Os progressos computacionais nas últimas décadas e a teoria matemática cada vez mais sólida têm possibilitado a resolução de problemas de alta complexidade, permitindo uma modelagem cada vez mais detalhada da realidade. Tal verdade aplica-se inclusive para os sistemas rígidos de Equações Diferencias Ordinárias (EDOs): existem métodos numéricos altamente performáticos para este tipo de problema, que permitem uma grande variação no tamanho do passo de integração sem impactar na sua convergência. Este trabalho apresenta um estudo sobre o conceito de rigidez e técnicas numéricas para resolução de problemas rígidos de EDOs. O que nos motivou a estudar tais técnicas foram problemas oriundos da Física Nuclear que envolvem cadeias de decaimento radioativo. Estes problemas podem ser modelados por uma cadeia fechada de compartimentos que se traduz em um sistema de EDOs. Os elementos destas cadeias podem possuir constantes de decaimento com ordens de grandeza muito distintas, caracterizando a sua rigidez e exigindo cautela na resolução das equações que as modelam. Embora seja possível determinar a solução analítica para estes problemas, o uso de métodos numéricos facilita a obtenção da solução quando consideramos sistemas com um número elevado de equações. Além disso, soluções numéricas permitem adaptações na modelagem ou em ajustes de dados com mais facilidade. Métodos implícitos são indicados para a resolução deste tipo de problema, pois possuem uma região de estabilidade ilimitada. Neste trabalho, implementamos dois métodos numéricos que possuem esta característica: o método de Radau II e o método de Rosenbrock. Estes métodos foram utilizados para obtenção de soluções numéricas robustas para problemas rígidos de decaimento radioativo envolvendo cadeias naturais e artificiais, considerando retiradas de elementos das cadeias durante o processo de decaimento e quando queremos determinar qual era o estado inicial de uma cadeia que está em decaimento. Ambos os métodos foram implementados com estratégias de controle do tamanho do passo de integração e produziram resultados consistentes dentro de uma precisão pré-fixada. / The computational progress in the last decades and the increasingly solid mathematical theory have made possible the resolution of highly complex problems allowing an ever more detailed modelling of reality. This is true even for the systems of stiff Ordinary Differential Equations (ODEs): there are highly performative numerical methods for this kind of problem which allow a wide variation in the size of integration step without impacting on their convergence. This thesis presents a study about the concept of stiffness and numerical techniques to solve stiff problems of ODEs. What motivated us to study these techniques were problems from the Nuclear Physics involving radioactive decay chains. These problems could be modelled by a closed chain of compartments which is translated into a system of ODEs. The elements of these chains could have decay constants with very different orders of magnitude which characterizes the stiffness of the problem and requires caution in solving the model equations. Although it is possible to determine the analytical solution to these problems when we consider systems with a high number of equations, calculate the solution by numerical methods becomes easier. Furthermore, numerical solutions allow adaptations in modelling or data adjustments more easily. Implicit methods are indicated to solve this kind of problem because they have an unlimited region of stability. In this study, we implemented two numerical methods which have this feature: Radau II method and Rosenbrock method. These methods were used to obtain robust numerical solutions for stiff problems of radioactive decay involving natural and artificial chains, considering the removal of elements during the decay process and when we want to determine what was the initial state of a chain which is decaying. Both methods were implemented with control strategies for integration step size providing consistent results within a pre-established accuracy.

