Conjugate Function Method and Conformal Mappings in Multiply Connected Domains

Conjugate Function Method and Conformal Mappings in Multiply Connected Domains
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  CONJUGATE FUNCTION METHOD AND CONFORMALMAPPINGS IN MULTIPLY CONNECTED DOMAINS HARRI HAKULA ∗ , TRI QUACH † ,  AND  ANTTI RASILA ‡ Abstract.  In this paper, we present a generalization of the algorithm from our earlier work, fornumerical computation of conformal mappings, on multiply connected domains. An implementationof the algorithm, along with several examples and illustrations are given. Key words.  numerical conformal mappings, conformal modulus, multiply connected domains,canonical domains 1. Introduction.  Conformal mappings play an important role in both theoret- 1 ical complex analysis and in certain engineering applications, such as electrostatics, 2 aerodynamics and fluid mechanics. Existence of conformal mappings of simply con- 3 nected domains onto the upper-half plane or the unit disk follows from the Riemann 4 mapping theorem, a well-known result in complex analysis [2], and there are gener- 5 alizations of this result for doubly and multiply connected domains. However, con- 6 structing such mappings analytically is usually very difficult, and use of numerical 7 methods is required. 8 There exists an extensive literature on numerical construction of conformal map- 9 pings for simply and doubly connected domains [23]. One popular method is based 10 on the Schwarz-Christoffel formula [11], and its implementation SC Toolbox is due to 11 Driscoll [9, 10]. SC Toolbox itself is based on earlier FORTRAN package by Trefethen 12 [26]. A new algorithm involving a finite element method and the conjugate harmonic 13 function was presented by the authors in [13]. The aim of this paper is to generalize 14 this algorithm to multiply connected domains. 15 While the study of numerical conformal mappings in multiply connected domains 16 dates back to 1980’s [21, 24], recently there has been significant interest in the subject. 17 DeLillo, Elcrat and Pfaltzgraff [8] were the first to give a Schwarz-Christoffel formula 18 for unbounded multiply connected domains. Their method relies on Schwarzian re- 19 flection principles. Crowdy [4] was the first to derive a Schwarz-Christoffel formula for 20 bounded multiply connected domains, which was based on the use of Schottky-Klein 21 prime function. The natural extension of this result to unbounded multiply connected 22 domains is given in [5]. It should be noted that a MATLAB implementation of com- 23 puting the Schottky-Klein prime function is freely available [6], and the algorithm is 24 described in [7]. A method involving the harmonic conjugate function is given in [20], 25 but the approach there differs from ours. 26 The main result (Proposition 3.1) of this paper is the connection between integra- 27 tion of the absolute value of gradient of harmonic solution  u  over equipotential curve 28 on domain Ω, the value of integration is denoted by  C  (Ω), and its conjugate problem 29 ˜Ω. These values are connected by the identity  C  (Ω)  ·  C  (˜Ω) = 1. Proposition forms 30 the theoretical foundation to our method. The presented algorithm itself is based on 31 solving numerically the Laplace equation subject to Dirichlet-Neumann mixed-type 32 ∗ Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, FINLAND( ) † Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, FINLAND( ) ‡ Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, FINLAND( )1  boundary conditions. This method is a generalization of the algorithm for simply 33 and doubly connected domains described in [13]. The foundation of the algorithm for 34 simply and doubly connected domains lies on properties of the (conformal) modulus, 35 which srcinates from the theory of quasiconformal mappings [1, 19, 23]. Our method 36 is suitable for a very general class of domains, which may have curved boundaries and 37 even cusps. The implementation of the algorithm is based on the  hp -FEM described 38 in [14], and in [15] it is generalized to cover unbounded domains. In [16], the method 39 has been used to compute a modulus of domains with strong singularities. 40 The rest of the paper is organized as follows: After a brief introduction to canon- 41 ical domains, the Conjugate Function Method is discussed in detail in Section 3, 42 followed numerical version, and a series of numerical experiments highlighting dif- 43 ferent features of the methodi in Section 5. Finally, some potential applications, 44 such as quadrilateral mesh generation and design of cubatures, are suggested before 45 concluding remarks. 