Biomedical Engineering Reference
In-Depth Information
Objective function evaluations at vertices p
W
and p
G
.
Go to step 9.
Equation (
3.378
) can be written in matrix notation using (
3.374
).
9. Step: Increment iteration counter k and perform convergence criteria checks
using the current objective function value U
k
, e.g.
U
k
tol
a
ð
3
:
379
Þ
j
U
k
U
k
þ
d
j
tol
b
d
2
N
ð
3
:
380
Þ
Case distinction
If (
3.379
) holds true the search terminates.
If (
3.380
) holds true no improvement is made and the search likely has become
stuck at some point. Go to step 10
If neither condition holds true go to step 3.
10. Step: (Automated) restart from the current position with appropriate side
length scaling.
In the above example one single objective function U was used. The algorithm
can instead be extended for multi-objective optimization as described in
Sect. 3.4.2
, by using a comparative function that combines (weighted) information
of the single objective functions. According to the sensitivity of changes in the
parameter values, a multiple increase in the initial simplex side length in the
direction of each base vector may be more advantageous than adding a particular
constant value. Addition of such a constant may result in a relatively large initial
geometry and more function calls for the simplex to reduce.
In practice, a visual check of convergence and status is essential to judge the
optimization quality.
3.4.4 Parameter Optimization
In parameter optimization, scalar values (parameters) p
i
*
are identified for which
the objective function value reaches an optimum:
p
T
¼½
p
1
U
ð
p
Þ¼
extr
f
U
ð
p
Þg
with
p
2
...
p
n
for
p
2
E
n
: ð
3
:
381
Þ
In engineering problems, prediction of the constitutive mechanical material
behaviour of real structural components under arbitrary loading is usually assessed
through simulation. The quality of such predictions is based on the (mathematical)
model
employed
in
the
simulation
process.
Following
a
phenomenological