Os solos sobre-adensados da Formação Guabirotuba principal unidade geológica de Curitiba e região metropolitana são estudados, buscando-se alcançar um melhor entendimento de seu comportamento geomecânico. Uma revisão de aspectos geológicos é realizada, caracterizando-se o contexto regional, a gênese e os tipos de sedimentos da Formação Guabirotuba. Os dados disponíveis e informações referentes a argilas rijas e duras são analisados, no intuito de se identificar possíveis causas de problemas em obras geotécnicas. Ênfase é dada ao papel que as fraturas exercem no comportamento do material. Um amplo trabalho experimental é conduzido no Sítio Experimental de Geotecnia da UFPR, através de ensaios de campo e laboratório. As investigações geotécnicas envolvem desde o tradicional SPT e abertura de poços até ensaios de cone, dilatômetro e pressiômetro autoperfurante. Os trabalhos de campo contemplam o estudo de efeitos de escala na resistência ao cisalhamento do terreno, conduzindo-se ensaios de carregamento dinâmico em estacas escavadas com diferentes diâmetros. No laboratório, a composição e a estrutura do solo são analisadas com técnicas de difratometria de raios-X e microscopia eletrônica. Ensaios de caracterização permitem delinear propriedades índice do Sítio Experimental. Curvas características e ensaios de adensamento de solo indeformado e remoldado fornecem dados que auxiliam no entendimento da história de tensões. Os efeitos de escala, que se devem ao fraturamento do material, são avaliados também com ensaios de cisalhamento direto e triaxiais, utilizando-se amostras de diferentes dimensões. As conclusões da pesquisa permitem identificar possíveis agentes de sobre-adensamento, efeitos do intemperismo químico, fatores que interferem no estado de tensões de campo e mecanismos associados à resistência ao cisalhamento. / The overconsolidated soils of the Guabirotuba Geological Formation are studied in order to achieve a better understanding of its geotechnical behavior. These soils are the main occurrence in the Curitiba Metropolitan Area. A revision of its geology is provided. The genesis and the sediments are described. The available data of the soils and published information about stiff and hard clays are analyzed in order to understand the unexpected behavior of foundations, excavations and slopes. It is given emphasis to the role of the fissures in the material behavior. A comprehensive experimental work is carried out at the Geotechnical Experimentation Site of the UFPR including field and laboratory studies. The geotechnical investigation comprises SPT, exploratory trenches, cone penetration, dilatometer and self boring pressuremeter tests. Scale effects in bored piles are assessed with high strain dynamic tests performed in elements with different cross sections. In the laboratory the composition and the microfabric are analyzed with X-ray diffraction and electron microscopy. Laboratory index soil tests are presented. Consolidation tests and characteristic curves of undisturbed and remolded samples provide information related with the stress history. The scale effects due to the fissures are assessed with direct shear and triaxial testing of specimens with different sizes. The conclusions indicate possible overconsolidation factors, chemical weathering effects and mechanisms affecting the in situ stress state and the shear strength.

Os solos sobre-adensados da Formação Guabirotuba principal unidade geológica de Curitiba e região metropolitana são estudados, buscando-se alcançar um melhor entendimento de seu comportamento geomecânico. Uma revisão de aspectos geológicos é realizada, caracterizando-se o contexto regional, a gênese e os tipos de sedimentos da Formação Guabirotuba. Os dados disponíveis e informações referentes a argilas rijas e duras são analisados, no intuito de se identificar possíveis causas de problemas em obras geotécnicas. Ênfase é dada ao papel que as fraturas exercem no comportamento do material. Um amplo trabalho experimental é conduzido no Sítio Experimental de Geotecnia da UFPR, através de ensaios de campo e laboratório. As investigações geotécnicas envolvem desde o tradicional SPT e abertura de poços até ensaios de cone, dilatômetro e pressiômetro autoperfurante. Os trabalhos de campo contemplam o estudo de efeitos de escala na resistência ao cisalhamento do terreno, conduzindo-se ensaios de carregamento dinâmico em estacas escavadas com diferentes diâmetros. No laboratório, a composição e a estrutura do solo são analisadas com técnicas de difratometria de raios-X e microscopia eletrônica. Ensaios de caracterização permitem delinear propriedades índice do Sítio Experimental. Curvas características e ensaios de adensamento de solo indeformado e remoldado fornecem dados que auxiliam no entendimento da história de tensões. Os efeitos de escala, que se devem ao fraturamento do material, são avaliados também com ensaios de cisalhamento direto e triaxiais, utilizando-se amostras de diferentes dimensões. As conclusões da pesquisa permitem identificar possíveis agentes de sobre-adensamento, efeitos do intemperismo químico, fatores que interferem no estado de tensões de campo e mecanismos associados à resistência ao cisalhamento. / The overconsolidated soils of the Guabirotuba Geological Formation are studied in order to achieve a better understanding of its geotechnical behavior. These soils are the main occurrence in the Curitiba Metropolitan Area. A revision of its geology is provided. The genesis and the sediments are described. The available data of the soils and published information about stiff and hard clays are analyzed in order to understand the unexpected behavior of foundations, excavations and slopes. It is given emphasis to the role of the fissures in the material behavior. A comprehensive experimental work is carried out at the Geotechnical Experimentation Site of the UFPR including field and laboratory studies. The geotechnical investigation comprises SPT, exploratory trenches, cone penetration, dilatometer and self boring pressuremeter tests. Scale effects in bored piles are assessed with high strain dynamic tests performed in elements with different cross sections. In the laboratory the composition and the microfabric are analyzed with X-ray diffraction and electron microscopy. Laboratory index soil tests are presented. Consolidation tests and characteristic curves of undisturbed and remolded samples provide information related with the stress history. The scale effects due to the fissures are assessed with direct shear and triaxial testing of specimens with different sizes. The conclusions indicate possible overconsolidation factors, chemical weathering effects and mechanisms affecting the in situ stress state and the shear strength.

