Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical biology, and the study of porous media.Comment: 58 pages, 1 figure. To appear in the Journal of Differential Equations. Incorporates final galley proof corrections corresponding to published versio

The Texture Animator

This paper discusses three distinct techniques for animation of procedural textures and describes the assisting software tool. Animation is attained by moving the rendered point before texture evaluation, changing the definition of texture space or changing the texture colour mapping. Examples are given for textures that base on noise and turbulence functions in order to simulate natural phenomena. Some phases of texture animation process can be automated by using the code generator supported by a library of animation effects. Aspects of practical implementation are discussed and Renderman compliant code is presented

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper, we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.Comment: 35 pages, 18 figure

Investigation on the aerodynamic performance of cycloidal rotors with active leading-edge morphing

A cycloidal rotor is a novel form of propulsion system which has a geometrical design differing completely from the conventional screw propeller. The blades of a cycloidal rotor rotate about the horizontal axis of rotation. A key advantage of this rotor system is the instantaneous control of the net thrust vector, meaning that the thrust can be adjusted to any desired direction, perpendicular to the rotor’s horizontal axis of rotation. Few investigations have been conducted to assess the negative impact dynamic stall has on the cycloidal rotor’s performance characteristics. Dynamic stall is a complex phenomenon associated with unsteady aerofoil pitching motion that generates large hysteresis effects on the blade’s aerodynamic characteristics during the pitch cycle. In this study, an investigation is conducted to assess the effect of active leading-edge morphing on alleviating the negative impact dynamic stall has on the aerofoil aerodynamic characteristics as well as the cycloidal rotor performance characteristics. Computational studies are performed for a large-scale cycloidal rotor and for a single pitch-oscillating symmetric aerofoil operating at a large Reynolds number, Re greater than 1,000,000. Dynamic stall wind tunnel testing of a single NACA0015 aerofoil with an active leading-edge flap is also conducted to validate the effects of leading-edge morphing from the single pitch-oscillating aerofoil’s CFD model. The main findings from this study showed that applying active leading-edge morphing resulted in significant improvements of both the single aerofoil’s aerodynamic characteristics and the cycloidal rotor’s performance characteristics. The results from the CFD for the single pitch-oscillating aerofoil showed that the negative effects of dynamic stall were alleviated when applying active leading-edge morphing. The results from the cycloidal rotor CFD simulations showed that the effects of dynamic stall were alleviated which led to a reduction in the level of blade-wake interference. This led to a significant improvement in the cycloidal rotor performance characteristics, such as a 4-blade cycloidal rotor with active leading-edge morphing applied producing less power dissipation in comparison to a rigid 2-blade cycloidal rotor. The main findings from the experimental analysis showed that active leading-edge morphing reduced negative effects of dynamic stall such as the level of aerodynamic hysteresis, as well as improving the aerodynamic efficiency