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Thesis (MScEng)-- Stellenbosch University, 2013. / ENGLISH ABSTRACT: Most of the parameters involved in the design and analysis of structures are of stochastic nature. This is, therefore, of paramount importance to be able to perform a fully stochastic analysis of structures both in component and system level to take into account the uncertainties involved in structural analysis and design. To the contrary, in practice, the (computerised) analysis of structures is based on a deterministic analysis which fails to address the randomness of design and analysis parameters. This means that an investigation on the algorithmic methodologies for a component and system reliability analysis can help pave the way towards the implementation of fully stochastic analysis of structures in a computer environment. This study is focused on algorithm development for component and system reliability analysis based on the various proposed methodologies. Truss structures were selected for this purpose due to their simplicity as well as their wide use in the industry. Nevertheless, the algorithms developed in this study can be used for other types of structures such as moment-resisting frames with some simple modi cations. For a component level reliability analysis of structures different methods such as First Order Reliability Methods (FORM) and simulation methods are proposed. However, implementation of these methods for the statistically indeterminate structures is complex due to the implicit relation between the response of the structural system and the load effect. As a result, the algorithm developed for the purpose of component reliability analysis should be based on the concepts of Stochastic Finite Element Methods (SFEM) where a proper link between the finite element analysis of the structure and the reliability analysis methodology is ensured. In this study various algorithms are developed based on the FORM method, Monte Carlo simulation, and the Response Surface Method (RSM). Using the FORM method, two methodologies are considered: one is based on the development of a finite element code where required alterations are made to the FEM code and the other is based on the usage of a commercial FEM package. Different simulation methods are also implemented: Direct Monte Carlo Simulation (DMCS), Latin Hypercube Sampling Monte Carlo (LHCSMC), and Updated Latin Hypercube Sampling Monte Carlo (ULHCSMC). Moreover, RSM is used together with simulation methods. Throughout the thesis, the effciency of these methods was investigated. A Fully Stochastic Finite Element Method (FSFEM) with alterations to the finite element code seems the fastest approach since the linking between the FEM package and reliability analysis is avoided. Simulation methods can also be effectively used for the reliability evaluation where ULHCSMC seemed to be the most efficient method followed by LHCSMC and DMCS. The response surface method is the least straight forward method for an algorithmic component reliability analysis; however, it is useful for the system reliability evaluation. For a system level reliability analysis two methods were considered: the ß-unzipping method and the branch and bound method. The ß-unzipping method is based on a level-wise system reliability evaluation where the structure is modelled at different damaged levels according to its degree of redundancy. In each level, the so-called unzipping intervals are defined for the identification of the critical elements. The branch and bound method is based on the identification of different failure paths of the structure by the expansion of the structural failure tree. The evaluation of the damaged states for both of the methods is the same. Furthermore, both of the methods lead to the development of a parallel-series model for the structural system. The only difference between the two methods is in the search approach used for the failure sequence identification. It was shown that the ß-unzipping method provides a better algorithmic approach for evaluating the system reliability compared to the branch and bound method. Nevertheless, the branch and bound method is a more robust method in the identification of structural failure sequences. One possible way to increase the efficiency of the ß-unzipping method is to define bigger unzipping intervals in each level which can be possible through a computerised analysis. For such an analysis four major modules are required: a general intact structure module, a damaged structure module, a reliability analysis module, and a system reliability module. In this thesis different computer programs were developed for both system and component reliability analysis based on the developed algorithms. The computer programs are presented in the appendices of the thesis. / AFRIKAANSE OPSOMMING: Meeste van die veranderlikes