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Algorithm for parallel Laplacian growth by iterated conformal maps.

Anders Levermann1, Itamar Procaccia

  • 1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 20, 2004
PubMed
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This study introduces a novel algorithm for generating Laplacian growth patterns via iterated conformal maps, overcoming previous limitations in layer formation. The generated patterns exhibit unique fractal dimensions, offering new insights into growth phenomena.

Area of Science:

  • Physics
  • Mathematics
  • Computational Science

Background:

  • Laplacian growth models describe natural phenomena like crystal formation and viscous fingering.
  • Previous algorithms struggled to achieve complete layers with width proportional to the Laplacian field gradient.

Purpose of the Study:

  • To develop and present a new algorithm for generating Laplacian growth patterns.
  • To overcome the challenge of creating complete layers in Laplacian growth simulations.
  • To analyze the characteristics, including fractal dimension, of the generated patterns.

Main Methods:

  • Utilized iterated conformal maps to generate Laplacian growth patterns.
  • Developed a novel algorithm to address limitations in layer formation.
  • Compared the algorithm's output with existing direct numerical solution methods.

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Main Results:

  • Successfully generated Laplacian growth patterns using the new algorithm.
  • Overcame the difficulty of growing complete layers with local width matching the Laplacian field gradient.
  • The generated patterns show distinct characteristics when compared to those from direct numerical solutions.

Conclusions:

  • The proposed algorithm offers an effective method for simulating Laplacian growth.
  • The approach provides a new way to study complex growth patterns and their fractal properties.
  • Further discussion on the fractal dimension of these patterns is warranted.