Multiply Connected Region

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The Experts below are selected from a list of 252 Experts worldwide ranked by ideXlab platform

Mohamed M. S. Nasser - One of the best experts on this subject based on the ideXlab platform.

Ali Hassan Mohamed Murid - One of the best experts on this subject based on the ideXlab platform.

  • Analytical Solution for Finding the Second Zero of the Ahlfors Map for an Annulus Region
    Journal of Mathematics, 2019
    Co-Authors: Nur Husna Abd Wahid, Ali Hassan Mohamed Murid, Mukhiddin I. Muminov
    Abstract:

    The Ahlfors map is a conformal mapping function that maps a Multiply Connected Region onto a unit disk. It can be written in terms of the Szego kernel and the Garabedian kernel. In general, a zero of the Ahlfors map can be freely prescribed in a Multiply Connected Region. The remaining zeros are the zeros of the Szego kernel. For an annulus Region, it is known that the second zero of the Ahlfors map can be computed analytically based on the series representation of the Szego kernel. This paper presents another analytical method for finding the second zero of the Ahlfors map for an annulus Region without using the series approach but using a boundary integral equation and knowledge of intersection points.

  • Conformal mapping of unbounded Multiply Connected Regions onto logarithmic spiral slit with infinite straight slit
    2017
    Co-Authors: Arif A. M. Yunus, Ali Hassan Mohamed Murid
    Abstract:

    This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for conformal mapping of unbounded Multiply Connected Regions. The canonical Region is the entire complex plane bounded by an infinite straight slit on the line Im ω = 0 and finite logarithmic spiral slits. Some linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on a Multiply Connected Region. These integral equations are uniquely solvable. The kernel involved in these integral equations is the adjoint generalized Neumann kernel.

  • Erratum to: Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe's Canonical Slit Regions
    Journal of Scientific Computing, 2016
    Co-Authors: Ali W. K. Sangawi, Ali Hassan Mohamed Murid, Lee Khiy Wei
    Abstract:

    This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded Multiply Connected Region onto a disk with spiral slits Region $$\varOmega _1$$Ω1. This extends the methods that have recently been given for mappings onto annulus with spiral slits Region $$\varOmega _2$$Ω2, spiral slits Region $$\varOmega _3$$Ω3, and straight slits Region $$\varOmega _4$$Ω4 but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded Multiply Connected Region onto all four Regions $$\varOmega _1$$Ω1, $$\varOmega _2$$Ω2, $$\varOmega _3$$Ω3, and $$\varOmega _4$$Ω4 as well as their inverses. The integral equations are solved numerically using combination of Nystrom method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is $$O((m + 1)n)$$O((m+1)n), where $$m+1$$m+1 is the multiplicity of the Multiply Connected Region and n is the number of nodes on each boundary component. Previous algorithms require $$O((m+1)^3 n^3)$$O((m+1)3n3) operations. The algorithm is tested on several test Regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.

  • Integral equation for the Ahlfors map on Multiply Connected Regions
    Jurnal Teknologi, 2015
    Co-Authors: Kashif Nazar, Ali Hassan Mohamed Murid, Ali W. K. Sangawi
    Abstract:

    This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with  where  is the boundary correspondence function of Ahlfors map of a bounded Multiply Connected Region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a Multiply Connected Region. The integral equation is solved numerically for  using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of  we solve for the boundary correspondence function  which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

  • Solving a Mixed Boundary Value Problem via an Integral Equation with Generalized Neumann Kernel on Unbounded Multiply Connected Region
    Malaysian Journal of Fundamental and Applied Sciences, 2014
    Co-Authors: Samer Abdo Ahmed Al-hatemi, Ali Hassan Mohamed Murid, Mohamed M. S. Nasser
    Abstract:

    In this paper, we solve the mixed boundary value problem on unbounded Multiply Connected Region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the Region. The kernel of this integral equation is the generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test Regions are presented.

Ali W. K. Sangawi - One of the best experts on this subject based on the ideXlab platform.

