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BloodFlowTrixi.jl

BloodFlowTrixi.jl is a Julia package that implements one-dimensional (1D) and two-dimensional (2D) blood flow models for arterial circulation. These models are derived from the Navier-Stokes equations and were developed as part of my PhD research in applied mathematics, focusing on cardiovascular pathologies such as aneurysms and stenoses.

Description

This package provides:

  • 1D Blood Flow Model: This model describes blood flow along compliant arteries in a single spatial dimension. It was derived under the assumption of axisymmetric flow and accounts for arterial compliance, inertia, and frictional losses.More details about this model can be found in my corresponding publication: [Article 1D]

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial x}(Q) &= 0 \\ \frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}\left(\frac{Q^2}{A} + A P(a)\right) &= P(a) \frac{\partial A}{\partial x} - 2 \pi R k \frac Q {A}\\ P(a) &= P_{ext} + \frac{Eh\sqrt{\pi}}{1-\xi^2}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{\frac{A}{\pi}} \end{aligned}\right.\]

  • 2D Blood Flow Model: The 2D model extends the Navier-Stokes equations under the thin-artery assumption, allowing for simulations in complex arterial geometries using curvilinear coordinates. It captures both longitudinal and angular dynamics, making it more accurate than classical 1D models while being less computationally expensive than full 3D models.

    This model is described in detail in: [Article 2D]

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial \theta}\left( \frac{Q_{R\theta}}{A} \right) + \frac{\partial}{\partial s}(Q_s) &= 0 \\ \frac{\partial Q_{R\theta}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta}^2}{2A^2} + A P(a)\right) + \frac{\partial}{\partial s}\left( \frac{Q_{R\theta}Q_s}{A} \right) &= P(a) \frac{\partial A}{\partial \theta} - 2 R k \frac{Q_{R\theta}}{A} + \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s^2}{A} \\ \frac{\partial Q_{s}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta} Q_s}{A^2} \right) + \frac{\partial}{\partial s}\left( \frac{Q_s^2}{A} - \frac{Q_{R\theta}^2}{2A^2} + A P(a) \right) &= P(a) \frac{\partial A}{\partial s} - R k \frac{Q_s}{A} - \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s Q_{R\theta}}{A^2} \\ P(a) &= P_{ext} + \frac{Eh}{\sqrt{2}\left(1-\xi^2\right)}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{2A} \end{aligned}\right.\]

Both models were designed to be used with Trixi.jl, a flexible and high-performance framework for solving systems of conservation laws using the Discontinuous Galerkin (DG) method.

Features

  • 1D and 2D models for arterial blood flow.
  • Derived from the Navier-Stokes equations with appropriate assumptions for compliant arteries.
  • To be used with Trixi.jl for DG-based numerical simulations.
  • Support for curvilinear geometries and compliant wall dynamics.

Installation

To install BloodFlowTrixi.jl, use the following commands in Julia:

julia> ]
pkg> add Trixi
pkg> add BloodFlowTrixi

Future Plans

short term

  • Add second order 1D model.
  • Design prim variables for 1D and 2D models.
  • Add proper tests for 1D and 2D models.
  • Add 3D representations of the solutions for 1D and 2D models.
  • Design easy to use interfaces for users to define their own initial and boundary conditions and source terms.

long term

  • Add 3D fluid-structure interaction models for complex arterial geometries.
  • Design support for artery networks and simulate vascular networks using the 2D and 1D model.
  • Autodiff support for 1D and 2D models for parameter optimization.

License

This package is licensed under the MIT license.

Acknowledgments

This package was developed as part of my PhD research in applied mathematics, focusing on mathematical modeling and numerical simulation of blood flow in arteries. Special thanks to the developers of Trixi.jl, whose framework was invaluable in implementing and testing these models.

BloodFlowTrixi.BloodFlowTrixi — Module
Package BloodFlowTrixi v0.1.5

This package implements 1D and 2D blood flow models for arterial circulation using Trixi.jl, enabling efficient numerical simulation and analysis.

