abstract: It is increasingly common to see machine learning techniques applied in conjunction with computational modeling for data-driven research in neuroscience. Such applications include using machine learning for model development, particularly for optimization of parameters based on electrophysiological constraints. Alternatively, machine learning can be used to validate and enhance techniques for experimental data analysis or to analyze model simulation data in large-scale modeling studies, which is the approach I apply here. I use simulations of biophysically-realistic cortical neuron models to supplement a common feature-based technique for analysis of electrophysiological signals. I leverage these simulated electrophysiological signals to perform feature selection that provides an improved method for neuron-type classification. Additionally, I validate an unsupervised approach that extends this improved feature selection to discover signatures associated with neuron morphologies - performing in vivo histology in effect. The result is a simulation-based discovery of the underlying synaptic conditions responsible for patterns of extracellular signatures that can be applied to understand both simulation and experimental data. I also use unsupervised learning techniques to identify common channel mechanisms underlying electrophysiological behaviors of cortical neuron models. This work relies on an open-source database containing a large number of computational models for cortical neurons. I perform a quantitative data-driven analysis of these previously published ion channel and neuron models that uses information shared across models as opposed to information limited to individual models. The result is simulation-based discovery of model sub-types at two spatial scales which map functional relationships between activation/inactivation properties of channel family model sub-types to electrophysiological properties of cortical neuron model sub-types. Further, the combination of unsupervised learning techniques and parameter visualizations serve to integrate characterizations of model electrophysiological behavior across scales. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

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La batterie lithium ion est la technologie de stockage d’énergie la plus répandue dans l'industrie automobile. Assurer sa haute efficacité, sa puissance, sa capacité, sa sécurité et son endurance présente un défi pour plusieurs chercheurs et industriels. En effet, une batterie est un système complexe renfermant plusieurs composants et soumis à divers risques de dégradations d’origines chimiques, mécaniques et électriques, se manifestant même dans les conditions normales de fonctionnement. Cependant, la batterie devrait assurer ses fonctions pour un grand nombre de cycles de charge et de décharge et continuer à servir sans que ces dégradations influencent sa performance globale. L’une des dégradations principales et inévitables est son gonflement qui induit une discontinuité électrique et une perte de sa capacité.En effet, le gonflement est un phénomène multi-physique qui fait intervenir l’électrochimie, la mécanique et la thermique. D’une part, une batterie lithium-ion est basée sur l’échange réversible de l’ion lithium entre une électrode positive et une électrode négative. Le processus d’insertion de l’ion dans les particules de l’électrode aboutit à un changement volumique significatif réversible de la batterie pour chaque cycle de charge/décharge. Cette variation de volume mène à la formation de contraintes quand la batterie est maintenue dans un pack rigide empêchant ou limitant sa déformation. D’autre part, la formation d’une couche à l’interface particule-électrolyte (SEI) suite aux réactions parasites se produisant à l’échelle de l’électrode constitue une cause principale d’un gonflement supplémentaire irréversible et de vieillissement de la batterie.Ainsi, le gonflement doit être pris en compte pendant la phase du dimensionnement mécanique de la batterie. Il est donc indispensable d’avoir un outil numérique fiable capable de prédire ce comportement mécanique pendant toutes les phases de fonctionnement de la batterie et de permettre aux concepteurs d’améliorer sa structure.Ce travail rentre dans le cadre d’une collaboration entre l’ENSTA ParisTech et le constructeur automobile Renault suite à un besoin industriel de comprendre et de maîtriser le gonflement des batteries utilisées dans les véhicules électriques et hybrides. Pour répondre à ce besoin, un modèle multi-physique et multi-échelle fondé sur la théorie de la thermodynamique des processus irréversibles, sur l’endommagement et sur la théorie de l’homogénéisation est développé. Il permet de décrire et de prédire la déformation d’une batterie lithium ion pendant son fonctionnement. Le modèle tient compte des phénomènes mécaniques, électrochimiques et thermiques qui se produisent à l’échelle locale des électrodes afin de calculer la déformation mécanique au niveau macroscopique de la batterie. / Lithium ion battery is the most popular energy storage technology in the automotive industry. Ensuring high efficiency, power, capacity, safety and endurance is a challenge for many researchers and manufacturers. Indeed, a battery is a complex system containing several components and subject to various risks of chemical, mechanical and electrical damage, manifesting even under normal operating conditions. However, the battery should perform its functions for a large number of charge and discharge cycles and continue to serve without these risks influencing its overall performance. One of the main and inevitable damage is its swelling, which induces an electrical discontinuity and a loss of its capacity.Indeed, swelling is a multi-physics phenomenon that involves electrochemistry, mechanics and heat. On the one hand, a lithium-ion battery is based on the reversible exchange of the lithium ion between a positive electrode and a negative electrode. The process of inserting the ion into the particles of the electrode results in a significant reversible volume change of the battery for each charge / discharge cycle. This variation in volume leads