Rozprawy doktorskie na temat „Spectral Stochastic Finite Element Method”

We propose a variant of the stochastic finite element method, where the random elements occuring in the problem formulation are approximated by simple random elements, i.e. random elements with only a finite number of possible values.

Following a stochastic approach, this thesis presents a numerical framework for elastostatics of random media. Firstly, after a mathematically rigorous investigation of the popular white noise model in an engineering context, the smooth spatial stochastic dependence between material properties is identified as a fundamental feature of practical random media. Based on the recognition of the probabilistic essence of practical random media and driven by engineering simulation requirements, a comprehensive random medium model, namely elementary random media (ERM), is consequently defined and its macro-scale properties including stationarity, smoothness and principles for material measurements are systematically explored. Moreover, an explicit representation scheme, namely the Fourier-Karhunen-Loeve (F-K-L) representation, is developed for the general elastic tensor of ERM by combining the spectral representation theory of wide-sense stationary stochastic fields and the standard dimensionality reduction technology of principal component analysis. Then, based on the concept of ERM and the F-K-L representation for its random elastic tensor, the stochastic partial differential equations regarding elastostatics of random media are formulated and further discretized, in a similar fashion as for the standard finite element method, to obtain a stochastic system of linear algebraic equations. For the solution of the resulting stochastic linear algebraic system, two different numerical techniques, i.e. the joint diagonalization solution strategy and the directed Monte Carlo simulation strategy, are developed. Original contributions include the theoretical analysis of practical random medium modelling, establishment of the ERM model and its F-K-L representation, and development of the numerical solvers for the stochastic linear algebraic system. In particular, for computational challenges arising from the proposed framework, two novel numerical algorithms are developed: (a) a quadrature algorithm for multidimensional oscillatory functions, which reduces the computational cost of the F-K-L representation by up to several orders of magnitude; and (b) a Jacobi-like joint diagonalization solution method for relatively small mesh structures, which can effectively solve the associated stochastic linear algebraic system with a large number of random variables.

Tato disertační práce vychází z výzkumu v rámci francouzsko-českého programu doktorátu pod dvojím vedením na pracovišti Institut français de mécanique avancée v Clermont-Ferrand a na Ústavu fyziky materiálu AV v Brně. Úvodní výzkumný úkol na brněnském pracovišti se zabýval numerickou analýzou pole napětí v okolí čela trhliny v tenké kovové fólii. Zvláštní pozornost byla zaměřena na vliv speciálního typu singularity v průsečíku čela trhliny s volným povrchem. Těžiště disertační práce spočívá v numerickém modelování a stochastické analýze problémů šíření trhlin se složitou geometrií v dvojrozměrném prostoru. Při analýze těchto problémů se dříve zřídka používaly numerické metody, a to z důvodu vysoké náročnosti na výpočtový čas. V této disertaci je ukázáno, že aplikací moderních metod numerické mechaniky a vhodných technik v analýze spolehlivosti lze tyto problémy řešit s pomocí numerických metod i na PC. Ve spolehlivostní analýze byla využita lineární aproximační metoda FORM. Pro rychlost šíření trhlin se vycházelo z Parisova-Erdoganova vztahu. Pro parametry tohoto vztahu byl použit dvourozměrný statistický model, který postihuje vysokou citlivost na korelaci obou parametrů. Mechanická odezva byla počítána rozšířenou metodou konečných prvků (XFEM), která eliminuje výpočetní náročnost a numerický šum související se změnou sítě v klasické metodě konečných prvků. Prostřednictvím přímé diferenciace bylo odvozeno několik vztahů pro derivace funkce odezvy, čímž se dosáhlo lepší numerické stability a konvergence spolehlivostní analýzy a výrazného zkrácení doby výpočtu. Problém zatížení s proměnou amplitudou byl řešen aplikací transformace zatížení metodou PREFFAS. Využití distribuce výpočtů v síti PC umožnilo další zrychlení analýzy.

Bibliography: p. 113-123.
This thesis investigates probabilistic and stochastic methods for structural analysis which can be integrated into existing, commercially available finite element programs. It develops general probabilistic finite element routines which can be implemented within deterministic finite element programs without requiring major code development. These routines are implemented in the general purpose finite element program ABAQUS through its user element subroutine facility and two probabilistic finite elements are developed: a three-dimensional beam element limited to linear material behaviour and a two-dimensional plane element involving elastic-plastic material behaviour. The plane element incorporates plane strain, plane stress and axisymmetric formulations. The numerical accuracy and robustness of the routines are verified and application of the probabilistic finite element method is illustrated in two case studies, one involving a four-story, two-bay frame structure, the other a reactor pressure vessel nozzle. The probabilistic finite element routines developed in this thesis integrate point estimate methods and mean value first order methods within the same program. Both methods require a systematic sequence involving the perturbation of the random parameters to be evaluated, although the perturbation sequence of the methods differ. It is shown that computer-time saving techniques such as Taylor series and iterative perturbation schemes, developed for mean value based methods, can also be used to solve point estimate method problems. These efficient techniques are limited to linear problems; nonlinear problems must use full perturbation schemes. Finally, it is shown that all these probabilistic methods and perturbation schemes can be integrated within one program and can follow many of the existing deterministic program structures and subroutines. An overall strategy for converting deterministic finite element programs to probabilistic finite element programs is outlined.

An efficient and accurate algorithm is developed to evaluate reliability in the time domain for nonlinear structures subjected to short duration dynamic loadings, including earthquake loading. The algorithm is based on the nonlinear stochastic finite element method (SFEM). Uncertainties in the dynamic and seismic excitation, and resistance-related parameters are incorporated by modeling them as realistically as possible. The uncertainty in them is explicitly addressed. The proposed algorithm intelligently integrates the concepts of response surface method (RSM), finite element method (FEM), first-order reliability method (FORM), and an iterative linear interpolation scheme. This leads to the stochastic finite element concept. It has the potential to estimate the risk associated with any linear or nonlinear structure that can be represented by a finite element algorithm subjected to seismic loading or any short duration dynamic loadings. In the context of the finite element method, the assumed stress-based finite element algorithm is used to increase its efficiency. Two iterative response surface schemes consisting of second order polynomials (with and without cross terms) are proposed. A mixture of saturated and central composite designs is used to assure both efficiency and accuracy of the algorithm. Sensitivity analysis is used to improve the efficiency further. The unique feature of the algorithm is that it is capable of calculating risk using both serviceability and strength limit states and actual earthquake loading time histories can be used to excite structures, enabling a realistic representation of the loading condition. The uncertainty in the amplitude of the earthquake is successfully considered in the context of RSM. Uncertainty in the frequency content of an earthquake is considered indirectly by conducting a parametric study to quantify the effect of uncertainty in the frequency content of earthquakes on the overall reliability of structures. The algorithm has been extensively verified using the Monte Carlo simulation technique. The verified algorithm is used to study the reliability of structures excited by actual earthquake time histories. The results of the numerical examples show that the proposed algorithm can be used accurately and efficiently to estimate the risk for nonlinear structures subjected to short duration time-variant loadings including seismic loading.

The far-field underwater explosion (UNDEX) problem is a complicated problem dominated by two phenomena: the shock wave traveling through the fluid and the cavitation in the fluid. Both of these phenomena have a significant effect on the loading of ship structures subjected to UNDEX. An approach to numerically modeling these effects in the fluid and coupling to a structural model is using cavitating acoustic finite elements (CAFE) and more recently cavitating acoustic spectral elements (CASE). The use of spectral elements in CASE has shown to offer the greater accuracy and reduced computational expense when compared to traditional finite elements. However, spectral elements also increase spurious oscillations in both the fluid and structural response. This dissertation investigates the application of CAFE, CASE, and a possible improvement to CAFE in the form of a finite element flux-corrected transport algorithm, to the far-field UNDEX problem by solving a set of simplified UNDEX problems. Specifically we examine the effect of increased oscillations on structural response and the effect of errors in cavitation capture on the structural response which have not been thoroughly explored in previous work. The main contributions of this work are a demonstration of the problem dependency of increased oscillations in the structural response when applying the CASE methodology, the demonstration of how the sensitivity of errors in the structural response changes with changes in the structural model, a detailed explanation of how error in cavitation capture influences the structural response, and a demonstration of the need to accurately capture the shape and magnitude of cavitation regions in the fluid in order to obtain accurate structural response results.
Ph. D.

An efficient stochastic finite element-based procedure is proposed for the reliability analysis of frame and truss structures with geometrical and material nonlinearities under static loading condition. The material properties, geometry and external loads of the structure are considered as random variables. The failure criteria of the structure are expressed in terms of limit-state functions. The method is based on the advanced first order second moment reliability analysis procedure. The assumed stress field approach is used in the finite element formulation to compute nonlinear structural responses and the corresponding response gradients. The proposed method is suitable for the reliability analysis of geometrically nonlinear frame structures with flexible connections. The mechanical properties of the nonlinear flexible connections can be deterministic or random. A random index parameter is introduced as a basic random variable to consider the uncertainties in the modeling of the connections. Structures with different types of connections can be handled by this method. An efficient method is also proposed for the reliability analysis of highly redundant elastic-perfectly-plastic frame structures with large deformations under proportional loading. The proposed method avoids dealing with the complicated failure mechanisms and stable configurations in the structure system reliability analysis, and has several advantages over the other available methods.

In this study, a stochastic finite element method is used to solve an inverse problem in groundwater flow. The adjoint states method combined with cokriging method is used to estimate the distribution of hydraulic conductivities in an area where hydraulic heads and hydraulic conductivities are measured at some locations. This method starts with obtaining the expected hydraulic heads in the entire study domain at different times. Then, the adjoint states at different times are calculated. Using both calculated values as input, the Jacobians that are needed for the development of covariance matrices of hydraulic heads at different times and the cross-covariance matrices between hydraulic heads at different times and hydraulic conductivities in the aquifer are calculated. Using the maximum likelihood estimate method (MLE), which utilizes all covariance and cross-covariance matrices obtained from the previous step, the statistical parameters (mean, variance, and correlation scale) of the model are estimated. Using the statistical parameter values and all observed values of hydraulic heads at different times and all measured hydraulic conductivities, the distribution of hydraulic conductivity in the entire study domain is estimated. An attempt is made in this thesis to verify the computer program by utilizing two hypothetical problems as verification cases. In some parts of the aquifer, mostly at locations around the observation wells, the resulting hydraulic conductivity distributions have the same pattern with the "true" distribution patterns in both cases of verification. The values of L2-norms calculated by using the "true" and estimated values of log hydraulic conductivity are 0.18 and 0.57 for Case 1 and 2, respectively. Data from two field problems are analyzed as an application of the computer program. The estimated values of hydraulic conductivity are found to be within the range of the observed values given in the original reports. These applications can be considered as a part of the validation of the method. Considering the results of both case studies, it appears that the computer program developed in this study can be used with reasonable success to estimate the hydraulic conductivity distribution in real aquifers. As will be explained in the discussion of the results, however, the effect of zonation needs to be investigated further. (Abstract shortened by UMI.)

