The research in this thesis is undertaken by observing that modern systems are becoming more and more complex and safety-critical due to the increasing requirements on system smartness and autonomy, and as a result health monitoring system needs to be developed to meet the requirements on system safety and reliability. The state-of-the-art approaches to monitoring system status are model based Fault Diagnosis (FD) systems, which can fuse the advantages of system physical modelling and sensors' characteristics. A number of model based FD approaches have been proposed. The conventional residual based approaches by monitoring system output estimation errors, however, may have certain limitations such as complex diagnosis logic for fault isolation, less sensitiveness to system faults and high computation load. More importantly, little attention has been paid to the problem of fault diagnosis system verification which answers the question that under what condition (i.e., level of uncertainties) a fault diagnosis system is valid. To this end, this thesis investigates the design and verification of fault diagnosis algorithms. It first highlights the differences between two popular FD approaches (i.e., residual based and fault estimation based) through a case study. On this basis, a set of uncertainty estimation algorithms are proposed to generate fault estimates according to different specifications after interpreting the FD problem as an uncertainty estimation problem. Then FD algorithm verification and threshold selection are investigated considering that there are always some mismatches between the real plant and the mathematical model used for FD observer design. Reachability analysis is drawn to evaluate the effect of uncertainties and faults such that it can be quantitatively verified under what condition a FD algorithm is valid. First the proposed fault estimation algorithms in this thesis, on the one hand, extend the existing approaches by pooling the available prior information such that performance can be enhanced, and on the other hand relax the existence condition and reduce the computation load by exploiting the reduced order observer structure. Second, the proposed framework for fault diagnosis system verification bridges the gap between academia and industry since on the one hand a given FD algorithm can be verified under what condition it is effective, and on the other hand different FD algorithms can be compared and selected for different application scenarios. It should be highlighted that although the algorithm design and verification are for fault diagnosis systems, they can also be applied for other systems such as disturbance rejection control system among many others.

L'assimilation de données pour les géosciences est une discipline cherchant à améliorer notre connaissance d'un système physique en se basant sur l'information issue de modèles numériques simulant ce système et sur l'information issue des mesures observant ce système. Les méthodes d'assimilation de données traditionnellement utilisées (e.g. le 4DVar ou les filtres de Kalman d'ensemble) reposent sur des hypothèses de Gaussianité des probabilités en jeu et de linéarité des modèles. Avec la complexification des modèles et des réseaux d'observations, ces hypothèses sont de plus en plus injustifiées et donc pénalisantes. Cette complexification est particulièrement forte en océanographie couplée à la biogéochimie marine.Les objectifs de cette thèse sont de mieux comprendre l'apparition des non-Gaussianités dans un problème d'estimation, d'envisager une méthode d'assimilation de données adaptée aux problèmes fortement non-Gaussiens et, dans le cadre du couplage de la dynamique océanique et de la biogéochimie marine, d'explorer la pertinence de l'utilisation de méthodes non-Gaussiennes.Dans un premier temps, une étude méthodologique est conduite. Cette étude, appuyé par des illustrations avec le modèle de Lorenz à trois variables, permet de mettre en évidence les limitations des méthodes traditionnellement utilisées, face à des problèmes non-Gaussiens. Cette étude aboutit sur le développement d'un filtre d'assimilation de données d'ensemble entièrement non-Gaussien : le Multivariate Rank Histogram Filter (MRHF).Il est montré que le MRHF est performant dans des régimes fortement non-Gaussiens (notamment dans un régime bimodal) pour un nombre de membres relativement faible.Dans un second temps, une étude numérique est conduite. Cette étude est réalisée aux travers d'expériences jumelles basées sur un modèle vertical 1D, ModECOGeL, couplant la dynamique et la biogéochimie en mer Ligure. Nous simulons différents réseaux d'observations combinant des profils in situ et des données satellites. Plusieurs méthodes d'assimilation sont alors comparées à l'aide de diagnostics d'évaluation d'ensemble avancés.Nos expériences montrent l'impact du réseau d'observations et des variables de contrôle, sur le degré de non-Gaussianité d'un problème d'estimation. Le contrôle de la partie dynamique du modèle par des observations de la dynamique à différentes fréquences est un problème quasi-Gaussien, qu'un filtre aux moindres carrés, tel l'Ensemble Transform Kalman Filter, résout bien. En revanche pour ces mêmes observations, le contrôle de la biogéochimie s'avère être un problème non-Gaussien et nécessite l'utilisation d'un filtre non-Gaussien.Enfin, il est montré que l'assimilation de la couleur de l'eau, pour le contrôle mixte de la dynamique et de la biogéochimie, est améliorée par des méthodes adaptées aux non-Gaussianités, tel l'Ensemble Kalman Filter anamorphosé. De plus, l'augmentation de la fréquence d'observation de la couleur de l'eau rend incontournable l'utilisation de filtres fondamentalement non-Gaussiens comme le MRHF. / Data assimilation for Geosciences is a discipline seeking to improve our knowledge of a physical system based on the information from numerical models simulating this system and the information from the measures observing this system. The data assimilation methods traditionally used (eg the 4DVAR or the ensemble Kalman filters) are based on assumptions of Gaussianity of the probabilities involved and linearity of the models. With the increasing complexity of models and observation networks, these assumptions are increasingly unjustified and therefore penalizing. This complexity is particularly strong in oceanography coupled with marine biogeochemistry.The objectives of this thesis are to understand the appearance of non Gaussianity in an estimation problem, to think out a data assimilation method adapted to highly non Gaussian problems and, in the coupling of ocean dynamics and marine biogeochemistry, to explore the relevance of the use of non Gaussian methods.At first, a methodological study is conducted. This study, supported by illustrations with the three variable Lorenz model, allows to highlight the limitations of traditional methods when facing non Gaussian problems. This study led to the development of a fully non Gaussian data assimilation filter : the Multivariate Rank Histogram Filter (MRHF).It is shown that the MRHF is efficient in highly non Gaussian regimes (including in a bimodal regime) for a relatively small number of members.Secondly, a numerical study is conducted. This study is conducted with twin experiments based on a 1D vertical model, ModECOGeL, coupling dynamics and biogeochemistry in the Ligurian Sea. We simulate different observation networks combining in situ profiles and satellite data. Several data assimilation methods are then compared using advanced ensemble evaluation diagnoses.Our experiments show the impact of observation networks and controled variables on the degree of non Gaussianity in an estimation problem. The control of the dynamic part of the model by observations of the dynamics at different frequencies is a quasi Gaussian problem, which a least squared filter such as the Ensemble Transform Kalman Filter solves well. In contrast, for the same observations, the control of biogeochemistry proves to be a non Gaussian problem and requires the use of a non Gaussian filter. Finally, it is shown that assimilation of ocean color data, for the joint control of the dynamic and the biogeochemistry, is improved by methods adapted for non Gaussianities such as the Anamorphosed Ensemble Kalman Filter. In addition, increasing the ocean color observation frequency makes unavoidable the use of fundamentally non Gaussian filters such as the MRHF.