to the formation of stresses when the battery is held in a rigid pack preventing or limiting its deformation. On the other hand, the formation of a layer at the particle-electrolyte interface (SEI) following parasitic reactions occurring at the electrode scale is a major cause of irreversible additional swelling and aging of the drums.Thus, the swelling must be taken into account during the mechanical sizing phase of the battery. It is therefore essential to have a reliable numerical tool able to predict this mechanical behavior during all phases of battery operation and to allow designers to improve its structure.This work is part of a collaboration between ENSTA ParisTech and the car manufacturer Renault following an industrial need to understand and control the swelling of batteries used in electric and hybrid vehicles. To meet this need, a multi-physics and multi-scale model based on the theory of the thermodynamics of irreversible processes, mechanical damage theory and the homogenization theory is developed. It allows to describe and predict the deformation of a lithium ion battery during its operation. The model takes into account the mechanical, electrochemical and thermal phenomena that occur at the local scale of the electrodes in order to calculate the mechanical deformation at the macroscopic level of the battery.

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A multiscale model is proposed to study the influence of interfacial interactions at the nanoscale in carbon nanotube(CNT)-polymer nanocomposites on the macroscale bulk elastic material properties. The efficiency of CNT reinforcement in terms of interfacial load transferring is assessed for the non-functionalized and functionalized interfaces between the CNTs and polymer matrix using force field based molecular dynamic simulations at the nanoscale. Polyethylene (PE) as a thermoplastic material is adopted and studied first because of its simplicity. Characterization of the nanoscale load transfer has been done through the identification of representative nanoscale interface elements for unfunctionalized CNT-PE interface models which are studied parametrically in terms of the length of the PE chains, the number of the PE chains and the "grip" position. Referring to the non-functionalized interface, CNTs interact with surrounding polymer only through weakly nonbonded van der Waals (vdW) forces in our study. Once appropriate values of these parameters are deemed to yield sufficiently converged results, the representative interface elements are subjected to normal and sliding mode simulations in order to obtain the force-separation responses at 100K and 300K for unfunctionalized CNT-PE interfaces. To study the functionalization effects, atomistic interface representative elements for functionalized CNT-PE interface are built based on non-functionalized interface models by grafting functional groups between the PE matrix and the graphene sheet. This introduces covalent bonding forces in addition to the non-bonded vdW forces. A modified consistent covalent force field (CVFF) and adaptive intermolecular reactive empirical bond order (AIREBO) potentials, both of which account for bond breaking, are applied to investigate the interfacial characteristic of functionalized CNT-PE interface in terms of the force-separation responses at 100K in both normal opening and sliding mode separations. In these studies, the focus has been on the influence of the functionalization density on the load transfer at the nanoscale interface. As an important engineering material, Epon 862/DETDA epoxy polymer,a thermoset plastic, has also been used as the polymer matrix material in order to see the difference in interfacial load transfer between a network structured polymer and the amorphous entangled structure of the PE matrix. As for thermoset epoxy polymer, emphasis has been put on investigating the effects of the crosslink density of the epoxy network on the interfacial load transfer ability for both non-functionalized and functionalized CNT-Epoxy interface at different temperatures(100K and 300K) and on the functionalization effect influenceing the interfacial interactions at the functionalized CNT-Epoxy interface. Cohesive zone traction-displacement laws are developed based on the force-separation responses obtained from the MD simulations for both non-functionalied and functionalized CNT-PE/epoxy interfaces. Using the cohesive zone laws, the influence of the interface on the effective elastic material properties of the nanocomposites are observed and determined in continuum level models using analytic and computational micromechanics approaches, allowing for the assessment of the improvement in reinforcement efficiency of CNTs due to the functionalization. It is found that the inclusion of the nanoscale interface in place of the perfectly bonded interface results in effective elastic properties which are dependent on the applied strain and temperature in accordance with the interface sensitivity to those effects, and which are significantly diminished from those obtained under the perfect interface assumption for non-functionalized nanocomposites. Better reinforcement efficiency of CNTs are also observed for the nanocomposites with the functionalized interface between CNTs and polymer matrix, which results in large increasing for the effective elastic material properties relative to the non-functionalized nanocomposites with pristine CNTs. Such observations indicates that trough controlling the degree of functionalization, i.e. the number and distribution of covalent bonds between the embedded CNTs and the enveloping polymer, one can tailor to some degree the interfacial load transfer and hence, the effective mechanical properties. The multiscale model developed in this study bridges the atomistic modeling and micromechanics approaches with cohesive zone models, which demonstrates to deepen the understanding of the nanoscale load transfer mechanism at the interface and its effects on the effective mechanical properties of the nanocomposites. It is anticipated that the results can offer insights about how to engineer the interface and improve the design of nanocomposites. / Ph. D.