A nonlinear stochastic finite element-based procedure is developed for reliability analyses of structures. The procedure is based on the First Order Reliability Method. The failure criteria of structures are expressed in terms of the ultimate and serviceability state functions. The adjoint variable method is used to formulate the computation of the gradient vector. The assumed stress-based finite element method is used to compute nonlinear structural responses and the corresponding response gradients for steel frames. Nonlinearities due to geometry, material and partially restrained connections are considered in the procedure. A computational model based on the Richard model is developed to address the uncertain properties of partially restrained connections. The material properties, geometric properties, connections parameters and external loads are considered as random variables. Several observations with design implications are made from numerical examples. Frames designed considering strength may not be acceptable when serviceability is considered. The presence of partially restrained connections changes the stress distribution in frames and makes frames more flexible so that serviceability could become the governing limit state. It is essential to properly consider the presence of partially restrained connections in the analysis and design of frames. The proposed method can be used as an alternative to the currently available methods to design a structure and evaluate the corresponding reliability. As an extended study, an efficient finite element-based procedure is also developed for estimating nonlinear responses of complex two or three dimensional steel frames with partially restrained connections under dynamic and seismic excitations. The hysteretic behavior of partially restrained connections are modeled by using the Masing rule combined with the Richard model to describe the loading, unloading and reverse loading paths for connections. Numerical examples show that this procedure is accurate and efficient compared with other existing nonlinear methods.

Structural members, such as, stiffeners are applied to aeronautical structures in order to promote the necessary global or local dynamic stiffness. The manufacture and assembly process of these parts and machines capability can introduce variability in the parts and thereby incorporating uncertainties in the structural performance. Tolerances known as GD&T (Geometrical dimensioning and Tolerance), must be specified for several dimensions during the engineering design, in order to enable the manufacture of the structure. In the same way, some material properties have variability, when different suppliers are involved. Thus, the uncertainties must be incorporated into the structural analysis in order to obtain a more reliable design. Therefore, the present thesis applies the SFEM to incorporate these uncertainties in a typical structural member and panel applied in the aeronautical industry. The approach used was Perturbation Technique using Taylor series expansions to incorporate the uncertainties in the structural members typically used in aircraft. Natural frequencies, frequency response functions and modal analysis are studied in order to understand the consequences of these uncertainties in a beam with hat and "Z" section normally applied as an aircraft panel stiffener. The stiffeners were modeled considering the Timoshenko Theory in Matlab software. Sensitivity analyses were applied for the correct interpretations and trends of the top contributors. Also an aircraft panel was performed using Nastran and Matlab software together to incorporate these uncertainties on the modal analysis. Final conclusions and issues of implementation and applicability are performed using MCS as a validation model for the stiffeners and the total range variation for the aircraft panel.

Previous modeling of wood materials has included many assumptions of unknown mechanical properties associated with the hierarchical structure of wood. The experimental validation of previous models did not account for the variation of mechanical properties present in wood materials. Little research has explored the uncertainties of mechanical properties in earlywood and latewood samples as well as wood strands. The goal of this study was to evaluate the effect of the intra-ring properties and grain angles on the modulus of elasticity (MOE) and ultimate tensile strength (UTS) of different orientation wood strands and to analyze the sensitivity of the MOE and UTS of wood strands with respect to these variables. Tension testing incorporating digital image correlation (DIC) was employed to measure the MOE and UTS of earlywood and latewood bands sampled from growth ring numbers 1-10 and growth ring numbers 11-20. A similar technique adjusted for strand size testing was also applied to measure the MOE and UTS of different orientation wood strands from the two growth ring numbers. The stochastic finite element method (SFEM) was used with the results from the earlywood and latewood testing as inputs to model the mechanical property variation of loblolly pine wood strands. A sensitivity analysis of the input parameters in the SFEM model was performed to identify the most important parameters related to mechanical response. Modulus of elasticity (MOE), Poisson ratio, and ultimate tensile strength (UTS) from earlywood and latewood generally increased as the growth ring number increased except for the UTS of latewood, which showed a slight decrease. MOE and UTS from radial, tangential, and angled grain orientation strands increased as the growth ring numbers increased while MOE and UTS from cross-grain strands decreased as the growth ring number increased. Shear modulus of wood strands increased as the growth ring number increased while shear strength decreased as the growth ring number increased. Poisson ratio from radial and angled grain strands decreased as the growth ring number increased while Poisson ratio from tangential and cross grain orientation strands increased as the growth ring number increased. The difference of average MOE from different grain strands between experimental results and SFEM results ranged from 0.96% to 22.31%. The cumulative probability distribution curves from experimental tests and SFEM results agreed well except for the radial grain models from growth ring numbers 11-20. From sensitivity analysis, earlywood MOE was the most important contributing factor to the predicted MOE from different grain orientation strand models. From the sensitivity analysis, earlywood and latewood participated differently in the computation of MOE of different grain orientation strand models. The predicted MOE was highly associated with the strain distribution caused by different orientation strands and interaction of earlywood and latewood properties. In general, earlywood MOE had a greater effect on the predicted MOE of wood strands than other SFEM input parameters. The difference in UTS between experimental and SFEM results ranged from 0.09% to 11.09%. Sensitivity analysis showed that grain orientation and growth ring number influenced the UTS of strands. UTS of strands from growth ring numbers 1-10 showed strength indexes (Xt, Yt, and S) to be the dominant factors while UTS of strands from growth ring numbers 11-20 showed both strength indexes and stress components (Ï 1, Ï 2, and Ï 12) to be the dominant factors. Grain orientations of strands were a strong indicator of mechanical properties of wood strands.
Ph. D.

Much of the vibration in fast moving vehicles is caused bydistributed random excitation, such as turbulent flow and roadroughness. Piping systems transporting fast flowing fluid isanother example, where distributed random excitation will causeunwanted vibration. In order to reduce these vibrations andalso the noise they cause, it is important to have accurate andcomputationally efficient prediction methods available.

The aim of this thesis is to present such a method. Thefirst step towards this end was to extend an existing spectralfinite element method (SFEM) to handle excitation of planetravelling pressure waves. Once the elementary response tothese waves is known, the response to arbitrary homogeneousrandom excitation can be found.

One example of random excitation is turbulent boundary layer(TBL) excitation. From measurements a new modified Chase modelwas developed that allowed for a satisfactory prediction ofboth the measured wall pressure field and the vibrationresponse of a turbulence excited plate. In order to model morecomplicated structures, a new spectral super element method(SSEM) was formulated. It is based on a waveguide formulation,handles all kinds of boundaries and its elements are easily putinto an assembly with conventional finite elements.

Finally, the work to model fluid-structure interaction withanother wave based method is presented. Similar to the previousmethods it seems to be computationally more efficient thanconventional finite elements.

Cette thèse traite de rupture localisée de structures construites en matériau composite hétérogène, comme le béton, à deux échelles différentes. Ces deux échelles sont connectées par le biais de la mise à l'échelle stochastique, où toute information obtenue à l'échelle méso est utilisée comme connaissance préalable à l'échelle macro. À l'échelle méso, le modèle de réseau est utilisé pour représenter la structure multiphasique du béton, à savoir le ciment et les granulats. L'élément de poutre représenté par une poutre Timoshenko 3D intégrée avec de fortes discontinuités assure un maillage complet indépendance de la propagation des fissures. La géométrie de la taille des agrégats est prise en accord avec la courbe EMPA et Fuller tandis que la distribution de Poisson est utilisée pour la distribution spatiale. Les propriétés des matériaux de chaque phase sont obtenues avec une distribution gaussienne qui prend en compte la zone de transition d'interface (ITZ) par l'affaiblissement du béton. À l'échelle macro, un modèle de plasticité multisurface est choisi qui prend en compte à la fois la contribution d'un écrouissage sous contrainte avec une règle d'écoulement non associative ainsi que des composants d'un modèle d'adoucissement de déformation pour un ensemble complet de différents modes de défaillance 3D. Le modèle de plasticité est représenté par le critère de rendement Drucker-Prager, avec une fonction potentielle plastique similaire régissant le comportement de durcissement tandis que le comportement de ramollissement des contraintes est représenté par le critère de St. Venant. La procédure d'identification du modèle macro-échelle est réalisée de manière séquentielle. En raison du fait que tous les ingrédients du modèle à l'échelle macro ont une interprétation physique, nous avons fait l'étalonnage des paramètres du matériau en fonction de l'étape particulière. Cette approche est utilisée pour la réduction du modèle du modèle méso-échelle au modèle macro-échelle où toutes les échelles sont considérées comme incertaines et un calcul de probabilité est effectué. Lorsque nous modélisons un matériau homogène, chaque paramètre inconnu du modèle réduit est modélisé comme une variable aléatoire tandis que pour un matériau hétérogène, ces paramètres de matériau sont décrits comme des champs aléatoires. Afin de faire des discrétisations appropriées, nous choisissons le raffinement du maillage de méthode p sur le domaine de probabilité et la méthode h sur le domaine spatial. Les sorties du modèle avancé sont construites en utilisant la méthode de Galerkin stochastique fournissant des sorties plus rapidement le modèle avancé complet. La procédure probabiliste d'identification est réalisée avec deux méthodes différentes basées sur le théorème de Bayes qui permet d'incorporer de nouvelles bservations générées dans un programme de chargement particulier. La première méthode Markov Chain Monte Carlo (MCMC) est identifiée comme mettant à jour la mesure, tandis que la deuxième méthode Polynomial Chaos Kalman Filter (PceKF) met à jour la fonction mesurable. Les aspects de mise en œuvre des modèles présentés sont donnés en détail ainsi que leur validation à travers les exemples numériques par rapport aux résultats expérimentaux ou par rapport aux références disponibles dans la littérature
This thesis deals with the localized failure for structures built of heterogeneous composite material, such as concrete, at two different scale. These two scale are latter connected through the stochastic upscaling, where any information obtained at meso-scale are used as prior knowledge at macro-scale. At meso scale, lattice model is used to represent the multi-phase structure of concrete, namely cement and aggregates. The beam element represented by 3D Timoshenko beam embedded with strong discontinuities ensures complete mesh independency of crack propagation. Geometry of aggregate size is taken in agreement with EMPA and Fuller curve while Poisson distribution is used for spatial distribution. Material properties of each phase is obtained with Gaussian distribution which takes into account the Interface Transition Zone (ITZ) through the weakening of concrete. At macro scale multisurface plasticity model is chosen that takes into account both the contribution of a strain hardening with non-associative flow rule as well as a strain softening model components for full set of different 3D failure modes. The plasticity model is represented with Drucker-Prager yield criterion, with similar plastic potential function governing hardening behavior while strain softening behavior is represented with St. Venant criterion. The identification procedure for macro-scale model is perfomed in sequential way. Due to the fact that all ingredients of macro-scale model have physical interpretation we made calibration of material parameters relevant to particular stage. This approach is latter used for model reduction from meso-scale model to macro-scale model where all scales are considered as uncertain and probability computation is performed. When we are modeling homogeneous material each unknown parameter of reduced model is modeled as a random variable while for heterogeneous material, these material parameters are described as random fields. In order to make appropriate discretizations we choose p-method mesh refinement over probability domain and h-method over spatial domain. The forward model outputs are constructed by using Stochastic Galerkin method providing outputs more quickly the the full forward model. The probabilistic procedure of identification is performed with two different methods based on Bayes’s theorem that allows incorporating new observation generated in a particular loading program. The first method Markov Chain Monte Carlo (MCMC) is identified as updating the measure, whereas the second method Polynomial Chaos Kalman Filter (PceKF) is updating the measurable function. The implementation aspects of presented models are given in full detail as well as their validation throughthe numerical examples against the experimental results or against the benchmarks available from literature

