桁架结构拓扑优化的功极射法及其全局收敛性

WORK-EXTREMUM-SCALE METHOD FOR TOPOLOGY OPTIMIZATION OF TRUSSES ANG ITS GLOBAL CONVERGENCE

功—重量分配准则是严格按照不等式Kuhn-Tucker的条件极值理论推导得到的。它说明,在功约束下对于节点受外力载荷作用的桁架结构,各分结构重量应按其承担外力功的大小来正比分配,才能达到最优。该准则能识别不承受外载荷的杆单元,适合用于桁架结构的拓扑优化。功极射法是利用功—重量分配准则以及设计变量的等比变换与功函数一阶偏导数所存在的特定关系构造的,它包括三个基本步骤,即确定当前乘子、求解功准则方程组和做射线步。由功—重量分配准则所导出的单元删除准则,是结合各杆承担功的比例以及相对于最大应力绝对值的应力偏差大小作为是否删除单元的判据。综合运用位移虚功法与矩阵位移法讨论设计变量的等比变换对求解式谱半径的影响,证明功极射法所用的迭代求解式具有全局收敛性,并用三杆桁架结构的解析实例加以验证。多工况下三杆和十杆桁架结构的算例表明,功极射法可有效地用于桁架结构的拓扑优化。

The work-weight-distribution criterion is derived from the Kuhn-Tucker extremum condition of work constraint. For a truss loaded external forces, the criterion implies that each of element weight can be best distributed according to the proportion of external work. It can be used to distinguish the unloaded bar-elements. The work-extremum-scale method（WESM） is derived from the criterion and the specific relations between a scale of design variables and work derivations. The WESM includes three basic steps that are to calculate the current multiplier, and solve the nonlinear equations and do the ratio-step. The percents of external work in bars and stress errors with regard to the maximum stress are used to cancel the spare elements. The influence of a scale of design variables on the spectral radius of solving equations is discussed by displacement-virtual-work method and matrix-displacement method. It verifies that iterative solution in the WESM is globally convergent. The analytic example of a three-bar truss. Numerical examples of three-bar and ten-bar trusses with multi-case indicate that the WESM is effective to topology optimization of trusses.