Polyetherimides are an important class of engineering thermoplastics used in a broad range of industries and applications because of their high heat resistance and stability, high strength and moduli, excellent electrical properties over a wide range of temperatures and frequencies, good processability, good adhesive properties, and chemical stability. All-atom molecular dynamics (MD) simulation is a useful tool to study polymers, but the accessible length and time scales are limited. In this thesis, we explore several computational methods that go beyond all-atom MD simulations to investigate polyetherimides. First, we have developed a transferable coarse-grained MD model of polyetherimides that captures their mechanical and thermal expansion properties. Our results show that in order to make the model transferable, it is critical to include an entropic correction term in the coarse-grained force field and require the coarse-grained model to capture the thermal expansion property of polyetherimides. Secondly, we have constructed a predictive model of the glass transition temperature (Tg) for polyimides by using machine-learning algorithms to analyze existing data on Tg reported in the literature. The predictive model is validated by comparing its predictions to experimental data not used in the training process of the model. We further demonstrate that the diffusion coefficients of small gas molecules can be quickly computed with all-atom MD simulations and used to determine Tg. Finally, we have developed a Monte Carlo (MC) program to model the polymerization process of branched polyetherimides and to compute their molecular weight distribution for a wide range of systems, including fully reacted, partially reacted, stoichiometric, and nonstoichiometric ones. The MC results are compared to the predictions of the Flory-Stockmayer theory of branched polymers and an excellent agreement is found below the gel point of the system under consideration. Above the gel point, the Flory- Stockmayer theory starts to fail but the MC method can still be used to quickly determine the molecular weight distribution of branched polyetherimides under very general conditions. / Doctor of Philosophy / Polyetherimides are an important category of engineering plastics with wide applications in many fields because of their superior mechanical, thermal, chemical, and electrical properties. All-atom molecular dynamics simulations serve as a useful tool to study the properties of polyetherimides in silico. However, such simulations are computationally expensive and therefore limited to small system sizes and short time scales. To overcome these issues, we employed various computational techniques in this thesis to model polyetherimides. First, we have developed a coarse-grained model of polyetherimides where atoms are grouped into beads. We show that molecular dynamics simulations on the basis of the coarse-grained model can be used to provide a reasonable description of the mechanical and thermal expansion properties of polyetherimides. Secondly, we have constructed a predictive model of the glass transition temperature, which is the temperature at which a material enters a glassy state when cooled rapidly, of polyimides using machine-learning algorithms. This model is capable of estimating the glass transition temperature of polyimides within an accuracy of ± 15 K even for those not synthesized yet. We further show that the diffusion coefficients of gas molecules, in addition to the polymer density, can be computed accurately with all-atom molecular dynamics simulations and used to determine the glass transition temperature of polyimides. Finally, we have developed a Monte Carlo scheme to efficiently model the polymerization and compute the chain-length distribution of branched polyetherimides under very general conditions. The results from Monte Carlo simulations are compared to the predictions of the Flory-Stockmayer theory of branched polymers. The range of applicability of the theory is revealed. Overall, we have demonstrated several computational techniques that can be used to efficiently model polyetherimides, potentially other polymers as well, beyond the widely-used all-atom molecular dynamics simulations.

