In this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings, we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving linear-elliptic operators for the extension and the effect of additional nonlinear advection on the set of reachable shapes. It is moreover discussed how the computational complexity of the proposed algorithm can be reduced. The benefit of the nonlinearity in the extension operator is substantiated by several numerical test cases of stationary, incompressible Navier–Stokes flows in 2d and 3d.

在本文中，我们提出了一种形状优化算法，该算法能够处理较大的变形，同时保持较高的网格质量。基于映射方法，我们引入了非线性扩展算子，该算子将边界控制链接到区域变形，从而确保了结果形状的可接受性。主要重点是在涉及线性椭圆算子进行扩展的完善方法与其他非线性对流对可到达形状集的影响之间进行比较。此外，讨论了如何减少所提出算法的计算复杂度。扩展算符中非线性的好处通过在2d和3d中固定的，不可压缩的Navier-Stokes流的几个数值测试案例得到了证实。