Ultra-high temperature ceramics (UHTCs) are a class of materials that maintain their structural integrity at high temperatures, e.g. 2000 °C. They have been limited in their aerospace applications because of their relatively high density and the difficulty involved in forming them into complex shapes, like leading edges and inlets. Recent advanced processing techniques have made significant headway in addressing these challenges, where the introduction of multiscale porosity has resulted in lightweight UHTCs dubbed multiscale porous UHTCs. The effect of multiscale porosity on material properties must be characterized to enable design, but doing so experimentally can be costly, especially when attempting to replicate hypersonic flight conditions for relevant testing of selected candidate samples. As such, this dissertation seeks to computationally characterize the thermomechanical properties of multiscale porous UHTCs, specifically titanium diboride, and validate those results against experimental results so as to build confidence in the model. An implicit quasi-static variant of the Material Point Method (MPM) is developed, whose capabilities include intrinsic treatment of large deformations and contact which are needed to capture the complex material behavior of the as-simulated porous UHTC microstructures. It is found that the MPM can successfully obtain the elastic thermomechanical properties of multiscale porous UHTCs over a wide range of temperatures. Furthermore, characterizations of post-elastic behavior are found to be qualitatively consistent with data obtained from uniaxial compression experiments and Brazilian disk experiments. / Doctor of Philosophy / This dissertation explores a class of materials called ultra-high temperature ceramics (UHTCs). These materials can sustain very high temperatures without degrading, and thus have the potential to be used on hypersonic aircraft which routinely experience high temperatures during flight. In lieu of performing experiments on physical UHTC specimens, one can perform a series of computer simulations to figure out how UHTCs behave under various conditions. This is done here, with a particular focus what happens when pores are introduced into UHTCs, thus rendering them more like a sponge than a solid block of material. Doing computer simulations instead of physical experiments is attractive because of the flexibility one has in a computational environment, as well as the significantly decreased cost associated with running a simulation vs. setting up and performing an experiment. This is especially true when considering challenging operating environments like those experienced by high-speed aircraft. The ultimate goal with this research is to develop a computational tool than can be used to design the ideal distribution of pores in UHTCs so that they can best perform their intended functions.