46 2. Canonical Domains.  The so called canonical domains play a crucial role 47 in the theory of quasiconformal mappings (cf. [19]). These domains have a simple 48 geometric structure. Let us consider a conformal mapping  f   :  D  →  Ω, where  D 49 is a canonical domain, and Ω is the domain of interest. The choice of the canonical 50 domain depends on the connectivity of the domain Ω, and both domains  D  and Ω have 51 the same connectivity. It should be noted that in the simply and doubly connected 52 cases, domains can be mapped conformally onto each other if and only if their moduli 53 agree, which is given by the energy norm (see e.g. [19]). In this sense, moduli divide 54 domains into conformal equivalence classes. For simply connected domains, natural 55 choices for canonical domains are the unit disk, the upper half-plane and a rectangle. 56 In the case of doubly connected domains an annulus is used as the canonical domain. 57 For  m -connected domains,  m >  2, we have 3 m  −  6 different moduli, which leads to 58 various choices of canonical domains. These domains have been studied in [12, 22]. 59 The generalization of Riemann mapping theorem onto multiply connected domains is 60 based on these moduli, see [12, Theorem 3.9.12, 3.9.14]. 61 f  Fig. 2.1: Conformal mappings of a triply connected domain onto an annulus withradial slits on the positive real axis and radial slits. 3. Conjugate Function Method.  In this section we introduce the algorithm 62 to construct a conformal mapping from a multiply connected domain onto an annulus 63 with  n  radial slits on the positive real axis. 64 2  Let  E  0 ,E  1 ,...,E  m  be disjoint and nondegenerate continua in the extended com- 65 plex plane  C ∞  =  C ∪ {∞} . Suppose that  E  j ,  j  = 1 , 2 ,...,m  are bounded, and that 66 the a set Ω m +1  =  C ∞ \  mj =0  E  j  is connected. Then Ω m +1  is an ( m  + 1) -connected  67 domain   and its  (conformal) capacity   is defined by 68 cap Ω m +1  = inf  u 0   Ω m +1 |∇ u 0 | 2 dxdy, where the infimum is taken over all non-negative, piecewise differentiable functions 69 u 0  with compact support in   mj =1  E  j  ∪ Ω m +1  such that  u 0  = 1 on  E  j ,  j  = 1 , 2 ,...,m . 70 Suppose that a function  u 0  is defined on Ω m +1  with 1 on  E  j ,  j  = 1 , 2 ,...,m  and 71 0 on  E  0 . Then if   u 0  is harmonic, it is unique, and it minimizes the above integral. 72 Then a modulus of Ω m +1  is defined by M(Ω m +1 ) = 2 π/ cap Ω m +1 . If the the degree 73 of connectivity does not play an important role, the subscript will be omitted and we 74 simply write Ω. 75 3.1. Cutting Process.  In order to explain the cutting process in multiply con- 76 nected domains similar to our work on doubly connected domains [13], we first describe 77 the process for triply connected domains. Let Ω =  C ∞ \ ( E  0  ∪  E  1  ∪  E  2 ) and suppose 78 that  u  is a harmonic solution in Ω such that it attains 0 on  E  0  and 1 on  E  1  ∪  E  2 . 79 First we cut the domain Ω from  ∂E  1  to  ∂E  2  along the steepest descent curve, which 80 is a flowline of the gradient of   u , such that it goes through the saddle point that lies 81 between the sets  E  1  and  E  2 . Then we cut from  ∂E  0  to  ∂E  1 , again along a flowline 82 of the gradient of   u . In triply connected domains, the cut between  E  1  and  E  2  always 83 exists. In general, if such a cut exists between  E  i  and  E  j ,  i   =  j , then we say that 84 E  i  and  E  j  are  conformally visible   to each other. It is easy to see that in a triply 85 connected domain, the sets  E  1  and  E  2  are always conformally visible. 86 In general, we cannot, a priori, say which of the sets  E  j ,  j  = 1 , 2 ,...,m  are 87 conformally visible to each other. Fortunately, without loss of generality, we can 88 rename  E  j ’s as  E  ′ j  such that  E  ′ j  can be considered as a binary tree of conformally 89 visible sets (cf. Subsection 3.2). The exact numbering of   E  ′ j  is done by using pre-order 90 method of tree search, see Figure 3.1 for an example. For more information about tree 91 search methods, we refer to [3, Sec. 12]. Note that, the rearrangement is not unique, 92 and any arrangement can be used in our algorithm. Obviously, a careful choice leads 93 to easier geometrical configurations and thus reduce the computational workload. For 94 the rest of the article, we assume that  E  j ,  j  = 1 , 2 ,...,m  are arranged such that  E  j 95 is a binary tree of conformally visible sets. 96 For  j  = 1 , 2 ,...,m , let  γ  j  denote the cut between  E  j  and its parent node, see 97 Figure 3.1. For  γ  1 , we use a following definition,  γ  +1  is the cut when travelling from 98 E  0  to  E  1  and likewise  γ  − 1  when we are travelling from  E  1  to  E  0 . This distinction is 99 used when the travelling direction between  E  0  and  E  1  does matter and becomes clear 100 when the Dirichlet-Neuman problem is defined in Section 3.4. 