Contents Conjugate Function Method for References 1 Numerical Conformal Mappings Introduction 2 Preliminaries 3 Tri Quach Quadrilateral Conformal Modulus Institute of Mathematics Aalto University School of Science Theorem 4 Joint work with Harri Hakula and Antti Rasila Algorithm 5 CCAAT - Protaras, Cyprus Examples 6 June 5–11, 2011 Quadrilaterals Ring Domains Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 1 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 2 / 28 References References [Hu] C. Hu, Algorithm 785: a software package for computing [BetSamVuo] D. Betsakos, K. Samuelsson and M. Vuorinen, The Schwarz-Christoffel conformal transformation for doubly connected computation of capacity of planar condensers , Publ. Inst. Math. 75 polygonal regions , ACM Transactions on Mathematical Software (89) (2004), 233-252. (TOMS), v.24 n.3, p.317-333, Sept. 1998 [DriTre] T.A. Driscoll and L.N. Trefethen, Schwarz-Christoffel [PapKok] N. Papamichael and C.A. Kokkinos, The use of singular Mapping . Cambridge Monographs on Applied and Computational functions for the approximate conformal mapping of doubly-connected Mathematics, 8. Cambridge University Press, Cambridge, 2002. domains , SIAM J. Sci. Stat. Comp. 5 (1984), 684-700. [HakQuaRas] H. Hakula, T. Quach, and A. Rasila, Conjugate function [PapSty] N. Papamichael and N.S. Stylianopoulos, Numerical method for numerical conformal mappings , arXiv math.NA 1103.4930, Conformal Mapping: Domain Decomposition and the Mapping of 2011. Quadrilaterals , World Scientific Publishing Company, 2010. [HakRasVuo] H. Hakula, A. Rasila, and M. Vuorinen, On moduli of [PapWar] N. Papamichael and M.K. Warby, Pole-type singularities rings and quadrilaterals: algorithms and experiments . SIAM J. Sci. and the numerical conformal mapping of doubly-connected domains , Comput. 33 (2011), no. 1, 279–309. J. Comp. Appl. Math. 10 (1984), 93-106. Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 3 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 4 / 28
Introduction - Motivation Introduction - Numerical Methods Conformal mappings can be applied, e.g., in aerodynamics and study of magnetic fields. Schwarz-Christoffel Mapping The cross-section of the cylinder with an orthogonal plane defines a Conformal mappings between polygons, circles and half-planes. two-dimensional ring domain and the expressions Widely used software 2 πε SC Toolbox implemented for Matlab by Driscoll [DriTre]. C = ln ( R / r ) , Hu’s [Hu] algorithm for doubly connected domains. define the capacity of this ring domain and ε is permittivity. Conjugate Function Method Based on the conjugate harmonic function and properties of R quadrilaterals. r Harmonic functions associated with Dirichlet-Neumann problems can be solved by any suitable methods. Other methods Circle Packing, Zipper algorithm. Figure: The cross-section of the cylinder. Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 5 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 6 / 28 Preliminaries - Generalized Quadrilateral Preliminaries – Modulus of Quadrilateral Definition (Geometric) Let Q be a quadrilateral. Let the function f = u + iv be a one-to-one conformal mapping of Q onto a rectangle R h = { z ∈ C : 0 < Re z < 1 , Definition (Generalized Quadrilateral) 0 < Im z < h } such that the image of z 1 , z 2 , z 3 , z 4 are 1 + ih , ih , 0 , 1, A Jordan domain Ω in C with marked (positively ordered) points respectively. Then the number h is called the (conformal) modulus of the z 1 , z 2 , z 3 , z 4 ∈ ∂ Ω is called a (generalized) quadrilateral , and is denoted by quadrilateral Q and is denoted by M ( Q ). Q := (Ω; z 1 , z 2 , z 3 , z 4 ). Note that the conformal modulus of a quadrilateral is unique. Denote the arcs of ∂ Ω between ( z 1 , z 2 ) , ( z 2 , z 3 ) , ( z 3 , z 4 ) , ( z 4 , z 1 ) , by Modulus of a quadrilateral is also conformally invariant, i.e., if f : Ω → Ω ′ then γ j , j = 1 , 2 , 3 , 4. Quadrilateral ˜ Q = (Ω; z 2 , z 3 , z 4 , z 1 ) is called the conjugate M (Ω; z 1 , z 2 , z 3 , z 4 ) = M (Ω ′ ; f ( z 1 ) , f ( z 2 ) , f ( z 3 ) , f ( z 4 )) . quadrilateral of Q . By the geometry [PapSty, pp. 53-54], we have the reciprocal identity: M ( Q ) · M ( ˜ Q ) = 1 , where Q = (Ω; z 1 , z 2 , z 3 , z 4 ) and ˜ Q = (Ω; z 2 , z 3 , z 4 , z 1 ). Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 7 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 8 / 28
Preliminaries – Dirichlet-Neumann Problem Preliminaries – Modulus of a Quadrilateral Let Ω be a Jordan domain such that the boundary ∂ Ω consist of finite number of regular Jordan curves and the normal of ∂ Ω is defined for Consider the following Dirichlet-Neumann problem for Laplace all points, except possibly at finitely many points. equation Let Ω = A ∪ B such that A ∩ B is finite. Suppose ψ A , ψ B be ∆ u = 0 , in Ω , real-valued continuous functions defined on A , B , respectively. u = 0 , on γ 2 , (1) Then the Dirichlet-Neumann problem is to find a function u satisfying the u = 1 , on γ 4 , following conditions: ∂ u ∂ n = 0 , on γ 1 ∪ γ 3 . 1 u is continuous and differentiable in Ω. 2 Dirichlet condition: u ( t ) = ψ A ( t ) , for all t ∈ A . If u is the solution to (1). Then by [PapSty, p. 63/Thm 2.3.3]: 3 Neumann condition: If ∂/∂ n denotes differentiation in the direction of ¨ |∇ u | 2 dx dy . M ( Q ) = (2) the exterior normal, then Ω ∂ ∂ nu ( t ) = ψ B ( t ) , for all t ∈ B . Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 9 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 10 / 28 Preliminaries – Laplace problem Preliminaries – Modulus of Ring Domain Let E and F be two disjoint and connected compact sets in the z 1 ∂ u ∂ n = 0 extended complex plane C ∞ . Then one of the sets E , F is bounded u = 1 and without loss of generality we may assume that it is E . The set R = C ∞ \ ( E ∪ F ) is connected and it is called a ring domain . The ∆ u = 0 capacity of R is defined by z 4 ¨ |∇ u | 2 dx dy , ∂ u ∂ n = 0 cap R = inf z 2 u R where the infimum is taken over all non-negative, piecewise differentiable functions u with compact support in R ∪ E such that z 3 u = 1 on E . u = 0 The harmonic function on R with boundary values 1 on E and 0 on F Figure: Laplace equation with Dirichlet-Neuman boundary conditions on a is the unique function that minimizes the above integral. quadrilateral Q = (Ω; z 1 , z 2 , z 3 , z 4 ), where Dirichlet and Neumann conditions are Conformal modulus: M ( R ) = 2 π/ cap R . mark with thin and thick lines, respectively. Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 11 / 28 Tri Quach (Aalto University) Conjugate Function Method June 5–11, 2011 12 / 28
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