Biomedical Engineering Reference
In-Depth Information
Stationarity of
L
in relation to the displacement field
u
gives
∂L
∂
u
∂
σ
(
u
)
∂
ε
∂
σ
(
u
)
∂
ε
=
ε
(
u
)
· ε
(
ξ
)
+
ε
(
ξ
)
·
ε
(
λ
)d
,
ξ
d
b
b
b
b
i
∂
σ
(
u
)
+
ε
(
ξ
)
·
ε
(
λ
)
d
i
∂
ε
∂
σ
(
u
)
∂
ε
+
ε
(
ξ
)
·
ε
(
λ
)
d
=
0
∀
ξ
∈
V
(2.16)
s
s
where the symmetry of the material tensor
∂
σ
(
u
)
∂
ε
= C
ρ
in
b
has been used. Since
for both
h
the tangent matrix is the linear elasticity tensor, the condition
can be rewritten as
b
and
∂
σ
(
u
)
∂
ε
T
b
C
ρ
(
ρ
)
ε
(
λ
)
·
ε
(
ξ
)
+
ε
(
λ
)
·
ε
(
ξ
)
d
d
b
i
i
s
b
C
ρ
(
+
s
C
ε
(
λ
)
·
ε
(
ξ
)
d
=−
ρ
)
ε
(
u
)
·
ε
(
ξ
)
d
∀
ξ
∈
V
(2.17)
s
b
This equation is called the adjoint problem, and it provides the adjoint solution
λ
.
Note that, in contrast to the state Eq. (2.18), this problem is linear in
λ
, allowing a
quite simple calculation.
Finally, the Lagrangian variation in relation to the design variable
ρ
takes the
form
∂L
∂ρ
∂
C
ρ
∂ρ
∂
C
ρ
∂ρ
1
2
,
β
=
ε
(
u
)
· ε
(
u
)d
+
ε
(
u
)
· ε
(
λ
)
d
b
b
b
b
∂
σ
(
u
)
∂ρ
+
·
ε
(
λ
)
d
+
α
d
=
0
(2.18)
i
b
Using conventional finite elements, Eqs (2.14) and (2.17) are easily discretized
leading to the following algebraic equations:
F
int
=
K
b
U
+
F
i
(
U
)
+
K
s
U
=
F
ext
(2.19)
K
b
+
K
s
+
K
i
(
U
)
T
=
F
adj
(
U
)
(2.20)
where
K
b
,
K
s
,and
F
ext
denote the conventional stiffness matrices and external
forces,
F
i
(
U
) denotes the nonlinear internal forces at the interface,
K
i
(
U
) denotes
the tangent matrix at the equilibrium configuration
U
,and
F
adj
(
U
) denotes the
adjoint forces from Eq. (2.17). Given
N
and
B
, the arrays of shape functions and
their derivatives, respectively, these matrices are computed as
B
T
C
ρ
B
d
s
B
T
s
K
b
=
K
s
=
C
B
d
(2.21)
b
i
B
T
∂
σ
(
U
)
B
T
F
i
(
U
)
=
σ
(
U
)d
K
i
=
B
d
(2.22)
∂
ε
b
B
T
C
ρ
(
ρ
)
ε
(
u
)d
b
F
adj
(
U
)
=−
(2.23)
b
