基于多相材料的连续体结构动态轻量化设计方法

Lightweight design method for continuum structure under vibration using multiphase materials

为了实现基于多相材料的结构轻量化设计,遵循独立连续映射法,提出了以结构总重最小化为目标和给定特征值为约束的拓扑优化模型。方法采用2类独立拓扑变量实现了单元刚度矩阵、质量矩阵和重量插值。推导了固有特征值和总重量的敏度表达式,通过1阶和2阶泰勒展开得到其近似表达式。对约束函数一次项过滤转换为偏微分方程的求解,消除了棋盘格现象和网格依赖性等数值不稳定性。通过二维数值算例验证了提出方法的可行性和优越性。结果表明,与单相组分材料拓扑优化结构相比,多相材料拓扑优化结构具有更轻的重量。通过附加相邻频率间隔约束或增加高阶频率约束,避免了模态交换现象。

To achieve light design of continuum structure containing multiphase materials, a topological optimization model for weight minimization with the given eigenvalue constraint is proposed using the independent continuous mapping method. Two sets of independent topological variables are employed to interpolate the elemental stiffness matrix, the mass matrix and weight. The sensitivity expressions for the eigenvalue and total weight are derived. The approximations of the eigenvalue and total weight can be obtained via the first-order and second-order Taylor expansion. The filtering technique for the first term of the constraint function is adopted as a solution to the partial differential equation. The numerical instabilities including check- erboard patterns and mesh dependence are removed. The feasibility and superiority of the proposed method are validated by two-dimensional numerical examples. The results show that the weight of the optimal structure constructed by multiphase ma- terials is lighter than that composed of constituent phase. The mode switch can be prevented by imposing the constraint on the gap of the adjacent frequency or additional high order frequency constraint.