Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method of freeform optics

The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory, and introduce an efficient approach for finding the optimal mapping of the LMK problem. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

The Monge-Ampére equation method in freeform optics design

The Monge-Ampére equation method could be the most advanced point source algorithm of freeform optics design. This paper introduces this method, and outlines two key issues that should be tackles to improve this method.

Characterizing chaos in a type of fractional duffing's equation

We characterize the chaos in a fractional Duffing’s equation computing the Lyapunov exponents and the dimension of the strange attractor in the effective phase space of the system. We develop a specific analytical method to estimate all Lyapunov exponents and check the results with the fiduciary orbit technique and a time series estimation method.

Estudio de validez del WAsP frente al WindSim en colinas Gaussinas

Este proyecto se basa en la comparación de un modelo de flujo lineal frente a un modelo de ecuaciones completas. Con esta motivación se empleará un programa de cada tipo. Los elegidos son el WAsP y el WindSim respectivamente. Tras una breve descripción de cada programa, estudiaremos los distintos elementos que los componen y su estructura. Entre todas las posibilidades que presentan ambos programas, el proyecto se centrará en la estimación del recurso eólico. En teoría, el programa que emplea un modelo lineal no será apto en terrenos complejos, por ello se tratará de estimar el error cometido por el modelo lineal tomando como referencia el modelo de ecuaciones completas. Con el objetivo de comparar ambos programas y poder distinguir sus diferencias, se plantea un caso común, en el cual se evaluarán distintas condiciones meteorológicas para colinas de forma gaussiana y distinta pendiente. Con ello se pretende medir la evolución de la precisión del WAsP conforme el terreno se va haciendo más complejo. Otras variables a tener en cuenta serán la variación de la velocidad del viento y la altura del punto de cálculo. Finalmente se analizan y explican los resultados obtenidos acompañados de elementos visuales proporcionados por los programas. 2. Abstract The main objective of this project is the comparison of two models, one based on the lineal flux and the other based on the complete equations. Thanks to two different computer programmes, WAsP and WindSim, the first one using a linear model and the second one using a complete equation model, we will be able to highlight the main differences between both models. Furthermore, a description of the structure and elements of each program will be outlined. This project will focus on the estimation of the wind resource. In theory, the program which uses a linear model will not be useful in complex terrains. Therefore, we will try to estimate the fault of the lineal model comparing it to the complete equation model. In order to be able to distinguish the differences between both programmes, the same exercise will be proposed to be solved by both of them. Here a range of meteorological conditions will be evaluated over a Gaussian hill with a slope that varies. Thereby, we will be able to measure the evolution of the precision of WAsP according to the increase of the slope. Finally, the results are analysed and explained with help of some visual characters.

Classical solutions for a logarithmic fractional diffusion equation

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion ?tu + (?)1/2 log(1 + u) = 0, posed for x ? R, with nonnegative initial data in some function space of LlogL type. The solutions are shown to become bounded and C? smooth in (x, t) for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.

An engineering modification of the blade element momentum equation for vertical descent : an autoration case study

An engineering modification of blade element/momentum theory is applied to describe the vertical autorotation of helicopter rotors. A full non-linear aerodynamic model is considered for the airfoils, taking into account the dependence of lift and drag coefficients on both the angle of attack and the Reynolds number. The proposed model, which has been validated in previous work, has allowed the identification of different autorotation modes, which depend on the descent velocity and the twist of the rotor blades. These modes present different radial distributions of driven and driving blade regions, as well as different radial upwash/downwash patterns. The number of blade sections with zero tangential force, the existence of a downwash region in the rotor disk, the stability of the autorotation state, and the overall rotor autorotation efficiency, are all analyzed in terms of the flight velocity and the characteristics of the rotor. It is shown that, in vertical autorotation, larger blade twist leads to smaller values of descent velocity for a given thrust generated by the rotor in the autorotational state.