Cellular mechanical metamaterials (CMMs) are a special class of materials that consist of microstructural architectures of macroscopic hierarchical frameworks that can have extraordinary properties. These properties largely depend on the topology and arrangement of the unit cells constituting the microstructure. The material hierarchy facilitates the synthesis and design of CMMs on the micro-scale to achieve enhanced properties (i.e., improved strength, toughness, low density) on the component (macro)-scale. However, designing on-demand cellular metamaterials usually requires solving a challenging inverse problem to explore the complex structure-property relations. The first part of this study (Ch. 3) proposes an experience-free and systematic design methodology for microstructures of CMMs using an advanced stochastic searching algorithm called micro-genetic algorithm (μGA). Locally, this algorithm minimizes the computational expense of the genetic algorithm (GA) with a small population size and a conditionally reduced parameter space. Globally, the algorithm employs a new search strategy to avoid local convergence induced by the small population size and the complexity of the parameter space. What's more, inspired by natural evolution in the GA, this study applies the inverse design method with the standard GA (sGA) as a sampling algorithm for intuitively mapping material-property spaces of CMMs, which requires the selection of objective properties and stochastic search of property points within the property space. The mapping methodology utilizing the sGA is proposed in the second part of the study (Ch. 4). This methodology involves a robust strategy that is shown to identify more comprehensive property spaces than traditional mapping approaches. The resulting property space allows designers to acknowledge the limitations of material performance, and select an appropriate class of CMMs based on the difficulty of the realization and fabrication of their microstructures. During the fabrication process, manufacturing defects cause uncertainty in the microstructures, and thus the structural properties. The third part of the study (Ch. 5) investigates the effects of the uncertainty stemming from manufacturing defects on the material property space. To accelerate the uncertainty quantification (UQ) via the Monte Carlo method, this study utilizes a machine learning technique to bypass the expensive simulations to compute properties. In addition to reducing the computational expense of the simulations, the deep learning method has been proven to be practical to accomplish non-intuitive design tasks. Due to the numerous combinations of properties and complex underlying geometries of metamaterials, it is numerically intractable to obtain optimal material designs that satisfy multiple user-defined performance criteria at the same time. Nevertheless, a deep learning method called conditional generative adversarial networks (CGANs) is capable of solving this many-to-many inverse problem. The fourth part of the study (Ch. 6) proposes a new inverse design framework using CGANs to overcome this challenge. Given combinations of target properties, the framework can generate a group of geometric patterns providing these target properties. Therefore, the proposed strategy provides alternative solutions to satisfy on-demand requirements while increasing the freedom in the fabrication process. Besides, with the advances in additive manufacturing (AM), the design space of an engineering material can be further enlarged by multi-scale topology optimization. As the interplay between microstructure and macrostructure drives the overall mechanical performance of engineering materials, it is necessary to develop a multi-scale design framework to optimize structural features in these two scales simultaneously. The final part of the study (Ch. 7) presents a concurrent multi-scale topology optimization method of CMMs. Structures in micro and macro scales are optimized concurrently by utilizing sequential quadratic programming (SQP) with the Solid Isotropic Material with Penalization (SIMP) method and a numerical homogenization approach. / Doctor of Philosophy / Cellular materials widely exist in natural biological systems such as honeycombs, bones, and wood. Recent advances in additive manufacturing have enabled us to fabricate these materials with high precision. Inspired by architectures in nature, cellular mechanical metamaterials (CMMs) have been introduced recently as a new class of architected systems. The materials are formed by hierarchical microstructural topologies, which have a decisive influence on the structural performance at the macro-scale. Therefore, the design of these materials primarily focuses on the geometric arrangement of their microstructures rather than the chemical composition of their base material. Tailoring the microstructures of these materials can lead to several outstanding features, such as high stiffness and strength, low density, and high energy absorption. However, it is challenging to design microstructures that satisfy user-defined requirements for properties and material costs. This is mainly due to the trade-off between the accuracy and computing times of the optimization process. In the first part of this study (Ch. 3), a design framework is proposed to overcome this issue. The framework employs a global search algorithm called the genetic algorithm (GA). With a newly designed search algorithm, the framework reduces errors between target and optimized material properties while improving computational efficiency. Inspired by the algorithm behind the GA, the second part of the study (Ch. 4) employs a similar algorithm to identify a material property chart demonstrating all possible combinations of mechanical properties of CMMs. Each axis of the material property chart corresponds to a selected mechanical property, such as Young's modulus or Poisson's ratio, along different directions. The boundary of the property space helps designers understand material performance limitations and make informed decisions in engineering practices. In the fabrication process, unexpected material properties might be achieved due to defects and tolerances in additive manufacturing (AM), such as uneven surfaces, shrinkage of pores, etc. The third part of the study (Ch. 5) investigates the uncertainty propagation on mechanical properties as a result of these manufacturing defects. To investigate the uncertainty propagation problem efficiently, the study uses a deep learning method to predict the variations (stochasticity) of properties. Consequently, the material property space boundary also varies with the uncertainty of properties. In addition to their computational efficiency, deep learning methods are beneficial for solving many-to-many inverse design problems. Traditionally, the global and local search/optimization methods retrieve alternative optimal solutions in their Pareto front set, where each solution is considered to be equally good. A deep learning method called conditional generative adversarial networks (CGANs) can bypass the property calculation to accelerate the simulation process while obtaining a group of candidates with on-demand properties. The fourth part of the study (Ch. 6) employs CGANs to build a new inverse design framework to increase flexibility in the fabrication process by generating alternative solutions for the microstructures of CMMs. Besides, as fabrication technologies have advanced, designing engineering systems has become increasingly complex. Material design is now not only focused on meeting micro-scale requirements but also addressing needs at multiple scales. The interaction between the microstructure (small-scale) and macrostructure (large-scale) significantly influences the overall performance of engineering systems. To optimize structures effectively, there is a need for a design framework that considers these two scales simultaneously. Thus, the final part of the study (Ch. 7) introduces a method called concurrent multi-scale topology optimization. To obtain the extreme performance of a multi-scale structure, this approach optimizes its structure at both micro- and macro-scales concurrently, using gradient-based optimization algorithms with density-based property determination methods in the two scales.

