Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Numerical simulations are widely used for predicting and optimising the exploitation of aquifers. They are also used to determine certain physical parameters, for example soil conductivity, by inverse calculations, where the model parameters are changed until the model results correspond optimally to measurements taken on site. The Richards’ equation describes the movement of an unsaturated fluid through porous media, and is characterised as a non-linear partial differential equation. The equation is subject to a number of parameters and is typically computationally expensive to solve. To determine the parameters in the Richards’ equation, inverse modelling studies often need to be undertaken. In these studies, the parameters of a numerical model are varied until the numerical response matches a measured response. Inverse modelling studies typically require 100’s of simulations, which implies that parameter optimisation in unsaturated case studies is common only in small or 1D problems in the literature. As a solution to overcome the computational expense incurred in inverse modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced Order Modelling (ROM) method is proposed in this thesis to speed-up individual simulations. An explanation of the Finite Element Method (FEM) is given using the Galerkin method, followed by a detailed explanation of the Galerkin POD approach. In the development of the Galerkin POD approach, the method of reducing matrices and vectors is shown, and the treatment of Neumann and Dirichlet boundary values is explained. The Galerkin POD method is applied to two case studies. The first case study is the Kogelberg site in the Table Mountain Group near Cape Town in South Africa. The response of the site is modelled at one well over the period of 2 years, and is assumed to be governed by saturated flow, making it a linear problem. The site is modelled as a 3D transient, homogeneous site, using 15 layers and ≈ 20000 nodes, using the FEM implemented on the open-source software FreeFem++. The model takes the evapotranspiration of the fynbos vegetation at the site into consideration, allowing the calculation of annual recharge into the aquifer. The ROM is created from high-fidelity responses taken over time at different parameter points, and speed-up times of ≈ 500 are achieved, corresponding to speed-up times found in the literature for linear problems. The purpose of the saturated groundwater model is to demonstrate that a POD-based ROM can approximate the full model response over the entire parameter domain, highlighting the excellent interpolation qualities and speed-up times of the Galerkin POD approach, when applied to linear problems. A second case study is undertaken on a synthetic unsaturated case study, using the Richards’ equation to describe the water movement. The model is a 2D transient model consisting of ≈ 5000 nodes, and is also created using FreeFem++. The Galerkin POD method is applied to the case study in order to replicate the high-fidelity response. This did not yield in any speed-up times, since the full matrices of non-linear problems need to be recreated at each time step in the transient simulation. Subsequently, a method is proposed in this thesis that adapts the Galerkin POD method by linearising the non-linear terms in the Richards’ equation, in a method named the Linearised Galerkin POD (LGP) method. This method is applied to the same 2D synthetic problem, and results in speed-up times in the range of 10 to 100. The adaptation, notably, does not use any interpolation techniques, favouring a code intrusive, but physics-based, approach. While the use of an intrusively linearised POD approach adds to the complexity of the ROM, it avoids the problem of finding kernel parameters typically present in interpolative POD approaches. Furthermore, the interpolation and possible extrapolation properties inherent to intrusive POD-based ROM’s are explored. The good extrapolation properties, within predetermined bounds, of intrusive POD’s allows for the development of an optimisation