  • Erratum to: Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe's Canonical Slit Regions
    Journal of Scientific Computing, 2016
    Co-Authors: Ali W. K. Sangawi, Ali Hassan Mohamed Murid, Lee Khiy Wei
    Abstract:

    This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded Multiply Connected Region onto a disk with spiral slits Region $$\varOmega _1$$Ω1. This extends the methods that have recently been given for mappings onto annulus with spiral slits Region $$\varOmega _2$$Ω2, spiral slits Region $$\varOmega _3$$Ω3, and straight slits Region $$\varOmega _4$$Ω4 but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded Multiply Connected Region onto all four Regions $$\varOmega _1$$Ω1, $$\varOmega _2$$Ω2, $$\varOmega _3$$Ω3, and $$\varOmega _4$$Ω4 as well as their inverses. The integral equations are solved numerically using combination of Nystrom method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is $$O((m + 1)n)$$O((m+1)n), where $$m+1$$m+1 is the multiplicity of the Multiply Connected Region and n is the number of nodes on each boundary component. Previous algorithms require $$O((m+1)^3 n^3)$$O((m+1)3n3) operations. The algorithm is tested on several test Regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.

  • Integral equation for the Ahlfors map on Multiply Connected Regions
    Jurnal Teknologi, 2015
    Co-Authors: Kashif Nazar, Ali Hassan Mohamed Murid, Ali W. K. Sangawi
    Abstract:

    This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with  where  is the boundary correspondence function of Ahlfors map of a bounded Multiply Connected Region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a Multiply Connected Region. The integral equation is solved numerically for  using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of  we solve for the boundary correspondence function  which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

  • Integral and differential equations for conformal mapping of bounded Multiply Connected Regions onto a disk with circular slits
    Malaysian Journal of Fundamental and Applied Sciences, 2014
    Co-Authors: Ali Hassan Mohamed Murid, Ali W. K. Sangawi, Mohamed M. S. Nasser
    Abstract:

    Conformal mapping is a useful tool in science and engineering. On the other hand exact mapping functions are unknown except for some special Regions. In this paper we present a new boundary integral equation with classical Neumann kernel associated to f f , where f is a conformal mapping of bounded Multiply Connected Regions onto a disk with circular slit domain. This boundary integral equation is constructed from a boundary relationship satisfied by a function analytic on a Multiply Connected Region. With f f known, one can then treat it as a differential equation for computing f .

  • Straight slits map and its inverse of bounded Multiply Connected Regions
    Advances in Computational Mathematics, 2014
    Co-Authors: Ali W. K. Sangawi
    Abstract:

    This paper presents a boundary integral equation method for computing numerical conformal mapping of bounded Multiply Connected Region onto a straight slits Region. The method is an extension of the author's method for computing the parallel slits map of bounded Multiply Connected Regions (Sangawi et al., J. Math. Anal. Appl. 389, 1280---1290 (2012)). Several numerical examples are given to prove the effectiveness of the proposed methods.

  • Spiral slits map and its inverse of bounded Multiply Connected Regions
    Applied Mathematics and Computation, 2014
    Co-Authors: Ali W. K. Sangawi
    Abstract:

    This paper presents a boundary integral equation method for computing numerical conformal mapping of bounded Multiply Connected Region onto a spiral slits Region. The method is an extension of the author's method for computing the circular slits map of bounded Multiply Connected Regions (see A.W.K. Sangawi et al. (2012) [16]). Several numerical examples are given to prove the effectiveness of the proposed methods.

Z X Wang - One of the best experts on this subject based on the ideXlab platform.

  • Properties of integral operators and solutions for complex variable boundary integral equation in plane elasticity for Multiply Connected Regions
    Engineering Analysis with Boundary Elements, 2015
    Co-Authors: Y Z Chen, Z X Wang
    Abstract:

    Abstract This paper studies properties of integral operators and solutions for CVBIE (complex variable boundary integral equation) in plane elasticity for Multiply Connected Regions. Four cases for considered Regions are studied. For the individual case, we study (a) the domain field equality, (b) the null field BIE and (c) the usual CVBIE. Properties of integral operators or the kernels are studied in detail, which is based on the properties of Cauchy type integral. The Neumann problem is considered. It is shown that for finite Region cases ( 2 Formulation and properties of solutions for CVBIE for the interior Region , 3 Properties for the kernels and the integral equations for finite Multiply Connected Region ) the CVBIE allow three modes of rigid motion along contours under the traction free condition. In addition, for infinite Region cases ( 4 Properties of the kernels and the CVBIE for an exterior Region , 5 Properties for the kernels and the integral equations for infinite Multiply-Connected Region ) the CVBIE does not allow three modes of rigid motion.

