Random heterogeneous microstructure construction of composites via fractal geometry

  • Siwen Wang
  • Zhansheng GuoEmail author


The microstructures of a composite determine its macroscopic properties. In this study, microstructures with particles of arbitrary shapes and sizes are constructed by using several developed fractal geometry algorithms implemented in MATLAB. A two-dimensional (2D) quadrilateral fractal geometry algorithm is developed based on the modified Sierpinski carpet algorithm. Square-, rectangle-, circle-, and ellipse-based microstructure constructions are special cases of the 2D quadrilateral fractal geometry algorithm. Moreover, a three-dimensional (3D) random hexahedron geometry algorithm is developed according to the Menger sponge algorithm. Cube- and sphere-based microstructure constructions are special cases of the 3D hexahedron fractal geometry algorithm. The polydispersities of fractal shapes and random fractal sub-units demonstrate significant enhancements compared to those obtained by the original algorithms. In addition, the 2D and 3D algorithms mentioned in this article can be combined according to the actual microstructures. The verification results also demonstrate the practicability of these algorithms. The developed algorithms open up new avenues for the constructions of microstructures, which can be embedded into commercial finite element method softwares.

Key words

microstructure fractal geometry algorithm arbitrary shape and size arbitrary quadrilateral arbitrary hexahedron 

Chinese Library Classification

O29 O341 

2010 Mathematics Subject Classification

68W99 82D20 


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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShanghai UniversityShanghaiChina
  2. 2.Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering ScienceShanghai UniversityShanghaiChina
  3. 3.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina

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