Chinese Journal of Theoretical and Applied Mechani ›› 2017, Vol. 49 ›› Issue (5): 1135-1144.DOI: 10.6052/0459-1879-17-176

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Sui Yunkang1, Peng Xirong2   

  1. 1 Numerical Simulation Center for Engineering, Beijing University of Technology, Beijing 100022, China;
    2 School of Civil Engineering, Hunan City University, Yiyang 413000, Hunan, China
  • Received:2017-05-13 Online:2017-09-15 Published:2017-09-27


An explicit exact formula is derived for the objective function of the dual model of a class of separable convex programming problems. It makes more mature and efficient methods can be chose to solve the dual model. Therefore, the advantage of applying the duality theory of nonlinear programming to efficiently solve structural topology optimization problems is fully exploited. The research work is rooted in that the gap of a nonlinear convex programming with its dual programming is zero. Solving original programming can be equivalently transformed into solving its dual programming. The scale of the solved programming can usually be reduced greatly. But an explicit relationship is not existed between the original programming and dual programming has affected the application of the dual solution algorithm. Fortunately, the programming models of a large class of structural optimization problems, including the continuum topology optimization, are convex and separable. And an explicit relationship between the original variables and their dual variables is existed; therefore, the dual solution algorithm has become one of the effective methods for 38 years. However, the objective function of the dual problem is not explicit for a long time. It is because the dual problem is a parametric minimization problem which leads to the objective function is expressed as an implicit expression. The common explicit expression for the dual objective function is a two-order approximation. The regular thinking tendency that the dual problem is too difficult to be expressed explicitly and can only be expressed approximately is breakthrough. A dual programming explicit model (DP-EM) method is put forward for the topology optimization of continuum structures. Comparison of computational efficiency among the DP-EM method, the dual sequential quadratic program (DSQP) method and the method of moving asymptotes (MMA) is presented. The results showed that:(1) more external iterations are needed for the MMA algorithm than the DP-EM algorithm and DSQP algorithm; (2) same external iterations are needed for the DP-EM algorithm and DSQP algorithm, but internal iterations is less for the DP-EM method. It shows the advantage of the DP-EM algorithm due to its explicit dual function.

Key words:

explicit dual objective function|separable convex programming|structural topology optimization|dual sequential quadratic program method|method of moving asymptotes

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