approach requiring a very small Design of Experiments (DOE) sets (e.g. with improved Latin Hypercube sampling). The optimisation method creates locally accurate models within the parameter space using Support Vector Classification (SVC). The region inside of the parameter space in which the optimiser is allowed to move is called the confidence region. This confidence region is chosen as the parameter region in which the ROM meets certain accuracy conditions. With the proposed optimisation technique, advantage is taken of the good extrapolation characteristics of the intrusive POD-based ROM’s. A further advantage of this optimisation approach is that the ROM is built on a set of high-fidelity responses obtained prior to the inverse modelling study, avoiding the need for full simulations during the inverse modelling study. In the methodologies and case studies presented in this thesis, initially infeasible inverse modelling problems are made possible by the use of the POD-based ROM’s. The speed up times and extrapolation properties of POD-based ROM’s are also shown to be favourable. In this research, the use of POD as a groundwater management tool for saturated and unsaturated sites is evident, and allows for the quick evaluation of different scenarios that would otherwise not be possible. It is proposed that a form of POD be implemented in conventional groundwater software to significantly reduce the time required for inverse modelling studies, thereby allowing for more effective groundwater management. / AFRIKAANSE OPSOMMING: Die Richards vergelyking beskryf die beweging van ’n vloeistof deur ’n onversadigde poreuse media, en word gekenmerk as ’n nie-lineêre parsiële differensiaalvergelyking. Die vergelyking is onderhewig aan ’n aantal parameters en is tipies berekeningsintensief om op te los. Om die parameters in die Richards vergelyking te bepaal, moet parameter optimering studies dikwels onderneem word. In hierdie studies, word die parameters van ’n numeriese model verander totdat die numeriese resultate die gemete resultate pas. Parameter optimering studies vereis in die orde van honderde simulasies, wat beteken dat studies wat gebruik maak van die Richards vergelyking net algemeen is in 1D probleme in die literatuur. As ’n oplossing vir die berekingskoste wat vereis word in parameter optimering studies, is die gebruik van Eie Ortogonale Ontbinding (POD) as ’n Verminderde Orde Model (ROM) in hierdie tesis voorgestel om individuele simulasies te versnel in die optimering konteks. Die Galerkin POD benadering is aanvanklik ondersoek en toegepas op die Richards vergelyking, en daarna is die tegniek getoets op verskeie gevallestudies. Die Galerkin POD metode word gedemonstreer op ’n hipotetiese gevallestudie waarin water beweging deur die Richards-vergelyking beskryf word. As gevolg van die nie-lineêre aard van die Richards vergelyking, het die Galerkin POD metode nie gelei tot beduidende vermindering in die berekeningskoste per simulasie nie. ’n Verdere gevallestudie word gedoen op ’n ware grootskaalse terrein in die Tafelberg Groep naby Kaapstad, Suid-Afrika, waar die grondwater beweging as versadig beskou word. Weens die lineêre aard van die vergelyking wat die beweging van versadigde water beskryf, is merkwaardige versnellings van > 500 in die ROM waargeneem in hierdie gevallestudie. Daarna was die die Galerkin POD metode aangepas deur die nie-lineêre terme in die Richards vergelyking te lineariseer. Die tegniek word die geLineariserde Galerkin POD (LGP) tegniek genoem. Die aanpassing het goeie resultate getoon, met versnellings groter as 50 keer wanneer die ROM met die oorspronklike simulasie vergelyk word. Al maak die tegniek gebruik van verder lineariseering, is die metode nogsteeds ’n fisika-gebaseerde benadering, en maak nie gebruik van interpolasie tegnieke nie. Die gebruik van ’n fisika-gebaseerde POD benaderings dra by tot die kompleksiteit van ’n volledige numeriese model, maar die kompleksiteit is geregverdig deur die merkwaardige versnellings in parameter optimerings studies. Verder word die interpolasie eienskappe, en moontlike ekstrapolasie