Explicit isogeometric topology optimization based on moving morphable voids with closed B-spline boundary curves

  • Yundong Gai
  • Xuefeng Zhu
  • Yongjie Jessica Zhang
  • Wenbin Hou
  • Ping HuEmail author
Research Paper


In this paper, we propose an explicit isogeometric topology optimization approach based on moving morphable voids (MMVs) with closed B-spline boundary curves, named TOP-IGA-MMV in short. We model the design domain by a non-uniform rational B-spline (NURBS) patch, and then employ the NURBS-based isogeometric analysis (IGA) for structural response and the well-established adjoint approach for sensitivity analysis. As for the geometry representation of structural topology, we utilize MMVs to describe boundaries of void material regions and model MMVs by closed star-like B-spline curves. Design variables consist of the coordinates of the central points and the distances from the central points to their corresponding independent control points of these closed B-spline MMVs. We perform structure analysis on a coarse mesh in which four identification schemes (control points, nodal points, Gaussian points, and Greville points) are adopted to form the Young’s modulus and volume fraction of a NURBS element and plot the structural topology on a high-resolution mesh after obtaining explicit boundaries. Three benchmark numerical examples are presented, demonstrating the effectiveness of TOP-IGA-MMV for topology optimization with different initial design and identification schemes and comparing numerical efficiency with the solid isotropic material with penalization (SIMP) method and the TOP-IGA-MMC method.


Topology optimization Moving morphable voids Moving morphable components NURBS Isogeometric analysis 



The authors would like to thank the constructive suggestions from Prof. Xu Guo and Prof. Weisheng Zhang and the valuable comments from anonymous reviewers on improving the quality of the present work.

Funding information

This work was sponsored by the National Natural Science Fund of China (No. 11272077), and the Research Funds of State Key Laboratory of Structural Analysis for Industrial Equipment (No. S15108). Y. Gai was also supported by the State Scholarship Fund of China Scholarship Council (No. 201806060022). Y. J. Zhang was supported in part by the PECASE award N00014-16-1-2254, NSF grant CBET-1804929 and CMU Manufacturing Futures Initiative.

Compliance with ethical standards

Conflict of interest

We declare that we 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 Automotive EngineeringDalian University of TechnologyDalianChina
  2. 2.Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA
  3. 3.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina

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