Abstract

We study the problem of constructing some optimal packings of simply-connected nonconvex plane domains with a union of congruent circles. We consider the minimization of the radius of circles if the number of the circles is fixed. Using subdifferential calculus, we develop theoretical methods for solution of the problem and propose an approach for constructing some suboptimal packings close to optimal. In the numerical algorithms, we use the iterative procedures and take into account mainly the location of the current center of a packing element, the centers of the nearest neighboring elements, and the points of the boundary of the domain. The algorithms use the same supergradient ascent scheme whose parameters can be adapted to the number of packing elements and the geometry of the domain. We present a new software package whose efficiency is demonstrated by several examples of numerical construction of some suboptimal packings of the nonconvex domains bounded by the Cassini oval, a hypotrochoid, and a cardioid.

中文翻译：

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构造具有弯曲边界的非凸域的次优堆积的数值方法

摘要

我们研究了用全等圆的联合构造一些简单连接的非凸平面域的最佳堆积的问题。如果圆的数目是固定的，我们考虑圆的半径的最小化。使用亚微积分，我们开发了解决问题的理论方法，并提出了一种构建一些接近最优的次优包装的方法。在数值算法中，我们使用迭代过程，并且主要考虑填充元素的当前中心的位置，最近的相邻元素的中心以及域边界的点。该算法使用相同的超梯度上升方案，其参数可以适应于填充元素的数量和域的几何形状。