Engineering with Computers

, Volume 34, Issue 2, pp 287–305 | Cite as

Multi-material proportional topology optimization based on the modified interpolation scheme

  • Mingtao CuiEmail author
  • Yifei Zhang
  • Xinfeng Yang
  • Chenchun Luo
Original Article


A multi-material proportional topology optimization (PTO) method based on the modified material interpolation scheme is proposed in this work. PTO method is a highly heuristic algorithm by which satisfactory results are obtained. When the proposed method is used to solve the minimum compliance problem, the design variables are assigned to elements proportionally by the value of compliance during the optimization process. It is worth mentioning that PTO algorithm does not incorporate sensitivities. Accordingly, there is nothing about sensitivity calculation but just a weighted density used as filtering in the proposed method. Hence, non-sensitivity is also one of the salient features of this method. According to the characteristics, a density interpolation approach based on the logistic function is introduced in the present study. This approach cannot only establish the relationship between the material densities and Young’s modulus more reasonably, but also effectively realize the polarization of the intermediate-density elements. The complication associated with sensitivities can be avoided by the complicated interpolation scheme in conjunction with PTO algorithm. The multi-material interpolation scheme is modified from the extended SIMP interpolation approach in three-phase topology optimization. A density-filter-based Heaviside threshold function combined with the modified interpolation is introduced in this work to obtain clear 0/1 optimal topology design. The effectiveness and feasibility of the proposed method are demonstrated by several typical numerical examples of multi-material topology optimization, in which the optimal design with distinct boundaries can be obtained.


Multi-material topology optimization PTO algorithm Logistic function Modified material interpolation scheme Heaviside threshold function 



This work was supported by the Project of China Scholarship Council (201506965015). The authors are also grateful to the anonymous reviewers for their valuable suggestions for improving the manuscript.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2017

Authors and Affiliations

  • Mingtao Cui
    • 1
    • 2
    Email author
  • Yifei Zhang
    • 1
  • Xinfeng Yang
    • 1
  • Chenchun Luo
    • 1
  1. 1.School of Mechano-electronic EngineeringXidian UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontrealCanada

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