eienskappe, inherent aan fisika-gebaseerde POD ROM tegnieke ondersoek in die navorsing. In die navorsing word ’n tegniek voorgestel waarin hierdie inherente eienskappe gebruik word om plaaslik akkurate modelle binne die parameter ruimte te skep. Die voorgestelde tegniek maak gebruik van ondersteunende vektor klassifikasie. Die grense van die plaaslik akkurate model word ’n vertrouens gebeid genoem. Hierdie vertrouens gebied is gekies as die parameter ruimte waarin die ROM voldoen aan vooraf uitgekiesde akkuraatheidsvereistes. Die optimeeringsbenadering vermy ook die uitvoer van volledige simulasies tydens die parameter optimering, deur gebruik te maak van ’n ROM wat gebaseer is op die resultate van ’n stel volledige simulasies, voordat die parameter optimering studie gedoen word. Die volledige simulasies word tipies uitgevoer op parameter punte wat gekies word deur ’n proses wat genoem word die ontwerp van eksperimente. Verdere hipotetiese grondwater gevallestudies is onderneem om die LGP en die plaaslik akkurate tegnieke te toets. In hierdie gevallestudies is die grondwater beweging weereens beskryf deur die Richards vergelyking. In die gevalle studie word komplekse en tyd-rowende modellerings probleme vervang deur ’n POD gebaseerde ROM, waarin individuele simulasies merkwaardig vinniger is. Die spoed en interpolasie/ekstrapolasie eienskappe blyk baie gunstig te wees. In hierdie navorsing is die gebruik van verminderde orde modelle as ’n grondwaterbestuursinstrument duidelik getoon, waarin voorsiening geskep word vir die vinnige evaluering van verskillende modellering situasies, wat andersins nie moontlik is nie. Daar word voorgestel dat ’n vorm van POD in konvensionele grondwater sagteware geïmplementeer word om aansienlike versnellings in parameter studies moontlik te maak, wat na meer effektiewe bestuur van grondwater sal lei.

The primary objective of this work is to develop an approach for evaluating GHG mitigation strategies that considers the detailed operation of the electricity system in question and to ascertain whether considering the detailed operation of the electricity system materially affects the assessment. A secondary objective is to evalute the potential benefit of flexible CO2 capture and storage. An electricity system simlator is developed based upon a deregulated electricity system containing markets for both real and reserve power. Using the IEEE RTS ???96 as a test case, the performance of the electricity system is benchmarked with GHG regulation. Two different implementations of CO2 capture are added to the electricity system ??? fixed CO2 capture and flexible CO2 capture ??? and the impact of having CCS is assessed. The results indicate that: - the assessment of GHG mtigation strategies for the electricity generation subsector should consider the detailed operation of the electricity system in question, - cost of generation alone is not necessarily a good indicator of the economic impact of GHG regulation or the deployment of a GHG mitigation strategy, - adding CCS, at even a single generating unit, can significantly reduce GHG emissions and moderate the ecnomic impact of GHG regulation relative to the cases where CCS is not present, and - a generating unit with a flexible CCS processes participates preferentially in the reserve market enabling it to increase its net energy benefit. It is conclued that there is a significant potential advantage to generating units with flexible CCS processes. The flexibiity of existing and novel CCS process should be an assessment and design criterion, respectively, and the development of novel CCS processes with optimial operability is a suggested area of future research activity. A reduced-order model of a coal-fired generating unit with flexible CO2 capture is developed and integrated into the MINLP formulation of an economic dispatch model. Both of these efforts, not observed previously in the literature, constitute an important contribution of the work as the methodology provides a template for future assessmments of CCS and other electricity mitigation strategies in the electricity generation subsector.

