Since the cost of petroleum fluctuates widely, it is advisable to optimize extraction of oil and other hydrocarbon products form existing oil reserves. Because of the costs involved in recovering oil from a reservoir, predicting reservoir performance can be a useful tool for determining whether continued extraction might be profitable. This can be done using computer simulations of the physical processes involved such as pressure/head, fluid velocities, and so forth. Fluid flow within a reservoir occurs at a very small scale relative to the size of the reservoir. This size difference makes performing simulations at the physically appropriate scale unfeasible. Homogenization is a technique used in reservoir simulation to upscale small scale dependent behavior, such as a permeability tensor, to make simulation feasible. To calculate a homogenized permeability tensor, the solution to a system of uncoupled elliptic partial differential equations must be found repeatedly throughout the reservoir. Generally, the solution to the system of differential equations is approximated numerically using finite element or finite difference methods. We explore using wavelets as a means of characterizing homogenization in reservoir simulations in the search for fast algorithms for computing equivalent tensors. In addition to the analogy developed between homogenization and wavelets, proofs of convergence results from homogenization within the wavelet characterization are considered.

Modern technologies require the materials with combinations of properties that can not be met by conventional single phase materials. This requirement leads to the development of composite materials or other materials with engineered microstructures, such as polymer composites and nanotube. Though the well-established finite element analysis (FEA) has the ability to analyze a small portion of such material, for the whole structure, the total degrees of freedom of a finite element model can easily exceed the bearable time in analysis or the capability of the best mainstream computers. To reduce the total degrees of freedom and save the computational efforts, an efficient way is to use a simpler and coarser mesh at the structure level with the micro level complexities captured by a homogenization method. Throughout the dissertation, the homogenization is carried on by variational asymptotic method which has been developed recently as the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH). This methodology is also expandable to the structure analysis as long as a representative structural element (RSE) can be obtained from structure. In the present research, the following problems are handled: (1) Maximizing the flexibility of choosing a RSE; (2) Bounding the effective properties of a random RSE; (3) Obtaining the equivalent plate stiffnesses for a corrugated plate from a RSE; (4) Extending the shell element of relative degree of freedom to analyze thin-walled RSE. These problems covered some important topics in homogenization theory. Firstly, the rules need to be followed when choosing a unit cell from a structure that can be homogenized. Secondly, for a randomly packed structure, the efficient way to predict effective material properties is to predict their bounds. Then, the composite material homogenization and the structural homogenization can be unied from a mathematical point of view, thus the repeating structure can be always simplified by the homogenization method. Lastly, the efficiency of analyzing thin-walled structures has been enhanced by the new type of shell element. In this research, the first two topics have been solved numerically through the finite element method under the framework of VAMUCH. The third one has been solved both analytically and numerically, and in the last, a new type of element has been implemented in VAMUCH to adapt the characteristics of a thin-walled problem. Numerous examples have demonstrated VAMUCH application and accuracy as a general-purpose analysis tool.

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The thesis addresses two topics in the study of material properties as determined by the microstructure. The first topic involves percolation as a tool in relating the grain boundary structure to global properties such as fracture and corrosion resistance. The second investigates optimization techniques in order to find the space of values that properties of a material can take, from consideration of the microstructure. In part I, the applicability of standard lattice percolation models to a random 2-D grain structure is explored. A random network based on the triangle lattice is proposed as a more appropriate model, and results in a higher percolation threshold (0.711 compared with 0.653 for the standard hexagonal lattice). The triple junction constraint inherent in grain boundary structures is subsequently applied to the new network. This results in a lowering of the percolation threshold to 0.686, which turns out to be very close to the value obtained from the hexagonal lattice under the same constraint. In Part II, an efficient method for finding the closure of a bi-objective optimization problem involving two material properties is formulated. The method is based upon two algorithms developed to find the Pareto front in multi-objective problems — the weighted sum, and the normal boundary intersection methods. The resultant procedure uses quadratic programming (QP) to find as many points on the closure as possible, changing to the less efficient sequential quadratic programming (SQP) only where necessary to find points on concave, or almost concave, regions. Improvements on the method are demonstrated using extra linear constraints in the generalized weighted sum (GWS) algorithm, and multiple GWS trials at each stage. Optimization using only the linear part of the problem is shown to give excellent results for the particular example used, and may help to achieve an approximate closure more quickly for larger problems. An adaptation of a common Pareto front method from genetic algorithms — the maxmin algorithm — is also demonstrated. The efficiency of the method is found to be reasonable for finding closures in the test case. Other general optimization techniques for the form of the problem in-hand are explored for completeness of the study. These include a survey of unconstrained techniques that might be useful in large-scale problems, and an in-depth application of QP for the constrained case.