Na última década foram descobertas jazidas de petróleo e gás em águas profundas ao longo da costa sudeste do Brasil, o que tem levado à reavaliação de conceitos e técnicas até então utilizados para sua exploração em pequenas profundidades. Parâmetros que anteriormente eram supostamente não críticos passaram a ser relevantes no dimensionamento estrutural dos risers. Efetivamente, a descoberta de novas jazidas na Bacia de Santos incentivou o desenvolvimento de pesquisas nesta área, no Brasil e no mundo. Os depósitos de petróleo e gás encontrados abaixo da camada de sal (daí serem referidas por pré-sal) ocorrem em áreas com lâmina dágua de mais de 2.000 metros, requerendo novas tecnologias para viabilizar sua extração. Os risers de produção nada mais são do que tubulações que levam petróleo e gás do fundo do oceano para a superfície. Nas suas diversas configurações geométricas (vertical, em catenária, lazy wave, entre outros), são elementos estruturais extremamente esbeltos, que devem suportar carregamentos dinâmicos oriundos da correnteza marítima em grande profundidade, ondas de superfície, escoamento interno e deslocamentos impostos, atendendo a exigentes critérios de projeto. O riser vertical será o foco deste trabalho, no qual se pretende utilizar modelos matemáticos com poucos graus de liberdade, denominados modelos de ordem reduzida (MOR), mas com adequada capacidade de representação qualitativa e quantitativa da resposta estrutural, fazendo uso de modos não lineares como funções de projeção, dentro do método de Galerkin não linear. Os modos não lineares, intensivamente estudados no Grupo de Pesquisa Dinâmica, Estabilidade e Controle de Sistemas Estruturais da Escola Politécnica da USP, por conterem intrinsecamente informações de harmônicos de ordem superior, são capazes de, em menor número do que os modos lineares utilizados no método da superposição modal clássico, descreverem acuradamente a resposta do sistema não linear. Serão utilizados procedimentos baseados tanto no método das variedades invariantes, quanto no método das múltiplas escalas, em modelos analíticos. Para a redução de graus de liberdade, será utilizado o método de projeção que se baseia na imposição da igualdade entre os trabalhos virtuais dos modelos de alta e baixa hierarquia (MOR), de sorte que o sistema sob carregamento dinâmico possa ser estudado em espaço de fase de baixa dimensão. A presente pesquisa, além dos desafios acadêmicos inerentes ao tema, apresenta evidente relevância econômica e estratégica para o País. / In the last decade, deposits of oil and gas under deep waters were discovered along the Brazilian Southeast coast, which led to reassessment of concepts and techniques previously used for their exploitation under shallow waters. Parameters that were not previously considered to be critical became relevant in the structural design of the risers. Indeed, the discovery of new deposits in the so-called Santos Basin encouraged the development of research in this area in Brazil and worldwide. The oil and gas deposits found in the pre-salt layer occur in waters deeper than 2,000 meters, requiring new technologies to facilitate their extraction. The risers of production are nothing more than pipes that carry oil and gas from the ocean to the surface. In its various geometric configurations (vertical, catenary, lazy waves, etc.), they are extremely slender structural elements, which must withstand dynamic loads from deep currents, surface waves, internal flow and imposed motions, observing the strict design criteria regarding ultimate and service limit states. The vertical riser will be the focus of this work, which uses mathematical models with few degrees of freedom, known as reduced-order models (ROM), but with adequate capacity to represent the structural response both qualitatively and quantitatively, using non-linear modes as projection functions within the non-linear Galerkin method. The non-linear modes were intensively studied in the research group \"Dynamics, Stability and Control of Structural Systems\" at the Escola Politécnica of USP. Because they contain information of higher-order harmonics, they are able to accurately describe the response of the nonlinear system, using a smaller number of modes than the linear modes used in the classical modal superposition method. Procedures based on the method of invariant manifold and the method of multiple scales alike will be applied to analytical continuum models (with infinite number of degrees of freedom). For the reduction of degrees of freedom, a method based on the identification of the virtual works in both the high-hierarchy and the ROM will be used, so that the system under dynamic loading can be studied in a low-dimension phase space. 13 This research, in addition to academic challenges inherent to the subject has obvious economic and strategic importance for the country.

