52 research outputs found
3D mappings by generalized joukowski transformations
The classical Joukowski transformation plays an important role in di erent applications of conformal mappings,
in particular in the study of
ows around the so-called Joukowski airfoils. In the 1980s H. Haruki and M.
Barran studied generalized Joukowski transformations of higher order in the complex plane from the view point
of functional equations. The aim of our contribution is to study the analogue of those generalized Joukowski
transformations in Euclidean spaces of arbitrary higher dimension by methods of hypercomplex analysis. They
reveal new insights in the use of generalized holomorphic functions as tools for quasi-conformal mappings.
The computational experiences focus on 3D-mappings of order 2 and their properties and visualizations for
di erent geometric con gurations, but our approach is not restricted neither with respect to the dimension
nor to the order.Financial support from "Center for Research and Development in Mathematics and Applications" of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), is gratefully acknowledged. The research of the first author was also supported by the FCT under the fellowship SFRH/BD/44999/2008. Moreover, the authors would like to thank the anonymous referees for their helpful comments and suggestions which improved greatly the final manuscript
On a quasi-conformal joukowski type transformation of second order in RM+1
The classical Joukowski transformation plays an important role in different applications
of conformal mappings, in particular in the study of flows around airfoils. Generalizations
of this transformation where used in the 1920s by J. L. Walsh in order to approximate a continuous
function on the boundary of its domain. Later, in the 1980s, H. Haruki and M. Barran
studied generalized Joukowski transformations of higher order in the complex plane but from
the perspective of functional equations. The aim of our contribution is to present a second order
Joukowski type transformation in Rm+1, but the construction also shows how to proceed in the
case of higher orders. Like in the complex plane it still preserves some of the main properties of
the ordinary Joukowski transformation (thereby justifying to be called a Joukowski type transformation),
but also reveals some new and less expected properties. We deal in some detail
only with the 3D-case corresponding to m = 2 and discuss its properties and visualizations for
different geometric configurations.Fundação para a Ciência e a Tecnologia (FCT
3D deformations by means of monogenic functions
In this paper, the authors compute the coefficient of quasiconformality for monogenic functions in an arbitrary ball of the Euclidean space . This quantification may be needed in applications but also appear to be of intrinsic interest. The main tool used is a 3D Fourier series development of monogenic functions in terms of a special set of solid spherical monogenics. Ultimately, we present some examples showing the applicability of our approach.info:eu-repo/semantics/publishedVersio
Aplicações numéricas e combinatórias de polinómios de Appell generalizados
Doutoramento em MatemáticaThis thesis studies properties and applications of different generalized Appell
polynomials in the framework of Clifford analysis.
As an example of 3D-quasi-conformal mappings realized by generalized Appell
polynomials, an analogue of the complex Joukowski transformation of order
two is introduced. The consideration of a Pascal n-simplex with hypercomplex
entries allows stressing the combinatorial relevance of hypercomplex Appell
polynomials.
The concept of totally regular variables and its relation to generalized Appell
polynomials leads to the construction of new bases for the space of
homogeneous holomorphic polynomials whose elements are all isomorphic to
the integer powers of the complex variable. For this reason, such polynomials
are called pseudo-complex powers (PCP). Different variants of them are
subject of a detailed investigation.
Special attention is paid to the numerical aspects of PCP. An efficient algorithm
based on complex arithmetic is proposed for their implementation. In this
context a brief survey on numerical methods for inverting Vandermonde
matrices is presented and a modified algorithm is proposed which illustrates
advantages of a special type of PCP.
Finally, combinatorial applications of generalized Appell polynomials are
emphasized. The explicit expression of the coefficients of a particular type of
Appell polynomials and their relation to a Pascal simplex with hypercomplex
entries are derived. The comparison of two types of 3D Appell polynomials
leads to the detection of new trigonometric summation formulas and
combinatorial identities of Riordan-Sofo type characterized by their expression
in terms of central binomial coefficients.Esta tese estuda propriedades e aplicações de diferentes polinómios de Appell
generalizados no contexto da análise de Clifford.