Understanding the bearing response of the footings on layered soils has always been a challenge for researchers. Due to the limitations of analytical and empirical solutions it had been difficult to understand the true bearing behavior. Some researchers have tried solving this problem by numerical analysis and have found some success. In this study the numerical analysis approach has been applied using a commercial tool FLAC (Fast Lagrangian Analysis of Continua) to study the bearing response of surface footings on layered clays. First, small deformation analyses were taken up to study the undrained bearing response of strip and circular footings resting on a horizontally layered strong over a soft clay foundation, and then over soft over strong clay foundation. In the end application of large strain mode of FLAC was explored to investigate the large deformation behavior of the strip footing resting on the surface of a strong over soft clay foundation. All models were run by applying velocity loading and a elastic-perfectly plastic Tresca yield criterion has been used. The results are compared with published Finite Element Method (FEM) results, and with analytical, empirical and semi-empirical solutions. It was found that bearing capacity results from the present small-strain FLAC analyses agree well with the FEM results. However, these results in most of the cases tend to differ (as much as 49% for certain layered clay foundations) from those predicted with analytical, empirical and semi-empirical solutions, mainly due to the assumptions made in these solutions. Since no such assumptions are made in the present FLAC analyses, the results and the methodology of this thesis can be applied to predict the bearing capacity of the practical problems. Application of the large-strain mode of FLAC to study the large deformation of shallow foundations has pointed out a limitation of FLAC in completing such analyses. However, it is observed from the early trends of these analyses that whereas the small deformation analysis may under estimate the ultimate bearing capacity for certain cases of layered foundations where the upper clay is moderately stiffer than the lower clay layer, it might also over predict the ultimate bearing capacity for other cases when the upper clay is very stiff in comparison to the lower clay layer. / February 2007

Electric power has become a basic necessity of human life. The major share of electric power comes from fossil fuel which results in global warming and pollution. A share of generated power comes from nuclear power which is equally dangerous. Big hydro projects take away lots of fertile land. The continuous usage of fossil fuel also poses a threat of petroleum and coal getting over in the near future. The only way out of this energy scarcity is to depend more and more on renewable sources like solar, wind and micro-hydro. At present, instead of having preference over any particular source of renewable energy, effort should be made to extract power from every possible energy source available in whatever form it is and use it in an optimal way. Like any renewable energy sources, the wind power contains large potential for harnessing energy that has been well understood hundreds of years ago. The importance of wind power generation has come to focus recently both at installation and research level and lot of activities are being carried out for efficient use of wind energy. There are different types of wind turbine designs available in the literature. But the most commercially used model is the two or three blade horizontal axis propeller type wind turbine. Research has shown that variable speed operation of this type of turbine is advantageous over fixed speed operation in terms of total energy synthesis. The most commonly used machines for wind power conversion are synchronous machine, squirrel cage induction machine and slip ring induction machine (SRIM). Variable speed operation using synchronous machine or squirrel cage induction machine requires large ratings of the power converters. However, SRIM based variable speed wind generator is advantageous over other schemes due to its inherent advantages like lower power rating for the converters, higher energy capture and the flexibility of sharing reactive power between the stator and the rotor. SRIM is used for both grid connected and stand alone applications and have been reported in the literature. The grid connected applications have received major attention in the literature whereas there are only a very few instances of its stand alone counterparts. There are many places both within and outside India where utility grid has not yet reached or the available grid is very weak. Moreover, in many of the places, the transmission line is so long that the losses in the system are extremely high. Isolated wind power generation can be of great advantage in such places where the available wind power is harnessed and utilized locally. This has been the motivation to go for proposing an isolated wind power generation scheme in this thesis. The proposed scheme is designed to supply power to the load even when very low or no wind power is available. Therefore, a battery bank is also a part of the system. The power converter assembly of the proposed scheme has three major components. One is the rotor side converter which is connected to the rotor terminals of the SRIM. The second one is the stator side converter with output LC filter which is connected to the stator side. These two converters share a common DC link which is interfaced to the battery bank through a multi phase bi-directional fly-back DC-DC converter. Fig. 1. Overall block diagram of the proposed stand alone wind power generator Functionally, this thesis proposes a system as shown in Fig. 1, which has primarily two components with multiple energy ports viz. (i) the SRIM is one triple energy port component and (ii) the proposed power conditioner is another triple energy port