Numerical simulations of large complex systems such as biomolecules often suffer from the full description of the system having too many dimensions for direct numerical calculations and Monte Carlo methods having trouble overcoming energy barriers. It is therefore desirable to formulate a description in lower dimension which captures the system’s macroscopic behaviour. Recently, Lindahl et al [1] proposed a metric, g(λ), on the extended space Λ based on the dynamics of the system to optimize Monte Carlo sampling within extended ensemble formalism. In this thesis, we formulate a low-dimensional effective coarse-grained dynamic on Λ as a diffusion process and ask if it is possible to use this metric to calculate thelocal effective diffusion matrix as D(λ) = g−1(λ). By testing various scenarios we conclude that computing D(λ) in this manner indeed gives a correct effective dynamic in most cases, where the scale of coarse-graining can be tuned. However, an incorrect dynamic is received for example when the scale of coarse-graining is comparable to the size of oscillations in the energy landscape. / Numeriska simuleringar av stora komplexa system såsom biomolekyler lider ofta av att den fulla beskrivningen av systemet har för många dimensioner för direkta numeriska beräkningar samt att Monte Carlo-metoder har svårt att komma över energibarriärer. Det är därför önskvärt att formulera en beskrivning i lägre dimension som fångar systemets makroskopiska beteende. Nyligen föreslog Lindahl et al [1] en metrik g(λ) på det utvidgade rummet Λ baserad på dynamiken av systemet för att optimera Monte Carlo-sampling inom formalismen av utvidgade ensembler. I den här uppsatsen formulerar vi en lågdimensionell effektiv grov dynamik på Λ som en diffusionsprocess och frågar om det är möjligt att använda den här metriken för att beräkna den lokala effektiva diffusionsmatrisen som D(λ) = g(λ)−1. Genom testning av flera scenarier drar vi slutsatsen att beräkna D(λ) på det här sättet ger en korrekt effektiv dynamik i de flesta fall, där skalan på förgrovningen kan ställas in. Däremot fås en inkorrekt dynamik till exempel när skalan på förgrovningen ärjämförbar med storleken på oscillationer i energilandskapet.

Le modèle de Poland-Scheraga (PS) est le modèle standard pour étudier la transition de dénaturation de deux brins d’ADN complémentaires et de même longueur. Ce modèle a fait l’objet d’une attention remarquable car il est exactement résoluble dans sa version homogène. Le caractère résoluble est lié au fait que le modèle PS homogène peut être mis en correspondance avec un processus de renouvellement discret. Dans la littérature biophysique une généralisation du modèle, obtenue en considérant des brins non complé- mentaires et de longueurs différentes, a été considérée et le caractère résoluble s’étend à cette généralisation substantielle. Dans cette thèse, nous présentons une analyse mathématique du modèle de Poland- Scheraga généralisé. Nous considérons d’abord le modèle homogène et nous exploitons que les deux brins de la chaîne peuvent être modélisés par un processus de renouvellement en deux dimensions. La distribution K(⋅) de l’emplacement (bidimensionnel) du premier contact entre les deux brins est supposée de la forme K(n+m) = (n+m)−α−2L(n+m) avec α ≥ 0 et L(⋅) à variation lente et correspond à une boucle avec n bases dans le premier brin et m dans le deuxième. Nous étudions la transition de localisation-délocalisation et nous montrons l’existence des transitions à l’intérieur de la phase localisée. Nous présentons ensuite des estimations précises sur les propriétés de chemin du modèle. Ensuite, nous étudions la version désordonnée du modèle en incluant une séquence de variables aléatoires indépendantes identiquement distribuées à deux indices. Nous nous concentrons sur l’influence du désordre sur la transition de dénaturation: nous voulons déterminer si la présence des inhomogénéités modifie les propriétés critiques du système par rapport au cas homogène. Nous prouvons que le désordre est non pertinent si α < 1 et nous montrons que pour α > 1, les points critiques gelés et recuits diffèrent (basant sur les techniques de coarse graining et la méthode des moments fractionnaires), ce qui prouve la présence d’un régime de désordre pertinent. / The Poland-Scheraga (PS) model is the standard basic model to study the denaturation transition of two complementary and equally long strands of DNA. This model has enjoyed a remarkable attention because it is exactly solvable in its homogeneous version. The solvable character is related to the fact that the homogeneous PS model can be mapped to a discrete renewal process. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. In this thesis we present a generalized version of the PS model that allows mismatches and non complementary strands (in particular, the two strands may be of different lengths). We consider first the homogeneous model and we exploit that this model can be mapped to a bivariate renewal process. The distribution K(⋅) of the location (in two dimensions) of the first contact between the two strands is assumed to be of the form K(n + m) = (n + m)−α−2L(n + m) with α ≥ 0 and L(⋅) slowly varying and corresponds to a loop with n bases in the first strand and m in the second. We study the localization-delocalization transition and we prove the existence of transitions inside the localized regime. We then present precise estimates on the path properties of the model. We then study the disordered version of the model by including a sequence of inde- pendent and identically distributed random variables with two indices. We focus on the influence of disorder on the denaturation transition: we want to determine whether the presence of randomness modifies the critical properties of the system with respect to the homogeneous case. We prove that the disorder is irrelevant if α < 1. We show also that for α > 1, the quenched and annealed critical points differ (basing on coarse graining techniques and fractional moment method), proving the presence of a relevant disorder regime.

