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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。
In order to realize the lightweight design based on multi-phase materials and follow the independent continuous mapping method, a topology optimization model is proposed which is based on the minimization of the total weight of the structure and the given eigenvalue. Methods Two kinds of independent topological variables were used to realize element stiffness matrix, mass matrix and weight interpolation. Derivation of the eigenvalues and the total weight of the sensitivity expression, through the first and second order Taylor expansion to obtain its approximate expression. One-term filtering of the constraint function is transformed into the solution of PDE, which eliminates the numerical instability such as checkerboard phenomenon and grid dependence. The feasibility and the superiority of the proposed method are verified by two-dimensional numerical examples. The results show that the multi-phase material topology has a lighter weight than the single-phase material topologies. By attaching adjacent frequency spacing constraints or adding higher order frequency constraints, modal switching is avoided.