component. The SRIM device consists of (i) a mechanical energy port that is interfaced with the windmill shaft (ii) an AC port through the stator windings that is interfaced with the micro-grid/load and (iii) a third port which is also an AC port through the rotor windings of the SRIM that interfaces with an AC port of the proposed power conditioner. The proposed power conditioner is another triple energy port device which consists of (i) a DC energy port that interfaces with a battery/accumulator, (ii) an AC port that interfaces with the rotor windings of the SRIM and (iii) another AC port that generates the micro-grid that is connected to the load and the stator port of the SRIM. The proposed power conditioner provides the frame work for managing the energy flow from the mechanical port of the SRIM to the rotor and accumulator as well as from the mechanical port to the stator/load and accumulator. Further, energy interaction can also take place between the stator and the rotor externally through the power conditioner. The power interfaces on all three energy ports of the proposed power conditioner poses several challenges that have been discussed in this thesis. This thesis focuses on developing schemes to solve these challenges as explained below. Speed sensorless control is a natural choice for slip ring induction machine because of the flexibility of sensing both stator and rotor currents. There are different methods proposed in the literature which deal with the speed sensorless control of slip ring induction machine. However, the elimination of the measurement noise in the flux position estimation is not sufficiently addressed. It is important to address this issue as this would lead to deterioration in rotor side control of SRIM if the measurement noise is not eliminated. Primarily, the schemes which use algebraic relation between the estimated rotor current in stator reference frame and the sensed rotor current, are prone to measurement noise. On the other hand, the schemes, which use rotor back-emf integration, are affected by DC drift problems, though they are not much affected by measurement noise. The proposed stator flux position estimation scheme incorporates the benefits obtained from both the above schemes while eliminating the disadvantages inherent to them. The rotor flux position is estimated by integrating the rotor back-electromotive force. The stator flux is then obtained from the rotor flux estimate. This integration mechanism leads to several problems like dc drift and lack of error decaying mechanism. This estimation scheme solves the above problems including reduction in the propagation of noise in the sensed current to the estimated rotor side unit vectors. On the implementation front, this scheme also eliminates the need for differentiating the unit vectors for estimating slip frequency. This makes the proposed flux estimator very robust. The proposed scheme is simulated and experimentally verified. There is an internal DC bus within the proposed power conditioner that manages the energy flow through the three energy ports. The internal DC bus is interfaced to an external accumulator or battery through a power interface called the multi phase bi-directional dc-dc converter. It is generally advantageous to have the motor rated for higher voltages in order to achieve better efficiencies for a given power rating as compared to low voltage motors. This implies higher DC bus voltage. On the other hand, it is advantageous to have the battery bank rated for low voltage in order to improve the volumetric efficiency which is better at lower battery bank voltages. Both these are contradictory requirements. The above problem is solved in this thesis by proposing a multi power port topology using a bidirectional fly-back converter that is capable of handling multiple power sources and still maintain simplicity and features like high gain, wide load variations and lower output current ripple. As a spin-off, the scheme can handle parallel energy transfer from even a eutectic combination of batteries without any additional control circuitry for parallel operation. Further, the scheme also incorporates a novel transformer winding technique which significantly reduces the leakage inductance of the coupled inductor. The proposed multi-port bidirectional converter is analyzed by including non-idealities like leakage inductance. The DC bus voltage regulation requirement is not very stringent because it is not directly fed to any load. Therefore, hysteresis voltage regulation with small proportional correction is used for DC bus voltage control. The proposed converter is built and experimentally verified in the proposed system as well as in a hybrid-electric vehicle prototype. The third port of the proposed power conditioner interfaces with the stator of the SRIM and the load. The stator/load needs to be connected to a stiff micro-grid. The control requirement of the micro-grid is very stringent because, even for a sudden variation in the wind speed or the load, the grid voltage magnitude and frequency should not change. The dynamic response of the grid voltage controller has to be very fast. Moreover, the grid voltage must be balanced in presence of unbalanced loading. This thesis proposes a converter called the stator side converter along with three phase L-C filter at its output to form the micro-grid. A generalized control scheme is proposed wherein the negative sequence components and the harmonics can be eliminated at the micro-grid by means of feed-forward compensators included in the fundamental positive synchronous reference frame alone. The theoretical foundation for this scheme is developed and discussed in the thesis. In isolated locations linear loads constitute a significant percentage of the total load. Therefore, on the implementation front, only the compensation of fundamental negative sequence is demonstrated. One more necessity for compensating the fundamental negative sequence is