betrokke by die ontwerp en analise van strukture is stogasties in hul aard. Om die onsekerhede betrokke in ontwerp en analise in ag te neem is dit dus van groot belang om 'n ten volle stogastiese analise te kan uitvoer op beide komponent asook stelsel vlak. In teenstelling hiermee is die gerekenariseerde analise van strukture in praktyk gebaseer op deterministiese analise wat nie suksesvol is om die stogastiese aard van ontwerp veranderlikes in ag te neem nie. Dit beteken dat die ondersoek na die algoritmiese metodiek vir komponent en stelsel betroubaarheid analise kan help om die weg te baan na die implementering van ten volle rekenaarmatige stogastiese analise van strukture. Di e studie se fokus is op die ontwikkeling van algoritmes vir komponent en stelsel betroubaarheid analise soos gegrond op verskeie voorgestelde metodes. Vakwerk strukture is gekies vir die doeleinde as gevolg van hulle eenvoud asook hulle wydverspreide gebruik in industrie. Die algoritmes wat in die studie ontwikkel is kan nietemin ook vir ander tipes strukture soos moment-vaste raamwerke gebruik word, gegewe eenvoudige aanpassings. Vir 'n komponent vlak betroubaarheid analise van strukture word verskeie metodes soos die "First Order Reliability Methods" (FORM) en simulasie metodes voorgestel. Die implementering van die metodes vir staties onbepaalbare strukture is ingewikkeld as gevolg van die implisiete verband tussen die gedrag van die struktuur stelsel en die las effek. As 'n gevolg, moet die algoritme wat ontwikkel word vir die doel van komponent betroubaarheid analise gebaseer word op die konsepte van stogastiese eindige element metodes ("SFEM") waar 'n duidelike verband tussen die eindige element analise van die struktuur en die betroubaarheid analise verseker is. In hierdie studie word verskeie algoritmes ontwikkel wat gebaseer is op die FORM metode, Monte Carlo simulasie, en die sogenaamde "Response Surface Method" (RSM). Vir die gebruik van die FORM metode word twee verdere metodologieë ondersoek: een gebaseer op die ontwikkeling van 'n eindige element kode waar nodige verandering aan die eindige element kode self gemaak word en die ander waar 'n kommersiële eindige element pakket gebruik word. Verskillende simulasie metodes word ook geïmplimenteer naamlik Direkte Monte Carlo Simulasie (DMCS), "Latin Hypercube Sampling Monte Carlo" (LHCSMC) en sogenaamde "Updated Latin Hypercube Sampling Monte Carlo" (ULHCSMC). Verder, word RSM tesame met die simulasie metodes gebruik. In die tesis word die doeltreffendheid van die bostaande metodes deurgaans ondersoek. 'n Ten volle stogastiese eindige element metode ("FSFEM") met verandering aan die eindige element kode blyk die vinnigste benadering te wees omdat die koppeling tussen die eindige element metode pakket en die betroubaarheid analise verhoed word. Simulasie metodes kan ook effektief aangewend word vir die betroubaarheid evaluasie waar ULHCSMC as die mees doeltre end voorgekom het, gevolg deur LHCSMC en DMCS. The RSM metode is die mees komplekse metode vir algoritmiese komponent betroubaarheid analise. Die metode is egter nuttig vir sisteem betroubaarheid analise. Vir sisteem-vlak betroubaarheid analise is twee metodes oorweeg naamlik die "ß-unzipping" metode and die "branch-and-bound" metode. Die "ß-unzipping" metode is gebaseer op 'n sisteem-vlak betroubaarheid ontleding waar die struktuur op verskillende skade vlakke gemodelleer word soos toepaslik vir die hoeveelheid addisionele las paaie. In elke vlak word die sogenaamde "unzipping" intervalle gedefinieer vir die identifikasie van die kritiese elemente. Die "branch-and-bound" metode is gebaseer op die identifikasie van verskillende faling roetes van die struktuur deur uitbreiding van die falingsboom. The ondersoek van die skade toestande vir beide metodes is dieselfde. Verder kan beide metodes lei tot die ontwikkeling van 'n parallelserie model van die strukturele stelsel. Die enigste verskil tussen die twee metodes is in die soek-benadering vir die uitkenning van falingsmodus volgorde. Dit word getoon dat die "ß-unzipping" metode 'n beter algoritmiese benadering is vir die ontleding van sisteem betroubaarheid vergeleke met die "branch-and-bound" metode. Die "branch-and- bound" metode word nietemin as 'n meer robuuste metode vir die uitkenning van die falings volgorde beskou. Een moontlike manier om die doeltre endheid van die "ß-unzipping" metode te verhoog is om groter "unzipping" intervalle te gebruik, wat moontlik is vir rekenaarmatige analise. Vir so 'n analise word vier hoof modules benodig naamlik 'n algemene heel-struktuur module, 'n beskadigde-struktuur module, 'n betroubaarheid analise module en 'n sisteem betroubaarheid analise module. In die tesis word verskillende rekenaar programme ontwikkel vir beide sisteem en komponent betroubaarheid analise. Die rekenaar programme word in die aanhangsels van die tesis aangebied.