To obtain predictive modeling of a spacecraft, the author investigates the effects of adding cables to a simple structure with the goal of developing an understanding of how cables interacting with a structure. In this research, the author presents predictive and accurate modeling of a cable-harnessed structure by means of the Spectral Element Method (SEM). A double beam model is used to emulate a cable-harnessed structure. SEM modeling can define the location and the number of connections between the two beams in a convenient fashion. The proposed modeling is applied and compared with the conventional FEM. The modeling approach was compared to and validated by measurement data. The validated modeling was implemented to investigate the effect of the number of connections, of the spring stiffness of interconnections, and of mass portion of an attached cable. Damping has an important role in structural design because it reduces the dynamic response, thereby avoiding excessive deflection or stress, fatigue loads, and settling times. Experimental results with some specimens indicate a clear change of damping on the main structure with the inclusion of cable dynamics. The author investigated the modification of the damping of the host structure induced by various attached cables. The identification of a damping matrix is performed using measured data. The effect of the flexibility of a cable harness on damping is observed through experiments with various types of cables. The effect of the number of connections on damping is also investigated by changing the number of connections. Moreover, to overcome the sensitivity to noise in measured data of damping matrix identification approach, various methods are compared with a simulated lumped model and real test results. An improved damping matrix identification approach is proposed and can generate the unique damping matrix over the full frequency range of interest.
Ph. D.

We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.

This research aims at understanding the conditions that lead to reaction initiation of polymer-bonded explosives (PBXs) as they undergo mechanical and thermal processes subsequent to impact. To analyze this issue, a cohesive finite element method (CFEM) based finite deformation framework is developed and used to quantify the thermomechanical response of PBXs at the microstructure level. This framework incorporates the effects of large deformation, thermomechanical coupling, failure in the forms of micro-cracks in both bulk constituents and along grain/matrix interfaces, and frictional heating. A novel criterion for the ignition of heterogeneous energetic materials under impact loading is developed, which is used to quantify the critical impact velocity, critical time to ignition, and critical input work at ignition for non-shock conditions as functions of microstructure of granular HMX and PBX. A threshold relation between impact velocity and critical input energy at ignition for non-shock loading is developed, involving an energy cutoff and permitting the effects of microstructure and loading to be accounted for. Finally, a novel approach for computationally predicting and quantifying the stochasticity of the ignition process in energetic materials is developed, allowing prediction of the critical time to ignition and the critical impact velocity below which no ignition occurs based on basic material properties and microstructure attributes. Results are cast in the form of the Weibull distribution and used to establish microstructure-ignition behavior relations.

Models which describe the performance of physical process are essential for quality prediction, experimental planning, process control and optimization. Engineering models developed based on the underlying physics/mechanics of the process such as analytic models or finite element models are widely used to capture the deterministic trend of the process. However, there usually exists stochastic randomness in the system which may introduce the discrepancy between physics-based model predictions and observations in reality. Alternatively, statistical models can be used to develop models to obtain predictions purely based on the data generated from the process. However, such models tend to perform poorly when predictions are made away from the observed data points. This dissertation contributes to model enhancement research by integrating physics-based model and statistical model to mitigate the individual drawbacks and provide models with better accuracy by combining the strengths of both models. The proposed model enhancement methodologies including the following two streams: (1) data-driven enhancement approach and (2) engineering-driven enhancement approach. Through these efforts, more adequate models are obtained, which leads to better performance in system forecasting, process monitoring and decision optimization. Among different data-driven enhancement approaches, Gaussian Process (GP) model provides a powerful methodology for calibrating a physical model in the presence of model uncertainties. However, if the data contain systematic experimental errors, the GP model can lead to an unnecessarily complex adjustment of the physical model. In Chapter 2, we proposed a novel enhancement procedure, named as "Minimal Adjustment", which brings the physical model closer to the data by making minimal changes to it. This is achieved by approximating the GP model by a linear regression model and then applying a simultaneous variable selection of the model and experimental bias terms. Two real examples and simulations are presented to demonstrate the advantages of the proposed approach. Different from enhancing the model based on data-driven perspective, an alternative approach is to focus on adjusting the model by incorporating the additional domain or engineering knowledge when available. This often leads to models that are very simple and easy to interpret. The concepts of engineering-driven enhancement are carried out through two applications to demonstrate the proposed methodologies. In the first application where polymer composite quality is focused, nanoparticle dispersion has been identified as a crucial factor affecting the mechanical properties. Transmission Electron Microscopy (TEM) images are commonly used to represent nanoparticle dispersion without further quantifications on its characteristics. In Chapter 3, we developed the engineering-driven nonhomogeneous Poisson random field modeling strategy to characterize nanoparticle dispersion status of nanocomposite polymer, which quantitatively represents the nanomaterial quality presented through image data. The model parameters are estimated through the Bayesian MCMC technique to overcome the challenge of limited amount of accessible data due to the time consuming sampling schemes. The second application is to calibrate the engineering-driven force models of laser-assisted micro milling (LAMM) process statistically, which facilitates a systematic understanding and optimization of targeted processes. In Chapter 4, the force prediction interval has been derived by incorporating the variability in the runout parameters as well as the variability in the measured cutting forces. The experimental results indicate that the model predicts the cutting force profile with good accuracy using a 95% confidence interval. To conclude, this dissertation is the research drawing attention to model enhancement, which has considerable impacts on modeling, design, and optimization of various processes and systems. The fundamental methodologies of model enhancement are developed and further applied to various applications. These research activities developed engineering compliant models for adequate system predictions based on observational data with complex variable relationships and uncertainty, which facilitate process planning, monitoring, and real-time control.

Thesis (Ph. D.)--Ohio State University, 2004.
Title from first page of PDF file. Document formatted into pages; contains xviii, 220 p.; also includes graphics (some col.). Includes abstract and vita. Advisor: R. Shivpuri, Dept. of Industrial, Welding and Systems Engineering. Includes bibliographical references.

Orientador: José Maria Campos dos Santos
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Este trabalho tem como objetivo geral investigar abordagens para a análise de incerteza em problemas de dinâmica estrutural, de forma computacionalmente eficiente, no contexto industrial. Neste sentido, utilizou-se um metamodelo, baseado no método da superfície de resposta, para simplificar a etapa do cálculo dos modos e das frequências naturais na análise de resposta em frequência da estrutura. Para viabilizar a análise de grandes modelos, a solução de elementos finitos foi realizada pelo Nastran®. O MatLab® foi utilizado para manipular os autovalores e autovetores, e calcular as FRFs. Já o processo de amostragem das variáveis, a preparação da superfície de resposta e a integração com os demais aplicativos, foram realizados por meio do Isight®. Inicialmente, a abordagem foi avaliada em um modelo simples de um para-brisa veicular, com espessura, modo de elasticidade e densidade como parâmetros incertos. Posteriormente, o método foi aplicado para um modelo de uma estrutura veicular com milhares graus de liberdade. Neste caso, as variáveis aleatórias consideradas foram espessuras de vinte peças estampadas. Todas as variáveis foram consideradas com distribuição normal. Para quantificar a incerteza na resposta dinâmica, a simulação por Monte Carlo foi conduzida em conjunto com o metamodelo. A variabilidade das frequências naturais e da FRF é comparada com o resultado do Monte Carlo direto
Abstract: This work has as general objective to investigate approaches for uncertainty analysis in structural dynamics problems in a computational efficient manner in an industrial context. In this sense, we used a metamodel based on the response surface method to simplify the process of modes and natural frequencies calculation for frequency response analysis of a structure. In order to make the process feasible for large models, the finite element solution was performed using Nastran®. MatLab® was used to manipulate the eigenvalues and eigenvectors and calculate the FRFs. Isight® was responsible for the variable sampling process, response surface preparation and integrating other applications as well. Initially, the approach was assessed in a simple model of a car windshield with its thickness, Young¿s modulus and material density as uncertain parameters. Later the method was applied to a vehicle structure model with thousands degrees of freedom. In this case, the random variables considered were thicknesses of twenty stamped parts. Gaussian distribution was considered for all variables. For the purpose of uncertainty quantification in the dynamic response, Monte Carlo simulation was performed over the metamodel. The variability of the natural frequencies and FRF is compared against to direct Monte Carlo results
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica

In our age in which the use of electronic devices is expanding all over the world, their reliability and miniaturization have become very crucial. The thesis is based on the study of one of the most frequent failure mechanisms in semiconductor packages, the delamination of interface or the separation of two bonded materials, in order to improve their adhesion and a fortiori the reliability of microelectronic devices. It focuses on the metal (-oxide) / polymer interfaces because they cover 95% of all existing interfaces. Since several years, research activities at mesoscopic scale (1-10µm) have proved that the more roughened the surface of the interface, i.e., presenting sharp asperities, the better the adhesion between these two materials. Because roughness exhibits extremely complex shapes, it is difficult to find a description that can be used for reliability analysis of interfaces. In order to investigate quantitatively the effect of roughness variation on adhesion properties, studies have been carried out involving analytical fracture mechanics; then numerical studies were conducted with Finite Element Analysis. Both were done in a deterministic way by assuming an ideal profile which is repeated periodically. With the development of statistical and stochastic roughness representation on the one hand, and with the emergence of probabilistic fracture mechanics on the other, the present work adds a stochastic framework to the previous studies. In fact, one of the Stochastic Finite Element Methods, the Perturbation method is chosen for implementation, because it can investigate the effect of the geometric variations on the mechanical response such as displacement field. In addition, it can carry out at once what traditional Finite Element Analysis does with numerous simulations which require changing geometric parameters each time. This method is developed analytically, then numerically by implementing a module in a Finite Element package MSc. Marc/Mentat. In order to get acquainted and to validate the implementation, the Perturbation method is applied analytically and numerically to the 3 point bending test on a beam problem, because the input of the Perturbation method in terms of roughness parameters is still being studied. The capabilities and limitations of the implementation are outlined. Finally, recommendations for using the implementation and for furture work on roughness representation are discussed.

The behavior of structures is totally different when they are exposed to fluctuating loading rather than static one which is a well known phenomenon in engineering called fatigue. When the loading is not static but dynamic, the dynamics of the structure should be taken into account since there is a high possibility to excite the resonance frequencies of the structure especially if the loading frequency has a wide bandwidth. In these cases, the structure&rsquo
s response to the loading will not be linear. Therefore, in the analysis of such situations, frequency domain fatigue analysis techniques are used which take the dynamic properties of the structure into consideration. Vibration fatigue method is also fast, functional and easy to implement. In this thesis, vibration fatigue theory is examined. Throughout the research conducted for this study, the ultimate aim is to find solutions to problems arising from test application for the loadings with nonzero mean value bringing a new perspective to mean stress correction techniques. A new method is developed to generate a modified input loading history with a zero mean value which leads in fatigue damage approximately equivalent to damage induced by input loading with a nonzero mean value. A mathematical procedure is proposed to implement mean stress correction to the output stress power spectral density data and a modified input loading power spectral density data is obtained. Furthermore, this method is improved for multiaxial loading applications. A loading history power spectral density set with zero mean but modified alternating stress, which leads in fatigue damage approximately equivalent to the damage caused by the unprocessed loading set with nonzero mean, is extracted taking all stress components into account using full matrixes. The proposed techniques&rsquo
efficiency is discussed throughout several case studies and fatigue tests.

Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. Analytical solutions can be derived only for the simplest models and numerical simulations are necessary in most cases to evaluate the models and their properties and to compare them with measured data. This thesis focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution. The second part of this thesis investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.

In this thesis, simulations are performed to study the motion ofindividual cells in flow, focusing on the hydrodynamics of actively swimming cells likethe self-propelling microorganisms, and of passively advected objects like the red bloodcells. In particular, we develop numerical tools to address the locomotion ofmicroswimmers in viscoelastic fluids and complex geometries, as well as the motion ofdeformable capsules in micro-fluidic flows. For the active movement, the squirmer is used as our model microswimmer. The finiteelement method is employed to study the influence of the viscoelasticity of fluid on theperformance of locomotion. A boundary element method is implemented to study swimmingcells inside a tube. For the passive counterpart, the deformable capsule is chosen as the modelcell. An accelerated boundary integral method code is developed to solve thefluid-structure interaction, and a global spectral method is incorporated to handle theevolving cell surface and its corresponding membrane dynamics. We study the locomotion of a neutral squirmer with anemphasis on the change of swimming kinematics, energetics, and flowdisturbance from Newtonian to viscoelastic fluid. We also examine the dynamics of differentswimming gaits resulting in different patterns of polymer deformation, as well as theirinfluence on the swimming performance. We correlate the change of swimming speed withthe extensional viscosity and that of power consumption with the phase delay of viscoelasticfluids. Moreover, we utilise the boundary element method to simulate the swimming cells in astraight and torus-like bent tube, where the tube radius is a few times the cell radius. Weinvestigate the effect of tube confinement to the swimming speed and power consumption. Weanalyse the motions of squirmers with different gaits, which significantly affect thestability of the motion. Helical trajectories are produced for a neutralsquirmer swimming, in qualitative agreement with experimental observations, which can beexplained by hydrodynamic interactions alone. We perform simulations of a deformable capsule in micro-fluidic flows. We look atthe trajectory and deformation of a capsule through a channel/duct with a corner. Thevelocity of capsule displays an overshoot as passing around the corner, indicating apparentviscoelasticity induced by the interaction between the deformable membrane and viscousflow. A curved corner is found to deform the capsule less than the straight one. In addition, we propose a new cell sorting device based on the deformability of cells. Weintroduce carefully-designed geometric features into the flow to excite thehydrodynamic interactions between the cell and device. This interaction varies andclosely depends on the cell deformability, the resultant difference scatters the cellsonto different trajectories. Our high-fidelity computations show that the new strategy achievesa clear and robust separation of cells. We finally investigate the motion of capsule in awall-bounded oscillating shear flow, to understand the effect of physiological pulsation to thedeformation and lateral migration of cells. We observe the lateral migration velocity of a cellvaries non-monotonically with its deformability.

QC 20140313

Uncertainties in load and system properties play a significant role in reliability analysis of vibrating structural systems. The subject of random vibrations has evolved over the last few decades to deal with uncertainties in external loads. A well developed body of literature now exists which documents the status of this subject. Studies on the influ­ence of system property uncertainties on reliability of vibrating structures is, however, of more recent origin. Currently, the problem of dynamic response characterization of sys­tems with parameter uncertainties has emerged as a subject of intensive research. The motivation for this research activity arises from the need for a more accurate assess­ment of the safety of important and high cost structures like nuclear plant installations, satellites and long span bridges. The importance of the problem also lies in understand­ing phenomena like mode localization in nearly periodic structures and deviant system behaviour at high frequencies. It is now well established that these phenomena are strongly influenced by spatial imperfections in the vibrating systems. Design codes, as of now, are unable to systematically address the influence of scatter and uncertainties. Therefore, there is a need to develop robust design algorithms based on the probabilistic description of the uncertainties, leading to safer, better and less over-killed designs. Analysis of structures with parameter uncertainties is wrought with diffi­culties, which primarily arise because the response variables are nonlinearly related to the stochastic system parameters; this being true even when structures are idealized to display linear material and deformation characteristics. The problem is further com­pounded when nonlinear structural behaviour is included in the analysis. The analysis of systems with parameter uncertainties involves modeling of random fields for the system parameters, discretization of these random fields, solutions of stochastic differential and algebraic eigenvalue problems, inversion of random matrices and differential operators, and the characterization of random matrix products. It should be noted that the mathematical nature of many of these problems is substantially different from those which are encountered in the traditional random vibration analysis. The basic problem lies in obtaining the solution of partial differential equations with random coefficients which fluctuate in space. This has necessitated the development of methods and tools to deal with these newer class of problems. An example of this development is the generalization of the finite element methods of structural analysis to encompass problems of stochastic material and geometric characteristics. The present thesis contributes to the development of methods and tools to deal with structural uncertainties in the analysis of vibrating structures. This study is a part of an ongoing research program in the Department, which is aimed at gaining insights into the behaviour of randomly parametered dynamical systems and to evolve computational methods to assess the reliability of large scale engineering structures. Recent studies conducted in the department in this direction, have resulted in the for­mulation of the stochastic dynamic stiffness matrix for straight Euler-Bernoulli beam elements and these results have been used to investigate the transient and the harmonic steady state response of simple built-up structures. In the present study, these earlier formulations are extended to derive the stochastic dynamic stiffness matrix for a more general beam element, namely, the curved Timoshenko beam element. Furthermore, the method has also been extended to study the mean and variance of the stationary response of built-up structures when excited by stationary stochastic forces. This thesis is organized into five chapters and four appendices. The first chapter mainly contains a review of the developments in stochas­tic finite element method (SFEM). Also presented is a brief overview of the dynamics of curved beams and the essence of the dynamic stiffness matrix method. This discussion also covers issues pertaining to modeling rotary inertia and shear deformations in the study of curved beam dynamics. In the context of SFEM, suitability of different methods for modeling system uncertainties, depending on the type of problem, is discussed. The relative merits of several schemes of discretizing random fields, namely, local averaging, series expansions using orthogonal functions, weighted integral approach and the use of system Green functions, are highlighted. Many of the discretization schemes reported in the literature have been developed in the context of static problems. The advantages of using the dynamic stiffness matrix approach in conjunction with discretization schemes based on frequency dependent shape functions, are discussed. The review identifies the dynamic analysis of structures built-up of randomly parametered curved beams, using dynamic stiffness matrix method, as a problem requiring further research. The review also highlights the need for studies on the treatment of non-Gaussian nature of system parameters within the framework of stochastic finite element analysis and simulation methods. The problem of deterministic analysis of curved beam elements is consid­ered first. Chapter 2 reports on the development of the dynamic stiffness matrix for a curved Timoshenko beam element. It is shown that when the beam is uniformly param-etered, the governing field equations can be solved in a closed form. These closed form solutions serve as the basis for the formulation of damping and frequency dependent shape functions which are subsequently employed in the thesis to develop the dynamic stiffness matrix of stochastically inhomogeneous, curved beams. On the other hand, when the beam properties vary spatially, the governing equations have spatially varying coefficients which discount the possibility of closed form solutions. A numerical scheme to deal with this problem is proposed. This consists of converting the governing set of boundary value problems into a larger class of equivalent initial value problems. This set of Initial value problems can be solved using numerical schemes to arrive at the element dynamic stiffness matrix. This algorithm forms the basis for Monte Carlo simulation studies on stochastic beams reported later in this thesis. Numerical results illustrating the formulations developed in this chapter are also presented. A satisfactory agreement of these results has been demonstrated with the corresponding results obtained from independent finite element code using normal mode expansions. The formulation of the dynamic stiffness matrix for a curved, randomly in-homogeneous, Timoshenko beam element is considered in Chapter 3. The displacement fields are discretized using the frequency dependent shape functions derived in the pre­vious chapter. These shape functions are defined with respect to a damped, uniformly parametered beam element and hence are deterministic in nature. Lagrange's equations are used to derive the 6x6 stochastic dynamic stiffness matrix of the beam element. In this formulation, the system property random fields are implicitly discretized as a set of damping and frequency dependent Weighted integrals. The results for a straight Timo- shenko beam are obtained as a special case. Numerical examples on structures made up of single curved/straight beam elements are presented. These examples also illustrate the characterization of the steady state response when excitations are modeled as stationary random processes. Issues related to ton-Gaussian features of the system in-homogeneities are also discussed. The analytical results are shown to agree satisfactorily with corresponding results from Monte Carlo simulations using 500 samples. The dynamics of structures built-up of straight and curved random Tim-oshenko beams is studied in Chapter 4. First, the global stochastic dynamic stiffness matrix is assembled. Subsequently, it is inverted for calculating the mean and variance, of the steady state stochastic response of the structure when subjected to stationary random excitations. Neumann's expansion method is adopted for the inversion of the stochastic dynamic stiffness matrix. Questions on the treatment of the beam characteris­tics as non-Gaussian random fields, are addressed. It is shown that the implementation of Neumann's expansion method and Monte-Carlo simulation method place distinc­tive demands on strategy of modeling system parameters. The Neumann's expansion method, on one hand, requires the knowledge of higher order spectra of beam properties so that the non-Gaussian features of beam parameters are reflected in the analysis. On the other hand, simulation based methods require the knowledge of the range of the stochastic variations and details of the probability density functions. The expediency of implementing Gaussian closure approximation in evaluating contributions from higher order terms in the Neumann expansion is discussed. Illustrative numerical examples comparing analytical and Monte-Carlo simulations are presented and the analytical so­lutions are found to agree favourably with the simulation results. This agreement lends credence to the various approximations involved in discretizing the random fields and inverting the global dynamic stiffness matrix. A few pointers as to how the methods developed in the thesis can be used in assessing the reliability of these structures are also given. A brief summary of contributions made in the thesis together with a few suggestions for further research are presented in Chapter 5. Appendix A describes the models of non-Gaussian random fields employed in the numerical examples considered in this thesis. Detailed expressions for the elements of the covariance matrix of the weighted integrals for the numerical example considered in Chapter 5, are presented in Appendix B; A copy of the paper, which has been ac­cepted for publication in the proceedings of IUTAM symposium on 'Nonlinearity and Stochasticity in Structural Mechanics' has been included as Appendix C.