Docs under https://yolhan83.github.io/BloodFlowTrixi.jl

source
BloodFlowTrixi.BloodFlowEquations1D — Type
BloodFlowEquations1D(;h,rho=1.0,xi=0.25,nu=0.04)

Blood Flow equations in one space dimension. This model describes the dynamics of blood flow along a compliant artery using one-dimensional equations derived from the Navier-Stokes equations. The equations account for conservation of mass and momentum, incorporating the effect of arterial compliance and frictional losses.

The governing equations are given by

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial x}(Q) &= 0 \\ \frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}\left(\frac{Q^2}{A} + A P(a)\right) &= P(a) \frac{\partial A}{\partial x} - 2 \pi R k \frac Q {A}\\ P(a) &= P_{ext} + \frac{Eh\sqrt{\pi}}{1-\xi^2}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{\frac{A}{\pi}} \end{aligned}\right.\]

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BloodFlowTrixi.BloodFlowEquations2D — Type
BloodFlowEquations2D(;h,rho=1.0,xi=0.25)

Defines the two-dimensional blood flow equations derived from the Navier-Stokes equations in curvilinear coordinates under the thin-artery assumption. This model describes the dynamics of blood flow along a compliant artery in two spatial dimensions (s, θ).

Parameters

  • h::T: Wall thickness of the artery.
  • rho::T: Fluid density (default 1.0).
  • xi::T: Poisson's ratio (default 0.25).
  • nu::T: Viscosity coefficient.

The governing equations account for conservation of mass and momentum, incorporating the effects of arterial compliance, curvature, and frictional losses.

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial \theta}\left( \frac{Q_{R\theta}}{A} \right) + \frac{\partial}{\partial s}(Q_s) &= 0 \\ \frac{\partial Q_{R\theta}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta}^2}{2A^2} + A P(a)\right) + \frac{\partial}{\partial s}\left( \frac{Q_{R\theta}Q_s}{A} \right) &= P(a) \frac{\partial A}{\partial \theta} - 2 R k \frac{Q_{R\theta}}{A} + \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s^2}{A} \\ \frac{\partial Q_{s}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta} Q_s}{A^2} \right) + \frac{\partial}{\partial s}\left( \frac{Q_s^2}{A} - \frac{Q_{R\theta}^2}{2A^2} + A P(a) \right) &= P(a) \frac{\partial A}{\partial s} - R k \frac{Q_s}{A} - \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s Q_{R\theta}}{A^2} \\ P(a) &= P_{ext} + \frac{Eh}{\sqrt{2}\left(1-\xi^2\right)}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{2A} \end{aligned}\right.\]

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Trixi.DissipationLocalLaxFriedrichs — Method
(dissipation::Trixi.DissipationLocalLaxFriedrichs)(u_ll, u_rr, orientation_or_normal_direction, eq::BloodFlowEquations1D)

Calculates the dissipation term using the Local Lax-Friedrichs method.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation_or_normal_direction: Orientation or normal direction.
  • eq: Instance of BloodFlowEquations1D.

Returns

Dissipation vector.

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BloodFlowTrixi.boundary_condition_outflow — Method
boundary_condition_outflow(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements the outflow boundary condition, assuming that there is no reflection at the boundary.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux.

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BloodFlowTrixi.boundary_condition_outflow — Method
boundary_condition_outflow(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations2D)

Applies an outflow boundary condition for the 2D blood flow model without reflecting any flux.

Parameters

  • u_inner: Inner state vector at the boundary.
  • orientation_or_normal: Orientation index or normal vector indicating the boundary direction.
  • direction: Index indicating the spatial direction (1 for ( \theta )-direction, otherwise ( s )-direction).
  • x: Position vector at the boundary.
  • t: Time value.
  • surface_flux_function: Function to compute the surface flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Boundary flux as an SVector.

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BloodFlowTrixi.boundary_condition_outflow — Method
boundary_condition_outflow(u_inner, orientation_or_normal, x, t, surface_flux_function, eq::BloodFlowEquations2D)

Applies an outflow boundary condition for the 2D blood flow model without reflecting any flux. This version does not use a specific direction parameter.

Parameters

  • u_inner: Inner state vector at the boundary.
  • orientation_or_normal: Orientation index or normal vector indicating the boundary direction.
  • x: Position vector at the boundary.
  • t: Time value.
  • surface_flux_function: Function to compute the surface flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Boundary flux as an SVector.