On fait l'analyse de la dynamique nonlinéaire d'une colonne de forage avec modélisation d'incertitudes. Une colonne de forage est une structure flexible mince qui tourne et fore des roches en cherchant du pétrole. Un modèle mathématique-mécanique est développeé pour cette structure en incluant de l'interaction fluide-structure, des impact, des nonlinéarités géométriques, et de l'interaction tête de la colonne-roche. Les équations de mouvement sont déduites, puis le système est discretisé par la méthode des éléments finis, et un code numérique est développeé pour les simulations numériques avec le logiciel MATLAB. Les modes normaux de la dynamique à la configuration précontrainte sont utilisés pour construire un modèle reduit pour le système. L'approche probabiliste nonparamétrique, qui est capable de prendre en compte les incertitude des paramètres du système et aussi les incertitude de modèle, est utilisée. Les fonctions densités de probabilité liées aux variables aléatoire sont construites par la méthode du Maximum d'Entropie et la réponse stochastique du système est calculée en utilisant la méthode de Monte Carlo. Une nouvelle approche pour prendre en compte des incertitudes de modèle dans une équation constitutive nonlinéaire (interaction tête de la colonne-roche) est développeé avec l'approche probabiliste nonparamétrique. La méthode du maximum de vraisemblance et la réduction statistique dans le domaine de la fréquence (Analyse de composantes principales) sont utilisés pour identifier le modèle probabiliste de l'interaction tête de la colonne-roche. Finalement, un problème d'optimisation robuste est analysé de façon à trouver les paramètres opérationnels du système qui maximise sa performance et respectent les limites d'intégrités du système, comme fatigue et instabilitée / This thesis analyzes the nonlinear dynamics of a drill-string including uncertainty modeling. A drill-string is a slender flexible structure that rotates and digs into the rock in search of oil. A mathematical-mechanical model is developed for this structure including fluid-structure interaction, impact, geometrical nonlinearities and bit-rock interaction. After the derivation of the equations of motion, the system is discretized by means of the finite element method and a computer code is developed for the numerical computations using the software MATLAB. The normal modes of the dynamical system in the prestressed configuration are used to construct a reduced order model for the system. To take into account uncertainties, the nonparametric probabilistic approach, which is able to take into account both system-parameter and model uncertainties, is used. The probability density functions related to the random variables are constructed using the maximum entropy principle and the stochastic response of the system is calculated using the Monte Carlo method. A novel approach to take into account model uncertainties in a nonlinear constitutive equation (bit-rock interaction model) is developed using the nonparametric probabilistic approach. To identify the probabilistic model of the bit-rock interaction model, the maximum likelihood method together with a statistical reduction in the frequency domain (using the Principal Component Analysis) is applied. Finally, a robust optimization problem is performed to find the operational parameters of the system that maximizes its performance, respecting the integrity limits of the system, such as fatigue and instability

Wir modellieren einen Finanzmarkt mit unendlich vielen Wertpapieren als stochastischen Prozeß X in stetiger Zeit mit Werten in einem separablen Hilbertraum H. In diesem Rahmen zeigen wir die Äquivalenz von Vollständigkeit des Marktes und der Eindeutigkeit des äquivalenten Martingalmaßes unter der Bedingung, daß X stetige Pfade besitzt. Weiter zeigen wir, daß (unter gewissen technischen Bedingungen) für X die Abwesenheit von asymptotischer Arbitrage der ersten/zweiten Art (im Sinne von Kabanov/Kramkov) äquivalent zur Absolutstetigkeit des Referenzmaßes zu einem eindeutigen, lokal äquivalenten Martingalmaß ist. Hat X stetige Pfade, so ist die Abwesenheit von allgemeiner asymptotischer Arbitrage äquivalent zur Existenz eines äquivalenten lokalen Martingalmaßes. Außerdem geben wir ein Kriterium für die Existenz einer optionalen Zerlegung von X an. Dies wenden wir auf das Problem der Risikominimierung bei vorgegebener Investitionsobergrenze (effizientes Hedgen (Föllmer/Leukert)) an, um dieses im unendlichdimensionalen Kontext zu behandeln. Außerdem stellen wir eine unendlichdimensionale Erweiterung des Heath-Jarrow-Morton-Modells vor und nutzen den Potentialansatz nach Rodgers, um zwei weitere Zinsstrukturmodelle zu konstruieren. Als Beitrag zur allgemeinen stochastischen Analysis in Hilberträumen beweisen wir eine pfadweise Version der Itoformel für stochastische Prozesse mit stetigen Pfaden in einem separablen Hilbertraum. Daraus läßt sich eine pfadweise Version des Satzes über die Vertauschbarkeit von stochastischem und Lebesgue-Integral ableiten. Außerdem zeigen wir eine Version der Clark-Formel für eine Brownsche Bewegung mit Werten in einem Hilbertraum. / We model a financial market with infinitely many assets as a stochastic process X with values in a separable Hilbert space H. In this setting we show the equivalence of market completeness and the uniqueness of the equivalent martingale measure, if X has continuous paths. Another result for our model is, that under some technical conditions, the absence of asymptotic arbitrage of the first/second kind (in the sense of Kabanov/Kramkov) is equivalent to the absolute continuity of the reference measure to a unique, locally equivalent, martingale measure. If X has continuous paths, the absence of general asymptotic arbitrage is equivalent to the existence of an equivalent local martingale measure. Furthermore, we give a sufficient condition for the existence of the optional decomposition of X. We apply this result to the problem of risk minimization with given upper limit for investion (efficient hedging (Föllmer/Leukert)). This allows us to solve this optimization problem in our infinite dimensional context. Another result is an infinite dimensional extension of the Heath-Jarrow-Morton term structure model. Two further term structure models are constructed, using the Markov potential approach developed by Rodgers. As a contribution to the theory of stochastic analysis in Hilbert spaces, we proof a pathwise version of the Ito formula for stochastic processes with continuous paths in a separable Hilbert space. This leads to a pathwise version of the interchangability theorem for stochastic and Lebesgue integrals. We also show a version of the Clark formula for Hilbert space valued Brownian motion.

A produção de frango de corte tem impressionado pelo dinamismo e pela competência conquistada nas últimas décadas, destacando-se o Brasil como o segundo maior produtor dessa proteína animal. O ganho de produtividade, associado à coordenação da cadeia avícola, colocou o País como um dos mais eficientes produtores. Entretanto, a significativa especialização da atividade tende a excluir do processo produtivo os pequenos avicultores e os produtores menos eficientes. Assim, o estudo buscou medir a eficiência econômica dos produtores de frango de corte das regiões Sul e Centro-Oeste do Brasil, identificando quais fatores influenciam essa medida de desempenho. Para isso, usou-se uma função fronteira de lucro estocástica em um estágio (modelo 2) em que os coeficientes da fronteira e os efeitos da ineficiência são obtidos simultaneamente, assumindo que os termos de erro não são identicamente distribuídos. Concluiu-se que para a região Sul, o preço da mão-de-obra contratada interfere significativamente na lucratividade das unidades produtivas, o que seria um dos fatores associados ao maior uso do trabalho familiar no desenvolvimento das atividades. Além disso, os resultados sugerem a presença de uma melhor utilização das áreas ocupadas com a produção avícola. Os efeitos da ineficiência são sentidos principalmente no baixo nível de educação dos que tomam as decisões e nos índices elevados de conversão alimentar. Já no Centro-Oeste, os coeficientes indicaram que o maior uso de mão-de-obra familiar poderia elevar a lucratividade dos produtores. Pelo fato de ser uma região relativamente nova e possuir condições favoráveis ao investimento em capital e tecnologia, estaria indicando um maior lucro na atividade. Não se identificou ganho de escala por meio do modelo estocástico. Embora não se observa uma tendência contínua associada à escala de eficiência, no Centro-Oeste parece existirem ganhos de eficiência relevante nos estratos de médio e alto escala de produção para os padrões regionais. / The dynamism and ability acquired through the last decade by the broiler production is very impressive. The productivity rate gain