Biomedical Engineering Reference
In-Depth Information
10.5.1
Optimization Algorithm
The shape optimization process is solved with a hybrid method that combines
the method of moving asymptotes (MMA) [38] with a gradient projection (GP)
method [2]. MMA is based on a convex approach for the cost function and this
monotonous approach gives MMA a fast convergence property. However, MMA is
not globally convergent and oscillations can appear near the optimal solution [39].
In fact, this behavior of the MMA is also present in the shape optimization process
presented in this work. Thus, a hybrid method that combines the MMA with GP
method) is used (see for instance [40]). Firstly, MMA approaches the solution to a
point near the optimum, and then the GP starts to avoid numerical oscillations. In
Figure 10.6, it is possible to observe MMA fast convergence and strong oscillation,
and also the soft, but slower, GP global convergence. This hybrid method has
proved to be a good strategy to solve this particular shape optimization problem.
Computationally, the optimization problem is solved following the flowchart
presented in Figure 10.7. First, for the initial stem geometry, the objective function
(
f
d
,
f
t
,
f
r
,or
f
mc
) is computed using the values of interface displacement, contact
stress, and strain energy density, which are the solution of the equilibrium problem
with contact conditions solved with the finite element program ABAQUS [41]. Next,
the sensitivity derivatives are obtained using forward finite differences with the
step size
10
−
5
δ
=
[2]. When the optimization method starts, the initial iterations
MMA
+
GP convergence
25
23
21
19
17
MMA
MMA + GP
15
13
11
9
7
5
0
2
4
6
8
10
12
14
16
18
20
Iteration
Figure 10.6
Convergence for MMA and MMA
+
GP hybrid method. Totally coated stem.
