Utility

InterpolationCollection

A collection of compatible interpolations over some (possilby different) cells.

ScalarInterpolationCollection

A collection of compatible scalar-valued interpolations over some (possilby different) cells.

VectorInterpolationCollection

A collection of compatible vector-valued interpolations over some (possilby different) cells.

VectorizedInterpolationCollection{order} <: InterpolationCollection

A collection of fixed-order vectorized Lagrange interpolations across different cell types.

LagrangeCollection{order} <: InterpolationCollection

A collection of fixed-order Lagrange interpolations across different cell types.

QuadratureRuleCollection(order::Int)

A collection of quadrature rules across different cell types.

CellValueCollection(::QuadratureRuleCollection, ::InterpolationCollection)

Helper to construct and query the correct cell values on mixed grids.

FacetValueCollection(::QuadratureRuleCollection, ::InterpolationCollection)

Helper to construct and query the correct face values on mixed grids.

QuadraturePoint{dim, T}

A simple helper to carry quadrature point information.

QuadratureIterator(::QuadratureRule)
QuadratureIterator(::FacetQuadratureRule, local_face_idx::Int)
QuadratureIterator(::CellValues)
QuadratureIterator(::FacetValues)

A helper to loop over the quadrature points in some rule or cache with type QuadraturePoint.

NodalIntergridInterpolation(dh_from::DofHandler{sdim}, dh_to::DofHandler{sdim}, field_name_from::Symbol, field_name_to::Symbol)
NodalIntergridInterpolation(dh_from::DofHandler{sdim}, dh_to::DofHandler{sdim}, field_name::Symbol)
NodalIntergridInterpolation(dh_from::DofHandler{sdim}, dh_to::DofHandler{sdim})

Construct a transfer operator to move a field field_name from dof handler dh_from to another dof handler dh_to, assuming that all spatial coordinates of the dofs for dh_to are in the interior or boundary of the mesh contained within dh_from. This is necessary to have valid interpolation values, as this operator does not have extrapolation functionality.

This is basically a fancy matrix-vector product to transfer the solution from one problem to another one.

PoissonECGReconstructionCache(fₑₚ::GenericSplitFunction, Ωₜ::AbstractMesh, κᵢ, κ, electrodes::AbstractVector{<:Vec}; ground, linear_solver, solution_vector_type, system_matrix_type)

Sets up a cache for calculating $\varphi_\mathrm{e}$ by solving the Poisson problem

\[\nabla \cdot (\boldsymbol{\kappa}_{\mathrm{i}} + \boldsymbol{\kappa}_{\mathrm{e}}) \nabla \varphi_{\mathrm{e}}=-\nabla \cdot\left(\boldsymbol{\kappa}_{\mathrm{i}} \nabla \varphi_\mathrm{m}\right)\]

as for example proposed in [23] and investigated in [24] (as well as other studies). Here κₑ is the extracellular conductivity tensor and κᵢ is the intracellular conductivity tensor. The cache includes the assembled stiffness matrix with applied homogeneous Dirichlet boundary condition at the first vertex of the mesh. As the problem is solved for each timestep with only the right hand side changing.

Keyword Arguments

• ground = Set([VertexIndex(1, 1)])
• linear_solver = LinearSolve.KrylovJL_CG()
• solution_vector_type = Vector{Float64}
• system_matrix_type = ThreadedSparseMatrixCSR{Float64,Int64}

Plonsey1964ECGGaussCache(op::AbstractBilinearOperator, φₘ::AbstractVector)

Here φₘ is the solution vector containing the transmembranepotential, op is the associated diffusion opeartor and κₜ is the torso's conductivity.

Returns a cache to compute the lead field with the form proposed in [25] with the Gauss theorem applied to it, as for example described in [24]. Calling evaluate_ecg with this method simply evaluates the following integral efficiently:

\[\varphi_e(x)=\frac{1}{4 \pi \kappa_t} \int_\Omega \frac{ \kappa_ ∇φₘ \cdot (\tilde{x}-x)}{||(\tilde{x}-x)||^3}\mathrm{d}\tilde{x}\]

The important simplifications taken are:

1. Surrounding volume is an infinite, homogeneous sphere with isotropic conductivity
2. The extracellular space and surrounding volume share the same isotropic, homogeneous conductivity tensor

Geselowitz1989ECGLeadCache(problem, κ, κᵢ, electordes, electrode_pairs, [ground, linear_solver, solution_vector_type, system_matrix_type])

Here the lead field, Z, is computed using the discretization of problem. The lead field is computed as the solution of

\[\nabla \cdot(\mathbf{\kappa} \nabla Z)=\left\{\begin{array}{cl} -1 & \text { at the positive electrode } \\ 1 & \text { at the negative electrode } \\ 0 & \text { else where } \end{array}\right.\]

Where $\kappa$ is the bulk conductivity tensor.

Returns a cache contain the lead fields that are used to compute the lead potentials as proposed in [26]. Calling reinit! with this method simply evaluates the following integral efficiently:

\[V(t)=\int \nabla Z(\boldsymbol{x}) \cdot \boldsymbol{\kappa}_\mathrm{i} \nabla \varphi_\mathrm{m} \,\mathrm{d}\boldsymbol{x}.\]

evaluate_ecg(method::Plonsey1964ECGGaussCache, x::Vec, κₜ::Real)

Compute the pseudo ECG at a given point x by evaluating:

\[\varphi_e(x)=\frac{1}{4 \pi \kappa_t} \int_\Omega \frac{ \kappa_ ∇φₘ \cdot (\tilde{x}-x)}{||(\tilde{x}-x)||^3}\mathrm{d}\tilde{x}\]

For more information please read the docstring for Plonsey1964ECGGaussCache