We introduce a new level-set shape optimization approach based on polytopic (i.e., polygonal in two and polyhedral in three spatial dimensions) discontinuous Galerkin methods. The approach benefits from the geometric mesh flexibility of polytopic discontinuous Galerkin methods to resolve the zero-level set accurately and efficiently. Additionally, we employ suitable Runge-Kutta discontinuous Galerkin methods to update the level-set function on a fine underlying simplicial mesh. We discuss the construction and implementation of the approach, explaining how to modify shape derivate formulas to compute consistent shape gradient approximations using discontinuous Galerkin methods, and how to recover dG functions into smoother ones. Numerical experiments on unconstrained and PDE-constrained test cases evidence the good properties of the proposed methodology.

翻译：本文提出了一种基于多面体（即在二维空间为多边形、三维空间为多面体）间断伽辽金方法的新型水平集形状优化方法。该方法充分利用多面体间断伽辽金方法的几何网格灵活性，能够精确高效地解析零水平集。此外，我们采用合适的龙格-库塔间断伽辽金方法在底层精细单纯形网格上更新水平集函数。本文详细讨论了该方法的构建与实现过程，阐释了如何修正形状导数公式以利用间断伽辽金方法计算一致的形状梯度近似，以及如何将间断伽辽金函数重构为更光滑的函数。通过对无约束和偏微分方程约束测试案例的数值实验，验证了所提方法具有良好的性能特性。