An understanding of the mechanical properties of viral capsids (protein assemblies forming shell containers) has become necessary as their perceived use as nano-materials for targeted drug delivery. In this thesis, a heterogeneous, spatially detailed model of the viral capsid is considered. This model takes into account the increased degrees of freedom between the capsomers (capsid sub-structures) and the interactions between them to better reflect their deformation properties. A spatially realistic finite element multi-domain decomposition of viral capsid shells is also generated from atomistic PDB (Protein Data Bank) information, and non-linear continuum elastic simulations are performed. These results are compared to homogeneous shell simulation re- sults to bring out the importance of non-homogenous material properties in determining the deformation of the capsid. Finally, multiscale methods in structural analysis are reviewed to study their potential application to the study of nanoindentation of viral capsids. / text

<p>This thesis consists of two papers considering different aspects of stochastic process modelling and the minimisation of computational cost.</p><p>In the first paper, we analyse statistical signal properties and develop a Gaussian pro- cess model for scenarios with a moving receiver in a scattering environment, as in Clarke’s model, with the generalisation that noise is introduced through scatterers randomly flip- ping on and off as a function of time. The Gaussian process model is developed by extracting mean and covariance properties from the Multipath Fading Channel model (MFC) through coarse graining. That is, we verify that under certain assumptions, signal realisations of the MFC model converge to a Gaussian process and thereafter compute the Gaussian process’ covariance matrix, which is needed to construct Gaussian process signal realisations. The obtained Gaussian process model is under certain assumptions less computationally costly, containing more channel information and having very similar signal properties to its corresponding MFC model. We also study the problem of fitting our model’s flip rate and scatterer density to measured signal data.</p><p>The second paper generalises a multilevel Forward Euler Monte Carlo method intro- duced by Giles [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. Giles work [1] proposed and analysed a Forward Euler Multilevel Monte Carlo method based on realsiations on a hierarchy of uniform time discretisations and a coarse graining based control variates idea to reduce the computa- tional effort required by a standard single level Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretisations generated by adaptive algorithms developed by Moon et al. [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time steps and are based on a posteriori error expansions first developed by Szepessy et al. [4]. Under sufficient regularity conditions, our numerical results, which include one case with singular drift and one with stopped dif- fusion, exhibit savings in the computational cost to achieve an accuracy of O(T ol), from O(T ol−3 ) to O (log (T ol) /T ol)2 . We also include an analysis of a simplified version of the adaptive algorithm for which we prove similar accuracy and computational cost results.</p><p> </p>

The aim of this work is to provide a mathematical and numerical analysis of the static quasicontinuum (QC) method. The QC method is, in essence, a finite element method for atomistic material models. By restricting the set of admissible deformations to linear splines with respect to a finite element mesh, the computational complexity of atomistic material models is reduced considerably. We begin with a general review of atomistic material models and the QC method and, most importantly, a thorough discussion of the correct concept of static equilibrium. For example, it is shown that, in contrast to global energy minimization, a ‘dynamic’ selection procedure based on gradient flows models the physically correct behaviour. Next, an atomistic model with long-range Lennard–Jones type interactions is analyzed in one dimension. A rigorous demonstration is given for the existence and stability of elastic as well as fractured steady states, and it is shown that they can be approximated by a QC method if the mesh is sufficiently well adapted to the exact solution; this can be measured by the interpolation error. While the a priori error analysis is an important theoretical step for understanding the approximation properties of the QC method, it is in general unclear how to compute the QC deformation whose existence is guaranteed by the a priori analysis. An a posteriori analysis is therefore performed as well. It is shown that, if a computed QC deformation is stable and has a sufficiently small residual, then there exists a nearby exact solution and the error is estimated. This a posteriori existence idea is also analyzed in an abstract setting. Finally, extensions of the ideas to higher dimensions are investigated in detail.