associated to the good management of poultry chain in Brazil led the country to be the second biggest producer of this animal’s protein. However, the significant specialization of this activity tend to exclude the smaller producers and those who are lesser efficient in the productive process. In view of that, this study intends to precise the economic efficiency of the broiler producers in the Southerner and Center-Southerner regions of Brazil. For that, the main factors that influence the economic efficiency were identified. To reach those results, it was considered the stochastic profit function in a stage (model 2) where the coefficients of the frontier and the inefficiency effects are obtained simultaneously, since the terms of error are not identical distributing. It was possible to conclude that in the Southerner region of Brazil, the contracted labor force influence significantly the profitability of the productive farms. That’s one of the factors that explain the great use of the familiar labor force in that region. Moreover, the results pointed a better utilization of the broiler producing areas in Brazil. The effects of the inefficiency are present mainly in the producers and players with low educational level, and also in the high food conversion ratio. In the Center-Westerner of Brazil, the coefficients pointed that the better use of the familiar manpower could raise the probability of the producers. For been a new region that present positive conditions for the investments in capital and technology, it also obtained a larger profit with the activity. It was not identified gains by the stochastic model. None successively trend associated to the efficiency scale were pointed in the Center-Westerner of Brazil. Even so, the region shows efficiency gains in the middle and high scales of production in regional terms.

Nous présentons des inégalités fonctionnelles pour les processus ponctuels. Nous prouvons une inégalité de Sobolev logarithmique modifiée, une inégalité de Stein et un théorème du moment quatrième sans terme de reste pour une classe de processus ponctuels qui contient les processus binomiaux et les processus de Poisson. Les preuves reposent sur des techniques inspirées de l'approche de Malliavin-Stein et du calcul avec l'opérateur $Gamma$ de Bakry-Émery. Pour mettre en œuvre ces techniques nous développons une analyse stochastique pour les processus ponctuels. Plus généralement, nous mettons au point une théorie d'analyse stochastique sans hypothèse de diffusion. Dans le cadre des processus de Poisson ponctuels, l'inégalité de Stein est généralisée pour étudier la convergence stable vers des limites conditionnellement gaussiennes. Nous appliquons ces résultats pour approcher des processus Gaussiens par des processus de Poisson composés et pour étudier des graphes aléatoires. Nous discutons d'inégalités de transport et de leur conséquence en termes de concentration de la mesure pour les processus binomiaux dont la taille de l'échantillon est aléatoire. Sur un espace métrique mesuré quelconque, nous présentons un développement de la concentration de la mesure qui prend en compte l'agrandissement parallèle d'ensembles disjoints. Cette concentration améliorée donne un contrôle de toutes les valeurs propres du Laplacien métrique. Nous discutons des liens de cette nouvelle notion avec une version de la courbure de Ricci qui fait intervenir le transport à plusieurs marginales / We present functional inequalities and limit theorems for point processes. We prove a modified logarithmic Sobolev inequalities, a Stein inequality and a exact fourth moment theorem for a large class of point processes including mixed binomial processes and Poisson point processes. The proofs of these inequalities are inspired by the Malliavin-Stein approach and the $Gamma$-calculus of Bakry-Emery. The implementation of these techniques requires a development of a stochastic analysis for point processes. As point processes are essentially discrete, we design a theory to study non-diffusive random objects. For Poisson point processes, we extend the Stein inequality to study stable convergence with respect to limits that are conditionally Gaussian. Applications to Poisson approximations of Gaussian processes and random geometry are given. We discuss transport inequalities for mixed binomial processes and their consequences in terms of concentration of measure. On a generic metric measured space, we present a refinement of the notion of concentration of measure that takes into account the parallel enlargement of distinct sets. We link this notion of improved concentration with the eigenvalues of the metric Laplacian and with a version of the Ricci curvature based on multi-marginal optimal transport