Uncertainties in load and system properties play a significant role in reliability analysis of vibrating structural systems. The subject of random vibrations has evolved over the last few decades to deal with uncertainties in external loads. A well developed body of literature now exists which documents the status of this subject. Studies on the influ­ence of system property uncertainties on reliability of vibrating structures is, however, of more recent origin. Currently, the problem of dynamic response characterization of sys­tems with parameter uncertainties has emerged as a subject of intensive research. The motivation for this research activity arises from the need for a more accurate assess­ment of the safety of important and high cost structures like nuclear plant installations, satellites and long span bridges. The importance of the problem also lies in understand­ing phenomena like mode localization in nearly periodic structures and deviant system behaviour at high frequencies. It is now well established that these phenomena are strongly influenced by spatial imperfections in the vibrating systems. Design codes, as of now, are unable to systematically address the influence of scatter and uncertainties. Therefore, there is a need to develop robust design algorithms based on the probabilistic description of the uncertainties, leading to safer, better and less over-killed designs. Analysis of structures with parameter uncertainties is wrought with diffi­culties, which primarily arise because the response variables are nonlinearly related to the stochastic system parameters; this being true even when structures are idealized to display linear material and deformation characteristics. The problem is further com­pounded when nonlinear structural behaviour is included in the analysis. The analysis of systems with parameter uncertainties involves modeling of random fields for the system parameters, discretization of these random fields, solutions of stochastic differential and algebraic eigenvalue problems, inversion of random matrices and differential operators, and the characterization of random matrix products. It should be noted that the mathematical nature of many of these problems is substantially different from those which are encountered in the traditional random vibration analysis. The basic problem lies in obtaining the solution of partial differential equations with random coefficients which fluctuate in space. This has necessitated the development of methods and tools to deal with these newer class of problems. An example of this development is the generalization of the finite element methods of structural analysis to encompass problems of stochastic material and geometric characteristics. The present thesis contributes to the development of methods and tools to deal with structural uncertainties in the analysis of vibrating structures. This study is a part of an ongoing research program in the Department, which is aimed at gaining insights into the behaviour of randomly parametered dynamical systems and to evolve computational methods to assess the reliability of large scale engineering structures. Recent studies conducted in the department in this direction, have resulted in the for­mulation of the stochastic dynamic stiffness matrix for straight Euler-Bernoulli beam elements and these results have been used to investigate the transient and the harmonic steady state response of simple built-up structures. In the present study, these earlier formulations are extended to derive the stochastic dynamic stiffness matrix for a more general beam element, namely, the curved Timoshenko beam element. Furthermore, the method has also been extended to study the mean and variance of the stationary response of built-up structures when excited by stationary stochastic forces. This thesis is organized into five chapters and four appendices. The first chapter mainly contains a review of the developments in stochas­tic finite element method (SFEM). Also presented is a brief overview of the dynamics of curved beams and the essence of the dynamic stiffness matrix method. This discussion also covers issues pertaining to modeling rotary inertia and shear deformations in the study of curved beam dynamics. In the context of SFEM, suitability of different methods for modeling system uncertainties, depending on the type of problem, is discussed. The relative merits of several schemes of discretizing random fields, namely, local averaging, series expansions using orthogonal functions, weighted integral approach and the use of system Green functions, are highlighted. Many of the discretization schemes reported in the literature have been developed in the context of static problems. The advantages of using the dynamic stiffness matrix approach in conjunction with discretization schemes based on frequency dependent shape functions, are discussed. The review identifies the dynamic analysis of structures built-up of randomly parametered curved beams, using dynamic stiffness matrix method, as a problem requiring further research. The review also highlights the need for studies on the treatment of non-Gaussian nature of system parameters within the framework of stochastic finite element analysis and simulation methods. The problem of deterministic analysis of curved beam elements is consid­ered first. Chapter 2 reports on the development of the dynamic stiffness matrix for a curved Timoshenko beam element. It is shown that when the beam is uniformly param-etered, the governing field equations can be solved in a closed form. These closed form solutions serve as the basis for the formulation of damping and frequency dependent shape functions which are subsequently employed in the thesis to develop the dynamic stiffness matrix of stochastically inhomogeneous, curved beams. On the other hand, when the beam properties vary spatially, the governing equations have spatially varying coefficients which discount the possibility of closed form solutions. A numerical scheme to deal with this problem is proposed. This consists of converting the governing set of boundary value problems into a larger class of equivalent initial value problems. This set of Initial value problems can be solved using numerical schemes to arrive at the element dynamic stiffness matrix. This algorithm forms the basis for Monte Carlo simulation studies on stochastic beams reported later in this thesis. Numerical results illustrating the formulations developed in this chapter are also presented. A satisfactory agreement of these results has been demonstrated with the corresponding results obtained from independent finite element code using normal mode expansions. The formulation of the dynamic stiffness matrix for a curved, randomly in-homogeneous, Timoshenko beam element is considered in Chapter 3. The displacement fields are discretized using the frequency dependent shape functions derived in the pre­vious chapter. These shape functions are defined with respect to a damped, uniformly parametered beam element and hence are deterministic in nature. Lagrange's equations are used to derive the 6x6 stochastic dynamic stiffness matrix of the beam element. In this formulation, the system property random fields are implicitly discretized as a set of damping and frequency dependent Weighted integrals. The results for a straight Timo- shenko beam are obtained as a special case. Numerical examples on structures made up of single curved/straight beam elements are presented. These examples also illustrate the characterization of the steady state response when excitations are modeled as stationary random processes. Issues related to ton-Gaussian features of the system in-homogeneities are also discussed. The analytical results are shown to agree satisfactorily with corresponding results from Monte Carlo simulations using 500 samples. The dynamics of structures built-up of straight and curved random Tim-oshenko beams is studied in Chapter 4. First, the global stochastic dynamic stiffness matrix is assembled. Subsequently, it is inverted for calculating the mean and variance, of the steady state stochastic response of the structure when subjected to stationary random excitations. Neumann's expansion method is adopted for the inversion of the stochastic dynamic stiffness matrix. Questions on the treatment of the beam characteris­tics as non-Gaussian random fields, are addressed. It is shown that the implementation of Neumann's expansion method and Monte-Carlo simulation method place distinc­tive demands on strategy of modeling system parameters. The Neumann's expansion method, on one hand, requires the knowledge of higher order spectra of beam properties so that the non-Gaussian features of beam parameters are reflected in the analysis. On the other hand, simulation based methods require the knowledge of the range of the stochastic variations and details of the probability density functions. The expediency of implementing Gaussian closure approximation in evaluating contributions from higher order terms in the Neumann expansion is discussed. Illustrative numerical examples comparing analytical and Monte-Carlo simulations are presented and the analytical so­lutions are found to agree favourably with the simulation results. This agreement lends credence to the various approximations involved in discretizing the random fields and inverting the global dynamic stiffness matrix. A few pointers as to how the methods developed in the thesis can be used in assessing the reliability of these structures are also given. A brief summary of contributions made in the thesis together with a few suggestions for further research are presented in Chapter 5. Appendix A describes the models of non-Gaussian random fields employed in the numerical examples considered in this thesis. Detailed expressions for the elements of the covariance matrix of the weighted integrals for the numerical example considered in Chapter 5, are presented in Appendix B; A copy of the paper, which has been ac­cepted for publication in the proceedings of IUTAM symposium on 'Nonlinearity and Stochasticity in Structural Mechanics' has been included as Appendix C.

