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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.

The finite difference method (FDM) is a conventional numerical method commonly used in computational science because partial differential equations can be directly discretized [

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 [

By applying the conformal mapping method to build an effective grid-generation system, Tsay et al. [

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 [

Although the applicability of the conformal transformation used by Tsay et al. [

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.

Based on the Cauchy-Riemann equations, the forward conformal transformation, from the physical domain (

By applying the divergence theorem, (

Configuration for

Definition of the elementwise local coordinates:

The inverse conformal transformation is also governed by Laplace equations, whose function values are

A sketch of the orthogonal transformation from the physical domain (

After

The Jacobian determinants for the forward transformation and inverse transformation are defined, respectively, as below:

The Jacobian determinant is a combination of partial derivatives. Since these derivatives are evaluated by the BIEM, singularities occur when the unknown derivatives are located on the boundaries. A previous research has suggested that these boundary derivatives can be approximated by evaluating the inner points that are very close to the boundaries [

The partial derivatives of

The discretization of these boundary integral equations is performed by linear approximations, and the potential and its normal derivative are subsequently approximated by

During the

The Jacobian determinant of the inverse transformation, from the rectangular region to the irregular region, defined as (

An illustration for the solution of the partial derivatives on the boundaries. The values of

Taking advantage of the transformed rectangular geometry, most singular terms in the conventional derivative formulations vanish, and the derivatives can then be evaluated on the boundaries. Furthermore, most terms in the reduced formulations are equal to zero, which implies far fewer computational errors than using conventional formulations. Although the present reduced formulations still cannot evaluate the derivatives at the source points, adopting approximations with the present formulations can yield accurate results. An illustrated example is provided in Section

To perform conformal mapping from a hyperrectangular region (

Let the width of the rectangular region (

The integral paths of the numerical integral method for finding the length-to-width ratio to apply conformal mapping from a hyperrectangular region onto a rectangular region.

This research proposes a numerical integral method to calculate the length-to-with ratio of a rectangular region when a conformal mapping is performed from a hyperrectangular region to the rectangular region. Besides, an improved scheme to directly solve for the partial derivatives on the boundaries when conformal mapping is performed using the BIEM is also proposed in this research. By applying the properties of rectangular geometry, the reduced formulations of BIEM discretization, (

In order to validate the improved boundary-derivative-solving scheme, three examples are provided in this section: the first is a transformation from a unit-square region into another unit-square region; the second is a transformation from an arc region into a rectangular region; the third is a transformation from a wave-block region (one side is a wave curve and other sides are straight lines) into a rectangular region.

The case of unit-square transformation and the error estimates at a source point are introduced first. When the calculation point is located at a source point, neither the conventional nor the present numerical scheme can provide the derivative. However, since the governing equation is the Laplace equation, the analytical solution should be continuous and smooth. These properties make it reasonable to approach the derivative at the source point using a numerical result from a nearby point. Tsay and Hsu [^{−6} from the inner domain be used to perform the derivative calculation, while in this research the approximation approach along the boundaries is used. To assess which approach can provide results that are more accurate, a unit-square region with two complicated Dirichlet and two Neumann boundary conditions is considered. The Dirichlet boundaries were set on the top and bottom and the Neumann boundaries were set on the right and left, as shown in Figure

The unit square with complicated boundary conditions. In order to evaluate the derivative on a source point, this research approached the point along the boundary, while Tsay and Hsu [

In order to capture the complicated analytical solution, 100 elements were applied on each side of the computational region, and then a series of computations in the derivative with respect to ^{−12}, Tsay and Hsu’s numerical result becomes singular, whereas the present scheme still provides an accurate result.

The percentage errors of the boundary derivatives near the source point

Unfortunately, a numerical result cannot be achieved at the corners using this boundary approach. The reason for this is as follows. Either (

For example, in order to calculate the derivative value on corner C in Figure

An illustration of the corner singularity. While evaluating the derivative of

The second example is a transformation from an arc region into a rectangular region, as shown in Figures

Conformal mapping from (a) an arc region to (b) a rectangular region, where

(a) The average length of rectangular region,

The percentage errors of the average length of rectangular region,

The governing equations and boundary conditions of (a)

Orthogonal

The numerical boundary derivatives and Jacobian determinants were examined with their analytical values. The analytical solutions of the forward transformation and the related derivatives are listed below:

Comparison of the percentage errors of (a)

Comparison of the percentage errors of (a)

The third example is a transformation from a wave-block region into a rectangular region, as shown in Figures ^{−6} and 3.23 × 10^{−4}, respectively. The small values of AREo and MREo indicate that

Conformal mapping from (a) a wave-block region to (b) a rectangular region, where

(a) The average length of the rectangular region transformed from the wave-block region. (b) The standard deviation of

The relative errors of orthogonality: (a) AREo and (b) MREo. The two errors are observed to linearly decrease with

To perform the conformal mapping with BIEM, the boundary is discretized into 1400 elements, where

The governing equations and boundary conditions of (a)

Orthogonal

The boundary derivatives of the conformal mapping are evaluated using the improved scheme proposed in this research. Since there is no analytical solution in this case, a conventional numerical method, FDM, is used to examine the correctness of the proposed scheme. It is applied in the rectangular region (

The derivatives of the conformal mapping,

The derivative,

The derivative,

A numerical integral method to find the length-to-width ratio of the rectangular region that fitted Cauchy-Riemann equations is proposed in this research. An arc example demonstrates that the numerical results can approach the analytical solution with negligible error (

By applying the geometric property of the transformed rectangular region and the Cauchy-Riemann equations, the derivatives and Jacobian determinant of the numerical conformal mapping method developed by Tsay and Hsu [

The authors declare that there is no conflict of interests regarding the publication of this paper.

Chuin-Shan Chen would like to acknowledge the grant support from the Ministry of Science and Technology in Taiwan.