On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method
On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method
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DOI：
10.1016/j.jcp.2022.111042
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发表时间：
2021-05
影响因子：
4.1
通讯作者：
Jae H. Lee;Boyce E. Griffith
中科院分区：
文献类型：
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作者：
Jae H. Lee;Boyce E. Griffith
浸没边界(IB)方法是流体-结构相互作用(FSI)的非体协调方法，它用欧拉描述流固耦合系统的动量、粘性和不可压缩性，用拉格朗日描述浸没结构的变形、应力和合力。具有狄拉克增量函数核的积分变换耦合了欧拉和拉格朗日变量，并且在实践中，这些积分变换的离散化使用正则化的增量函数核。人们提出了许多不同的核函数，但以往研究核函数选择对方法精度影响的数值工作往往局限于简化的测试案例或Stokes流动条件，这些条件可能不能反映方法在应用中的性能，特别是在中高雷诺数或不同加载条件下。本文系统地研究了浸没有限元/差分(IFED)方法在几个流固耦合基准试验中正则化增量函数选取的影响。IFED方法是IB方法的扩展，该方法采用有限元结构离散化和笛卡尔网格有限差分相结合的方法来求解不可压缩的N-S方程。传统的IB方法将力从结构网格的节点散布，并将速度内插到这些节点上，而IFED公式计算的是一组可选择为比拉格朗日网格的节点更密集的相互作用点上的正则化Delta函数。这打开了使用具有较大节点间距的结构离散化的可能性，这将在节点耦合格式中产生欧拉力的间隙(例如，如果节点间距与正则化的Delta函数的支持相当或更宽)。这一方法的早期工作表明，对于剪切为主的情况，这种粗糙的结构网格可以产生更高的精度，并且进一步发现，随着结构网格间距的增加，精度也会提高。然而，这些结果仅限于不包括结构上的大量压力载荷的简单测试用例。这项研究在更广泛的试验范围内研究了改变拉格朗日离散和欧拉离散的相对网格宽度的效果。我们的结果表明，满足通常强加的奇偶条件的核需要更高的分辨率才能达到与不满足该条件的核相似的精度。我们还发现，较窄的核函数更稳健，因为它们产生的结果对欧拉和拉格朗日网格间距的相对变化不那么敏感，并且比笛卡尔网格粗糙得多的结构网格对于剪切为主的情况可以产生高精度，但对于法向力较大的情况则不是这样。我们在脉冲复制器中的牛心包生物瓣膜的大规模FSI模型中验证了我们的结果。
The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta function). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing isincreased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even–odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.