  • degenerate scale problem for plane elasticity in a Multiply Connected Region with outer elliptic boundary
    Archive of Applied Mechanics, 2010
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper investigates the degenerate scale problem for plane elasticity in a Multiply Connected Region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the non-trivial solution along the elliptic boundary in the problem are same as in the case of a single elliptic contour without voids. The present study mainly depends on integrations of several integrals, which can be integrated in a closed form.

  • Formulation of indirect BIEs in plane elasticity using single or double layer potentials and complex variable
    Engineering Analysis with Boundary Elements, 2010
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper provides a formulation for indirect BIEs in plane elasticity using single or double layer potentials and complex variable. There are two ways to obtain two kinds of layer and the relevant indirect BIEs. In the first way, the displacement expression at domain point is directly obtained from the Somigliana identity with necessary modification. In the second way, after placing some density functions, for example, the body force or the dislocation doublet, along the layers, one can obtain the displacement expression at domain point. For both single and double layers, the continuous or discontinuous properties for the displacement and traction are studied in detail when a moving point is passing through the boundaries. Formulations of the Dirichlet and the Neumann problems are proposed. The ranges for solving the boundary value problem by using the single or double layer potentials are clearly indicated. For the case of single layer, the degenerate scale problems for the finite Multiply Connected Region and infinite Multiply Connected Region are studied. For the case of double layer, a hypersingular BIE for crack can be formulated by assuming that the density functions are vanishing along a portion of boundary.

  • numerical solution for degenerate scale problem for exterior Multiply Connected Region
    Engineering Analysis With Boundary Elements, 2009
    Co-Authors: Y Z Chen, Z X Wang
    Abstract:

    Based on some previous publications, this paper investigates the numerical solution for degenerate scale problem for exterior Multiply Connected Region. In the present study, the first step is to formulate a homogenous boundary integral equation (BIE) in the degenerate scale. The coordinate transform with a magnified factor, or a reduced factor h is performed in the next step. Using the property ln(hx)=ln(x)+lg(h), the new obtained BIE equation can be considered as a non-homogenous one defined in the transformed coordinates. The relevant scale in the transformed coordinates is a normal scale. Therefore, the new obtained BIE equation is solvable. Fundamental solutions are introduced. For evaluating the fundamental solutions, the right-hand terms in the non-homogenous equation, or a BIE, generally take the value of unit or zero. By using the obtained fundamental solutions, an equation for evaluating the magnified factor “h” is obtained. Finally, the degenerate scale is obtainable. Several numerical examples with two ellipses in an infinite plate are presented. Numerical solutions prove that the degenerate scale does not depend on the normal scale used in the process for evaluating the fundamental solutions.

  • the degenerate scale problem for the laplace equation and plane elasticity in a Multiply Connected Region with an outer circular boundary
    International Journal of Solids and Structures, 2009
    Co-Authors: Y Z Chen, Z X Wang, X Y Lin
    Abstract:

    This paper investigates the degenerate scale problem for the Laplace equation and plane elasticity in a Multiply Connected Region with an outer circular boundary. Inside the boundary, there are many voids with arbitrary configurations. The problem is analyzed with a relevant homogenous BIE (boundary integral equation). It is assumed that all the inner void boundary tractions are equal to zero, and tractions on the outer circular boundary are constant. Therefore, all the integrations in BIE are performed on the outer circular boundary only. By using the relation z * conjg(z) = a * a, or conjg(z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form. The degenerate scale a = 1 is found in the Laplace equation and in plane elasticity regardless of the void configuration.

Y Z Chen - One of the best experts on this subject based on the ideXlab platform.