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BloodFlowTrixi.boundary_condition_pressure_in — Method
boundary_condition_pressure_in(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a pressure inflow boundary condition where the inflow pressure varies with time.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the boundary direction.
  • x: Position vector.
  • t: Time scalar.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux with inflow pressure specified by:

\[P_{in} = \begin{cases} 2 \times 10^4 \sin^2(\pi t / 0.125) & \text{if } t < 0.125 \\ 0 & \text{otherwise} \end{cases}\]

The corresponding inflow area A_{in} is computed using the inverse pressure relation, and the boundary state is constructed accordingly.

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BloodFlowTrixi.boundary_condition_pressure_in — Method
boundary_condition_pressure_in(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations2D)

Applies an inflow boundary condition with a prescribed pressure for the 2D blood flow model.

Parameters

  • u_inner: Inner state vector at the boundary.
  • orientation_or_normal: Orientation index or normal vector indicating the boundary direction.
  • direction: Index indicating the spatial direction (1 for ( \theta )-direction, otherwise ( s )-direction).
  • x: Position vector at the boundary.
  • t: Time value.
  • surface_flux_function: Function to compute the surface flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Boundary flux as an SVector.

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BloodFlowTrixi.boundary_condition_pressure_in — Method
boundary_condition_pressure_in(u_inner, normal, x, t, surface_flux_function, eq::BloodFlowEquations2D)

Applies an inflow boundary condition with a prescribed pressure for the 2D blood flow model. This version does not use a specific direction parameter.

Parameters

  • u_inner: Inner state vector at the boundary.
  • normal: Normal vector indicating the boundary direction.
  • x: Position vector at the boundary.
  • t: Time value.
  • surface_flux_function: Function to compute the surface flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Boundary flux as an SVector.

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BloodFlowTrixi.boundary_condition_slip_wall — Method
boundary_condition_slip_wall(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a slip wall boundary condition where the normal component of velocity is reflected.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux at the slip wall.

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BloodFlowTrixi.boundary_condition_slip_wall — Method
boundary_condition_slip_wall(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations2D)

Applies a slip-wall boundary condition for the 2D blood flow model by reflecting the normal component of the velocity at the boundary.

Parameters

  • u_inner: Inner state vector at the boundary.
  • orientation_or_normal: Orientation index or normal vector indicating the boundary direction.
  • direction: Index indicating the spatial direction (1 for ( \theta )-direction, otherwise ( s )-direction).
  • x: Position vector at the boundary.
  • t: Time value.
  • surface_flux_function: Function to compute the surface flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Boundary flux as an SVector.

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BloodFlowTrixi.curvature — Method
curvature(x)

Returns a constant curvature for the 2D blood flow model.

Parameters

  • x: Position vector.

Returns

Curvature as a scalar.

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BloodFlowTrixi.flux_nonconservative — Method
flux_nonconservative(u_ll, u_rr, normal, eq::BloodFlowEquations2D)

Computes the non-conservative flux for the 2D blood flow model based on a normal vector.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • normal: Normal vector indicating the direction of the flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Non-conservative flux vector.

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BloodFlowTrixi.flux_nonconservative — Method
flux_nonconservative(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Computes the non-conservative flux for the model, used for handling discontinuities in pressure.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Non-conservative flux vector.

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BloodFlowTrixi.flux_nonconservative — Method
flux_nonconservative(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations2D)

Computes the non-conservative flux for the 2D blood flow model based on the orientation.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Direction index for the flux (1 for ( \theta )-direction, otherwise ( s )-direction).
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Non-conservative flux vector.

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BloodFlowTrixi.friction — Method
friction(u, x, eq::BloodFlowEquations1D)

Calculates the friction term for the blood flow equations, which represents viscous resistance to flow along the artery wall.

Parameters

  • u: State vector containing cross-sectional area and flow rate.
  • x: Position along the artery.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Friction coefficient as a scalar.

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BloodFlowTrixi.friction — Method
friction(u, x, eq::BloodFlowEquations2D)

Computes the friction term for the 2D blood flow model.

Parameters

  • u: State vector.
  • x: Position vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Friction term as a scalar.

