Contents Conjugate Function Method for References 1 Numerical - PowerPoint PPT Presentation

Contents Conjugate Function Method for References 1 Numerical Conformal Mappings Motivation 2 Preliminaries 3 Tri Quach Quadrilateral Conformal Modulus of a Quadrilateral Institute of Mathematics Aalto University School of Science Theorem 4 Joint work with Harri Hakula and Antti Rasila Algorithm 5 KAUS 2011 – Gothenburg, Sweden Examples 6 21 January 2011 Quadrilaterals Ring Domains Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 1 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 2 / 22 References References [Ahlfors1] L.V. Ahlfors, Conformal invariants: topics in geometric [HakQuaRas] H. Hakula, T. Quach, and A. Rasila, Conjugate function method for numerical conformal mappings , Manuscript. function theory , McGraw-Hill Book Co., 1973. [Ahlfors2] L.V. Ahlfors, Complex Analysis , An introduction to the [HakRasVuo] H. Hakula, A. Rasila, and M. Vuorinen, On moduli of theory of analytic functions of one complex variable, Third edition. rings and quadrilaterals: algorithms and experiments . arXiv math.NA International Series in Pure and Applied Mathematics. McGraw-Hill 0906.1261, 2009. TKK-A575 (2009). SIAM J. Sci. Comput. (to Book Co., New York, 1978. appear). [Crowdy] D. Crowdy, The Schwarz-Christoffel mapping to bounded [Hu] C. Hu, Algorithm 785: a software package for computing multiply connected polygonal domains . Proc. R. Soc. Lond. Ser. A Schwarz-Christoffel conformal transformation for doubly connected Math. Phys. Eng. Sci. 461 (2005), no. 2061, 2653–2678. polygonal regions , ACM Transactions on Mathematical Software (TOMS), v.24 n.3, p.317-333, Sept. 1998 [CroMar] D. Crowdy and J. Marshall, Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. [LehVir] O. Lehto and K.I. Virtanen, Quasiconformal mappings in the Theory 6 (2006), no. 1, 59–76. plane , Springer, Berlin, 1973. [DriTre] T.A. Driscoll and L.N. Trefethen, Schwarz-Christoffel [PapSty] N. Papamichael and N.S. Stylianopoulos, Numerical Mapping . Cambridge Monographs on Applied and Computational Conformal Mapping: Domain Decomposition and the Mapping of Mathematics, 8. Cambridge University Press, Cambridge, 2002. Quadrilaterals , World Scientific Publishing Company, 2010. Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 3 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 4 / 22

Motivation Preliminaries - Generalized Quadrilateral Definition (Generalized Quadrilateral) Conformal mappings can be applied in electrostatics, aerodynamics, A Jordan domain Ω in C with marked (positively ordered) points etc. z 1 , z 2 , z 3 , z 4 ∈ ∂ Ω is called a (generalized) quadrilateral , and denoted by Numerical methods are considered since the analytical solution exists Q := (Ω; z 1 , z 2 , z 3 , z 4 ). only for few domains. Schwarz–Christoffel (SC) toolbox by Driscoll [DriTre]. Denote the arcs of ∂ Ω between ( z 1 , z 2 ) , ( z 2 , z 3 ) , ( z 3 , z 4 ) , ( z 4 , z 1 ) , by Hu’s [Hu] SC algorithm for doubly connected domains. γ j , j = 1 , 2 , 3 , 4. For multiply connected domains, see eg. [Crowdy, CroMar]. Finite element methods (FEM) approach, see eg. [HakRasVuo]. Quadrilateral ˜ Q = (Ω; z 2 , z 3 , z 4 , z 1 ) is called the conjugate quadrilateral of Q . Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 5 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 6 / 22 Preliminaries – Modulus of a Quadrilateral Preliminaries – Modulus of a Quadrilateral Consider the following Laplace equation Definition (Geometric) Let Q be a quadrilateral. Let the function f = u + iv be a one-to-one  ∆ u = 0 , in Ω , conformal mapping of the domain Ω onto a rectangle    u = 0 , on γ 2 ,  R h = { z ∈ C : 0 < Re z < 1 , 0 < Im z < h } such that the image of   (1) u = 1 , on γ 4 , z 1 , z 2 , z 3 , z 4 are 1 + ih , ih , 0 , 1, respectively. Then the number h is called   ∂ u the (conformal) modulus of the quadrilateral Q and we will denote it by   ∂ n = 0 , on γ 1 ∪ γ 3 .   M ( Q ). If u is the solution to (1). Then by [Ahlfors1, p. 65/Thm 4.5], Note that the conformal modulus of a quadrilateral is unique. [PapSty, p. 63/Thm 2.3.3]: By the geometry [LehVir, p. 15], [PapSty, pp. 53-54], we have the ¨ |∇ u | 2 dx dy . reciprocal identity: M ( Q ) = (2) M ( Q ) · M ( ˜ Q ) = 1 . Ω Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 7 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 8 / 22