On considère un écoulement compressible bidimensionnel, autour d'une cavité ouverte. Des d'instabilité, auto-entretenues par l'effet de rétroaction de l'écrasement de la couche de cisaillement sur le bord aval de la cavité, génèrent des émissions acoustiques qu'il faut réduire. Des simulations numériques directes (DNS) permettent d'obtenir, avec ou sans actionnement, un modèle précis de l'écoulement. A partir des champs issus de la simulation, des décompositions orthogonales de modes propres (POD) sont proposées pour bâtir, par projection de Galerkin sur les équations isentropiques, des modèles d'ordre réduit non linéaires en prenant en compte l'actionnement (le contrôle). Pour éviter la divergence temporelle, les coefficients du système dynamique non forcé sont calibrés par diverses approches originales dont une basée sur la sensiblité modale. A partir du système dynamique forcé par un actionnement multifréquentiel (présent aussi dans les DNS), un contrôle en boucle fermée linéaire quadratique gaussien est proposé sur un système linéarisé. La reconstruction de l'état est basée sur une estimation stochastique linéaire sur 6 points de pression. Le contrôle optimal obtenu s'avère être périodique à la fréquence du second mode de Rossiter, qui est exactement celles des instabilits auto-entretenues dans la cavité. Par introduction de ce contrôle dans les simulations numériques directes, nous avons obtenu une réduction du bruit (faible) sur la fréquence du contrôle. / We consider a two dimensional compressible flow around an open cavity. The Flow around a cavity is characterised by a self-sustained mechanism in which the shear layer impinges on the downstream edge of the cavity resulting in an acoustic feedback mechanism which must be reduced. Direct Numerical Simulations (DNS) of the flow at a representative Reynolds number has been carried to obtain pressure and velocity fields, both for the case of unactuated and multi frequency actuation. These fields are then used to extract energy ranked coherent structures also called as the Proper Orthogonal Decomposition (POD) modes. A Reduced Order Model is constructed by a Galerkin projections of the isentropic compressible equations. The model is then extended to include the effect of control. To avoid the divergence of the model while integrating in time various calibration techniques has been utillized. A new method of calibration which minimizes a linear functional of error, based on modal sensitivity is proposed. The calibrated low order model is used to design a feedback control of the Linear Quadratic Gaussian (LQG) type, coupled with an observer. For the experimental implementation of the controller, a state estimate based on the observed pressure measurements at 6 different locations, is obtained through a Linear Stochastic Estimation (LSE). The optimal control obtained is periodic with a frequency corresponding to the second Rossiter mode of the cavity. Finally the control obtained is introduced into the DNS to obtain a decrease in spectra of the cavity acoustic mode.

Detonation is a supersonic mode of combustion that is modeled by a system of conservation laws of compressible fluid mechanics coupled with the equations describing thermodynamic and chemical properties of the fluid. Mathematically, these governing equations admit steady-state travelling-wave solutions consisting of a leading shock wave followed by a reaction zone. However, such solutions are often unstable to perturbations and rarely observed in laboratory experiments. The goal of this work is to study the stability of travelling-wave solutions of detonation models by the following novel approach. We linearize the governing equations about a base travelling-wave solution and solve the resultant linearized problem using high-order numerical methods. The results of these computations are postprocessed using dynamic mode decomposition to extract growth rates and frequencies of the perturbations and predict stability of travelling-wave solutions to infinitesimal perturbations. We apply this approach to two models based on the reactive Euler equations for perfect gases. For the first model with a one-step reaction mechanism, we find agreement of our results with the results of normal-mode analysis. For the second model with a two-step mechanism, we find that both types of admissible travelling-wave solutions exhibit the same stability spectra. Then we investigate the Fickett’s detonation analogue coupled with a particular reaction-rate expression. In addition to the linear stability analysis of this model, we demonstrate that it exhibits rich nonlinear dynamics with multiple bifurcations and chaotic behavior.

