Immersed Methods for Fluid-Structure Interaction

Abstract

Fluid-structure interaction is ubiquitous in nature and occurs at all biological scales. Immersed methods provide mathematical and computational frameworks for modeling fluid-structure systems. These methods, which typically use an Eulerian description of the fluid and a Lagrangian description of the structure, can treat thin immersed boundaries and volumetric bodies, and they can model structures that are flexible or rigid or that move with prescribed deformational kinematics. Immersed formulations do not require body-fitted discretizations and thereby avoid the frequent grid regeneration that can otherwise be required for models involving large deformations and displacements. This article reviews immersed methods for both elastic structures and structures with prescribed kinematics. It considers formulations using integral operators to connect the Eulerian and Lagrangian frames and methods that directly apply jump conditions along fluid-structure interfaces. Benchmark problems demonstrate the effectiveness of these methods, and selected applications at Reynolds numbers up to approximately 20,000 highlight their impact in biological and biomedical modeling and simulation.

Keywords: applications in medicine and biology; fluid–structure interaction; immersed boundary method; immersed finite-element method; immersed interface method.

Figure 1

(a) An immersed boundary in the domain Ω (left) and potential corresponding Eulerian and Lagrangian discretizations (right). In formulations that use a regularized delta function, forces are applied in a region with finite thickness about the interface, and velocities are interpolated to the interface from the same region. (b) In volumetric formulations, the coordinate mapping χ connects reference coordinates $\mathbf{X}\in {\Omega }_0^{\text{s}}$ to current coordinates $\mathbf{x}\in {\Omega }_t^{\text{s}}$.

Figure 2

Two-, three-, four-, and five-point B-spline kernel functions. The limiting function is a Gaussian. A one-dimensional regularized delta function δ_h(x) may be defined in terms of a basic kernel function φ(r) via ${\delta }_h(x)=\frac{1}{h}\phi (\frac{x}{h})$. The three-dimensional regularized delta function δ_h(x) can then be obtained by δ_h(x) = δ_h(x)δ_h(y)δ_h(z).

Figure 3

Vorticity fields for viscous flow over a circular cylinder at Reynolds number Re = 200 by immersed boundary (left) and immersed interface (right) methods. Both methods produce essentially identical large-scale flow features in the wake behind the cylinder, but the immersed boundary formulation generates spurious interior motion that is eliminated in the immersed interface calculation.

Figure 4

Torsion test demonstrating the effect of the form of the elasticity model on the accuracy of the immersed finite-element method (FEM). The face of an incompressible elastic beam with a Mooney–Rivlin material model is twisted 450°. Four elasticity formulations are considered. “Stabilized” indicates that a volumetric energy, U(J), is included in the strain energy Ψ; “unstabilized” indicates that this energy is not included in Ψ. “Unmodified” indicates that the shearing energy W is expressed in terms of the invariants of the right Cauchy–Green tensor, $\mathbb{C}={\mathbb{F}}^T\mathbb{F}$, in which $\mathbb{F}$ is the deformation gradient tensor; “modified” indicates that the invariants of the modified Cauchy–Green strain $\overline{\mathbb{C}}={\overline{\mathbb{F}}}^T\overline{\mathbb{F}}$ are used, with $\overline{\mathbb{F}}=J^{-\frac{1}{3}}\mathbb{F}$. The immersed formulation yields an accuracy comparable to benchmark FEM results when using both modified invariants and volumetric stabilization.

Figure 5

(a) Three-dimensional simulation of neurally activated swimming eel using a constraint-based immersed method (Patel et al. 2018). (Left) Fluid vorticity field; (right) color contour of the cross-sectional moment of the body. Magenta lines and white lines show the progression of the moment wave and the curvature wave, respectively. The difference in slopes of the two lines is the neuromechanical phase lag consistent with experimental observations. Panel adapted with permission from Patel et al. (2018). (b) Immersed simulation of jellyfish turning using a hyperelastic structural model similar to those of Hoover et al. (2017, 2019).

Figure 6

(a) Immersed simulation of bolus transport in an esophagus (Kou et al. 2015b). The brown region is the downward-moving bolus, and the blue and gray meshes represent the muscle layers. (b) Experimental (left) and simulated (right) esophageal pressure topography (EPT) of a normal subject. The fiber architecture, showing circular (green) and longitudinal (red) muscles that transition from proximal to distal ends, is shown in the center. Kou et al. (2017a) showed that the change in fiber architecture can lead to the well-known pressure transition zone in a normal patient. Panels adapted with permission from (a) Kou et al. (2015b) and (b) Kou et al. (2017a).

Figure 7

Model of a porcine bioprosthetic heart valve in an experimental pulse duplicator (Lee et al. 2019). (a) Simulated axial flow in the aortic test section. (b) Comparison of computational and experimental flow rates, upstream and downstream pressures, and leaflet kinematics quantified by valve open area. Shaded regions show the 95% confidence intervals in multicycle experimental data. Confidence intervals for the pressure data are extremely narrow and are not shown in panel b.

Figure 8

(a) Idealized inferior vena cava (IVC) model developed at the US Food and Drug Administration. (b) Simulation of flow in the idealized IVC model under exercise conditions (Kolahdouz et al. 2019). Subpanels i–iv show flow patterns along the cross-sections indicated on the three-dimensional model. Results were obtained using an immersed interface method (subpanels i–iii) and by a body-conforming method in OpenFOAM (subpanel iv) (Craven et al. 2018).

Figure 9

Fluid–structure interaction models of whole-heart dynamics. (a) Patient-specific fiber-based model of a failing heart in systole. Note mitral regurgitation has developed after several computed cardiac cycles during which the left ventricle enlarges slightly on each beat because more blood is returning to it than can be pumped away. (b) A healthy heart model that uses hyperelastic constitutive models fit to biaxial tensile test data, here shown in atrial systole (left) and ventricular systole (right). The figure highlights the configurations of the mitral valve, including model chordae tendenae and papillary muscles, and the aortic valve. Transparent views are provided for the remaining structures of the heart and nearby great vessels.