Geometrically optimized projection bases for parametric model reduction for large-scale fluid dynamics systems

流体动力学在模拟各种物理系统中起着重要的作用。实际感兴趣的例子包括汽车和航空空气动力学、流体动力学以及天气和气候模拟。与使用比例模型或真实规模原型的风洞测试活动相关的巨额成本使计算流体动力学成为处理工业流体流动问题的不可或缺的工具。然而，在一个连续的参数范围内计算流动解，例如从飞机开始到着陆或在设计优化循环中重复进行，会导致数值问题很容易达到几周的计算时间。因此，高维系统的参数降维建模变得至关重要。这里，术语参数是指降阶模型可以有效地适应参数的变化。最主要的模型降阶方法都有一个共同点，那就是将原始的大规模模型投影到由低维基跨越的合适的子空间上。投影基本质上决定了所得到的降阶模型的逼近质量。为了解释潜在的参数依赖性，在现有文献中，建议在矩阵流形上应用内插技术来计算任意参数条件下的投影基。建议的研究项目的主要目标是通过在所讨论的矩阵流形上优化合适的目标函数来增强投影基的预测能力，该目标函数实际上衡量了逼近质量。这个多学科的问题只能通过结合数值和微分几何的工具来解决，同时还要考虑到问题的工程方面。计划在实际测试用例中演示所提出的方法的适用性和益处。

Fluid dynamics play a fundamental role in modeling a large variety of physical systems. Examples of practical interest include automotive and aeronautical aerodynamics, hydrodynamics and weather and climate modeling. The enormous costs associated with wind tunnel test campaigns using scaled models or real-scale prototypes render the deployment of Computational Fluid Dynamics an indispensable tool for treating industrial fluid flow problems. However, computing flow solutions over a continuous parameter range, say from start to landing of an aircraft or repeatedly within a design optimization loop, leads to numerical problems reaching easily computation times of several weeks. Hence, parametric reduced order modeling of high-dimensional systems becomes essential. Here, the term parametric means that the reduced order model can be adapted efficiently to parameter changes. The most prominent approaches to model order reduction all have in common that the original large-scale model is projected onto suitable subspaces spanned by low-dimensional bases. The projection bases essentially determine the approximation quality of the resulting reduced order models. In order to account for the underlying parametric dependency, it is suggested in the current literature to apply interpolation techniques on matrix manifolds for computing projection bases at arbitrary parameter conditions. The main objective of the proposed research project is to enhance the prediction capabilities of the projection bases by optimizing a suitable goal function on the matrix manifold in question, which actually measures the approximation quality. This multidisciplinary problem can be tackled only by combining tools from numerics and differential geometry, while additionally taking the engineering aspects of the problem into account. It is planned to demonstrate the applicability and benefit of the proposed method for real-life test cases.