This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers. The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.      In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities. / <p>QC 20130926</p>

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.

Based on the underlying structure of fabric materials, a three-scale model is constructed to describe the mechanical behavior of fabric materials. The current model assumes that fabric materials take on an overall behavior of anisotropic membranes, so membrane scale is taken as the macroscopic or continuum scale of the model. Following the membrane scale, yarn scale is introduced, in which yarns and their weaving structure are accounted for explicitly and the yarns are modeled as extensible elasticae. A unit cell consisting of two overlapping yarns is used to formulate the weaving patterns of yarns, which governs the constitutive nonlinear behavior of fabric materials. The third scale, named fibril scale, zooms to the fibrils inside a yarn and incorporates its material properties. Via a coupling process between these three scales, the overall behavior and performance of the complex fabric products become predictable by knowing the material properties of a single fibril and the weaving structure of the fabrics. In addition, potential damage during deformation is also captured in the current model through tracking the deformation of yarns in fibril scale. Based on the multi-scale model, both static and dynamic simulations were implemented. Comparison between the static simulations and experiment demonstrates the model abilities as desired. Through the dynamic simulations, parameter research was conducted and indicates the ballistic performance and mechanical behavior of the fabric materials are determined by a combination of various factors and conditions rather than the material properties alone. Factors such as boundary conditions, material orientation and projectile shapes etc. affect the damage patterns and energy absorption of the fabric. / Mechanical Engieering

A multiple scale theory is developed for the prediction of damage evolution in heterogeneous viscoelastic media. Asymptotic expansions of the field variables are used to derive a global scale viscoelastic constitutive equation that includes the effects of local scale damage. Damage, in the form discrete cracks, is allowed to grow according to a micromechanically-based viscoelastic traction-displacement law. Finite element formulations have been developed for both the global and local scale problems. These formulations have been implemented into a two-scale computational model Numerical results are given for several example problems in order to demonstrate the effectiveness of the technique.

In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.

Image processing applications such as image denoising, image segmentation, object detection, object recognition and texture synthesis often require a multi-scale analysis of images. This is useful because different features in the image become prominent at different scales. Traditional imaging models, which have been used for multi-scale analysis of images, have several limitations such as high sensitivity to noise and structural degradation observed at higher scales. Parametric models make certain assumptions about the image structure which may or may not be valid in several situations. Non-parametric methods, on the other hand, are very flexible and adapt to the underlying image structure more easily. It is highly desirable to have effi cient non-parametric models for image analysis, which can be used to build robust image processing algorithms with little or no prior knowledge of the underlying image content. In this thesis, we propose a non-parametric pixel neighbourhood based framework for multi-scale image analysis and apply the model to build image denoising and saliency detection algorithms for the purpose of illustration. It has been shown that the algorithms based on this framework give competitive results without using any prior information about the image statistics.

In this dissertation we develop, analyze and implement effective numerical methods for multiscale phenomena arising from flows in heterogeneous porous media. The main purpose is to develop innovative numerical and analytical methods that can capture the effect of small scales on the large scales without resolving the small scale details on a coarse computational grid. This research activity is strongly motivated by many important practical applications arising in contaminant transport in heterogeneous porous media, oil reservoir simulations and subsurface characterization. In the work, we investigate three main multiscale numerical methods, i.e., multiscale finite element method, partition of unity method and mixed multiscale finite element method. These methods employ limited single or multiple global information. We apply these numerical methods to partial differential equations (elliptic, parabolic and wave equations) with continuum scales. To compute the solution of partial differential equations on a coarse grid, we define global fields such that the solution smoothly depends on these fields. The global fields typically contain non-local information required for achieving a convergence independent of small scales. We present a rigorous analysis and show that the proposed global multiscale numerical methods converge independent of small scales. In particular, a global mixed multiscale finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The numerical results demonstrate that the global multiscale numerical methods achieve high accuracy.

This dissertation describes a new version of the Generalized Finite Element Method (GFEM), which is well suited for problems set in domains with a large number of internal features (e.g. voids, inclusions, etc.), which are practically impossible to solve using the standard FEM. The main idea is to employ the mesh-based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the Partition of Unity Method (PUM). It is shown that the p-version of the Generalized FEM using mesh-based handbook functions is capable of achieving very high accuracy. It is also analyzed that the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. The robustness of the method is illustrated by several model problems defined in domains with a large number of closely spaced voids and/or inclusions with various shapes, including the heat conduction problem defined on domains with porous media and/or a real composite material.

A multiple scale theory is developed for the prediction of damage evolution in heterogeneous viscoelastic media. Asymptotic expansions of the field variables are used to derive a global scale viscoelastic constitutive equation that includes the effects of local scale damage. Damage, in the form discrete cracks, is allowed to grow according to a micromechanically-based viscoelastic traction-displacement law. Finite element formulations have been developed for both the global and local scale problems. These formulations have been implemented into a two-scale computational model Numerical results are given for several example problems in order to demonstrate the effectiveness of the technique.