Layout optimization of continuum structures embedded with movable components and holes simultaneously

  • Xuan Wang
  • Ping HuEmail author
  • Zhan Kang
Research Paper


In this paper, the layout optimization problem of continuum structures embedded with movable components and holes simultaneously is solved for the first time. We propose a new methodology for the embedding problem under SIMP-based computational framework, where the positions and orientations of embedded components and holes and the topology of connecting structure are optimized concurrently to maximize the overall stiffness. To this end, the parameterized topology description function, combining the Kreisselmeier-Steinhauser (KS) function, is used to construct the geometric shapes of embedded components and holes. The material density defining the topology of connecting structure and the geometric parameters used to describe the location and orientation of embedded components and holes are considered as design variables of the optimization problem. To unite these two seemingly different representations into a single computational framework, we first project the embedded components and holes into two density fields on a fixed grid using a smoothed Heaviside function, then introduce a new SIMP-motivated material interpolation scheme invoked at the finite element level for material parameterization. The material parameterization scheme supports full analytical sensitivities, which can greatly improve the accuracy and efficiency of sensitivity analysis, and make it suitable for use with efficient gradient-based algorithms. Finally, the effectiveness of the proposed method is illustrated by several numerical examples.


Layout optimization Embedded components Embedded holes SIMP 



These supports are gratefully acknowledged. The authors are also grateful to Prof. Krister Svanberg for providing the matlab MMA code, and the anonymous reviewers for their insightful suggestions and comments on the early version of this paper.

Funding information

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11872140, 11872017) and the Fundamental Research Funds for the Central Universities (Grant Nos. JZ2019HGBZ0127, PA2019GDPK0039, and PA2019GDQT0016).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringHefei University of TechnologyHefeiChina
  2. 2.Anhui Key Laboratory of Civil Engineering Structures and MaterialsHefei University of TechnologyHefeiChina
  3. 3.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina

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