Exemplificando uma transformação realizada por polinómios de Appell
generalizados, é introduzida uma transformação análoga à transformação de
Joukowski complexa de ordem dois. A análise de um n- simplex de Pascal com
entradas hipercomplexas permite sublinhar a relevância combinatória de
polinómios hipercomplexos de Appell.
O conceito de variáveis totalmente regulares e a sua relação com polinómios
de Appell generalizados conduz à construção de novas bases para o espaço
dos polinómios homogéneos holomorfos cujos elementos são todos isomorfos
às potências inteiras da variável complexa. Por este motivo, tais polinómios
são chamados de potências pseudo-complexas (PCP). Diferentes variantes de
PCP são objeto de uma investigação detalhada.
É dada especial atenção aos aspectos numéricos de PCP. Um algoritmo
eficiente baseado em aritmética complexa é proposto para a sua
implementação. Neste contexto, é apresentado um breve resumo de métodos
numéricos para inverter matrizes de Vandermonde e é proposto um algoritmo
modificado para ilustrar as vantagens de um tipo especial de PCP.
Finalmente, são enfatizadas aplicações combinatórias de polinómios de Appell
generalizados. A expressão explícita dos coeficientes de um tipo particular de
polinómios de Appell e a sua relação com um simplex de Pascal com entradas
hipercomplexas são obtidas. A comparação de dois tipos de polinómios de
Appell tridimensionais leva à deteção de novas fórmulas envolvendo somas
trigonométricas e de identidades combinatórias do tipo de Riordan – Sofo,
caracterizadas pela sua expressão em termos de coeficientes binomiais
centrais
Quaternions in Joukowski Transformation
Conformal mappings have been exploited for a long time in a number of physical problems arising in aerodynamics, thermal equilibrium, electrostatics, fluid flow and so on. These complex-valued functions are implemented with just a single complex variable z. However, in this study, quaternions are introduced into a Joukowski transformation, a conformal map used in the study of fluid flow around airfoils. Analysis is effected so as to determine the properties of this transformation function and spheres of both Euclidean and hyperbolic geometry are executed in this expedition
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
Numerical procedure for three-dimensional hypersonic viscous flow over aerobrake configuration
A numerical method, which is simpler and more efficient than others currently in use, is proposed for the computation of the full viscous flow over an aerobrake body in hypersonic stream at high altitude. It treats the shock layer surrounding the blunt forebody and the near wake behind the base simultaneously by formulating the Navier-Stokes equations in conformal and azimuthal-angle coordinates. The computational domain is confined by the body wall, outflow surface and the shock, which is adjusted along the coordinate normal to the wall in the course of iterations. Because of the optimal grid and a well developed alternating direction implicit factorization technique for the governing equations, reasonably accurate results can be obtained with a 28 x 36 x 7 grid and 400 time-marching iterations. Excellent agreement of shock location is found between the present result and the schlieren photograph. Details of the base flow and shear layer impingement on the cylindrical aft body are presented for an adiabatic wall case
A note on a one-parameter family of non-symmetric number triangles
The recently growing interest in special Cliff ord Algebra valued polynomial solutions of generalized Cauchy-Riemann systems in (n + 1)-dimensional Euclidean spaces suggested
a detailed study of the arithmetical properties of their coe fficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coe cient's set can be treated as a one-parameter family of non-symmetric triangles of fractions. The discussion of its properties, similar to those
of the ordinary Pascal triangle (which itself does not belong to the family), is carried out in this paper.Fundação para a Ciência e a Tecnologia (FCT
Developments and trends in three-dimensional mesh generation
An intense research effort over the last few years has produced several competing and apparently diverse methods for generating meshes. Recent progress is reviewed and the central themes are emphasized which form a solid foundation for future developments in mesh generation
Combinatorial Identities Associated with a Multidimensional Polynomial Sequence
In this paper we combine the knowledge of different structures of a special Appell
multidimensional polynomial sequence with the problem of establishing combinatorial
identities. The elements of this special polynomial sequence have values in a Clifford
algebra, are homogeneous hypercomplex differentiable functions of different degrees and their coefficients properties can be used to stress interesting matrix and combinatorial relations
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