Inframonogenic functions and their applications in 3-dimensional elasticity theory

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector-valued) monogenic, an anti-monogenic and a ψ-hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Decomposition of inframonogenic functions with applications in elasticity theory

Unlike in complex analysis, in all hypercomplex function theories a hypercomplex variable is not monogenic (regular) and there exists a problem to define analogues of positive and negative powers of the complex variable. R. Fueter firstly introduces a system of symmetric regular quaternion polynomials as analogues of positive powers of a complex variable and proves the Taylor theorem in his theory. In Clifford analysis an analogical idea is used. The Fueter symmetric polynomials are both left- and right-regular, the symmetric polynomials in Clifford analysis are also both left- and right-monogenic. In this paper we construct only left-regular quaternion polynomials and show that the theory of regular quaternion functions of a vector valued quaternion variable can be developed using these polynomials.

Regular quaternionic polynomials and their properties

For an algebra $$ {\mathbbm{B}}_0:= \left\{{c}_1e+{c}_2\omega :{c}_k\in \mathbb{C},k=1,2\right\} $$
 , e2 = ω2 = e, eω = ωe = ω, over the field of complex numbers ℂ, we consider arbitrary bases (e, e2) such that $$ e+2p{e}_2^2+{e}_2^4=0 $$
 for any fixed p > 1. We study $$ {\mathbbm{B}}_0 $$
 -valued “analytic” functions
 $$ \Phi \left( xe+y{e}_2\right)={U}_1\left(x,y\right)e+{U}_2\left(x,y\right) ie+{U}_3\left(x,y\right){e}_2+{U}_4\left(x,y\right)i{e}_2 $$
 such that their real-valued components Uk,
 $$ k=\overline{1,4} $$
 , satisfy the equation for the stress function u in the case of orthotropic plane deformations $$ \left(\frac{\partial^4}{\partial {x}^4}+2p\frac{\partial^4}{\partial {x}^2\partial {y}^2}+\frac{\partial^4}{\partial {y}^4}\right)u\left(x,y\right)=0 $$
 , where x and y are real variables. All functions Φ for which U1 ≡ u are described in the case of a simply connected domain. Particular solutions of the equilibrium system of equations in displacements are found in the form of linear combinations of the components Uk,
 $$ k=\overline{1,4} $$
 , of the function Φ for some plane orthotropic media.

Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. II

In complex analysis, every harmonic function admits a decomposition as a sum of a holomorphic and an anti-holomorphic function. However, this fact does not hold for paravector-valued harmonic functions, or so-called &#x1d49c;-valued harmonic functions, in quaternion function theory. In previous articles, the authors proved that by taking into account ψ-hyperholomorphic functions, where ψ is a structural set different from the standard one and its conjugation, every &#x1d49c;-valued harmonic function can be written as a sum of a monogenic, an anti-monogenic and a ψ-hyperholomorphic function. In this paper, we revisit such a decomposition and apply it to study problems in elasticity.

ψ-hyperholomorphic functions and an application to elasticity problems

S. G. Georgiev, Complete orthogonal systems of monogenic polynomials over 3D prolate spheroids have recently experienced an upsurge of interest because of their many remarkable properties. These generalized polynomials and their applications to the theory of quasi-conformal mappings and approximation theory have played a major role in this development. In particular, the underlying functions of three real variables take on values in the reduced quaternions (identified with ) and are generally assumed to be null-solutions of the well-known Riesz system in . The present paper introduces and explores a new complete orthogonal system of monogenic functions as solutions to this system for the space exterior of a 3D prolate spheroid. This will be made in the linear spaces of square integrable functions over . The representations of these functions are explicitly given. Some important properties of the system are briefly discussed, from which several recurrence formulae for fast computer implementations can be derived. Copyright © 2015 John Wiley & Sons, Ltd.

On 3D orthogonal prolate spheroidal monogenics

Additive decompositions of harmonic functions play an important role in function theory, and in its application to the solution of partial differential equations. One of the most fundamental results is the decomposition of a harmonic function into a sum of a holomorphic and an anti-holomorphic function. This decomposition can be generalized to the case of quaternion valued harmonic function, where the summands are then monogenic and anti-monogenic, respectively. For paravector-valued functions, sometimes called $\mathcal{A}$-valued functions, this decomposition is not possible. In previous articles it was shown that a harmonic function can be represented by a linear combination of monogenic, anti-monogenic and ψ-hyperholomorphic functions, where ψ = {1,e 2, − e 1} is the structural set. In this paper we will study the question of such a representation for a general structural set ψ.

An Additive Decomposition of Harmonic Functions in \rm{I\!R}^{3}

This paper studies a modified economic dispatch problem (EDP) of an energy network when both power generation and transmission costs are taken into account. The considered network consists of power stations and energy consuming systems connecting via transmission lines. Due to the differences in generation costs and capacities of generators and the demand of consumers, it is highly required to have an optimal coordination in power generation distribution and power transmission. By assuming the cost functions have convex quadratic forms and applying Douglas–Rachford alternating direction method of multipliers (ADMM), we design a simple iteration update for each agent (corresponding to one power generators or one energy consumer systems) to determine the optimal power generation distribution (for generator nodes) and the optimal power transmitted to its neighbors in a distributed manner. The effectiveness of the proposed method is verified via numerical simulations for a modified IEEE 30-bus test network. We also remark that it can be used to solve the traditional EDP with a small modification in problem setup.

Hung Manh Nguyen

Papers

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

ψ-hyperholomorphic functions and an application to elasticity problems

On 3D orthogonal prolate spheroidal monogenics

An Additive Decomposition of Harmonic Functions in \rm{I\!R}^{3}

Distributed coordination of power generation and transmission for an energy network