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8.7.6 Fluid-Structure Interactions
Fluid-structure interactions (FSI) may be modeled in FEBio using the fluid-FSI module (Section 3.2↑). In an FSI analysis the finite element mesh defining the fluid domain is deformable, so that the shape of the fluid domain does not have to remain fixed over time. The fluid flows through the deforming mesh, just as it does in standard computational fluid dynamics (CFD) analyses with fixed meshes. The deformation of the fluid domain may be controlled by prescribing suitable boundary conditions (so-called moving boundary problems) and by interactions with solid domains. For example, fluid may flow inside a flexible tube and the fluid pressure may cause the tube to expand or contract; the tube itself may be pinched at some location, reducing the cross-section through which fluid flows. The tube wall is a deformable solid domain which may be modeled in FEBio using any of the solid material models described in Sections 4.1↑ through 4.7↑, or it may be modeled as a porous deformable domain containing a viscous fluid, in which case one should use the biphasic-FSI material for the tube wall (Section 4.14.4↑). The inner tube (the fluid domain) should be modeled as a fluid-FSI material (Section 4.14.3↑).
In FEBio, a fluid-FSI material is modeled as a specialized fluid-solid mixture where the solid matrix density is zero and its stiffness should be set to a negligible value. In contrast, a biphasic-FSI material is a full-fledged fluid-solid mixture, where the solid may have mass and exhibit any desired nonlinear elastic or inelastic response. Both FSI materials require the user to define a fluid material from the same list as those available for CFD analyses (Section 4.14↑). It also requires the user to define a solid material to regularize the mesh deformation, which may be selected from Sections 4.1↑ through 4.7↑ for a biphasic-FSI material. In contrast, the solid material of a fluid-FSI material is massless (the density value entered by the user is ignored and reset internally to zero). Among all the choices available in Section 4.1↑, it is recommended to use the neo-Hookean elastic solid (Section 220.127.116.11↑), setting Poisson's ratio to zero and Young's modulus to a very low value (a few orders of magnitude smaller than the moduli of surrounding solid domains).
The degrees of freedom at each node of a FSI domain consist of the components of the solid displacement (which describe the mesh deformation), the fluid velocity relative to the solid, and the fluid dilatation . Essential boundary conditions may be prescribed on any of these degrees of freedom. For example, in a moving boundary problem, the motion of a boundary may be prescribed by setting the relevant components of . A no-slip boundary between an FSI material and a solid domain may be defined by simply setting , regardless of the motion of that boundary. In general, all boundary conditions described for CFD analyses in Section 8.7.1↑ may be applied identically to FSI analyses.
Currently in FEBio, the mesh of a fluid/biphasic-FSI material domain must be continuous with the mesh of surrounding solid domains. The surface between domains, which consists of the shared faces of adjoining finite elements from each domain, is called the FSI interface. Because the mesh is continuous, the solid displacement and the solid component of the traction are automatically continuous across FSI interfaces; when fluid is present on both sides of the interface, the relative fluid flux is also automatically continuous. In other words, when two FSI domains are adjoining (e.g., fluid-FSI with fluid-FSI, or fluid-FSI with biphasic-FSI), all boundary conditions are automatically satisfied across those interfaces. However, when an FSI domain interfaces with a solid domain, the fluid component of the traction in the fluid-FSI and biphasic-FSI domains must be explicitly transferred to the solid domain across that interface. This is done by prescribing a fluid-FSI traction or a biphasic-FSI traction on that interface (Section 18.104.22.168↑). This surface load does not require any user-defined parameters; its sole purpose is to identify those FSI interfaces where the fluid traction must be properly transferred to the surrounding solid domain. Omitting this boundary condition means that the fluid traction (i.e., the fluid pressure and the normal and tangential viscous components of the fluid stress) have no effect on this interface; the interface may still deform in response to the motion of the solid domain. For example, when a fluid interacts with a rigid solid domain whose motion is entirely prescribed, it is not necessary to apply a fluid-FSI traction at the interface between the fluid and rigid solid domains.
A fluid-FSI traction is also required on free fluid surfaces, such as the surface of a fluid in open channel flow, in order to properly capture the motion of that surface (such as waves). This condition is required since the net traction on a free surface is zero. As the fluid-FSI domain is modeled as a specialized solid-fluid mixture, the net traction consists of the sum of solid and fluid tractions, . Setting to zero produces a traction on the solid mesh, allowing it to deform to the proper shape.
Here are a few cautionary notes to keep in mind when running FSI analyses: (1) Depending on the problem being analyzed, fluid-structure interactions may cause significant mesh deformations that lead to mesh distortion. Currently, adaptive meshing is not yet available to relieve this mesh distortion, therefore analyses and meshes need to be designed to minimize this effect or to terminate before the solution becomes too corrupted. (2) It is possible to pinch a solid domain that surrounds a fluid domain until the flow cross-section has reduced considerably; however it is not possible to completely shut off the flow, as this would require reducing the volume of some finite elements to zero. (3) While it is possible to model the deformation of free fluid surfaces, such as the formation of surface waves in open channel flow or the deformation of fluid blobs floating in a gravity-free environment, it is not possible to model the separation of such fluid domains into sub-domains such as spraying fluid droplets; this means that analyses where such phenomena are expected to happen will likely fail. (4) Whereas standard CFD analyses may be jump-started by prescribing a relatively large fluid velocity at the very first time step of an analysis, these types of boundary conditions may lead to dynamic oscillations and potential instabilities in an FSI analysis; therefore, users must be more considerate when prescribing boundary conditions for FSI analyses.