Multiply-connected Regions Research Articles

Global Dynamics of an Oligopoly Game Model with Nonlinear Costs and Strategic Delegation

A dynamic oligopoly game model with nonlinear cost and strategic delegation is built on the basis of isoelastic demand in this paper. And the dynamic characteristics of this game model are investigated. The local stability of the boundary equilibrium points is analyzed by means of the stability theory and Jacobian matrix, and the stability region of the Nash equilibrium point is obtained by Jury criterion. It is concluded that the system may lose stability through Flip bifurcation and Neimark–Sacker bifurcation. And the effects of speed of adjustment, price elasticity, profit weight coefficient and marginal cost on the system stability are discussed through numerical simulation. After that, the coexistence of attractors is analyzed through the basin of attraction, where multiple stability always means path dependence, implying that the long-term behavior of enterprises is strongly affected by historical contingency. In other words, a small perturbation of the initial conditions will have a significant impact on the system. In addition, the global dynamical behavior of the system is analyzed by using the critical curves, the basin of attraction, absorbing areas and a noninvertible map, revealing that three global bifurcations, the first two of which are caused by the interconversion of simply-connected and multiply-connected regions in the basin of attraction, and the third global bifurcation, that is, the final bifurcation is caused by the contact between attractors and the boundary of the basin of attraction.

The method of fundamental solutions for two-dimensional elasticity problems based on the Airy stress function

This paper presents a new version of the method of fundamental solutions (MFS) for two-dimensional linear elasticity problems based on the stress function (Airy stress function), which is different from the MFS utilizing the fundamental solutions of displacement. The displacement compatibilities are derived by the single-valuedness of displacements in multiply-connected region. Based on the strain and rotation of the line element on the boundary, the displacement boundary conditions are deduced in forms of the Airy stress function. Furthermore, the displacement conditions exclude pure rigid body motion, and are in the derivative forms instead of the integral ones. The interpolation equations are also reconstructed by the single-valuedness conditions of the displacements and the approximate solutions in a multiply-connected region. The numerical examples show that the proposed method has good effect and keeps high accuracy in all kinds of boundary conditions.

3-D BEM Formulations for Eddy-Current Problems With Multiply-Connected Domains and Circuit Coupling

Quasi-static linear problems can be solved efficiently with Boundary Element Method (BEM). This method is based on surface integral equations dealing with equivalent magnetic and electric surface current densities. Many works have shown the potentiality of BEM especially for the modeling of non-destructive testing devices. In this paper, after selecting formulations enabling the modeling of multiply-connected regions, an original coupling is proposed in order to take into account external electric circuit in the problem.

Study on placement optimization of solar arrays at multiply-connected region

Study on placement optimization of solar arrays at multiply-connected region

H-$\Phi$ Field Formulation With Lumped Sources and Unbounded Domains

We consider a H-Φ field formulation to solve 3-D frequency-domain eddy-current problems. This formulation uses vector and scalar tetrahedral finite elements within, respectively, the conductive and nonconductive domains. It can handle multiply-connected regions and eliminates the need to compute the source current density and source magnetic field before the actual simulation. We propose three ways to couple finite elements with the multiple multipole program (MMP) and solve this H-Φ variational form on an unbounded domain. The MMP is a method that uses exact solutions of the homogeneous equations as basis functions (the so-called “multipoles”). The desired behavior at infinity is given by the chosen multipoles-this eliminates the need for artificially truncating the computational domain. Interface conditions between the finite-element method (FEM) and MMP domains allow to express the coupled FEM-MMP problem.

A Complex Variable Solution for Lining Stress and Deformation in a Non-Circular Deep Tunnel II Practical Application and Verification

A new complex variable method is presented for stress and displacement problems in a non-circular deep tunnel with certain given boundary conditions at infinity. In order to overcome the complex problems caused by non-circular geometric configurations and the multiply-connected region, a complex variable method and continuity boundary conditions are used to determine stress and displacement within the tunnel lining and within the surrounding rock. The coefficients in the conformal mapping function and stress functions are determined by the optimal design and complex variable method, respectively. The new method is validated by FLAC3D finite difference software through an example. Both the new method and the numerical simulation obtained similar results for the stress concentration and the minimum radial displacement occurred at a similar place in the tunnel. It is demonstrated that the new complex variable method is reliable and reasonable. The new method also provides another way to solve non-circular tunnel excavation problems in a faster and more accurate way.

