In this work, the Immersed Boundary Method (IBM) with feedback forcing introduced by Goldstein et al. (1993) and often referred in the literature as the Virtual Boundary Method (VBM), is addressed. The VBM has been extensively applied both within a Spectral and a Finite Difference (FD) framework. Here, we propose to combine the VBM with a computationally efficient Finite Volume (FV) method. We will show that for similar computational configurations, FV and FD methods provide significantly different results. Furthermore, we propose to modify the standard feedback forcing scheme, based on a Proportional-Integral (PI) controller, with the introduction of a derivative action, in order to obtain a Proportial-Integral-Derivative (PID) controller. The stability analysis for the Backward Differentiation Formula of order 1 (BDF1) time scheme is modified accordingly, and extended to the Backward Differentiation Formula of order 2 (BDF2) time scheme. We will show that, for the BDF2 time scheme, the derivative action allows to improve the stability characteristics of the system. Our approach is validated against numerical data available in the literature for a stationary/rigidly moving 2D circular cylinder in several configurations. Finally, a Fluid-Structure Interaction (FSI) benchmark, related to the frequency response of a cantilever beam coupled with a fluid, is presented: we numerically demonstrate that the introduction of the derivative action plays an important role in order to properly detect the fluid-structure interaction coupling.
翻译:在这项工作中,讨论了由Goldstein等人(1993年)采用并经常在文献中称为虚拟边界方法(VBM)的反馈力驱动的混合边界方法(IBM),在虚拟边界方法(VBM)框架内广泛应用。在这里,我们提议将VBM与计算高效的Finite Vol(FV)方法相结合。我们将表明,对于类似的计算配置、FV和FD方法,基于Goldstein等人(1993年)采用并经常在文献中被称作虚拟边界方法(VBM)的反馈力驱动标准办法提供了显著不同的结果。此外,我们提议修改基于比例-Intragal(PI)控制器的标准反馈力驱动办法,并引入衍生行动,以便获得一个Proportal-Intergal-Derivation(PID)控制器。我们的提议是,对顺序1(BDFFS)的后向后调制公式进行稳定化分析,并推广到顺序2(BDFFS2)的后调制时间方案。我们提议,衍生行动使得衍生工具能够改善系统的稳定特性特性特性-DRILBLBS的周期作用作用作用。最后,我们的方法是根据一个可用的数字-RIFS的变压数据基数调制, 向一个可变压的变压的变压的基调制。