Les suspensions colloïdales se trouvent un peu partout autour de nous, dans les matériauxde constructions, en cosmétique, dans l'alimentation, en biologie. Elles sont composésde particules nanométriques ou micrométriques dispersés dans un gaz, un liquide ou unsolide.Cette thèse porte sur les suspensions colloïdales dans des solutions ioniques,où les colloïdes portent une charge électriques, par exemple des particules de silicedans une solution aqueuse de chlorure de sodium, à un pH basique. Les colloïdes,ici approximés par des sphères, peuvent varier significativement en taille,ce qui peut avoir un effet important sur le comportement de ces systèmes.Cette étude vise à améliorer la compréhension de ces suspensions colloïdales chargéespar des modèles théoriques résolus par des simulations numériques.Un des défis de ces simulations est le grand nombre de degrés de libertés. Pour chaque(micro-)ion il y a des centaines de molécules de solvant, et pour chaque colloïdedes centaines voire des milliers d'ions. Pour s'en sortir, nous avons calculéles interactions effectives à l'échelle colloïdale. Nous avons repris et développéplusieurs approches, présentant chacune un compromis en terme de temps de calcul etprécision.La variation en taille des colloïdes peut introduire des effets intéressants,observés expérimentalement, notamment le fractionnement des suspensions en plusieursphases cristallines quand on augmente la concentration en colloïdes.Des techniques de simulations Monte-Carlo simples associées aux interactions inter-colloïdescalculées précédemment ont permis d'obtenir des résultats en bon accord avec l'expérience. / Colloidal suspensions are found a bit everywhere around us, in construction materials,in cosmetics, in food, in biology. They are composed of nanometric or micrometric particlesdispersed in a gas, a liquid or sometimes a solid.This thesis is about colloidal suspensions in ionic solutions, where colloids bear anelectric charge, for example silica particles in an aqueous solution of sodium chloride,at a basic pH. The colloids, here approximated by spheres, can vary significantly in size,which can have an important effect on the behavior of these systems.This study aims at improving the understanding of these charged colloidal suspensionsby theoretical models solved by numerical simulations.of these charged colloidal suspensionsby theoretical models solved by numerical simulations.One of the challenge of theses simulations is the huge number of degrees of freedom.For each (micro-)ion there is hundreds of solvent molecules, and for each colloidthere is hundreds if not thousands of ions. To get away with it, we calculated theeffective interactions at the colloidal scale. We took and developed several approaches,each showing a trade-off in terms of computational time and accuracy.The size variation of colloids may introduce interesting effects, experimentallyobserved, notably the fractionation of suspensions in several crystalline phaseswhen the colloidal concentration is increased. Some simple Monte-Carlo simulationtechniques in combination with the inter-colloid interactions computed previouslyallowed us to obtain results in good agreement with experiments.