Application of iterative algorithms for gas-turbine blades cooling optimization

This paper presents two iterative algorithms used to solve the inverse problem of gas-turbine blade cooling. First of them was obtained by variational calculus methods, and the second one is reduced to solving the least-square approximation problem. To determine the temperature distribution on the inner boundary of the blade (of the multiply-connected region), a combination of trigonometric functions defined on the boundary of a unit circle was used. Both algorithms are presented in two variants: the first one is about reducing the number of trigonometric functions, and in the second one, coefficients of trigonometric functions are regularized in the process of iteration. Results of calculations indicate that both algorithms operate more effectively in the variant where coefficients of linear combination of trigonometric functions are regularized.

Optimizing velocities and transports for complex coastal regions and archipelagos

We derive and apply a methodology for the initialization of velocity and transport fields in complex multiply-connected regions with multiscale dynamics. The result is initial fields that are consistent with observations, complex geometry and dynamics, and that can simulate the evolution of ocean processes without large spurious initial transients. A class of constrained weighted least squares optimizations is defined to best fit first-guess velocities while satisfying the complex bathymetry, coastline and divergence strong constraints. A weak constraint towards the minimum inter-island transports that are in accord with the first-guess velocities provides important velocity corrections in complex archipelagos. In the optimization weights, the minimum distance and vertical area between pairs of coasts are computed using a Fast Marching Method. Additional information on velocity and transports are included as strong or weak constraints. We apply our methodology around the Hawaiian islands of Kauai/Niihau, in the Taiwan/Kuroshio region and in the Philippines Archipelago. Comparisons with other common initialization strategies, among hindcasts from these initial conditions (ICs), and with independent in situ observations show that our optimization corrects transports, satisfies boundary conditions and redirects currents. Differences between the hindcasts from these different ICs are found to grow for at least 2–3weeks. When compared to independent in situ observations, simulations from our optimized ICs are shown to have the smallest errors.

THE SECOND ORDER FINITE ELEMENT ANALYSIS OF EDDY CURRENTS BASED ON THE T-Ω METHOD

Based on a proposed inexact Hodge decomposition, this paper describes a viable scheme using the second order finite elements in the T-Ω method considering multiply-connected regions for the eddy current problems. Several numerical examples have been presented to demonstrate the effectiveness of this scheme.

A conformal mapping approach to a root-clustering problem

This paper presents a new approach for matrix root-clustering in sophisticated and multiply-connected regions of the complex plane. The parametric sweeping method and a concept of the closed forbidden region covered by a set of modified three-parametrical Cassini regions are used. A conformal mapping approach was applied to formulate the main results of the paper. An application of the developed method to the problem of matrix root-clustering in a multiply connected region is shown for illustration.

Potential fields induced by point sources in assemblies of shells weakened with apertures

Well-posed boundary-value problems in multiply-connected regions are targeted for some sets of two-dimensional Laplace equations written in geographical coordinates on joint surfaces of revolution. Those are problems that simulate potential fields induced by point sources in joint perforated thin shell structures consist of fragments of different geometry. A semi-analytical approach is proposed to accurately compute solutions of such problems. The approach is based on the matrix of Green's type formalism. The elements of required matrices of Green's type are obtained analytically and expressed in closed computer-friendly form. This makes it possible to efficiently deal with the targeted class of problems. Copyright © 2014 John Wiley & Sons, Ltd.

A finite element method for the Saint-Venant torsion and bending problems for prismatic beams

In this work, we present a finite element formulation for the Saint-Venant torsion and bending problems for prismatic beams. The torsion problem formulation is based on the warping function, and can handle multiply-connected regions (including thin-walled structures), compound and anisotropic bars. Similarly, the bending formulation, which is based on linearized elasticity theory, can handle multiply-connected domains including thin-walled sections. The torsional rigidity and shear centers can be found as special cases of these formulations. Numerical results are presented to show the good coarse-mesh accuracy of both the formulations for both the displacement and stress fields. The stiffness matrices and load vectors (which are similar to those for a variable body force in a conventional structural mechanics problem) in both formulations involve only domain integrals, which makes them simple to implement and computationally efficient.

