Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1283–1295 | Cite as

Local optimum in multi-material topology optimization and solution by reciprocal variables

  • Kai Long
  • Xuan Wang
  • Xianguang Gu


It is revealed that the local optimum is particularly prone to occur in multi-material topology optimization using the conventional SIMP method. To overcome these undesirable phenomena, reciprocal variables are introduced into the formulation of topology optimization for minimization of total weight with the prescribed constraint of various structural responses. The SIMP scheme of multi-phase materials is adopted as the interpolation of the elemental stiffness matrix, mass matrix and weight. The sensitivities of eigenvalue and weight with respect to design variables are derived. Explicit approximations of natural eigenvalue and weight are given with the help of the first and second order Taylor series expansion. Thus, the optimization problem is solved using a sequential quadratic programming approach, by setting up a sub-problem in the form of a quadratic program. The filtering technique by solving the Helmholtz-type partial differential equation is performed to eliminate the checkerboard patterns and mesh dependence. Numerical analysis indicates that it is beneficial to avoid the local optimum by using the reciprocal SIMP formulation. Besides, the structure composed of multi-materials can achieve a lighter design than that made from the exclusive base material. The effectiveness and capability of the proposed method are also verified by nodal displacement constraint and multiple constraints.


Topology optimization Local optimum Multi-phase materials Natural frequency Sequential quadratic programming 



This research was financially supported by the National Natural Science Foundation of China (No. 51405123, 51775308). the Fundamental Research Funds for the Central Universities (2017MS077). We are thankful for Professor Krister Svanberg for MMA program made freely available for research purposes.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Beijing Key Laboratory of Energy Safety and Clean UtilizationNorth China Electric Power UniversityBeijingChina
  2. 2.State Key Laboratory for Alternate Electrical Power System with Renewable Energy SourcesNorth China Electric Power UniversityBeijingChina
  3. 3.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering MechanicsDalian University of TechnologyDalianChina
  4. 4.School of Automobile and Traffic EngineeringHefei University of TechnologyHefeiChina

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