Typically 1D arterial networks are made up of a connection of different base units: segments, bifurcations and merging junctions. The input format in the PulseWaveSolver means these connections are handle naturally from the mesh topology; hence care must be taken when designing the 1D domain. The figure below outlines the structure of a bifurcation, which is a common reoccurring structure in the vasculature.
To represent this topology in the xml file we specify the following vertices under the section
VERTEX (the extents are: -100 ≥ x ≤ 100 and -100 ≥ y ≤ 100 )
The elements from these vertices are then constructed under the section
ELEMENT by defining
The composites, which represent groups of elements and boundary regions are defined under
Each of the segments can be then represented under the section
We will use the different domains later to define variable material properties and cross-sectional areas.
Methodthe time-stepping method.
Variantthe variant to the method.
Orderthe order of the method.
FreeParametersany free parameters required.
The PulseWaveSolver is specified through the
EquationType option in the session file. This
can be set as follows:
Projection: Only a discontinuous projection can be specified using the following option:
Discontinuousfor a discontinous Galerkin (DG) projection.
Truereturns the wave speed and both characteristics
The following parameters can be specified in the
PARAMETERS section of the session file.
TimeStepis the time-step size;
FinTimeis the final physical time at which the simulation will stop;
NumStepsis the equivalent of
FinTimebut instead of specifying the physical final time the number of time-steps is defined;
IO_CheckStepssets the number of steps between successive checkpoint files;
IO_InfoStepssets the number of steps between successive info stats are printed to screen;
rhodensity of the fluid. Default value = 1.0;
nuePoisson’s ratio. Default value = 0.5 ;
pextexternal pressure. Default value = 0;
poutoutflow pressure to the venous system for the terminal boundary conditions. Default value = 0;
h0wall thickness. Default value = 1.0;
In this section we can specify the boundary conditions for our problem. First we need to define
the variables under the section
The composites that we want to apply out boundary conditions then need to be defined
BOUNDARYREGIONS, for example if we had three composites (C, C and
C) that correspond to three vertices of the computational mesh we would define:
Finally we can specify the boundary conditions on the regions specified under
The Pulse Wave Solver comes with a number of boundary conditions that are unique to this solver. Boundary conditions must be provided for both the area and velocity at the inlets and outlets of the domain. Examples of the different boundary conditions will be provided in the following.
Inlet boundary condition:
The inlet condition may be specified algebraically in four different ways: as an area
A-inflow); a velocity profile (
U-inflow); a volume flux (
Q-inflow); or by
prescribing the forward characteristic (
TimeDependent). When prescribing a volume flux,
it must be specified in the input file via the area, as illustrated below. Note that
u = 1.0.
Terminal boundary conditions:
At the outlets of the domain there are four possible boundary conditions: reflection
Terminal), terminal resistance
R-terminal, Two element windkessel (CR)
three element windkessel (RCR)
RCR-terminal. An example of the outflow boundary
condition of the RCR terminal is given below
RT is the total peripheral resistance used in the the
The following functions can be specified inside the
CONDITIONS section of the session file:
MaterialProperties: specifies β for each domain.
A_0: specifies A0 for each domain as used in the tube law.
Viscoelasticity: specifies Γ for each domain. Defaults to zero for every artery if not included.
StrainStiffening: specifies α for each domain for Eq. (13.3). Defaults to 0.5 for every artery if not included.
AdvectionVelocity: specifies the advection velocity v.
InitialConditions: specifies the initial condition for unsteady problems.
Forcing: specifies the forcing function f
As an example to specify the material properties for each domain in the previous bifurcation example we would enter:
The values of
beta are used in the pressure-area relationship (Eq. (13.2)).