Multiply-connected Domains Research Articles

Compact operators on the Bergman space of multiply-connected domains

If Q is a smoothly bounded multiply-connected domain in the complex plane and A = Z> ii7k T1j k, where fj, E L(Q, dv), we show that A is compact if and only if its Berezin transform vanishes at the boundary.

A Maximum Principle for Optimal Boundary Control of Multiply-Connected Vibrating Plates

A maximum principle is formulated for the active vibration control of plates of general shape with the control forces acting on the boundary. The applicability of the maximum principle is shown for plates that can have multiply-connected domains. The performance index is specified as a quadratic functional of displacement and velocity with a suitable penalty term involving the control forces. An explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions and the proposed maximum principle. The implementation of the theory is presented in two examples, and the effectiveness of the boundary control is investigated by a numerical example.

A revisit of a cylindrically anisotropic tube subjected to pressuring, shearing, torsion, extension and a uniform temperature change

The problem of a cylindrically anisotropic tube or bar was seemed to be first examined by Lekhnitskii (1981) [Lekhnitskii, S.G., 1981. Theory of Elasticity of an Anisotropic Body. (Trans. from the revised 1977 Russian edition.) Mir, Moscow]. Recently, a thorough investigation of the subject was performed by Ting (1996) [Ting, T.C.T., 1996. Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. Roy. Soc. Lond. A452, 2397–2421] in which a formulation akin to that of Stroh’s formalism is employed to resolve the boundary value problem subjected to a uniform pressure, shearing, torsion and uniform extension. In a continuing paper, Ting (1999) [Ting, T.C.T., 1999. New solutions to pressuring, shearing, torsion and extension of a cylindrically anisotropic elastic circular tube or bar. Proc. Roy. Soc. Lond, to appear.] rederived the solutions based on a modified formalism of Lekhnitskii, in which the solutions are in terms of elastic compliances, reduced elastic compliances as well as doubly reduced compliance. The results are much more compact and simpler than those of the earlier one. Independently, in this work, we construct the governing system also under the Lekhnitskii’s framework. Nevertheless, the present work and Ting’s formulation (1999) are not alike. Besides the loads considered in Ting (1996, 1999), we add the effect of a uniform temperature change in the formulation. The assumption that the stresses depend only on r makes it possible to incorporate the various loading cases considered. In addition to the explicit forms of admissible stresses, we derive the admissible displacements which are ensured to be single-valued for a multiply-connected domain. In contrast to the Ting’s works (1996, 1999), which often require superpositions of two or more basic solutions, the present solutions offer complete forms of solutions ready for direct calculations. We also report that, as in rectilinearly anisotropic solids, an entire analogy is observed between the fields of a uniform axial extension and a uniform temperature change in cylindrically anisotropic solids.

Moduli of multiply-connected domains on a Riemannian Mobius strip

We investigate the modulus problem for families of curves in multiply-connected nonorientable domains on a Riemannian Mobius strip. We determine the extremal metric and the modulus of a “cross” family of arcs.

Analytic self-maps of the punctured plane

Let f be a non-Möbius self-map of C *. Write J *(f) for the Julia Set of f in Ĉ and J(f) for the closure of J *(f) in Č. Then we have one of the three cases (i)J(f), J *(f) are connected, (ii)J(f) is connected, J *(f) has infinitely many components, (iii)J(f) has two components, J *(f) has infinitely many components. All three cases can occur. It is known that F(f) has at most one multiply-connected domain A whose connectivity must in fact be two. If in addition A is relatively compact in nC *, then either (i) A is a Herman ring, (ii) A is pre-periodic but nor periodic, or (iii) A is a wandering component Example of all three cases are constructed.

Electromagnetic 2.5-dimensional forward modelling with a boundary integral formulation

An effective and accurate technique for the numerical solution of 2-D electromagnetic scattering problems with 3-D sources is presented. This solution introduces a set of the usual boundary integral equations and uses a scalar Green’s function. In this scalar version, the unknowns of the problem are the boundary values of the longitudinal fields and their normal derivatives in the Fourier domain. A generalization of the usual boundary integral formulation enables us to handle a large class of models composed of piecewise homogeneous domains, including contiguous domains, multiply-connected domains and unbounded domains. This formulation involves the solution of a system of linear equations, and results in a significant saving in computation time in comparison with other rigorous methods. The requirements for the numerical implementation of this solution are described in detail. Numerical tests were carried out using the important example of electromagnetic tomography. The specific symmetry properties of the response function in this case are illustrated. Numerical accuracy is verified over a large frequency range, up to 1 MHz.