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BloodFlowTrixi.get3DData — Method
 get3DData(eq::BloodFlowEquations2D,curve::F1,er::F2,semi,sol,time_index ::Int = 1;vtk ::Bool=false,out ::T="./datas") where {T<:AbstractString,F1<:Function,F2<:Function}

Generates 3D spatial data from a 2D blood flow model for visualization. This function extracts unique node coordinates, computes relevant flow parameters, and generates a 3D representation of the arterial domain using cylindrical coordinates. Optionally, it can export the data in VTK format.

Parameters

  • eq::BloodFlowEquations1D: Instance of BloodFlowEquations1D representing the blood flow model.
  • curve::F1: Function representing the curve of the vessel (s)->curve(s).
  • er::F2: Function representing the radial vector (theta,s)->er(theta,s).
  • semi: Semi-discretization structure containing mesh and numerical information.
  • sol: Solution array containing the numerical state variables.
  • time_index::Int=1: Time step index for extracting the solution (default: 1).
  • vtk::Bool=false: Whether to export data to VTK format (default: false).
  • out::T="./datas": Output directory for VTK files (default: "./datas").

Returns

Named tuple containing:

  • x: X-coordinates of the generated 3D points.
  • y: Y-coordinates of the generated 3D points.
  • z: Z-coordinates of the generated 3D points.
  • A: Cross-sectional areas at each point.
  • wtheta: Flow angular velocity at each point.
  • ws: Flow axial velocity at each point.
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BloodFlowTrixi.get3DData — Method
get3DData(eq::BloodFlowEquations2D,curve::F1,tanj::F2,nor::F3,semi,sol,time_index ::Int = 1;vtk ::Bool=false,out ::T="./datas") where {T<:AbstractString,F1<:Function,F2<:Function,F3<:Function}

Generates 3D spatial data from a 2D blood flow model for visualization. This function extracts unique node coordinates, computes relevant flow parameters, and generates a 3D representation of the arterial domain using cylindrical coordinates. Optionally, it can export the data in VTK format.

Parameters

  • eq::BloodFlowEquations1D: Instance of BloodFlowEquations1D representing the blood flow model.
  • curve::F1: Function representing the curve of the vessel (s)->curve(s).
  • tanj::F2: Function representing the tanjent vector (s)->tanj(s).
  • nor::F2: Function representing the normal vector (s)->nor(s).
  • semi: Semi-discretization structure containing mesh and numerical information.
  • sol: Solution array containing the numerical state variables.
  • time_index::Int=1: Time step index for extracting the solution (default: 1).
  • vtk::Bool=false: Whether to export data to VTK format (default: false).
  • out::T="./datas": Output directory for VTK files (default: "./datas").

Returns

Named tuple containing:

  • x: X-coordinates of the generated 3D points.
  • y: Y-coordinates of the generated 3D points.
  • z: Z-coordinates of the generated 3D points.
  • A: Cross-sectional areas at each point.
  • wtheta: Flow angular velocity at each point.
  • ws: Flow axial velocity at each point.
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BloodFlowTrixi.get3DData — Method
get3DData(eq::BloodFlowEquations1D,semi,sol,time_index ::Int = 1;theta_disc ::Int = 32,vtk ::Bool=false,out ::T="./datas") where T<:AbstractString

Generates 3D spatial data from a 1D blood flow model for visualization. This function extracts unique node coordinates, computes relevant flow parameters, and generates a 3D representation of the arterial domain using cylindrical coordinates. Optionally, it can export the data in VTK format.

Parameters

  • eq::BloodFlowEquations1D: Instance of BloodFlowEquations1D representing the blood flow model.
  • semi: Semi-discretization structure containing mesh and numerical information.
  • sol: Solution array containing the numerical state variables.
  • time_index::Int=1: Time step index for extracting the solution (default: 1).
  • theta_disc::Int=32: Number of angular discretization points for the cylindrical representation (default: 32).
  • vtk::Bool=false: Whether to export data to VTK format (default: false).
  • out::T="./datas": Output directory for VTK files (default: "./datas").

Returns

Named tuple containing:

  • x: X-coordinates of the generated 3D points.
  • y: Y-coordinates of the generated 3D points.
  • z: Z-coordinates of the generated 3D points.
  • A: Cross-sectional areas at each point.
  • w: Flow velocities at each point.
  • P: Pressure values at each point.

