约束优化理论和动力松弛技术在张拉整体结构找形分析中的联合应用

Combined application of constrained optimization theory and dynamic relaxation technique to form finding of tensegrity structures

在诸多具有代表性的张拉整体结构找形方法中,一个有效的思路是将其归结为约束优化问题,从而运用非线性规划的方法进行求解。文中首先证明了约束优化问题的最优性条件与张拉整体结构平衡方程的等价性。随后,在优化模型的求解问题上,利用目标函数的梯度向量即为节点不平衡力这一重要关系,在梯度法的基础上引入动力松弛思想,让非平衡形态下的质点系在不平衡力的作用下开始运动,并通过"运动阻尼"使结构收敛至平衡位形。计算实践表明,引入动力松弛技术可确保迭代点在无需步长调整的情况下稳定而迅速地逼近极小点的邻域,改善梯度法在极小点附近收敛缓慢,须逐步减小步长以提高收敛精度的不足。由于目标函数中幂指数的取值对找形的成败影响较大,文中对此进行了重点探讨,建议其合理取值为3或4。在幂指数取定的情况下,通过调整权重系数和目标杆长控制结构位形,从而获取理想的找形结果。对于存在固定节点的索杆体系,还给出边界条件的处理方法,为开发新型空间结构提供有效的工具。

In various proposed typical form-finding methods for tensegrity structures,one effective approach is to view the problem as a constrained optimization problem,which can be solved by the nonlinear programming method. Firstly,the equivalence between optimality conditions of constrained optimization problem and the equilibrium equations of tensegrity structures was validated,providing the theoretical foundation of using constrained optimization theory for the form-finding analysis. Since the gradient vector of objective function can be viewed as the nodal unbalanced force vector,the dynamic relaxation technique was introduced into the gradient method,so that the system of particles in unbalanced state can move under the action of unbalanced force and make the structure converge to the equilibrium state through kinetic damping. Numerical results show that after the introduction of dynamic relaxation technique,the iteration point can approach the neighborhood of minimum point stably and rapidly and eventually converge to it without adjusting step length. Hence,the convergent rate is remarkably accelerated. Since the value of power exponent in objective function has great influence on the form-finding results,it was seriously investigated and the value of power exponent was recommended as 3 or 4. Provided that the value of power exponent is given,the structural configuration can be controlled by adjusting weight coefficients and objective strut lengths so that the ideal form-finding results can be obtained. For cable-strut system with fixed nodes,the treatment method of boundary conditions was proposed,providing an effective tool for developing novel spatial structure.