From virtual clustering analysis to self-consistent clustering analysis: a mathematical study

  • Shaoqiang Tang
  • Lei Zhang
  • Wing Kam Liu
Original Paper


In this paper, we propose a new homogenization algorithm, virtual clustering analysis (VCA), as well as provide a  mathematical framework for the recently proposed self-consistent clustering analysis (SCA) (Liu et al. in Comput Methods Appl Mech Eng 306:319–341, 2016). In the mathematical theory, we clarify the key assumptions and ideas of VCA and SCA, and derive the continuous and discrete Lippmann–Schwinger equations. Based on a key postulation of “once response similarly, always response similarly”, clustering is performed in an offline stage by machine learning techniques (k-means and SOM), and facilitates substantial reduction of computational complexity in an online predictive stage. The clear mathematical setup allows for the first time a convergence study of clustering refinement in one space dimension. Convergence is proved rigorously, and found to be of second order from numerical investigations. Furthermore, we propose to suitably enlarge the domain in VCA, such that the boundary terms may be neglected in the Lippmann–Schwinger equation, by virtue of the Saint-Venant’s principle. In contrast, they were not obtained in the original SCA paper, and we discover these terms may well be responsible for the numerical dependency on the choice of reference material property. Since VCA enhances the accuracy by overcoming the modeling error, and reduce the numerical cost by avoiding an outer loop iteration for attaining the material property consistency in SCA, its efficiency is expected even higher than the recently proposed SCA algorithm.


Numerical homogenization Virtual and self-consistent clustering analysis Convergence Machine learning Lippmann–Schwinger equation 



We would like to thank Dr. Zeliang Liu, Dr. Modesar Shakoor, Mr. Cheng Yu, Mr. Hengyang Li, and Mr. Jiaying Gao for stimulating discussions and helps in editing the manuscript. This work is partially supported by NSFC under Grant No. 11521202. W.K.L. thanks National Institute of Standards and Technology and Center for Hierarchical Materials Design (CHiMaD) under Grant Nos. 70NANB13Hl94 and 70NANB14H012; W.K.L also acknowledges the support of the AFOSR.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.HEDPS and LTCS, College of EngineeringPeking UniversityBeijingChina
  2. 2.College of EngineeringPeking UniversityBeijingChina
  3. 3.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

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