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Discrete Optimum Design Method Based on S-type Curve Integral Function

DOI：

 作者 单位 邮编 孙　蓉 山东科技大学能源与矿业工程学院 266590 谭　涛* 山东科技大学能源与矿业工程学院 266590 李艳艳 山东科技大学数学与系统科学学院

离散变量结构优化设计的结果相较于连续优化设计更加符合工程实际的需求，然而设计变量的不连续性，导致许多有效的解析数学的优化算法和条件不再适用。因此本文提出了一种连续化处理离散变量结构优化设计问题的新函数，称为S型曲线积分函数，该函数在自变量取0或1时，函数值为零且函数具有连续光滑的性质。通过引入新的0-1设计变量，离散变量优化设计问题可以转化成等价的0-1规划，令S型曲线积分函数值为零取代新设计变量取0-1离散值的约束条件，以实现离散变量优化问题的连续化处理。建立起与原问题等价的非线性规划问题的数学模型，用数学规划的方法进行求解，并给出了相应的算法。最后，基于MATLAB数值分析软件计算数值算例和经典桁架结构截面尺寸优化问题，数值结果表明所提方法是有效的且对问题规模不敏感。

Compared with the continuous optimization design method, the results of the discrete optimum design are more in line with the actual needs of projects. However, because of the discontinuity of design variables, many effective analytical mathematical optimization algorithms and conditions are no longer applicable. Therefore, a new function for the continuous method of structural optimization design with discrete variables is proposed in this paper, which is called S-type curve integral function. The function has the property of continuous differentiability, and when the independent variable is zero or one, the corresponding function value is zero. According to the properties of the proposed function, new zero-one design variables are introduced to replace original design variables, and discrete optimum design is transformed into an equivalent zero-one programming. Then, the constraint condition that the new design variable is zero or one is replaced by the condition that the value of S-type curve integral function is zero, and the continuous processing of the discrete variable optimization problem is realized in this way. In addition, the mathematical model and the corresponding algorithm of the nonlinear programming problem equivalent to the original problem are given, which can be solved by the general mathematical programming method. Finally, a numerical example and several classical truss structure section size optimization problems are calculated by MATLAB numerical analysis software. Numerical results show that the proposed method is effective and has the advantage of being insensitive to the scale of the problem.
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