The central theme of this thesis is the design of optimal balanced black-box stopping criteria in iterative solvers of symmetric positive-definite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimally-by avoiding unnecessary computations and premature stopping-algebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no user-defined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a black-box iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced black-box stopping tests in MINRES solver for solving symmetric positive-definite and symmetric indefinite linear systems can be estimated cheaply on-the- fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced black-box stopping test in a memory-expensive Krylov solver like GMRES but it also presents an optimal balanced black-box stopping test in memory-inexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memory-inexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced black-box stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional Navier-Stokes equations. The optimal balanced black-box stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.

The diploma thesis deals with penalty approach to stochastic optimization with chance constraints which are applied to structural mechanics. The problem of optimal design of beam dimensions is modeled and solved. The uncertainty is involved in the form of random load. The corresponding mathematical model contains a condition in the form of ordinary differencial equation that is solved by finite element method. The probability condition is approximated by several types of penalty functions. The results are obtained by computations in the MATLAB software.

Orientador: José Maria Campos dos Santos
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
Made available in DSpace on 2018-08-27T14:11:06Z (GMT). No. of bitstreams: 1 Nobrega_EdilsonDantas_M.pdf: 12912450 bytes, checksum: 790ec696a10ed45d5c22747b666dea61 (MD5) Previous issue date: 2015
Resumo: A fim de superar as limitações atuais na análise dinâmica de estruturas em médias e altas frequências e tirando proveito da natureza periódica de muitas destas estruturas, nos últimos anos, foram desenvolvidos métodos de guia de ondas. São modelos obtidos a partir de fatias das guia de ondas e modeladas pelo Método dos Elementos Finitos (MEF) e pelo Método do Elemento Espectral (SEM), também conhecidos como Método de Propagação de Ondas por Elementos Finitos (Wave Finite Element Method - WFEM) e Método de Propagação de Ondas por Elementos Espectrais (Wave Spectral Element Method - WSEM), respectivamente. Exemplos de guia de ondas podem ser encontrados em diferentes tipos de estruturas tais como, os trilhos de trem, tubulações e até mesmo em estruturas complexas tipo a fuselagem de um avião e outras. Este trabalho apresenta uma extensão destes métodos de guias de ondas para a modelagem com elementos de barra de alta ordem. Os métodos foram implementados computacionalmente em códigos Matlab e os resultados são comparados com os do Método do Elemento Espectral (Spectral Element Method - SEM), do Método dos Elementos Finitos e com os do modelo analítico da Placa de Rayleigh-Lamb. Três elementos de barra de alta ordem são formulados: o modelo de Love (ou de Um modo), o modelo de Mindlin-Herrmann (ou de Dois modos) e o modelo de Doyle (ou de Três modos). O método é avaliado através de exemplos simulados computacionalmente e os resultados são analisados e comparados com aqueles da literatura
Abstract: In order to overcome the current limitations in the dynamic analysis of structures at middle and high frequencies and taking advantage of the periodic nature of many of these structures, in recent years, waveguide methods were developed. Models are obtained from slices of patterned waveguides, were developed by Finite Element Method (FEM) and Spectral Element Method, also known as Wave Finite Element Method - WFEM and Wave Spectral Element Method - WSEM. Examples of waveguides can be found in different types of structures such as the railroad tracks, pipelines and even complex structures like the fuselage of an airplane and others. This work presents an extension of these waveguides methods to model high order rod elements. The methods were implemented in Matlab codes and the results are compared with Spectral Element Method - SEM, Finite Element Method and the analytical Rayleigh-Lamb plate model. Three high order bar elements are formulated: the Love's model (or one mode), the Mindlin-Herrmann's model (or two modes) and the Doyle's model (or three modes). The method is evaluated through computationally simulated examples and the results are analyzed and compared with those of the literature
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica

Two-stage stochastic programming problem with PDE constraint, specially elliptic equation is formulated. The computational scheme is proposed, whereas the emphasis is put on approximation techniques. We introduce method of approximation of random variables of stochastic problem and utilize suitable numerical methods, finite difference method first, then finite element method. There is also formulated a mathematical programming problem describing a membrane deflection with random load. It is followed by determination of the acceptableness of using stochastic optimization rather than deterministic problem and assess the quality of approximations based on Monte Carlo simulation method and the theory of interval estimates. The resulting mathematical models are implemented and solved in the general algebraic modeling system GAMS. Graphical and numerical results are presented.

Tissue Engineering is a very promising research field for the development of natural biological substitutes that restore damaged tissue functions. Cells play a crucial role in tissue regeneration and repair due to their characteristics of proliferation and differentiation, cell-to-cell interaction, biomolecular production and extracellular matrix formation. In particular cell migration is a phenomenon that is involved in different physiological processes such as morphogenesis, wound healing and new tissue deposition. In the absence of external guiding factors it is essentially a phenomenon that shares quite a few analogies with Brownian motion. The presence of biochemical or biophysical cues, on the other hand, can influence cell migration in terms of speed, direction and persistence, transforming it in a biased random movement. Recent studies have shown that cells, in particular fibroblasts, are able to recognize the mechanical properties of a substratum over which they move and that these properties direct the motion through a phenomenon called durotaxis. The aim of this thesis is to study this phenomenon for a better understanding of cell behaviour in durotaxis conditions and for Tissue Engineering applications. In order to do that, in the first part of the work a mathematical model for the description of durotaxis is presented. The model is based on a stochastic differential equation for the cell velocity which is derived from the Langevin equation: cell movement is affected by two forces, namely a deterministic one representing the dissipative effects of the system, and a stochastic one which is due to all the probabilistic processes that might affect cell motility (random fluctuations in motile sensing, response mechanisms, etc.). The original contribution of this work concerns the stochastic force, which has been modified to account for the directions of highest perceived local stiffness through a finite element scheme that reminds the cellular probing mechanism. Numerical simulations of the model provide individual cell tracks that can be qualitatively compared with experimental observations. The present model is solved for two important cases that are reported in literature and a comparison with experimental data obtained on PDMS substrata is presented. The degree of agreement is satisfactory thus the model could be utilized to quantify relevant parameters of cell migration as a function of substratum mechanical properties. The second part of the work is concerned on the study and development of a durotaxis-based substratum, able to guide cells in their migration and in particular, able to guide cells along straight path. It was proved, in fact, that a relation exist between the alignment of collagen produced by fibroblasts or others tissue cells and their migration. Thus, the idea is to obtain an aligned tissue made of new collagen, giving to the cells the conditions to move along straight-lines through the mechanical properties of the substratum. To realize this substratum Polyethylenglycole (PEG) was used. First, smooth PEG was synthesized and cell migration experiments was performed over it to better understand its response. Then a specific technique was developed to produce durotaxis-based PEG substrata, and preliminary experiments of cell adhesion over it were performed showing aligned adhesion of cells over them.

Orientador: José Maria Campos dos Santos
Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
Made available in DSpace on 2018-08-27T18:23:30Z (GMT). No. of bitstreams: 1 Moraes_ElsonCesar_D.pdf: 11465202 bytes, checksum: 1ad71136d449760c1a551cda489beaca (MD5) Previous issue date: 2015
Resumo: Neste trabalho, estudam-se fenômenos de propagação de ondas acústicas em cavidades unidimensionais e bidimensionais e se propõe uma metodologia para a análise do comportamento acústico com o objetivo de identificar a propagação de onda nessas estruturas em médias e altas frequências. Para isso se utilizou os métodos originados da Análise de Fluxo de Energia (EFA) que têm como variáveis primárias a densidade de energia e fluxo de energia. Duas formulações foram usadas para resolver as equações diferenciais de energia aproximada derivada do EFA: o Método dos Elementos Finitos de Energia (EFEM) e o Método do Elemento Espectral de Energia (ESEM). O Método do Elemento Espectral (SEM) foi outra formulação estudada para resolver as equações diferenciais de propagações de ondas em cavidades acústicas unidimensionais, bem como para validar a metodologia proposta. Diferentes tipos de cavidades acústicas unidimensionais foram investigadas e implementas tais como: guia de onda simples, guia de onda com descontinuidade geométrica e guia de onda com descontinuidade das propriedades médias. Para isso, desenvolveu-se relações de acoplamento que descrevem essas descontinuidades acústicas. Um exemplo simulado envolvendo uma cavidade acústica retangular fechada é analisado com ESEM e os resultados são comparados com aqueles obtidos pelo EFA, EFEM e a Solução Analítica Acústica (SAA) da equação forçada de Helmholtz para cavidades retangulares bidimensionais fechadas. Uma investigação experimental do comportamento acústico de uma cavidade acústica bidimensional fechada foi realizada para verificar e validar os resultados do ESEM. De maneira geral, se observou que os resultados obtidos pelos modelos simulados apresentaram um comportamento semelhante ao resultado encontrado experimentalmente
Abstract: In this research, propagation phenomena of acoustic waves in one-dimensional and two-dimensional cavities is investigated and a methodology, based on energy methods, is proposed to predict acoustic response of these structures, at mid and high frequencies. These methods, originated from Energy Flow Analysis, use as primary variables the energy density and the energy flow. Two methodologies are used to solve approximated energy differential equations derived from EFA: Energy Finite Element Method (EFEM) and Energy Spectral Element Method (ESEM). The Spectral Element Method (SEM), is used to solve the wave propagation differential equations of acoustic in one-dimensional cavities, moreover this method is applied to validate the proposed methodology. Different acoustics cavities were investigated, such as simple waveguide, waveguide with geometric discontinuity and waveguide with discontinuity of the average properties. To this purpose, coupling relationships that represent those discontinuities are developed. A simulated example involving a closed rectangular acoustic cavity is analyzed with ESEM and the results are compared with those obtained by EFA, the Energy Finite Element Method (EFEM) and the acoustic analytical solution (SAA) of forced Helmholtz equation for rectangular two-dimensional closed cavities. An experimental investigation of acoustic response of a closed two-dimensional acoustic cavity was performed to verify the results of ESEM. The simulated models results, calculated in terms of energy variables, to analyze cavity, showed a good agreement with the experimental result
Doutorado
Mecanica dos Sólidos e Projeto Mecanico
Doutor em Engenharia Mecânica