Thesis (Ph. D.)--Joint Program in Oceanography/Applied Ocean Science and Engineering (Massachusetts Institute of Technology, Dept. of Mechanical Engineering; and the Woods Hole Oceanographic Institution), 2007. / Includes bibliographical references (p. 313-325). / This thesis presents a stochastic mapping framework for autonomous robotic chemical plume source localization in environments with multiple sources. Potential applications for robotic chemical plume source localization include pollution and environmental monitoring, chemical plant safety, search and rescue, anti-terrorism, narcotics control, explosive ordinance removal, and hydrothermal vent prospecting. Turbulent flows make the spatial relationship between the detectable manifestation of a chemical plume source, the plume itself, and the location of its source inherently uncertain. Search domains with multiple sources compound this uncertainty because the number of sources as well as their locations is unknown a priori. Our framework for stochastic mapping is an adaptation of occupancy grid mapping where the binary state of map nodes is redefined to denote either the presence (occupancy) or absence of an active plume source. A key characteristic of the chemical plume source localization problem is that only a few sources are expected in the search domain. The occupancy grid framework allows for both plume detections and non-detections to inform the estimated state of grid nodes in the map, thereby explicitly representing explored but empty portions of the domain as well as probable source locations. / (cont.) However, sparsity in the expected number of occupied grid nodes strongly violates a critical conditional independence assumption required by the standard Bayesian recursive map update rule. While that assumption makes for a computationally attractive algorithm, in our application it results in occupancy grid maps that are grossly inconsistent with the assumption of a small number of occupied cells. To overcome this limitation, several alternative occupancy grid update algorithms are presented, including an exact solution that is computationally tractable for small numbers of detections and an approximate recursive algorithm with improved performance relative to the standard algorithm but equivalent computational cost. Application to hydrothermal plume data collected by the autonomous underwater vehicle ABE during vent prospecting operations in both the Pacific and Atlantic oceans verifies the utility of the approach. The resulting maps enable nested surveys for homing-in on seafloor vent sites to be carried out autonomously. This eliminates inter-dive processing, recharging of batteries, and time spent deploying and recovering the vehicle that would otherwise be necessary with survey design directed by human operators. / by Michael V. Jakuba. / Ph.D.

Microplitis croceipes, one of the nectar feeding parasitoid wasps, has been found to associatively learn chemical cues through feeding. The experiments on M. croceipes are performed and recorded by a Sony camcorder in the USDA-ARS Biological Control Laboratory in Tifton, GA, USA. The experimental videos have shown that M. croceipes can respond to Coffee odour in this study. Their detection capabilities and the behaviour of M. croceipes with different levels of coffee odours were studied. First, the data that are related to trained M. croceipes behaviour was extracted from the experimental videos and stored in a Microsoft Excel database. The extracted data represent the behaviour of M. croceipes trained to 0.02g and then exposed to 0.001g, 0.005g, 0.01g, 0.02g and 0.04g of coffee. Secondly, indices were developed to uniquely characterise the behaviour of trained M. croceipes under different coffee concentrations. Thirdly, a preliminary model and its parameters were developed to classify the response of trained wasps when exposed to these five different coffee odours. In summary, the success of this thesis demonstrates the usefulness of data analysis for interpreting experimental data, developing indices, as well as understanding the design principles of a simple model based on trained wasps.

