This dissertation focuses on inverse problems for partial differential equations with multiscale coefficients in which the goal is to determine the coefficients in the equation using solution data. Such problems pose a huge computational challenge, in particular when the coefficients are of multiscale form. When faced with balancing computational cost with accuracy, most approaches only deal with models of large scale behavior and, for example, account for microscopic processes by using effective or empirical equations of state on the continuum scale to simplify computations. Obtaining these models often results in the loss of the desired fine scale details. In this thesis we introduce ways to overcome this issue using a multiscale approach. The first part of the thesis establishes the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. The theory developed is based on the mathematical analysis of multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f (x, x/[epsilon]), (0 < [epsilon] ≪ 1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from nonuniform samples of optimal rate. In the second part, we present a new multiscale approach for inverse homogenization problems. We prove that in certain cases where the specific form of the multiscale coefficients is known a priori, imposing an additional constraint of a microscale parametrization results in a well-posed inverse problem. The mathematical analysis is based on homogenization theory for partial differential equations and classical theory of inverse problems. The numerical analysis involves the design of multiscale methods, such as the heterogeneous multiscale method (HMM). The use of HMM solvers for the forward model has unveiled theoretical and numerical results for microscale parameter recovery, including applications to inverse problems arising in exploration seismology and medical imaging. / text

Microstructure determines fracture toughness of materials through the activation of different fracture mechanisms. To tailor the fracture toughness through microstructure design, it is important to establish relations between microstructure and fracture toughness. To this end, systematic characterization of microstructures, explicit tracking of crack propagation process and realistic representation of deformation and fracture at different length scales are required. A cohesive finite element method (CFEM) based multiscale framework is proposed for analyzing the effect of microstructural heterogeneity, phase morphology, texture, constituent behavior and interfacial bonding strength on fracture toughness. The approach uses the J-integral to calculate the initiation/propagation fracture toughness, allowing explicit representation of realistic microstructures and fundamental fracture mechanisms. Both brittle and ductile materials can be analyzed using this framework. For two-phase Al₂O₃/TiB₂ ceramics, the propagation fracture toughness is improved through fine microstructure size scale, rounded reinforcement morphology and appropriately balanced interphase bonding strength and compliance. These microstructure characteristics can promote interface debonding and discourage particle cracking induced catastrophic failure. Based on the CFEM results, a semi-empirical model is developed to establish a quantitative relation between the propagation toughness and statistical measures of microstructure, fracture mechanisms, constituent and interfacial properties. The analytical model provides deeper insights into the fracture process as it quantitatively predicts the proportion of each fracture mechanism in the heterogeneous microstructure. Based on the study on brittle materials, the semi-analytical model is extended to ductile materials such as AZ31 Mg alloy and Ti-6Al-4V alloy. The fracture resistance in these materials not only depends on the crack surfaces formed during the failure process, but also largely determined by the bulk plastic energy dissipation. The CFEM simulation permits surface energy release rate to be quantified through explicit tracking of crack propagation in the microstructure. The plastic energy dissipation rate is evaluated as the difference between the predicted J value and the surface energy release rate. This method allows competition between material deformation and fracture as well as competition between transgranular and intergranular fracture to be quantified. The methodology developed in this thesis is potentially useful for both the selection of materials and tailoring of microstructure to improve fracture resistance.

The Effective Continuum Micromechanics Analysis Code (EC-MAC) was developed for predicting effective properties of composites containing multiple distinct nanoheterogeneities (fibers, spheres, platelets, voids, etc.) each with an arbitrary number of coating layers based upon either the modified Mori-Tanaka method (MTM) and self consistent method (SCM). This code was used to investigate the effect of carbon nanofiber morphology (i.e., hollow versus solid cross-section), nanofiber waviness, and both nanofiber-resin interphase properties and dimensions on bulk nanocomposite elastic moduli. For a given nanofiber axial force-displacement relationship, the elastic modulus for hollow nanofibers can significantly exceed that for solid nanofibers resulting in notable differences in bulk nanocomposite properties. The development of a nanofiber-resin interphase had a notable effect on the bulk elastic moduli. Consistent with results from the literature, small degrees of nanofiber waviness resulted in a significant decrease in effective composite properties. Key aspects of nanofiber morphology were characterized using transmission electron microscopy (TEM) images for VGCNF/vinyl ester (VE) nanocomposites. Three-parameter Weibull probability density functions were generated to describe the statistical variation in nanofiber outer diameters, wall thicknesses, relative wall thicknesses, visible aspect ratios, and visible waviness ratios. Such information could be used to establish more realistic nanofiber moduli and strengths obtained from nanofiber tensile tests, as well as to develop physically motivated computational models for predicting nanocomposite behavior. This study represents one of the first attempts to characterize the distribution of VGCNF features in real thermoset nanocomposites. In addition, the influence of realistic nanoreinforcement geometries, distinct elastic properties, and orientations on the effective elastic moduli was addressed. The effect of multiple distinct heterogeneities, including voids, on the effective elastic moduli was investigated. For the composites containing randomly oriented wavy vapor grown carbon nanofibers (VGCNFs) and voids, the predicted moduli captured the essential character of the experimental data, where the volume fraction of voids was approximated as a nonlinear function of the volume fraction of reinforcements. This study should facilitate the development of multiscale materials design by providing insight into the relationships between nanomaterial morphology and properties across multiple spatial scales that lead to improved macroscale performance.

Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations. To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales.

Wafer bonding is a process by which two or more mirror-polished flat surfaces are joined together. This process is increasingly used in microelectronics and microsystems industries as a key fabrication technique for various applications: production of SOI wafers, pressure sensors, accelerometers and all sorts of advanced MEMS. Unfortunately, the lack of reliability of these systems does not allow them to enter the production market. This lack of reliability is often related to the lack of understanding and control of the thermo-mechanical properties of materials used for the fabrication of MEMS (indeed, at this small scale, properties of materials are sometimes quite different than at large scale) but it is also due to the limited knowledge of the different phenomena occurring during the working of these devices, the most detrimental of them being fracture. Among all of these fracture processes, the integrity of the interfaces and, particularly, the interfaces created by wafer bonding is a generic problem with significant technological relevance. In order to understand the bonding behavior of silicon wafers, the interface chemistry occurring during the different steps of the bonding process has been detailed. The formation of strong covalent bonds across the two surfaces is responsible of the high fracture resistance of gwafer bondingh interfaces after appropriate surface treatments and annealing. The bonding process (surface treatments and annealing step) has been optimized toward reaching the best combination of interface toughness and bonding uniformity. The fracture resistance of gwafer bondingh interfaces or interface toughness has been determined using a steady-state method developed in the framework of this thesis. The high sensitivity to geometrical and environmental factors of gwafer bondingh interfaces has been quantified and related to the interface chemistry. A new technique involving the insertion of a dissipative ductile interlayer between the silicon substrate and the top silicon oxide has been proposed in order to increase the overall fracture resistance. A multiscale modeling strategy which involves the description of the interface fracture at the atomic scale, of the plasticity in the thin interlayer at the microscopic scale, and of the macroscopic structure of specimen has been used to guide the optimization of this technique. Numerical simulations have shown the influence of the ductile interlayer parameters (yield strength, workhardening exponent and thickness) and the critical strength of the interface on the overall toughness of such assemblies. A first set of experimental data has allowed increasing the interface toughness by 70%. The critical strength of the interface is finally determined by inverse identification and turns out to be in the expected range of theoretical strength. The knowledge of the strength and the fracture toughness of gwafer bondingh interfaces is of practical importance because these two values can be used in a simple fracture model (e.g. cohesive-zone model) in order to observe the behavior of such interfaces under complex loading using finite element simulations.

The flagellar motor of Escherichia coli is a bidirectional rotary nano-motor, powered by a transmembrane influx of protons. The maximum speed of rotation is about 300 Hz. The motor rotates either counterclockwise (CCW) or clockwise (CW) and the rotation direction is controlled by a chemotactic protein, CheY-P. Despite extensive structural and functional studies, precise mechanisms regarding the torque generation and the directional switching processes remain unclear. In this work, a bottom-up strategy is proposed and followed to investigate this motor. This strategy, named as a multiscale modeling approach, integrates various publicly available experimental data and ‘state-of-the-art’ molecular modeling methods to build structural models for the two most important parts of the motor, the C ring and the stator. Starting from the primary sequences of the composition proteins of these two substructures, tertiary structures are predicted by means of comparative modeling or de novo prediction when the comparative modeling is not available. Quaternary structures of these proteins’s complexes are then predicted by data-driven protein-protein docking or multiscale molecular dynamics (MD) simulations. Finally, structural models of the C ring at CCW and CW rotational states are constructed by cryo-EM aided assembling methods (constraint search that is based on the multiscale modeling tools and under the constraint of the EM images). In the case of the stator, its composition proteins, MotA and MotB, are assembled by coarse-grained MD simulations. This is the first molecular model based atomistic details for the stator. A new molecular mechanism for rotational switching is proposed based on the structural models of the C ring and the stator. The two states of the C ring display significant differences in the interfaces among the self-assembled FliMs and the orientations of FliG C-terminal domain. Based on protein docking results, a binding site of CheY-P is identified on FliM which is close to the interfaces of FliMs for self-assembling. Thereby, I propose that the CheY-P binding interferes with the interactions between neighboring FliM proteins, and thus, induces ~65° rotation of the FliGc domain with respect to FliM. Subsequently, the interactions between the stator and FliGc domains are altered and the rotation direction is changed. Furthermore, a mechanism accounting for the directed rotation of the flagellar motor is proposed based on systematic MD studies on the dynamics of FliGc. It is found that the C-terminal subdomain of FliGc is capable of rotating by ~180° with respect to the N-terminal subdomain of FliGc. If this dynamics is considered in the framework of the whole C ring, the rotation of the C-terminal subdomain of FliGc exhibits an asymmetric feature. As a result, the C ring is decorated with asymmetric teeth on the outer periphery and hence similar with Feynman’s ratchet. The preference of the motor in CCW rotation or in CW rotation is then explained based on the Feynman’s ratchet model. / published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

