Biomedical Engineering Reference
In-Depth Information
The last term,
l
(
λ
)
=
b
·
λ
d
+
t
·
λ
d
t
accounts for the work of the external forces.
Takingtherelativedensitiy
ρ
as a design variable, the minimization of the
strain energy in
b
subject to constraints on the bone mass and satisfaction of
equilibrium is then proposed:
nlc
1
2
min
b
ω
σ
j
(
u
)
·
ε
j
(
u
)
d
ρ
∈
K
subject to
(2.9)
j
b
j
=
1
b
ρ
d
=
V
(2.10)
b
σ
j
(
u
)
· ε
j
(
λ
)d
+
i
σ
j
(
u
)
· ε
j
(
λ
)d
b
i
+
s
σ
j
(
u
)
·
ε
j
(
λ
)d
−
l
j
(
λ
)
=
0
∀
λ
∈
V
(2.11)
s
where nlc is the number of load cases and
ω
is a weighting factor for each load case.
The lateral constraints
[0, 1] are excluded from the equations just for clarity
reasons and will be reintroduced at the algorithmic level. Allowing, for simplicity,
asingleloadcase(nlc
ρ
(
x
)
∈
=
1,
ω
=
1), the Lagrangian of the problem is defined as
1
1
2
L
(
u
,
λ
,
α
,
ρ
)
=
b
σ
(
u
)
· ε
(
u
)d
+
b
σ
(
u
)
· ε
(
λ
)
d
b
b
+
i
σ
(
u
)
·
ε
(
λ
)d
+
s
σ
(
u
)
·
ε
(
λ
)d
−
l
(
λ
)
i
s
V
o
+
α
b
ρ
d
−
(2.12)
where
λ
and
play the role of Lagrange multipliers of the equilibrium equation
and mass constraint, respectively. The Lagrangian stationarity condition is
d
α
,
d
d
c
=
0
(2.13)
where
c
=
).
The stationarity with respect to variables
λ
and
(
u
,
λ
,
α
,
ρ
)and
d
=
(
ξ
,
η
,
γ
,
β
α
, recovers the nonlinear state
equation and mass constraint:
∂L
∂
λ
i
σ
(
u
)
·
ε
(
η
)d
i
,
η
=
b
σ
(
u
)
·
ε
(
η
)d
b
+
s
σ
j
(
u
)
·
ε
(
η
)d
+
−
l
(
η
)
=
0
∀
η
∈
V
(2.14)
s
∂L
∂α
,
γ
=
γ
b
ρ
d
b
−
V
b
=
0
∀
γ
∈
R
(2.15)