This paper will analyze the effects of using hybrid optimization methods for optimizing objective functions that are determined by computational fluid dynamics solvers for compressible viscous flow for optimal design of airfoils. Previous studies on this topic by the authors had examined the application of deterministic optimization methods and stochastic optimization methods such as Simulated Annealing and Simultaneous Perturbation Stochastic Analysis (SPSA). The studies indicated that SPSA method has a greater or equal efficiency as compared with SA method in reaching optimal airfoil designs for the design problem in question. However, in some situations SPSA method has a tendency to demonstrate an oscillatory behavior in the vicinity of a local optima. To overcome this tendency, a hybrid method designed to take full advantage of SPSA’s high rate of reduction of the objective function at the inception of the design process to drive the design cycles towards the optimal zone at first, and then combining with other methods to perform the final stages of the convergence towards the optimal solutions is considered. SPSA method has been combined with the gradient-based Broydon-Fletcher-Goldfarb-Shanno (BFGS) method as well as Simulated Annealing method for the transonic inverse airfoil design problem that is concerned with the specification of a target airfoil surface pressure distribution and starting from an initial guess of an airfoil shape, the target airfoil shape is reached by way of minimization of a quantity that depends on the difference between the target and current airfoil surface pressure distribution. For a typical transonic flow test case, the effects of using hybrid optimization techniques such as SPSA+BFGS and SPSA+SA as opposed to using SPSA alone can be seen in Figure 1. After 800 design cycles using SPSA, the hybrid SPSA+SA method took 2521 function evaluations of SA while the SPSA+BFGS method took 271 function evaluations to reach similar values which are much better than that reached by using SPSA alone in the entire minimization process. Results indicate that both of the two hybrid methods have capability to find a global optimum more efficiently than the SPSA method. The paper will address issues related to hybridization and its impact on the optimal airfoil shape designs in various contexts. / Singapore-MIT Alliance (SMA)

Analysis of flow and solute transport problem in porous media are affected by uncertainty inbuilt both in boundary conditions and spatial variability in system parameters. The experimental investigation reveals that the parameters may vary in various scales by several orders. These affect the solute plume characteristics in field-scale problem and cause uncertainty in the prediction of concentration. The main focus of the present thesis is to analyze the probabilistic behavior of solute concentration in three dimensional(3-D) heterogeneous porous media. The framework for the probabilistic analysis has been developed using perturbation approach for both spectral based analytical and finite element based numerical method. The results of the probabilistic analysis are presented either in terms of solute plume characteristics or prediction uncertainty of the concentration. After providing a brief introduction on the role of stochastic analysis in subsurface hydrology in chapter 1, a detailed review of the literature is presented to establish the existing state-of-art in the research on the probabilistic analysis of flow and transport in simple and complex heterogeneous porous media in chapter 2. The literature review is mainly focused on the methods of solution of the stochastic differential equation. Perturbation based spectral method is often used for probabilistic analysis of flow and solute transport problem. Using this analytical method a nonlocal equation is solved to derive the expression of the spatial plume moments. The spatial plume moments represent the solute movement, spreading in an average sense. In chapter 3 of the present thesis, local dispersivity if also assumed to be random space function along with hydraulic conductivity. For various correlation coefficients of the random parameters, the results in terms of the field scale effective dispersivity are presented to demonstrate the effect of local dispersivity variation in space. The randomness of local dispersivity is found to reduce the effective fields scale dispersivity. The transverse effective macrodispersivity is affected more than the longitudinal effective macrodispersivity due to random spatial variation of local dispersivity. The reduction in effective field scale longitudinal dispersivity is more for positive correlation coefficient. The applicability of the analytical method, which is discussed in earlier chapter, is limited to the simple boundary conditions. The solution by spectral method in terms of statistical moments of concentration as a function of space and time, require higher dimensional integration. Perturbation based stochastic finite element method(SFEM) is an alternative method for performing probabilistic analysis of concentration. The use of this numerical method for performing probabilistic analysis of concentration. The use of this numerical method is non common in the literature of stochastic subsurface hydrology. The perturbation based SFEM which uses FEM for spatial discretization of the steady state flow and Laplace transform for the solute transport equation, is developed in chapter 4. The SFEM is formulated using Taylor series of the dependent variable upto second-order term. This results in second-order accurate mean and first-order accurate standard deviation of concentration. In this study the governing medium properties viz. hydraulic Conductivity, dispersivity, molecular diffusion, porosity, sorption coefficient and decay coefficient are considered to vary randomly in space. The accuracy of results and computational