that, the SRIM offers only leakage impedance to the fundamental negative sequence components resulting in high fundamental negative sequence current even for a small fundamental negative sequence voltage present at the micro-grid. The proposed scheme ensures balanced three phase currents at the SRIM windings and the full unbalanced current is provided from the stator side converter. This scheme is validated both by simulation and experimentation. The proposed power conditioner is integrated and used in the implementation of the entire wind power generation scheme that is proposed in the thesis. The maximum power point tracking of the wind power unit is also incorporated in the proposed system. The simulation and experimental results are also presented. Finally, the engineering issues involved in the implementation of the proposed scheme are discussed in detail highlighting the hardware configuration and the equipments used. The wind turbine is emulated using a chopper controlled DC motor. The shaft torque of the DC motor is controlled to give the Cp−λ characteristic of a typical windmill. The control issues of the DC machine to behave as a wind turbine are also explained. Finally the thesis is concluded by a statement of potentials and possibilities for future work in this research area.

Understanding the bearing response of the footings on layered soils has always been a challenge for researchers. Due to the limitations of analytical and empirical solutions it had been difficult to understand the true bearing behavior. Some researchers have tried solving this problem by numerical analysis and have found some success. In this study the numerical analysis approach has been applied using a commercial tool FLAC (Fast Lagrangian Analysis of Continua) to study the bearing response of surface footings on layered clays. First, small deformation analyses were taken up to study the undrained bearing response of strip and circular footings resting on a horizontally layered strong over a soft clay foundation, and then over soft over strong clay foundation. In the end application of large strain mode of FLAC was explored to investigate the large deformation behavior of the strip footing resting on the surface of a strong over soft clay foundation. All models were run by applying velocity loading and a elastic-perfectly plastic Tresca yield criterion has been used. The results are compared with published Finite Element Method (FEM) results, and with analytical, empirical and semi-empirical solutions. It was found that bearing capacity results from the present small-strain FLAC analyses agree well with the FEM results. However, these results in most of the cases tend to differ (as much as 49% for certain layered clay foundations) from those predicted with analytical, empirical and semi-empirical solutions, mainly due to the assumptions made in these solutions. Since no such assumptions are made in the present FLAC analyses, the results and the methodology of this thesis can be applied to predict the bearing capacity of the practical problems. Application of the large-strain mode of FLAC to study the large deformation of shallow foundations has pointed out a limitation of FLAC in completing such analyses. However, it is observed from the early trends of these analyses that whereas the small deformation analysis may under estimate the ultimate bearing capacity for certain cases of layered foundations where the upper clay is moderately stiffer than the lower clay layer, it might also over predict the ultimate bearing capacity for other cases when the upper clay is very stiff in comparison to the lower clay layer.

Understanding the bearing response of the footings on layered soils has always been a challenge for researchers. Due to the limitations of analytical and empirical solutions it had been difficult to understand the true bearing behavior. Some researchers have tried solving this problem by numerical analysis and have found some success. In this study the numerical analysis approach has been applied using a commercial tool FLAC (Fast Lagrangian Analysis of Continua) to study the bearing response of surface footings on layered clays. First, small deformation analyses were taken up to study the undrained bearing response of strip and circular footings resting on a horizontally layered strong over a soft clay foundation, and then over soft over strong clay foundation. In the end application of large strain mode of FLAC was explored to investigate the large deformation behavior of the strip footing resting on the surface of a strong over soft clay foundation. All models were run by applying velocity loading and a elastic-perfectly plastic Tresca yield criterion has been used. The results are compared with published Finite Element Method (FEM) results, and with analytical, empirical and semi-empirical solutions. It was found that bearing capacity results from the present small-strain FLAC analyses agree well with the FEM results. However, these results in most of the cases tend to differ (as much as 49% for certain layered clay foundations) from those predicted with analytical, empirical and semi-empirical solutions, mainly due to the assumptions made in these solutions. Since no such assumptions are made in the present FLAC analyses, the results and the methodology of this thesis can be applied to predict the bearing capacity of the practical problems. Application of the large-strain mode of FLAC to study the large deformation of shallow foundations has pointed out a limitation of FLAC in completing such analyses. However, it is observed from the early trends of these analyses that whereas the small deformation analysis may under estimate the ultimate bearing capacity for certain cases of layered foundations where the upper clay is moderately stiffer than the lower clay layer, it might also over predict the ultimate bearing capacity for other cases when the upper clay is very stiff in comparison to the lower clay layer.