Shocks form in the solutions of first-order nonlinear hyperbolic PDEs with constant co-efficients. Where solitary waves arise in the solutions of first-order nonlinear hyperbolic PDEs with variable coefficients, those solitary waves occur due to the coupling of nonlinearity and dispersive effects that comes from the medium’s heterogeneity. In this thesis, we study a fluid that propagates in a narrow pipe with periodically-varying cross-sectional area described by a system of first-order nonlinear hyperbolic PDEs. Multiple-scale perturbation theory is applied to derive homogenized effective equations, which take the form of a constant-coefficient system including higher-order dispersive terms. We investigate the behavior of the solution by deriving the linear dispersion relation of the homogenized system. The homogenized equations are solved using a psuedospectral discretization in space and explicit Runge-Kutta method in time. Lastly, we develop a Riemann solver in Clawpack to solve the variable coefficients system and compare the obtained solution with the homogenized equations solution.

Additives used in the cathode of a Lithium-ion (Li-ion) battery to improve electrical conductivity can negatively impact the ionic conductivity and specific capacity. Therefore, recent focus on the design of Li-ion battery is on the additive-free cathodes. This research work aims to provide a simple rule for the design of cathode microstructure using extensive study of the effect of particle size and volume fraction on effective electrical conductivity. Most design methods used to model the effective transport properties of lithium ion battery electrodes utilize the approximations based on Bruggeman’s formula. However, this formula does not consider the microstructure geometry and hence cannot accurately predict the effective transport properties of complicated microstructure like those of Li-ion battery electrodes. In this thesis, based on the principles of mathematical homogenization, an extensive analysis of randomly generated two-phase microstructures idealized for li-ion battery cells is carried out to obtain more accurate estimates of the effective electrical conductivity. To this end, a wide range of values of particle size, volume fraction and conductivity ratios are considered to evaluate the effective conductivity values. From these results, an explicit formulation based on these three parameters to predict the effective conductivity is provided to establish a framework for a simple design rule for additive-free cathode microstructures. Finally, the significance of the microstructural information is highlighted by studying the discharge characteristics of a battery for a theoretical battery model using the Brugemman’s formulation as well as the proposed formulation based on the mathematical homogenization technique. / Thesis / Master of Applied Science (MASc)

Torsion of cylindrical shafts has long been a basic subject in the classical theory of elasticity. In 1998 Swanson proposed a theoretical solution for the torsion problem of laminated composites. He adopted the traditional formulation of the torsion problem based on Saint Venant's torsion theory. The eigenfunction expansion method was employed to solve the formulated problem. The analytical method is proposed in this study enabling one to solve the torsion problem of laminated composite beams. Instead of following the classical Saint Venant theory formulation, the notion of effective elastic constant is utilized. This approach uses the concept of elastic constants, and in this context the three-dimensional non-homogeneous orthotropic laminate is replaced by an equivalent homogeneous orthotropic material. By adopting the assumptions of constant stress and constant strain, the effective shear moduli of the composite laminates are then derived. Upon obtaining the shear moduli of the equivalent homogeneous material, the effective torsional rigidity of the laminated composite rods can be determined by employing the theory developed by Lekhnitskii in 1963. Finally, the predicted results based on the present analytical approach are compared with those by the finite element, the finite difference method and Swanson's results. / Master of Science

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed in this work, we are able to numerically compute the approximations of the pressure variables. This algorithm can directly be applied to elliptic problems with discontinuous coefficients.We also implemented Java codes to solve the two dimensional elliptic problems using our methods of solutions. Furthermore, we propose homogenization wavelet reconstruction algorithm, fast transform and the inverse transform algorithms that use the results from the solutions of the local problems and the partial derivatives of the pressure variables to reconstruct the solutions.

