Page 65 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 35

sum of the forces on the right hand side. The second equation ex-
presses the conservation of mass for the case of an incompressible
fluid.
In a Newtonian fluid, the stress tensor takes the form:

T = 2μD(u) −∇p, (1.23)

where μ is the fluid viscosity and p its pressure, while D(u) is
∇u+∇u T
the strain rate tensor defined as D(u) = . Substituting in
2
Eq. (1.21) we obtain the formulation of the Navier–Stokes equa-
tions for Newtonian incompressible fluids:
∂u
ρ + u ·∇u − μ u +∇p = b, (1.24)
∂t
div u = 0. (1.25)

This is a system of equations in variables u and p, valid in a portion
3
of the 3D space Ω ∈ R , with proper initial and boundary condi-
tions.
As mentioned above, in a fluid-structure interaction problem
the interface between fluid and solid is free to move subject to
the forces exchanged by the two bodies. This means that at least
part of the boundary Γ w ∈ ∂Ω changes position over time, result-
ing in a deformation of Ω. This is naturally described in the elas-
todynamics problem, when Γ w is considered as part of the solid
body and its motion is modeled as the motion of all other mate-
rial points in the solid. The Navier–Stokes equations are instead
usually posed in an Eulerian frame, so that the fluid is observed
as it flows through a fixed region of space. To handle the deforma-
tion of the fluid domain Ω, different approaches can be followed
depending on the method used to discretize the equations. For in-
stance, in the case of finite elements methods, care has to be taken
so that the motion of the domain does not cause degeneration of
the volumetric mesh used to discretize it. Mesh update can be per-
formed using smooth operators, that extend the velocity of Γ w to
the entire fluid domain. In case of large displacements, grid dis-
tortion may be unavoidable: in which case the mesh may have to
be re-computed adding computational complexity to the method.
Methods relying on fixed fluid grids such as the immersed bound-
ary and fictitious domain methods remove the complexity asso-
ciated with continuous updates to the computational domain,
howevertendtosufferfromlargerapproximationerrorsinthe
neighborhood of Γ w due to interpolation effects. More details and
comments on numerical discretization techniques are provided in
Section 2.4.