13.3 Session file configuration

13.3.1 Pulse Wave Solver mesh connectivity

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 the section COMPOSITE by

Each of the segments can be then represented under the section DOMAIN by

We will use the different domains later to define variable material properties and cross-sectional areas.

13.3.2 Time Integration Scheme

• Method the time-stepping method.
• Variant the variant to the method.
• Order the order of the method.
• FreeParameters any free parameters required.

13.3.3 Session Info

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:
  • Discontinuous for a discontinous Galerkin (DG) projection.
• UpwindTypePulse:
  • UpwindPulse
• OutputExtraFields:
  • True returns the wave speed and both characteristics
• PressureArea:
  • Beta for Eq. (13.2)
  • Empirical for Eq. (13.3)

13.3.4 Parameters

The following parameters can be specified in the PARAMETERS section of the session file.

• TimeStep is the time-step size;
• FinTime is the final physical time at which the simulation will stop;
• NumSteps is the equivalent of FinTime but instead of specifying the physical final time the number of time-steps is defined;
• IO_CheckSteps sets the number of steps between successive checkpoint files;
• IO_InfoSteps sets the number of steps between successive info stats are printed to screen;
• rho density of the fluid. Default value = 1.0;
• nue Poisson’s ratio. Default value = 0.5 ;
• pext external pressure. Default value = 0;
• pout outflow pressure to the venous system for the terminal boundary conditions. Default value = 0;
• h0 wall thickness. Default value = 1.0;

13.3.5 Boundary conditions

In this section we can specify the boundary conditions for our problem. First we need to define the variables under the section VARIABLES.

The composites that we want to apply out boundary conditions then need to be defined in the BOUNDARYREGIONS, for example if we had three composites (C[1], C[4] and C[8]) that correspond to three vertices of the computational mesh we would define:

Finally we can specify the boundary conditions on the regions specified under BOUNDARYREGIONS.

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 variation (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) CR-terminal, and three element windkessel (RCR) RCR-terminal. An example of the outflow boundary condition of the RCR terminal is given below

Where RT is the total peripheral resistance used in the the R-terminal, CR-terminal and RCR-terminal models

13.3.6 Functions

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)).