• Proceedings of the Korea Information Processing Society Conference
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• 2010.11a
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• pp.1839-1842
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• 2010

등각사상은 함수론의 기본적인 문제로 2차원 Laplace방정식이 나타나는 열전도, 정전(靜電) potential, 유체의 문제에 이용되는 등 공학이나 물리학에서 그 응용분야가 넓다. 수치등각사상의 목적은 보다 빠르고, 보다 정확하며, 보다 적용범위가 넓은 계산법을 연구하는데 있다. 단위원 내부로부터 Jordan 영역 내부로의 등각사상을 구하는 문제는 비선형 방정식인 Theodorsen 방정식을 푸는 것으로 귀결된다. Theodorsen 방정식에 관해서는 여러 가지 수치해법이 제안되어 있는데 본 논문에서는 그 중 SOR 법인 Niethammer와 Newton법인 Vertgeim의 방법을 다루어 비교, 분석하였다. 이 2가지 방법을 실제 계산기상에 실현시켜 수치실험을 하여 그 유효성을 비교, 분석한 결과 난이도가 낮은 문제에서는 Niethammer의 방법이 난이도가 높은 문제에서는 Vertgeim이 제안한 방법이 유효함을 알게 되었다.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2009.10a
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• pp.821-824
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• 2009

Conformal mapping is useful to solve problems in physics, engineering and so on. This paper is to discuss the numerical conformal mapping from the unit disk onto Jordan region, which can be solved by Theodorsen equation. Wegmann's method has been known as the most efficient one for the Theodorsen equation. However, we found divergence through numerical experiments by the iterative method of Wegmann. The divergence occurs especially when some degree of difficulty is high. We analyze the cause of divergence and propose an improved method by applying a low frequency pass filter to Wegmann's method. By this proposed method we can get a stable convergence for all the problems which was unstable with the Wegmann's method.

• The KIPS Transactions:PartA
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• v.8A no.4
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• pp.503-508
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• 2001

Conformal mapping is useful to solve problems in heat conduction, electrostatic potential and fluid flow involving Laplace's equation in two independent variables. Determinations of conformal maps from the unit disk onto a Jordan region eventually requires solving the Theodorsen equation which is in general nonlinear with respect to the boundary correspondence function. H bner's method which has been well known for the efficient method among the many suggestions for the Theodorsen equation, was improved in early study[1, 2]. In this paper Wegmann's method is treated that is more efficient in computation cost rather than H bner's. But we found that a question which is divergent in some difficult problems by numerical experiment of Wegmann's iteration. We analyze theoretically the cause of divergence and propose an improved method by applying a low frequency filter to the Wegmann's method. Numerical experiments by our improved method show convergence for all divergent problems by Wegmann's method.

• Proceedings of the Korea Information Processing Society Conference
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• 2002.11a
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• pp.419-422
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• 2002

저자는 등각사상을 추하기 위한 기존의 여러 Theodorsen 방정식의 해법 중 가장 유효한 해법으로 알려져 있는 Wegmann의 방법을 다룬바 있다. Wegmann의 방법으로 수치실험을 한 결과 난이도가 높다고 예상되는 문제에 있어 수렴했다가 발산을 하는 불안정현상이 나타났으며 수렴하지 않는 불안정현상의 원인을 분석하여 저주파필터를 적용한 새로운 반복법을 제안하여 Wegmann 방법으로는 발산하는 모든 문제에 있어서 수렴하는 수치실험 결과를 얻었다[1]. 본 논문에서는 저주파필터를 적용한 해법에 의해 수치적으로 수렴한 결과를 이론적으로 증명한다.

• The KIPS Transactions:PartA
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• v.14A no.1 s.105
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• pp.73-76
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• 2007

The determination of the conformal maps from the unit disk onto a Jordan region has been completed by solving the Theodorsen equation which is an nonlinear equation for the boundary correspondence function. Wegmann's method has been well known for the efficient mothed among the many suggestions for the Theodorsen equation. We proposed an improved method for convergence by applying a low-frequency pass filter to the Wegmann's method and theoretically proved convergence of improved iteration[1, 2]. And we proposed an effective method which makes it possible to estimate an error even if the real value is nut acquired[3]. In this paper, we propose an automatic algorithm for numerical conformal mapping bared on this error analysis in our early study. By this algorithm numerical conformal mapping is determined automatically according to the given domain of problem and the required accuracy. The discrete numbers and parameters of the low-frequency filter were acquired only by experience. This algorithm, however, is able to determine the discrete numbers and parameters of the low-frequency filter automatically in accordance with the given region This results from analyzing the function, which may decide the shape of the given domain under the assumption that the degree of the problem depends of the transformation of a given domain, as seen in the Fourier Transform. This proposed algorithm is also ploved by numerical experience.

• The KIPS Transactions:PartA
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• v.11A no.2
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• pp.203-206
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• 2004

Wegmann's method has been known as the most efficient one for the Theodorsen equation that is needed to solve conformal mapping. It was researched in the earlier studies (1). However divergence was revealed in some difficult problems by numerical experiment using Wegmann's method. We analyzed the cause of divergence and proposed an improved method by applying a low frequency pass filter to Wegmann's method. Numerical experiments using the improved method showed convergence for all divergent problems using the Wegmann's method. In this paper, we prove theroretically the cause of convergence in the Numerical experiment using the improved method by applying a low frequency pass filter to Wegmann's method. We make use of Fourier transforms in this theoretical proof of convergence.

• Proceedings of the Korea Information Processing Society Conference
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• 2005.05a
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• pp.1053-1056
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• 2005

단위원의 내부로부터 Jordan 영역으로의 등각사상을 구하는 것은 일반적으로 경계대응함수에 관한 Theodorsen 방정식을 푸는 것으로 귀결된다. 저자는 이 비선형 방정식의 수치적 해법 중 가장 효율적인 방법으로 알려진 Wegmann의 해법을 저주파 필터를 적용하여 개량한 바 있다. 또한 이 해법에 있어 참값을 모르더라도 오차평가가 가능한 방법을 제안하였다 [1,2]. 본 논문에서는 참값을 모르더라도 오차평가가 가능한 연구결과를 이용하여 저주파필터를 적용한 Wegmann 방법에서 지금까지 경험에 의존했었던 표본수와 저주파필터의 파라메터가 주어진 문제 영역에 따라 자동적으로 결정되는 알고리즘을 제안한다. 이것은 문제의 난이도가 문제영역의 변형에 의존한다는 전제로 문제영역의 모양을 결정하는 함수를 Fourier 급수로 분석하여 얻을 수 있다. 수치실험을 통해 그 유효성을 입증한다.