Bien que les méthodes de calcul peuvent donner de bons résultats, ils ne sont généralement pas en accord avec exactement les données d'essais en vol en raison des incertitudes dans les modelé de calcul de structure et aérodynamiques. Une méthode efficace est nécessaire pour la mise à jour des modelé aeroelastiques en utilisant les données d'essais en vol, les données d'essais de vibration au sol ( GVT ) et les données de soufflerie. Tout d'abord tous les développements ont été valides sur une section de l'aile 2D et sur un modèle 3D simple et après l'approche POD a été applique= a une configuration industrielle (modèle de soufflerie aile-fuselage et modèle d' avions complète). / Although computational methods can provide good results, they usually do not agree exactly with the flight test data due to uncertainties in structural and aerodynamic computational models. An effective method is required for updating computational aeroelastic models using the flight test data along with Ground Vibration Test (GVT) data and wind tunnel data. Firstly all developments have been validated on a 2D wing section and on a simple 3D model and after the POD approach has been applied to an industrial configuration (wing-fuselage wind tunnel model and complete aircraft model).

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

Ce travail de recherche présente une extension de la construction classique des modèles réduits (ROMs) obtenus par analyse modale, en dynamique numérique des structures linéaires. Cette extension est basée sur une stratégie de projection multi-niveau, pour l'analyse dynamique des structures complexes en présence d'incertitudes. De nos jours, il est admis qu'en dynamique des structures, la prévision sur une large bande de fréquence obtenue à l'aide d'un modèle éléments finis doit être améliorée en tenant compte des incertitudes de modèle induites par les erreurs de modélisation, dont le rôle croît avec la fréquence. Dans un tel contexte, l'approche probabiliste non-paramétrique des incertitudes est utilisée, laquelle requiert l'introduction d'un ROM. Par conséquent, ces deux aspects, évolution fréquentielle des niveaux d'incertitudes et réduction de modèle, nous conduisent à considérer le développement d'un ROM multi-niveau, pour lequel les niveaux d'incertitudes dans chaque partie de la bande de fréquence peuvent être adaptés. Dans cette thèse, on s'intéresse à l'analyse dynamique de structures complexes caractérisées par la présence de plusieurs niveaux structuraux, par exemple avec un squelette rigide qui supporte diverses sous-parties flexibles. Pour de telles structures, il est possible d'avoir, en plus des modes élastiques habituels dont les déplacements associés au squelette sont globaux, l'apparition de nombreux modes élastiques locaux, qui correspondent à des vibrations prédominantes des sous-parties flexibles. Pour ces structures complexes, la densité modale est susceptible d'augmenter fortement dès les basses fréquences (BF), conduisant, via la méthode d'analyse modale, à des ROMs de grande dimension (avec potentiellement des milliers de modes élastiques en BF). De plus, de tels ROMs peuvent manquer de robustesse vis-à-vis des incertitudes, en raison des nombreux déplacements locaux qui sont très sensibles aux incertitudes. Il convient de noter qu'au contraire des déplacements globaux de grande longueur d'onde caractérisant la bande BF, les déplacements locaux associés aux sous-parties flexibles de la structure, qui peuvent alors apparaître dès la bande BF, sont caractérisés par de courtes longueurs d'onde, similairement au comportement dans la bande hautes fréquences (HF). Par conséquent, pour les structures complexes considérées, les trois régimes vibratoires BF, MF et HF se recouvrent, et de nombreux modes élastiques locaux sont entremêlés avec les modes élastiques globaux habituels. Cela implique deux difficultés majeures, concernant la quantification des incertitudes d'une part et le coût numérique d'autre part. L'objectif de cette thèse est alors double. Premièrement, fournir un ROM stochastique multi-niveau qui est capable de rendre compte de la variabilité hétérogène introduite par le recouvrement des trois régimes vibratoires. Deuxièmement, fournir un ROM prédictif de dimension réduite par rapport à celui de l'analyse modale. Une méthode générale est présentée pour la construction d'un ROM multi-niveau, basée sur trois bases réduites (ROBs) dont les déplacements correspondent à l'un ou l'autre des régimes vibratoires BF, MF ou HF (associés à des déplacements de type BF, de type MF ou bien de type HF). Ces ROBs sont obtenues via une méthode de filtrage