Preliminaries – Modulus of a Ring Domain Theorem – Illustration of the Problem Let E and F be two disjoint compact sets in the extended complex Find f such that f : Ω → R h . plane C ∞ . Then one of the sets E , F is bounded and without loss of generality we may assume that it is E . If both E and F are connected y γ ′ v and the set R = C ∞ \ ( E ∪ F ) is connected, then R is called a ring 1 z 1 γ 4 γ 1 domain . In this case R is a doubly connected plane domain. The ih 1 + ih capacity of R is defined by f ( z ) Ω R h ¨ γ ′ γ ′ |∇ u | 2 dx dy , x 2 4 cap R = inf γ 3 z 4 u R z 2 where the infimum is taken over all nonnegative, piecewise z 3 u differentiable functions u with compact support in R ∪ E such that 0 1 γ ′ γ 2 3 u = 1 on E . The harmonic function on R with boundary values 1 on E and 0 on F Figure: Dirichlet-Neumann boundary value problem. Dirichlet and Neumann boundary conditions are mark with thin and thick lines, respectively. is the unique function that minimizes the above integral. Conformal modulus: M ( R ) = 2 π/ cap R . Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 9 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 10 / 22 Theorem - Formulation Proof of the Theorem – Part 1 Lemma Theorem (Conjugate Function Method) Let Q , h , u , v be as before. If ˜ u is the solution to the Dirichlet-Neumann problem associated with the conjugate quadrilateral ˜ Q. Then v = h 2 ˜ u. Let Q be a quadrilateral with modulus h and let u 1 satisfy (1). Suppose that u 2 is the solution to Dirichlet–Neumann boundary value problem v := h 2 ˜ associated with the conjugate quadrilateral ˜ It is clear that v , ˜ u are harmonic. Thus ˜ u is harmonic. Q. Then f = u 1 + ihu 2 is the Since M ( ˜ conformal mapping that maps maps Ω onto a rectangle R h such that the Q ) = 1 / h , therefore v and ˜ v have the same values on γ 1 , γ 3 . image of the points z 1 , z 2 , z 3 , z 4 are 1 + ih , ih , 0 , 1 , respectively. The On γ 2 , γ 4 we have mapping f maps the boundary curves γ 1 , γ 2 , γ 3 , γ 4 onto curves ∂ v γ ′ 1 , γ ′ 2 , γ ′ 3 , γ ′ 4 , respectively. ∂ n = �∇ v , n � = v x n 1 + v y n 2 = u y n 1 − u x n 2 = 0 , Proof is based on the following facts: because u is constant on γ 2 , γ 4 , and therefore u x = u y = 0. 1 There is a connection between the harmonic conjugate v 1 of u 1 and Thus v and ˜ v also have same values on γ 2 , γ 4 . the solution u 2 . Then by the uniqueness theorem for harmonic functions [Ahlfors2, p. 2 There is a confomal mapping f = u + iv that maps Ω onto R h . 166] we have v = ˜ v . Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 11 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 12 / 22

Proof of the Theorem – Part 2 Algorithm – Conjugate Function Method Algorithm (HakQuaRas) Let u satisfy (1) and suppose that v is the harmonic conjugate 1 Solve the Dirichlet-Neumann problem to obtain u 1 and compute the function of u . Then f = u + iv is analytic. modulus h. Re f = u and u = 0 on γ ′ 2 and u = 1 on γ ′ 4 . (Dirichlet boundary) 2 Solve the conjugate problem for u 2 . On γ ′ 1 , γ ′ 3 , we use Lemma from previous slide. Since v = 0 on γ ′ 3 , we 3 Then the conformal mapping is given by f = u 1 + ihu 2 . have v = h on γ ′ 1 . (Neumann boundary) For univalency, suppose that f is not univalent, i.e., there exists Note: z 1 , z 2 ∈ Ω, z 1 � = z 2 such that f ( z 1 ) = f ( z 2 ). The solution u can be obtained by any standard numerical methods. Thus Re f ( z 1 ) = Re f ( z 2 ), so z 1 , z 2 are on the same equivpotential In our examples the hp -FEM software by H. Hakula, [HakRasVuo], is curve C of u . used. Similarly for the imaginary part, we have that z 1 = z 2 . The reciprocal identity is used for the error analysis. Draw the rectangular grid on R h and map it onto Ω. Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 13 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 14 / 22 Example – Analytic Example – Schwarz (1869) (Ω; z 1 , z 2 , z 3 , z 4 ), where z j = e i θ j , θ j = ( j − 1) π/ 2. 1.5 2.5 2.0 1.0 1.5 1.0 0.5 0.5 0.5 1.0 1.5 2.0 2.5 0.2 0.4 0.6 0.8 1.0 Figure: Conformal mapping from a quadrilateral onto a rectangle. Figure: Error of the conformal mapping is 10 − 13 . Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 15 / 22 Tri Quach (Aalto University) Conjugate Function Method 21 January 2011 16 / 22