Biomedical Engineering Reference
In-Depth Information
Having provided comparable (force) values, the least-square problem for the
illustrated case can be formulated as follows
h
i
2
!
min
: ð
3
:
390
Þ
U
k
ð
a
k
;
p
;
m
k
;
p
;
...
Þ
:
¼
X
n
F
sim
k
;
i
ð
a
k
;
p
;
m
k
;
p
;
...
Þj
u
si
i
F
exp
k
;
i
j
u
sim
i
i
¼
1
Performing interpolation, the following case distinctions should be covered in
the program code to exclude ambiguity:
(i) points, i.e. function arguments, are identical, u
ex
n
¼
u
si
m
: no interpolation is
needed, and (force) deviation can be deduced directly
(ii) points, i.e. function arguments, are unequal, u
ex
n
6¼
u
si
m
: interpolation is
required.
(iii) both points, i.e. function arguments, are unequal, u
ex
n
6¼
u
si
m
and one set
max
[ u
si
max
: interpolation of the
function value corresponding to the smaller argument on the interpolant
associated to the curve containing the larger argument.
Remark: As described in
Sects. 3.3
and
3.4.1
, the A
BAQUS
finite element solver
was used to solve the boundary value problem for each iteration step, cf. Figs.
3.31
and
3.32
. In this process, the ask_delete parameter in the A
BAQUS
environment
file is set to off in order to run simulations without prompting for attributes.
contains larger valued function arguments u
exp
3.4.8 Optimization Constraints: Material Stability
As described in
Sects. 4.3
and
5.3
, long-term human soft tissue and long-term soft
foam material behavior were modeled using hyperelastic material models. These
models assume that the material behavior can be derived from a strain-energy
potential. Such a potential is non-dissipative, path independent and reversible. It is
commonly referred to as a strain-(stored)-energy (density) function, which rep-
resents the strain-energy stored in the material per unit of reference volume. Fur-
thermore, elastic materials for which a strain-energy function can be formulated are
referred to as G
REEN
-elastic or hyperelastic materials. Restrictions are imposed on
the form of the strain-energy function based on physical considerations, and the
strain-energy function must thus be consistent with the following issues:
(i) w must be positive for any deformation
w
¼
w
ð
k
i
Þ
[ 0
for
0 \ k
i
\
1
with
k
i
6¼
1
and
ð
i
¼
1
;
2
;
3
Þ
ð
3
:
391
Þ
where the k
i
denote the principal stretches.
(ii) w must be equal to zero in the strainless initial state where no strain-energy
is stored
