Lee Jae H, Griffith Boyce E
Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA.
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, USA.
J Comput Phys. 2022 May 15;457. doi: 10.1016/j.jcp.2022.111042. Epub 2022 Feb 9.
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 functions). 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 is . 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.
浸入边界（IB）方法是一种用于流固耦合（FSI）的非贴合体方法，它对耦合流固系统的动量、粘性和不可压缩性采用欧拉描述，而对浸入结构的变形、应力和合力采用拉格朗日描述。带有狄拉克δ函数核的积分变换将欧拉变量和拉格朗日变量耦合起来，在实际应用中，这些积分变换的离散化使用正则化δ函数核。人们已经提出了许多不同的核函数，但之前研究核函数选择对该方法精度影响的数值工作通常局限于简化的测试案例或斯托克斯流条件，这些可能无法反映该方法在实际应用中的性能，特别是在中高雷诺数或不同载荷条件下的性能。这项工作使用浸入有限元/差分（IFED）方法，在多个流固耦合基准测试中系统地研究了正则化δ函数选择的影响，IFED方法是IB方法的一种扩展，它对不可压缩纳维 - 斯托克斯方程采用有限元结构离散化与笛卡尔网格有限差分方法相结合。与传统的IB方法将力从结构网格的节点扩散并将速度插值到这些节点不同，IFED公式在一组相互作用点上评估正则化δ函数，这些相互作用点可以选择比拉格朗日网格的节点更密集。这就使得使用具有宽节点间距的结构离散化成为可能，而在节点耦合方案中（例如，如果节点间距与正则化δ函数的支撑相当或更宽），这会在欧拉力中产生间隙。早期使用该方法的工作表明，对于剪切主导的情况，这种粗结构网格可以提高精度，并且进一步发现当结构网格间距为[此处原文缺失具体内容]时精度会提高。然而，这些结果仅限于不包括对结构施加大量压力载荷的简单测试案例。本研究在更广泛的测试范围内研究了拉格朗日和欧拉离散化相对网格宽度变化的影响。我们的结果表明，满足通常施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，较窄的核更稳健，从这个意义上说，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太敏感，并且比笛卡尔网格粗得多的结构网格对于剪切主导的情况可以产生高精度，但对于具有大法向力的情况则不然。我们在脉动复制器中的牛心包生物人工心脏瓣膜的大规模FSI模型中验证了我们的结果。
