In this paper we use level set-based topology optimisation to design three-dimensional periodic piezoelectric materials with enhanced properties. Our methodology is fully memory-distributed and written in Julia using the package GridapTopOpt. We compare and assess several existing iterative solvers with respect to their weak scalability and find that an approximate Schur complement preconditioned GMRES method demonstrates the best performance and scalability for solving the piezoelectric homogenisation equations. We use the developed techniques to computationally design high-resolution piezoelectric metamaterials with enhanced stiffness and piezoelectric properties that yield new insights into material design for sensor, hydrophone, and actuator applications. We suggest two robust structures with simple geometric features that exhibit enhanced piezoelectric properties several times larger than those of the base material. We find that level set-based topology optimisation is well suited to problems involving piezoelectricity and has the advantage of avoiding large regions of intermediate density material.

翻译：本文采用基于水平集的拓扑优化方法，设计具有增强性能的三维周期性压电材料。我们的方法实现了完全内存分布式并行计算，并基于Julia语言的GridapTopOpt软件包开发实现。我们比较评估了多种现有迭代求解器在弱可扩展性方面的表现，发现采用近似Schur互补预处理的GMRES方法在求解压电均匀化方程时展现出最佳的性能与可扩展性。利用所发展的技术，我们通过计算设计出具有增强刚度和压电特性的高分辨率压电超材料，这些结果为传感器、水听器和致动器应用的材料设计提供了新的见解。我们提出了两种具有简单几何特征的稳健结构，其压电性能相比基体材料提升数倍。研究发现，基于水平集的拓扑优化方法非常适用于压电相关问题，并具有避免产生大范围中间密度材料的优势。