La fabrication des structures composites conduit à une variabilité élevée de ses paramètres mécaniques. La thèse a comme objectif global de développer des méthodes économiques et robustes pour étudier la variabilité de la réponse statique ou dynamique des structures composites modélisées par éléments finis, prenant en compte les propriétés matériaux (modules d'élasticité, coefficients de Poisson, masses volumiques...) et physiques (épaisseurs et orientations des fibres) incertaines. Deux méthodes stochastiques : Certain Generalized Stresses Method (CGSM) et Modal Stability Procedure (MSP), sont développées. La méthode CGSM considère une hypothèse mécanique, les efforts généralisés sont supposés indépendants des paramètres incertains. Elle permet d'évaluer la variabilité de la réponse statique. La méthode MSP, proposée pour étudier la variabilité d'une structure en dynamique, est basée sur l'hypothèse considérant que les modes propres sont peu sensibles aux paramètres incertains. Les hypothèses mécaniques et une unique analyse par éléments finis permettent de construire un méta-modèle exploité dans une simulation de Monte Carlo. Le coût de calcul de ces méthodes stochastiques est donc réduit considérablement. De plus, elles présentent les avantages de ne pas limiter le nombre de paramètres incertains ou le niveau de variabilité d'entrée, et d'être compatibles avec tout code éléments finis standard. Quatre exemples académiques de plaque et coque composite sont traités avec la méthode CGSM, deux exemples académiques de plaque composite carrée et un exemple de plaque raidie sont traités avec la méthode MSP. La variabilité de la réponse statique (déplacement et critère de rupture) et dynamique (fréquence propre), soit la moyenne, l'écart-type, le coefficient de variation et la distribution, est évaluée. Les résultats statistiques obtenus par les méthodes proposées sont comparés avec ceux obtenus par une simulation de Monte Carlo directe, considérée comme la méthode de référence. La comparaison montre que les méthodes développées fournissent des résultats de bonne qualité et qu'elles sont très performantes en temps de calcul. Un indicateur d'erreur est également proposé, permettant de donner une estimation du niveau d'erreur des résultats obtenus par les méthodes CGSM ou MSP par rapport à la méthode de référence, sans réaliser un grand nombre d'analyses par éléments finis
The manufacture of composite structures leads to a high variability of mechanical parameters. The objective of this work is to develop economic and robust methods to study the variability of the static or dynamic response of composite structures modeled by finite elements, taking into account uncertain material (elastic moduli, Poisson's ratios, densities... ) and physical (thicknesses and fiber orientations) properties. Two methods are developed: the Certain Generalized Stresses Method (CGSM) and the Modal Stability Procedure (MSP). The CGSM considers a mechanical assumption, the generalized stresses are assumed to be independant of uncertain parameters. lt allows to evaluate the variability of static response. The MSP, proposed to study the variability of structures in dynamics, is based on the assumption that the modes shapes are insensitive to uncertain parameters. These mechanical assumptions and only one fïnite element analysis allow to construct a metamodel used in a Monte Carlo simulation. As a result, the computational cost is reduced considerably. Moreover, they are not limited by the number of considered parameters or the level of input variability, and are compatible with standard finite element software. Four academic examples of composite plate and shell are treated with the CGSM, while two academic examples of composite square plate and an example of stiffened plate are studied by the MSP. The variability of static response (displacement and failure criterion) and dynamic response (natural frequency), namely mean value, standard deviation, coefficient of variation and distribution, is evaluated. The results obtained by the proposed methods are compared with those obtained by the direct Monte Carlo simulation, considered as the reference method. The comparison shows that the proposed methods provide quite accurate results and highlights their high computational efficiency. An error indicator is also proposed, which allows to provide an estimation of the error level of the results obtained by the CGSM or MSP compared to the reference method, without performing a large number of finite element analyses

Concrete-filled steel tubular (CFST) truss is a type of composite structure with CFST chords and hollow tubular braces. CFST trusses have been increasingly used in large-scale structures such as towers, bridge girders, piers and arch ribs. The compression and flexural behaviour of CFST trusses are greatly improved compared to hollow tubular trusses due to the concrete infill in chords. For such complex composite structural systems, traditional structural analysis approaches are readily applied for the safety checks of individual members and connections, whilst system-level inelastic analysis and reliability calibration are very limited. In the past, relevant studies on the reliability of CFST structures mostly focused on structural components such as individual columns and beams, and there is no mature system-based design regulations nor reliability evaluations on this complex composite system. This thesis aims to address this gap by adopting stochastic finite element analysis (FEA) modellings of CFST truss systems through numerical approach considering both the structural nonlinearities and random uncertainties. With the complex configuration, nonlinear material interaction and sophisticated construction process, initial imperfections may largely affect the strength and stability of a CFST truss structure. The deterministic and probabilistic studies of CFST trusses with random imperfections based on the statistics of imperfections obtained through experiments and on-site measurements are carried out first, which lays a foundation for further study on the comprehensive reliability analysis of composite truss structures with uncertainties in material properties, structural configurations, initial imperfections and model uncertainties. The verified FEA models of CFST truss systems (two-chord, three-chord and four-chord) are then further developed to implement random uncertainties, including the elastic modulus of steel (Es), the yield strength of steel (fy), the thickness of steel (ts), the cylinder strength of concrete (fc’), the initial steel imperfection (χs) and the initial concrete imperfection (χc1, χc2), to facilitate a comprehensive reliability analysis. To consider the random variation of these uncertainties based on statistics, the Monte Carlo (MC) simulation and the Latin Hypercube Sampling (LHS) technique are incorporated on the basis of the above-described Abaqus modelling. A large number of stochastic FEA models of CFST truss samples are randomly analysed using Abaqus-Python technique with random variables considered to estimate the statistics of the flexural strengths of CFST trusses. Using the obtained statistics of system resistance, reliability analysis is then undertaken to calculate the system reliability indices (β) of the three typical CFST truss systems. Random dead loads and live loads are implemented to the FEA models. The structure reliability of the CFST trusses is evaluated in MATLAB using the First-Order Reliability Method (FORM). The relationship between reliability indices (β) and system resistance factors (ϕs) under various load cases is obtained. Finally, a comprehensive reliability-based design guideline of CFST truss systems by advanced analysis is suggested. The proposed novel computational approach and the reliability analysis contribute to both the practical design and the standard drafting for CFST truss systems.

The research presented in this thesis focuses on the probabilistic modelling of soil consolidation via prefabricated vertical drains (PVDs) considering soil spatial variability. Soils are highly variable from one point to another in the ground and yet this is often coupled with inadequate site data, probabilistic analysis is a more rational approach to assess the behaviour of soil consolidation by PVDs. Although the fact that spatial variation of soil properties can affect soil consolidation has long been realized, the design of soil consolidation via PVDs has been traditionally carried out deterministically and thus can be misleading due to the ignorance of the uncertainty associated with the inherent spatial variation of soil properties. One of the major advantages of probabilistic modelling over deterministic approach is that it can explicitly incorporate soil spatial variability in the analysis and design of a geotechnical problem and subsequently provides much physical insight into the impact of soil spatial variability on the behaviour of the problem under consideration.However, owing to the complexity of the stochastic problem, available research into consolidation of highly variable soils has been limited. The review of relevant literature has indicated that soil spatial variability in relation to ground improvement by PVDs has never been previously considered in a systematic, scientific manner in design and little research has been done in this area. Therefore, to obtain a more realistic measure of the degree of consolidation at any specified time, the effect of soil spatial variability needs to be taken into account by employing probabilistic modelling approach.Among several available methods of stochastic modelling, the random finite element method (RFEM) using random variable soil input properties in a Monte Carlo framework has gained much popularity in recent years. The same approach is adopted in the present research for modelling soil spatial variability in soil consolidation by PVDs. The soil permeability, k, and volume compressibility, mv, are considered as random variables and the variability of both k and mv is characterised statistically in terms of the mean, standard deviation, lognormal probability distribution and scale of fluctuation (SOF).The random fields of k and mv are generated using 2D local average subdivision (LAS) method developed by Fenton and Vanmarcke (1990). The generated random fields are then used as inputs in a stochastic finite element modelling of soil consolidation by PVDs. In this research, all numerical analyses are carried out using the 2D finite element computer program AFENA (Carter and Balaam 1995), in which the consolidation process of soil is treated as a coupled transient problem governed by the Biot’s consolidation theory (Biot 1941).