La structure de l’oeil permet d’observer la microcirculation, grâce aux caméras de fond d’oeil. Ces appareils sont bon marché et couramment utilisés dans la pratique clinique, permettant le dépistage de maladies oculaires. La capacité des vaisseaux à adapter leur diamètre (autorégulation) afin de réguler le débit sanguin est importante dans la microcirculation. L’hémodynamique de l’oeil est impactée par la pression à l’intérieur du globe oculaire (IOP), qui est à son tour influencée par le flux sanguin oculaire. Les altérations de l’autorégulation et l’IOP jouent un rôle dans les maladies oculaires. La modélisation mathématique peut aider à interpréter l’interaction entre ces phénomènes et à mieux exploiter les données médicales disponibles. Dans la première partie, nous présentons un modèle simplifié d’interaction fluidestructure qui inclut l’autorégulation, appliqué à un reseau 3D obtenu par imagerie médicale. Les cellules musculaires lisses regulant le diamètre du vaisseau sont modélisés dans la structure. Ensuite, nous utilisons des équations de poroélasticité pour décrire le flux sanguin dans la choroïde, dans un modèle multi-compartiments de l’oeil. Cette approche permet de rendre compte de la transmission de la pulsatilité de la choroïde à la chambre antérieure, où l’IOP est mesurée. Nous présentons des résultats préliminaires sur la choroïde, l’humeur aqueuse et sur la choroïde couplée avec la vitrée. Enfin, nous présentons un modèle d’ordre réduit pour accélérer des simulations multi-physique. Des modèles de haute précision sont utilisés pour les compartiments d’intérêt et une représentation réduite de l’opérateur de Steklov-Poincaré est utilisée pour les autres compartiments. / The structure of the eye offers a unique opportunity to directly observe the microcirculation, by means, for instance, of fundus camera, which are cheap devices commonly used in the clinical practice. This can facilitate the screening of systemic deseases such as diabetes and hypertension, or eye diseases such as glaucoma. A key phenomenon in the microcirculation is the autoregulation, which is the ability of certain vessels to adapt their diameter to regulate the blood flow rate in response to changes in the systemic pressure or metabolic needs. Impairments in autoregulation are strongly correlated with pathological states. The hemodynamics in the eye is influenced by the intraocular pressure (IOP), the pressure inside the eye globe, which is in turn influenced by the ocular blood flow. The interest in the IOP stems from the fact that it plays a role in several eye-diseases, such as glaucoma. Mathematical modelling can help in interpreting the interplay between these phenomena and better exploit the available data. In the first part of the thesis we present a simplified fluid-structure interaction model that includes autoregulation. A layer of fibers in the vessel wall models the smooth muscle cells that regulate the diameter of the vessel. The model is applied to a 3D image-based network of retinal arterioles. In the second part, we propose a multi-compartments model of the eye. We use the equations of poroelasticity to model the blood flow in the choroid. The model includes other compartments that transmit the pulsatility from the choroid to the anterior chamber, where the measurements of the IOP are actually performed. We present some preliminary results on the choroid, the aqueous humor and on the choroid coupled with the vitreous. Finally, we present a reduced order modelling technique to speed up multiphysics simulations. We use high fidelity models for the compartments of particular interest from the modelling point of view. The other compartments are instead replaced by a reduced representation of the corresponding Steklov-Poincaré operator.

Membrane Distillation (MD) is an emerging sustainable desalination technique. While MD has many advantages and can be powered by solar thermal energy, its main drawback is the low water production rate. However, the MD process has not been fully optimized in terms of its manipulated and controlled variables. This is largely due to the lack of adequate dynamic models to study and simulate the process. In addition, MD is prone to membrane fouling, which is a fault that degrades the performance of the MD process. This work has three contributions to address these challenges. First, we derive a mathematical model of Direct Contact Membrane Distillation (DCMD), which is the building block for the next parts. Then, the proposed model is extended to account for membrane fouling and an observer-based fouling detection method is developed. Finally, various control strategies are implemented to optimize the performance of the DCMD solar-powered process. In part one, a reduced-order dynamic model of DCMD is developed based on lumped capacitance method and electrical analogy to thermal systems. The result is an electrical equivalent thermal network to the DCMD process, which is modeled by a system of nonlinear differential algebraic equations (DAEs). This model predicts the water-vapor flux and the temperature distribution along the module length. Experimental data is collected to validate the steady-state and dynamic responses of the proposed model, with great agreement demonstrated in both. The second part proposes an extension of the model to account for membrane fouling. An adaptive observer for DAE systems is developed and convergence proof is presented. A method for membrane fouling detection is then proposed based on adaptive observers. Simulation results demonstrate the performance of the membrane fouling detection method. Finally, an optimization problem is formulated to maximize the process efficiency of a solar-powered DCMD. The adapted method is known as Extremum Seeking (ES). A Newton-based ES is designed and the proposed model is used to accurately forecast the distilled water flux. Although good results are obtained with this method, a practical modification to the ES scheme is proposed to enhance the practical stability.

