<div>The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solutions of the Boltzmann equation present a huge computational challenge, and often the collision operator is simplified for practical reasons, hereby, referred to as linear kinetic models. These models utilize the moment of the underlying probability distribution to mimic some properties of the original collision operator. But, just because we know the moments of a distribution, doesn't mean we know the actual distribution. The approximation of reality can never supersede the reality itself. Because, all the facts (moments) about the world (distribution) cannot explain the world. The premise lies not in the fact that a certain flow behavior can be correctly predicted; but rather that the Boltzmann equation can reveal and explain previously unsuspected aspects of reality.</div><div><br></div><div>Therefore, in this work, we introduce accurate, efficient, and robust numerical schemes for solving the multi-species Boltzmann equation which can model general repulsive interactions. These schemes are high order spatially and temporally accurate, spectrally accurate in molecular velocity space, exhibit nearly linear parallel efficiency on the current generation of processors; and can model a wide-range of rarefied flows including flows involving momentum, heat, and diffusive transport. The single-species variant formed the basis of author's Masters' thesis.</div><div><br></div><div>While the first part of the dissertation is targeted towards multi-species flows that exhibit rich non-equilibrium phenomenon; the second part focuses on single-species flows that do not depart significantly from equilibrium. This is the case, for example, in micro-nozzles, where a portion of flow can be highly rarefied, whereas others can be in near-continuum regime. However, when the flow is in near-continuum regime, the traditional deterministic numerical schemes for kinetic equations encounter two difficulties: a) since the near-continuum is essentially an effect of large number of particles in an infinitesimal volume, the average time between successive collisions decrease, and therefore the discrete simulation timestep has to be made smaller; b) since the number of molecular collisions increase, the flow acquires steady state slowly, and therefore one needs to carry out time integration for large number of time steps. Numerically, the underlying issue stems from stiffness of the discretized ordinary differential equation system. This situation is analogous to low Reynolds number scenario in traditional compressible Navier-Stokes simulations. To circumvent these issues, we introduce a class of high order spatially and temporally accurate implicit-explicit schemes for single-species Boltzmann equation and related kinetic models with the following properties: a) since the Navier-Stokes can be derived from the asymptotics of the Boltzmann equation (using Chapman-Enskog expansion~\cite{cercignani2000rarefied}) in the limit of vanishing rarefaction, these schemes preserve the transition from Boltzmann to Navier-Stokes; b) the timestep is independent of the rarefaction and therefore the scheme can handle both rarefied and near-continuum flows or combinations thereof; c) these schemes do not require iterations and therefore are easy to scale to large problem sizes beyond thousands of processors (because parallel efficiency of Krylov space iterative solvers deteriorate rapidly with increase in processor count); d) with use of high order multi-stage time-splitting, the time integration over sufficiently long number of timesteps can be carried out more accurately. The extension of the present methodology to the multi-species case can be considered in the future. </div><div><br></div><div>A series of numerical tests are performed to illustrate the efficiency and accuracy of the proposed methods. Various benchmarks highlighting different scattering models, different mass ratios, momentum transport, heat transfer, and diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method. As an engineering use-case, the developed methodology is applied for the study of thermal processes in micro-systems, such as heat transfer in electronic-chips; and primarily, the ingeniously Purdue-developed, Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) sensor, which can be used for flow actuation and measurement.</div>

This dissertation focuses on the development of adaptive time-stepping and high-order parallel stages exponential Runge–Kutta methods for discretizing stiff partial differential equations (PDEs). The design of exponential Runge–Kutta methods relies heavily on the existing stiff order conditions available in the literature, primarily up to order 5. It is well-known that constructing higher-order efficient methods that strictly satisfy all the stiff order conditions is challenging. Typically, methods up to order 5 have been derived by relaxing one or more order conditions, depending on the desired accuracy level. Our approach will be based on a comprehensive investigation of these conditions. We will derive novel and efficient exponential Runge–Kutta schemes of orders up to 5, which not only fulfill the stiff order conditions in a strict sense but also support the implementation of variable step sizes. Furthermore, we develop the first-ever sixth-order exponential Runge–Kutta schemes by leveraging the exponential B-series theory. Notably, all the newly derived schemes allow the efficient computation of multiple stages, either simultaneously or in parallel. To establish the convergence properties of the proposed methods, we perform an analysis within an abstract Banach space in the context of semigroup theory. Our numerical experiments are given on parabolic PDEs to confirm the accuracy and efficiency of the newly constructed methods.