<p align="justify">In a context of restoration of historical masonry structures, it is crucial to properly estimate the residual strength and the potential structural failure modes in order to assess the safety of buildings. Due to its mesostructure and the quasi-brittle nature of its constituents, masonry presents preferential damage orientations, strongly localised failure modes and damage-induced anisotropy, which are complex to incorporate in structural computations. Furthermore, masonry structures are generally subjected to complex loading processes including both in-plane and out-of-plane loads which considerably influence the potential failure mechanisms. As a consequence, both the membrane and the flexural behaviours of masonry walls have to be taken into account for a proper estimation of the structural stability.</p> <p align="justify">Macrosopic models used in structural computations are based on phenomenological laws including a set of parameters which characterises the average behaviour of the material. These parameters need to be identified through experimental tests, which can become costly due to the complexity of the behaviour particularly when cracks appear. The existing macroscopic models are consequently restricted to particular assumptions. Other models based on a detailed mesoscopic description are used to estimate the strength of masonry and its behaviour with failure. This is motivated by the fact that the behaviour of each constituent is a priori easier to identify than the global structural response. These mesoscopic models can however rapidly become unaffordable in terms of computational cost for the case of large-scale three-dimensional structures.</p> <p align="justify">In order to keep the accuracy of the mesoscopic modelling with a more affordable computational effort for large-scale structures, a multi-scale framework using computational homogenisation is developed to extract the macroscopic constitutive material response from computations performed on a sample of the mesostructure, thereby allowing to bridge the gap between macroscopic and mesoscopic representations. Coarse graining methodologies for the failure of quasi-brittle heterogeneous materials have started to emerge for in-plane problems but remain largely unexplored for shell descriptions. The purpose of this study is to propose a new periodic homogenisation-based multi-scale approach for quasi-brittle thin shell failure.</p> <p align="justify">For the numerical treatment of damage localisation at the structural scale, an embedded strong discontinuity approach is used to represent the collective behaviour of fine-scale cracks using average cohesive zones including mixed cracking modes and presenting evolving orientation related to fine-scale damage evolutions.</p> <p align="justify">A first originality of this research work is the definition and analysis of a criterion based on the homogenisation of a fine-scale modelling to detect localisation in a shell description and determine its evolving orientation. Secondly, an enhanced continuous-discontinuous scale transition incorporating strong embedded discontinuities driven by the damaging mesostructure is proposed for the case of in-plane loaded structures. Finally, this continuous-discontinuous homogenisation scheme is extended to a shell description in order to model the localised behaviour of out-of-plane loaded structures. These multi-scale approaches for failure are applied on typical masonry wall tests and verified against three-dimensional full fine-scale computations in which all the bricks and the joints are discretised.</p>

This dissertation examines the development of coarse-grained models of atomistic systems for the purpose of predicting target quantities of interest in the presence of uncertainties. It addresses fundamental questions in computational science and engineering concerning model selection, calibration, and validation processes that are used to construct predictive reduced order models through a unified Bayesian framework. This framework, enhanced with the concepts of information theory, sensitivity analysis, and Occam's Razor, provides a systematic means of constructing coarse-grained models suitable for use in a prediction scenario. The novel application of a general framework of statistical calibration and validation to molecular systems is presented. Atomistic models, which themselves contain uncertainties, are treated as the ground truth and provide data for the Bayesian updating of model parameters. The open problem of the selection of appropriate coarse-grained models is addressed through the powerful notion of Bayesian model plausibility. A new, adaptive algorithm for model validation is presented. The Occam-Plausibility ALgorithm (OPAL), so named for its adherence to Occam's Razor and the use of Bayesian model plausibilities, identifies, among a large set of models, the simplest model that passes the Bayesian validation tests, and may therefore be used to predict chosen quantities of interest. By discarding or ignoring unnecessarily complex models, this algorithm contains the potential to reduce computational expense with the systematic process of considering subsets of models, as well as the implementation of the prediction scenario with the simplest valid model. An application to the construction of a coarse-grained system of polyethylene is given to demonstrate the implementation of molecular modeling techniques; the process of Bayesian selection, calibration, and validation of reduced-order models; and OPAL. The potential of the Bayesian framework for the process of coarse graining and of OPAL as a means of determining a computationally conservative valid model is illustrated on the polyethylene example. / text

In the nucleus of eukaryotic cells, DNA wraps around histone proteins to form nucleosomes, which in turn associate in a compact and dynamic fiber called chromatin. The physical properties of this fiber at different lengthscales, from the DNA double-helix to micrometer-sized chromosomes, are essential to the complex mechanisms of gene expression and its regulation. The present thesis is a contribution to the development of physical models, which are able to link different scales and to interpret and integrate data from a wide range of experimental and computational approaches. In the first part, we use Molecular Dynamics simulations of DNA oligomers to study doublehelical DNA at different temperatures. We estimate the sequence-dependent contribution of entropy to DNA elasticity, in relation with recent experiments on DNA persistence length. In the second part, we model the DNA-histone interactions within the nucleosome core particle,using DNA nanomechanics to extract a force field from a set of crystallographic nucleosome structures and Molecular Dynamics snapshots. In the third part, we consider the softer part of the nucleosome, the linker DNA between coreparticles which transiently associates with the histone H1 to form a "stem".We combine existing structural knowledge with experimental data at two different resolutions (DNA footprints and electro-micrographs) to develop a nanoscale model of the stem.