Pour l'analyse de durabilité et la conception économique (moins de matériel) de structures en matériaux ressemblant à du béton, la modélisation de la rupture est essentielle. Dans le cadre de la mécanique des milieux continus, une longueur interne est introduite dans les modèles non locaux pour remédier au problème lié à la sensibilité du maillage qui est une pathologie des modèles d'endommagement classiques , lorsqu'il s'agit de matériaux adoucissantes. Toutefois, l'évaluation de la longueur interne de hétérogénéités du matériau est toujours une question difficile, ce qui rend un problème obscur en utilisant des modèles non locaux. Nos travaux portent sur le développement d'un outil numérique basée sur la méthode des éléments en treillis (LEM) qui est un modèle discret pour la simulation et la prévision de la rupture des structures en béton. En utilisant le modèle de réseau à l'échelle mésoscopique, il n'est pas nécessaire d'introduire une longueur interne dans la loi de comportement, comme cela se fait dans les modèles non locaux, et nous pouvons affranchir ce paramètre en introduisant explicitement la mesotructure matérielle via une description géométrique. Basé sur l'outil numérique développé, nous avons étudié, en effectuant des tests numériques de traction uniaxiale, l'influence géométrique de la mesotructure du matériau ainsi que l'influence des conditions aux limites et de tailles d'échantillons (qui se traduisent par le gradient de sollicitation et le champ de rotation de matériel différents) sur le taille de la FPZ (fracture process zone) et sur la longueur caractéristique du matériau quasi-fragile homogénéisé. Ces études fournissent des recommandations/avertissements lors de l'extraction d'une longueur interne nécessaire pour les modèles nonlocaux à partir de la microstructure du matériau. Par ailleurs, les études contribuent un aperçu direct de l'origine mésoscopic de la taille FPZ et la longueur de la caractéristique du matériau, et par conséquent sur l'origine et la nature du comportement non linéaire du matériau. Ensuite, nous avons implanté le modèle du treillis dans la bibliothèque de SOFA développé par l'INRIA pour réaliser le couplage avec la méthode des éléments finis (MEF) afin de faire face avec des structures à grande échelle. Nous avons proposé un algorithme de couplage entre une approche macroscopique représentée par MEF et une approche mésoscopique infligés par LEM au sein d'une manière adaptative. Le modèle de couplage est d'abord utilisée pour valider l'approche multi-échelle proposée sur des simulations heuristiques. Et à long terme, il fournit un outil prometteur pour des simulations de grandes structures en matériaux quasi-fragiles de la vie réelle. / For durability analysis and economic design (less material) of structures made of concrete-like materials, modeling of cracking process and failure is essential. In the framework of continuum mechanics, an internal length is introduced in nonlocal models to remedy the problem related to mesh sensitivity which is a pathology of classical damage models, when dealing with softening materials. However, the assessment of the internal length from heterogeneities of the material is still a difficult question, which makes an obscure issue in using nonlocal models. Our work concerns developing of a numerical tool based on the Lattice Element Method (LEM) which is a discrete model for simulating and predicting fracture in concrete(-like) material. Using the lattice model at the mesoscopic scale, there is no need to introduce any internal length in the constitutive law, as done in nonlocal models, and we can enfranchise this parameter by explicitly introducing the material mesotructure via geometric description. Based on the developed numerical tool, we studied, by performing numerical uniaxial tensile tests, the geometric influence of the material mesotructure as well as the influence of the boundary conditions and specimen sizes (that result in different stress gradient and material rotation field) on the size of the FPZ (Fracture Process Zone) and on the characteristic length of the homogenized quasi-brittle material. These studies provide recommendations/warnings when extracting an internal length required for nonlocal damage models from the material microstructure. Moreover, the studies contribute a direct insight into the mesoscale origin of the FPZ size and the material characteristic length, and consequently into the origin and nature of the nonlinear behavior of the material. Then, we implemented the lattice model into SOFA library developed by INRIA for realizing the coupling with the Finite Element Method (FEM) in order to deal with large-scale structures. We proposed a strong coupling algorithm between a macroscopic approach represented by FEM and a mesoscopic approach dealt by LEM within an adaptive manner. The coupling model is first used to validate the multiscale approach proposed on heuristic simulations. And in the long term, it provides a promising tool for simulations of large-scale structures made of quasi-brittle materials of real life.