  • Properties of integral operators and solutions for complex variable boundary integral equation in plane elasticity for Multiply Connected Regions
    Engineering Analysis with Boundary Elements, 2015
    Co-Authors: Y Z Chen, Z X Wang
    Abstract:

    Abstract This paper studies properties of integral operators and solutions for CVBIE (complex variable boundary integral equation) in plane elasticity for Multiply Connected Regions. Four cases for considered Regions are studied. For the individual case, we study (a) the domain field equality, (b) the null field BIE and (c) the usual CVBIE. Properties of integral operators or the kernels are studied in detail, which is based on the properties of Cauchy type integral. The Neumann problem is considered. It is shown that for finite Region cases ( 2 Formulation and properties of solutions for CVBIE for the interior Region , 3 Properties for the kernels and the integral equations for finite Multiply Connected Region ) the CVBIE allow three modes of rigid motion along contours under the traction free condition. In addition, for infinite Region cases ( 4 Properties of the kernels and the CVBIE for an exterior Region , 5 Properties for the kernels and the integral equations for infinite Multiply-Connected Region ) the CVBIE does not allow three modes of rigid motion.

  • degenerate scale problem for plane elasticity in a Multiply Connected Region with outer elliptic boundary
    Archive of Applied Mechanics, 2010
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper investigates the degenerate scale problem for plane elasticity in a Multiply Connected Region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the non-trivial solution along the elliptic boundary in the problem are same as in the case of a single elliptic contour without voids. The present study mainly depends on integrations of several integrals, which can be integrated in a closed form.

  • Degenerate scale problem for the Laplace equation in the Multiply Connected Region with outer elliptic boundary
    Acta Mechanica, 2010
    Co-Authors: Y Z Chen, X Y Lin
    Abstract:

    This paper investigates the degenerate scale problem for the Laplace equation in a Multiply Connected Region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the eigenfunction along the elliptic boundary in the problem is the same as in the case of a single elliptic contour without voids, or the involved voids have no influence on the degenerate scale. The present study mainly depends on the integrations of two integrals, which can be integrated in closed form.

  • Formulation of indirect BIEs in plane elasticity using single or double layer potentials and complex variable
    Engineering Analysis with Boundary Elements, 2010
    Co-Authors: Y Z Chen, X Y Lin, Z X Wang
    Abstract:

    This paper provides a formulation for indirect BIEs in plane elasticity using single or double layer potentials and complex variable. There are two ways to obtain two kinds of layer and the relevant indirect BIEs. In the first way, the displacement expression at domain point is directly obtained from the Somigliana identity with necessary modification. In the second way, after placing some density functions, for example, the body force or the dislocation doublet, along the layers, one can obtain the displacement expression at domain point. For both single and double layers, the continuous or discontinuous properties for the displacement and traction are studied in detail when a moving point is passing through the boundaries. Formulations of the Dirichlet and the Neumann problems are proposed. The ranges for solving the boundary value problem by using the single or double layer potentials are clearly indicated. For the case of single layer, the degenerate scale problems for the finite Multiply Connected Region and infinite Multiply Connected Region are studied. For the case of double layer, a hypersingular BIE for crack can be formulated by assuming that the density functions are vanishing along a portion of boundary.

  • numerical solution for degenerate scale problem for exterior Multiply Connected Region
    Engineering Analysis With Boundary Elements, 2009
    Co-Authors: Y Z Chen, Z X Wang
    Abstract:

    Based on some previous publications, this paper investigates the numerical solution for degenerate scale problem for exterior Multiply Connected Region. In the present study, the first step is to formulate a homogenous boundary integral equation (BIE) in the degenerate scale. The coordinate transform with a magnified factor, or a reduced factor h is performed in the next step. Using the property ln(hx)=ln(x)+lg(h), the new obtained BIE equation can be considered as a non-homogenous one defined in the transformed coordinates. The relevant scale in the transformed coordinates is a normal scale. Therefore, the new obtained BIE equation is solvable. Fundamental solutions are introduced. For evaluating the fundamental solutions, the right-hand terms in the non-homogenous equation, or a BIE, generally take the value of unit or zero. By using the obtained fundamental solutions, an equation for evaluating the magnified factor “h” is obtained. Finally, the degenerate scale is obtainable. Several numerical examples with two ellipses in an infinite plate are presented. Numerical solutions prove that the degenerate scale does not depend on the normal scale used in the process for evaluating the fundamental solutions.

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