Abstract

Shape optimization based on isogeometric analysis (IGA) has gained popularity in recent years. Performing shape optimization directly over parameters defining the computer-aided design (CAD) geometry, such as the control points of a spline parametrization, opens up the prospect of seamless integration of a shape optimization step into the CAD workflow. One of the challenges when using IGA for shape optimization is that of maintaining a valid geometry parametrization of the interior of the domain during an optimization process, as the shape of the boundary is altered by an optimization algorithm. Existing methods impose constraints on the Jacobian of the parametrization, to guarantee that the parametrization remains valid. The number of such validity constraints quickly becomes intractably large, especially when 3D shape optimization problems are considered. An alternative, and arguably simpler, approach is to formulate the isogeometric shape optimization problem in terms of both the boundary and the interior control points. To ensure a geometric parametrization of sufficient quality, a regularization term, such as the Winslow functional, is added to the objective function of the shape optimization problem. We illustrate the performance of these methods on the optimal design problem of electromagnetic reflectors and compare their performance. Both methods are implemented for multipatch geometries, using the IGA library G+Smo and the optimization library Ipopt. We find that the second approach performs comparably to a state-of-the-art method with respect to both the quality of the found solutions and computational time, while its performance in our experience is more robust for coarse discretizations.

中文翻译：

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实用的等几何形状优化：通过正则化进行参数化

摘要

近年来，基于等几何分析（IGA）的形状优化已广受欢迎。直接在定义计算机辅助设计（CAD）几何形状的参数（例如样条参数化的控制点）上执行形状优化，打开了将形状优化步骤无缝集成到CAD工作流程中的前景。使用IGA进行形状优化时的挑战之一是在优化过程中保持区域内部的有效几何参数化，因为边界的形状会通过优化算法进行更改。现有方法对参数化的雅可比行列式施加约束，以确保参数化仍然有效。这样的有效性约束的数量很快就变得很大，特别是在考虑3D形状优化问题时。另一种方法（可能是更简单的方法）是根据边界和内部控制点来制定等几何形状优化问题。为了确保具有足够质量的几何参数化，将正则化项（例如Winslow函数）添加到形状优化问题的目标函数中。我们说明了这些方法在电磁反射器的最佳设计问题上的性能，并比较了它们的性能。两种方法都使用IGA库G + Smo和优化库Ipopt针对多面体几何实现。我们发现，就找到的解决方案的质量和计算时间而言，第二种方法的性能与最新方法相当。