Biomedical Engineering Reference
In-Depth Information
human detailed anatomical structures could be a useful tool for establishing injury criteria based on
evaluation of the failure level of bone and ligament (Untaroiu, Darvish, and Crandall 2005).
Explicit techniques were widely used because they were easier to compute than implicit tech-
niques. However, simulations based on explicit formulation without considering physiological mus-
cle action may not accurately predict internal stress/strain of the foot and footwear.
Most existing FE models are based on static or quasi-static simulation. Since sole materials,
such as elastomers, and biological tissues exhibit both viscoelasticity and nonlinear elasticity, an
FE model with an implicit dynamic simulation can aid in evaluating these responses, based on
time-dependent or velocity-related material properties. It is necessary to establish a dynamic FE
model simulation of the foot for heel impact, using an implicit dynamic FE solver. A comprehensive
dynamic three-dimensional FE model of the foot and footwear implemented by implicit formula-
tion, incorporating realistic geometrical properties of bone and soft tissues, taking the viscoelas-
ticity of soft tissue and sole into consideration, is still lacking and the biomechanically dynamic
response of the foot to external impact forces has not been well addressed.
5.2 ImPlICIt dynamIC method
Both explicit and implicit methods are algorithms used in numerical analysis for obtaining solutions
of time-dependent ordinary and partial differential equations. The explicit method calculates the
state of a system at a later time from the state of the system at the current time, while the implicit
method finds a solution by solving an equation involving both the current state of the system and the
later one. In mathematics, if
Y
(
t
) is the current system state and
Y
(
t
+ Δ
t
) is the state at the later time
(Δ
t
is a small time step), then, for an explicit method:
Yt
(
+∆
t
) (())
=
FYt
(5.1)
while for an implicit method one solves the equation
GY tYt
((),
(
+∆
t
))0
=
(5.2)
to find
Y
(
t
+ Δ
t
).
It is clear that mplicit methods may require extra computational efforts to solve the above equa-
tions, which can be much harder to implement. The implicit method is used because many problems
arising in practice are stiff, for which the use of an explicit method would require impractically
small time steps Δ
t
to keep the error in the result tolerance. Therefore, for such problems, to achieve
a given accuracy, it may take much less computational time to use an implicit method with larger
time steps.
The implicit method is unconditionally stable, but it will encounter some difficulties when a
complicated three-dimensional model is considered. Two main reasons are as follows: (1) with
continuing reduction of the time increment, the computational costs in the tangent stiffness matrix
will dramatically increase and even cause divergence; and (2) local instabilities will make it difficult
to achieve equilibrium. Only direct integration methods, including implicit dynamics and explicit
methods, are suitable for nonlinear problems. Most of the reported works on the comparison of
implicit and explicit methods are on quasi-static nonlinear problems (Sun, Lee, and Lee 2000).
5.2.1 S
olution
p
rocedureS
The implicit solution procedure uses an automatic increment method based on the success rate of a
full Newton iterative solution method (ABAQUS manual):
∆
u
(i 1)
+
∆+ ⋅
u
(i)
KF I
−
t
1
(
(i)
−
(i)
)
(5.3)
Search WWH ::
Custom Search