Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5072309, 18 pages

http://dx.doi.org/10.1155/2016/5072309

## Exact Boundary Derivative Formulation for Numerical Conformal Mapping Method

^{1}Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan^{2}Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi City 60004, Taiwan

Received 10 October 2015; Accepted 11 January 2016

Academic Editor: Ivano Benedetti

Copyright © 2016 Wei-Lin Lo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Conformal mapping is a useful technique for handling irregular geometries when applying the finite difference method to solve partial differential equations. When the mapping is from a hyperrectangular region onto a rectangular region, a specific length-to-width ratio of the rectangular region that fitted the Cauchy-Riemann equations must be satisfied. In this research, a numerical integral method is proposed to find the specific length-to-width ratio. It is conventional to employ the boundary integral method (BIEM) to perform the conformal mapping. However, due to the singularity produced by the BIEM in seeking the derivatives on the boundaries, the transformation Jacobian determinants on the boundaries have to be evaluated at inner points instead of directly on the boundaries. This approximation is a source of numerical error. In this study, the transformed rectangular property and the Cauchy-Riemann equations are successfully applied to derive reduced formulations of the derivatives on the boundaries for the BIEM. With these boundary derivative formulations, the Jacobian determinants can be evaluated directly on the boundaries. Furthermore, the results obtained are more accurate than those of the earlier mapping method.

#### 1. Introduction

The finite difference method (FDM) is a conventional numerical method commonly used in computational science because partial differential equations can be directly discretized [1–3]. However, when the computational region is irregular, boundary values must be evaluated through interpolation, extrapolation, or both, which results in greater computation costs and decreases accuracy [4, 5].

A common solution to circumvent this problem is to transform the irregular region into a rectangular region and solve partial differential equations on the rectangular region [6–10]. According to a study by Thompson et al. [11], a system of elliptic partial differential equations can guarantee a one-to-one transformation between the physical and computational domains. When the transformation is conformal, the number of additional terms introduced by the transformation of the governing equation is the minimum. Thus, inaccuracy can be avoided.

By applying the conformal mapping method to build an effective grid-generation system, Tsay et al. [12] employed the boundary integral equation method (BIEM) (also called the boundary element method, BEM) to solve the Laplace equations. The BIEM is preferable to the Laplace-equation-governed transformation process for a number of reasons: (1) it is not necessary to discretize the entire domain; (2) solutions can be evaluated exactly; (3) derivatives of variables can be evaluated directly without difference schemes; and (4) once the solutions at all the discretized boundary nodes are obtained, at any given point in the domain the solution as well as the partial derivatives can be easily found without remeshing. When the conformal mapping is performed, the governing equation of the physical domain will be transformed into a formulation in terms of derivatives with respect to the coordinates. Although the derivatives in the domain can be directly evaluated by using the BIEM, it suffers from the fact that the derivatives cannot be evaluated exactly on the boundaries [13, 14]. This drawback makes one be only able to evaluate the derivatives close to the boundaries as the boundary derivatives.

A limitation of the Laplace-equation-governed transformation is that the region to be transformed should be a quadrilateral with right-angled corners, which is known as a hyperrectangular region [12]. When a hyperrectangular region is transformed into a rectangular region by using conformal mapping, the length-to-width ratio of the rectangular region is a particular value [15]. To find this ratio, Seidl and Klose [15] converted the Cauchy-Riemann equations into a set of two Laplace equations and proposed an iterative procedure with a finite difference solution. In this research, an explicit integral method was proposed to find this ratio, without the time consumption of solving a system of linear equations.

Although the applicability of the conformal transformation used by Tsay et al. [12] to arbitrary irregular regions could be extended by using a sequence of transformations [16], the applicability of the grid-generation system proposed by Tsay et al. [12, 14, 16] is still limited due to the above-mentioned problem of inaccurate boundary partial derivatives. This problem not only precludes the derivatives and the transformation Jacobian determinants from being evaluated exactly on the boundaries, but also decreases the accuracy of the finite difference computation after the transformation.

In this paper, by applying the rectangular properties of the transformed region and the Cauchy-Riemann equations, the derivatives and the Jacobian determinants of the transformation can be evaluated on the boundaries. This evaluation significantly improves the transformation accuracy.

#### 2. Mathematical Description of the Coordinate Transformation

Based on the Cauchy-Riemann equations, the forward conformal transformation, from the physical domain ( coordinates) to the computational domain ( coordinates), can be reduced to a problem governed by two Laplace equations:with the boundary conditionswhere is the outward normal vector of the boundaries and is the gradient operator. Here, the range of either or can be chosen arbitrarily while the range of the other coordinate is restricted by the satisfaction of the Cauchy-Riemann equations:That is, when satisfying the Cauchy-Riemann equations, there is always a specific value of that corresponds to a given , and vice versa. To find this specific value, Seidl and Klose [15] proposed an iterative scheme. Besides, Wu et al. [17] showed that the transformation is not conformal but still orthogonal if one simply chooses both values of and arbitrarily. In Section 4, we will propose a numerical integral method to find the specific ratio of to which satisfies the Cauchy-Riemann equations.

By applying the divergence theorem, (1a) and (1b) can be represented as an integral equation [13]:where represents or ; is the bounding contour of the problem domain ; is the out-normal direction of ; represents a source point; represents a boundary point; is the distance from to ; when is located inside the domain; and is or when is located on the corners or the straight boundaries, respectively, as shown in Figure 1. An elementwise local coordinate system is adopted to discretize the geometry as linear elements, as shown in Figure 2, where the subscripts and represent the starting and ending nodes of an element and the subscript represents the th source point. After discretizing the boundary and choosing the source points as the nodal points of the elements, the unknown function values, , and their normal derivatives, , can be formulated as simultaneous equation system and then be solved [13, 16]. To ease the explanation, the above solution for the unknowns of boundary elements is named* boundary-unknown solving step* in this paper.