Κατά τις υβριδικές δοκιμές δύσκαμπτων κατασκευών, παρουσιάζονται σημαντικά προβλήματα όταν αυτές διεξάγονται με έλεγχο της μετακίνησης των υδραυλικών εμβόλων. Καθ’ ότι στις υβριδικές δοκιμές η δύναμη αντίδρασης σε κάθε βήμα χρησιμοποιείται για την εύρεση της μετακίνησης του επόμενου, το οιοδήποτε λάθος μέτρησής της (εγγενές στις δύσκαμπτες κατασκευές), δρα σωρευτικά οδηγώντας σε αμφιβόλου αξιοπιστίας αποτελέσματα. Επιπροσθέτως, σε πολλές περιπτώσεις δοκιμών δύσκαμπτων κατασκευών παρουσιάζονται φαινόμενα αλληλεπίδρασης μεταξύ των εμβόλων που δυσχεραίνουν τη διεξαγωγή τους. Λύση στο πρόβλημα των δοκιμών δύσκαμπτων κατασκευών μπορεί να δοθεί εφόσον η διεξαγωγή τους γίνεται σε έλεγχο δύναμης αντί της μετακίνησης. Αυτό όμως προϋποθέτει την προγενέστερη γνώση των δυνάμεων που θα επιβληθούν από τα έμβολα σε κάθε βήμα. Οι επικρατέστερες προσεγγίσεις για την εύρεση των δυνάμεων κάθε βήματος βασίζονται είτε στον προσδιορισμό του εφαπτομενικού μητρώου δυσκαμψίας, ή σε διαδοχικές προσεγγίσεις. Στην παρούσα μελέτη παρουσιάζεται μία καινοτόμος προσέγγιση η οποία αποφεύγει τον προσδιορισμό του εφαπτομενικού μητρώου δυσκαμψίας. Χρησιμοποιούνται δύο ελεγκτές PID (αντί για έναν όπως γίνεται συνήθως) για τον έλεγχο κάθε εμβόλου. Ο πρώτος μετατρέπει την στοχευόμενη μετακίνηση σε στοχευόμενη δύναμη, ενώ ο δεύτερος είναι υπεύθυνος για τον έλεγχο του εμβόλου με βάση τη δύναμη που προέρχεται από τον πρώτο. Επίσης, με την υιοθέτηση αυτής της στρατηγικής (στρατηγική διπλού τύπου ελεγκτή) σε συνδυασμό με τη συνεχή ψευδοδυναμική μέθοδο αποφεύγονται οι διαδοχικές προσεγγίσεις που μπορούν να οδηγήσουν σε παρασιτικούς κύκλους φόρτισης – αποφόρτισης. Η εγκυρότητα της στρατηγικής ελέγχθηκε με υβριδικές δοκιμές σε δύσκαμπτο δοκίμιο οπλισμένου σκυροδέματος μεγάλης κλίμακας. Το δοκίμιο χρησιμοποιήθηκε και για την αξιολόγηση της στρατηγικής για την εφαρμογή της σε απλές κυκλικές δοκιμές. Τέλος, παρουσιάζεται το σύστημα τηλεπαρακολούθησης του Εργαστηρίου Κατασκευών, το οποίο αναβαθμίζει το σύστημα ελέγχου δοκιμών και συνδράμει αφενός στη διασφάλιση της ποιότητας των δοκιμών και αφετέρου στη διάχυση των παραγόμενων αποτελεσμάτων στην επιστημονική κοινότητα. / Hybrid simulation of stiff structures encounters significant difficulties when relying on the displacement control of hydraulic actuators. The reason being that in hybrid tests the restoring force in each step is used to calculate the displacement of the next step; therefore a potential error in the calculation (intrinsic in the case of stiff structures) has a cumulative effect, leading to results of questionable validity. Furthermore, in many cases interaction between actuators hinders the execution of tests of stiff structures. A solution to this problem encountered when testing stiff structures may be given if tests are conducted in force control rather than displacement control. This, however, presupposes that the forces to be applied by actuators in each step are known beforehand. Dominant approaches in terms of calculating forces in each step are either based on the secant stiffness matrix, or on iterations. The present doctoral thesis introduces an innovative approach which bypasses the secant stiffness matrix calculation. Two PID controllers are used (instead of one, which is the standard practice) in order to control each actuator. The first controller converts the target displacement to target force, whilst the second one controls the actuator based on the force calculated by the first controller. Moreover, when combining this strategy (dual type control strategy) with the continuous pseudodynamic method, iterations that may lead to unwanted loading-unloading cycles are avoided. The validity of the proposed strategy was assessed by conducting hybrid tests on a large-scale stiff reinforced concrete specimen. The same specimen was also used to assess the application of the strategy to cyclic tests. Lastly, the telepresence system of the Structures Laboratory is presented, which upgrades the test control system and at the same time contributes to the quality of testing and to the distribution of results to the scientific community.

