Partial Differential Equations - Modelling and ... - ResearchGate

218 J. Hao et al.
and solve for V n+5/6
G
and ω n+5/6 from
⎧
⎪⎨ 〈e i , u n+5/6 − V n+5/6 −−−−−→
n+5/6
G
− ω G n+4/6 ⊥
x 〉 n+4/6 B
=0, for i =1, 2,
h
⎪⎩ 〈 −−−−−→
G n+4/6 ⊥
x , u n+5/6 − V n+5/6 −−−−−→
n+5/6
G
− ω G n+4/6 ⊥
x 〉 n+4/6 B
=0.
h
(30)
Step 7. Finally, take V n+1,0
G
= V n+5/6
G
and G n+1,0 = G n+4/6 . Then predict
the final position and translation velocity as follows:
For k =1,...,N,
Call Adhesive Dynamics Algorithm,
̂V n+1,k
G
= V n+1,k−1
G
+ F r (G n+1,k−1 )△t/2N, (31)
Ĝ n+1,k = G n+1,k−1 +(̂V n+1,k
G
V n+1,k
G
= V n+1,k−1
G
G n+1,k = G n+1,k−1 +(V n+1,k
G
+ V n+1,k−1
G
)△t/4N, (32)
+(F r (Ĝn+1,k )+F r (G n+1,k−1 ))△t/4N,
(33)
+ V n+1,k−1
G
)△t/4N, (34)
enddo;
and let V n+1
G
= Vn+1,N G
, G n+1 = G n+1,N ; and set u n+1 = u n+5/6 ,
ω n+1 = ω n+5/6 .
In Algorithm 1, we have t n+s =(n + s)△t, W n+1
g = W 0,h g 0,h(t n+1 ), Λ n+s
0,h =
Λ 0,h (t n+s ), B n+s
h
is the region occupied by the particle centered at G n+s ,
and F r is the combination of a short range repulsion force which prevents the
particle/particle and particle/wall penetration (see, e.g., [GPHJ99, GPH + 01])
and the force obtained from the adhesive dynamics algorithm for the cell
adhesion. Finally, α and β verify α + β =1;wehavechosenα =1andβ =0
in the numerical simulations discussed later.
The degenerated quasi-Stokes problem (21) is solved by a preconditioned
conjugate gradient method introduced in [GPP98], in which discrete elliptic
problems from the preconditioning are solved by a matrix-free fast solver from
FISHPAK by Adams et al. in [ASS80]. The advection problem (22) for the
velocity field is solved by a wave-like equation method as in [DG97]. The
problem (24) is a classical discrete elliptic problem which can be solved by
the same matrix-free fast solver. To enforce the rigid body motion inside
the region occupied by the particles, we have applied the conjugate gradient
method discussed in [PG02, PG05].
4 Numerical Results and Discussion
We consider the detachment of 20 cells in shear flow as the test problem for
cell adhesion model at the initial stage of the adhesion. The computational
domain is Ω =(0, 23)×(0, 10) (unit: 10 µm). Cells have the shape of an ellipse,