Une modélisation d'ordre réduit basée sur le partitionnement des données (cluster-based reduced-order modelling ou CROM) est développée pour identifier de manière non supervisée des mécanismes d'interaction non linéaires. La connaissance de ces mécanismes permet de pronostiquer la formation d’événements souhaitables ou non. L’approche proposée adopteun point de vue probabiliste en mettant à profit la linéarité de l’équation d’évolution de probabilité qui tient cependant compte d'éventuelles actions non linéaires des actionneurs. Le cadre est appliqué à l’attracteur de Lorenz, aux données numériques de la couche de mélange, à la turbulence tridimensionnelle du sillage d’un corps non profilé, d’un train, et aux données expérimentales d’un moteur à combustion.Pour ces exemples, le CROM permettait l'identification des quasi-attracteurs par exemple les deux régimes d’écoulement de la couche de mélange ou les états bimodaux du corps Ahmed . Les transitions principales entre ces quasi-attracteurs sont caractérisées par des regroupements de données appelé « flipper cluster ». L'identification de ces « flipper cluster » peut servir pour le contrôle des écoulements en utilisant le partitionnement des données obtenues par exemple de l'évolution temporelle de la traînée ou de la portance.Un contrôle en boucle fermé basé sur la CROM est appliqué à un écoulement le long d'une rampe courbée en vue de diminuer les extensions de la zone de recirculation par rapport à la meilleure excitation périodique en boucle ouverte. L'actionneur est mis en marche en fonction des regroupements préalablement observés. Le résultat est comparé à l’ensemble des lois de contrôle définies par toutes les combinaisons possibles des « on » et « of » par les regroupements de données. Bien quele contrôle basé sur la CROM ne permet pas de réduire la zone de recirculation par rapport à la réduction maximale en boucle ouverte, 28 % de l'apport d 'énergie nécessaire et 81 % pour une loi de contrôle particulière peuvent être économisé. / A cluster-based reduced-order modeling strategy is developed for the unsupervised identification of nonlinear flow mechanisms and precursors to desirable or undesirable events. The proposed approach assumes a probabilistic viewpoint taking advantage of the linearity of the evolution equation for the probability while including nonlinear actuation dynamics.The framework is applied to the Lorenz attractor, numerical data of the spatially evolving mixing layer, the three-dimensional turbulent wake of a bluf body, of a train, and experimental data of a combustion engine.For these examples, CROM has been shown to identify quasi-attractors such as the two shedding regimes of the mixing layer or the bimodal states of the Ahmed body; main transition processes between those quasiattractors are characterized by branching regions or flipper cluster; desirable phase space regions and possible actuation mechanisms areindicated by analysis of cluster features like drag and lift forces which can be further exploited for control purposes.In particular, a CROM-based feedback control is applied to a separating flow over a smooth ramp to examine whether the recirculation area can be diminished compared to the best open-loop periodic excitation by turning the actuation on or of depending on the applicable cluster. The CROMbased control is compared to the complete set of control laws defined byall possible combinations of 'on' and 'of' for the given set of clusters.While the recirculation area cannot be further decreased compared to the best open-loop forcing, a similar size can be achieved for 28% (CROMbased control) or 81% (one particular control law) savings in the control input energy.

No description available.

