Conjugate function method for numerical conformal mappings

Hakula, Harri; Quach, Tri; Rasila, Antti

doi:10.1016/j.cam.2012.06.003

Cited by 17 publications

(17 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we introduce concepts from function theory and review the algorithm for simply or doubly connected domains. For details and references we refer to [14].…”

Section: Preliminariesmentioning

confidence: 99%

“…SC Toolbox itself is based on earlier FORTRAN package by Trefethen [29]. A new algorithm involving a finite element method and the harmonic conjugate function was presented by the authors in [14].…”

mentioning

confidence: 99%

“…The main result of this paper is the generalization of the of the algorithm for simply and doubly connected domains described in [14] to multiply connected ones with different boundary conditions. To our knowledge this is the first method for problems of type R. More specifically, the fundamental new result is the definition of the conjugate problem for multiply connected domains.…”

mentioning

confidence: 99%

See 2 more Smart Citations

The Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

Hakula

Quach

Rasila

2019

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains. In this paper the conjugate function method is generalized for multiply connected domains. The key challenge addressed here is the construction of the conjugate domain and the associated conjugate problem. All variants of the method preserve the so-called reciprocal relation of the moduli. An implementation of the algorithm, along with several examples and illustrations are given.

show abstract

“…In this section we introduce concepts from function theory and review the algorithm for simply or doubly connected domains. For details and references we refer to [14].…”

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

The Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

Hakula

Quach

Rasila

2019

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our current research efforts are aligned in the following directions: 1) Indoor network planning. The goal is to evolve the proposed framework for planning and optimization of indoor deployments, using cutting-edge mapping techniques developed by the authors, such as [31].…”

Section: Conclusion and Research Directionsmentioning

confidence: 99%

Spatial Mappings for Planning and Optimization of Cellular Networks

Gonzalez

Hakula

Rasila

et al. 2018

IEEE/ACM Trans. Networking

Self Cite

View full text Add to dashboard Cite

In cellular networks, users are grouped into different cells and served by different access points (base stations) that provide wireless access to services and applications. In general, the service demand is very heterogeneous, non-uniformly distributed, and dynamic. Consequently, radio access networks create very irregular topologies with more access points where service demand is concentrated. While this dynamism requires networks with the ability to adapt to time-varying conditions, the non-uniformity of the service demand makes the planning, analysis, and optimization difficult. In order to help with these tasks, a framework based on canonical domains and spatial mappings (e.g., conformal mapping) have recently been proposed. The idea is to carry out part of the planning in a canonical (perfectly symmetric) domain that is connected to the physical one (real-scenario) by means of a spatial transformation designed to map the access points consistently with the service demand. This paper continues the research in that direction by introducing additional tools and possibilities to that framework, namely the use of centroidal Voronoi algorithms and non-conformal composite mappings. Moreover, power optimization is also introduced to the framework. The results show the usability and effectiveness of the proposed method and its promising research perspectives. I. INTRODUCTION A. Context and motivationRadio access planning and optimization are fundamental tasks in cellular networks. Broadly speaking, planning refers to the tasks of determining the number, location, and configuration of access points to provide wireless access to users (and things) to services and applications, with a certain targeted Quality of Service (QoS). In particular, the problem of finding the number of access points is also referred to as dimensioning [1], and this initial step aims at providing the required capacity for the service demand volume that is expected. However, in practice, both dimensioning and sites positioning are very difficult problems because the service demand is not uniformly distributed and it is quite diverse and dynamic. Nowadays, taking into account the continuous evolution of radio access technologies, and the new concepts and paradigms that are expected for the fifth generation (5G) of cellular networks, the boundary between planning and optimization tasks becomes blurred. Indeed, according to the excellent work presented in [1], planning and optimization are iterative tasks that go hand-in-hand. In this line of thinking, the authors of [2] also pose the need for re-thinking planning. They emphasize the importance of distributing the service demand as evenly as possible among cells as a key criterion to achieve effective planning; a goal that in the opinion of the authors of [2] (and in our's) is a very valid way to enhance system performance.In our previous work [3], also motivated by the aforementioned ideas, a novel framework for planning and optimization based on the use of canonical domains and spatial transforma...

show abstract

“…can be solved with an iterative method between even and odd points [15]. The value of can be determined according to the corresponding condition of Γ 2 and Γ 2 .…”

Section: Conformal Mapping Functionmentioning

confidence: 99%

A Complex Variable Solution for Rectangle Pipe Jacking in Elastic Half‐Plane

Liu

2017

Mathematical Problems in Engineering

View full text Add to dashboard Cite

In mechanics, the solution of soil stresses and displacements field caused by shallow rectangular jacking pipe construction can be simplified as half-plane problem. Both the boundary conditions of the surface and the cavity boundary must be taken into account. It is the essential prerequisite for mechanical analysis of the pipe jacking with the complex variable theory that the mechanical boundary must be transformed from the half-plane with a rectangle cavity to the concentric ring. According to Riemann's existence theorem and basic complex variable theory, a conformal mapping function is established. Both sides of boundary conditions equation are developed into Laurent series, and then the coefficients of complex stress function are solved by power series method. The derived solution is applied to an example and a comparison is made using FEM method to show the accuracy of the methods. The result shows the following: (1) the method presented in this paper is applicable to a shallow-buried rectangular tunnel; (2) the complex function method proposed in this paper is characterized by clear steps, fast convergence, and simple operation.

show abstract

Conjugate function method for numerical conformal mappings

Cited by 17 publications

References 15 publications

The Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

The Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

Spatial Mappings for Planning and Optimization of Cellular Networks

A Complex Variable Solution for Rectangle Pipe Jacking in Elastic Half‐Plane

Contact Info

Product

Resources

About