Notes

  • The function first extracts unique spatial positions from the mesh.
  • The blood flow variables (A, Q, E, A0) are obtained from the solution array.
  • Pressure is computed using the pressure function.
  • A cylindrical coordinate transformation is applied to represent the vessel cross-section.
  • If vtk is true, the function writes the data to VTK format using vtk_grid.
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BloodFlowTrixi.get3DData — Method
get3DData(eq::BloodFlowEquations2D,semi,sol,time_index ::Int = 1;vtk ::Bool=false,out ::T="./datas") where {T<:AbstractString,F1<:Function,F2<:Function,F3<:Function}

Generates 3D spatial data from a 2D blood flow model for visualization. This will use a straight vessel. This function extracts unique node coordinates, computes relevant flow parameters, and generates a 3D representation of the arterial domain using cylindrical coordinates. Optionally, it can export the data in VTK format.

Parameters

  • eq::BloodFlowEquations1D: Instance of BloodFlowEquations1D representing the blood flow model.
  • semi: Semi-discretization structure containing mesh and numerical information.
  • sol: Solution array containing the numerical state variables.
  • time_index::Int=1: Time step index for extracting the solution (default: 1).
  • vtk::Bool=false: Whether to export data to VTK format (default: false).
  • out::T="./datas": Output directory for VTK files (default: "./datas").

Returns

Named tuple containing:

  • x: X-coordinates of the generated 3D points.
  • y: Y-coordinates of the generated 3D points.
  • z: Z-coordinates of the generated 3D points.
  • A: Cross-sectional areas at each point.
  • wtheta: Flow angular velocity at each point.
  • ws: Flow axial velocity at each point.
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BloodFlowTrixi.initial_condition_simple — Method
initial_condition_simple(x, t, eq::BloodFlowEquations1D; R0=2.0)

Generates a simple initial condition with a specified initial radius R0.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.
  • R0: Initial radius (default: 2.0).

Returns

State vector with zero initial area perturbation, zero flow rate, constant elasticity modulus, and reference area computed as A_0 = \pi R_0^2.

This initial condition is suitable for basic tests without complex dynamics.

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BloodFlowTrixi.initial_condition_simple — Method
initial_condition_simple(x, t, eq::BloodFlowEquations2D; R0=2.0)

Defines a simple initial condition for the 2D blood flow model.

Parameters

  • x: Position vector.
  • t: Initial time.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.
  • R0: Initial radius (default is 2.0).

Returns

State vector as an SVector.

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BloodFlowTrixi.inv_pressure — Method
inv_pressure(p, u, eq::BloodFlowEquations1D)

Computes the inverse relation of pressure to cross-sectional area.

Parameters

  • p: Pressure.
  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Cross-sectional area corresponding to the given pressure.

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BloodFlowTrixi.inv_pressure — Method
inv_pressure(p, u, eq::BloodFlowEquations2D)

Computes the inverse of the pressure function for the 2D blood flow model.

Parameters

  • p: Pressure value.
  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Inverse pressure as a scalar.

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BloodFlowTrixi.pressure — Method
pressure(u, eq::BloodFlowEquations1D)

Computes the pressure given the state vector based on the compliance of the artery.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Pressure as a scalar.

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BloodFlowTrixi.pressure — Method
pressure(u, eq::BloodFlowEquations2D)

Computes the pressure for the 2D blood flow model.

Parameters

  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Pressure as a scalar.

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BloodFlowTrixi.pressure_der — Method
pressure_der(u, eq::BloodFlowEquations1D)

Computes the derivative of pressure with respect to cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Derivative of pressure.

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BloodFlowTrixi.pressure_der — Method
pressure_der(u, eq::BloodFlowEquations2D)

Computes the derivative of the pressure with respect to the cross-sectional area for the 2D blood flow model.

Parameters

  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Derivative of pressure as a scalar.

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BloodFlowTrixi.radius — Method
radius(u, eq::BloodFlowEquations1D)

Computes the radius of the artery based on the cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Radius as a scalar.