Le désaccordage des roues aubagées est une thématique de recherche d’un intérêt tout particulier pour l’industrie aéronautique, en recherche constante d’outils de calcul toujours plus prédictifs et performants pour répondre aux exigences croissantes des organismes de certification. Si le phénomène est aujourd’hui relativement bien maîtrisé dans un cadre linéaire, la prise en compte des non-linéarités dans l’étude du désaccordage reste encore problématique, notamment en raison du manque de méthode adaptée pour mener ce type d’analyses sur des modèles industriels. L’objectif principal de ce travail de thèse est de proposer une nouvelle méthode de calcul permettant de déterminer efficacement la réponse forcée d’une roue aubagée désaccordée, en tenant compte de l’impact des non-linéarités sur la dynamique de la structure à l’échelle macroscopique. La méthode développée repose sur le concept de sous-structuration, et exploite la notion de mode complexe non-linéaire pour capturer les non-linéarités dans l’espace de réduction de chaque sous-structure. En adoptant une approche fréquentielle, les sous-structures sont représentées par des super-éléments non-linéaires, dont l’assemblage conduit au modèle réduit de la roue désaccordée. La résolution du système mathématique obtenu est ensuite réalisée numériquement par des techniques itératives. La méthode développée a pu être testée et validée sur différents systèmes soumis à des non-linéarités de frottement, allant du simple modèle phénoménologique à un modèle éléments finis de roue aubagée industrielle. Sur des modèles à paramètres concentrés de taille relativement faible, les performances très intéressantes de cette méthode permettent de conduire des études statistiques quantitatives sur l’impact du désaccordage en présence de non-linéarités. Les résultats obtenus suggèrent que le comportement du système non-linéaire face au désaccordage est susceptible d’être significativement différent du comportement de son homologue linéaire, d’où l’intérêt de mener ce type d’investigations. Les performances de cette méthode ont également pu être confirmées sur des modèles éléments finis de grande taille, en permettant de réaliser à un coût raisonnable des simulations de réponse forcée non-linéaire sur une roue industrielle désaccordée. / Mistuning of bladed disks has been a key topic of research for the aeronautics industry. To get accreditation for their engines, manufacturers must comply with evermore stringent requirements, and thus constantly seek for better simulation tools. Even though the phenomenon is well understood nowadays for linear systems, nonlinearities are still seldom taken into account when dealing with the mistuning of industrial structures, partly due to the lack of a dedicated method to tackle such a complex problematic. The main objective of this work is to develop a novel method allowing to compute efficiently the forced response of a mistuned bladed disk, while taking into account the impact of nonlinearities on the vibrations at a macroscopic scale. The method derived relies on a substructuring approach, and uses the concept of nonlinear complex modes to capture the nonlinearities in the reduction basis of each substructure. In the frequency domain, the substructures take the form of nonlinear superelements, which once assembled lead to the reduced-order model of the mistuned bladed disk. The resulting mathematical system is then solved by means of iterative solvers. This new method is tested and validated on different systems subjected to dry friction nonlinearities, from basic phenomenological models to large-scale finite element models of industrial structures. On lumped-parameter models, the performance of this method allows to investigate the statistical impact of mistuning in the presence of nonlinearities, by performing thousands of simulations. The results suggest that the behaviour of the nonlinear model can be significantly different from that of the linear one, hence the importance to carry out such investigations. The capabilities of the method have also been confirmed on large-scale models, by performing several forced response computations on a nonlinear and mistuned finite element model, at a reasonable cost

<div>Excessive deformation and stress of skin following reconstructive surgery plays a crucial role in wound healing, often leading to complications. Yet, despite of this concern, surgeries are still planned and executed based on each surgeon's training and experience rather than quantitative engineering tools. The limitations of current treatment planning and execution stem in part from the difficulty in predicting the mechanical behavior of skin, challenges in directly measuring stress in the operating room, and inability to predict the long term adaptation of skin following reconstructive surgery. Computational modeling of soft tissue mechanics has emerged as an ideal candidate to determine stress contours over sizable skin regions in realistic situations. Virtual surgeries with computational mechanics tools will help surgeons explore different surgeries preoperatively, make prediction of stress contours, and eventually aid the surgeon in planning for optimal wound healing. While there has been significant progress on computational