ACME 2011 Proceedings of the 19 UK National Conference of the ...

discontinuities are permitted to share the same location at the start of the branch in human arterial tree [4].
2 METHODOLOGY
The 1D first order hyperbolic governing equations of an elastic cylindrical vessel of incompressible and
Newtonian fluid, based on the mass and momentum conservation laws can be written as [4]:
where A is the cross-sectional area of the vessel, u is mean velocity of the fluid, p is the internal pressure
of the cross-section , is the blood density (assumed to be constant due to incompressibility of blood) and
is the shear stress where is the dynamic viscosity (constant), r is the radial direction in
3D cylindrical coordinates and R is the vessel radius. In the shear stress term, the inertia component will
be neglect while the viscous component is assumed as a fully developed Newtonian steady flow with a
parabolic velocity profile (Poiseuille flow) which can be expressed as:
In order to close the system, equations (1) and (2) have to be supplemented with a commonly used
pressure-area relation [7] which takes the form:
where is the external pressure from the surrounding tissues, is the unstressed cross-sectional area
of the vessel, is the material properties of the elastic vessel, h is the vessel wall thickness, E is the
Young’s modulus and is the Poisson’s ratio.
The explicit semi-discrete form can then be achieved by using the standard second-order Taylor series
expansion and the partial differential equation is used to replace the time derivatives with spatial
derivatives which can be shown as:
where U, F and S are vectors of the primitive variables, conservative variables and source term
respectively. The semi-discrete form will then be further discretised using the LCG finite element method.
As with the global Galerkin method, each of the variables is approximated by the standard finite element
spatial discretisation. The interface flux can be reintroduced via integration by parts and contained
contributions from convective component. In order to achieve local conservation, a simple averaging
post-processing procedure is introduced at each time step. The post-processing procedure uses the nodal
values from the continuous solution obtained from the previous time-step to provide an accurate interface
flux and also establishes connectivity between elements at the next time-step. Full discrete form of the
governing equations is shown in [4, 5, 6].
,
(1)
(2)
(3)
(4)
(5)
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