Dans ce travail de thèse nous nous sommes intéressés aux phénomènes de transport au sein du réseau lacuno-canaliculaire (RLC) et de l'ostéon dans le tissu osseux cortical. Pour étudier la mécanotransduction ostéocytaire amenant au remodelage osseux, nous avons développé un modèle à trois échelles où sont pris en : l'électrcompte ostatique (modélisée par l'équation de Poisson Boltzmann), l'écoulement du fluide (représenté textit{via} une équation de Stokes modifiée et la conservation de la masse fluide) et le transport ionique (régi par l'équation de Nernst-Planck). L'étude de la distribution du potentiel électrique, a mis en exergue l'importance des double-couches électriques au voisinage des parois chargées des pores. Ces double-couches électriques, ainsi que la composition chimique du fluide donnent lieu à des phénomènes d'osmose et d'électroosmose intervenant dans l'écoulement du fluide interstitiel, et influençant la diffusion efficace des ions dans les pores. L'étude a démarré à l'échelle du pore canaliculaire pour être propagée à l'échelle du canalicule puis de l'ostéon, en utilisant une procédure d'homogénéisation périodique asymptotique. Une étude paramétrique nous a permis de cibler les paramètres agissant sur les phénomènes de transport et pouvant faire réagir les ostéocytes. Il est ressorti de cette étude que les effets électro-chimiques jouent rôle important. Nous avons donc choisi de nous focaliser sur la chimie et plus particulièrement sur les effets des flux ioniques physiologiques sur les ostéocytes dans le RLC. Des expériences, mises en place pour étayer ces aspects ont souligné l'importance des échanges chimiques entre les cellules et le fluide qui les entoure. Finalement, nous avons montré que les phénomènes de transports ayant lieu dans le RLC et dans l'ostéon interagissent les uns les autres, parachevant ainsi la description à trois échelles du tissu cortical / Transport phenomena appearing within the cortical bone lacuno-canalicular network (LCN) and the osteon were the objective of this study. We developped a three-scale model to investigate the osteocyte mechanotranduction which is at the origin of the bone remodeling process. This model took into account three physical phenomena : the electrostatics (through the Poisson-Boltzman equation), the interstitial fluid flow (modeled by a modified Stokes equation) and the ionic transport (governed by a Nernst-Planck equation). Analysis of the electrical potential distribution highlighted the importance of the electrical double layers close to the pore charged surface. These electrical double layers, as well as the interstitial fluid chemical composition, induce osmotic and electroosmotic fluid flows and affect the ionic effective diffusion within the pores. Using a periodic asymptotic homogeneisation procedure, the model at the canalicular pore scale was upscaled at the canalicular scale and then at the osteonal scale. A parametric study pointed out the relevant parameters acting on the transport phenomena and possibly affecting osteocyte mecanosensitivity. Our results emphasized the importance of the electro-chemical effects. We thus focused on the chemistry and more especially on the effects of the physiological ionic fluxes on the osteocyte. In vitro experiments and numrical simulations were performed to elucidate these questions. Our results underlined the importance of the chemical exchanges between the osteocyte and the surrounding fluid. Finally, we showed that fluid flow and chemical transport occuring within the LCN and the osteon interact with each other, thus achieving a three-scale description of the transport phenomena in the osteon