Prev Section 3.7: Weak Formulation for BFSI Up Section 3.7: Weak Formulation for BFSI Subsection 3.7.2: BFSI Linearization Next
3.7.1 Virtual Work and Weak Form
The virtual work statement is used to enforce the three governing equations needed to solve for the nodal DOFs , and , namely the mixture mass balance (2.13.1-6), the fluid momentum balance (2.13.1-7), and the solid momentum balance (2.13.1-8). We may rewrite the momentum balance equations to facilitate the enforcement of natural traction boundary conditions given in (2.13.2-4) and (2.13.2-5). Using , these become The virtual work statement for a Galerkin finite element formulation  is , where These integrals are evaluated in the current configuration of . Here, is the virtual solid velocity, is the virtual relative fluid volumetric flux, and is the virtual fluid energy density. Integrating by parts and using the divergence theorem, the weak form of this statement may be written as where the internal virtual work is and the external part is where Here, is the elastic traction, is the true fluid viscous traction, and is the normal component of the relative fluid flux on the boundary , whose outward unit normal is . The traction emerges from the jump condition in (2.13.2-5). The integrands of the surface integrals represent the natural boundary conditions for this formulation. If boundary conditions are not set explicitly on , the natural boundary conditions are , , and . These natural boundary conditions are consistent with the jump conditions presented above. Essential boundary conditions are prescribed on the solid displacement , relative volumetric fluid flux , and fluid dilatation , which are also consistent with the above jump conditions. In particular, an essential no-slip boundary condition may be prescribed on by setting . A symmetry plane may be prescribed with the essential boundary condition and the natural boundary conditions and .
In this formulation, the mixture traction is defined as , which may also be written as . Because of the way we chose to split the internal and external virtual work in (3.7.1-3)-(3.7.1-4), is not a natural boundary condition in this formulation. In this expression for , , and may be prescribed as natural boundary conditions, whereas may be prescribed as an essential boundary condition on , using (2.13.1-9). However, there are two general scenarios where needs to be prescribed on a region of the boundary with incomplete prior knowledge of , , or : (1) When a BFSI boundary represents a free surface (such as the fluid surface in channel flow), the mixture traction boundary condition requires that , in which case it is necessary to explicitly enforce as a constraint equation on that boundary, to impart the free surface its natural shape. (2) At a biphasic-solid interface , must balance the traction acting on the adjoining solid domain. Since is continuous across due to shared nodes, the solid natural boundary condition of is already accounted for by the deformation, so that it is only necessary to prescribe the portion of on the solid domain, thus where is the (equal and opposite) traction acting on the solid domain. In both cases, the form of this traction boundary condition is the same, with and representing the tractions acting on and , respectively.
For both of these cases, the resulting virtual work on the free surface or the interface takes the form where the elemental area on may be evaluated from the covariant basis vectors ( ), where and is the parametric representation of , defined on the solid constituent. The outward normal to on is evaluated from As a result, the virtual work can be rewritten as In FEBio this boundary condition is called BFSI traction, which the user must explicitly prescribe on free surfaces and deformable interfaces . The code automatically determines which of these two types of interfaces is being considered.