This thesis developed a novel theory of fiber jamming and experimentally verified it. The theory relates the performance, which is the ratio between the stiff and soft states of a fiber jamming chamber, to three relative design parameters: the ratio of the wall thickness to the membrane inner diameter, the ratio of the fiber diameter to membrane inner diameter, and the number of fibers. These three parameters, when held constant across different chamber sizes, hold the performance constant. To test the theory, three different types of fiber jamming chambers were built in three different sizes. Each chamber was set up as a cantilever beam and deflected 10mm in both the un-jammed (soft) and jammed (stiff) states. When the three design parameters were held constant, the performance of the chamber was consistent within 10\%. In contrast, when the parameters were altered, there was a statistically significant $p < .0001$ and noticeable effect on chamber performance. These two results can be used in tandem to design miniaturized fiber jamming chambers. These results also have a direct application in soft robots designed for minimally invasive surgery.

This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)

Glycine receptor antibodies have been identified in a few patients with progressive encephalomyelitis with rigidity and myoclonus (PERM), a highly disabling disorder characterised by rigidity, spasm and brainstem symptomatology. The clinical characteristics of patients with glycine receptor antibodies have not yet been fully described and it is not clear whether GlyR-Abs are pathogenic or just an epiphenomenon. This study examined the clinical features and immunotherapy responses of 45 patients; characterised the GlyR-Ab pathogenicity, subunit specificity and binding to different brain region in vitro, and examined mice injected with GlyR-Abs to model the disease in vivo. Most of the patients were classified as PERM but some patients had symptomatology beyond the classical motor manifestations and there were four patients with tumours (thymomas and lymphomas). GlyR-Ab titres were varied in serum and CSF, but there was intrathecal synthesis in the six patients with suitable samples. Most patients were very disabled but almost all showed excellent responses to immunotherapies. The antibodies were mainly IgG1 and IgG3 subclasses, activated complement on glycine receptor-transfected HEK cells at room temperature, and caused internalisation and lysosomal degradation of the glycine receptors at 37°C. GlyR-Abs bound to rodent spinal cord and brainstem co-localising with monoclonal antibodies to GlyRα1 on the surface of neurons. GlyR-IgG injected intra-peritoneally led to impairment in forced walking ability, sensorimotor function and coordination. Analysis of the brain showed that animals injected with patients' IgG, but not control IgG, had antibodies bound to the brainstem, spinal cord, cerebellum and caudate, co-localising with GlyRα1 monoclonal antibody. Intra-cerebroventricular injection of GlyR-IgG caused an anxiety-like behaviour in mice but no evident motor disturbances. These results provide the first evidence of in vitro and in vivo pathogenicity of the GlyR-Abs, supporting the use of long term immunosuppression in these patients to provide them with a good prognosis.

During shearing of stiff clays, plate-shaped clay particles are parallel-oriented in the direction of shear reaching the minimum resistance of &ldquo / residual shear strength&rdquo / . The residual shear strength envelopes of stiff clays are curved, but for practical purposes represented by linear envelopes. This study investigates the nonlinearity of the residual shear strength envelope using experimental evidence (i) from laboratory reversal direct shear tests on two stiff clays (Ankara clay and kaolinite) at 25 to 900 kPa effective normal stresses and (ii) from laboratory data collected from literature. To evaluate the importance of nonlinearity of the envelope for geotechnical engineering practice, by limit equilibrium method, (a) case histories of reactivated landslides are analyzed and (b) a parametric study is carried out. Conclusions of this study are: (1) The residual shear strength envelopes of both Ankara clay and kaolinite are nonlinear, and can be represented by a power function (cohesion is zero). (2) At least 3 reversals or cumulative 20 mm shear displacement of direct shear box is recommended to reach residual condition. (3) Empirical relations between plasticity index and residual friction angle can accurately estimate the residual strength of stiff clays. (4) Nonlinearity is especially important for landslides where average effective normal stress on the shear plane is less than 50 kPa, both for translational and rotational failures. For such slopes using a linear strength envelope overestimates the factor of safety (more significantly for the case of high pore pressures). (5) As the plasticity index increases, the power &ldquo / b&rdquo / of the nonlinear shear strength envelope decreases, indicating more significant nonlinearity. For less plastic materials, using linear and nonlinear shear strength envelopes does not affect the factor of safety.

