RECIPROCAL PROGRAMMING THEORY AND ITS APPLICATION TO ESTABLISH A REASONABLE MODEL OF STRUCTURAL TOPOLOGY OPTIMIZATION

Based on mathematical programming theory, the reciprocal programming is defined as a pair of programming which objective and constraint functions are changed each other. After that, it is pointed out that reciprocal programming and dual programming seem similar but there are five differences between them: (1) the difference in whether they are the same problem or not; (2) the difference in whether there exists a dual gap or not; (3) the difference in the number of design variables; (4) the difference between single-objective and multi-objective problems; (5) the difference between reasonable and unreasonable problems. Finally, based on the definition of the reciprocal programming, the structural topology optimization model is examined; and following results are obtained: (1) from this perspective, it is clear that there exists indeed unreasonable models that have been used in structural optimization; (2) a way to correct the unreasonable model and make it reasonable is put forward; (3) the reasons that the minimizing compliance model with volume constraint (MCVC for short) is unreasonable are presented; (4) according to the theory presented in this paper, the MCVC model is actually the m-aspect of reciprocal programming, so its corresponding s-aspect is established, that is, the minimizing volume model with multiple compliance constraints (MVCC for short); (5) the physical interpretation and algorithm of structural compliance constraints in the MVCC model are presented; and the concepts and methods of global stress constraint in the ICM (Independent continuous and mapping ) method are demonstrated; (6) numerical examples show the differences between the MCVC and MVCC model as a pair of reciprocal programming and verify the rationality of the MVCC model. Different optimized topologies are obtained for the MCVC model with different volume constraints and different weighting coefficients for multiple load cases. But a unique optimized topology can be achieved by the MVCC model with compliance constraints under multiple load case.