Os progressos computacionais nas últimas décadas e a teoria matemática cada vez mais sólida têm possibilitado a resolução de problemas de alta complexidade, permitindo uma modelagem cada vez mais detalhada da realidade. Tal verdade aplica-se inclusive para os sistemas rígidos de Equações Diferencias Ordinárias (EDOs): existem métodos numéricos altamente performáticos para este tipo de problema, que permitem uma grande variação no tamanho do passo de integração sem impactar na sua convergência. Este trabalho apresenta um estudo sobre o conceito de rigidez e técnicas numéricas para resolução de problemas rígidos de EDOs. O que nos motivou a estudar tais técnicas foram problemas oriundos da Física Nuclear que envolvem cadeias de decaimento radioativo. Estes problemas podem ser modelados por uma cadeia fechada de compartimentos que se traduz em um sistema de EDOs. Os elementos destas cadeias podem possuir constantes de decaimento com ordens de grandeza muito distintas, caracterizando a sua rigidez e exigindo cautela na resolução das equações que as modelam. Embora seja possível determinar a solução analítica para estes problemas, o uso de métodos numéricos facilita a obtenção da solução quando consideramos sistemas com um número elevado de equações. Além disso, soluções numéricas permitem adaptações na modelagem ou em ajustes de dados com mais facilidade. Métodos implícitos são indicados para a resolução deste tipo de problema, pois possuem uma região de estabilidade ilimitada. Neste trabalho, implementamos dois métodos numéricos que possuem esta característica: o método de Radau II e o método de Rosenbrock. Estes métodos foram utilizados para obtenção de soluções numéricas robustas para problemas rígidos de decaimento radioativo envolvendo cadeias naturais e artificiais, considerando retiradas de elementos das cadeias durante o processo de decaimento e quando queremos determinar qual era o estado inicial de uma cadeia que está em decaimento. Ambos os métodos foram implementados com estratégias de controle do tamanho do passo de integração e produziram resultados consistentes dentro de uma precisão pré-fixada. / The computational progress in the last decades and the increasingly solid mathematical theory have made possible the resolution of highly complex problems allowing an ever more detailed modelling of reality. This is true even for the systems of stiff Ordinary Differential Equations (ODEs): there are highly performative numerical methods for this kind of problem which allow a wide variation in the size of integration step without impacting on their convergence. This thesis presents a study about the concept of stiffness and numerical techniques to solve stiff problems of ODEs. What motivated us to study these techniques were problems from the Nuclear Physics involving radioactive decay chains. These problems could be modelled by a closed chain of compartments which is translated into a system of ODEs. The elements of these chains could have decay constants with very different orders of magnitude which characterizes the stiffness of the problem and requires caution in solving the model equations. Although it is possible to determine the analytical solution to these problems when we consider systems with a high number of equations, calculate the solution by numerical methods becomes easier. Furthermore, numerical solutions allow adaptations in modelling or data adjustments more easily. Implicit methods are indicated to solve this kind of problem because they have an unlimited region of stability. In this study, we implemented two numerical methods which have this feature: Radau II method and Rosenbrock method. These methods were used to obtain robust numerical solutions for stiff problems of radioactive decay involving natural and artificial chains, considering the removal of elements during the decay process and when we want to determine what was the initial state of a chain which is decaying. Both methods were implemented with control strategies for integration step size providing consistent results within a pre-established accuracy.

