论文标题
稳定的降低基础方法,用于参数化的稳定stokes和Navier-Stokes方程
Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations
论文作者
论文摘要
在减少的鞍点问题的基础近似值中众所周知,即使使用稳定的高富达方法来生成快照,也不能保证降低空间上的盖尔金投影。对于计算流体动力学中的问题,无法准确近似压力场反映了INF-SUP稳定性。在这种情况下,通常通过合适的冠军函数的速度空间的富集来恢复INF-SUP稳定性。这项工作的主要目标是提出一种替代方法,该方法依赖于有限元文献中通常采用的基于残留的稳定技术,例如Brezzi-Pitkaranta,Franca-Hughes,Streamline Upwind Petrov-Galerkar,Galerkin Lives Square。本着\ textIt {offline-online}的精神减少了基础计算分解,提出了两个这样的选项,即\ textit {offline-ofline-holly稳定}和\ textIt {离线内线稳定}。然后将这些方法与(并与)最高侵占富集方法的状态进行比较。讨论了数值结果,强调了所提出的方法可以获得较小的减小基础空间(即忽略至上的富集),该空间仍在降低的订单水平上保留了修改的INF-SUP稳定性。
It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of \textit{offline-online} reduced basis computational splitting, two such options are proposed, namely \textit{offline-only stabilization} and \textit{offline-online stabilization}. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.