Finite Element Method (FEM) is a powerful numerical tool which, combined with the fast development of Computer Science in the lastdecades, had made possible to perform mechanical analysis of a widerange of bodies and boundary conditions. However, the complexity of some cases may turn the calculationprocess too slow and sometimes even unaffordable for most computers. This work aims to simplify an intricate system of layers withdifferent geometries and material properties by approximating itthrough a homogeneous material, with unique mechanical parameters.Besides the Finite Element analysis, a theoretical model is created, in order to understand the basis of the problem, and, as a firstapproach, check whether the assumptions made in the FEM model areacceptable or not. This work intends to make a small contribution to the understandingof the mechanical behaviour of the Vasa vessel, which will eventuallylead to the design of a new support structure for the ship. The preservation of the Vasa is a priority for the Swedish Property Board, as it is one of the main monuments of Sweden.

<p align="justify">The conservation of structures of the historical heritage is an increasing concern nowadays for public authorities. The technical design phase of repair operations for these structures is of prime importance. Such operations usually require an estimation of the residual strength and of the potential structural failure modes of structures to optimize the choice of the repairing techniques.</p> <p align="justify">Although rules of thumb and codes are widely used, numerical simulations now start to emerge as valuable tools. Such alternative methods may be useful in this respect only if they are able to account realistically for the possibly complex failure modes of masonry in structural applications.</p> <p align="justify">The mechanical behaviour of masonry is characterized by the properties of its constituents (bricks and mortar joints) and their stacking mode. Structural failure mechanisms are strongly connected to the mesostructure of the material, with strong localization and damage-induced anisotropy.</p> <p align="justify">The currently available numerical tools for this material are mostly based on approaches incorporating only one scale of representation. Mesoscopic models are used in order to study structural details with an explicit representation of the constituents and of their behaviour. The range of applicability of these descriptions is however restricted by computational costs. At the other end of the spectrum, macroscopic descriptions used in structural computations rely on phenomenological constitutive laws representing the collective behaviour of the constituents. As a result, these macroscopic models are difficult to identify and sometimes lead to wrong failure mode predictions.</p> <p align="justify">The purpose of this study is to bridge the gap between mesoscopic and macroscopic representations and to propose a computational methodology for the analysis of plane masonry walls. To overcome the drawbacks of existing approaches, a multi-scale framework is used which allows to include mesoscopic behaviour features in macroscopic descriptions, without the need for an a priori postulated macroscopic constitutive law. First, a mesoscopic constitutive description is defined for the quasi-brittle constituents of the masonry material, the failure of which mainly occurs through stiffness degradation. The mesoscopic description is therefore based on a scalar damage model. Plane stress and generalized plane state assumptions are used at the mesoscopic scale, leading to two-dimensional macroscopic continuum descriptions. Based on periodic homogenization techniques and unit cell computations, it is shown that the identified mesoscopic constitutive setting allows to reproduce the characteristic shape of (anisotropic) failure envelopes observed experimentally. The failure modes corresponding to various macroscopic loading directions are also shown to be correctly captured. The in-plane failure mechanisms are correctly represented by a plane stress description, while the generalized plane state assumption, introducing simplified three-dimensional effects, is shown to be needed to represent out-of-plane failure under biaxial compressive loading. Macroscopic damage-induced anisotropy resulting from the constituents' stacking mode in the material, which is complex to represent properly using macroscopic phenomenological constitutive equations, is here obtained in a natural fashion. The identified mesoscopic description is introduced in a scale transition procedure to infer the macroscopic response of the material. The first-order computational homogenization technique is used for this purpose to extract this response from unit cells. Damage localization eventually appears as a natural outcome of the quasi-brittle nature of the constituents. The onset of macroscopic localization is treated as a material bifurcation phenomenon and is detected from an eigenvalue analysis of the homogenized acoustic tensor obtained from the scale transition procedure together with a limit point criterion. The macroscopic localization orientations obtained with this type of detection are shown to be strongly related to the underlying mesostructural failure modes in the unit cells.</p> <p align="justify">A well-posed macroscopic description is preserved by embedding localization bands at the macroscopic localization onset, with a width directly deduced from the initial periodicity of the mesostructure of the material. This allows to take into account the finite size of the fracturing zone in the macroscopic description. As a result of mesoscopic damage localization in narrow zones of the order of a mortar joint, the material response computationally deduced from unit cells may exhibit a snap-back behaviour. This precludes the use of such a response in the standard strain-driven multi-scale scheme.</p> <p align="justify">Adaptations of the multi-scale framework required to treat the mesostructural response snap-back are proposed. This multi-scale framework is finally applied for a typical confined shear wall problem, which allows to verify its ability to represent complex structural failure modes.</p>