101 3.2. Common Saddle Point.  In some domains, three or more sets have one 102 common saddle point. Thus, for these sets, the cutting process will produce a cycle 103 and not the binary tree, which we would be expecting. This kind of anomaly can 104 be seen, e.g., in fully symmetric domains, see Figure 3.2, where the common saddle 105 point lies in the middle. Note that, nodes around the common saddle point can form 106 a binary tree or even a cycle of their own depending on their child nodes. 107 3.3. Jump Between Cutting Curves.  Integrating the absolute value of gra- 108 dient of   u  through equipotential curves starting from  γ  +1  to  γ  − 1  , we encounter two 109 3  E  0 E  1 E  2  E  3 E  4  E  5  E  6 E  7  E  8 renaming γ  1 γ  4 γ  8 γ  5 γ  7 γ  6 γ  2 γ  3 E  ′ 0 E  ′ 1 E  ′ 2 E  ′ 3 E  ′ 6 E  ′ 5 E  ′ 4 E  ′ 7 E  ′ 8 Fig. 3.1: In above illustration, on the right hand side we have a binary tree of confor-mally visible sets  E  j . By renaming the sets, we have  E  ′ j  shown on the left hand sidewith the cutting curves  γ  j . E  0 E  1 E  2 E  3 E  4 E  5 Fig. 3.2: Example of a fully symmetric domain. A simple symmetric case and insteadof a binary tree, we obtain a cycle, which is due to the common saddle point in themiddle.situations. If we take an equipotential curve near  ∂E  0  then it will enclose all the  E  j , 110  j  = 1 , 2 ,...,m . On the other hand, if we choose an equipotential curve near  ∂E  1  then 111 it will enclose only  E  1 . In the latter case, to overcome this enclosing problem we will 112 integrate only to cutting curves defined in Section 3.1. If the integration process hits 113 a cutting curve, we will move along the cutting curve past the saddle point and start 114 the integration again. For ( m  + 1)-connected domain, the process has up to 2 m  −  1 115  jumps, see Figure 3.3 for an example. In the process we form a curve which encloses 116 E  j ,  j  = 1 , 2 ,...,m . Note that, e.g., in the case of symmetric domains, we have less 117 than 2 m  −  1 jumps. 118 Let Γ 0  be an equipotential curve near the  ∂E  0 , then we define a constant  d  as 119 follows 120 C  (Ω) =  d  =   Γ 0 |∇ u | ds.  (3.1)Let us assume that we have  n  jumps, then Γ j ,  j  = 1 ,...,n  are each a part of an 121 4  γ  3 γ  2 γ  1 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 γ  1 Γ 1 Γ 2  Γ 3 Γ 4 Fig. 3.3: Jump process with 5 total of jumps. The number of jumps depends onsymmetricity of the domain, and jump between the cutting curves depends on thebranching of the cutting process.equipotential curves and we define 122 d j  =   Γ j |∇ u | ds.  (3.2)By the properties of   u , we obtain 123 d  = n  j =1 d j . 3.4. Dirichlet-Neumann Problem.  Let Ω be a domain in the complex plane 124 whose boundary  ∂  Ω consists of a finite number of regular Jordan curves, so that at 125 every point, except possibly at finitely many points of the boundary, a normal is 126 defined. Let  ∂  Ω =  A  ∪  B  where  A,B  both are unions of regular Jordan arcs such 127 that  A ∩ B  is finite. Let  ψ A ,  ψ B  be real-valued continuous functions defined on  A,B , 128 respectively. Find a function  u  satisfying the following conditions: 129 1.  u  is continuous and differentiable in Ω. 130 2.  u ( t ) =  ψ A ( t ) ,  for all  t  ∈  A . 131 3. If   ∂/∂n  denotes differentiation in the direction of the exterior normal, then 132 ∂ ∂nu ( t ) =  ψ B ( t ) ,  for all  t  ∈  B. Suppose that  u  is the (unique) harmonic solution of the Dirichlet-Neumann problem 133 with mixed boundary values of   u  equal to 0 on  ∂E  0 , equal to 1 on  ∂E  j ,  u  =  u 0  on  γ  j , 134  j  = 2 , 3 ,...,m , and  ∂u/∂n  = 0 on  γ  1 . Let  v  be a conjugate harmonic function of   u 135 such that  v (Re ˜ z, Im ˜ z ) = 0, where ˜ z  is the intersection point of   E  0  and  γ  +1  . 136 Then  ϕ 1  =  u  +  iv  is an analytic function, and it maps Ω onto a rectangle  R d  = 137 { z  ∈ C : 0  <  Re z <  1 ,  0  <  Im z < d }  minus  n  −  2 line-segments, parallel to real axis, 138 between points ( u (˜ z j ) ,d j ) and (1 ,d j ), where ˜ z j  is the saddle point of the corresponding 139  j th jump. In the process we have total of   n  jumps. See Figure 3.4 for an illustration 140 of a triply connected example. 141 5
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