基于空间连续尺寸场的动态均匀化方法

摘 要：提出了基于空间连续尺寸场的动态均匀化方法。方法以均匀化理论为基础，将各向同性蜂窝状微孔结构作为材料描述方式，合理的物理描述自然克服了动态优化中的局部模态现象。不同于将微孔结构尺寸变量依附于单元或节点，采用物质点对应的微孔尺寸为设计变量，基于修正过滤公式的形函数，构造了具有空间连续性的尺寸场，从而克服了棋盘格等数值不稳定性问题。基于复合函数求导法则，推导了总刚度阵、总质量阵等敏度表达式。以动态结构响应量最小化或最大化为目标，体积比为约束，建立了动态结构拓扑优化模型，通过二维结构数值算例对理论方法进行验证。结果表明，方法在连续体结构动态拓扑优化设计中具有可行性和有效。

HOMOGENIZATION METHOD FOR DYNAMIC PROBLEMS BASED ON CONTINUOUS SIZE FIELD

Abstract: A new topological optimization method of continuum structure for dynamic problems is presented based on con-tinuous size field. Based on homogenization theory, hexagonal microstructure with isotropic behavior is adopted as descrip-tion of the macroscopic material. The proper character of the microcosmic cell is numerically validated to avoid localized modes. The size of the hexagonal cell for the material point instead of element or node is accepted as design variables. Con-tinuity of design variables field is ensured by interpolation of the modified filtering interpolation functions. Checkerboard patterns that are a concern in most topological optimization methods are avoided naturally. Sensitivities of global stiffness matrix, global mass matrix and so on are derived according to calculation method for partial derivative of compound function. Topological optimization models are established where dynamic structural responses are taken as objective and prescribed volume fraction is referred to as constraint conditions. Numerical examples show that the proposed method is feasible and robust in dynamic topological optimization design of continuum structure.