Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 182

For block-structured and composite grids, this structure is
preserved within each block, and the solvers for regular structured
grids may be used. For block-structured grids, the coeficient matrix
remains sparse, but it no longer has banded structure. For a 2D grid
of quadrilaterals and approximations that use only the four nearest
neighbor nodes, there are only ive non-zero coeficients that lie
within a certain range of the main diagonal but not necessarily on
deinite diagonals. A different type of iterative solver must be used
for such matrices.

5.2.6 Finite-Volume Method

Finite-volume method basically consists of the integral form of the
conversation equation as follows:
±

± ±

( 8 (5.64)
The solution area is subdivided into a inite number of small
control volume (CVs) by a grid, compared with FDM. Control volume
has volume boundaries, not the computational node. The integral
conservation equation (5.76) is applied to not only each CVs but
also whole solution area. If whole CVs equations are summed up,
the global conservation equation is obtained, since surface integrals
over inner CV faces cancel out. The global conservation is built into
the method automatically because of its integral form and this is the
advantage of FVM.
Figures 5.12 and 5.13 are typical 2D and 3D control volumes,
respectively. The CV surface consists of four (in 2D) or six (3D) plane
faces, denoted by lower case index corresponding to their direction
(e, w, n, s, b, and t) with respect to the central point P . The net lux
thorough CV boundary is the sum of integrals over the four (2D) or
six (3D) CV faces.
± ¤ ±

SG – G –

vn dS grad n dS qd

S S

fdS fds (5.65)

S S

k

k

Here f is the component of the convective ( ρ G v ⋅ n ) or the diffusive
(Γgrad G

⋅

n) lux vector in the direction normal to CV face. For
maintaining conservation, it is important that CVs do not overlap.
Each CV face is independent on the two CVs, which lie on either side of
it. To calculate the surface integral in Eq. 5.77, the integral f needs to
be known on the surface S e This information is not available, as only
the nodal (CV center) values of G are calculated, so an approximation
must be introduced.