Adaptive hybrid (prismatic-tetrahedral) grids for incompressible flows

Hybrid grids consisting of prisms and tetrahedra are employed for the solution of the 3-D Navier–Stokes equations of incompressible flow. A pressure correction scheme is employed with a finite volume–finite element spatial discretization. The traditional staggered grid formulation has been substituted with a collocated mesh approach which uses fourth-order artificial dissipation. The hybrid grid is refined adaptively in local regions of appreciable flow variations. The scheme operations are performed on an edge-wise basis which unifies treatment of both types of grid elements. The adaptive method is employed for incompressible flows in both single and multiply-connected domains. © 1998 John Wiley & Sons, Ltd.

Sur les champs de Beltrami linéaires dans des domaines tridimensionnels

Linear Beltrami fields (or Force-free fields) are three-components divergence-free fields solutions to the equation curl B = αB, where n is a real number. Recently, they have received enormous attention in many research fields, especially in fluid mechanics, astrophysics, and plasma physics. In this Note, we shall give some existence, uniqueness, and regularity results for some corresponding houndary value problems in a 3-D multiply-connected domain and in a half-cylinder.

Some identity relations between plane problems for visco- and elasticity

In this paper, the boundary value problems of plane problems with a simply- or multiply-connected domain for isotropic linear visco-elasticity are first established by terms of Airy stress function F(x0 t). Secondly some identity relations between displacements and stresses for plane problems of visco- and elasticity are discussed in detail and some meaningful conclusions are obtained. As an example, the deformation response for viscoelastic plate with a small circular hole at the center is analyzed under a uniaxial uniform extension.

A GENERALIZED BOUNDARY-ELEMENT METHOD FOR STEADY-STATE HEAT CONDUCTION IN HETEROGENEOUS ANISOTROPIC MEDIA

A generalized boundary integral equation (BIE) is formulated for heat conduction problems in anisotropic media with spatially varying thermal conductivities arising from material heterogeneity. The generalized integral equation is expressed in terms of contour integrals only. This is accomplished with the aid of a generalized fundamental solution E and with the definition of a singular nonsymmetric generalized forcing function, D. Generalized fundamental solutions, E, are derived as locally radially symmetric responses to this nonsymmetric singular forcing function, D. Both E and D are defined in terms of the thermal conductivity of the medium. Although it is locally radially symmetric about the source point, E varies within the domain as the source point changes position. Examples of generalized fundamental solutions are provided for various thermal conductivities along with the corresponding forcing function, D. Numerical implementation is discussed. Numerical results are provided for several examples with spatially variable thermal conductivity in isotropic, orthotropic, and fully anisotropic heterogeneous media. Problems ore solved in 2-D and 3-D regular and irregular regions with single- and multiply-connected domains. Close agreement found between the generalized BIE solution and exact solutions validates the formulation developed in this article.

Computation of electric charges and eddy-currents with an E-formulation

We present the modelling of eddy-currents and surface electric charges created by an electromagnetic field in a conductor. We used an electrical field formulation in which a finite element method inside the conductor is coupled with an integral method for the outside. The induced current is then directly given by Ohm's law. Multiply-connected domains are considered. The electric charges which appear at the surface of the conductor are calculated and capacitive effects are evaluated.

