*Forma,* Vol. 17 (No. 3), pp. 253-274, 2002

*Original Paper*

## Evolution towards the Solution of a Shape Optimization Problem

Kuniko Satake* and Hisao Honda

Hyogo University, Kakogawa, Hyogo 675-0195, Japan

*E-mail address: satake@hyogo-dai.ac.jp

(Received September 9, 2002; Accepted November 26, 2002)

**Keywords: **
Bending Energy, Intermediate Pattern, Optimization Problem, Shape of Closed Curve, Vertex Dynamics

**Abstract. **
In problems to find the minimum or maximum of a target function under given constraints, attention has been paid to the final solution. Processes leading to the solution are often not of interest. Here we examine the paths and their variations leading to the solution of an optimization problem of the shape of a closed curve in a plane. The problem we deal with is to find the smoothest curve among curves whose length and enclosed area are given. We use the method of vertex dynamics where behavior of vertex coordinates is described by simultaneous partial differential equations including a potential term. We obtain, through intermediate patterns consisting of five, four and three lobes, finally a pattern consisting of two lobes. In the process of the pattern change, we observe fusing lobes (two neighboring lobes merge into one), retracting lobes (a lobe is drawn back into a body) and their intermediate types. Mechanisms of fusion and retraction are discussed with respect to the minimum bending energy under the given constraints.

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