en.txt After an introduction into current sheet physics, with emphasis to solar physics we showed some formulations of finite element method. We implemented and evaluated new discontinuous finite element with Taylor basis and introduced deal.II library with an example of burgers equation. While the program is dimension independent, we compared our solution with a one-dimensional analytical solution. Finally, using previously derived LSFEM formulation, we solved simple current sheet problem using deal.II. Stránka 1

The aim of this project is to study the viscous Burgers' equation for the case where the viscosity is constant, but also when it contains a jump in viscosity. In the first case where the viscosity is constant, Burgers' is simply solved on a singular domain. For the case with jump in viscosity, Burgers' is solved on multiple domains with different viscosity. The different domains are then connected by applying inner boundary conditions at an interface in order to produce a singular solution. The inner boundary conditions are imposed using three different methods; simultaneous approximation term (SAT), projection and hybrid method, where the hybrid method is a combination of both the SAT and projection method. These methods are used in combination with a stable and high-order accurate summation by parts (SBP) finite difference approximation in MATLAB. The three methods are then compared to each other with respect to the least square error and the corresponding convergence rate to determine which method is the most preferable to use. The methods resulting in the highest convergence rates are the projection and the hybrid methods. These methods manage to live up to the expected convergence rates for all operators with different orders of accuracy and are therefore both good methods to use. However, the best method to use is the projection method since it is much simpler to implement than the two other methods but still achieves just as good convergence rates as the hybrid method.

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Multiple strain path changes during forming lead to complex geometrically necessary dislocation (GND) development in strain gradient fields, inducing internal stresses that contribute to the Bauschinger effect, residual stresses, and springback which alters the final geometrical shape of the part. In order to analyze and design improved processing routes, models must capture the evolution of these internal stresses. However, most models capture the effects of these stresses via phenomenological approaches that require calibration to each new material and strain path. The development of models that capture the underlying physics at the sub-grain level is underway but requires in-depth studies of dislocation behavior (at the relevant meso length scale) in order to guide and validate them. The novel experimental campaign central to this thesis aims to tackle this problem by capturing unprecedented data of dislocation activity for several sheet metals during multiple strain path deformation. The resultant insights provide a new window into multi-path forming of metals, while also aiding the development and validation of two crystal plasticity (CP) models by collaborators at the University of New Hampshire (UNH). The models incorporate internal stresses at the grain and sub-grain levels, respectively. The hardening response due to strain path change during forming of AA6016-T4 was studied at the macro- and micro-level via combined experiments and an elasto-plastic self-consistent (EPSC) model. The experiments demonstrated that possible recombination and/or redirection of dislocations onto different slip systems under strain path change allowed for a gradual elasto-plastic transition, in comparison to a much sharper response upon continued deformation under the same strain path due to buildup and immediate activation of backstresses. The phenomenological backstress law of the EPSC model underpredicted the yield stress response for the strain path change deformations, possibly due to missing sub-grain GND development and an accurate description of associated backstresses. A more detailed experimental study of multi-path deformation for the AA6016-T4 was required in order to guide development of a strain gradient elasto-visco plasticity self-consistent model (SG-EVPSC); the model includes sub-grain strain gradient fields, and related internal stress fields. Total dislocation and GND density were tracked at various points of the deformation, and a complete 3D statistical volume element was characterized, to enable accurate modeling of the microstructure. The tests revealed a relatively lower yield stress response following strain path change, presumably aided by lower latent hardening than self hardening; the tests then showed a rapid accumulation of dislocations on the newly activated slip systems resulting in much higher final dislocation density without affecting the ductility of the pre-strained material. Interestingly, GND development was dominated by the precipitates instead of grain boundaries. These observations are vital for an accurate forming prediction from CPFEA models. Finally, optimized forming conditions of continuous bending under tension produced a ratcheting strain path resulting in a gradual GND development and a more complete retained austenite transformation in quenched-&-partitioned- and TRIP-assisted bainitic ferritic-1180 steels increasing their ductility by at least 360%.

