声子晶体能带结构仿真的CS-FEM方法及其在拓扑优化设计中的应用

Smoothed finite element method and an application in topological optimization design for phononic crystals

针对声子晶体拓扑优化设计时能带结构和灵敏度计算效率低的缺陷,构造了一种Cell-based光滑有限元法模型(CS-FEM),并结合双向渐进结构优化法(BESO)实现了声子晶体的带隙最大化设计。基于四边形单元构造光滑子域,将梯度光滑技术与Bloch定理结合,构建了声子晶体能带结构计算的CS-FEM数值模型,并将其用于正问题的仿真模拟。在渐进优化准则下,通过BESO算法完成了声子晶体的优化设计。数值算例表明:CS-FEM能够适当软化离散系统的刚度,提供更加准确、高效的能带结构仿真结果;基于CS-FEM进行正问题的计算,在优化设计中得到了最优的拓扑构型,并有效地提高了优化效率,对于实现声子晶体的高效设计具有重要的参考价值。

In order to solve the problems of low computational efficiency for band structures simulation and sensitivity gradient calculation in phononic crystals topological optimization,a theoretical framework of Cell-based finite element method(CS-FEM)was formulated.The design of band gap maximization of phononic crystals was realized by combining Bi-directional evolutionary structural optimization(BESO).Based on the quadrilateral elements,a series of smoothing do-mains are further formed.The numerical algorithm of CS-FEM was established for band structures simulation by combin-ing gradient smoothing technique(GST)and Bloch theories,which is used in the simulation of forward problems.On the progressive optimization scheme,the design of phononic band gap crystals was finished by BESO method.Numerical ex-amples show that the CS-FEM can soften the stiffness of discrete system properly and provide more accurate and efficient simulation results of band structures;Based on CS-FEM,the optimal topological configuration is obtained in the optimiza-tion design,and the optimization efficiency is effectively improved,which has important reference value for the efficient design of phononic crystals.