modeling of both reconstructive surgery and skin mechanical and mechanobiological behavior, there remain major gaps preventing computational mechanics to be widely used in the clinical setting. At the preoperative stage, better calibration of skin mechanical properties for individual patients based on minimally invasive mechanical tests is still needed. One of the key challenges in this task is that skin is not stress-free in vivo. In many applications requiring large skin flaps, skin is further grown with the tissue expansion technique. Thus, better understanding of skin growth and the resulting stress-free state is required. The other most significant challenge is dealing with the inherent variability of mechanical properties and biological response of biological systems. Skin properties and adaptation to mechanical cues changes with patient demographic, anatomical location, and from one individual to another. Thus, the precise model parameters can never be known exactly, even if some measurements are available. Therefore, rather than expecting to know the exact model describing a patient, a probabilistic approach is needed. To bridge the gaps, this dissertation aims to advance skin biomechanics and computational mechanics tools in order to make virtual surgery for clinical use a reality in the near future. In this spirit, the dissertation constitutes three parts: skin growth and its incompatibility, acquisition of patient-specific geometry and skin mechanical properties, and uncertainty analysis of virtual surgery scenarios.</div><div>Skin growth induced by tissue expansion has been widely used to gain extra skin before reconstructive surgery. Within continuum mechanics, growth can be described with the split of the deformation gradient akin to plasticity. We propose a probabilistic framework to do uncertainty analysis of growth and remodeling of skin in tissue expansion. Our approach relies on surrogate modeling through multi-fidelity Gaussian process regression. This work is being used calibrate the computational model against animal model data. Details of the animal model and the type of data obtained are also covered in the thesis. One important aspect of the growth and remodeling process is that it leads to residual stress. It is understood that this stress arises due to the nonhomogeneous growth deformation. In this dissertation we characterize the geometry of incompatibility of the growth field borrowing concepts originally developed in the study of crystal plasticity. We show that growth produces unique incompatibility fields that increase our understanding of the development of residual stress and the stress-free configuration of tissues. We pay particular attention to the case of skin growth in tissue expansion.</div><div>Patient-specific geometry and material properties are the focus on the second part of the thesis. Minimally invasive mechanical tests based on suction have been developed which can be used in vivo, but these tests offer only limited characterization of an individual's skin mechanics. Current methods have the following limitations: only isotropic behavior can be measured, the calibration problem is done with inverse finite element methods or simple analytical calculations which are inaccurate, the calibration yields a single deterministic set of parameters, and the process ignores any previous information about the mechanical properties that can be expected for a patient. To overcome these limitations, we recast the calibration problem in a Bayesian framework. To sample from the posterior distribution of the parameters for a patient given a suction test, the method relies on an inexpensive Gaussian process surrogate. For the patient-specific geometry, techniques such as magnetic resonance imaging or computer tomography scans can be used. Such approaches, however, require specialized equipment and set up and are not affordable in many scenarios. We propose to use multi-view stereo (MVS) to capture patient-specific geometry.</div><div>The last part of the dissertation focuses on uncertainty analysis of the reconstructive procedure itself. To achieve uncertainty analysis in the clinical setting we propose to create surrogate and reduced order models, especially principal component analysis and Gaussian process regression. We first show the characterization of stress profiles under uncertainty for the three most common flap designs. For these examples we deal with idealized geometries. The probabilistic surrogates enable not only tasks such as fast prediction and uncertainty quantification, but also optimization. Based on a global sensitivity analysis we show that the direction of anisotropy of skin with respect to the flap geometry is the most important parameter controlled by the surgeon, and we show hot to optimize the flap in this idealized setting. We conclude with the application of the probabilistic surrogates to perform uncertainty analysis in patient-specific geometries. In summary, this dissertation focuses on some of the fundamental challenges that needed to be addressed to make virtual surgery models ready for clinical use. We anticipate that our results will continue to shape the way computational models continue to be incorporated in reconstructive surgery plans.</div>