The Aharonov-Bohm effect with a twist: Electron transport through finite-width Möbius rings

The Aharonov-Bohm effect manifests itself in the interference of electron waves traveling through multiply-connected regions and the flux linked with the paths for the electron. Here, we consider the transmission of electrons in the presence of a magnetic field through a finite-width Möbius ring structure which displays a nontrivial topology. The results are compared with the transport through a flat annular ring and a cylindrical ring, with finite-width input and output contacts attached at the periphery in order to highlight the differences in the transmission and conductance patterns. We develop a model to account for the main features associated with the interference effects for propagating states on the Möbius ring. We demonstrate that the periodicity in the magnetic flux, in units of h/e, is weakly broken on 2D rings of finite width, so that the simple treatment proposed here is sufficient for interpreting results. The unusual states with half-integer values of ⟨Lz⟩ present on Möbius rings display a different characteristic in transmission. Such resonant states are in constructive interference for transmission at magnetic fields where the contribution from ordinary states with integer ⟨Lz⟩ is in destructive interference, and vice versa. This leads to an alternating dominance of the set of half-integer ⟨Lz⟩ states and the set of integer ⟨Lz⟩ states in transport with increasing magnetic fields. We calculate the conductance of the rings, using the Landauer-Büttiker formula, as a function of the magnetic field and the applied bias at contact reservoirs. The differences in the structures considered lend hope for possible for magnetic sensor applications.

Fast integral equation methods for the Laplace-Beltrami equation on the sphere

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple islands are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples.

A Sweeping Method for Matrix Root Clustering

Abstract This paper presents a new criteria for matrix root-clustering problem. It provides generalizations of the existing theorems onto multiply-connected and non-convex regions of the complex plane.

2221 ミクロ構造を有する傾斜機能材料からなる多重連結領域のミクロ-マクロ熱応力解析(S14-4 傾斜機能性や不均質性を有する材料の特性への取り組み,S14 先進材料の材料特性における弾性数理解析の新たな取り組み)

2221 ミクロ構造を有する傾斜機能材料からなる多重連結領域のミクロ-マクロ熱応力解析(S14-4 傾斜機能性や不均質性を有する材料の特性への取り組み,S14 先進材料の材料特性における弾性数理解析の新たな取り組み)

Effectiveness of a Spectral Finite Difference Method to a Multiply-connected Region

A method using a spectral finite difference method to a multiply-connected region in an infinite extension of natural convection is considered. A boundary-fitted coordinate system to a boundary value problem is formed, using a Jacobian elliptic function (sn). Fluid is assumed to be rest and at an uniform temperature at infinity, which can be satisfied automatically by a charicteristic of the mapping function. A significant difference between the surface temperatures of bodies is assigned. Presented are streamlines, isotherms, drag & lift coefficients, and mean Nusselt numbers in a steady-state for a variety of dimensionless parameters such as a Grashof number, a Prandtl number, and fin length. Extensive effectiveness of a spectral finite difference method is established also in a multiply-connected region, taking into account the condition of multiply-connected regions and an infinite extension.

A novel hybrid FEM-BEM method for 3D eddy current field calculation using current density J

This paper introduces a novel hybrid FEM-BEM method for calculating 3D eddy current field. In the eddy current region, the eddy current density J is solved by the finite element method (FEM) which is discretized by brick finite element mesh, while in the eddy current free region, the magnetic field intensity H is solved by the boundary element method (BEM) which is discretized by rectangular boundary element mesh. Under the boundary conditions, an algebraic equation group is obtained that only includes J by eliminating H. This method has many advantages over traditional ones, such as fewer variables, more convenient coupling between the FEM and the BEM and wider application to multiply-connected regions. The calculated values of two models are in good agreement with experimental results. This shows the validity of our method.

Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.