In this thesis, we study the degree of the canonical map of surfaces of general type. In particular, we give the first examples known in the literature of surfaces having degree d=10,11, 13, 14, 15, and 18 of the canonical map. They are presented in a self-contained and independent way from the rest of the thesis. We show also how we have discovered them. These surfaces are product-quotient surfaces. In this thesis, we study the theory of product-quotient surfaces giving also some new results and improvements. As a consequence of this, we have written and run a MAGMA script to produce a list of families of product-quotient surfaces having geometric genus three and a self-intersection of the canonical divisor large. After that, we study the canonical map of product-quotient surfaces and we apply the obtained results to the list of product-quotient surfaces just mentioned. In this way, we have discovered the examples of surfaces having degree d=10,11,14, and 18 of the canonical map. The remaining ones with degrees 13 and 15 do not satisfy the assumptions to compute the degree of the canonical map directly. Hence we have had to compute the canonical degree of these two families of product-quotient surfaces in a very explicit way through the equations of the pair of curves defining them. Another work of this thesis is the classification of all smooth surfaces of general type with geometric genus three which admits an action of a group G isomorphic to \mathbb Z_2^k and such that the quotient is a projective plane. This classification is attained through the theory of abelian covers. We obtained in total eleven families of surfaces. We compute the canonical map of all of them, finding in particular a family of surfaces with a canonical map of degree 16 not in the literature. We discuss the quotients by all subgroups of G finding several K3 surfaces with symplectic involutions. In particular, we show that six families are families of triple K3 burgers in the sense of Laterveer. Finally, in another work we study also the possible accumulation points for the slopes K^2/ \chi of unbounded sequences of minimal surfaces of general type having a degree d of the canonical map. As a new result, we construct unbounded families of minimal (product-quotient) surfaces of general type whose degree of the canonical map is 4 and such that the limits of the slopes K^2/ \chi assume countably many different values in the closed interval [6+2/3, 8].

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This thesis examines the historical connecting threads between legacy and contemporary forms of television animation. I argue that Bob’s Burgers has been neglected by animation studies due to its incomparability to the established fulcrum of television animation: The Simpsons (1989-). However, by analyzing Bob's Burgers through a historical lens that reaches beyond The Simpsons, we can begin to see what the field of animation studies has overlooked when it has dismissed animated texts that were not comparable to The Simpsons. This thesis will utilize Bob's Burgers as a primary case study and draws connecting threads between the series and Hanna-Barbera cartoons to make this argument.

Motivated by the results concerning the regularity of solutions to the fractional Navier-Stokes system and questions about the influence of noise on the formation of singularities in hydrodynamic models, we have explored these two problems in the context of the fractional 1D Burgers equation. First, we performed highly accurate numerical computations to characterize the dependence of the blow-up time on the the fractional dissipation exponent in the supercritical regime. The problem was solved numerically using a pseudospectral method where integration in time was performed using a hybrid method combining the Crank-Nicolson and a three-step Runge-Kutta techniques. A highlight of this approach is automated resolution refinement. The blow-up time was estimated based on the time evolution of the enstrophy (H1 seminorm) and the width of the analyticity strip. The consistency of the obtained blow-up times was verified in the limiting cases. In the second part of the thesis we considered the fractional Burgers equation in the presence of suitably colored additive noise. This problem was solved using a stochastic Runge-Kutta method where the stochastic effects were approximated using a Monte-Carlo method. Statistic analysis of ensembles of stochastic solutions obtained for different noise magnitudes indicates that as the noise amplitude increases the distribution of blow-up times becomes non-Gaussian. In particular, while for increasing noise levels the mean blow-up time is reduced as compared to the deterministic case, solutions with increased existence time also become more likely. / Thesis / Master of Science (MSc)

Numerical models of PDE systems can involve very large matrix equations, but feedback controllers for these systems must be computable in real time to be implemented on physical systems. Classical control design methods produce controllers of the same order as the numerical models. Therefore, reduced order control design is vital for practical controllers. The main contribution of this research is a method of control order reduction that uses a newly developed low order basis. The low order basis is obtained by applying Proper Orthogonal Decomposition (POD) to a set of functional gains, and is referred to as the functional gain POD basis. Low order controllers resulting from the functional gain POD basis are compared with low order controllers resulting from more commonly used time snapshot POD bases, with the two dimensional heat equation as a test problem. The functional gain POD basis avoids subjective criteria associated with the time snapshot POD basis and provides an equally effective low order controller with larger stability radii. An efficient and effective methodology is introduced for using a low order basis in reduced order compensator design. This method combines "design-then-reduce" and "reduce-then-design" philosophies. The desirable qualities of the resulting reduced order compensator are verified by application to Burgers' equation in numerical experiments. / Ph. D.