Biomedical Engineering Reference
In-Depth Information
it. Passive stress along the fiber direction was made to increase exponentially with
constitutive model were taken to be consistent with Blemker et al. [
27
]:
λ
ⓦ
=
1
.
4
(the along fiber stretch at which collagen fibers are straightened),
C
3
=
0
.
05 (scales
the exponential stresses) and
C
4
=
6 (rate of uncrimping of the collagen fibers).
The maximum active fiber stress was 100kPa.
6
.
11.3 Coupled Rigid-Body/FE Modeling
Simulating orofacial biomechanics is particularly challenging because of the
mechanical coupling between relatively hard structures (such as the jaw, skull, and
teeth) and relatively soft structures (the face, tongue, soft-palate, and pharyngeal
tract). Previous models of the face, jaw, and tongue have largely neglected these cou-
pling effects, but we have shown these effects to be significant [
8
]. In this section,
we discuss the simulation methods that we have developed in ArtiSynth for coupled
simulation of our face-jaw-tongue model. The main components of the simulator
necessary for face and vocal tract simulations are: (1) FE simulation, (2) coupling
and (3) contact handling.
11.3.1 Finite-Element Simulation
ArtiSynth is an interactive simulation platform that combines multibody models,
composed of rigid bodies connected by joints, with FE models composed of nodes
and elements. The physics solver is described in detail in Sect.
11.4
of Lloyd et al.
[
34
].
The positions, velocities, and forces for all rigid bodies (6 DOF) and FE nodes
(3 DOF) are described respectively by the composite vectors
q, u
, and
f
. Likewise,
we have a composite mass matrix
M
. The forces
f
are the sum of external forces
and internal forces due to damping and elastic deformation. Simulation consists of
advancing
q
and
u
through a sequence of time steps
k
with step size
h
. The velocity
update is determined from Newton's Law, which leads to update rules such as the
first order Euler step
Mu
k
+
1
Mu
k
h
f
k
. In addition, we enforce both bilateral
constraints (such as joints or incompressibility) and unilateral constraints (such as
contact and joint limits), which respectively lead to constraints on the velocities
given by
Gu
k
+
1
=
+
0 and
Nu
k
+
1
0, where
G
and
N
are the (sparse) bilateral and
unilateral constraint matrices. These constraints are enforced over each time step by
impulses
=
ₒ
and
z
acting on
G
T
and
N
T
, so that the velocity update becomes
λ
Mu
k
+
1
Mu
k
h
f
k
G
T
N
T
z
=
+
+
λ
+
.
(11.1)
