compuestos han aparecido como un nuevo concepto dentro de la industria aeroespacial, y disponen de un vasto potencial para reducir los enormes costes de certificación y desarrollo asociados con las tediosas campañas experimentales, que incluyen un gran número de paneles, subcomponentes y componentes. El objetivo de los ensayos virtuales es sustituir algunos ensayos por simulaciones computacionales con alta fidelidad. Esta tesis es una contribución a la aproximación multiescala desarrollada en el Instituto IMDEA Materiales para predecir el comportamiento mecánico de un laminado de material compuesto dadas las propiedades de la lámina y la intercara. La mecánica de daño continuo (CDM) formula el daño intralaminar a nivel constitutivo de material. El modelo de daño intralaminar se combina con elementos cohesivos para representar daño interlaminar. Se desarrolló e implementó un modelo de daño continuo, y se aplicó a configuraciones simples de ensayos en laminados: impactos de baja y alta velocidad, ensayos de tracción, tests a cortadura. El análisis del método y la correlación con experimentos sugiere que los métodos son razonablemente adecuados para los test de impacto, pero insuficientes para el resto de ensayos. Para superar estas limitaciones de CDM, se ha mejorado la aproximación discreta de elementos finitos enriqueciendo la cinemática para incluir discontinuidades embebidas: el método extendido de los elementos finitos (X-FEM). Se adaptó X-FEM para un esquema explícito de integración temporal. El método es capaz de representar cualitativamente los mecanismos de fallo detallados en laminados. Sin embargo, los resultados muestran inconsistencias en la formulación que producen resultados cuantitativos erróneos. Por último, se ha revisado el método tradicional de X-FEM, y se ha desarrollado un nuevo método para superar sus limitaciones: el método cohesivo X-FEM estable. Las propiedades del nuevo método se estudiaron en detalle, y se concluyó que el método es robusto para implementación en códigos explícitos dinámicos escalables, resultando una nueva herramienta útil para la simulación de daño en composites. Virtual testing of composite materials has emerged as a new concept within the aerospace industry. It presents a very large potential to reduce the large certification costs and the long development times associated with the experimental campaigns, involving the testing of a large number of panels, sub-components and components. The aim of virtual testing is to replace some experimental tests by high-fidelity numerical simulations. This work is a contribution to the multiscale approach developed in Institute IMDEA Materials to predict the mechanical behavior of a composite laminate from the properties of the ply and the interply. Continuum Damage Mechanics (CDM) formulates intraply damage at the the material constitutive level. Intraply CDM is combined with cohesive elements to model interply damage. A CDM model was developed, implemented, and applied to simple mechanical tests of laminates: low and high velocity impact, tension of coupons, and shear deformation. The analysis of the results and the comparison with experiments indicated that the performance was reasonably good for the impact tests, but insuficient in the other cases. To overcome the limitations of CDM, the kinematics of the discrete finite element approximation was enhanced to include mesh embedded discontinuities, the eXtended Finite Element Method (X-FEM). The X-FEM was adapted to an explicit time integration scheme and was able to reproduce qualitatively the physical failure mechanisms in a composite laminate. However, the results revealed an inconsistency in the formulation that leads to erroneous quantitative results. Finally, the traditional X-FEM was reviewed, and a new method was developed to overcome its limitations, the stable cohesive X-FEM. The properties of the new method were studied in detail, and it was demonstrated that the new method was robust and can be implemented in a explicit finite element formulation, providing a new tool for damage simulation in composite materials.