utilisant des fonctions de forme globales pour l'énergie cinétique (par opposition aux fonctions de forme locales des éléments finis). L'implémentation de l'approche probabiliste non-paramétrique dans le ROM multi-niveau permet d'obtenir un ROM stochastique multi-niveau avec lequel il est possible d'attribuer un niveau d'incertitude spécifique à chaque ROB. L'application présentée est relative à une automobile, pour laquelle le ROM stochastique multi-niveau est identifié par rapport à des mesures expérimentales. Le ROM proposé permet d'obtenir une dimension réduite ainsi qu'une prévision améliorée, en comparaison avec un ROM stochastique classique / This work deals with an extension of the classical construction of reduced-order models (ROMs) that are obtained through modal analysis in computational linear structural dynamics. It is based on a multilevel projection strategy and devoted to complex structures with uncertainties. Nowadays, it is well recognized that the predictions in structural dynamics over a broad frequency band by using a finite element model must be improved in taking into account the model uncertainties induced by the modeling errors, for which the role increases with the frequency. In such a framework, the nonparametric probabilistic approach of uncertainties is used, which requires the introduction of a ROM. Consequently, these two aspects, frequency-evolution of the uncertainties and reduced-order modeling, lead us to consider the development of a multilevel ROM in computational structural dynamics, which has the capability to adapt the level of uncertainties to each part of the frequency band. In this thesis, we are interested in the dynamical analysis of complex structures in a broad frequency band. By complex structure is intended a structure with complex geometry, constituted of heterogeneous materials and more specifically, characterized by the presence of several structural levels, for instance, a structure that is made up of a stiff main part embedding various flexible sub-parts. For such structures, it is possible having, in addition to the usual global-displacements elastic modes associated with the stiff skeleton, the apparition of numerous local elastic modes, which correspond to predominant vibrations of the flexible sub-parts. For such complex structures, the modal density may substantially increase as soon as low frequencies, leading to high-dimension ROMs with the modal analysis method (with potentially thousands of elastic modes in low frequencies). In addition, such ROMs may suffer from a lack of robustness with respect to uncertainty, because of the presence of the numerous local displacements, which are known to be very sensitive to uncertainties. It should be noted that in contrast to the usual long-wavelength global displacements of the low-frequency (LF) band, the local displacements associated with the structural sub-levels, which can then also appear in the LF band, are characterized by short wavelengths, similarly to high-frequency (HF) displacements. As a result, for the complex structures considered, there is an overlap of the three vibration regimes, LF, MF, and HF, and numerous local elastic modes are intertwined with the usual global elastic modes. This implies two major difficulties, pertaining to uncertainty quantification and to computational efficiency. The objective of this thesis is thus double. First, to provide a multilevel stochastic ROM that is able to take into account the heterogeneous variability introduced by the overlap of the three vibration regimes. Second, to provide a predictive ROM whose dimension is decreased with respect to the classical ROM of the modal analysis method. A general method is presented for the construction of a multilevel ROM, based on three orthogonal reduced-order bases (ROBs) whose displacements are either LF-, MF-, or HF-type displacements (associated with the overlapping LF, MF, and HF vibration regimes). The construction of these ROBs relies on a filtering strategy that is based on the introduction of global shape functions for the kinetic energy (in contrast to the local shape functions of the finite elements). Implementing the nonparametric probabilistic approach in the multilevel ROM allows each type of displacements to be affected by a particular level of uncertainties. The method is applied to a car, for which the multilevel stochastic ROM is identified with respect to experiments, solving a statistical inverse problem. The proposed ROM allows for obtaining a decreased dimension as well as an improved prediction with respect to a classical stochastic ROM

No description available.