Anisotropy in engineering properties of soils occurs due to the depositional process forming the soil fabric and/or different directional stresses in soil history. This study investigates the anisotropy in undrained shear strength and drained compressibility of preconsolidated, stiff and fissured Ankara Clay. The compressibility behavior is determined using standard oedometer testing while the shear strength anisotropy is investigated through large diameter unconsolidated-undrained triaxial testing on undisturbed samples taken in vertical and horizontal directions from several deep excavation sites along the Konya Road in &Ccedil / ukurambar-Balgat Area, Ankara. According to the results achieved, Ankara Clay is slightly anisotropic in compressibility, with an anisotropy ratio between 0.72 and 1.17 in terms of coefficient of volume compressibility for several pressure ranges between 50 kPa and 1600 kPa. On the other hand, while a slight anisotropy in undrained shear strength at a ratio ranging between 0.87 and 1.19 in terms of deviator stress can be observed in Ankara Clay, considering the great variation in the test results of samples in same direction which mostly overlaps with the range of results obtained in the other direction, it has been concluded that the Ankara Clay located in this area can be regarded as isotropic in terms of shear strength for practical purposes.

We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as φ-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed.

The fully softened shear strength was defined by Skempton (1970) as the peak drained shear strength of a clay in a normally consolidated state. All the experience available on the applicability of the fully softened shear strength for slopes is based on back-analyses. Back-analyses of first-time failures in cuts in stiff-fissured clays and embankments constructed of fat clays have shown that, over a long period of time, the shear strength gets reduced from what is measured in the laboratory using undisturbed samples to the fully softened shear strength. These back-analyses require knowledge or assumption of pore pressures in the slope, which will have a significant influence on the shear strength obtained. Karl Terzaghi, in 1936, was the first person that qualitatively explained the behavior of cut slopes in stiff-fissured clays. According to Terzaghi (1936), a softening process is initiated by the water percolating into the fissures causing swelling and decreasing the overall shear strength of the clay mass. Investigations presented later by Skempton and his colleagues showed that the controlling shear strength for cuts in stiff-fissured clays was equal to the fully softened shear strength and recommended this shear strength to be used for design (Skempton 1970; Chandler and Skempton 1974; Chandler 1974; Skempton 1977). Skempton (1977) concluded that displacements caused by progressive failure decrease the shear strength of stiff clays toward the fully softened shear strength. At first, it was believed that only stiff-fissured clays were subjected to softening and that intact clays should be designed using the peak shear strength measured using undisturbed samples (Skempton and Brown 1961; Skempton 1964, 1970). Recent publications have showed that the likelihood of a clay experiencing softening is more dependent on the plasticity of the clay rather than the fissures (Bjerrum 1967; Chandler 1984a; Mesri and Abdel-Ghaffar 1993). Fat clays, when compared to lean clays, tend to be more brittle. This means that fat clays have a more pronounced decrease in shear strength after the peak shear strength is achieved and for this reason are more susceptible to progressive failure. First-time failures in stiff clays usually occur a long period of time after construction. For this reason, steady state seepage was used in the back-analyses of the case histories presented by Skempton and his colleagues. They found that a pore pressure ratio of 0.3 was applicable to first-time failures in cuts in stiff-fissured clays (James 1970; Vaughan and Walbancke 1973; Chandler 1974; Skempton 1977). Investigations presented by Professor Steve Wright and his colleagues of the University of Texas at Austin showed, based on back-analyses, that the fully softened shear strength is also the controlling shear strength of compacted embankments constructed of highly plastic clays (Green and Wright 1986; Kayyal and Wright 1991; Wright 2005; Wright et al. 2007). Steve Wright and his colleagues concluded that weathering, expressed in cycles of wetting and drying, was the main mechanism decreasing the shear strength of compacted clay embankments toward the fully softened shear strength. Failures in this type of projects were found to be shallow (less than 10 ft deep) and to occur numerous years after construction (USACE 1983; Stauffer and Wright 1984; Kayyal and Wright 1991; Wright et al. 2007). A pore pressure ratio ranging from 0.4 to 0.6 was found to be applicable for the case histories analyzed by Wright and his colleagues. Day and Axten (1989) recommended the use of the infinite slope method with seepage parallel to the slope face for slope stability analyses. This same recommendation was presented by Lade (2010). A seepage parallel to the slope face corresponds to a pore pressure ratio ranging from 0.4 to 0.5 for slopes with ratios of 2H:1V to 5H:1V. Failures on compacted clay embankments related to softening have been reported in Texas (Stauffer and Wright 1984; Kayyal and Wright 1991; Wright 2005; Wright et al. 2007), and Mississippi (USACE 1983). According to McCook (2012), softening of this type of structures also occur in Louisiana To perform slope stability