Une forte concurrence avec d'autres moyens de transport a poussé l’industrie ferroviaire à se réinventer et rechercher des performances toujours plus élevées. De nos jours, l’obtention de vitesses chaque fois plus élevées exige le développement de modèles numériques précis pour concevoir et prédire le comportement des voies ferrées sous les contraintes mécaniques imposées par le passage du convoi. Dans cette thèse, nous avons concentré l'étude sur la couche de ballast. Ce composant présente un comportement mécanique complexe, lié à la nature granulaire de ses composants, il peut être solide, liquide ou gazeux.Ce comportement dépend de l'état de contrainte et de l'historique de déformation du milieu.Deux classes de modèles numériques sont couramment utilisées pour prédire le comportement de ces systèmes : (1) les approches discrètes et (2) les approches continues. Pour ces premières, chaque grain du ballast est représenté par un corps rigide et interagit avec ses voisins parle biais de forces de contact non linéaires en utilisant, par exemple, la méthode de dynamique non régulière des contacts. En raison des limites de calcul, ce type de méthode ne peut résoudre que quelques mètres de longueur de ballast. Le couplage avec le sol sous la couche de ballast et avec les traverses reste également un problème non résolu dans la littérature. Pour les approches continues, le ballast est remplacé par un milieu continu homogénéisé, de façon à permettre l’utilisation de la méthode par éléments finis classique (EF). Cependant, ces modèles sont normalement utilisés avec des paramètres mécaniques homogènes, de sorte qu'ils ne représentent pas complètement l'hétérogénéité des déformations et des contraintes dans la couche de ballast.Nous étudions dans cette thèse une approche alternative, utilisant un modèle de continuum hétérogène stochastique, qui peut être résolu avec une méthode par éléments finis tout en conservant dans une large mesure l'hétérogénéité des champs de contrainte et de déformation.L'objectif de ce modèle continu est de représenter statistiquement l'hétérogénéité du champ de contraintes dans un modèle de milieu continu ainsi que dans un modèle granulaire discret. Pour ce faire, les propriétés mécaniques sont représentées à l'aide de champs aléatoires. La présente thèse est divisée en trois parties: (1) la construction du modèle et l'identification des paramètres du matériau continuum (densité marginale de premier ordre, moyenne, variance, modèle de corrélation) ;(2) la propagation des ondes dans une voie ferrée ballastée et (3) l’exploration préliminaire de deux ensembles de données expérimentales. La première partie définit le modèle du continuum à fluctuations aléatoires et identifie les paramètres de notre modèle de continuum sur de petits échantillons cylindriques de ballast discret. Des modèles continus équivalents aux échantillons discrets sont générés et résolus en utilisant la méthode EF, et le champ stochastique utilisé pour fournir les propriétés mécaniques. Un processus d'optimisation est utilisé pour trouver une variance normalisée pour le matériau hétérogène stochastique. La deuxième partie de ce travail se concentre sur la résolution des équations dynamiques sur un modèle à grande échelle d'une voie ferrée ballastée utilisant la méthode des éléments spectraux. L'influence de l'hétérogénéité est mise en évidence et étudiée. En conséquence,des courbes de dispersion sont obtenues. Enfin, la troisième partie présente deux jeux de données distincts de mesures expérimentales sur le matériau de ballast : (1) une boîte de ballast ; (2) un passage de train dans un segment de voie ferrée ballastée.Les courbes de mobilité ont été extraites de l'expérience sur les ballasts. Un problème inverse a été résolu afin d'estimer la vitesse de l'onde homogénéisée et la vitesse de l'onde locale dans le milieu. Les passages de trains enregistrés pour l'analyse de la vibration à moyenne fréquences
The stronger competition with other means of transportation has increased the demand for performance in the railway industry. One way to achieve higher performance is using accurate numerical models to design/predict railways tracks behaviour. Two classes of numerical models are commonly used to predict the behaviour of these systems: (i) discrete approaches and (ii) continuum approaches. In the former, each grain of the ballast is represented by a rigid body and interacts with its neighbours through nonlinear contact forces using, for example, the nonsmooth contact dynamics method. Due to computational limits, this kind of method can only solve a few meters-length of ballast. The coupling with the soil under the ballast layer and with the sleepers also remains an open problem. In continuum approaches, the ballast is replaced by a homogenized continuum and the classical Finite Element (FE) Method (or similar) is used. However, they are normally used with homogeneous mechanical parameters, so that they do not represent fully the heterogeneity of the strains and stresses within the ballast layer. We investigate in this thesis an alternative approach using a stochastic heterogeneous continuum model, that can be solved with a FElike method while retaining to a large degree the heterogeneity of the stress and strain fields. The objective of this continuous model is to represent statistically the heterogeneity of the stress field in a continuum model as well as in a discrete granular model. To do this, the mechanical properties are represented using random fields. The present thesis is divided into three parts: (1) the construction of the model and the identification of the parameters of the continuum material (first-order marginal density, mean, variance, correlation model, and correlation length); (2) wave propagation in a ballasted railway track. (3) preliminary exploration of two experimental datasets. The first part sets the randomly-fluctuating continuum model and identifies the parameters of our continuum model on small cylindrical samples of discrete ballast. Continuum models equivalent to the discrete samples are generated and solved using the FE method, and the stochastic field used as mechanical properties. An optimization process is used to find a normalized variance for the stochastic heterogeneous material. The second part of this work concentrates on the solution of the dynamical equations on a large-scale model of a ballasted railway track using the Spectral Element Method. The influence of the heterogeneity is highlighted and studied. As a result, dispersion curves are obtained. Finally, the third part presents two distinct datasets of experimental measurements on ballast material: (1) a ballast box; (2) a train passage in a segment of ballasted railway track. Mobility curves were extracted from the ballast box experiment. An inverse problem was solved in order to estimate the homogenized wave velocity and local wave velocity in the medium. The trains pass-by recorded for the analysis of the vibration at medium frequencies

Cette thèse est consacrée à la modélisation de la propagation des ondes planes dans les plaques multicouches infinies, dans le cadre de l'élasticité linéaire. L’objet du travail est de trouver une approximation analytique ou semi-analytique des relations de dispersion des ondes lorsque le rapport de l'épaisseur de la plaque sur la longueur d'onde est petit. Ces relations de dispersion, liant la fréquence angulaire et le nombre d'onde, fournissent des informations clés sur les caractéristiques de propagation des différents modes. On propose dans cette thèse deux modélisations : le modèle du Bending-Gradient et la méthode des développements asymptotiques. La pertinence de ces méthodes est testée en comparant leurs prédictions à celles des théories de plaques bien connues, et à des résultats de référence obtenus par la méthode des éléments finis. Au préalable, dans la première partie de la thèse, une justification mathématique de la théorie du Bending-Gradient dans le cadre statique est réalisée à l’aide des méthodes variationnelles. Il s'agit d'abord d'identifier les espaces mathématiques dans lesquels les problèmes variationnels du Bending-Gradient sont bien posés. Puis, des théorèmes d'existence et d'unicité des solutions correspondantes sont ensuite formulés et prouvés. La deuxième partie est consacrée à la formulation des équations du mouvement du Bending-Gradient. Des simulations numériques sont effectuées pour plusieurs types d'empilements, permettant ainsi de tester la validité du modèle pour la modélisation de la propagation des ondes de flexion. La troisième partie est dédiée à l'analyse asymptotique des équations tridimensionnelles du mouvement, menée à bien grâce à la méthode des développements asymptotiques, le petit paramètre étant le rapport de l'épaisseur sur la longueur d'onde. En supposant que les champs tridimensionnels s'écrivent comme des séries en puissance du petit paramètre, on obtient une succession de problèmes à résoudre en cascade. La validité de cette méthode est évaluée par comparaison avec la méthode des éléments finis
This thesis is dedicated to the modelling of plane wave propagation in infinite multilayered plates, in the context of linear elasticity. The aim of this work is to find an analytical or semi-analytical approximation of the wave dispersion relations when the ratio of the thickness to the wavelength is small. The dispersion relations, linking the angular frequency and the wave number, provide key information about the propagation characteristics of the wave modes. Two methods are proposed in this thesis: the Bending-Gradient model and the asymptotic expansion method. The relevance of these methods is tested by comparing their predictions to those of well-known plate theories, and to reference results computed using the finite element method. Preliminarily, the first part of the thesis is devoted to the mathematical justification of the Bending-Gradient theory in the static framework using variational methods. The first step is to identify the mathematical spaces in which the variational problems of the Bending-Gradient are well posed. A series of existence and uniqueness theorems of the corresponding solutions are then formulated and proved. The second part is dedicated to the formulation of the equations of motion of the Bending-Gradient theory. Numerical simulations are realized for different types of layer stacks to assess the ability of this model to correctly predict the propagation of flexural waves. The third part is concerned with the asymptotic analysis of the three-dimensional equations of motion, carried out using the asymptotic expansion method, the small parameter being the ratio of the thickness to the wavelength. Assuming that the three-dimensional fields can be written as expansions in power of the small parameter, a series of problems which can be solved recursively is obtained. The validity of this method is evaluated by comparison with the finite element method

Dans cette thèse, une nouvelle approche est développée pour les problèmes de viscoplasticité et de dynamique non linéaire. En particulier, les équations variationnelles sont élaborées selon le principe de Helligner-Reissner, de sorte que les champs de contrainte et de déplacement apparaissent comme des champs inconnus sous la forme faible. Trois nouveaux éléments finis sont développés. Le premier élément fini est formulé pour le problème axisymétrique, dans lequel le champ de contraintes est approximé par des polynômes d’ordre inférieur tels que des fonctions linéaires. Cette approche donne des solutions précises spécifiquement dans les problèmes incompressibles et rigides. De plus, un élément fini de flexion de membrane et de plaque est nouvellement conçu en discrétisant le champ de contraintes en utilisant l’espace vectoriel de Raviart-Thomas d’ordre le plus bas RT0. Cette approche garantit la continuité du champ de contraintes sur tout un domaine discret, ce qui est un avantage significatif dans la méthode numérique, notamment pour les problèmes de propagation des ondes. Les développements sont effectués pour le comportement constitutif visco-plastique des matériaux, où les équations d’évolution correspondantes sont obtenues en faisant appel au principe de dissipation maximale. Pour résoudre les équations d’équilibre dynamique, des schémas de conservation et de décroissance de l’énergie sont formulés en conséquence. Le schéma de conservation de l’énergie est inconditionnellement stable, car il peut préserver l’énergie totale d’un système donné sous une vibration libre, tandis que le schéma décroissant peut dissiper des modes de vibration à plus haute fréquence. La dernière partie de cette thèse présente les procédures d’upscaling du comportement des matériaux visco-plastiques. Plus précisément, la mise à l’échelle est effectuée par une méthode d’identification stochastique via une mise à jour baysienne en utilisant le filtre de Gauss-Markov-Kalman pour l’assimilation des propriétés importantes des matériaux dans les régimes élastique et inélastique
In this thesis, a novel approach is developed for visco-plasticity and nonlinear dynamics problems. In particular, variational equations are elaborated following the Helligner-Reissner principle, so that both stress and displacement fields appear as unknown fields in the weak form. Three novel finite elements are developed. The first finite element is formulated for the axisymmetric problem, in which the stress field is approximated by low-order polynomials such as linear functions. This approach yields accurate solutions specifically in incompressible and stiff problems. In addition, a membrane and plate bending finite element are newly designed by discretizing the stress field using the lowest order Raviart-Thomas vector space RT0. This approach guarantees the continuity of the stress field over an entire discrete domain, which is a significant advantage in the numerical method, especially for the wave propagation problems. The developments are carried out for the viscoplastic constitutive behavior of materials, where the corresponding evolution equations are obtained by appealing to the principle of maximum dissipation. To solve the dynamic equilibrium equations, energy conserving and decaying schemes are formulated correspondingly. The energy conserving scheme is unconditional stable, since it can preserve the total energy of a given system under a free vibration, while the decaying scheme can dissipate higher frequency vibration modes. The last part of this thesis presents procedures for upscaling of the visco-plastic material behavior. Specifically, the upscaling is performed by stochastic identification method via Baysian updating using the Gauss-Markov-Kalman filter for assimilation of important material properties in the elastic and inelastic regimes