This example illustrates simulation of flow in a simple bifurcation network consisting of both arterial and venous trees. The geometries of the trees are non-identical. Also illustrated is the discretisation of the physical domain with uniform spatial steps. If the discretisation of the element (1D arc) is performed with equal xi spacing, one would end up with non-uniform grid spacing in the physical space, specially in the vicinity of sharp changes within and between the elements. The difference in the two approaches is depicted in figures 1 & 2. As we implement fully explicit finite difference scheme, which is conditionally stable, the ratio between the time step and the spatial step plays a pivotal role in the stability and convergence of the numerical method. It is therefore important to have uniform grid spacing to ensure convergence of the algorithm as well as the accuracy of the results for a given time step. Discretisation with uniform arc length spacing is done using the Trapezoidal rule.

The pressure at the arterial entry is linearly increased from 12.5 to 14.0 kPa over a period of 0.1 seconds while the pressure at the venous exit is held at 12.5 kPa constant. The points where the transition occurs from arteries to veins are treated as reservoirs and represented by a simple lumped parameter model with a capacitance term and two resistance terms(up-stream & down-stream).

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NOTE : For the grid points in the vicinity of the transition points, NXQ is used to determine neighbouring grid points instead of CONECT in the XPFD30 (fe30.f)

Fig. 1 Discretisation based on uniform xi spacing Fig. 2 Discretisation based on uniform arc spacing (approximately 1.0 mm) Fig. 3 Pressure variation