analyses using fully softened shear strength parameter, the type of soils, type of projects, and depths where this shear strength is applicable, and the pore pressures and factor of safety to be used in design should be determined. As stated above, the fully softened shear strength has been found to be the controlling shear strength of cuts in stiff clays and compacted embankments constructed of highly plastic clays. Steady state seepage conditions should be used to design cuts in stiff clays, and a pore pressure ratio ranging from 0.4 to 0.6 or a phreatic surface at the surface of the slope should be used to design compacted embankments made of fat clays. In cuts in stiff clays, both shallow and deep failures related to fully softened shear strength have been observed. For this type of project, the recommended methodology for design is to assign a curved fully softened failure envelope to the whole slope, search for the critical failure surface, and obtain the factor of safety. This approach will provide the correct factor of safety but the critical surface obtained might not be what is expected to occur in situ. Pore pressures corresponding to steady state seepage should be used for design. It should be emphasized that the recommendation to use fully softened shear strength in first-time failures in stiff clays is based on the back-analyses of case histories. Research is required to better understand progressive failure and its influence on the shear strength mobilized in situ. In compacted embankments constructed of fat clays, only shallow failures related to fully softened shear strength have been observed. For this type of projects, the recommended methodology for design is to assign a curved fully softened failure envelope to the whole embankment, search for the critical failure surface, and obtain the factor of safety. If for any reason deep failures are to be considered in designing compacted embankments constructed of fat clays, based on the fact that failures in this type of projects are usually shallow, the first 10 ft below the surface of the slope should be assumed to have a shear strength equal to the fully softened shear strength. Pore pressures should be calculated based on a water table coincident with the slope face. The fully softened shear strength should not be used in the foundation soil. If any softening occurred in the foundation soil, this should be reflected in the shear strength measured using undisturbed samples. Softening of the foundation soil is not expected to occur after the embankment is constructed. The consequences of shallow and a deep failures are usually not the same. For this reason, is reasonable that the same factor of safety should not be required for both cases. A shallow failure may be considered by some agencies solely as a maintenance issue. The factor of safety should be based on the uncertainties in the parameters being used for design and the consequences of failure of the structure (Duncan and Wright 2005). The parameters that have more impact on the factor of safety obtained for slope stability are shear strength and pore pressures. The fully softened shear strength is the lowest shear strength expected to be mobilized in first-time slides. This shear strength, coupled with a conservative assumption of pore pressure gives a low uncertainty in the parameters that have the most influence in the factor of safety. For shallow failures, the consequences of failure are very low. For this reason, if the fully softened shear strength is used, coupled with a water table corresponding to the worst case scenario possible, a factor of safety as low as 1.25 can be used. For deep failures, the consequences of failure will vary depending on the structure. The pore pressure for this type of analyses should be based on the worst seepage condition expected throughout the life of the project. In this case, for structures with low to mid consequences of failure, a factor of safety of 1.35 can be used. For structures with a high consequence of failure, a factor of safety of 1.50 can be used. These factors of safety are based on the recommendations presented by Duncan and Wright (2005) for factors of safety based on uncertainties in the parameters and consequences of failures. The fully softened shear strength should be measured using normally consolidated remolded specimens as recommended by Skempton (1977). Soil samples should be hydrated for two days using distilled or site-specific water. The soil sample should then be washed or pushed through a No. 40 (425 µm) sieve. To achieve the desired water content, the soil sample cab be air-dried or more water should be added. Water contents equal to or higher than the liquid limit should be used to prepare test specimens for fully softened shear strength measurements. The direct shear device is recommended for fully softened shear strength measurements. The Bromhead ring shear device does not provide accurate values of fully softened shear strength. The triaxial device requires more time and effort to measure the fully softened shear strength and provides about the same fully softened shear strength as the direct shear device. The fully softened shear strength failure envelope can be estimated using the correlation presented in Figure 6.59 for the parameters required for Equation 4.1. This correlation is only intended to be used in preliminary design or if better information is not available. Laboratory determination of fully softened shear strength is always recommended for final designs. If this is not possible, the confidence limits presented in Figure 6.59 should be used to determine the fully softened shear strength parameters. / Ph. D.