Page 103 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 73

ulate neighboring organs and the stiff pericardial sac. Finally, the
total boundary condition force is given by f b = f base + f peri .

2.3.4 Putting it all together: a fast computational
framework for cardiac biomechanics

Combining all the forces derived in the previous sections and
distributing them to the nodes of the mesh leads to the final TLED
formulation. Time integration is done using the central difference
scheme:
1 2
u(t + δt) = u(t) + δt ˙ u(t) + δt ¨ u(t). (2.25)
2
Let us assume that the displacements u(t) and u(t −δt) are known,
and that the total nodal forces f(t) have been computed (including
internal and external forces). The central difference method then
yields a node-wise equation for computing displacements at the
next time step:

u i (t + δt) = A i f i (t) + B i u i (t) + C i u i (t − δt), (2.26)

1 2M ii D ii B i
A i = , B i = A i , C i = A i − . (2.27)
D ii + M ii δt 2 2δt 2
2δt δt 2
As any explicit time integration, δt must be small enough to guar-
antee stability and accuracy. In particular, δt must be smaller than
the critical limit δt cr = L e /c,where L e is the smallest characteris-
tic element length in the assembly and c is the dilatational wave
speed of the material.

Description of the TLED finite elements algorithm
One of the advantages of TLED algorithm is that all the vari-
ables are referred to the initial configuration, which allows very
efficient updates during the main iteration loop. The algorithm
starts with a pre-computation stage to initialize all variables (Al-
gorithm 3). Before starting the main iteration loop, the solver and
constraints are initialized (Algorithm 4), followed by the main it-
eration, which consists in first calculating the net nodal force (Al-
gorithm 5) and then updating the displacement (Algorithm 6).

2.3.5 Evaluation of the TLED algorithm
Validation against analytical solution
As shown in [120], simple shear in different planes can be used
to verify the implementation of any constitutive law. Indeed, this