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8.7.1 Degrees of Freedom and Boundary Conditions
FEBio's fluid solver differs from most computational fluid dynamics programs by its use of fluid dilatation , instead of fluid pressure , as a nodal degree of freedom. In the fluid module, these two variables are related by the constitutive relation, , where is the fluid's bulk modulus. The reason for selecting instead of is that the dilatation is a kinematic measure, just like the fluid velocity (the other nodal degree of freedom), whereas the pressure is a function of state (it needs to be formulated as a function of the deformation, just like the viscous fluid stress ). The fluid Cauchy stress is given by . The viscous stress depends on the fluid rate of deformation tensor according to a user-selected constitutive relation, such as Newtonian fluid.
In a fluid analysis, the user may prescribe any of the components of as an essential boundary condition, or the corresponding component of the viscous traction as a natural boundary condition ( being the normal to the boundary surface). When neither a component of nor the corresponding component of is prescribed explicitly, the latter is naturally assumed to be zero. Similarly, the user may prescribe (or , see Section 126.96.36.199↑) as an essential boundary condition, or as a natural boundary condition. When neither is prescribed explicitly, it is naturally assumed that . Natural boundary conditions are useful for creating symmetry planes in fluid analyses, where the normal fluid velocity and the viscous traction are zero: On those symmetry planes, no boundary conditions should be specified.
It is noteworthy that the fluid formulation in FEBio allows the prescription of the nodal value of as an essential boundary condition, and the surface value of as a natural boundary condition. On boundaries where the fluid velocity is completely prescribed, i.e., when normal and tangential components are known (and not zero), prescribing and instead of just produces the best computational outcome. In those cases, the user should use the fluid velocity surface load, which combines both boundary conditions (Section 188.8.131.52↑). If the user only wants to prescribe a known normal velocity and leave the tangential velocity unspecified, use the fluid normal velocity surface load (Section 184.108.40.206↑); this surface load also provides the option of setting the tangential fluid velocity to zero. Finally, if the user wants the velocity to remain normal to the selected surface (e.g., an inlet or outlet surface on which is prescribed or fixed), but does not know the velocity magnitude a priori, use the normal fluid velocity constraint (Section 3.14.2↑) to enforce this condition.
A viscous fluid satisfies the no-slip condition on a physical boundary surface. This means that on no-slip boundaries. This boundary condition can be enforced simply by fixing the three components of (Section 3.10.2↑) and not specifying any value for on that surface (which naturally enforces ).
Computational fluid dynamics analyses are often performed over a mesh that truncates the real fluid domain. Thus, fluid may enter the finite element domain across some upstream inlet boundary and leave the domain across some downstream outlet boundary. The exact flow conditions on these boundaries may not be known, thus they have to be approximated using best guesses. Arguably, the viscous traction is the least intuitive boundary condition to guess. On an outlet boundary, it is necessary to prescribe or fix the dilatation to overcome the natural boundary condition . However, in most cases the tangential components of on that boundary are not known, therefore they cannot be fixed or prescribed, leading to the natural condition . This natural condition may work fine in some cases, but it will often fail numerically when vortices being shed inside the finite element domain try to cross the outlet boundary. In those cases, consider using the fluid backflow stabilization (Section 220.127.116.11↑) and fluid tangential stabilization (Section 18.104.22.168↑) surface loads, which prescribe a velocity-dependent viscous traction .
Similarly, on an inlet boundary that shares an edge with a no-slip surface, prescribing the fluid velocity on the inlet boundary may not accurately reproduce the velocity profile in the boundary layer that would be produced on the adjoining no-slip surface. This potential mismatch could result in numerical instabilities; in such cases, prescribing a dilatation on the shared edge may mitigate this issue.