Page 327 - Advances in Biomechanics and Tissue Regeneration

16.5 IMPLEMENTATION 325

0r
P N r 0 P N f ϕ becomes
Finally, Eq. (16.34) with ¼ s ,S i ¼ r¼1 i r i r¼1 i r
S ϕ , and f ¼
r
S ¼ f , i ¼ 1,…,m (16.56)
0
i
i
N
P r
S ϕ :
and Eq. (16.35) with ϕ ¼ ϕ s and S i ¼ r¼1 i r
0 1
N Z Z Z
A r
X ∂S i
κ ϕ ϕ dΓ S + ϕ ϕ g dΓ, i ¼ 1,…,m
0
0
@ i s r i s dΓ ¼ s i (16.57)
r¼1 Γ 0 R Γ 0 ∂n Γ 0
R
R
i i i
16.5.1.2.3 COMPACT FORM
1 N 1 N 1 N 1 N
i
i
i
i
For notation purposes, we define the vectors C i ¼ðC ,…,C Þ, S i ¼ðS ,…,S Þ, θ ¼ (θ , …, θ ), V ¼ (V , …, V ), W ¼
N
1
1
N
(v , …,v ), and P k ¼ðp ,…,p Þ.
k k
With these notations, Eq. (16.37)with ϕ ¼ ϕ s , and using Eq. (16.47), becomes
m
ðiÞ _
X
 C i +  C i +  S j ¼ F , i ¼ 1,…,n (16.58)
ðiÞ
ðiÞ
ðijÞ
j¼1
with
Z Z
^
 ðiÞ  dΩ + κ i ϕ ϕ dΓ, i ¼ 1,…,n
ðiÞ
sr ¼ sr s r
Ω Γ R i
Z Z
^
ðijÞ  ðijÞ
 sr ¼ sr dΩ + κ j ϕ ϕ dΓ, i ¼ 1,…,n
s r
Ω Γ R j
Z
^
 ðiÞ  sr dΩ, i ¼ 1,…,n
sr ¼
Ω
Z Z Z
F dΩ, i ¼ 1,…,n
F ðiÞ ¼ f i ϕ dΩ + ϕ g i dΓ ^ ðiÞ (16.59)
s
s
s
s
Ω Γ R i Ω
n ðiÞ
Defining C ¼ (C 1 , …,C n ), S ¼ (S 1 , …,S m ), P ¼ (P 1 , …,P k ),  C ¼ i¼1  ,
2 3
 ð1Þ 0 ⋯ 0
6 0  ð2Þ ⋯ 0 7
6 7
⋯ ⋯ ⋯ ⋯
 C ¼
4 5
0 0 ⋯  ðnnÞ (16.60)
2 3
 ð11Þ  ð12Þ ⋯  ð1mÞ
  ⋯ 
6 ð21Þ ð22Þ ð2mÞ 7
⋯ ⋯ ⋯ ⋯
6 7
 CS ¼
4 5
 ðn1Þ  ðn2Þ ⋯  ðnmÞ (16.61)
(1)
(n)
and F C ¼ (F , …,F ), Eq. (16.58) for i ¼ 1, …, n can be expressed in a compact form as:
_ (16.62)
 C C +  C C ++  CS S ¼ F C
If we make explicit the functional dependencies we have
_ (16.63)
 C C +  C ðC,S,P,W,θÞC +  CS SðC,S,P,θÞ¼ F C ðC,S,P,V,θÞ
Using the same vectorial notations, Eq. (16.48) becomes
ðiÞ _
 S i +  S i ¼ F , i ¼ 1,…,m (16.64)
0
0
0
ðiÞ
ðiÞ
where we have denoted
Z Z
^
 0ðiÞ  0ðiÞ dΩ + κ ϕ ϕ dΓ, i ¼ 1,…,m
0
sr ¼ sr i s r (16.65)
Ω
R
Γ 0
i
Z
^
 0ðiÞ  sr dΩ, i ¼ 1,…,m
sr ¼
Ω
Z Z (16.66)
F 0ðiÞ ¼ ^ 0ðiÞ dΩ + ϕ g dΓ, i ¼ 1,…,m
F
0
s
s i
s
Ω
R
Γ 0
i
II. MECHANOBIOLOGY AND TISSUE REGENERATION