In this work, the topological derivatives of L² and energy norms associated with the solution to Kirchhoff and Reissner-Mindlin plate bending models are introduced. Based on existing theoretical results, closed forms of the sensitivities are presented. A zero-order term is introduced in the equilibrium equations, which allows for adapting the obtained sensitivities to the context of topology optimization of plates under elastic support and free vibration condition as well. The resulting analytical formulae are used together with a level-set domain representation method to devise a simple topology design algorithm. Several finite element-based representative numerical experiments are presented showing its applications to the compliance minimization and eigenvalue maximization of Kirchhoff as well as Reissner-Mindlin plate structures under bending effects.

在这项工作中，L ²的拓扑导数并介绍了与Kirchhoff和Reissner-Mindlin板弯模型的解相关的能量准则。基于现有的理论结果，提出了敏感性的封闭形式。在平衡方程中引入了一个零阶项，该项可以使获得的灵敏度适应弹性支撑和自由振动条件下板的拓扑优化。所得的分析公式与水平集域表示方法一起使用，以设计一种简单的拓扑设计算法。提出了几个基于有限元的代表性数值实验，表明了其在弯曲作用下对基尔霍夫以及Reissner-Mindlin板结构的柔度最小化和特征值最大化的应用。