SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS
SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS
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雪花谐波和计算机图形学：分形鼓上光谱的数值计算
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
C. A. Griffith
中科院分区：
文献类型：
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作者：
M. Lapidus;J. Neuberger;R. Renka;C. A. Griffith
在这项工作中，我们研究了“科赫雪花鼓”的稳态振动，数值和计算机图形学的手段。特别是，我们近似最小的50个特征值（或鼓的“频率”），以及相应的特征函数（称为“雪花谐波”）在Koch雪花域上的狄利克雷拉普拉斯函数。我们描述了在计算中使用的数值方法，并展示了一组选定的特征函数（以及它们的梯度）的图形表示。在一次谐波的情况下，图形结果与数学推导的结果（Lapidus和Pang）关于边界上的梯度行为（膜的“爆炸”或“无限应力”）一致，并提出了关于更高特征函数的进一步猜想。根据物理学家Sapoval和他的合作者的早期工作，这项研究可能有助于更好地理解自然界分形结构（如海岸线、树木和血管）的形成和“稳定”。
In this work, we study the steady-states vibrations of the “Koch snowflake drum”, both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or “frequencies” of the drum), along with the corresponding eigenfunctions (called “snowflake harmonics”) of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (“blow up” or “infinite stress” of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions. According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and “stabilization” of fractal structures (e.g., coastlines, trees and blood vessels) in nature.