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BloodFlowTrixi.radius — Method
radius(u, eq::BloodFlowEquations2D)

Computes the radius based on the cross-sectional area for the 2D blood flow model.

Parameters

  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Radius as a scalar.

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BloodFlowTrixi.source_term_simple — Method
source_term_simple(u, x, t, eq::BloodFlowEquations1D)

Computes a simple source term for the blood flow model, focusing on frictional effects.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector where:

  • s_1 = 0 (no source for area perturbation).
  • s_2 represents the friction term given by s_2 = \frac{2 \pi k Q}{R A}.

Friction coefficient k is computed using the friction function, and the radius R is obtained using the radius function.

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BloodFlowTrixi.source_term_simple — Method
source_term_simple(u, x, t, eq::BloodFlowEquations2D)

Computes a simple source term for the 2D blood flow model, including friction and curvature effects.

Parameters

  • u: State vector.
  • x: Position vector.
  • t: Time value.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Source term as an SVector.

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Trixi.cons2entropy — Method
Trixi.cons2entropy(u, eq::BloodFlowEquations1D)

Converts the conserved variables to entropy variables.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Entropy variable vector.

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Trixi.cons2entropy — Method
Trixi.cons2entropy(u, eq::BloodFlowEquations2D)

Converts the conservative variables to entropy variables for the 2D blood flow model.

Parameters

  • u: State vector in conservative form.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

State vector in entropy form as an SVector.

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Trixi.cons2prim — Method
Trixi.cons2prim(u, eq::BloodFlowEquations1D)

Converts the conserved variables to primitive variables.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Primitive variable vector.

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Trixi.cons2prim — Method
Trixi.cons2prim(u, eq::BloodFlowEquations2D)

Converts the conservative variables to primitive variables for the 2D blood flow model.

Parameters

  • u: State vector in conservative form.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

State vector in primitive form as an SVector.

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Trixi.entropy — Method
Trixi.entropy(u, eq::BloodFlowEquations1D)

Computes the entropy of the system for the given state vector.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Entropy as a scalar value.

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Trixi.entropy — Method
Trixi.entropy(u, eq::BloodFlowEquations2D)

Computes the entropy for the 2D blood flow model.

Parameters

  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Entropy as a scalar.

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Trixi.flux — Method
Trixi.flux(u, normal, eq::BloodFlowEquations2D)

Computes the flux vector for the conservation laws of the 2D blood flow model based on a normal vector.

Parameters

  • u: State vector.
  • normal: Normal vector indicating the direction of the flux.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Flux vector as an SVector.

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Trixi.flux — Method
Trixi.flux(u, orientation::Integer, eq::BloodFlowEquations1D)

Computes the flux vector for the conservation laws of the blood flow model.

Parameters

  • u: State vector.
  • orientation::Integer: Orientation index for flux computation.
  • eq: Instance of BloodFlowEquations1D.

Returns

Flux vector as an SVector.

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Trixi.flux — Method
Trixi.flux(u, orientation::Integer, eq::BloodFlowEquations2D)

Computes the flux vector for the conservation laws of the 2D blood flow model in either the ( \theta )-direction or the ( s )-direction, depending on the specified orientation.

Parameters

  • u: State vector.
  • orientation::Integer: Direction of the flux computation (1 for ( \theta )-direction, otherwise ( s )-direction).
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Flux vector as an SVector.

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Trixi.initial_condition_convergence_test — Method
initial_condition_convergence_test(x, t, eq::BloodFlowEquations1D)

Generates a smooth initial condition for convergence tests of the blood flow equations.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Initial condition state vector with zero initial area perturbation, sinusoidal flow rate, a constant elasticity modulus, and reference area.

Details

The returned initial condition has:

  • Zero perturbation in area (a = 0).
  • A sinusoidal flow rate given by Q = sin(\pi x t).
  • A constant elasticity modulus E.
  • A reference cross-sectional area A_0 = \pi R_0^2 for R_0 = 1.

This initial condition can be used to verify the accuracy and stability of numerical solvers.

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Trixi.max_abs_speed_naive — Method
Trixi.max_abs_speed_naive(u_ll, u_rr, normal, eq::BloodFlowEquations2D)

Computes the maximum absolute speed for wave propagation in the 2D blood flow model using a naive approach, based on a normal vector.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • normal: Normal vector indicating the direction of wave propagation.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Maximum absolute speed.