Shape optimisation by the boundary element method: a comparison between mathematical programming and normal movement approaches

The aim of this work is to find the best boundary shape of a structural component under certain loading, to have minimum weight, or uniformly distributed equivalent stresses. Two shape optimisation algorithms are developed. One of them is a mathematical programming method, and considers nodal coordinates on the design boundary directly as the design variables, while the other one is rather an optimality criterion approach, based on normal movement of the design boundary. Solving the optimisation problem of stress concentration for a perforated plate which has an analytical solution, shows that the presented mathematical programming method results in almost the same as the analytical solution. Nevertheless, increasing the number of design variables to find more smooth shapes in mathematical methods can cause severe programming problems. Comparing the result of this method with that of the optimality criterion indicates that the latter is much easier to apply without any limit on the number of design variables. To calculate stresses at every iteration, the boundary element method (BEM) is used. Therefore both algorithms benefit from a simple mesh generation based on equal length elements, which provides the possibility of solving multiply-connected domains or geometrically complicated mechanical components. Both methods are used to find the optimum shape of a circular plate under radial loading with four design holes. Finally, the problem of the best topology and shape of circular disks is solved by the optimality criterion approach. Also it is proposed that ‘a fully stressed design algorithm which starts from the best topology design, has the best shape for weight optimisation.

Schrodinger operators in spaces of multifunctions defined in multiply-connected domains

Certain problems of quantum physics (for example, the Aharonov-Bohm effect) lead to the eigenvalue problem for a Schrodinger operator with wave multifunctions. For a multiply-connected configuration space with a simplest topology (for example, for a n-dimensional torus) this problem was considered by several authors. In the present paper, by using rigorous mathematical methods we investigate this problem on an arbitrary multi-dimensional smooth manifold (possibly, with a boundary). We carefully define the concept of multifunctions, then we introduce spaces of these objects similar to L2 and H01. Finally, we present a spectral theorem on the existence of a self-adjoint extension of a Schrodinger operator in the introduced spaces which implies the completeness of the system of eigenfunctions of this operator in the considered functional spaces.

An Inexact Newton Algorithm for Solving the Tokamak Edge Plasma Fluid Equations on a Multiply-Connected Domain

Newton's method is combined with a preconditioned conjugate gradient-like algorithm and finite volume discretization to solve the steady-state two-dimensional tokamak edge plasma fluid equations. A numerical evaluation of the Jacobian is employed. Mesh sequencing, pseudo-transient continuation, and adaptive damping are used to increase the radius of convergence. The computations are performed on a multiply-connected curvilinear geometry in a fully coupled manner. The preconditioned conjugate gradient-like algorithm is shown to have a significant storage advantage over the previously used banded Gaussian elimination, while maintaining the excellent convergence characteristics of the overall algorithm. Simulations of a high recycling divertor and a gaseous divertor on the DIII-D tokamak geometry are used to demonstrate algorithm performance.

Generalized Stokes Eigenfunctions: A New Trial Basis for the Solution of Incompressible Navier-Stokes Equations

The present study focuses on the solution of the incompressible Navier-Stokes equations in general, non-separable domains, and employs a Galerkin projection of divergence-free vector functions as a trial basis. This basis is obtained from the solution of a generalized constrained Stokes eigen-problem in the domain of interest. Faster convergence can be achieved by constructing a singular Stokes eigen-problem in which the Stokes operator is modified to include a variable coefficient which vanishes at the domain boundaries. The convergence properties of such functions are advantageous in a least squares sense and are shown to produce significantly better approximations to the solution of the Navier-Stokes equations in post-critical states where unsteadiness characterizes the flowfield. Solutions for the eigen-systems are efficiently accomplished using a combined Laoczos-Uzawa algorithm and spectral element discretizations. Results are presented for different simulations using these global spectral trial basis on non-separable and multiply-connected domains. It is confirmed that faster convergence is obtained using the singular eigen-expansions in approximating stationary Navier-Stokes solutions in general domains. It is also shown that 100-mode expansions of time-dependent solutions based on the singular Stokes eigenfunctions are sufficient to accurately predict the dynamics of flows in such domains, including Hopf bifurcations, intermittency, and details of flow structures.

