The paper presents a level set based topology optimization method for unidirectional phononic structures with finite layers of lattice cells. Boundary element method(BEM) is employed as the numerical approach to solve the acoustic problems governed by Helmholtz equation. A sized reduced coefficient matrix is derived due to the iteration forms for input and output quantities on the periodic boundary of unit cells. Topological derivatives are formulated by boundary integral equation combined with adjoint variable method and computed for each layer. An average topological sensitivity of a single design domain is proposed for the updating of the level set function(LSF) governed by an evolution equation. Numerical models with different number of layers are considered and several optimized structures of unit cells are obtained in concerned frequencies. A further investigation into the transmission of acoustic waves is carried out by employing more layers of the periodic structures between the input and output domains. The results demonstrate the effectiveness of the proposed optimization method for the finite unidirectional phononic structures.

本文提出了一种基于水平集的拓扑优化方法，用于有限单元格单元的单向声子结构。采用边界元法（BEM）求解Helmholtz方程所控制的声学问题。由于单位单元的周期性边界上的输入和输出量的迭代形式，得出了大小减小的系数矩阵。通过边界积分方程结合伴随变量法来构造拓扑导数，并针对每一层进行计算。提出了单个设计域的平均拓扑敏感性，以更新由演化方程控制的水平集函数（LSF）。考虑具有不同层数的数值模型，并在相关频率下获得了多个优化的晶胞结构。通过在输入域和输出域之间使用更多层的周期性结构，可以进一步研究声波的传输。