Ce travail de recherche est une contribution au développement de la méthode Décomposition Propre Généralisée (PGD) à la résolution de problèmes de diffusion-réaction raides dédiés à la mécanique des matériaux. Ce type d’équations est notamment rencontré lors de l’oxydation des matériaux polymères et il est donc nécessaire de mettre en place un outil pour simuler ce phénomène afin de prédire numériquement le vieillissement de certains matériaux composites à matrice organique utilisés dans l’aéronautique. La méthode PGD a été choisie dans cette thèse car elle permet un gain en temps de calcul notable par rapport à la méthode des éléments finis. Néanmoins cette famille d’équations n’a jamais été traitée avec cette méthode. Cette dernière se résume à la recherche de solutions d’Équations aux Dérivées Partielles sous forme séparée. Dans le cas d’un problème 1D transitoire, cela revient à chercher la solution sous la forme d’une représentation séparée espace-temps. Dans le cadre de cette thèse, un outil numérique a été mis en place permettant une flexibilité telle que différents algorithmes peuvent être testés. La diffusion Fickienne 1D est tout d’abord évaluée avec en particulier une discussion sur l’utilisation d’un schéma de type Euler ou Runge-Kutta à pas adaptatif pour la détermination des fonctions temporelles. Le schéma de Runge-Kutta permet de réduire notablement le temps de calcul des simulations.Ensuite, la mise en place de l’outil pour les systèmes d’équation de type diffusion-réaction nécessite des algorithmes de résolution de systèmes non linéaires, couplés et raides. Pour cela, différents algorithmes ont été implémentés et discutés.Dans le cas d’un système non linéaire, l’utilisation de la méthode de Newton-Raphson dans les itérations pour la recherche du nouveau mode permet de réduire le temps de calcul en limitant le nombre de modes à considérer pour une erreur donnée. En ce qui concerne les couplages, deux stratégies de résolution ont été évaluées. Le couplage fort mène aux mêmes conclusions que dans le cas non linéaire. Les systèmes raides mais linéaires ont ensuite été traités en implémentant l’algorithme de Rosenbrock pour la détermination des fonctions temporelles. Cet algorithme permet contrairement à Euler et à Runge-Kutta de construire une solution avec un temps de calcul raisonnable liée à l’adaptation du maillage temporel sous-jacent à l’utilisation de cette méthode. La résolution d’un système d’équations de diffusion-réaction raides non linéaires utilisée pour la prédiction de l’oxydation d’un composite issu de la littérature a été testée en utilisant les différents algorithmes mis en place. Néanmoins, les non linéarités et la raideur du système génèrent des équations différentielles intermédiaires à coefficients variables pour lesquelles la méthode de Rosenbrock montre ses limites. Il sera donc nécessaire de tester ou développer d’autres algorithmes pour lever ce verrou.Mots / This work presents the development of the Proper Generalized Decomposition (PGD) method for solving stiff reaction-diffusion equations in the framework of mechanics of materials. These equations are particularly encountered in the oxidation of polymers and it is therefore necessary to develop a tool to simulate this phenomenon for example for the ageing of organic matrix composites in aircraft application. The PGD method has been chosen in this work since it allows a large time saving compared to the finite element method. However this family of equations has never been dealt with this method. The PGD method consists in approximating a solution of a Partial Differential Equation with a separated representation. The solution is sought under a space-time separated representation for a 1D transient equation.In this work, a numerical tool has been developed allowing a flexibility to test different algorithms. The 1D Fickian diffusion is first evaluated and two numerical schemes, Euler and Runge-Kutta adaptive methods, are discussed for the determination of the time modes. The Runge-Kutta method allows a large time saving. The implementation of the numerical tool for reaction-diffusion equations requires the use of specific algorithms dedicated to nonlinearity, couplingand stiffness. For this reason, different algorithms have been implemented and discussed. For nonlinear systems, the use of the Newton-Raphson algorithm at the level of the iterations to compute the new mode allows time saving by decreasing the number of modes required for a given precision. Concerning the couplings, two strategies have been evaluated. The strong coupling leads to the same conclusions as the nonlinear case. The linear stiff systems are then studied by considering a dedicated method, the Rosenbrock method, for the determination of the time modes. This algorithm allows time saving compared to the Runge-Kutta method. The solution of a realistic nonlinear stiff reaction-diffusionsystem used for the prediction of the oxidation of a composite obtained from the literature has been tested by using the various implemented algorithms. However, the nonlinearities and the stiffness of the system generate differential equations with variable coefficients for which the Rosenbrock method is limited. It will be necessary to test or develop other algorithms to overcome this barrier.