Biomedical Engineering Reference
In-Depth Information
consider Dirichlet boundary conditions (i.e., the values u p in boundary voxels are
not considered as unknown), then i = i l ,...,i r , j = j l ,...,j r , k = k l ,...,k r ,
where i r
N 1 2, j r
N 2 2, k r
N 3 2. We define the
i l
j l
k l
N 1
space discretization step h =
, and for simplicity we assume that voxels have
cubic shape. For every co-volume p , the set C p
consists of
6 neighbours, west u i− 1 ,j,k , east u i +1 ,j,k , south u i,j− 1 ,k , north u i,j +1 ,k , bottom
u i,j,k− 1 , and top u i,j,k +1 , and the set
= { w, e, s, n, b, t }
N p consists of 24 tetrahedra.
In every discrete time step n =1 ,...,N and for every i, j, k , we compute
the absolute value of gradient
u n− 1
T
|∇
|
on these 24 tetrahedra. We denote by
G z,l
C p the square of the gradient on the tetrahedra crossing
thewest, east, south, north, bottom, and top co-volume faces. If we define (omitting
upper index n
i,j,k ,l =1 ,..., 4 ,z
1)
s i,j,k
= u i,j,k + u i− 1 ,j,k + u i,j− 1 ,k + u i− 1 ,j− 1 ,k +
+
u i,j,k− 1 + u i− 1 ,j,k− 1 + u i,j− 1 ,k− 1 + u i− 1 ,j− 1 ,k− 1 ) / 8 ,
the value at the left-south-bottom NDF node of the co-volume, then for the west
face we get
u i,j,k
2
s i,j,k +1
2
u i− 1 ,j,k
h
s i,j,k
G w, 1
i,j,k
=
+
+
h
u i,j,k + u i− 1 ,j,k
2 ,
s i,j,k +1
s i,j,k
h
u i,j,k
2
s i,j +1 ,k +1
2
u i− 1 ,j,k
h
s i,j,k +1
G w, 2
i,j,k
=
+
+
h
s i,j +1 ,k +1 + s i,j,k +1 u i,j,k u i 1 ,j,k
h
2 ,
(24)
u i,j,k
2
s i,j +1 ,k +1
2
u i− 1 ,j,k
h
s i,j +1 ,k
G w, 3
i,j,k
=
+
+
h
s i,j +1 ,k +1 + s i,j +1 ,k
2 ,
u i,j,k
u i− 1 ,j,k
h
u i,j,k
2
s i,j +1 ,k
2
u i− 1 ,j,k
h
s i,j,k
G w, 4
i,j,k
=
+
+
h
u i,j,k + u i− 1 ,j,k
2 ,
s i,j +1 ,k
s i,j,k
h
and correspondingly we get all G z,l
i,j,k for the further co-volume faces.
In the same way, but only once at the beginning of the algorithm, we compute
values G σ,z,l
C p , changing u by I σ in the previous expressions,
and we apply function g to all these values to get discrete values of g T .
i,j,k , l =1 ,..., 4 ,z
 
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