ON MURRAY LAW FOR OPTIMAL BRANCHING RATIO

Fractals Pub Date : 2024-07-02 DOI:10.1142/s0218348x24500920
Yu-Ting Zuo
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Abstract

Tree-like branching networks are widespread in nature and have found wide applications in engineering, where Murray’s law is generally adopted to optimally design tree-like systems, but it may become invalid in some cases. Here we give an energy approach to the analysis of the law and re-find Li–Yu’s law for the optimal ratio of the square root of 2 with a suitable constraint. When the cross-section of each branch is considered as a fractal pattern, a modified Murray’s law is obtained, which includes the original Murray’s law for a Peano-like pore and Li–Yu’s law for cylindrical branches, furthermore a useful relationship between the diameter and length of each hierarchy is obtained, which is contrary to the tree-like fractal patterns, and the new hierarchy is named as “fractal Murray tree”, which also has many potential applications in science, engineering, social science and economics. This paper is intended to serve as a foundation for further research into the fractal Murray tree and its applications in various fields.
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关于最佳分支率的穆雷定律
树状分支网络在自然界中广泛存在,在工程领域也有广泛应用,一般采用默里定律来优化设计树状系统,但在某些情况下可能会失效。在此,我们给出了一种分析该定律的能量方法,并在合适的约束条件下重新找到了 2 的平方根的最优比值的李-尤定律。当把每个树枝的横截面看作分形图案时,就得到了修正的墨累定律,其中包括了原来的皮诺类孔隙的墨累定律和圆柱形树枝的李-尤定律,而且还得到了每个层次的直径和长度之间的有用关系,这与树状分形图案是相反的,新的层次结构被命名为 "分形墨累树",它在科学、工程、社会科学和经济学中也有许多潜在的应用。本文旨在为进一步研究分形墨累树及其在各个领域的应用奠定基础。
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