Les simulations numériques représentent un élément central du processus de conception d’un avion complétant les tests physiques et essais en vol. Elles peuvent notamment bénéficier de méthodes innovantes, telle que l’intelligence artificielle qui se diffuse largement dans l’aviation. Simuler une mission de vol complète pour plusieurs disciplines pose d’importants problèmes à cause des coûts de calcul et des conditions d’opérations changeantes. De plus, des phénomènes complexes peuvent se produire. Par exemple, des chocs peuvent apparaître sur l’aile pour l’aérodynamique alors que le mélange entre les écoulements du moteur et de l’air extérieur impacte fortement l’aérothermie autour de la nacelle et du mât. Des modèles de substitution peuvent être utilisés pour remplacer les simulations haute-fidélité par des approximations mathématiques afin de réduire le coût de calcul et de fournir une méthode construite autour des données de simulations. Deux développements sont proposés dans cette thèse : des modèles de substitution utilisant l’apprentissage automatique pour approximer des calculs aérodynamiques et l’intégration de modèles de substitution classiques dans un processus aérothermique industriel. La première approche sépare les solutions en sous-ensembles selon leurs formes grâce à de l’apprentissage automatique. En outre, une méthode de reéchantillonnage complète la base d’entrainement en ajoutant de l’information dans des sous-ensembles spécifiques. Le deuxième développement se concentre sur le dimensionnement du mât moteur en remplaçant les simulations aérothermiques par des modèles de substitution. Ces deux développements sont appliqués sur des configurations avions afin de combler l’écart entre méthode académique et industrielle. On peut noter que des améliorations significatives en termes de coût et de précision ont été atteintes. / Numerical simulations provide a key element in aircraft design process, complementing physical tests and flight tests. They could take advantage of innovative methods, such as artificial intelligence technologies spreading in aviation. Simulating the full flight mission for various disciplines pose important problems due to significant computational cost coupled to varying operating conditions. Moreover, complex physical phenomena can occur. For instance, the aerodynamic field on the wing takes different shapes and can encounter shocks, while aerothermal simulations around nacelle and pylon are sensitive to the interaction between engine flows and external flows. Surrogate models can be used to substitute expensive high-fidelitysimulations by mathematical and statistical approximations in order to reduce overall computation cost and to provide a data-driven approach. In this thesis, we propose two developments: (i) machine learning-based surrogate models capable of approximating aerodynamic experiments and (ii) integrating more classical surrogate models into industrial aerothermal process. The first approach mitigates aerodynamic issues by separating solutions with very different shapes into several subsets using machine learning algorithms. Moreover, a resampling technique takes advantage of the subdomain decomposition by adding extra information in relevant regions. The second development focuses on pylon sizing by building surrogate models substitutingaerothermal simulations. The two approaches are applied to aircraft configurations in order to bridge the gap between academic methods and real-world applications. Significant improvements are highlighted in terms of accuracy and cost gains

In der vorliegenden Arbeit wird eine Prozedur zur Ordnungsreduktion von Finite Elemente Modellen mikromechanischer Struktur mit elektrostatischem Wirkprinzip entwickelt und analysiert. Hintergrund der Ordnungsreduktion ist eine Koordinatentransformation von lokalen Finite Elemente Koordinaten in globale Koordinaten. Die globalen Koordinaten des reduzierten Modells werden durch einige wenige Formfunktionen beschrieben. Damit wird das Makromodell nicht mehr durch lokale Knotenverschiebungen beschrieben, sondern durch globale Formfunktionen, welche die gesamte Deformation der Struktur beeinflussen. Es wird gezeigt, dass Eigenvektoren der linearisierten mechanischen Struktur einfache und effiziente Formfunktionen darstellen. Weiterhin kann diese Methode für bestimmte Nichtlinearitäten und für verschiedene in Mikrosystemen auftretende Lasten angewendet werden. Das Ergebnis sind Makromodelle, die über Klemmen in Systemsimulatoren eingebunden werden können, die Genauigkeiten einer Finite Elemente Analyse erreichen und für Systemsimulationen typische Laufzeitverhalten besitzen.

<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>