Concurrent optimization design of axial shape and cross-sectional topology for beam structures

  • Ji Liu
  • Quhao Li
  • Shutian LiuEmail author
  • Liyong Tong
Research Paper


This paper presents a new method for simultaneously optimizing both axial shape and cross-sectional topology of a non-uniform beam with geometrically similar cross-sections. One challenge in this concurrent shape and topology optimization is the high cost associated with calculating sectional properties of different cross-sections. In resolving this, a mapping function is proposed to calculate the sectional properties of all other cross-sections by using those of one reference cross-section. Two types of design variables are introduced, and they are the mapping parameter for each beam element for defining varying axial shape and the density for each cross-sectional element for describing the topology of the reference cross-section. Three objective functions considered are minimum compliance, maximum fundamental frequency, and the maximum gap between two adjacent frequencies, and the associated sensitivity analyses and algorithms are also developed. Several numerical examples involving a rectangular and an airfoil cross-sections are presented to illustrate the capability and efficiency of the present method.


Concurrent optimization Non-uniform beam Mapping function Beam shape Cross-sectional topology 


Funding information

The authors gratefully acknowledge the financial support to this work by the National Natural Foundation of China (Grant Nos. 11332004, 11572073, and 11802164), the 111 Project (B14013), the Fundamental Research Funds for the Central Universities of China (DUT18ZD103), the Fundamental Research Funds of Shandong University (31360078614014), and the Australian Research Council via Discovery-Project Grants (DP140104408).


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Copyright information

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

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina
  2. 2.School of Mechanical EngineeringShandong UniversityJinanChina
  3. 3.School of Aerospace, Mechanical and Mechatronic EngineeringThe University of SydneySydneyAustralia

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