Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

翻译：拓扑优化是工程领域中一种重要的设计工具，能够辅助实现结构的最优设计。然而，拓扑变化通常需要重新划分网格，这一步骤往往具有挑战性。本文提出了一种基于拓扑导数驱动的等几何拓扑优化方法。通过将水平集方法与浸入式等几何框架相结合，可以实现几何形状的无缝更新，而无需重新划分网格。同时，拓扑导数能够在无需定义初始孔洞的情况下实现拓扑修改[7]。我们研究了高阶基函数在水平集表示和求解近似中的影响。两个数值算例验证了所提出的方法，结果表明：采用高阶基函数近似求解可以提高精度，而线性基函数对于水平集函数的表示仍然足够。