Hybrid finite difference/finite element immersed boundary method

Abstract

SUMMARY The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

Bristles reduce the force required to ‘fling’ wings apart in the smallest insects

“…More details on the method can be found in Griffith and Luo (2012) and Jones (2016). The finite element bristle meshes were constructed using Gmsh (Geuzaine and Remacle, 2009), and each individual bristle consisted of at least 16 triangular elements.…”

“…The simulation code used in this paper has been validated for standard fluid structure interaction problems (Griffith et al, 2007;Griffith and Luo, 2012). In addition to performing the above convergence study, we wanted to further validate our model by comparing the results of the model with those from a previously published analytical model given by Cheer and Koehl (1987).…”

Studies of abnormalities of the lower esophageal sphincter during esophageal emptying based on a fully coupled bolus–esophageal–gastric model

“…We used the immersed-boundary-finite-element (IB-FE) method, which is suitable for applications with fluid-structure interactions (Peskin 2002; Griffith and Luo 2017). This method adopts an Eulerian description of the momentum and continuity equations for fluid-structure systems, and a Lagrangian description of the deformation and stresses of the structure.…”

Multi-scale modelling of the human left ventricle

“…心脏疾病是世界范围内影响人类寿命和生活质 量的最主要疾病之一．对心脏功能进行数学建模和 力学仿真 [1][2][3][4][5][6] ，对心脏疾病的早期检测和预防具有指 导意义，对于心脏病的治疗和减少其发病率、死亡 率也是至关重要的． 心 脏 核 磁 共 振 图 像 ( Cardiac Magnetic Resonance，CMR)能够提供高分辨率、高品质的图 像，对心脏的解剖结构和功能进行准确的描述，是 心脏疾病诊断的重要辅助手段，对心血管疾病的早 期无创诊断和准确预后评估具有重要意义 [7] ． 而 Level Set 方法已被广泛地应用于医学图像分割等领 域 [8,9] ，其中比较经典的方法是 Chan 的主动轮廓线方 法 [10] 和 Li 的无重新初始化方法 [11] ．本文将基于改进 的 Level Set 方法来分割 CMR 图像，进而可获得相 应三维计算区域重构． Immersed Boundary(IB)法则主要用于解决弹 性固体浸入粘性不可压流体对应的流固耦合问题 [12] ：弹 性结 构定 义在 拉 格 朗日 坐标 系， 而流 体粘 性、动量和不可压条件则定义在欧拉坐标系，最后 两个坐标系下的变量通过狄拉克 Delta 函数的积分变 换来实现耦合．在 [13]中 Griffith 和 Luo 提出了基于 有限元离散的 IB 模型(IB/FE) ，使得 IB 方法在实 施过程中，不再要求拉格朗日坐标系下的网格比欧 拉坐标系下的网格更加细密，而且非线性超弹性体 能量方程能够应用于有限元离散结构体模拟．本文 将基于 [13,14]中 IB/FE 方法来模拟左心室从舒张末 期到收缩末期这个过程．其中，左心室的被动响应 将基于 Holzapfel-Ogden(HO)非线性各向异性模型 [1] ．左心室的主动收缩应力则通过优化带激励电流项 的 Fenton-Karma ( FK-S ) 细 胞 电 势 模 型 [15] 和 Niederer-Hunter-Smith(NHS)模型 [16] 来模拟． 基于上述方法，本文最终给出了关于左心室多 尺度模拟的整体框架，成功模拟了个性化的左心室 从舒张末期到收缩末期对应的心脏力学行为．通过 与基于 CMR 图像的结果和前人已有结果相比较，验 证了本文方法的可行性和有效性． 中国科学: 物理学 力学 天文学 2 左心室计算模型的建立 为了进行相关力学计算，本文将基于一个健康 人 类 心 脏 舒 张 早 期 的 CMR 图 像 ( University of Glasgow 提供) ，先分割出左心室的内、外壁边界， 再进行三维重构。在舒张早期(在等容舒张之后) ， 二尖瓣刚刚打开，此时左心室内压力最低，其形状 可近似认为接近于零载荷下的形态 [2] 。采用 [17]中改 进的 Level Set 图像分割模型(请见附录 A 中的 A5 式)便可完成左心室一系列 CMR 图像中的内、外壁 的分割(请见图 1(a)) ．将一系列分割结果先后导入 [13,19,20]，整个流固耦合系统的柯西 应力可以表述为 …”