efficiency of the SFEM are compared with Monte Carle Simulation method(MCSM) for both I-D and 3-D problems. The comparison of results obtained hby SFEM and MCSM indicates that SFEM is capable in providing reasonably accurate mean and standard deviation of concentration. The Laplace transform based SFEM is simpler and advantageous since it does not require any stability criteria for choosing the time step. However it is not applicable for nonlinear transport problems as well as unsteady flow conditions. In this situation, finite difference method is adopted for the time discretization. The first part of the Chapter 5, deals with the formulation of time domain SFEM for the linear solute transport problem. Later the SFEM is extended for a problem which involve uncertainty of both system parameters and boundary/source conditions. For the flow problem, the randomness in the boundary condition is attributed by the random spatial variation of recharge at the top of the domain. The random recharge is modeled using mean, standard deviation and 2-D spatial correlation function. It is observed that even for the deterministic recharge case, the behavior of prediction uncertainty of concentration in the space is affected significantly due to the variation of flow field. When the effect of randomness of recharge condition is included, the standard deviation of concentration increases further. For solute transport, the concentration input at the source is modeled as a time varying random process. Two types of random source at the source is modeled as a time varying random process. Two types of random source condition are considered, firstly the amount of solute mass released at uniform time interval is random and secondly the source is treated as a Poission process. For the case of multiple random mass releases, the stochastic response function due to stochastic system is obtained by using SFEM. Comparing the results for the two type of random sources, it sis found that the prediction uncertainty is more when it is modeled as a Poisson process. The probabilistic analysis of nonlinear solute transport problem using MCSM is often requires large computational cost. The formulation of the alternative efficient method, SFEM, for nonlinear solute transport problem is presented in chapter 6. A general Langmuir-Freundlich isotherm is considered to model the equilibrium mass transfer between aqueous and sorbed phase. In the SFEM formulation, which uses the Taylor Series expansion, the zeroth-order derivatives of concentration are obtained by solving nonlinear algebraic equation. The higher order derivatives are obtained by solving linear equation. During transport, the nonlinear sorbing solutes is characterized by sharp solute fronts with a traveling wave behavior. Due to this the prediction uncertainty is significantly higher. The comparison of accuracy and computational efficiency of SFEM with MCSM for I-D and 3-D problems, reveals that the performance of SFEM for nonlinear problem is good and similar to the linear problem. In Chapter 7, the nonlinear SFEM is extended for probabilistic analysis of biodegrading solute, which is modeled by a set of PDEs coupled with nonlinear Monod type source/sink terms. In this study the biodegradation problem involves a single solute by a single class of microorganisms coupled with dynamic microbial growth is attempted using this methods. The temporal behavior of mean and standard deviation of substrate concentration are not monotonic, they show peaks before reaching lower steady state value. A comparison between the SFEM and MCSM for the mean and standard deviation of concentration is made for various stochastic cases of the I-D problem. In most of the cases the results compare reasonably well. The analysis of probabilistic behavior of substrate concentration for different correlation coefficient between the physical parameters(hydraulic conductivity, porosity, dispersivity and diffusion coefficient) and the biological parameters(maximum substrate utilization rate and the coefficient of cell decay) is performed. It is observed that the positive correlation between the two sets of parameters results in a lower mean and significantly higher standard deviation of substrate concentration. In the previous chapters, the stochastic analysis pertaining to the prediction uncertainty of concentration has been presented for simple problem where the system parameters are modeled as statistically homogeneous random. The experimental investigations in a small watershed, point towards a complex in geological substratum. It has been observed through the 2-D electrical resistivity imaging that the interface between the layers of high conductive weathered zone and low conductive clay is very irregular and complex in nature. In chapter 8 a theoretical model based on stochastic approach is developed to stimulate the complex geological structure of the weathered zone, using the 2-D electrical image. The statistical parameters of hydraulic conductivity field are estimated using the data obtained from the Magnetic Resonance Sounding(MRS) method. Due to the large complexity in the distribution of weathered zone, the stochastic analysis of seepage flux has been carried out by using MCSM. A batter characterization of the domain based on sufficient experimental data and suitable model of the random conductivity field may help to use the efficient SFEM. The flow domain is modeled as (i) an unstructured random field consisting of a single material with spatial heterogeneity, and (ii) a structured random field using 2-D electrical imaging which is composed of two layers of different heterogeneous random hydraulic properties. The simulations show that the prediction uncertainty of seepage flux is comparatively less when structured modeling framework is used rather than the unstructured modeling. At the end, in chapter 9 the important conclusions drawn from various chapters are summarized.