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Trixi.max_abs_speed_naive — Method
Trixi.max_abs_speed_naive(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Calculates the maximum absolute speed for wave propagation in the blood flow model using a naive approach.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Maximum absolute speed.

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Trixi.max_abs_speed_naive — Method
Trixi.max_abs_speed_naive(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations2D)

Computes the maximum absolute speed for wave propagation in the 2D blood flow model using a naive approach, based on the given orientation.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Direction index for the speed computation (1 for ( \theta )-direction, otherwise ( s )-direction).
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Maximum absolute speed.

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Trixi.max_abs_speeds — Method
Trixi.max_abs_speeds(u, eq::BloodFlowEquations2D)

Computes the maximum absolute speeds for wave propagation in the 2D blood flow model in both ( \theta )- and ( s )-directions.

Parameters

  • u: State vector.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Tuple containing the maximum absolute speeds in the ( \theta )- and ( s )-directions.

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Trixi.prim2cons — Method
Trixi.prim2cons(u, eq::BloodFlowEquations1D)

Converts the primitive variables to conserved variables.

Parameters

  • u: Primitive variable vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Conserved variable vector.

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Trixi.prim2cons — Method
Trixi.prim2cons(u, eq::BloodFlowEquations2D)

Converts the primitive variables to conservative variables for the 2D blood flow model.

Parameters

  • u: State vector in primitive form.
  • eq::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

State vector in conservative form as an SVector.

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Trixi.source_terms_convergence_test — Method
source_terms_convergence_test(u, x, t, eq::BloodFlowEquations1D)

Computes the source terms for convergence tests of the blood flow equations.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector.

Details

The source terms are derived based on the smooth initial condition and friction effects:

  • s_1 represents the source term for area perturbation and is given by s_1 = \pi t \cos(\pi x t).
  • s_2 represents the source term for the flow rate and includes contributions from spatial and temporal variations as well as friction effects.

The radius R is computed using the radius function, and the friction coefficient k is obtained using the friction function.

This function is useful for evaluating the correctness of source term handling in numerical solvers.

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2cons), ::BloodFlowEquations1D)

Returns the variable names corresponding to the conserved variables in the blood flow model.

Parameters

  • ::typeof(cons2cons): Type indicating conserved to conserved variable conversion.
  • ::BloodFlowEquations1D: Instance of BloodFlowEquations1D.

Returns

A tuple of variable names: ("a", "Q", "E", "A0").

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2cons), ::BloodFlowEquations2D)

Returns the variable names in conservative form for the 2D blood flow model.

Parameters

  • ::typeof(cons2cons): Type representing the conservative variables.
  • ::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Tuple containing the names of the conservative variables.

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2entropy), ::BloodFlowEquations1D)

Returns the variable names corresponding to the entropy variables in the blood flow model.

Parameters

  • ::typeof(cons2entropy): Type indicating conserved to entropy variable conversion.
  • ::BloodFlowEquations1D: Instance of BloodFlowEquations1D.

Returns

A tuple of variable names: ("A", "w", "En", "A0", "P").

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2entropy), ::BloodFlowEquations2D)

Returns the variable names in entropy form for the 2D blood flow model.

Parameters

  • ::typeof(cons2entropy): Type representing the entropy variables.
  • ::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Tuple containing the names of the entropy variables.

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2prim), ::BloodFlowEquations1D)

Returns the variable names corresponding to the primitive variables in the blood flow model.

Parameters

  • ::typeof(cons2prim): Type indicating conserved to primitive variable conversion.
  • ::BloodFlowEquations1D: Instance of BloodFlowEquations1D.

Returns

A tuple of variable names: ("A", "w", "P", "A0", "P").

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Trixi.varnames — Method
Trixi.varnames(::typeof(cons2prim), ::BloodFlowEquations2D)

Returns the variable names in primitive form for the 2D blood flow model.

Parameters

  • ::typeof(cons2prim): Type representing the primitive variables.
  • ::BloodFlowEquations2D: Instance of BloodFlowEquations2D.

Returns

Tuple containing the names of the primitive variables.

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