Organic electronic materials, possessing conjugated π-systems, are extensively used as the active layers in organic electronic devices, where they are responsible for charge transport. In this dissertation, we employ a combination of quantum-mechanical and molecular- mechanics methods to provide insight into how molecular structure, orientation, packing, and local molecular environment influence the energetic landscape experienced by an excess charge in these organic electronic materials. We begin with an overview of charge transport in organic electronic materials with a focus on electronic polarization while discussing recent models, followed by a review of the computational methods employed throughout our investigations. We provide a bottom-up approach to the problem of describing electronic polarization by first laying the framework of our model and comparing calculated properties of bulk materials to available experimental data and previously proposed models. We then explore the effects of changing the electronic structure of our systems though perfluorination, and investigate the effects of modifying the crystalline packing through the addition of bulky functional groups while investigating how the non-bonded interactions between molecular neighbors change in different packing motifs. As interfaces are common in organic electronics and important processes such as charge transport and charge separation occur at these interfaces, we model organic-vacuum and organic-organic interfaces to determine the effect changing the environment from bulk to interface has on the electronic polarization. We first investigate the effects of removing polarizable medium adjacent to the charge carrier and then, by modeling a realistic organic- organic interface in a model solar cell, probe the environment of each molecular site at the interface to gain a more complete understanding of the complex energetic landscape. Finally, we conclude with a study of the non-bonded interactions in linear oligoacene dimers, model π-conjugated materials, to assess the impact of dimer configuration and acene length on the intermolecular interaction energy, and highlight the importance of dispersion and charge penetration to these systems.

Quite often, the data we observe can be effectively represented using graphs. The underlying structure of the resulting graph, however, might contain noise and does not always hold constant across scales. With the right tools, we could possibly address these two problems. This thesis focuses on developing the right tools and provides insights in looking at them. Specifically, I study several problems that incorporate network data within the multi-scale framework, aiming at identifying common patterns and differences, of signals over networks across different scales. Additional topics in network denoising and network bootstrapping will also be discussed. The first problem we consider is the connectivity changes in dynamic networks constructed from multiple time series data. Multivariate time series data is often non-stationary. Furthermore, it is not uncommon to expect changes in a system across multiple time scales. Motivated by these observations, we in-corporate the traditional Granger-causal type of modeling within the multi-scale framework and propose a new method to detect the connectivity changes and recover the dynamic network structure. The second problem we consider is how to denoise and approximate signals over a network adjacency matrix. We propose an adaptive unbalanced Haar wavelet based transformation of the network data, and show that it is efficient in approximation and denoising of the graph signals over a network adjacency matrix. We focus on the exact decompositions of the network, the corresponding approximation theory, and denoising signals over graphs, particularly from the perspective of compression of the networks. We also provide a real data application on denoising EEG signals over a DTI network. The third problem we consider is in network denoising and network inference. Network representation is popular in characterizing complex systems. However, errors observed in the original measurements will propagate to network statistics and hence induce uncertainties to the summaries of the networks. We propose a spectral-denoising based resampling method to produce confidence intervals that propagate the inferential errors for a number of Lipschitz continuous net- work statistics. We illustrate the effectiveness of the method through a series of simulation studies.

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Full-domain multiscale analyses of unidirectional AS4/H3502 open-hole composite tensile specimens were performed to assess the effect of microscale progressive fiber failures in regions with large stress/strain gradients on macroscale composite strengths. The effect of model discretization at the microscale and macroscale on the calculated composite strengths and analysis times was investigated. Multiple sets of microscale analyses of repeating unit cells, each containing varying numbers of fibers with a distinct statistical distribution of fiber strengths and fiber volume fractions, were used to establish the microscale discretization for use in multiscale calculations. In order to improve computational times, multiscale analyses were performed over a reduced domain of the open-hole specimen. The calculated strengths obtained using reduced domain analyses were comparable to those for full-domain analyses, but at a fraction of the computational cost. Such reduced domain analyses likely are an integral part of efficient adaptive multiscale analyses of large all-composite air vehicles.