Differential boundary value problems of elliptic systems

This paper deals with differential boundary value problems of elliptic systems, regular solutions of which are called analytic vectors [11]. These problems, in some particular cases, are considered in [1–9]. In [10] the author has studied the differential boundary value problem of linear conjugacy with displacements in simply-connected domain. The purpose of this paper is to extend the study of the problem to the case of multiply-connected domains. First, we solve this problem for Q-holomorphic vectors. In this case a representation formula of solutions to the problem is constructed. In view of this formula we reduce the problem to an equivalent system of singular integral equations and deduce necessary and sufficient conditions for solvability and a formula for the index of the problem for (2-holornorphic vectors. Using Bojarski's formula we reduce the problem in the general case to an analogous problem for Q-holomorphic vectors. The same result about solvability conditions and the index of the problem for generalized analytic vectors can be obtained.

Hele–Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply-connected region

We consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap. A blob of Newtonian fluid is sandwiched between the planes, and we suppose its plan-view to occupy a bounded, multiply-connected domain; physically, we have a viscous fluid with the holes giving rise to the multiple connectivity occupied by relatively inviscid air. The relevant free boundary condition is taken to be one of constant pressure, but we allow different pressures to act within the different holes, and at the outer boundary. The motion is driven either by injection of further fluid into the blob at certain points, or by injection of air into the holes to change their area, or by a combination of these; suction, instead of injection, is also contemplated. A general mathematical theory of the above class of problems is developed, and applied to the particular situation that arises when fluid is injected into an initially empty gap bounded by two straight, semi-infinite barriers meeting at right-angles: injection into a quarterplane. For a range of positions of the injection point, air is trapped in the corner and, invoking images, the problem is equivalent to one involving a doubly-connected blob. When there is an air vent in the corner, so that the pressure is the same on the two free boundaries in these circumstances, the air hole rapidly disappears, as might be expected. If, however, there is no air vent and we suppose the air to be incompressible, so that the area of the region occupied by the air in the plan-view remains constant, we find there to be no solution within the framework of our model. Other scenarios within this same geometry, involving both suction and injection of fluid at the injection point, and air at the corner, are also examined.

The Two-Phase Method for Finding a Great Number of Eigenpairs of the Symmetric or Weakly Non-symmetric Large Eigenvalue Problems

Although it has been stated that "an attempt to solve (very large problems) by subspace iterations seems futile" (H. G. Matthies, Comput. Struct.21 (1985), p. 324), we will show that the statement is not true, especially for extremely large eigenproblems. In this paper a new two-phase subspace iteration/Rayleigh quotient/conjugate gradient method for generalized, large, symmetric eigenproblems Ax = λBx is presented. It has the ability of solving extremely large eigenproblems, N = 216,000, for example, and finding a large number of leftmost or rightmost eigenpairs, up to 1000 or more. Multiple eigenpairs, even those with multiplicity 100, can be easily found. The use of the proposed method for solving the big full eigenproblems (N ∼ 103), as well as for large weakly non-symmetric eigenproblems, have been considered also. The proposed method is fully iterative; thus the factorization of matrices is avoided. The key idea consists in joining two methods: subspace and Rayleigh quotient iterations. The systems of indefinite and almost singular linear equations (A - σB) x = By are solved by various iterative conjugate gradient/Lanczos methods. It will be shown that the standard conjugate gradient method can be used without danger of breaking down due to its property that may be called "self-correction towards the eigenvector," discovered recently by us. The use of various preconditioners (SSOR and IC) has also been considered. The main features of the proposed method have been analyzed in detail. Comparisons with other methods, such as, accelerated subspace iteration, Lanczos, Davidson, TLIME, TRACMN, and SRQMCG, are presented. The results of numerical tests for various physical problems (acoustic, vibrations of structures, quantum chemistry) are presented as well. The final conclusion is that our new method is usually much faster than other iterative methods, especially for very large eigenproblems arising from 3D elliptic or biharmonic problems defined on irregular, multiply-connected domains, discretized by the finite element (FEM) or finite difference (FDM) methods.

An Eddy current Calculation on a Multiply-connected Domain by a Finite Element Method

An Eddy current Calculation on a Multiply-connected Domain by a Finite Element Method

First-order elliptic systems degenerated at the boundary

SynopsisIn this paper we state that the bounded solutions of general first order linear elliptic systems of two equations in bounded multiply-connected plane domains, degenerated at the boundary, are determined in domains without any boundary conditions, provided the boundary is not characteristic for this system. The explicit formula for calculating the index of the system is derived.