Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Edge function method applied to thin plates resting on Winkler foundation

The Edge Function method formerly developed by Quinlan⁽²⁵⁾ is applied to solve the problem of thin elastic plates resting on spring supported foundations subjected to lateral loads the method can be applied to plates of any convex polygonal shapes, however, since most plates are rectangular in shape, this specific class is investigated in this thesis. The method discussed can also be applied easily to other kinds of foundation models (e.g. springs connected to each other by a membrane) as long as the resulting differential equation is linear. In chapter VII, solution of a specific problem is compared with a known solution from literature. In chapter VIII, further comparisons are given. The problems of concentrated load on an edge and later on a corner of a plate as long as they are far away from other boundaries are also given in the chapter and generalized to other loading intensities and/or plates springs constants for Poisson's ratio equal to 0.2

High-resolution automatic water quality monitoring systems applied to catchment and reservoir monitoring

Automatic recording instruments provide the ideal means of recording the responses of rivers, lakes and reservoirs to short-term changes in the weather. As part of the project ‘Using Automatic Monitoring and Dynamic Modelling for the Active Management of Lakes and Reservoirs', a family of three automatic monitoring stations were designed by engineers at the Centre for Ecology and Hydrology in Windermere to monitor such responses. In this article, the authors describe this instrument network in some detail and present case studies that illustrate the value of high resolution automatic monitoring in both catchment and reservoir applications.

Comparison of marketing policies applied by the same company in different countries and analysis of the reasons for the differences

The focus of this project is going to be, as the title indicates, on the comparison of marketing policies applied by the same company in different countries and analysis of the reasons for the differences. In order to do that, I have selected the company Nestlé to analyze the marketing decisions it makes across national boundaries to market the brand of Kit Kat and keep it as a leader snack worldwide. After having analyzed the brand in all continents, I can say the execution of the strategy used by Nestlé with Kit Kat really matches the planning of the strategy which is to think globally and act locally. Nestlé uses global brand identity but, from the internal point of view, it uses local ingredients and gives autonomy to its local branches based in different countries to make pricing and distributions decisions and therefore satisfy different consumers’ needs and preferences in local environments where changes happen very rapidly. The “glocal” approach to marketing is an effective way for Nestlé Kit Kat to stay focused on the consumer in worldwide markets.

Digital holographic interferometry and the angular spectrum method applied to measuring the domain inversion in ferroelectric crystal

The application of digital holographic interferometry on the quantitative measurement of the domain inversion in a RuO2: LiNbO3 crystal wafer is presented. The recorded holograms are reconstructed by the angular spectrum method. From the reconstructed phase distribution we can clearly observe the boundary between the inverted and un-inverted domain regions. Comparisons with the results reconstructed by use of the Fresnel transform method are given. Factors that influence the measurement include the spectrum filter size and the spectrum movement are discussed. The spectrum filter size has an effect on the measurement of the details. Although the spectrum movement affects every single reconstructed image, it has no influence on the final measurement.

Aspects of the washout of salmonid eggs. 4. Effects of a standard mechanical shock, applied at different stage of development, upon survival and development of eggs of brown trout (Salmo trutta L.)

It is generally accepted by fish culturists that salmonid eggs are sensitive to mechanical shock and that the sensitivity varies with the stage of development of the eggs. In general, the period of greatest sensitivity is thought to occur between fertilization and ”eyeing”. However, it is reasonable to expect that, during a period (perhaps of several hours) following fertilization, sensitivity will be low because in nature during this period the eggs may be subject to some mechanical shock caused by the parent fish covering them with gravel. In 1983-4 and 1984-5 experiments were performed on brown trout (Salmo trutta L.) eggs to examine the effect of a standard mechanical shock (c. 2,500 eggs in 1983-4 and c. 8,400 eggs in 1984-5) at various stages of development upon survival to hatching and time of hatching.The results of these experiments are reported in this study.