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<p>Topology optimization (TO) of smart materials and structures (SMSs) enhances mechanical-to-physical energy conversion efficiency. However, special grayscale issues induced by multi-material TO with nonmonotonic objective functions distort optimized configurations, constrain the design domain, and compromise manufacturability. This study elucidates how nonmonotonic objective functions cause this grayscale issue and the design domain constraint in multi-material TO of SMSs and develops an effective grayscale suppression strategy into a solid isotropic material with penalization&#xa0;(SIMP)-based TO framework through a sequential projection of elastic/piezoelectric densities, demonstrated via a piezoelectric energy harvester (PEH) for electrical energy conversion efficiency (EECE) optimization. Heaviside projection as the first projection converts elastic density to physical density, followed by piezoelectric density projection as a penalty function of elastic physical density for the second projection. Numerical examples demonstrate that this dual-projection approach can effectively eliminate grayscale issues compared with the traditional Heaviside projection, direct penalization of grayness (DGP), and discrete material optimization (DMO), leading to a larger design domain with strategic piezoelectric material distribution near loading zones and prevent grayscale-induced EECE overestimation. Compared with multi-material TO using ordered SIMP reported in literature, our method achieves superior constraints over material volume fractions.</p>

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Suppressing grayscale for multiphysics topology optimization of multiphase piezoelectric materials and structures via a dual projection

  • 期刊论文

  • 发表日期：

    2026年2月23日

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  • Cheng Liu,
  • Xiaobing He,
  • Bo Yang,
  • Zhelong He,
  • Guannan Wang

摘要

Topology optimization (TO) of smart materials and structures (SMSs) enhances mechanical-to-physical energy conversion efficiency. However, special grayscale issues induced by multi-material TO with nonmonotonic objective functions distort optimized configurations, constrain the design domain, and compromise manufacturability. This study elucidates how nonmonotonic objective functions cause this grayscale issue and the design domain constraint in multi-material TO of SMSs and develops an effective grayscale suppression strategy into a solid isotropic material with penalization (SIMP)-based TO framework through a sequential projection of elastic/piezoelectric densities, demonstrated via a piezoelectric energy harvester (PEH) for electrical energy conversion efficiency (EECE) optimization. Heaviside projection as the first projection converts elastic density to physical density, followed by piezoelectric density projection as a penalty function of elastic physical density for the second projection. Numerical examples demonstrate that this dual-projection approach can effectively eliminate grayscale issues compared with the traditional Heaviside projection, direct penalization of grayness (DGP), and discrete material optimization (DMO), leading to a larger design domain with strategic piezoelectric material distribution near loading zones and prevent grayscale-induced EECE overestimation. Compared with multi-material TO using ordered SIMP reported in literature, our method achieves superior constraints over material volume fractions.